diff --git a/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_001.md b/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_001.md new file mode 100644 index 0000000..a03b9e3 --- /dev/null +++ b/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_001.md @@ -0,0 +1,955 @@ + + +# A continuum mechanics based four-node shell element for general non-linear analysis + +Eduardo N. Dvorkin and Klaus-Jürgen Bathe + +Department of Mechanical Engineering, + +Massachusetts Institute of Technology, Cambridge, MA 02139, USA + +(Received December 1983) + +# ABSTRACT + +A new four-node (non-flat) general quadrilateral shell element for geometric and material non-linear analysis is presented. The element is formulated using three-dimensional continuum mechanics theory and it is applicable to the analysis of thin and thick shells. The formulation of the element and the solutions to various test and demonstrative example problems are presented and discussed. + +# INTRODUCTION + +The finite element analysis of general shell structures has been a very active field of research for a large number of years $^{14,29}$ . However, despite the fact that many different shell elements have already been proposed, the search for a shell element capable of representing the general nonlinear behaviour of shells with arbitrary geometry and loading conditions in an effective and reliable manner is still continuing very actively. + +During recent years it has become apparent that two approaches for the development of shell elements are very appropriate: (1) the use of simple elements, based on the discrete-Kirchhoff approach for the analysis of thin shells $^{2,5-9}$ ; (2) the use of degenerated isoparametric elements in which fully three-dimensional stress and strain conditions are degenerated to shell behaviour $^{2,3,5,7,17,19,24,29}$ . + +The latter approach has the advantage of being independent of any particular shell theory, and this approach was used by Bathe and Bolourchi $^{3}$ to formulate a general shell element for geometric and material non-linear analysis. This element has been employed very successfully when used with 9 or, in particular, 16 nodes. However, the 16-node element is quite expensive, and although it is possible to use in some analyses only a few elements to represent the total structure (see later examples) in other analyses still a fairly large number of elements need by employed $^{5}$ . + +Considering general shell analyses, much emphasis has been placed onto the development of a versatile, reliable and cost-effective 4-node shell element $^{16,17,22,28}$ . Such element would complement the above high-order 16-node element and may be more effective in certain analyses. The difficulties in the development of such element lie in that the element should be applicable in a reliable manner to thin and thick shells of arbitrary geometries for general non-linear analysis. + +The objective in this paper is to present a simple 4-node general shell element with the following properties: the element is formulated using three-dimensional stress and strain conditions without use of a shell theory; the element is applicable to thin and thick shells and can be employed to model arbitrary geometries; the element is applicable to the conditions of large displacements and rotations but small strains, and can be used effectively in materially non-linear analysis. + +The formulation of the element is quite simple and transparent, and the element has good predictive capability without containing spurious zero energy modes. + +In the next section of the paper we discuss some basic considerations with respect to the assumptions used, and in the following section we present the element formulation for non-linear analysis. The results obtained in numerical solutions that demonstrate the properties of the element are given in the final section. + +# BASIC CONSIDERATIONS + +The formulation of the 4-node shell element represents an extension of the shell element discussed previously $^{2,3}$ , and we therefore use the same notation as in those references. Also, to focus attention onto some key issues of the formulation, we consider in this section only linear analysis conditions. + +The geometry of the element (see Figure 1) is described using $^{2}$ : + +$$ +{ } ^ { l } x _ { i } = \sum _ { k = 1 } ^ { 4 } h _ { k } { } ^ { l } x _ { i } ^ { k } + \frac { r _ { 3 } } { 2 } \sum _ { k = 1 } ^ { 4 } a _ { k } h _ { k } { } ^ { l } V _ { n i } ^ { k } \tag {1} +$$ + +![](images/page-001_d5a8b8dfa6fbec52dcaa04baf84bcf6c3e6094dcbdc7affb122aa54bf942eed1.jpg) + +
+text_image + +r2 +2 +node 1 +r1 +3 +g3 +g2 +g1 +oVn4 +a4 +4 +oVn^k +u3^k +u2^k +oV2^k +node k +oV1^k +αk +oV1^k = e2 × 0Vn^k / |e2 × 0Vn^k | +oV2^k = 0Vn^k × 0V1^k +
+ +Figure 1 Four-node shell element + + + +![](images/page-002_ff020b598ebf94ae65e6bc78efebdc2f3b74a4d3a7eec6bb1611ec0d8c770c3e.jpg) + +
+text_image + +r₃ +r₂ +2 +A +I +q₃ +q₂ +q₁ +B +D +r₁ +3 +C +4 +
+ +![](images/page-002_6cb23535ce5ce32d6a992b2e0ab5048f0b4e374b4f3923a30cde0f385d0c7b5a.jpg) + +![](images/page-002_d08941e2c28a89d87966f44f673a97c182c7f8e7e7a8463c4dc64d3a59eda4f5.jpg) +$\tilde{\varepsilon}_{13}$ interpolation + +![](images/page-002_f0e3ac3a321ba459c27856613fb953d2e49581e7c39cd42eb6a6c0cd5f67612c.jpg) + +![](images/page-002_a5d95902e898986b8afdfd449bb9b0b86be37c1ae18a1cc360bfc197ca4bb023.jpg) +$\tilde{\varepsilon}_{23}$ interpolation +Figure 2 Interpolation functions for the transverse shear strains + +where the $h_{k}(r_{1},r_{2})$ are the two-dimensional interpolation functions corresponding to node k; the $r_{i}$ are the natural coordinates; and $^{l}x_{i}=$ Cartesian coordinates of any point in the element; $^{l}x_{i}^{k}=$ Cartesian coordinates of nodal point k; $^{l}V_{nl}^{k}=$ components of director vector at node k (which is not necessarily normal to the midsurface of the element); and $a_{k}$ is the shell thickness at node k, measured along the vector $^{l}V_{n}^{k}$ . The left superscript is zero for the initial geometry of the element and is equal to 1 for the deformed element geometry. Note that the thickness of the element varies and the element is in general non-flat. + +The displacements of any particle with natural coordinates $r_{i}$ of the shell element in the stationary Cartesian coordinate system are: + +$$ +u _ {i} = \sum_ {k = 1} ^ {4} h _ {k} u _ {i} ^ {k} + \frac {r _ {3}}{2} \sum_ {k = 1} ^ {4} a _ {k} h _ {k} \left(- ^ {0} V _ {2 i} ^ {k} \alpha_ {k} + ^ {0} V _ {1 i} ^ {k} \beta_ {k}\right) \tag {2} +$$ + +where the $u_{i}^{k}$ are the nodal point displacements into the Cartesian coordinate directions, and the $\alpha_{k}$ and $\beta_{k}$ are the rotations of the director vector ${}^{0}V_{n}^{k}$ about the ${}^{0}V_{1}^{k}$ and ${}^{0}V_{2}^{k}$ axes (see Figure 1). + +A basic problem inherent in the use of the above interpolation of the displacements, and the derivation of the strain-displacement matrices therefrom, is that the element 'locks' when it is thin. This is due to the fact that with these interpolations the transverse shear strains cannot vanish at all points in the element, when it is subjected to a constant bending moment. Hence, although the basic continuum mechanics assumptions contain the Kirchhoff shell assumptions, the finite element discretization is not able to represent these assumptions rendering the element not applicable to the analysis of thin plates or shells $^{2,5,7}$ . To solve this deficiency, various remedies based on selective and reduced integration have been proposed $^{17,22,23}$ but there is still much room for more effective and reliable elements for general non-linear analysis. + +Considering our element formulation - because the problem lies in the representation of the transverse shear strains - we proceed to not evaluate these shear strains from the displacements in (2), but to introduce separate interpolations for these strain components. Since we consider non-flat shell elements, the separate interpolations are performed effectively in a convected coordinate system†. + +The choice of the interpolation for the transverse shear strain components is the key assumption in our element formulation, because adequate coupling between the element displacements and rotations must be introduced and the element should not exhibit any spurious zero energy modes. For our element we use (see Figure 2): + +$$ +\begin{array}{l} \tilde {\varepsilon} _ {1 3} = \frac {1}{2} (1 + r _ {2}) \tilde {\varepsilon} _ {1 3} ^ {\mathrm{A}} + \frac {1}{2} (1 - r _ {2}) \tilde {\varepsilon} _ {1 3} ^ {\mathrm{C}} \\ \tilde {\varepsilon} _ {2 3} = \frac {1}{2} \left(1 + r _ {1}\right) \tilde {\varepsilon} _ {2 3} ^ {\mathrm{D}} + \frac {1}{2} \left(1 - r _ {1}\right) \tilde {\varepsilon} _ {2 3} ^ {\mathrm{B}} \tag {3} \\ \end{array} +$$ + +Since the kinematic relations for the above shear strains are not satisfied using (3), we impose them using Lagrange multipliers $^{2,27}$ to obtain, + +$$ +\Pi^ {*} = \frac {1}{2} \int_ {V} \tilde {\tau} ^ {i j} \tilde {\varepsilon} _ {i j} \mathrm{d} V + \int_ {V} \lambda^ {1 3} \left(\tilde {\varepsilon} _ {1 3} - \tilde {\varepsilon} _ {1 3} ^ {\mathrm{DI}}\right) \mathrm{d} V + \tag {4} +$$ + +$$ +\int_ {V} \lambda^ {2 3} \left(\tilde {\varepsilon} _ {2 3} - \tilde {\varepsilon} _ {2 3} ^ {\mathrm{DI}}\right) \mathrm{d} V - \mathscr {W} +$$ + +where the $\tilde{\tau}^{ij}$ are the contravariant components of the Cauchy stress tensor $^{13,15}$ , the $\tilde{\varepsilon}_{ij}$ are the covariant components of the infinitesimal strain tensor, the $\lambda^{13}$ and $\lambda^{23}$ are the Lagrange multipliers, the $\tilde{\varepsilon}_{13}^{DI}$ and $\tilde{\varepsilon}_{23}^{DI}$ are the transverse shear strains evaluated using the displacement interpolations in (2), and W is the potential of the external loads. For the Lagrange multipliers we choose the following interpolations, + +$$ +\lambda^ {1 3} = \lambda^ {A} \delta (r _ {1}) \delta (1 - r _ {2}) + \lambda^ {C} \delta (r _ {1}) \delta (1 + r _ {2}) +$$ + +$$ +\lambda^ {2 3} = \lambda^ {\mathrm{D}} \delta \left(r _ {2}\right) \delta \left(1 - r _ {1}\right) + \lambda^ {\mathrm{B}} \delta \left(r _ {2}\right) \delta \left(1 + r _ {1}\right) \tag {5} +$$ + +where $\delta(\ldots)$ is the Dirac-delta function. This represents a weakening of the Lagrange multiplier constraint in (4) $^{10}$ . Substituting from (5) into (4) and invoking that $\delta\Pi^{*}=0$ gives the distinct constrains: + +$$ +\left. \tilde {\varepsilon} _ {1 3} \right| _ {\text { at A }} = \left. \tilde {\varepsilon} _ {1 3} ^ {\mathrm{DI}} \right| _ {\text { at A }} \quad \left. \tilde {\varepsilon} _ {1 3} \right| _ {\text { at C }} = \left. \tilde {\varepsilon} _ {1 3} ^ {\mathrm{DI}} \right| _ {\text { at C }} \tag {6} +$$ + +$$ +\tilde {\varepsilon} _ {2 3} \left| _ {\text {at D}} = \tilde {\varepsilon} _ {2 3} ^ {\mathrm{DI}} \right| _ {\text {at D}} \quad \tilde {\varepsilon} _ {2 3} \left| _ {\text {at B}} = \tilde {\varepsilon} _ {2 3} ^ {\mathrm{DI}} \right| _ {\text {at B}} +$$ + +Hence, the complete element stiffness matrix is calculated using the functional: + +$$ +\Pi^ {*} = \frac {1}{2} \int_ {V} \tilde {\tau} ^ {i j} \tilde {\varepsilon} _ {i j} \mathrm{d} V - \mathcal {W} \tag {7} +$$ + + + +![](images/page-003_f95c601761b2d3e08c15c9560b9b44d0e52d2e50af8c3719ef6481f69875f563.jpg) + +
+text_image + +r2 +r3 +g2 +g3 +r1 +e3 +e2 +e1 +g1 +e3 = g3 / |g3|; e1 = g2 × e3 / |g2 × e3|; e2 = e3 × e1 +
+ +Figure 3 Local Cartesian coordinate system used + +with stress and strain components in convected coordinates and (1) and (2) to evaluate the strain components $\tilde{\varepsilon}_{11}$ , $\tilde{\varepsilon}_{22}$ and $\tilde{\varepsilon}_{12}$ ; (3) to evaluate the strain components $\tilde{\varepsilon}_{13}$ , $\tilde{\varepsilon}_{23}$ ; and (6) to express the variables $\tilde{\varepsilon}_{13}^{\mathrm{A}}$ , $\tilde{\varepsilon}_{13}^{\mathrm{C}}$ , $\tilde{\varepsilon}_{23}^{\mathrm{D}}$ , and $\tilde{\varepsilon}_{23}^{\mathrm{B}}$ in terms of the nodal point displacements and rotations of (2). + +Considering the representation that we have chosen for the transverse shear strains, we can make the following three important observations: + +(1) The element is able to represent the six rigid body modes. The element contains the rigid body modes because zero strains are calculated in the formulation when the element nodal point displacements and rotations correspond to an element rigid body displacement. This can be verified by using (1) to (6) to evaluate the strains, but more easily we can use the fact that the 4-node shell element of reference 3 satisfies the rigid body mode criterion. Hence, for a rigid body displacement the $\tilde{\varepsilon}_{13}^{DI}$ and $\tilde{\varepsilon}_{23}^{DI}$ are zero, from which it follows that also the shear strains in (3) are zero, and the rigid body mode criterion is satisfied. + +(2) The element can approximate the Kirchhoff–Love hypothesis of negligible shear deformation effects and can be used for thin shells. Various demonstrative solutions are given in the fourth section. + +(3) Based on our studies the element does not contain any spurious zero energy modes (using a 'full' numerical integration). We reach this observation by studying the strains along the element sides. If the element were to contain a spurious zero energy mode, the strains along every side should vanish for a displacement pattern (to be identified) other than the displacements corresponding to a true rigid body mode. However, such displacement pattern could not be identified. + +Considering the practical use of the element the interpolation employed for the transverse shear strains shows that $\tilde{\varepsilon}_{13}$ is constant with $r_{1}$ and in general discontinuous at $r_{1}=\pm1$ (between elements), and similarly $\tilde{\varepsilon}_{23}$ is constant with $r_{2}$ and in general discontinuous at $r_{2}=\pm1$ . As a consequence, the accuracy with which transverse shear stresses are predicted depends to a significant degree on the mesh used and the geometric distortions of the elements. However, our experience is + +that the bending stress predictions are relatively little affected by element distortions (see examples). + +To employ (7), we also need to use the appropriate constitutive relations: + +$$ +\tilde {\tau} ^ {i j} = \tilde {C} ^ {i j k l} \tilde {\varepsilon} _ {k l} \tag {8} +$$ + +where $\tilde{C}^{ijkl}$ is the fourth-order contravariant constitutive tensor in the convected coordinates $r_{i}$ . The constitutive law is known in the local Cartesian system of orthonormal base vectors $\hat{e}_{i}, i=1,2,3$ , with the condition $\hat{\tau}^{33}$ equal to zero $^{2}$ , (see Figure 3). Denoting this constitutive tensor by $\hat{C}^{mnop}$ , the constitutive tensor for (8) is obtained using the transformation: + +$$ +\tilde {C} ^ {i j k l} = \left(\mathbf {g} ^ {i} \cdot \hat {\mathbf {e}} _ {m}\right) \left(\mathbf {g} ^ {j} \cdot \hat {\mathbf {e}} _ {n}\right) \left(\mathbf {g} ^ {k} \cdot \hat {\mathbf {e}} _ {0}\right) \left(\mathbf {g} ^ {l} \cdot \hat {\mathbf {e}} _ {p}\right) \hat {C} ^ {m n o p} \tag {9} +$$ + +where the $g^{i}$ are the contravariant base vectors of the convected coordinates $r_{i}$ . These vectors are calculated using the covariant base vectors $g_{i}$ , where: + +$$ +\mathbf {g} _ {i} = \frac {\partial^ {0} \mathbf {x}}{\partial r _ {i}} \tag {10} +$$ + +with $^{0}x$ from (1) and the following relations, + +$$ +g _ {i j} = \mathbf {g} _ {i} \cdot \mathbf {g} _ {j} \tag {11} +$$ + +and + +$$ +\mathbf {g} ^ {i} = g ^ {i j} \mathbf {g} _ {j} \tag {12} +$$ + +$$ +g ^ {i j} = \frac {D ^ {i j}}{| \mathbf {J} | ^ {2}} +$$ + +where $D^{ij}$ is the cofactor of the term $g_{ij}$ in the matrix of the metric tensor and $|J|$ is the determinant of the Jacobian matrix at the point considered. + +# TOTAL LAGRANGIAN FORMULATION + +The large displacement formulation of the shell element is based on the derivation given in ref. 2 (Section 6.3.5), and the concepts and interpolations presented in the previous section. + +The geometry of the element at any time t is defined as in (1) but using the nodal point coordinates, $^{t}x_{i}^{k}$ , and director vectors $^{t}V_{n}^{k}$ , at time $t,\dagger$ + +$$ +{ } ^ { t } x _ { i } = h _ { k } { } ^ { t } x _ { i } ^ { k } + \frac { r _ { 3 } } { 2 } a _ { k } h _ { k } { } ^ { t } V _ { n i } ^ { k } \tag {13} +$$ + +where we imply summation over k. The displacements, $u_{i}$ , and incremental displacements, $u_{i}$ , of a particle of the element at time t are hence given by: + +$$ +{ } ^ { t } u _ { i } = h _ { k } { } ^ { t } u _ { i } ^ { k } + \frac { r _ { 3 } } { 2 } a _ { k } h _ { k } ( { } ^ { t } V _ { n i } ^ { k } - { } ^ { 0 } V _ { n i } ^ { k } ) \tag {14} +$$ + +$$ +u _ {i} = h _ {k} u _ {i} ^ {k} + \frac {r _ {3}}{2} a _ {k} h _ {k} \left(- ^ {t} V _ {2 i} ^ {k} \alpha_ {k} + ^ {t} V _ {1 i} ^ {k} \beta_ {k}\right) +$$ + +where the $^{t}u_{i}^{k}$ are the nodal point displacements at time $t$ , the $u_{i}^{k}$ are the incremental nodal point displacements from the configuration at time $t$ , and the variables $^{t}V_{2i}^{k}, ^{t}V_{1i}^{k}, \alpha_{k}$ and $\beta_{k}$ are defined as in (2) but referred to the configuration at time $t$ . + +This kinematic description implies the following hy- + + + +potheses: the director vectors remain straight during the deformations; the 'thickness' of the element measured along the director vectors remains constant during the deformations; hence only small strain conditions are considered. + +Using the assumptions in (13) and (14) the geometric and material non-linear response is analysed using an incremental formulation $^{2}$ , in which the configuration is sought for time (load step) ' $t+\Delta t$ ', when the configuration for time t is known. The basis of this incremental formulation is the use of the virtual work principle applied to the configuration at time $t+\Delta t$ . In essence, two approaches can be employed leading to the updated Lagrangian and the total Lagrangian formulations. These approaches are, from a continuum mechanics point of view, equivalent, and in the following we develop the governing finite element relations for the total Lagrangian formulation. + +The principle of virtual work applied to the configuration at time $t + \Delta t$ is: + +$$ +\int_ {0 V} ^ {t + \Delta t} \tilde {S} _ {0} ^ {i j} \delta_ {0} ^ {t + \Delta t} \tilde {\varepsilon} _ {i j} ^ {0} \mathrm{d} V = ^ {t + \Delta t} \mathcal {R} \tag {15} +$$ + +where the $^{t+\Delta t}_{0}\tilde{S}^{ij}$ are the contravariant components of the second Piola-Kirchhoff stress tensor at time $t+\Delta t$ and referred to the configuration at time 0, and the $^{t+\Delta t}_{0}\tilde{E}_{ij}$ are the covariant components of the Green-Lagrange strain tensor at time $t+\Delta t$ and referred to time 0. Both sets of tensor components are measured in the convected coordinate system $r_{i}, i=1,2,3$ . The external virtual work is given by $^{t+\Delta t}\mathcal{R}$ and includes the work due to the applied surface tractions and body forces. + +For the incremental solution, the stresses and strains are decomposed into the known quantities, ${}_{0}^{t}\tilde{S}^{ij}$ and ${}_{0}^{t}\tilde{e}_{ij}$ , and unknown increments, ${}_{0}\tilde{S}^{ij}$ and ${}_{0}\tilde{e}_{ij}$ , so that + +$$ +{ } _ { 0 } ^ { t + \Delta t } \tilde { S } ^ { i j } = { } _ { 0 } ^ { t } \tilde { S } ^ { i j } + { } _ { 0 } \tilde { S } ^ { i j } \tag {16} +$$ + +$$ +{ } ^ { t + \Delta t } _ { 0 } \tilde { \varepsilon } _ { i j } = { } _ { 0 } ^ { t } \tilde { \varepsilon } _ { i j } + { } _ { 0 } \tilde { \varepsilon } _ { i j } \tag {17} +$$ + +In addition, the strain increment can be written as a linear part, $_{0}\tilde{e}_{ij}$ , and a non-linear part, $_{0}\tilde{\eta}_{ij}$ , hence + +$$ +_ 0 \tilde {\varepsilon} _ {i j} = _ {0} \tilde {e} _ {i j} + _ {0} \tilde {\eta} _ {i j} \tag {18} +$$ + +Substituting from (16) to (18) into (15) and using the linearized expressions $_{0}\bar{S}^{ij}=_{0}\bar{C}^{ijkl}_{0}\tilde{e}_{kl}$ and $\delta_{0}\tilde{\varepsilon}_{ij}=\delta_{0}\tilde{e}_{ij}$ we obtain the linearized equation of motion: + +$$ +\begin{array}{l} \int_ {0 _ {V}} ^ {0} \tilde {C} ^ {i j k l} _ {0} \tilde {e} _ {k l} \delta_ {0} \tilde {e} _ {i j} ^ {0} \mathrm{d} V + \int_ {0 _ {V}} ^ {t} \tilde {S} ^ {i j} \delta_ {0} \tilde {\eta} _ {i j} ^ {0} \mathrm{d} V \tag {19} \\ = ^ {t + \Delta t} \mathcal {R} - \int_ {0 V} ^ {t} \tilde {S} ^ {i j} \delta_ {0} \tilde {e} _ {i j} ^ {0} \mathrm{d} V \\ \end{array} +$$ + +This equation is the basic equilibrium relation employed to develop the governing finite element matrices. For the actual solution of problems it is frequently important to use equilibrium iterations, but the finite element matrices and vectors used in these iterations can be derived directly from the matrices obtained using (19) $^{2}$ . Note that $_{0}\tilde{C}^{ijkl}$ is now obtained using (9) with the condition $_{0}^{t}\hat{S}^{33}=0$ , which implies the more natural condition $^{t}\hat{\tau}^{33}=0$ only in the small strain case. + +The basic problem of the finite element discretization of (19) lies in expressing the strain terms of (19) in terms of the finite element interpolations. Using the definition of the Green-Lagrange strain components: + +$$ +{ } _ { 0 } ^ { t } \tilde { \varepsilon } _ { i j } = \frac { 1 } { 2 } ( { } ^ { t } \mathbf { g } _ { i } \cdot { } ^ { t } \mathbf { g } _ { j } - { } ^ { 0 } \mathbf { g } _ { i } \cdot { } ^ { 0 } \mathbf { g } _ { j } ) \tag {20} +$$ + +and the relations in (13) and (14) we obtain: + +$$ +{ } _ { 0 } \tilde { e } _ { i i } = h _ { k , i } { } ^ { t } \mathbf { g } _ { i } \cdot \mathbf { u } _ { k } + \frac { r _ { 3 } } { 2 } a _ { k } h _ { k , i } ( - \alpha _ { k } { } ^ { t } \mathbf { g } _ { i } \cdot { } ^ { t } \mathbf { V } _ { 2 } ^ { k } + \beta _ { k } { } ^ { t } \mathbf { g } _ { i } \cdot { } ^ { t } \mathbf { V } _ { 1 } ^ { k } ) \tag {21a} +$$ + +$$ +_ {0} \tilde {\eta} _ {i i} = \frac {1}{2} h _ {k, i} h _ {p, i} \mathbf {u} _ {k} \cdot \mathbf {u} _ {p} + \frac {r _ {3}}{2} h _ {k, i} h _ {p, i} a _ {p} \left(- \alpha_ {p} ^ {t} \mathbf {V} _ {2} ^ {p} \cdot \mathbf {u} _ {k} + \beta_ {p} ^ {t} \mathbf {V} _ {1} ^ {p} \cdot \mathbf {u} _ {k}\right) + +$$ + +$$ +\begin{array}{r l} \frac {(r _ {3}) ^ {2}}{8} h _ {k, i} h _ {p, i} a _ {k} a _ {p} (- \alpha_ {k} ^ {t} \mathbf {V} _ {2} ^ {k} + \beta_ {k} ^ {t} \mathbf {V} _ {1} ^ {k}) \cdot (- \alpha_ {p} ^ {t} \mathbf {V} _ {2} ^ {p} + \beta^ {p t} \mathbf {V} _ {1} ^ {p}) & (i = 1, 2) \\ & (2 1 b) \end{array} +$$ + +with the notation $h_{k,i} = \frac{\partial h_k}{\partial r_i}, \mathbf{u}_k^{\mathrm{T}} = [u_1^k \quad u_2^k \quad u_3^k]$ , and + +$$ +{ } _ { 0 } \tilde { e } _ { 1 2 } = \frac { 1 } { 2 } \left[ h _ { k , 2 } { } ^ { t } \mathbf { g } _ { 1 } \cdot \mathbf { u } _ { k } + h _ { k , 1 } { } ^ { t } \mathbf { g } _ { 2 } \cdot \mathbf { u } _ { k } + \right. +$$ + +$$ +\frac {r _ {3}}{2} h _ {k, 2} a _ {k} \left(- \alpha_ {k} ^ {t} \mathbf {V} _ {2} ^ {k} \cdot^ {t} \mathbf {g} _ {1} + \beta_ {k} ^ {t} \mathbf {V} _ {1} ^ {k} \cdot^ {t} \mathbf {g} _ {1}\right) + +$$ + +$$ +\frac {r _ {3}}{2} h _ {k, 1} a _ {k} (- \alpha_ {k} ^ {t} \mathbf {V} _ {2} ^ {k} \cdot^ {t} \mathbf {g} _ {2} + \beta_ {k} ^ {t} \mathbf {V} _ {1} ^ {k} \cdot^ {t} \mathbf {g} _ {2}) ] \tag {22a} +$$ + +$$ +_ 0 \tilde {\eta} _ {1 2} = \frac {1}{2} \left[ h _ {k, 1} h _ {p, 2} \mathbf {u} _ {k} \cdot \mathbf {u} _ {p} + \right. +$$ + +$$ +\frac {r _ {3}}{2} h _ {k, 1} h _ {p, 2} a _ {p} \left(- \alpha_ {p} ^ {\prime} \mathbf {V} _ {2} ^ {p} \cdot \mathbf {u} _ {k} + \beta_ {p} ^ {\prime} \mathbf {V} _ {1} ^ {p} \cdot \mathbf {u} _ {k}\right) + +$$ + +$$ +\frac {r _ {3}}{2} h _ {k, 1} h _ {p, 2} a _ {k} \left(- \alpha_ {k} ^ {t} \mathbf {V} _ {2} ^ {k} \cdot \mathbf {u} _ {p} + \beta_ {k} ^ {t} \mathbf {V} _ {1} ^ {k} \cdot \mathbf {u} _ {p}\right) + +$$ + +$$ +\frac {(r _ {3}) ^ {2}}{4} h _ {k, 1} h _ {p, 2} a _ {k} a _ {p} \left(- \alpha_ {k} ^ {t} \mathrm{V} _ {2} ^ {k} + \beta_ {k} ^ {t} \mathrm{V} _ {1} ^ {k}\right) \cdot \left(- \alpha_ {p} ^ {t} \mathrm{V} _ {2} ^ {p} + \beta_ {p} ^ {t} \mathrm{V} _ {1} ^ {p}\right) ] \tag {22b} +$$ + +Further, we obtain for the transverse shear strains, using (3) and (6): + +$$ +{ } _ { 0 } \tilde { e } _ { 1 3 } = \frac { 1 } { 8 } ( 1 + r _ { 2 } ) \left[ ^ { t } g _ { 3 i } ^ { \mathrm{A} } ( u _ { i } ^ { 1 } - u _ { i } ^ { 2 } ) + \right. +$$ + +$$ +\frac {1}{2} ^ {t} g _ {1 i} ^ {A} \left(- \alpha_ {1} a _ {1} ^ {t} V _ {2 i} ^ {1} + \beta_ {1} a _ {1} ^ {t} V _ {1 i} ^ {1} - \alpha_ {2} a _ {2} ^ {t} V _ {2 i} ^ {2} + \beta_ {2} a _ {2} ^ {t} V _ {1 i} ^ {2}\right) ] + +$$ + +$$ +\frac {1}{8} (1 - r _ {2}) \left[ ^ {t} g _ {3 i} ^ {C} \left(u _ {i} ^ {4} - u _ {i} ^ {3}\right) + \frac {1}{2} ^ {t} g _ {1 i} ^ {C} \left(- \alpha_ {4} a _ {4} ^ {t} V _ {2 i} ^ {4} + \right. \right. +$$ + +$$ +\left. \beta_ {4} a _ {4} ^ {t} V _ {1 i} ^ {4} - \alpha_ {3} a _ {3} ^ {t} V _ {2 i} ^ {3} + \beta_ {3} a _ {3} ^ {t} V _ {1 i} ^ {3}) \right] \tag {23a} +$$ + +$$ +{ } _ { 0 } \tilde { \eta } _ { 1 3 } = \frac { 1 } { 3 2 } ( 1 + r _ { 2 } ) \left[ ( - \alpha _ { 1 } a _ { 1 } { } ^ { t } V _ { 2 i } ^ { 1 } + \beta _ { 1 } a _ { 1 } { } ^ { t } V _ { 1 i } ^ { 1 } - \right. +$$ + +$$ +\left. \alpha_ {2} a _ {2} ^ {t} V _ {2 i} ^ {2} + \beta_ {2} a _ {2} ^ {t} V _ {1 i} ^ {2}) \left(u _ {i} ^ {1} - u _ {i} ^ {2}\right) \right] + +$$ + +$$ +\frac {1}{3 2} (1 - r _ {2}) \left[ \left(- \alpha_ {4} a _ {4} ^ {\prime} V _ {2 i} ^ {4} + \beta_ {4} a _ {4} ^ {\prime} V _ {1 i} ^ {4} - \right. \right. +$$ + +$$ +\left. \alpha_ {3} a _ {3} ^ {t} V _ {2 i} ^ {3} + \beta_ {3} a _ {3} ^ {t} V _ {1 i} ^ {3}) \left(u _ {i} ^ {4} - u _ {i} ^ {3}\right) \right] \tag {23b} +$$ + +and + +$$ +{ } _ { 0 } \tilde { e } _ { 2 3 } = \frac { 1 } { 8 } ( 1 + r _ { 1 } ) \left[ ^ { t } g _ { 3 i } ^ { \mathrm{D} } ( u _ { i } ^ { 1 } - u _ { i } ^ { 4 } ) + \right. +$$ + +$$ +\frac {1}{2} ^ {t} g _ {2 i} ^ {\mathrm{D}} \left(- \alpha_ {1} a _ {1} ^ {t} V _ {2 i} ^ {1} + \beta_ {1} a _ {1} ^ {t} V _ {1 i} ^ {1} - \alpha_ {4} a _ {4} ^ {t} V _ {2 i} ^ {4} + \beta_ {4} a _ {4} ^ {t} V _ {1 i} ^ {4}\right) ] + +$$ + +$$ +\frac {1}{8} (1 - r _ {1}) _ {L} ^ {t} g _ {3 i} ^ {B} (u _ {i} ^ {2} - u _ {i} ^ {3}) + \frac {1}{2} ^ {t} g _ {2 i} ^ {B} (- \alpha_ {2} a _ {2} ^ {t} V _ {2 i} ^ {2} + +$$ + +$$ +\left. \beta_ {2} a _ {2} ^ {t} V _ {1 i} ^ {2} - \alpha_ {3} a _ {3} ^ {t} V _ {2 i} ^ {3} + \beta_ {3} a _ {3} ^ {t} V _ {1 i} ^ {3}) \right] \tag {24a} +$$ + + + +$$ +\begin{array}{l} _ 0 \tilde {\eta} _ {2 3} = \frac {1}{3 2} (1 + r _ {1}) \left[ \left(- \alpha_ {1} a _ {1} ^ {\prime} V _ {2 i} ^ {1} + \beta_ {1} a _ {1} ^ {\prime} V _ {1 i} ^ {1} - \right. \right. \\ \left. \alpha_ {4} a _ {4} ^ {t} V _ {2 i} ^ {4} + \beta_ {4} a _ {4} ^ {t} V _ {1 i} ^ {4}) \left(u _ {i} ^ {1} - u _ {i} ^ {4}\right) \right] + \\ \frac {1}{3 2} (1 - r _ {1}) \left[ \left(- \alpha_ {2} a _ {2} ^ {t} V _ {2 i} ^ {2} + \beta_ {2} a _ {2} ^ {t} V _ {1 i} ^ {2} - \right. \right. \\ \alpha_ {3} a _ {3} ^ {\prime} V _ {2 i} ^ {3} + \beta_ {3} a _ {3} ^ {\prime} V _ {1 i} ^ {3}) (u _ {i} ^ {2} - u _ {i} ^ {3}) ] \\ \end{array} +$$ + +(24b) + +Note that, since we assume the thickness of the shell to be constant, the strain $t_{0}\tilde{\varepsilon}_{33}$ through the element thickness is zero. + +The expressions in (21) to (24) are substituted into (19) which in the standard manner yields the linear strain incremental stiffness matrix ${}^{t}_{0}K_{L}$ , the non-linear strain (or geometric) incremental stiffness matrix ${}^{t}_{0}K_{NL}$ and the nodal point force vector ${}^{t}_{0}F$ in the finite element incremental equilibrium relations $^{2}$ , + +$$ +(_ {0} ^ {t} \mathbf {K} _ {L} + _ {0} ^ {t} \mathbf {K} _ {N L}) \mathbf {u} = ^ {t + \Delta t} \mathbf {R} - _ {0} ^ {t} \mathbf {F} \tag {25} +$$ + +The element matrices in (25) correspond to five degrees of freedom per node (see Figure 1) but in some applications it is convenient to use instead of $\alpha_{k}$ and $\beta_{k}$ three rotations about the global coordinate axes (see examples). In this case, we simply transform the matrices of (25) in the standard manner $^{2}$ . + +# NUMERICAL TESTS AND EXAMPLE SOLUTIONS + +We have implemented our shell element in the ADINA computer program and have performed various numerical tests to study the predictive capabilities of the element. The following solutions were all obtained using $2 \times 2$ Gauss integration in the $r_{3}=0$ surface of the element, and 2 and 4 point Gauss integration in the $r_{3}$ direction, for elastic and elastoplastic analyses, respectively. + +# Some simple tests + +As a first step to test the element, the eigenvalues of the stiffness matrices of undistorted and distorted elements were calculated. In all cases, as expected, the element displayed the six rigid body modes and no spurious zero energy modes. + +Patch tests. For the patch test $^{2,18}$ the mesh shown in Figure 4a was used. In the first analysis (Figure 4b) the mesh was loaded with the constant moment indicated and a constant curvature (linear distribution of rotations) was obtained for both plate thicknesses in the two plate directions. The transverse displacements predicted by the model were, as expected, those of Kirchhoff–Love plate theory at nodes 7 and 8. + +In the second analysis (Figure 4c) the rotational degrees of freedom were deleted and the mesh was subjected to shear forces. As expected, for both plate thicknesses a linear distribution of transverse displacements was obtained. + +In the third analysis (Figure 4d) the mesh was subjected to an external twisting moment. In the thin plate analysis, constant curvatures were obtained in both plate directions and the transverse displacements agreed with the analytical thin plate theory solution. In the thick plate analysis, a slight non-symmetry in the displacement response (the third digit) was obtained due to the unsymmetric representation of the transverse shear deformations. This non-symmetry is not observed, if the shear deformations are suppressed (which corresponds to thin + +![](images/page-005_a48de6bf3f03750b20738b9903df25da0d15abf6d3d92458cdd4f606a0f75c3d.jpg) + +
+text_image + +x₂ +1 +(Q,1Q) +7(10,1Q) +3(4,7.) +5(8,7.) +10. +4(2,2) +6 +(8,3) +2 +(Q,Q) +8(10,Q) +x₁ +10. +
+ +(a) Patch test mesh + +![](images/page-005_00510176490cef82f49b70141ea71894cb95c13d476fc9a8c2bcced4d290419c.jpg) + +
+text_image + +u_{1-2-3}=0 +β=0 +BENDING +u_{1-3}=0 +β=0 +
+ +(b) Constant curvature patch test + +![](images/page-005_578dbb8081c2c49793ec95f7bdcfd44b9148f8302223b63bb98f8236dc704aa6.jpg) + +
+text_image + +u₃=0 +SHEAR +u₁₋₂=0 +α=Ω=0 +u₃=0 +
+ +(c) Constant shear patch test (zero rotations) + +![](images/page-005_1012548cf5134eca3c0e9354dd072c4ec5a4b225b6cc50fdc277d9688d9c026e.jpg) + +
+text_image + +u₃=0 +TWISTING +u₁₋₂ = 0 +u₃=0 +u₃=0 +
+ +(d) Constant twist patch test +Figure 4 Patch tests. $E = 2.1 \times 10^{6}$ ; $v = 0.3$ ; thickness $= \begin{cases} 1.0 \\ 0.001 \end{cases}$ + + + +![](images/page-006_561f992b28f0957f27f23d84b3ef820569f70b8004e6f194f1f363d45439d344.jpg) + +
+text_image + +x₃ +4 +3 +x₂ +M/2 +1.0 +1 +2 +M/2 +x₁ +L={100,10.} +
+ +(a) One element case. Node 1: $x = 0$ ; $u_{2-3} = 0$ . Node 4: $x = 0$ ; $u_{1-2-3} = 0$ + +![](images/page-006_16f6a8ca84210938f4fae40ea7c42718158fa4014f8f772cadfa18f393160368.jpg) + +
+text_image + +x₃ +L +0.3L +x₂ +x₁ +0.3L +
+ +(b) Two element case +Figure 5 Cantilever subjected to tip bending moment. $E=2.1 \times 10^{6}$ ; v=0.3; thickness=0.1. + +plate theory) by choosing a large value for the shear correction factor $k$ (or when using rectangular elements in the mesh) $^2$ . + +Finally, it should be noted that the patch test is of course passed for the three membrane stress states ( $\tau_{11}$ , $\tau_{22}$ and $\tau_{12}$ constants). + +Cantilever linear analyses. A cantilever of unit width, thickness 0.1 and lengths 10 and 100 was subjected to a tip bending moment. The structure was modelled using one single element and two distorted elements as shown in Figure 5. The results obtained in these analyses for the displacements and rotations at the cantilever tip and the stresses were those of Bernoulli beam theory. + +Next, the cantilever in Figure 6a was analysed for the transverse tip load shown. Using 4 equal size elements to idealize the cantilever, again good results were obtained when compared with beam theoretical results (see Figure 6b and Table 1). + +Finally, the elements modelling the cantilever were distorted as shown in Figure 6c for a thin and a thick cantilever. The results given in Figure 6d and Table 2 show that the transverse displacements and normal bending stresses are almost insensitive to the element distortions. However, the calculated transverse shear stresses (not shown in the Figure) are not accurate. + +Linear analyses of a simply-supported plate. A simply-supported plate was considered for a static and a frequency analysis using a consistent mass matrix. To model one quarter of the plate the $4 \times 4$ mesh of equal elements (Figure 7a) was used. Figure 7b and Tables 3 and 4 give a comparison of the numerically and analytically predicted results. The same plate was also analysed using the distorted element mesh also shown in Figure 7a and the results of Figure 7b and Tables 3 and 4 were obtained. + +$$ +E = 2. 1 \times 1 0 ^ {6}; v = 0. 0; \text { thickness } = 0. 1; P = 1. 0 +$$ + +![](images/page-006_38ebcfb83a18e241db3810921bae82e46545bac6903cd665b2e6790bf82066be.jpg) + +
+text_image + +x3 +elem. 1 +elem. N +P/2 +x2 +α=0 +u1-2-3=0 +α=0 +u2-3=0 +P/2 +1.0 +x1 +10. +
+ +(a) Cantilever subjected to transverse tip load + +![](images/page-006_dec6db2921847506cee29537cb7e1fb8f97613d01edbef9a15d4bcd2bdcf06ef.jpg) + +
+line +| x₂ | τ₂₂ | τ₂₃ | +| ---- | ------- | ------- | +| 0 | 3464.10 | 10 | +| L | 0 | 10 | +| x₂ | 0 | 10 | +
+ +(b) Solution using non-distorted elements + +![](images/page-006_572bd2c8413d9ea55265b5ffb1ebc5848a2ae8c33b16ef03b5392bb163b374c0.jpg) + +
+flowchart + +```mermaid +graph TD + A["0. 2.5 4.5 7.5 10. B"] -->|x₁| B["0. 2. 5. 7. 10. A"] + B --> C["x₂"] +``` +
+ +(c) Distorted mesh - plan view + +![](images/page-006_b868b07d372b77a5c662be3a2db817454282936b40b91ee4720442982a1fad6a.jpg) +(d) Solution using distorted mesh - two thicknesses and loads +Figure 6 Response of a cantilever subjected to transverse tip load, stresses shown are those at the Gauss integration stations $r_{3}=0.57735$ ; $\tau_{pp}$ , is the principal stress in the distorted mesh, and its direction was always less than 11 degrees from the $x_{2}$ axis. ——, Analytical (Bernoulli); ○, shell element (N=4) + +Table 1 Cantilever tip transverse displacement: non-distorted meshes of N elements + +
N $u_{3\text{TIP}}^{\text{FEM}} \left/ \left( \frac{\text{PL}^3}{3\text{EI}} + \frac{\text{PL}}{\text{AG}} \right) \right.$
10.750
40.984
+ +Table 2 Cantilever tip transverse displacements + +
Thickness $\eta|_{point B}$ $\eta|_{point A}$
0.10.9890.996
2.01.00130.995
+ +$\eta = (u_{3}$ distorted mesh)/(u3 non-distorted mesh) + + + +![](images/page-007_fc638e677479023b3329468c72e192e8c07bc681f3891a39caebca185648552b.jpg) + +
+text_image + +L/2 +12.50 +x₂ +x₁ +12.50 +L/2 +
+ +(a) Non-distorted and distorted meshes ( $\Delta=2.50$ ) + +![](images/page-007_26fe881157d94642d5935f90d7e3b59d6a7fe50f64a26e0ab1352a0b0c6aae9e.jpg) + +
+line + +| x₁ | τ₂₂/qL² | +|----|---------| +| 0 | 20 | +| 10 | 18 | +| 20 | 16 | +| 30 | 14 | +| 40 | 12 | +| 50 | 0 | +
+ +![](images/page-007_1df20afc31504b59377219a264749fb20ccc8fa41de4382d1727d6d7363d97fc.jpg) + +
+line + +| x₁ | τ₁₁/qL² | +| --- | ------- | +| 0 | 20 | +| 10 | 18 | +| 20 | 15 | +| 30 | 12 | +| 40 | 8 | +| 50 | 4 | +
+ +(b) Static response due to constant pressure loading, stresses are given along line $x_{2}=0$ , $x_{3}=0.028868$ . ——, analytical (Kirchhoff plate); ○, non-distorted mesh; □, distorted mesh. +Figure 7 Linear analysis of a simply-supported plate + +Table 3 Non-dimensional displacements at centre of simply-supported plate: distorted and non-distorted meshes + +
Model $u_{3}^{\text{FEM}}/u_{3}^{\text{thin plate}}$ at centre
non-dist.0.995
dist.0.992
+ +Table 4 Non-dimensional frequencies f (cycles/sec) for a simply-supported plate: distorted and non-distorted meshes + +
Mode shape $f^{FEM}/f^{thin plate}$
1-11.02
1-31.18
3-31.17
+ +Analysis of a rhombic cantilever. The rhombic cantilever shown in Figure 8, fixed at one side and subjected to constant pressure was analysed using a $4 \times 4$ element mesh. In Table 5, the results for the transverse displacements at six locations are compared against the solutions obtained using the DKT triangular element $^{6}$ , experimental measurements $^{1}$ and using the 16-node isoparametric element (with $4 \times 4 \times 2$ Gauss integration). In all cases a one step geometric non-linear analysis with equilibrium iterations was performed. Good correspondence between the experimental results and the solution obtained using our new 4-node element is observed. + +# Linear analysis of a cylindrical (Scordelis–Lo) shell + +The shell structure shown in Figure 9a has frequently been used to test the performance of shell elements $^{12}$ . Figure 9b shows the solutions obtained with our elements. In each of the solutions uniform meshes with equal sized elements were employed over one-quarter of the shell. Solutions obtained using the 3-node DKT triangular element $^{25}$ and the 16-node isoparametric element $^{25}$ are also shown. + +# Linear analysis of a pinched cylinder + +The pinched cylinder problem shown in Figure 10a was also frequently analysed to test shell elements. Figure 10b and Tables 6 and 7 show the convergence behaviour obtained with our new element, when comparing the finite element solutions $^{11,21}$ . Note that using the isoparametric shell element $^{3}$ also a fairly large number of degrees of freedom are required to predict the response of the cylinder accurately. + +# Large deflection analysis of a cantilever + +The cantilever shown in Figure 11a was analysed for its large displacement and large rotation response. This is a typical problem considered to test the geometric nonlinear behaviour of beam and shell elements $^{25}$ . Figure 11a also shows the models used in the analysis. + +The first two models are single element, cubic and parabolic isoparametric degenerate shell element models. Model I predicts the response of the cantilever very accurately, whereas model II yields an accurate response solution in linear analysis but locks once the element is curved in the non-linear response solution. This observation is in accordance with the results reported elsewhere $^{5}$ . + +The same nodal point layouts were next employed for models III and IV using our new 4-node shell element. Figures 11b–11d give the results obtained with these models. It is seen that model III yields an accurate large displacement response prediction, and even model IV yields quite accurate results up to about 60 degrees of rotation. The computer time required in these analyses were only little different using models I, III and IV. + +Another important result is shown in Table 8. As reported earlier $^{5}$ , the cubic shell element is sensitive to 'in-plane' distortions, and hence it is interesting to study the effect of using a distorted element mesh in the analysis of the cantilever (see Figures 12a and 12b). Table 8 summarizes the results obtained using the one cubic element and three 4-node elements with a nodal layout that corresponds to distorting the elements. It is seen that the predictive capability of our new 4-node element is considerably less sensitive to the element distortions. + + + +![](images/page-008_5da32209f9cc0a6196f1164fc9ce754f4ae6c38f136e9ce5053c93e0be7c0c81.jpg) + +
+text_image + +x₂ +3 +2 +1 +45° +6 +5 +4 +12 +x₁ +12 +u₁₋₂₋₃ = α = β = 0 +
+ +4 x 4 mesh - 4-node elements + +![](images/page-008_340afbf55b4f809ae3ac12c13d5aba569d93c5d818819279e228ce3c5951471c.jpg) + +
+text_image + +4 x 4 mesh - DKT elements +2 x +
+ +![](images/page-008_b658945d9e061f5d0f1bcb162247b3e5701759c6319100d3bed3030913ba93e2.jpg) + +
+text_image + +2 x 2 mesh - 16-node elements +(Int. 4x4x2) +
+ +Figure 8 Response of rhombic cantilever subjected to constant pressure. q=0.26066; $E=10.5\times10^{6}$ ; thickness=0.125; r=0.3 +Table 5 + +
ElementMeshCPU timeCPU time of DKTDeflection at location
123456
DKT4×41.000.2930.1960.1140.1180.0550.024
4-node4×4approx. 20.2720.1830.1060.1020.0460.019
16-node2×2approx. 6 $\frac{1}{2}$ 0.2660.1820.1100.1050.0480.019
Experimental $^1$ 0.2970.2040.1210.1290.0560.022
+ +![](images/page-008_e1cf53288253eab932778ca78bfedae1ad8dc7144444b1e7b165c443e4bf982d.jpg) + +
+text_image + +diaphragm +φ +R +A +D +B +C +L +y +z +
+ +(a) Cylindrical shell + +![](images/page-008_18f29259871d03f105eb75c2aa4c103e6d193978c0ab18cd8c222680ec1d647e.jpg) + +
+line + +| Number of d.o.f. | w_B | Grid Size | +| ---------------- | ---- | --------- | +| 2 x 1 | 3.45 | (2 x 1) | +| 5 x 5 | 3.45 | (5 x 5) | +| 8 x 8 | 3.50 | (8 x 8) | +| 12 x 12 | 3.55 | (12 x 12) | +
+ +(b) Convergence of displacement at point B +Figure 9 Linear analysis of a cylinder shell subjected to dead weight. The $2 \times 1$ result refers to the solution obtained with two 16-node shell elements spanning from C to B. The $16 \times 16$ result refers to the use of 512 equal triangular DKT elements. R=300; L=600; $\phi=40^{\circ}$ ; thickness=3.0; $E=3 \times 10^{6}$ ; v=0.0; specific weight=0.208333, ——, reference solutions; ●—●, present study; □, 16-node element (Int. $4 \times 4 \times 2$ ); ∇, DKT element + +Geometric non-linear response of a shallow spherical shell + +Figure 13a shows the spherical shell that was also analysed $^{3}$ with one cubic shell element, modelling one-quarter of the shell. To test our new 4-node shell element, the same nodal point layout was used $^{3}$ , giving a mesh of nine elements. Figure 13b shows the response calculated, including the post-buckling response (not reported in ref. 3) with the automatic load stepping algorithm $^{4}$ . Good correspondence with the analytical solution of Leicester $^{20}$ and the solution of Horrigmoe $^{16}$ was obtained. The solution with the 16-node element was almost twice as expensive as the 4-node element solution (using in both cases the same parameters for the automatic step-by-step solution algorithm). + +Linear buckling analysis and large deflection response of a simply-supported stiffened plate + +The stiffened plate shown in Figure 14a was analysed for its buckling reresponse. Since we expect the buckling mode to be symmetric $^{26}$ only one-quarter of the plate is modelled using symmetry boundary conditions. The model consists of nine 4-node shell elements and three 2-node isoparametric beam elements. At the nodes where a shell element connects to a beam element, three rotational degrees of freedom aligned with the global axes are considered for the shell element. In order to avoid locking of the isoparametric beam elements, one point Gauss integration along the beam axes was used. This does not introduce spurious zero energy modes in the model although the bending stiffness of the beam is underestimated. + +The linearized buckling problem was solved as described in reference 4(37) and we obtained: + +$$ +\frac {\sigma_ {\mathrm{cr}} (\text { finite element solution })}{\sigma_ {\mathrm{cr}} (\text { analytical solution })} = 1. 0 2 +$$ + + + +![](images/page-009_caf3c4b08aca827d2c811c83e3da163e5ea94265b15cea98c674f800ed5888ca.jpg) + +
+text_image + +L/2 +P +L/2 +D +C +R +A +B +end +diaphragm +end +diaphragm +P +
+ +(a) Pinched cylinder. $R / t = 100, L / R = 2$ + +![](images/page-009_8cf79691768477d08f742f3a168d99cbba823b5ec17c33c1039cfaa10981a892.jpg) + +
+line + +| Time Point | Etw/P (Top) | Etw/P (Bottom) | Etu/P (Top) | Etu/P (Bottom) | +| ---------- | ----------- | -------------- | ----------- | -------------- | +| D | 0 | 0 | 0 | 0 | +| C | -50 | -150 | 0 | 0 | +| A | -100 | -150 | 0 | 0 | +| C | -150 | -150 | 0 | 0 | +
+ +(b) Displacements: —, analytical solution; +, present study (20×20 mesh). +Figure 10 Linear analysis of a pinched cylinder; u=axial displacement, w=radial displacement + +Table 6 Convergence study for 4-node element: pinched cylinder + +
Mesh for 1/8th of shellNumber of d.o.f. $\hat{w}_{C}^{FEM}/\hat{w}_{C}^{analyt}$
5×51300.51
10×105100.83
20×2020200.96
+ +$\hat{w}_{C}$ (series solution) = -164.24 by Lindberg et al. $\hat{w}_{C} = \frac{w_{C}Et}{P}$ + +Table 7 Comparison between displacements for 4-node and 16-node elements: pinched cylinder + +
ElementMesh for $\frac{1}{8}$ th of shellNumber of d.o.f. $\hat{w}_{C}^{FEM}/\hat{w}_{C}^{analyt}$
4-node20×2020200.96
16-node10×1045300.98
+ +Next, an initial imperfection with the shape of the first buckling mode and a maximum amplitude of 1/5 of the plate thickness was introduced. Figure 14b shows the large deflection response of this model as calculated using the automatic load stepping scheme of reference 4 with a tight energy convergence tolerance. + +# Analysis of elastoplastic response of a circular plate + +The thin circular plate shown in Figure 15a was analysed for its elastoplastic response, when subjected to a concentrated load at its centre. The plate is simply-supported with its edges restrained from moving in its plane. + +In a first solution, the plate model shown in Figure 15a was used to analyse the plate assuming small displacements (materially-non-linear-only conditions). Figure 15c shows that the theoretical collapse load is overestimated, but for the coarse mesh used, the predicted response is quite reasonable. + +In a second solution, large displacements and elastoplastic conditions were assumed and in this case the stiffening behaviour of the plate shown in Figure 15c was predicted. In order to have a comparison, also the model of five axisymmetric 8-node elements shown in Figure 15b was solved. Figure 15c shows that both models predict in essence the same response; however, in this case relatively little plasticity was developed for the range of displacements considered. + +# CONCLUSIONS + +A new four-node non-flat general non-linear shell element has been presented with the following important element properties: (1) the element is formulated using three-dimensional continuum mechanics theory; hence the use of the element is not restricted by application of a specific shell theory; (2) the element is reliable and has good predictive capability in the analysis of thick and thin shells; (3) the amount of computations required to calculate the element stiffness matrix are very closely those that are used in standard isoparametric formulations. The computer time used could be reduced considerably in elastic analysis by using analytical integration through the element thickness. + +In this paper we have presented the formulation and some applications of the element. The solution results obtained are most encouraging, but a formal mathematical convergence study of the element would be very valuable, and we are currently pursuing such research. + +Finally, it should be noted that the element presented here provides a very attractive basic formulation that could be extended to large strain analysis and analysis of composite shells. Also, the concepts applied here to formulate a 4-node element could equally well be employed in an effective manner to formulate higher-order shell elements. + +# ACKNOWLEDGEMENTS + +We are grateful for the financial support by the U.S. Army contract no. DAAK11-82-K-0005 and the ADINA users group for this work. + +Note added in proof. — We have just learned — and regret not to have known of it earlier — that R. H. MacNeal [J. Nucl. Eng. Design, 70, 3–12 (1982)] proposed a plate element for linear analysis that is very close to the element presented above. + + + +![](images/page-010_1a4e5447eff6132d258a7e2b08738cb779d2a112055027502b4da88578c508e0.jpg) + +
+text_image + +z +b +y +u +φ +w +M +x +L +
+ +![](images/page-010_2fca9c66a0ea26af571892007ae9ad3e046dc7a6fe618069b5b1cf18dfffd4d7.jpg) + +
+text_image + +Int 4x2x2 +I +Int 3x2x2 +II +III +IV +
+ +(a) Finite element models: $b = 1.0$ ; $t = 1.0$ ; $L = 12.0$ ; $E = 1800$ ; $v = 0.0$ + +![](images/page-010_a9a3dc752fe05a0fe3390b5c9d12147c22e06c9b8be587266fdf0659be489412.jpg) + +
+line + +| η = ML / 2π EI | u/L | w/L | φ/2π | +| -------------- | ------ | ------ | ------ | +| 0.0 | 0.0000 | 0.0000 | 0.0000 | +| 0.05 | 0.0500 | 0.1000 | 0.0250 | +| 0.10 | 0.1000 | 0.2000 | 0.0500 | +| 0.15 | 0.1500 | 0.3000 | 0.0750 | +| 0.20 | 0.2000 | 0.4000 | 0.1000 | +| 0.25 | 0.2500 | 0.5000 | 0.1250 | +| 0.30 | 0.3000 | 0.6000 | 0.1500 | +
+ +(c) Response of model III + +![](images/page-010_ac31feb014ad68fd431cdc04e17196a071969d0a0ad53fb93a246a6fd68c1b00.jpg) + +
+line + +| η = ML / 2πEI | u/L | w/L | φ/2π | +| ------------- | ------ | ------ | ------ | +| 0.00 | 0.0000 | 0.0000 | 0.0000 | +| 0.05 | 0.0500 | 0.1000 | 0.0250 | +| 0.10 | 0.1000 | 0.2000 | 0.0500 | +| 0.15 | 0.1500 | 0.3000 | 0.0750 | +| 0.20 | 0.2000 | 0.4000 | 0.1000 | +| 0.25 | 0.2500 | 0.5000 | 0.1250 | +| 0.30 | 0.3000 | 0.6000 | 0.1500 | +
+ +(b) Response of model I + +![](images/page-010_f956249974be0a7100fffa5b018c13399784ee4d88f2f9c71e73ed966eb2ddf5.jpg) + +
+line + +| η = ML/2πEI | u/L | w/L | φ/2π | +| ----------- | ------ | ------ | ------ | +| 0.0 | 0.0000 | 0.0000 | 0.0000 | +| 0.05 | 0.0500 | 0.1000 | 0.0200 | +| 0.10 | 0.1000 | 0.2000 | 0.0500 | +| 0.15 | 0.1500 | 0.3000 | 0.1000 | +| 0.20 | 0.2000 | 0.4000 | 0.1500 | +| 0.25 | 0.2500 | 0.5000 | 0.2000 | +| 0.30 | 0.3000 | 0.6000 | 0.2500 | +
+ +(d) Response of model IV +Figure 11 Large deflection analysis of a cantilever using non-distorted elements. —, Analytical solution, ●, □, ▽, respective model response + +# REFERENCES + +1 Adini, A. Analysis of shell structures by the finite element method, PhD Dissertation, Department of Civil Engineering, University of California, Berkeley (1961) +2 Bathe, K. J. Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, New Jersey (1982) + +3 Bathe, K. J. and Bolourchi, S. A geometric and material nonlinear plate and shell element, J. Comput. Struct., 11, 23–48 (1979) +4 Bathe, K. J. and Dvorkin, E. N. On the automatic solution of nonlinear finite element equations, J. Comput. Struct. 17, (5–6), 871–879 (1983) +5 Bathe, K. J., Dvorkin, E. N. and Ho, L. W. Our discrete-Kirchhoff and isoparametric shell elements for nonlinear analysis – an assessment, J. Comput. Struct., 16, (1–4), 89–98 (1983) diff --git a/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_002.md b/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_002.md new file mode 100644 index 0000000..8e5bdd9 --- /dev/null +++ b/docs/reference-papers/MITC4/AContinuumMechanicsBasedFourNodeShell/AContinuumMechanicsBasedFourNodeShell_002.md @@ -0,0 +1,173 @@ + + +![](images/page-011_ce5ce4625df64c46433bc92128a90eb39be61afbd4c2315e7e0c0071db43588a.jpg) + +
+text_image + +3. +Int 4x2x2 +
+ +(a) +Model I - distorted + +![](images/page-011_039915932f73557d586a445907a46b35c323592070cc7fd250afef7c523be1ae.jpg) + +
+text_image + +4. +4. +
+ +(b) +Model III - distorted +Figure 12: Large deflection analysis of a cantilever using distorted elements + +Table 8 Results for large deflection analysis of a cantilever using distorted elements + +
Model I (distorted)Model III (distorted)
step 2step 5step 8step 2step 5step 8
$\phi^{FEM}/\phi^{analyt}$ 0.130.130.130.950.840.76
$u^{FEM}/u^{analyt.}$ 0.010.010.010.890.680.56
$w^{FEM}/w^{analyt}$ 0.100.110.120.950.860.81
$\phi^{analyt}$ 18°45°72°18°45°72°
+ +![](images/page-011_23b0df71e88de62769bd055734ee603af4fcfa7adf7da03a944a9206496022b7.jpg) + +
+text_image + +P +2a +h +2a +R1 +R2 +
+ +(a) Spherical shell + +![](images/page-011_6d1d03aec07b27283cc63c39e2ea41ab7c89aa78366687854f02d2fe26fcfd54.jpg) + +
+line + +| Central deflection, Wc | Central load, (P/1000) | +| ---------------------- | ---------------------- | +| 0 | 0 | +| 50 | 30 | +| 100 | 45 | +| 150 | 50 | +| 200 | 40 | +| 250 | 35 | +| 300 | 55 | +
+ +(b) Non-linear load displacement curve. +Figure 13 Geometric non-linear response of a spherical shell. O, Horrigmoe; —, Leicester; ●, nine 4-node elements; □, one 16-node element Int 4×4×2 + +![](images/page-011_c2174153f8463396ae83e28f9b093efd21de5102a9dd440d8d175f192ab5eff0.jpg) + +
+text_image + +ε +ε +102. +54. +0.54 +0.5 +4 +
+ +(a) Stiffened plate + +![](images/page-011_cf304ccd3c6278299800e4191548a141c23c268e4c96d4d53b4accd146b583fb.jpg) + +
+line + +| Vertical displac. of center | τ/τ_CR | +| --------------------------- | ------ | +| 0.004 | 0.95 | +| 0.008 | 1.00 | +| 0.012 | 1.00 | +| 0.016 | 1.00 | +| 0.020 | 1.00 | +
+ +(b) Large deflection response +Figure 14 Non-linear response of a stiffened plate. $E=2.1\times10^{6}$ ; v=0.3 + +6 Bathe, K. J. and Ho, L. W. A simple and effective element for analysis of general shell structures, J. Comput. Struct., 13, 673–682 (1980) +7 Bathe, K. J. and Hô, L. W. Some results in the analysis of thin shell structures, Nonlinear Finite Element Analysis in Structural Mechanics, (Ed. W. Wunderlich et al.), Springer-Verlag, Berlin (1981) +8 Batoz, J. L., Bathe, K. J. and Ho, L. W. A study of three-node triangular plate bending elements, Int. J, Num. Meth. Eng., 15, 1771–1812 (1980) +9 Batoz, J. L. and Ben Tahar, M. Evaluation of a new quadrilateral plate bending element, Int. J. Num. Meth. Eng., 18, 1655–1677 (1982) +10 Bercovier, M., Hasbani, Y., Gilon, Y., and Bathe, K., J., On a finite element procedure for nonlinear incompressible elasticity, Hybrid and Mixed Finite Element Methods, (Ed, S. M. Atluri et al.), John Wiley, New York (1983) +11 Flügge, W. Stresses in Shells, 2nd edn, Springer-Verlag, Berlin (1973) +12 Forsberg, K. and Hartung, R. An evaluation of finite difference and finite element techniques for analysis of general shells, Symp. High Speed Computing of Elastic Structures, IUTAM, Liège (1970) +13 Fung, Y. C. Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey (1965) +14 Gallagher, R. H. Problems and progress in thin shell finite element analysis, Finite Elements in Thin Shells and Curved Members, (Ed. D. G. Ashwell and R. H. Gallagher), John Wiley, New York (1976) +15 Green, A. E. and Zerna, W. Theoretical Elasticity, 2nd edn, Oxford University Press (1968) +16 Horrigmoe, G. Finite element instability analysis of free-form shells, Report 77-2, Division of Structural Mechanics, The Norwegian Institute of Technology, University of Trondheim, Norway (1977) +17 Hughes, T. J. R. and Liu, W. K. Nonlinear finite element analysis of shells: Part I, Three-dimensional shells, J. Comput. Meth. Appl. Mech. Eng., 26, 331–362 (1981) + + + +![](images/page-012_4cb42a5782816bc4cedbd4393a3f6987776bde88664fcddcdc6d74f32812e067.jpg) + +
+text_image + +hinged +immovable edge +
+ +(a) 4-node shell model + +![](images/page-012_7914d5c2d5d8cf5d37f7c6a7e743ff25301e4b7687afbf07d510d3d86e254568.jpg) + +
+text_image + +t +R +
+ +(b) Axisymmetric model + +18 Irons, B. M. and Razzaque, A. Experience with the patch test for convergence of finite elements. The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, (Ed. A. K. Aziz), Academic Press, New York (1972) +19 Kråkeland, B. Nonlinear analysis of shells using degenerate isoparametric elements, Finite Elements in Nonlinear Mechanics, Vol. 1, (Ed. P. G. Bergan et al.), Tapir Publishers (Norwegian Institute of Technology, Trondheim, Norway) (1978) +20 Leicester, R. H. Finite deformations of shallow shells, Proc. Am. Soc. Civil Eng., 94, (EM6), 1409–1423 (1968) +21 Lindberg, G. M., Olson, M. D. and Cowper, G. R. New developments in the finite element analysis of shells, Q. Bull. Div. Mech. Eng. and the National Aeronautical Establishment, National Research Council of Canada, Vol. 4 (1969) +22 MacNeal, R. H. A simple quadrilateral shell element, J. Comput. Struct. 8, 175–183 (1978) +23 Noor, A. K. and Peters, J. M. Mixed models and reduced/selec- + +![](images/page-012_3a106631a364ac4a83c9b22dfb1ea729baf8c2940fd538340267a37203fe1421.jpg) + +
+line + +| Vertical displac. of center | P | +| --------------------------- | ----- | +| 0 | 0 | +| 1 | 1500 | +| 2 | 2500 | +| 3 | 1000 | +| 4 | 1200 | +| 5 | 1300 | +| 6 | 1400 | +| 7 | 1500 | +| 8 | 1600 | +| 9 | 1700 | +| 10 | 1700 | +
+ +(c) Elastoplastic load-displacement curve +Figure 15 Response of elastic-perfectly plastic circular plate subjected to a concentrated load, P, at its centre. TLF abbreviates use of total Lagrangian formulation and MNO abbreviates use of materially non-linear-only formulation. R=100, t=1; $E=2.1\times10^{6}$ ; $E_{T}=0.0$ ; $\nu=0.3$ ; $\sigma_{\nu}=1000$ . Circular plate response; —, axisymmetric model; +●, 4-node shell model + +tive integration displacement models for nonlinear analysis of curved beams, Int. J. Num. Meth. Eng., 17, 615–631 (1981) +24 Ramm, E. and Sattele, J. M. Elasto-plastic large deformation shell analysis using degenerated elements, Nonlinear Finite Element Analysis of Plates and Shells, (Ed. T. J. R. Hughes), AMD-Vol. 48, Am. Soc. Mech. Eng., New York (1981) +25 Report AE 83-5, ADINA System Verification Manual, ADINA Engineering, Västerås, Sweden and Watertown, Mass. (1983) +26 Timoshenko, S. P. and Gere, J. M. Theory of Elastic Stability, 2nd edn, McGraw-Hill, New York (1961) +27 Washizu, K. Variational Methods in Elasticity and Plasticity, Pergamon Press, Oxford and New York (1968) +28 Wempner, G., Talaslidis, D. and Hwang, C.-M. A simple and efficient approximation of shells via finite quadrilateral elements, J. Appl. Mech., 49, 115–120 (1982) +29 Zienkiewicz, O. C. 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a/docs/reference-papers/MITC4/FourNodeQuadrilateralShellElementMITC4/FourNodeQuadrilateralShellElementMITC4_001.md b/docs/reference-papers/MITC4/FourNodeQuadrilateralShellElementMITC4/FourNodeQuadrilateralShellElementMITC4_001.md new file mode 100644 index 0000000..01fd8e5 --- /dev/null +++ b/docs/reference-papers/MITC4/FourNodeQuadrilateralShellElementMITC4/FourNodeQuadrilateralShellElementMITC4_001.md @@ -0,0 +1,249 @@ + + +# Four-Node Quadrilateral Shell Element MITC4 + +DVORˇ AKOV´ A Edita´ 1,a,∗, PATZAK Bo´ ˇrek1,b + +1Department of Mechanics, Faculty of Civil Engineering, CTU in Prague, Thakurova 7, ´ 160 00 Prague, Czech Republic + +aedita.dvorakova@fsv.cvut.cz, bborek.patzak@fsv.cvut.cz + +Keywords: Finite Elements; MITC4; Scordelis-Lo Shell; Shear Locking; Shell Structures. + +Abstract. Four-node quadrilateral element MITC4 applicable to both thick and thin shells is presented. The element formulation starts from three-dimensional continuum description degenerated to shell behavior. Shear locking, which is common problem in analysis of thin shells, is overcome by the use of MITC (Mixed Interpolation of Tensorial Components) approach. Element has been implemented into finite element code OOFEM and its performance is demonstrated on Scordelis-Lo shell, a benchmark problem frequently used in the evaluation of shell elements. + +# Introduction + +Shell structures are widely used in structural engineering for their load-carrying efficiency. The finite element method appears to be the most powerful tool in analysis of shell structures, however the accuracy of the method is significantly influenced by the choice of the particular element. Many shell elements have been developed over the years. Existing elements can be divided into two main groups. One group consists of elements based on some particular shell theory, the other group of the elements is based on three-dimensional analysis degenerated to shell behavior. + +Presented quadrilateral element has been proposed by Dvorkin and Bathe [1] and is based on the second approach, therefore it is independent of any particular shell theory. The authors aimed to formulate element applicable to both thick and thin shells of arbitrary geometries with all degrees of freedom concentrated to the vertices of the element. However high-order elements with 9 and 16 nodes have been already successfully employed by Bathe and Bolourchi [2], this 4-node element suffers from the deficiency of shear locking. To prevent the element from the shear locking the use of MITC approach has been proposed. + +# Element Formulation + +The element is shown in Fig. 1 and its geometry is described using + +$$ +x _ {i} = \sum_ {k = 1} ^ {4} h _ {k} x _ {i} ^ {k} + \frac {r _ {3}}{2} \sum_ {k = 1} ^ {4} a _ {k} h _ {k} V _ {n i} ^ {k}, \tag {1} +$$ + +where $h _ { k } ( r _ { 1 } , r _ { 2 } )$ are the two-dimensional interpolation functions corresponding to node $k , r _ { i }$ are the natural coordinates, $x _ { i }$ are the Cartesian coordinates of any point in the element, $x _ { i } ^ { k }$ are the Cartesian coordinates of node $k , V _ { n i } ^ { k }$ are the components of director vector at node k and $a _ { k }$ is the thickness of structure measured along the director vector. The displacement of any point in the element can be described using + +$$ +u _ {i} = \sum_ {k = 1} ^ {4} h _ {k} u _ {i} ^ {k} + \frac {r _ {3}}{2} \sum_ {k = 1} ^ {4} a _ {k} h _ {k} \left(- V _ {2 i} ^ {k} \alpha_ {k} + V _ {1 i} ^ {k} \beta_ {k}\right), \tag {2} +$$ + + + +![](images/page-002_e6a485030491b89d22a5ccc1e2c171ae773fae1da876ddf8ca524da4398a7728.jpg) + +
+text_image + +Vₙ³ +3 +B +Vₙ² +C +Vₙ⁴ +r₃ +r₂ +r₁ +2 +4 +D +Vₙ¹ +A +1 +x₃ +x₂ +x₁ +
+ +Fig. 1: Geometry of quadrilateral shell element MITC4. + +where $u _ { i }$ are the displacements in direction of Cartesian coordinates and $\alpha _ { k }$ and $\beta _ { k }$ are the rotations of director vector at node k around $V _ { 1 } ^ { k }$ and $V _ { 2 } ^ { k }$ , where + +$$ +V _ {1} ^ {k} = \frac {\boldsymbol {e} _ {2} \times V _ {n} ^ {k}}{\left\| \boldsymbol {e} _ {2} \times V _ {n} ^ {k} \right\|}, \tag {3} +$$ + +$$ +V _ {2} ^ {k} = V _ {n} ^ {k} \times V _ {1} ^ {k}. \tag {4} +$$ + +Vector $e _ { 2 }$ is the basis vector of Cartesian coordinate system. Considering linear interpolation of displacement field, we introduce the vector of unknown displacements + +$$ +\boldsymbol {r} _ {e} = \left\{u _ {1} ^ {1}, u _ {2} ^ {1}, u _ {3} ^ {1}, \alpha_ {1}, \beta_ {1}, u _ {1} ^ {2}, u _ {2} ^ {2}, u _ {3} ^ {2}, \alpha_ {2}, \beta_ {2}, u _ {1} ^ {3}, u _ {2} ^ {3}, u _ {3} ^ {3}, \alpha_ {3}, \beta_ {3}, u _ {1} ^ {4}, u _ {2} ^ {4}, u _ {3} ^ {4}, \alpha_ {4}, \beta_ {4} \right\} ^ {T}. \tag {5} +$$ + +The biggest drawback of this formulation of the element is that the element locks when the thickness of shell is small. Using the interpolation (2), non-zero transverse shear strains are obtained even when the structure is subjected to the constant bending moment. The remedy is to construct an assumed transverse strain, such that + +$$ +\tilde {\varepsilon} _ {1 3} = \frac {1}{2} (1 + r _ {2}) \tilde {\varepsilon} _ {1 3} ^ {A} + \frac {1}{2} (1 - r _ {2}) \tilde {\varepsilon} _ {1 3} ^ {C}, \tag {6} +$$ + +$$ +\tilde {\varepsilon} _ {2 3} = \frac {1}{2} (1 + r _ {1}) \tilde {\varepsilon} _ {2 3} ^ {D} + \frac {1}{2} (1 - r _ {1}) \tilde {\varepsilon} _ {2 3} ^ {B}, +$$ + +where ∼ over the quantities emphasizes the formulation in convected coordinate system and components $\tilde { \varepsilon } _ { 1 3 } ^ { A } , \tilde { \varepsilon } _ { 2 3 } ^ { B } , \tilde { \varepsilon } _ { 1 3 } ^ { C }$ and $\tilde { \varepsilon } _ { 2 3 } ^ { D }$ are equal to the values of corresponding strains in the middle of element edges calculated from displacements. This assumed transversal strain $\tilde { \varepsilon } _ { 1 3 }$ is constant along $r _ { 1 }$ direction and linear along $r _ { 2 }$ direction. + +To transform formulas for shear strain components to Cartesian coordinate system, it is convenient to use covariant and contravariant bases in convected coordinate system of $r _ { 1 } , r _ { 2 }$ and $r _ { 3 }$ . For the element with midsurface in plane $x , y$ the covariant basis vectors ${ \pmb g } _ { i } ( i = 1 , 2 , 3 )$ are given by + + + +$$ +\boldsymbol {g} _ {i} = \frac {\partial \boldsymbol {x}}{\partial r _ {i}}. \tag {7} +$$ + +Contravariant basis vectors $\pmb { g } ^ { j } \left( j = 1 , 2 , 3 \right)$ can be then calculated using + +$$ +\boldsymbol {g} _ {i} \boldsymbol {g} ^ {j} = \delta_ {i} ^ {j}, \tag {8} +$$ + +where $\delta _ { i } ^ { j }$ is Kronecker delta, $\delta _ { i } ^ { j } = 1$ for $i = j$ and $\delta _ { i } ^ { j } = 0$ otherwise. Components $\tilde { \varepsilon } _ { i j }$ can be evaluated using + +$$ +\tilde {\varepsilon} _ {i j} = \frac {1}{2} \left(\frac {\partial \boldsymbol {u}}{\partial r _ {i}} \boldsymbol {g} _ {j} + \frac {\partial \boldsymbol {u}}{\partial r _ {j}} \boldsymbol {g} _ {i}\right), \tag {9} +$$ + +$$ +\tilde {\varepsilon} _ {i j} = \frac {1}{2} \left(\frac {\partial \boldsymbol {u}}{\partial r _ {i}} \frac {\partial \boldsymbol {x}}{\partial r _ {j}} + \frac {\partial \boldsymbol {u}}{\partial r _ {j}} \frac {\partial \boldsymbol {x}}{\partial r _ {i}}\right). \tag {10} +$$ + +By substituting from (1) and (2) into equation (10) and evaluating at points $A , B , C , D$ the relations for $\tilde { \varepsilon } _ { 1 3 } ^ { A } , \tilde { \varepsilon } _ { 2 3 } ^ { B } , \tilde { \varepsilon } _ { 1 3 } ^ { C }$ and $\tilde { \varepsilon } _ { 2 3 } ^ { D }$ can be obtained + +$$ +\begin{array}{l} \tilde {\varepsilon} _ {1 3} ^ {A} = \frac {1}{8} \left[ \left(\boldsymbol {u} ^ {1} - \boldsymbol {u} ^ {2}\right) \cdot \frac {1}{2} \left(a _ {1} V _ {n} ^ {1} + a _ {2} V _ {n} ^ {2}\right) + \right. \\ \left. \left(\boldsymbol {x} ^ {1} - \boldsymbol {x} ^ {2}\right) \cdot \frac {1}{2} \left(a _ {1} \left(- V _ {2} ^ {1} \alpha_ {1} + V _ {1} ^ {1} \beta_ {1}\right) + a _ {2} \left(- V _ {2} ^ {2} \alpha_ {2} + V _ {1} ^ {2} \beta_ {2}\right)\right) \right], \\ \tilde {\varepsilon} _ {1 3} ^ {C} = \frac {1}{8} \left[ \left(\boldsymbol {u} ^ {4} - \boldsymbol {u} ^ {3}\right) \cdot \frac {1}{2} \left(a _ {3} V _ {n} ^ {3} + a _ {4} V _ {n} ^ {4}\right) + \right. \\ \left. \left(\boldsymbol {x} ^ {4} - \boldsymbol {x} ^ {3}\right) \cdot \frac {1}{2} \left(a _ {3} \left(- V _ {2} ^ {3} \alpha_ {3} + V _ {1} ^ {3} \beta_ {3}\right) + a _ {4} \left(- V _ {2} ^ {4} \alpha_ {4} + V _ {1} ^ {4} \beta_ {4}\right)\right) \right], \tag {11} \\ \tilde {\varepsilon} _ {2 3} ^ {B} = \frac {1}{8} \left[ \left(\boldsymbol {u} ^ {1} - \boldsymbol {u} ^ {4}\right) \cdot \frac {1}{2} \left(a _ {1} V _ {n} ^ {1} + a _ {4} V _ {n} ^ {4}\right) + \right. \\ \left. \left(\boldsymbol {x} ^ {1} - \boldsymbol {x} ^ {4}\right) \cdot \frac {1}{2} \left(a _ {1} \left(- V _ {2} ^ {1} \alpha_ {1} + V _ {1} ^ {1} \beta_ {1}\right) + a _ {4} \left(- V _ {2} ^ {4} \alpha_ {4} + V _ {1} ^ {4} \beta_ {4}\right)\right) \right], \\ \tilde {\varepsilon} _ {2 3} ^ {D} = \frac {1}{8} \left[ \left(\boldsymbol {u} ^ {2} - \boldsymbol {u} ^ {3}\right) \cdot \frac {1}{2} \left(a _ {2} V _ {n} ^ {2} + a _ {3} V _ {n} ^ {3}\right) + \right. \\ \left. \left(\boldsymbol {x} ^ {2} - \boldsymbol {x} ^ {3}\right) \cdot \frac {1}{2} \left(a _ {2} \left(- V _ {2} ^ {2} \alpha_ {2} + V _ {1} ^ {2} \beta_ {2}\right) + a _ {3} \left(- V _ {2} ^ {3} \alpha_ {3} + V _ {1} ^ {3} \beta_ {3}\right)\right) \right]. \\ \end{array} +$$ + +Final equations for $\tilde { \varepsilon } _ { 1 3 }$ and $\tilde { \varepsilon } _ { 2 3 }$ are then derived by substituting (11) back into the equations (6). Transformation to the Cartesian coordinates is performed using + +$$ +\tilde {\varepsilon} _ {i j} \boldsymbol {g} ^ {i} \boldsymbol {g} ^ {j} = \varepsilon_ {k l} \boldsymbol {e} _ {k} \boldsymbol {e} _ {l}, \tag {12} +$$ + +where $\varepsilon _ { k l }$ are components of strain tensor in Cartesian coordinates with basis vectors $\boldsymbol { e } _ { k }$ and $e _ { l }$ . By considering the symmetry of strain tensor, the shear components of strain vector $\gamma _ { x z }$ and $\gamma _ { y z }$ are obtained + +$$ +\gamma_ {x y} = 2 \tilde {\varepsilon} _ {1 3} \left(\boldsymbol {g} ^ {1} \cdot \boldsymbol {e} _ {1}\right) \left(\boldsymbol {g} ^ {3} \cdot \boldsymbol {e} _ {3}\right) + 2 \tilde {\varepsilon} _ {2 3} \left(\boldsymbol {g} ^ {2} \cdot \boldsymbol {e} _ {1}\right) \left(\boldsymbol {g} ^ {3} \cdot \boldsymbol {e} _ {3}\right), \tag {13} +$$ + +$$ +\gamma_ {y z} = 2 \tilde {\varepsilon} _ {1 3} (\pmb {g} ^ {1} \cdot \pmb {e} _ {2}) (\pmb {g} ^ {3} \cdot \pmb {e} _ {3}) + 2 \tilde {\varepsilon} _ {2 3} (\pmb {g} ^ {2} \cdot \pmb {e} _ {2}) (\pmb {g} ^ {3} \cdot \pmb {e} _ {3}) +$$ + +Substituting from (6) and (11) into (13) the final formulas for transverse shear strains are derived. The remaining components of strain vector are calculated directly from the displacements. + + + +![](images/page-004_b59ba642a14f205b5ede6bcbde1874b06756a0b760891d0a63ff604ebd6c3217.jpg) + +
+text_image + +rigid diaphragm +z +y +x +R +40° +B +L +E = 3.10⁶ +v = 0.0 +L = 600.0 +R = 300.0 +thickness = 3.0 +specific weight = 0.208333 +
+ +NOSTI AUTODESK Č ENO VE VÝUKOVÉM PRODUKTU SPOLE Ř VYTVOFig. 2: Cylindrical Scordelis-Lo shell subjected to dead weight. + +Stiffness matrix is evaluated using + +$$ +\boldsymbol {K} = \int_ {V} \boldsymbol {B} ^ {T} \boldsymbol {D} \boldsymbol {B} d V, \tag {14} +$$ + +where B is a strain-displacement matrix resulting from derived formulas for strain components and D is a material matrix for three-dimensional problem degenerated to shell behaviour, where the condition of $\sigma _ { z } = 0$ is enforced. + +# Scordelis-Lo Shell Problem + +Element has been implemented into finite element code OOFEM [3, 4] and implementation has been verified using classical patch tests. It has been proven, that the patch tests are passed for the case of pure bending, pure shear and pure twist and also for three membrane stress states. + +To test the element performance on more complex structure and to compare its results with other available elements and with the analytical solution the analysis of Scordelis-Lo Shell [1] is presented. The structure is loaded by its dead weight and is shown in Fig. 2. Due to the symmetry only one quarter of the structure has been analyzed. Deformed shape of the structure is shown in Fig. 3, obtained results are shown in Fig. 4. Solution obtained using the element composed of plane-stress element with rotational degrees of freedom and Discrete Kirchhoff Triangle plate element, is also shown for reference, labeled as RDKT. + +# Summary + +Element for shells has been implemented into existing finite element code OOFEM and its performance has been verified. Shear locking, which can cause highly inaccurate results in analysis of thin shells, has been overcome by MITC approach. This approach seems to be sufficient as appropriate results were obtained while testing the element implementation. Numerical results has shown that the element is competitive with the performance of thin-plate elements. Possibility of use of MITC4 element for both thick and thin shells can be seen as advantage over the RDKT and other shell theory based elements, which are usually limited by thickness/span ratio to either thin or thick shells. The element shows very good convergence to the reference solution [1], even outperforming the planar shell element composed of thin Kirchhoff plate element and membrane element. + + + +![](images/page-005_5264a113c58e729f97b7450003ab398d57a3a2bc1d6b8609fabcfeee66be49af.jpg) + +
+heatmap + +| u3 | Value | +|-------|--------| +| 0.529 | -3.53 | +
+ +Fig. 3: Cylindrical Scordelis-Lo shell subjected to dead weight. Only one quarter of the structure has been analyzed, mesh of 8×8 elements and deformed shape are shown. + +![](images/page-005_565a63add77008dc0e0fec449464747117a8c7e2ce311984a3e7f5e83f429837.jpg) + +
+line + +| Number of quadrilateral (triangular) elements | MITC4 | RDKT | +| --------------------------------------------- | ----- | ---- | +| 4x4(x2) | 3.43 | 3.35 | +| 8x8(x2) | 3.53 | 3.51 | +| 16x16(x2) | 3.59 | 3.58 | +| 32x32(x2) | 3.61 | 3.60 | +| 64x64(x2) | 3.61 | 3.60 | +
+ +Fig. 4: Analysis of cylindrical Scordelis-Lo shell subjected to dead weight. Convergence of displacement at point B is studied, solutions obtained with MITC4 and RDKT elements are shown as well as analytical solutions. + + + +# Acknowledgement + +The financial support of this research by the Grant Agency of the Czech Technical University in Prague (SGS project No. 15/031/OHK1/1T/11) and by the Technology Agency of the Czech Republic (TACR project No. TA02011196) is gratefully acknowledged. ˇ + +# References + +[1] K. J. Bathe, E. Dvorkin, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput., Vol.1, March (1984) 77-88. +[2] K. J. Bathe, S. Bolourchi, A geometric ad material nonlinear plate and shell element, Computers & Structures, Vol.11, (1980) 23-48. +[3] B. Patzak, Z. Bittnar, Design of object oriented finite element code, Advances in Engineering ´ Software 32 (10-11) (2001) 759-767. +[4] B. Patzak, OOFEM project home page, http://www.oofem.org, 2014. ´ +[5] K. J. Bathe, Finite Element Procedures, Prentice Hall International, Inc., 1996. +[6] M. L. Bucalem, K. J. Bathe, Finite element analysis of shell structures, Archives of Computational Methods in Engineering, Vol.4, 1 (1997) 3-61. + + + +# Modern Methods of Experimental and Computational Investigations in Area of Construction + +10.4028/www.scientific.net/AMM.825 + +# Four-Node Quadrilateral Shell Element MITC4 + +10.4028/www.scientific.net/AMM.825.99 + +# DOI References + +[1] K. J. Bathe, E. Dvorkin, A continuum mechanics based four-node shell element for general nonlinear analysis, Eng. Comput., Vol. 1, March (1984) 77-88. +10.1108/eb023562 +[2] K. J. Bathe, S. Bolourchi, A geometric ad material nonlinear plate and shell element, Computers & Structures, Vol. 11, (1980) 23-48. +10.1016/0045-7949(80)90144-3 +[3] B. Patz´ak, Z. Bittnar, Design of object oriented finite element code, Advances in Engineering Software 32 (10-11) (2001) 759-767. +10.1016/s0965-9978(01)00027-8 +[5] K. J. Bathe, Finite Element Procedures, Prentice Hall International, Inc., (1996). +10.1002/nme.1620190115 +[6] M. L. Bucalem, K. J. 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MITC shell element + +- 3차원 솔리드 형상으로부터 쉘형상을 표현 (유한요소 정식화가 다른 쉘요소에 비해 간단) +- 쉘 이론을 사용하지 않고 3차원 응력, 변형률을 사용하여 쉘을 표현할 수 있다. +임의의 형상에 대한 두꺼운 쉘과 얇은 쉘 모두 적용 가능 +Locking을 방지하기 위해 횡방향 전단 변형률에 보간법 사용 + +2. Kinematics + +![](page-001_64f5f5a6fb0b89f088f64198f2101fe1662d688eb8c02d8fa6f1a855f09aaed4.jpg) + +
+flowchart + +```mermaid +graph TD + A["0 Ω"] --> B["0 v_n²"] + B --> C["t u"] + C --> D["t + Δt u"] + D --> E["4"] + E --> F["3"] + F --> G["2"] + G --> H["1"] + H --> I["t + Δt v_n²"] + I --> J["t + Δt Ω"] + J --> K["t + Δt u"] + K --> L["t + Δt v_n²"] + L --> M["t + Δt u"] + M --> N["t + Δt v_n²"] + N --> O["t + Δt u"] + O --> P["t + Δt v_n²"] + P --> Q["t + Δt u"] + Q --> R["t + Δt v_n²"] + R --> S["t + Δt u"] + S --> T["t + Δt v_n²"] + T --> U["t + Δt u"] + U --> V["t + Δt v_n²"] + V --> W["t + Δt u"] + W --> X["t + Δt v_n²"] + X --> Y["t + Δt u"] + Y --> Z["t + Δt v_n²"] + Z --> AA["t + Δt u"] + AA --> AB["t + Δt v_n²"] + AB --> AC["t + Δt u"] + AC --> AD["t + Δt v_n²"] + AD --> AE["t + Δt u"] + AE --> AF["t + Δt v_n²"] + AF --> AG["t + Δt u"] + AG --> AH["t + Δt v_n²"] + AH --> AI["t + Δt u"] + AI --> AJ["t + Δt v_n²"] + AJ --> AK["t + Δt u"] + AK --> AL["t + Δt v_n²"] + AL --> AM["t + Δt u"] + AM --> AN["t + Δt v_n²"] + AN --> AO["t + Δt u"] + AO --> AP["t + Δt v_n²"] + AP --> AQ["t + Δt u"] + AQ --> AR["t + Δt v_n²"] + AR --> AS["t + Δt u"] + AS --> AT["t + Δt v_n²"] + AT --> AU["t + Δt u"] + AU --> AV["t + Δt v_n²"] + AV --> AW["t + Δt u"] + AW --> AX["t + Δt v_n²"] + AX --> AY["t + Δt u"] + AY --> AZ["t + Δt v_n²"] + AZ --> BA["t + Δt u"] + BA --> BB["t + Δt v_n²"] + BB --> BC["t + Δt u"] + BC --> BD["t + Δt v_n²"] + BD --> BE["t + Δt u"] + BE --> BF["t + Δt v_n²"] + BF --> BG["t + Δt u"] + BG --> BH["t + Δt v_n²"] + BH --> BI["t + Δt u"] + BI --> BJ["t + Δt v_n²"] + BJ --> BK["t + Δt u"] + BK --> BL["t + Δt v_n²"] + BL --> BM["t + Δt u"] + BM --> BN["t + Δt v_n²"] + BN --> BO["t + Δt u"] + BO --> BP["t + Δt v_n²"] + BP --> BQ["t + Δt u"] + BQ --> BR["t + Δt v_n²"] + BR --> BS["t + Δt u"] + BS --> BT["t + Δt v_n²"] + BT --> BU["t + Δt u"] + BU --> BV["t + Δt v_n²"] + BV --> BW["t + Δt u"] + BW --> BX["t + Δt v_n²"] + BX --> BY["t + Δt u"] + BY --> BZ["t + Δt v_n²"] + BZ --> CA["t + Δt u"] + CA --> CB["t + Δt v_n²"] + CB --> CC["t + Δt u"] + CC --> CD["t + Δt v_n²"] + CD --> CE["t + Δt u"] + CE --> CF["t + Δt v_n²"] + CF --> CG["t + Δt u"] + CG --> CH["t + Δt v_n²"] + CH --> CI["t + Δt u"] + CI --> CJ["t + Δt v_n²"] + CJ --> CK["t + Δt u"] + CK --> CL["t + Δt v_n²"] + CL --> CM["t + Δt u"] + CM --> CN["t + Δt v_n²"] + CN --> CO["t + Δt u"] + CO --> CP["t + Δt v_n²"] + CP --> CQ["t + Δt u"] + CQ --> CR["t + Δt v_n²"] + CR --> CS["t + Δt u"] + CS --> CT["t + Δt v_n²"] + CT --> CU["t + Δt u"] + CU --> CV["t + Δt v_n²"] + CV --> CW["t + Δt u"] + CW --> CX["t + Δt v_n²"] + CX --> CY["t + Δt u"] + CY --> CZ["t + Δt v_n²"] + CZ --> DA["t + Δt u"] + DA --> DB["t + Δt v_n²"] + DB --> DC["t + Δt u"] + DC --> DD["t + Δt v_n²"] + DD --> DE["t + Δt u"] + DE --> DF["t + Δt v_n²"] + DF --> DG["t + Δt u"] + DG --> DH["t + Δt v_n²"] + DH --> DI["t + Δt u"] + DI --> DJ["t + Δt v_n²"] + DJ --> DK["t + Δt u"] + DK --> DL["t + Δt v_n²"] + DL --> DM["t + Δt u"] + DM --> DN["t + Δt v_n²"] + DN --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + D --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + Do --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + EO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + O --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + AO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + BO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + OD --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DN --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO + DO --> DO +``` +
+ +![](page-001_65026891a36c5d50bc54d22bb4fc099887273632fcbc2746d2786df72ef41bb2.jpg) + +
+text_image + +mid surface +t+Δt E₃ +t+Δt Vₙ² +α₁² +t+Δt E₂ +α₂² +t+Δt V₂² +t+Δt E₁ +h₂ +t+Δt x +
+ + + +Shell의 초기 위치 벡터는 다음과 같이 shape function으로 나타낼 수 있다. + +$$ +{ } ^ { 0 } \mathbf { X } = \sum _ { i = 1 } ^ { 4 } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) { } ^ { 0 } \mathbf { X } _ { i } + \frac { \xi ^ { 3 } } { 2 } \sum _ { i = 1 } ^ { 4 } h _ { i } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) { } ^ { 0 } \mathbf { V } _ { n } ^ { i } +$$ + +마찬가지로 시간이 $t, t + \Delta t$ 일 때 위치벡터는 다음과 같다. + +$$ +{ } ^ { t } \mathbf { x } = \sum _ { i = 1 } ^ { 4 } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) { } ^ { t } \mathbf { x } _ { i } + \frac { \xi ^ { 3 } } { 2 } \sum _ { i = 1 } ^ { 4 } h _ { i } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) { } ^ { t } \mathbf { V } _ { n } ^ { i } +$$ + +$$ +{ } ^ { t + \Delta t } \mathbf { x } = \sum _ { i = 1 } ^ { 4 } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) ^ { t + \Delta t } \mathbf { x } _ { i } + \frac { \xi ^ { 3 } } { 2 } \sum _ { i = 1 } ^ { 4 } h _ { i } \phi _ { i } \left( \xi ^ { 1 } , \xi ^ { 2 } \right) ^ { t + \Delta t } \mathbf { V } _ { n } ^ { i } +$$ + +시간이 t일 때와 $t+\Delta t$ 일 때 변위 벡터는 다음과 같이 계산 할 수 있다. + +$$ +\begin{array}{l} { } ^ { t } \mathbf { u } = { } ^ { t } \mathbf { x } - { } ^ { 0 } \mathbf { X } \\ = \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t} \mathbf {x} _ {i} + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t} \mathbf {V} _ {n} ^ {i} - \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {X} _ {i} - \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {V} _ {n} ^ {i} \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t} \mathbf {x} _ {i} - ^ {0} \mathbf {X} _ {i}\right) + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t} \mathbf {V} _ {n} ^ {i} - ^ {0} \mathbf {V} _ {n} ^ {i}\right) \\ { } ^ { t + \Delta t } \mathbf { u } = { } ^ { t + \Delta t } \mathbf { x } - { } ^ { 0 } \mathbf { X } \\ = \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t + \Delta t} \mathbf {x} _ {i} + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {X} _ {i} - \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {V} _ {n} ^ {i} \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t + \Delta t} \mathbf {x} _ {i} - ^ {0} \mathbf {X} _ {i}\right) + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - ^ {0} \mathbf {V} _ {n} ^ {i}\right) \\ \end{array} +$$ + +$$ +\begin{array}{l} { } ^ { t + \Delta t } \mathbf { u } = { } ^ { t + \Delta t } \mathbf { x } - { } ^ { 0 } \mathbf { X } \\ = \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t + \Delta t} \mathbf {x} _ {i} + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - \sum_ {i = 1} ^ {4} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {X} _ {i} - \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} (\xi^ {1}, \xi^ {2}) ^ {0} \mathbf {V} _ {n} ^ {i} \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t + \Delta t} \mathbf {x} _ {i} - ^ {0} \mathbf {X} _ {i}\right) + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - ^ {0} \mathbf {V} _ {n} ^ {i}\right) \\ \end{array} +$$ + +따라서 시간 t와 $t + \Delta t$ 사이의 incremental displacement는 다음과 같이 나타낼 수 있다. + +$$ +\begin{array}{l} \Delta^ {t} \mathbf {u} = ^ {t + \Delta t} \mathbf {u} - ^ {t} \mathbf {u} \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t + \Delta t} \mathbf {x} _ {i} - ^ {0} \mathbf {X} _ {i}\right) + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - ^ {0} \mathbf {V} _ {n} ^ {i}\right) - \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t} \mathbf {x} _ {i} - ^ {0} \mathbf {X} _ {i}\right) - \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t} \mathbf {V} _ {n} ^ {i} - ^ {0} \mathbf {V} _ {n} ^ {i}\right) \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \left(^ {t + \Delta t} \mathbf {x} _ {i} - ^ {t} \mathbf {x} _ {i}\right) + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(^ {t + \Delta t} \mathbf {V} _ {n} ^ {i} - ^ {t} \mathbf {V} _ {n} ^ {i}\right) \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \Delta^ {t} \mathbf {u} _ {i} + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \left(- \alpha_ {1} ^ {i} {} ^ {t} \mathbf {V} _ {2} ^ {i} + \alpha_ {2} ^ {i} {} ^ {t} \mathbf {V} _ {1} ^ {i}\right) \\ = \sum_ {i = 1} ^ {4} \phi_ {i} \Delta^ {t} \mathbf {u} _ {i} + \frac {\xi^ {3}}{2} \sum_ {i = 1} ^ {4} h _ {i} \phi_ {i} \Delta^ {t} \mathbf {V} _ {n} ^ {i} \\ \end{array} +$$ + +위치 벡터와 변위 벡터를 matrix form으로 나타내면 + + + +$$ +^ 0 \mathbf {X} = ^ {0} \mathbf {N} ^ {0} \mathbf {X} _ {n} +$$ + +$$ += \left[ \begin{array}{l l l l} \phi_ {1} & \phi_ {2} & \phi_ {3} & \phi_ {4} \end{array} \right] \left[ \begin{array}{l} ^ {0} \mathbf {X} _ {1} \\ ^ {0} \mathbf {X} _ {2} \\ ^ {0} \mathbf {X} _ {3} \\ ^ {0} \mathbf {X} _ {4} \end{array} \right] + \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{l} ^ {0} \mathbf {V} _ {n} ^ {1} \\ ^ {0} \mathbf {V} _ {n} ^ {2} \\ ^ {0} \mathbf {V} _ {n} ^ {3} \\ ^ {0} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ += \left[ \begin{array}{c c c c c c c c} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \phi_ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \phi_ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \phi_ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ 0 \mathbf {X} _ {1} \\ ^ 0 \mathbf {V} _ {n} ^ {1} \\ ^ 0 \mathbf {X} _ {2} \\ ^ 0 \mathbf {V} _ {n} ^ {2} \\ ^ 0 \mathbf {X} _ {3} \\ ^ 0 \mathbf {V} _ {n} ^ {3} \\ ^ 0 \mathbf {X} _ {4} \\ ^ 0 \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +{ } ^ { t } \mathbf { x } = { } ^ { 0 } \mathbf { N } ^ { t } \mathbf { x } _ { n } +$$ + +$$ += \left[ \begin{array}{c c c c c c c c c} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \phi_ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \phi_ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \phi_ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t} \mathbf {x} _ {1} \\ ^ {t} \mathbf {V} _ {n} ^ {1} \\ ^ {t} \mathbf {x} _ {2} \\ ^ {t} \mathbf {V} _ {n} ^ {2} \\ ^ {t} \mathbf {x} _ {3} \\ ^ {t} \mathbf {V} _ {n} ^ {3} \\ ^ {t} \mathbf {x} _ {4} \\ ^ {t} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +\mathbf {\Lambda} ^ {t + \Delta t} \mathbf {x} = \mathbf {\Lambda} ^ {0} \mathbf {N} ^ {t + \Delta t} \mathbf {x} _ {n} +$$ + +$$ += \left[ \begin{array}{c c c c c c c c c} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \phi_ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \phi_ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \phi_ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} t + \Delta t \mathbf {X} _ {1} \\ t + \Delta t \mathbf {V} _ {n} ^ {1} \\ t + \Delta t \mathbf {X} _ {2} \\ t + \Delta t \mathbf {V} _ {n} ^ {2} \\ t + \Delta t \mathbf {X} _ {3} \\ t + \Delta t \mathbf {V} _ {n} ^ {3} \\ t + \Delta t \mathbf {X} _ {4} \\ t + \Delta t \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +\Delta^ {t} \mathbf {x} = ^ {0} \mathbf {N} \Delta^ {t} \mathbf {x} _ {n} +$$ + +$$ += \left[ \begin{array}{c c c c c c c c} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \phi_ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \phi_ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \phi_ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{l} \Delta^ {t} \mathbf {x} _ {1} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {1} \\ \Delta^ {t} \mathbf {x} _ {2} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {2} \\ \Delta^ {t} \mathbf {x} _ {3} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {3} \\ \Delta^ {t} \mathbf {x} _ {4} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + + + +변위 벡터도 마찬가지로 나타낼 수 있다. + +$$ +{ } ^ { t } \mathbf { u } = ^ { 0 } \mathbf { N } \left( { } ^ { t } \mathbf { x } _ { n } - ^ { 0 } \mathbf { X } _ { n } \right) +$$ + +$$ += \left[ \begin{array}{c c c c c c c c} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \phi_ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \phi_ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \phi_ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t} \mathbf {x} _ {1} - ^ {0} \mathbf {X} _ {1} \\ ^ {t} \mathbf {V} _ {n} ^ {1} - ^ {0} \mathbf {V} _ {n} ^ {1} \\ ^ {t} \mathbf {x} _ {2} - ^ {0} \mathbf {X} _ {2} \\ ^ {t} \mathbf {V} _ {n} ^ {2} - ^ {0} \mathbf {V} _ {n} ^ {2} \\ ^ {t} \mathbf {x} _ {3} - ^ {0} \mathbf {X} _ {3} \\ ^ {t} \mathbf {V} _ {n} ^ {3} - ^ {0} \mathbf {V} _ {n} ^ {3} \\ ^ {t} \mathbf {x} _ {4} - ^ {0} \mathbf {X} _ {4} \\ ^ {t} \mathbf {V} _ {n} ^ {4} - ^ {0} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +t + \Delta t _ {\mathbf {u}} +$$ + +$$ += \left[ \begin{array}{c c c c} \phi_ {1} & \phi_ {2} & \phi_ {3} & \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t + \Delta t} \mathbf {x} _ {1} - ^ {0} \mathbf {X} _ {1} \\ ^ {t + \Delta t} \mathbf {x} _ {2} - ^ {0} \mathbf {X} _ {2} \\ ^ {t + \Delta t} \mathbf {x} _ {3} - ^ {0} \mathbf {X} _ {3} \\ ^ {t + \Delta t} \mathbf {x} _ {4} - ^ {0} \mathbf {X} _ {4} \end{array} \right] + \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t + \Delta t} \mathbf {V} _ {n} ^ {1} - ^ {0} \mathbf {V} _ {n} ^ {1} \\ ^ {t + \Delta t} \mathbf {V} _ {n} ^ {2} - ^ {0} \mathbf {V} _ {n} ^ {2} \\ ^ {t + \Delta t} \mathbf {V} _ {n} ^ {3} - ^ {0} \mathbf {V} _ {n} ^ {3} \\ ^ {t + \Delta t} \mathbf {V} _ {n} ^ {4} - ^ {0} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ += \left[ \begin{array}{c c c c} \phi_ {1} & \phi_ {2} & \phi_ {3} & \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t + \Delta t} \mathbf {u} _ {1} \\ ^ {t + \Delta t} \mathbf {u} _ {2} \\ ^ {t + \Delta t} \mathbf {u} _ {3} \\ ^ {t + \Delta t} \mathbf {u} _ {4} \end{array} \right] + \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} \Delta^ {t} \mathbf {V} _ {n} ^ {1} + \Delta^ {0} \mathbf {V} _ {n} ^ {1} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {2} + \Delta^ {0} \mathbf {V} _ {n} ^ {2} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {3} + \Delta^ {0} \mathbf {V} _ {n} ^ {3} \\ \Delta^ {t} \mathbf {V} _ {n} ^ {4} + \Delta^ {0} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +\left(^ {t + \Delta t} \mathbf {V} _ {n} ^ {1} - ^ {0} \mathbf {V} _ {n} ^ {1} = ^ {t + \Delta t} \mathbf {V} _ {n} ^ {1} - ^ {t} \mathbf {V} _ {n} ^ {1} + ^ {t} \mathbf {V} _ {n} ^ {1} - ^ {0} \mathbf {V} _ {n} ^ {1} = \Delta^ {t} \mathbf {V} _ {n} ^ {1} + \Delta^ {0} \mathbf {V} _ {n} ^ {1}\right) +$$ + +$$ += \left[ \begin{array}{l l l l} \phi_ {1} & \phi_ {2} & \phi_ {3} & \phi_ {4} \end{array} \right] \left[ \begin{array}{c} ^ {t + \Delta t} \mathbf {u} _ {1} \\ ^ {t + \Delta t} \mathbf {u} _ {2} \\ ^ {t + \Delta t} \mathbf {u} _ {3} \\ ^ {t + \Delta t} \mathbf {u} _ {4} \end{array} \right] + \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} - \alpha_ {1} ^ {1 t} \mathbf {V} _ {2} ^ {1} + \alpha_ {2} ^ {1 t} \mathbf {V} _ {1} ^ {1} \\ - \alpha_ {1} ^ {2 t} \mathbf {V} _ {2} ^ {2} + \alpha_ {2} ^ {2 t} \mathbf {V} _ {1} ^ {2} \\ - \alpha_ {1} ^ {3 t} \mathbf {V} _ {2} ^ {3} + \alpha_ {2} ^ {3 t} \mathbf {V} _ {1} ^ {3} \\ - \alpha_ {1} ^ {4 t} \mathbf {V} _ {2} ^ {4} + \alpha_ {2} ^ {4 t} \mathbf {V} _ {1} ^ {4} \end{array} \right] +$$ + +$$ ++ \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} \Delta^ {0} \mathbf {V} _ {n} ^ {1} \\ \Delta^ {0} \mathbf {V} _ {n} ^ {2} \\ \Delta^ {0} \mathbf {V} _ {n} ^ {3} \\ \Delta^ {0} \mathbf {V} _ {n} ^ {4} \end{array} \right] +$$ + +$$ +\Rightarrow^ {t + \Delta t} \mathbf {u} = ^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} ^ {n} +$$ + +$$ += \left[ \begin{array}{c c c c c c c c c c c c c c c} \phi_ {1} & - \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} \mathbf {v} _ {2} ^ {1} & \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} \mathbf {v} _ {1} ^ {1} & \phi_ {2} & - \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} \mathbf {v} _ {2} ^ {2} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} \mathbf {v} _ {1} ^ {2} & \phi_ {3} & - \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} \mathbf {v} _ {2} ^ {3} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} \mathbf {v} _ {1} ^ {3} & \phi_ {4} & - \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \mathbf {v} _ {2} ^ {4} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \mathbf {v} _ {1} ^ {4} \end{array} \right] \left[ \begin{array}{c} t + \Delta t \mathbf {u} _ {1} \\ \alpha_ {1} ^ {1} \\ \alpha_ {2} ^ {1} \\ t + \Delta t \mathbf {u} _ {2} \\ \alpha_ {1} ^ {2} \\ \alpha_ {2} ^ {2} \\ t + \Delta t \mathbf {u} _ {3} \\ \alpha_ {1} ^ {3} \\ \alpha_ {2} ^ {3} \\ t + \Delta t \mathbf {u} _ {4} \\ \alpha_ {1} ^ {4} \\ \alpha_ {2} ^ {4} \end{array} \right] +$$ + + + +$$ ++ \left[ \begin{array}{c c c c} \frac {\xi^ {3}}{2} h _ {1} \phi_ {1} & \frac {\xi^ {3}}{2} h _ {2} \phi_ {2} & \frac {\xi^ {3}}{2} h _ {3} \phi_ {3} & \frac {\xi^ {3}}{2} h _ {4} \phi_ {4} \end{array} \right] \left[ \begin{array}{c} \Delta^ {0} \mathbf {v} _ {n} ^ {1} \\ \Delta^ {0} \mathbf {v} _ {n} ^ {2} \\ \Delta^ {0} \mathbf {v} _ {n} ^ {3} \\ \Delta^ {0} \mathbf {v} _ {n} ^ {4} \end{array} \right] +$$ + +# 3. FE Formulation + +현재 형상(t+Δt)에서의 평형방정식은 다음과 같다. + +$$ +\nabla_ {X} \cdot^ {t + \Delta t} \boldsymbol {\sigma} + \rho^ {t + \Delta t} \mathbf {f} = \rho^ {t + \Delta t} \ddot {\mathbf {u}} +$$ + +가상일 원리를 적용하면 + +$$ +\begin{array}{l} \int_ {V} \delta^ {t + \Delta t} \mathbf {u} \cdot \left(\nabla_ {X} \cdot^ {t + \Delta t} \boldsymbol {\sigma} + \rho^ {t + \Delta t} \mathbf {f}\right) d V = \int_ {V} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho^ {t + \Delta t} \ddot {\mathbf {u}} d V \\ \Rightarrow \int_ {V} \delta u _ {i} \left(\frac {\partial \sigma_ {i j}}{\partial x _ {j}} + \rho f _ {i}\right) d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \quad \left(\delta u _ {i} \frac {\partial \sigma_ {i j}}{\partial x _ {j}} = \frac {\partial \left(\delta u _ {i} \sigma_ {i j}\right)}{\partial x _ {j}} - \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j}\right) \\ \Rightarrow \int_ {V} \left\{\frac {\partial \left(\delta u _ {i} \sigma_ {i j}\right)}{\partial x _ {j}} - \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i} \right\} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \quad \rightarrow \text { Gauss's divergence theorem } \\ \end{array} +$$ + +$$ +\Rightarrow \int_ {\partial V} \delta u _ {i} \sigma_ {i j} n _ {j} d A + \int_ {V} \left\{- \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i} \right\} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \quad \left(\text { geometric B . C 瓦 natural B . C }\right) +$$ + +$$ +\Rightarrow \int_ {\partial V _ {g}} \delta u _ {i} \sigma_ {i j} n _ {j} d A + \int_ {\partial V _ {m}} \delta u _ {i} \bar {t} _ {i} d A + \int_ {V} \left\{- \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i} \right\} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V +$$ + +$$ +\left(\int_ {\partial V _ {g}} \delta u _ {i} \sigma_ {i j} n _ {j} d A = 0, \text { 기하학적 경계조건에서 가상변위가 } 0\right) +$$ + +$$ +\Rightarrow \int_ {\partial V _ {m}} \delta u _ {i} \bar {t} _ {i} d A + \int_ {V} \left\{- \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i} \right\} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V +$$ + +$$ +\left(\frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} = \frac {1}{2} \left(\frac {\partial \delta u _ {i}}{\partial x _ {j}} + \frac {\partial \delta u _ {i}}{\partial x _ {j}}\right) \sigma_ {i j} = \delta \varepsilon_ {i j} \sigma_ {i j}\right) +$$ + +$$ +\Rightarrow \int_ {\partial V _ {m}} \delta u _ {i} \bar {t} _ {i} d A + \int_ {V} \delta u _ {i} \rho f _ {i} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V + \int_ {V} \delta \varepsilon_ {i j} \sigma_ {i j} d V +$$ + +비선형 해석을 위해 하중을 조금씩 증가시켜 물체의 변형을 순차적으로 구해 나가며 이러한 반복계산에 있어 기준이 되는 물체의 형상을 설정하는 방법에는 크게 Total Lagrange formulation과 Updated Lagrange formulation이 있다. Total Lagrange formulation은 변형과 관련된 변수들을 초기형상을 이용하여 정의하고 Updated Lagrange formulation은 변수들을 현재 변형된 형상으로 정의한다. + + + +$$ +\int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V + \int_ {V} \delta \varepsilon_ {i j} \sigma_ {i j} d V = \int_ {\partial V _ {m}} \delta u _ {i} \overline {{t}} _ {i} d A + \int_ {V} \delta u _ {i} \rho f _ {i} d V +$$ + +$$ +\left( \begin{array}{l} \int_ {V} (\bullet) d V = \int_ {V _ {0}} (\bullet) \det (\mathbf {F}) d V _ {0} = \int_ {V _ {0}} (\bullet) J d V _ {0} \\ \int_ {\partial V} (\bullet) \mathbf {n} d A = \int_ {\partial V} (\bullet) J \mathbf {F} ^ {- T} \tilde {\mathbf {n}} d A (N a n s o n ^ {\prime} s f o r m u l a) \\ \int_ {V _ {0}} \delta \varepsilon_ {i j} \sigma_ {i j} d V _ {0} = \int_ {V _ {0}} \delta F _ {i j} P _ {i j} d V _ {0} = \int_ {V _ {0}} \delta E _ {i j} S _ {i j} d V _ {0} \end{array} \right) +$$ + +$$ +\Rightarrow \int_ {V _ {0}} \delta u _ {i} \cdot \rho J \ddot {u} _ {i} d V _ {0} + \int_ {V _ {0}} \delta E _ {i j} S _ {i j} d V _ {0} = \int_ {\partial V _ {0 m}} \delta u _ {i} \sigma_ {i j} J \left[ F ^ {- T} \right] _ {k j} \tilde {n} _ {k} d A _ {0} + \int_ {V _ {0}} \delta u _ {i} \rho J f _ {i} d V _ {0} +$$ + +$$ +\Rightarrow \int_ {V _ {0}} \delta u _ {i} \cdot \rho_ {0} \ddot {u} _ {i} d V _ {0} + \int_ {V _ {0}} \delta E _ {i j} S _ {i j} d V _ {0} = \int_ {\partial V _ {0 m}} \delta u _ {i} \sigma_ {i j} J \left[ F ^ {- T} \right] _ {k j} \tilde {n} _ {k} d A _ {0} + \int_ {V _ {0}} \delta u _ {i} \rho_ {0} f _ {i} d V _ {0} +$$ + +$$ +\Rightarrow \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho_ {0} ^ {t + \Delta t} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} _ {0} \mathbf {E} \cdot {} _ {0} ^ {t + \Delta t} \mathbf {S} d V _ {0} = \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} J ^ {t + \Delta t} \boldsymbol {\sigma} F ^ {- T t + \Delta t} \tilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \rho_ {0} ^ {t + \Delta t} \mathbf {f} d V _ {0} +$$ + +$$ +\left(J \boldsymbol {\sigma} F ^ {- T} = \mathbf {P} = \mathbf {F S}\right) +$$ + +위 식을 정리하면 다음과 같다. + +$$ +\int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho_ {0} ^ {t + \Delta t} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} _ {0} \mathbf {E} \cdot {} _ {0} ^ {t + \Delta t} \mathbf {S} d V _ {0} = \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} ^ {t + \Delta t} _ {0} \mathbf {F} ^ {t + \Delta t} _ {0} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \rho_ {0} ^ {t + \Delta t} \mathbf {f} d V _ {0} +$$ + +$$ +\Leftrightarrow \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho_ {0} ^ {t + \Delta t} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta_ {0} ^ {t + \Delta t} \mathbf {F}: _ {0} ^ {t + \Delta t} \mathbf {P} d V _ {0} = \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} _ {0} ^ {t + \Delta t} \mathbf {P} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \rho_ {0} ^ {t + \Delta t} \mathbf {f} d V _ {0} +$$ + +E 는 Green-Lagrange strain, S 는 2 $^{nd}$ PK stress, P 는 1 $^{st}$ PK stress, F 는 deformation gradient를 나타낸다. 여기에서는 Green-Lagrange strain과 2 $^{nd}$ PK stress를 사용한다. + +$$ +\int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho_ {0} ^ {t + \Delta t} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta_ {0} ^ {t + \Delta t} \mathbf {E} \cdot {} _ {0} ^ {t + \Delta t} \mathbf {S} d V _ {0} = \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} _ {0} ^ {t + \Delta t} \mathbf {F} _ {0} ^ {t + \Delta t} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \rho_ {0} ^ {t + \Delta t} \mathbf {f} d V _ {0} +$$ + +먼저 좌변을 살펴보면 첫 번째 항을 다음과 같이 정리 할 수 있다. + +$$ +\int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho^ {t + \Delta t} \ddot {\mathbf {u}} J d V _ {0} = \int_ {V _ {0}} \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right) \cdot \rho^ {t} \mathbf {N} ^ {t + \Delta t} \ddot {\mathbf {u}} J d V _ {0} +$$ + +$$ += \int_ {V _ {0}} \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}\right) \cdot \rho \left(^ {t} \mathbf {N} ^ {t + \Delta t} \ddot {\mathbf {u}}\right) J d V _ {0} +$$ + +$$ += \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \int_ {V _ {0}} \rho \left[ ^ {t} \mathbf {N} \right] ^ {T} \left[ ^ {t} \mathbf {N} \right] J d V _ {0} \left[ ^ {t + \Delta t} \ddot {\mathbf {u}} _ {n} \right] +$$ + +두 번째 항도 마찬가지로 다음과 같이 정리 할 수 있다. 먼저 Green-Lagrange strain은 다음과 같이 나타낼 수 있다. + + + +$$ +\begin{array}{l} { } _ { 0 } ^ { t + \Delta t } \mathbf { E } = \frac { 1 } { 2 } \Big ( { } _ { 0 } ^ { t + \Delta t } \mathbf { g } _ { i } \cdot { } _ { 0 } ^ { t + \Delta t } \mathbf { g } _ { j } - { } _ { 0 } \mathbf { G } _ { i } \cdot { } _ { 0 } \mathbf { G } _ { j } \Big ) \Big ( { } _ { 0 } \mathbf { G } ^ { i } \otimes _ { 0 } \mathbf { G } ^ { j } \Big ) \\ = \frac {1}{2} \left(\frac {\partial_ {0} ^ {t + \Delta t} \mathbf {x}}{\partial \xi^ {i}} \cdot \frac {\partial_ {0} ^ {t + \Delta t} \mathbf {x}}{\partial \xi^ {j}} - \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ = \frac {1}{2} \left(\frac {\partial \left(^ {0} \mathbf {X} + ^ {t + \Delta t} \mathbf {u}\right)}{\partial \xi^ {i}} \cdot \frac {\partial \left(^ {0} \mathbf {X} + ^ {t + \Delta t} \mathbf {u}\right)}{\partial \xi^ {j}} - \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} - \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \left(^ {t} \mathbf {u} + \Delta^ {t} \mathbf {u}\right)}{\partial \xi^ {j}} + \frac {\partial \left(^ {t} \mathbf {u} + \Delta^ {t} \mathbf {u}\right)}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial \left(^ {t} \mathbf {u} + \Delta^ {t} \mathbf {u}\right)}{\partial \xi^ {i}} \cdot \frac {\partial \left(^ {t} \mathbf {u} + \Delta^ {t} \mathbf {u}\right)}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ \end{array} +$$ + +정리하면 Green-Lagrange strain을 Δu에 대한 상수 term, 선형 term, 비선형 term으로 나눌 수 있다. + +$$ +{ } _ { 0 } ^ { t + \Delta t } \mathbf { E } = { } _ { 0 } ^ { t + \Delta t } \mathbf { E } _ { 0 } + { } _ { 0 } ^ { t + \Delta t } \mathbf { E } _ { C } + { } _ { 0 } ^ { t + \Delta t } \mathbf { E } _ { L } +$$ + +$$ +\left( \begin{array}{l} ^ {t + \Delta t} _ {0} \mathbf {E} _ {0} = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ ^ {t + \Delta t} _ {0} \mathbf {E} _ {C} = \frac {1}{2} \left(\frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \\ ^ {t + \Delta t} _ {0} \mathbf {E} ^ {L} = \frac {1}{2} \left(\frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) \end{array} \right) +$$ + +와 같다. 두 번째 항을 위의 표현으로 나타내면 + +$$ +\int_ {V _ {0}} \delta^ {t + \Delta t} _ {0} \mathbf {E}: ^ {t + \Delta t} _ {0} \mathbf {S} d V _ {0} = \int_ {V _ {0}} \delta^ {t + \Delta t} _ {0} \mathbf {E}: ^ {t + \Delta t} _ {0} \mathbf {C}: ^ {t + \Delta t} _ {0} \mathbf {E} d V _ {0} +$$ + +$$ += \int_ {V _ {0}} \left(\underbrace {\delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {0}} _ {0} + \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} + \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L}\right): _ {0} ^ {t + \Delta t} \mathbf {C}: \left(_ {0} ^ {t + \Delta t} \mathbf {E} _ {0} + _ {0} ^ {t + \Delta t} \mathbf {E} _ {C} + _ {0} ^ {t + \Delta t} \mathbf {E} _ {L}\right) d V _ {0} +$$ + +$$ += \int_ {V _ {0}} \underbrace {\delta^ {t + \Delta t} \mathbf {E} _ {C} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {0}} _ {\text {constant term}} d V _ {0} + \int_ {V _ {0}} \underbrace {\delta^ {t + \Delta t} \mathbf {E} _ {C} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {C} + \delta^ {t + \Delta t} \mathbf {E} _ {L} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {0}} _ {\text {linear term}} d V _ {0} +$$ + +$$ ++ \int_ {V _ {0}} \underbrace {\delta^ {t + \Delta t} \mathbf {E} _ {C} : {} _ {0} ^ {t + \Delta t} \mathbf {C} : {} _ {0} ^ {t + \Delta t} \mathbf {E} _ {L} + \delta^ {t + \Delta t} \mathbf {E} _ {L} : {} _ {0} ^ {t + \Delta t} \mathbf {C} : {} _ {0} ^ {t + \Delta t} \mathbf {E} _ {C} + \delta^ {t + \Delta t} \mathbf {E} _ {L} : {} _ {0} ^ {t + \Delta t} \mathbf {C} : {} _ {0} ^ {t + \Delta t} \mathbf {E} _ {L}} _ {\text {high order term}} d V _ {0} +$$ + +와 같다. 이후 선형화 시키고(High order term 무시) component형태로 나타내면 아래와 같다. + +$$ +\int_ {V _ {0}} \underbrace {\delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {0}} _ {\text {constant term}} d V _ {0} + \int_ {V _ {0}} \underbrace {\delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {C} + \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} : _ {0} ^ {t + \Delta t} \mathbf {C} : _ {0} ^ {t + \Delta t} \mathbf {E} _ {0}} _ {\text {linear term}} d V _ {0} +$$ + +$$ += \int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {E} _ {0} \right] _ {k l}} _ {\text { constant term }} d V _ {0} +$$ + +$$ ++ \int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {E} _ {C} \right] _ {k l} + \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {E} _ {0} \right] _ {k l}} _ {\text {linear term}} d V _ {0} +$$ + +$$ += \int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j}} _ {\text {constant term}} d V _ {0} + \int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {C} \right] ^ {i j} + \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j}} _ {\text {linear term}} d V _ {0} +$$ + + + +Green-Lagrange strain의 변분을 구해보면 다음과 같다. 변분은 incremental displacement에만 적용되는데 이는 incremental displacement가 정의되지 않았기 때문이다. 따라서 $\delta^{t+\Delta t}\mathbf{u}=\delta\left(^{t}\mathbf{u}+\Delta^{t}\mathbf{u}\right)=\delta\Delta^{t}\mathbf{u}$ 이고 $\delta^{t+\Delta t}_{0}\mathbf{E}_{0}=0$ 이 성립한다. + +$$ +\delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) +$$ + +$$ +\delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} = \frac {1}{2} \left(\frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}}\right) \left(_ {0} \mathbf {G} ^ {i} \otimes_ {0} \mathbf {G} ^ {j}\right) +$$ + +Component form으로 나타내면 + +$$ +\begin{array}{l} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {E} _ {0} \right] _ {i j} = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) \\ = \left[ ^ {0} \mathbf {X} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] \\ + \frac {1}{2} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] \\ = \left[ ^ {0} \mathbf {X} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] \\ + \frac {1}{2} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] \\ = \frac {1}{2} \left[ ^ {t} \mathbf {x} _ {n} + ^ {0} \mathbf {X} _ {n} \right] ^ {T} \underbrace {\frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\}} _ {[ \mathbf {e} ] _ {i j}} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] \\ \end{array} +$$ + +$$ +\left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {E} _ {C} \right] _ {i j} = \frac {1}{2} \left(\frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) +$$ + +$$ += \left[ ^ {0} \mathbf {X} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ \Delta^ {t} \mathbf {u} _ {n} \right] +$$ + +$$ ++ \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ \Delta^ {t} \mathbf {u} _ {n} \right] +$$ + +$$ += \left[ ^ {t} \mathbf {x} _ {n} \right] ^ {T} \underbrace {\frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\}} _ {[ \mathbf {a} ] _ {i j}} \left[ \Delta^ {t} \mathbf {u} _ {n} \right] +$$ + +$$ +\begin{array}{l} \left[ \begin{array}{c} ^ {t + \Delta t} _ {0} \mathbf {E} _ {L} \end{array} \right] _ {i j} = \frac {1}{2} \left(\frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta^ {t} \mathbf {u}}{\partial \xi^ {j}}\right) \\ = \left[ \Delta^ {t} \mathbf {u} _ {n} \right] ^ {T} \frac {1}{2} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] \left[ \Delta^ {t} \mathbf {u} _ {n} \right] \\ \end{array} +$$ + + + +$$ +\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} = \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}}\right) +$$ + +$$ += \frac {1}{2} \left( \begin{array}{c} \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {j}} + \frac {\partial \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} \\ + \frac {\partial \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {j}} \end{array} \right) +$$ + +$$ += \frac {1}{2} \left(\frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {j}} + \frac {\partial \delta^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {i}} \cdot \frac {\partial^ {0} \mathbf {X}}{\partial \xi^ {j}} + \frac {\partial^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial^ {t + \Delta t} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {j}}\right) +$$ + +$$ += \left[ \boldsymbol {\delta} ^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ ^ {0} \mathbf {X} _ {n} \right] +$$ + +$$ ++ \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \frac {1}{2} \left\{\left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] \right\} \left[ ^ {t} \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \right] +$$ + +$$ += \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \underbrace {\frac {1}{2} \left\{\left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \left[ \frac {\partial^ {0} \mathbf {N}}{\partial \xi^ {j}} \right] \right\}} _ {[ \mathbf {a} ] _ {i j}} \left[ ^ {t} \mathbf {x} _ {n} \right] +$$ + +$$ +\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} \right] _ {i j} = \frac {1}{2} \left(\frac {\partial \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {i}} \cdot \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right)}{\partial \xi^ {j}}\right) +$$ + +$$ += \frac {1}{2} \left(\frac {\partial \delta^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {i}} \cdot \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {j}} + \frac {\partial \Delta_ {0} \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \delta^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}}{\partial \xi^ {j}}\right) +$$ + +$$ += \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \underbrace {\frac {1}{2} \left(\left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \cdot \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right] + \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {i}} \right] ^ {T} \cdot \left[ \frac {\partial^ {t} \mathbf {N}}{\partial \xi^ {j}} \right]\right)} _ {[ \mathbf {c} ] _ {i j}} \left[ \Delta^ {t} \mathbf {u} _ {n} \right] +$$ + +다시 가상일 항으로 돌아와서 위에 구한 Green-Lagrange strain을 대입하면 + +$$ +\int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j}} _ {\text {constant term}} d V _ {0} + \int_ {V _ {0}} \underbrace {\left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {C} \right] ^ {i j} + \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j}} _ {\text {linear term}} d V _ {0} +$$ + +$$ +\left( \begin{array}{l} \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j} = \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \left[ \mathbf {a} \right] _ {i j} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \frac {1}{2} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} + ^ {0} \mathbf {X} _ {n} \end{array} \right] ^ {T} \left[ \mathbf {e} \right] _ {k l} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \end{array} \right] \\ \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {C} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {C} \right] ^ {i j} = \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \left[ \mathbf {a} \right] _ {i j} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] \left[ \begin{array}{c} t + {\Delta t} \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] ^ {T} \left[ \mathbf {a} \right] _ {k l} \left[ \begin{array}{c} \Delta^ {t} \mathbf {u} _ {n} \end{array} \right] \\ \left[ \delta_ {0} ^ {t + \Delta t} \mathbf {E} _ {L} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j} = \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \left[ \mathbf {c} \right] _ {i j} \left[ \begin{array}{c} \Delta^ {t} \mathbf {u} _ {n} \end{array} \right] \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \frac {1}{2} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} + ^ {0} \mathbf {X} _ {n} \end{array} \right] ^ {T} \left[ \mathbf {e} \right] _ {k l} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} - ^ {0} \mathbf {X} _ {n} \end{array} \right] \end{array} \right) +$$ + +$$ += \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \left\{ \begin{array}{l} \left(\int_ {V _ {0}} \left[ \mathbf {a} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j} d V _ {0}\right) \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] \\ + \left(\int_ {V _ {0}} \left[ \mathbf {a} \right] _ {i j} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] ^ {i j k l} \left[ \begin{array}{c} t \\ \mathbf {x} _ {n} \end{array} \right] ^ {T} \left[ \mathbf {a} \right] _ {k l} + \left[ \mathbf {c} \right] _ {i j} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {S} _ {0} \right] ^ {i j} d V _ {0}\right) \left[ \Delta^ {t} \mathbf {u} _ {n} \right] \end{array} \right\} +$$ + +와 같다. 이제 우변의 항들을 정리해보면 아래와 같다. 여기서 body force에 대한 영향은 무시한다. + + + +$$ +\begin{array}{l} \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} _ {0} ^ {t + \Delta t} \mathbf {F} _ {0} ^ {t + \Delta t} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} = \int_ {\partial V _ {0 m}} \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n} + ^ {0} \tilde {\mathbf {N}} \Delta^ {0} \tilde {\mathbf {X}} _ {n}\right) _ {0} ^ {t + \Delta t} \mathbf {F} _ {0} ^ {t + \Delta t} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} \\ = \int_ {\partial V _ {0 m}} \delta \left(^ {t} \mathbf {N} ^ {t + \Delta t} \mathbf {u} _ {n}\right) _ {0} ^ {t + \Delta t} \mathbf {F} _ {0} ^ {t + \Delta t} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} \\ = \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \int_ {\partial V _ {0 m}} \left[ ^ {t} \mathbf {N} \right] _ {0} ^ {T t + \Delta t} \mathbf {F} _ {0} ^ {t + \Delta t} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} \\ = \left[ \delta^ {t + \Delta t} \mathbf {u} _ {n} \right] ^ {T} \int_ {\partial V _ {0 m}} \left[ ^ {t} \mathbf {N} \right] ^ {T} [ \mathbf {t} ] d A _ {0} \\ \end{array} +$$ + +따라서 가상변위를 지워 모든 식을 정리하면 + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \cdot \rho_ {0} ^ {t + \Delta t} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} _ {0} \mathbf {E}: ^ {t + \Delta t} _ {0} \mathbf {S} d V _ {0} = \int_ {\partial V _ {0 m}} \delta^ {t + \Delta t} \mathbf {u} ^ {t + \Delta t} _ {0} \mathbf {F} ^ {t + \Delta t} _ {0} \mathbf {S} ^ {t + \Delta t} \tilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta^ {t + \Delta t} \mathbf {u} \rho_ {0} ^ {t + \Delta t} \mathbf {f} d V _ {0} \\ \Rightarrow \underbrace {\int_ {V _ {0}} \rho \left[ ^ {t} \mathbf {N} \right] ^ {T} \left[ ^ {t} \mathbf {N} \right] J d V _ {0}} _ {\mathbf {M}} ^ {t + \Delta t} \ddot {\mathbf {u}} + \underbrace {\int_ {V _ {0}} \left[ \mathbf {a} \right] _ {i j} \left[ ^ {t + \Delta t} {} _ {0} \mathbf {S} _ {0} \right] ^ {i j} d V _ {0} {} ^ {t} \mathbf {x}} _ {\mathbf {f} _ {\text {int}}} \\ + \underbrace {\int_ {V _ {0}} \left[ \mathbf {a} \right] _ {i j} \left[ ^ {t} \mathbf {x} _ {n} \right] \left[ ^ {t + \Delta t} _ {0} \mathbf {C} \right] ^ {i j k l} \left[ ^ {t} \mathbf {x} _ {n} \right] ^ {T} \left[ \mathbf {a} \right] _ {k l} + \left[ \mathbf {c} \right] _ {i j} \left[ ^ {t + \Delta t} _ {0} \mathbf {S} _ {0} \right] ^ {i j} d V _ {0}} _ {\mathbf {K} _ {t}} \Delta^ {t} \mathbf {u} = \underbrace {\int_ {\partial V _ {0 m}} \left[ ^ {t} \mathbf {N} \right] ^ {T} [ \mathbf {t} ] d A _ {0}} _ {\mathbf {P} _ {d i s t}} + \mathbf {P} _ {c o n} \\ \Rightarrow \mathbf {M} ^ {t + \Delta t} \ddot {\mathbf {u}} + \mathbf {K} _ {t} \Delta^ {t} \mathbf {u} = \mathbf {P} _ {\text { dist }} + \mathbf {P} _ {\text { con }} - \mathbf {f} _ {\text { int }} \\ \end{array} +$$ + +와 같이 정리할 수 있다. 여기서 M은 mass matrix, $K_{t}$ 는 tangent stiffness matrix, $P_{dist}$ 는 분포하중에 의한 힘, $P_{con}$ 는 집중하중, $f_{int}$ 는 변형에 의한 힘을 나타낸다. + +# 4. Constitutive matrix + +Plane stress 가정을 사용하는 구성행렬은 다음과 같다. + +$$ +\left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] _ {x ^ {1} x ^ {2} x ^ {3}} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c c c c} 1 & \nu & 0 & 0 & 0 & 0 \\ n & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \kappa \frac {1 - \nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \kappa \frac {1 - \nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \kappa \frac {1 - \nu}{2} \end{array} \right] +$$ + +위의 구성 행렬은 local Cartesian coordinate에서 정의되었기 때문에 transformation matrix를 이용하여 natural coordinate로 바꾸어 줄 수 있다. + +$$ +\left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] _ {\xi^ {1} \xi^ {2} \xi^ {3}} = \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {T} \right] ^ {T} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {C} \right] _ {x ^ {1} x ^ {2} x ^ {3}} \left[ \begin{array}{c} t + \Delta t \\ 0 \end{array} \mathbf {T} \right] +$$ + +이 때 전단보정계수 $\kappa$ 는 $\frac{5}{6}$ 를 사용하였다. diff --git a/docs/reference-papers/MITC4/MITC공부/MITC공부_002.md b/docs/reference-papers/MITC4/MITC공부/MITC공부_002.md new file mode 100644 index 0000000..e2a2a34 --- /dev/null +++ b/docs/reference-papers/MITC4/MITC공부/MITC공부_002.md @@ -0,0 +1,122 @@ + + +$$ +E _ {k l} = \tilde {E} _ {m n} \underbrace {\left(\mathbf {E} _ {k} \cdot \mathbf {G} ^ {m}\right) \left(\mathbf {E} _ {l} \cdot \mathbf {G} ^ {n}\right)} _ {\equiv \mathbf {T}} = \tilde {E} _ {m n} \left(\mathbf {E} _ {k} \cdot \frac {\partial X ^ {a}}{\partial \xi^ {m}} \mathbf {E} _ {a}\right) \left(\mathbf {E} _ {l} \cdot \frac {\partial X ^ {b}}{\partial \xi^ {n}} \mathbf {E} _ {b}\right) = \frac {\partial X ^ {k}}{\partial \xi^ {m}} \frac {\partial X ^ {l}}{\partial \xi^ {n}} +$$ + +$$ +\left[ \mathbf {T} \right] = \left[ \begin{array}{c c c c c c c c c c c c} \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {1}}{\partial \xi^ {1}} & \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {1}}{\partial \xi^ {2}} & \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {1}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {1}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {1}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {1}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {1}}{\partial \xi^ {2}} \\ \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {1}} & \frac {\partial X ^ {2}}{\partial \xi^ {2}} \frac {\partial X ^ {2}}{\partial \xi^ {2}} & \frac {\partial X ^ {2}}{\partial \xi^ {3}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} & \frac {\partial X ^ {2}}{\partial \xi^ {2}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {2}} \\ \frac {\partial X ^ {3}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & \frac {\partial X ^ {3}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} & \frac {\partial X ^ {3}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {3}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {3}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {3}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {3}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} \\ \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & 2 \frac {\partial X ^ {2}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} & 2 \frac {\partial X ^ {2}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {2}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} + \frac {\partial X ^ {2}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} + \frac {\partial X ^ {2}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} + \frac {\partial X ^ {2}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & \frac {\partial X ^ {2}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\bar {\partial} \xi^ {2}} + \frac {\partial X ^ {2}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} \\ \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & 2 \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} & 2 \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} + \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {3}} + \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {3}}{\partial \xi^ {2}} + \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {3}}{\partial \xi^ {1}} \\ \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {1}} & 2 \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {2}}{\partial \xi^ {2}} & 2 \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} & \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} + \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {2}}{\partial \xi^ {2}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {3}} + \frac {\partial X ^ {1}}{\partial \xi^ {3}} \frac {\partial X ^ {2}}{\partial \xi^ {1}} & \frac {\partial X ^ {1}}{\partial \xi^ {1}} \frac {\partial X ^ {2}}{\partial \xi^ {2}} + \frac {\partial X ^ {1}}{\partial \xi^ {2}} \frac {\partial X ^ {2}}{\partial \xi^ {1}} \end{array} \right] +$$ + +# 5. Nonlinear Newmark- $\beta$ integration method + +먼저 물체의 비선형 운동방정식은 다음과 같다. + +$$ +\mathbf {M} \ddot {\mathbf {u}} + \mathbf {C} (\dot {\mathbf {u}}) \dot {\mathbf {u}} + \mathbf {K} (\mathbf {u}) \mathbf {u} = \mathbf {P} +$$ + +여기서 밀도는 시간이나 변위에 따라 변화하지 않는다고 가정하면 M은 항상 일정하다. 또한 구조물의 동적문제이기 때문에 C는 없다고 생각 할 수 있다. $n+1$ 시간에서 평형방정식을 생각하면 + +$$ +\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} + \mathbf {K} \left(\mathbf {u} _ {n + 1}\right) \mathbf {u} _ {n + 1} = \mathbf {P} _ {n + 1} +$$ + +와 같다. 운동방정식이 비선형이기 때문에 $n+1$ 시간에서 평형을 만족하는 변위와 가속도를 계산하기 위해서 반복 계산이 필요하다. 따라서 Newton-Raphson method를 사용하여 반복계산을 수행하였다. $k+1$ 번째 반복에서 평형이 이루어졌다면 식은 다음과 같다. + +$$ +\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k + 1} + \mathbf {K} \left(\mathbf {u} _ {n + 1} ^ {k + 1}\right) \mathbf {u} _ {n + 1} ^ {k + 1} = \mathbf {P} _ {n + 1} ^ {k + 1} +$$ + +위 식을 정리하면 + +$$ +\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k + 1} + \mathbf {K} \left(\mathbf {u} _ {n + 1} ^ {k + 1}\right) \mathbf {u} _ {n + 1} ^ {k + 1} - \mathbf {P} _ {n + 1} ^ {k + 1} = 0 = \mathbf {R} _ {n + 1} ^ {k + 1} +$$ + +와 같고 Taylor series expansion을 통해 선형화 시키면 + +$$ +\mathbf {R} _ {n + 1} ^ {k + 1} = \mathbf {R} _ {n + 1} ^ {k} + \frac {\partial \mathbf {R} _ {n + 1} ^ {k}}{\partial \mathbf {u} _ {n + 1} ^ {k}} \Delta \mathbf {u} _ {n + 1} ^ {k} + \frac {\partial \mathbf {R} _ {n + 1} ^ {k}}{\partial \dot {\mathbf {u}} _ {n + 1} ^ {k}} \Delta \dot {\mathbf {u}} _ {n + 1} ^ {k} + \frac {\partial \mathbf {R} _ {n + 1} ^ {k}}{\partial \ddot {\mathbf {u}} _ {n + 1} ^ {k}} \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} +$$ + +와 같다. 이를 풀어 쓰면 + + + +$$ +\begin{array}{l} 0 = \mathbf {P} _ {n + 1} ^ {k} + \frac {\partial \mathbf {P} _ {n + 1} ^ {k}}{\partial \mathbf {u} _ {n + 1} ^ {k}} \Delta \mathbf {u} _ {n + 1} ^ {k} - \left\{\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \underbrace {\mathbf {K} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \mathbf {u} _ {n + 1} ^ {k}} _ {\mathbf {f} _ {\text {int}} \left(\mathbf {u} _ {n + 1} ^ {k}\right)} \right\} - \left\{\mathbf {M} \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \underbrace {\frac {\partial \left(\mathbf {K} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \mathbf {u} _ {n + 1} ^ {k}\right)}{\partial \mathbf {u} _ {n + 1} ^ {k}}} _ {\mathbf {K} _ {t}} \Delta \mathbf {u} _ {n + 1} ^ {k} \right\} \\ \Rightarrow \mathbf {M} \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \mathbf {K} _ {t} \Delta \mathbf {u} _ {n + 1} ^ {k} - \mathbf {P} _ {t} \Delta \mathbf {u} _ {n + 1} ^ {k} = \mathbf {P} _ {n + 1} ^ {k} - \left\{\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \mathbf {f} _ {\text { int }} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \right\} \\ \end{array} +$$ + +와 같다. Newmark-β method를 적용하면 $n+1$ 시간에서 변위와 속도를 구할 수 있다. + +$$ +\mathbf {u} _ {n + 1} = \mathbf {u} _ {n} + h \dot {\mathbf {u}} _ {n} + h ^ {2} \left(\frac {1}{2} - \beta\right) \ddot {\mathbf {u}} _ {n} + h ^ {2} \beta \ddot {\mathbf {u}} _ {n + 1} = \mathbf {u} _ {n} + h \dot {\mathbf {u}} _ {n} + \frac {h ^ {2}}{2} \ddot {\mathbf {u}} _ {n} - h ^ {2} \beta \ddot {\mathbf {u}} _ {n} + h ^ {2} \beta \ddot {\mathbf {u}} _ {n + 1} +$$ + +$$ +\dot {\mathbf {u}} _ {n + 1} = \dot {\mathbf {u}} _ {n} + h (1 - \gamma) \ddot {\mathbf {u}} _ {n} + h \gamma \ddot {\mathbf {u}} _ {n + 1} = \dot {\mathbf {u}} _ {n} + h \ddot {\mathbf {u}} _ {n} + \gamma h \ddot {\mathbf {u}} _ {n + 1} - \gamma h \ddot {\mathbf {u}} _ {n} +$$ + +위의 식을 가속도와 속도로 나타내면 + +$$ +h ^ {2} \beta \ddot {\mathbf {u}} _ {n + 1} = \mathbf {u} _ {n + 1} - \mathbf {u} _ {n} - h \dot {\mathbf {u}} _ {n} - \frac {h ^ {2}}{2} \ddot {\mathbf {u}} _ {n} + h ^ {2} \beta \ddot {\mathbf {u}} _ {n} +$$ + +$$ +\dot {\mathbf {u}} _ {n + 1} = \dot {\mathbf {u}} _ {n} + h \ddot {\mathbf {u}} _ {n} + \gamma h \ddot {\mathbf {u}} _ {n + 1} - \gamma h \ddot {\mathbf {u}} _ {n} +$$ + +여기서 마찬가지로 $k+1$ 반복에서 평형을 이룬다면 + +$$ +h ^ {2} \beta \ddot {\mathbf {u}} _ {n + 1} ^ {k + 1} = \mathbf {u} _ {n + 1} ^ {k + 1} - \mathbf {u} _ {n} - h \dot {\mathbf {u}} _ {n} - \frac {h ^ {2}}{2} \ddot {\mathbf {u}} _ {n} + h ^ {2} \beta \ddot {\mathbf {u}} _ {n} +$$ + +$$ +\dot {\mathbf {u}} _ {n + 1} ^ {k + 1} = \dot {\mathbf {u}} _ {n} + h \ddot {\mathbf {u}} _ {n} + \gamma h \ddot {\mathbf {u}} _ {n + 1} ^ {k + 1} - \gamma h \ddot {\mathbf {u}} _ {n} +$$ + +와 같고 반복에 대한 항을 선형화 시키면 + +$$ +h ^ {2} \beta \ddot {\mathbf {u}} _ {n + 1} ^ {k} + h ^ {2} \beta \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} = \mathbf {u} _ {n + 1} ^ {k} + \Delta \mathbf {u} _ {n + 1} ^ {k} - \mathbf {u} _ {n} - h \dot {\mathbf {u}} _ {n} - \frac {h ^ {2}}{2} \ddot {\mathbf {u}} _ {n} + h ^ {2} \beta \ddot {\mathbf {u}} _ {n} +$$ + +$$ +\dot {\mathbf {u}} _ {n + 1} ^ {k} + \Delta \dot {\mathbf {u}} _ {n + 1} ^ {k} = \dot {\mathbf {u}} _ {n} + h \ddot {\mathbf {u}} _ {n} + \gamma h \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \gamma h \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} - \gamma h \ddot {\mathbf {u}} _ {n} +$$ + +위 식을 다음과 같이 k 번째 반복의 가속도와 속도, k 번째 반복의 미소 가속도와 미소 속도 항으로 분리 할 수 있다. + +$$ +\ddot {\mathbf {u}} _ {n + 1} ^ {k} = \frac {1}{h ^ {2} \beta} \mathbf {u} _ {n + 1} ^ {k} - \frac {1}{h ^ {2} \beta} \mathbf {u} _ {n} - \frac {1}{h \beta} \dot {\mathbf {u}} _ {n} - \frac {1}{2 \beta} \ddot {\mathbf {u}} _ {n} + \ddot {\mathbf {u}} _ {n} +$$ + +$$ +\dot {\mathbf {u}} _ {n + 1} ^ {k} = \dot {\mathbf {u}} _ {n} + h \ddot {\mathbf {u}} _ {n} + \gamma h \ddot {\mathbf {u}} _ {n + 1} ^ {k} - \gamma h \ddot {\mathbf {u}} _ {n} = \frac {\gamma}{h \beta} \mathbf {u} _ {n + 1} ^ {k} - \frac {\gamma}{h \beta} \mathbf {u} _ {n} + \left(1 - \frac {\gamma}{\beta}\right) \dot {\mathbf {u}} _ {n} + h \left(1 - \frac {\gamma}{2 \beta}\right) \ddot {\mathbf {u}} _ {n} +$$ + +$$ +\Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} = \frac {1}{h ^ {2} \beta} \Delta \mathbf {u} _ {n + 1} ^ {k} +$$ + +$$ +\Delta \dot {\mathbf {u}} _ {n + 1} ^ {k} = \gamma h \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} = \frac {\gamma}{h \beta} \Delta \mathbf {u} _ {n + 1} ^ {k} +$$ + +위의 미소 가속도, 미소 속도를 대입하면 + + + +$$ +\begin{array}{l} \mathbf {M} \Delta \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \mathbf {K} _ {t} \Delta \mathbf {u} _ {n + 1} ^ {k} - \mathbf {P} _ {t} \Delta \mathbf {u} _ {n + 1} ^ {k} = \mathbf {P} _ {n + 1} ^ {k} - \left\{\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \mathbf {f} _ {\text { int }} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \right\} \\ \Rightarrow \left[ \frac {1}{h ^ {2} \beta} \mathbf {M} + \mathbf {K} _ {t} \left(\mathbf {u} _ {n + 1} ^ {k}\right) - \mathbf {P} _ {t} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \right] \Delta \mathbf {u} _ {n + 1} ^ {k} = \underbrace {\mathbf {P} _ {n + 1} ^ {k} - \left\{\mathbf {M} \ddot {\mathbf {u}} _ {n + 1} ^ {k} + \mathbf {f} _ {\text {int}} \left(\mathbf {u} _ {n + 1} ^ {k}\right) \right\}} _ {\mathbf {R} \left(\mathbf {u} _ {n + 1} ^ {k}\right)} \\ \end{array} +$$ + +여기서 $\mathbf{f}_{\mathrm{int}}\left(\mathbf{u}_{n+1}^{k}\right)$ 와 $\mathbf{K}_{t}\left(\mathbf{u}_{n+1}^{k}\right)$ 는 $u_{n+1}^{k}$ 의 함수이기 때문에 반복이 수행될 때마다 다시 계산해 주어야 한다. +이후 다음 반복에 대한 변위, 속도, 가속도는 다음과 같다. + +$$ +\begin{array}{l} \mathbf {u} _ {n + 1} ^ {k + 1} = \mathbf {u} _ {n + 1} ^ {k} + \Delta \mathbf {u} _ {n + 1} ^ {k} \\ \dot {\mathbf {u}} _ {n + 1} ^ {k + 1} = \frac {\gamma}{h \beta} \mathbf {u} _ {n + 1} ^ {k + 1} - \left(\frac {\gamma}{h \beta} \mathbf {u} _ {n} - \left(1 - \frac {\gamma}{\beta}\right) \dot {\mathbf {u}} _ {n} - h \left(1 - \frac {\gamma}{2 \beta}\right) \ddot {\mathbf {u}} _ {n}\right) \\ \ddot {\mathbf {u}} _ {n + 1} ^ {k + 1} = \frac {1}{h ^ {2} \beta} \mathbf {u} _ {n + 1} ^ {k + 1} - \left(\frac {1}{h ^ {2} \beta} \mathbf {u} _ {n} + \frac {1}{h \beta} \dot {\mathbf {u}} _ {n} + \frac {1}{2 \beta} \ddot {\mathbf {u}} _ {n} - \ddot {\mathbf {u}} _ {n}\right) \\ \end{array} +$$ diff --git a/docs/reference-papers/MITC4/MITC공부/images/page-001_64f5f5a6fb0b89f088f64198f2101fe1662d688eb8c02d8fa6f1a855f09aaed4.jpg b/docs/reference-papers/MITC4/MITC공부/images/page-001_64f5f5a6fb0b89f088f64198f2101fe1662d688eb8c02d8fa6f1a855f09aaed4.jpg new file mode 100644 index 0000000..5b1b5d1 Binary files /dev/null and b/docs/reference-papers/MITC4/MITC공부/images/page-001_64f5f5a6fb0b89f088f64198f2101fe1662d688eb8c02d8fa6f1a855f09aaed4.jpg differ diff --git a/docs/reference-papers/MITC4/MITC공부/images/page-001_65026891a36c5d50bc54d22bb4fc099887273632fcbc2746d2786df72ef41bb2.jpg b/docs/reference-papers/MITC4/MITC공부/images/page-001_65026891a36c5d50bc54d22bb4fc099887273632fcbc2746d2786df72ef41bb2.jpg new file mode 100644 index 0000000..2f46992 Binary files /dev/null and b/docs/reference-papers/MITC4/MITC공부/images/page-001_65026891a36c5d50bc54d22bb4fc099887273632fcbc2746d2786df72ef41bb2.jpg differ 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therelations among themare presented inthis paper.Wereviewthebasicshellmathematical model, which is theunderlyingmathematicalmodel of the contiuum mechanics based shellfiniteelements.Theasymptotic theoryof sellstructures thenis reviewed andwe presenthowtoevaluate theasymptoticbehavior in finiteelementsolutions.S-norm is introducedasaneror measureoffinite elementsolutionsandweshow“locking”in theconvergencecurves ofsellfinite element solutions.Wediscusstheconceptof “uniformoptimalconvergence”infiniteelementanalysisofshels.Wefnallysummarizerequirementsonideal shellfiniteelements and propose how to perform benchmark tests of shell finite elements. + +Keywords :shellstructures,finite elements，asymptotic behavioruniformoptimal convergence,benchmark tests + +# 刀 + +본 논문에서는 최근 주요 연구들을 토대로 쉘 구조물의 유한요소해석에 대하여 중요한 개념들과 그 연관관계를 고찰한다감절점 쉘 유한요소의 수학모델인 기본쉘수학모델을 살펴본다 쉘 구조물의 두께가 얇아짐에 따라 일어나는 쉘 구조문제의세가지 극한거동들휨지배거동 막지배거동 혼합지배거동에 대한 쉘의 점근거동 이론을 소개하고 점근거동을 유한요소해석을 통해 찾아내는 방법을 알아본다 유한요소해의 오차를 으로 평가하는 방법을 소개하고 이를 이용하여 쉘 유한요소의 잠김현상이 유한요소해의 수렴곡선에 어떻게 나타나는지 살펴본다 쉘 구조물의 유한요소해석에서 균일최적수렴의 개념을논의한다 마지막으로 이상적인 쉘 유한요소의 조건을 알아보고 쉘 유한요소의 성능평가를 위한 방법론을 제시한다 + +쉘 구조물 유한요소해석 점근거동 균일최적수렴 성능시험 + +# 1. + +계란의 외피가 얼마나 큰 외력에 견딜 수 있는가 하는 것은 일반 대중들에게도 잘 알려져 있는 사실이다 그 밖에도우리는 자연 속에서 갑각류의 표피나 조개 껍질과 같은 여러 종류의 쉘 구조물들 을 접할 수 있다 이런 자연의 예들은 쉘 구조가 매우 이상적이고 효과적인 구조물의 한 형태라는 것을 간접적으로 증명하고 있다 인간이만들어낸 쉘 구조물들도 셀 수 없이 다양하고 많으며 우리는 쉘 구조물들 속에서 살고 있다고 해도 과언이 아니다 + +유한요소법 은 쉘 구조물의 선형및 비선형 해석에 가장 널리 쓰이는 방법으로 지난 수십 년동안 쉘 구조물의 유한요소해석에 대한 연구가 활발하게 이루어져 오고 있다 최창근그러나 쉘 구조물은 쉘의형상 경계조건 하중 등에 따라 아주 다양한 거동을 보이며특히 쉘의 두께가 얇을 경우 거동의 특성이 매우 민감해지기 때문에 쉘 구조물의 유한요소해석에 앞서 근본적인 쉘··········································································································이론 및 물리적 거동에 대한 이해가 필수적이다and Bathe,1998；Lee and Bathe,2002；Chapelle and이러한 쉘 구조물의 거동에 대한 이해의 핵심은 쉘 구조물의 두께가 얇아짐에 따라 나타나는 점근거동을 연구하는 것이다2002；Bathe，Chapelle and Lee，2003). 圣号거동은 일반적으로 휨지배 거동 막지배거동 혼합 지배거동 등으로 나누어진다 + +일반적으로 공학 에서 주어진 문제를 풀기 위한 접근방법은 실험 과 관찰 을 통하여 물리적 거동을 살펴본 후 그 물리문제의 중요한 특성인자를 찾아내고 그에 따른 여러 가지 가정들을 사용하여 문제를 단순화시켜 수학모델을 만든다 수학모델은 주어진 물리문제를 이론적으로접근할 수 있는 방법을 제공해 준다 수학모델의 해는 이론 + + + +적인 방법으로 구해질 수 있으나 풀고자 하는 문제가 매우복잡할 때는 해를 구하기가 거의 불가능하다. 수치해석(numerical analysis)을 이용하면 복잡한 물리문제의 근사해(approximation)를 구할 수 있고 그 결과를 실험/관찰 결과와 비교하면 수학모델이나 수치해석의 적정성을 평가할 수있다. + +이와 같은 일련의 작업들, 즉, 물리적 거동, 수학모델, 수치해석은 서로 밀접하게 연관되어 있다. 그러므로, 쉘 구조물의 유한요소해석을 명확하게 이해하기 위해서는 쉘 구조물의 물리적 거동에 대한 이해, 쉘의 수학모델(mathematicalshell model)에 대한 이해와 쉘 유한요소에 대한 이해가 동시에 체계적이고 심도 있게 이루어져야 한다. 세가지 모두에대한 통합적인 이해가 없을 경우 쉘 구조물의 유한요소해석에 있어서 중대한 오류를 범할 수 있다. 본 논문의 목적은이 각각의 세가지 부분에 대한 이해와 이들이 서로 어떻게유기적으로 관계를 맺고 있는지를 최근 주요 연구들(Leeand Bathe, 2002; Chapelle and Bathe, 2003; Bathe,Chapelle and Lee, 2003; Hiller and Bathe, 2003; Leeand Bathe, 2005)을 중심으로 정리하여 고찰해 보고 이상적인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가 방법을제시하는 것이다. 특히 쉘 구조물의 설계를 위하여 유한요소해석을 수행하는 기술자들(engineers)이나 유한요소법을 공부/연구하는 학생/연구자들의 “쉘 구조물의 유한요소해석에 대한 이해”를 돕고자 하는데 글의 초점을 맞추었다. + +유한요소법에 의해 쉘 구조물을 효과적으로 해석하기 위해서는 신뢰할 만한 쉘 유한요소를 사용해야 한다는 것은 자명하다. 일반적으로 변위법에 의해 정식화(displacementbased formulation)된 쉘 유한요소는 사용된 근사함수(interpolation function)의 차수(order)에 상관없이 휨지배 및혼합지배거동을 하는 쉘 구조물에 대하여 과도한 강성을 나타낸다(Bathe, 1996; Chapelle and Bathe, 2003). 이를 잠김현상(locking phenomenon)이라 하며 효과적인 쉘 유한요소의 개발에 있어서 극복해야 할 어려운 문제들 중의 하나이다. 이상적인 쉘 유한요소는 여러 가지 점근거동과 다양한형상의 쉘 구조문제에 있어서 균일최적수렴(uniform optimalconvergence)을 보여야 하며 이상적인 쉘 유한요소를 개발하는 것은 매우 어려운 일이다. + +본 논문에서는 먼저 쉘의 대표적인 수학모델을 설명하고쉘 구조물의 점근거동에 대한 기본이론을 살펴본 후 임의의쉘 구조물의 점근거동을 어떻게 알아낼 수 있는지를 알아본다. 또한 쉘 유한요소해의 오차(error)를 평가하는 규준을 살펴보고 이를 바탕으로 잠김현상 발생 시의 쉘 유한요소의수렴과 균일최적수렴에 대해 알아본다. 마지막으로 이상적인쉘 유한요소와 쉘 유한요소의 성능평가에 대하여 논한다. 앞으로 논의하는 내용은 등방성재료(isotropic material)에 대한선형탄성(linear elastic)이론에 국한된다. + +# 2. 早g(mathematical shell model) + +기본쉘수학모델(basic shell mathematical model)은 쉘 구조물의 휨(bending)거동, 면내(membrane)거동, 면외전단(transverse shearing)거동과 그 상호연관(coupling)관계를 모두 표현할 수 있는 가장 일반적인 쉘 수학모델로 3차원 연속체 역학으로부터 유도된 감절점 쉘 유한요소(Ahmad,Irons and Zienkiewicz, 1970; Bathe, 1996)와 동일한 변형률 항들을 가지고 있다(Chapelle and Bathe, 1998; Chapelleand Bathe, 2003; Lee and Bathe, 2005). 즉, 기본쉘수학모델은 감절점 쉘 유한요소의 수학모델인 것이다. 본 장에서는 미분기하학(differential geometry)을 이용하여 쉘의 형상(geometry)과 변형거동(kinematics)을 살펴보고 기본쉘수학모델의 유도를 보여준다. + +# 2 . 1 (s h e l l g eo m etry) + +쉘은 두께(thickness)가 얇은 3차원 구조물이다. 두께가 얇다는 특징 때문에 쉘의 형상은 쉘의 중심면으로 이루어진 2차원 영역과 두께에 의하여 정의 수 있다. 여기서는 기본쉘수학모델에서 쓰이는 쉘의 형상에 관한 개 을 미분기하학을 통하여 보여준다. + +![](images/page-002_34c197da10900820920b2de2867c9e962cee63d23bf27916c5234ee81406902c.jpg) + +
+text_image + +Z +Y +X +S +ξ³ - curve +ξ² - curve +midsurface +Φ +ξ² +ω +ξ¹ +ξ¹ - curve +Φ̄(ξ¹, ξ², ξ³) = φ̄(ξ¹, ξ²) + ξ³ ā₃(ξ¹, ξ²) +
+ +1. + + + +(Einstein summation convention) }.αβ，μi,j,미분기하학의 정 + +(midsurface는 에서 지 변하며 는 에서 지 변하는. (covariant base vector)쉘의 중심면 + +$$ +\vec {a} _ {\alpha} = \frac {\partial \vec {\phi} (\xi^ {1} , \xi^ {2})}{\partial \xi^ {\alpha}} \tag {1} +$$ + +』(covariant) (contravariant base vector). + +$$ +\vec {a} ^ {\alpha} \cdot \vec {a} _ {\beta} = \delta_ {\beta} ^ {\alpha} \tag {2} +$$ + +王 $\delta _ { \beta } ^ { \alpha _ { 1 } } \equiv \ \alpha _ { } \mathcal { A }$ 는 다 의 관계에 의하여 어진다βKronecker sybolo．(covariant)]画E] 画音(vector product)여기서 + +$$ +\vec {a} _ {3} = \frac {\vec {a} _ {1} \times \vec {a} _ {2}}{\left\| \vec {a} _ {1} \times \vec {a} _ {2} \right\|} \tag {3} +$$ + +$$ +\vec {\Phi} \left(\xi^ {1}, \xi^ {2}, \xi^ {3}\right) = \vec {\phi} \left(\xi^ {1}, \xi^ {2}\right) + \xi^ {3} \vec {a} _ {3} \left(\xi^ {1}, \xi^ {2}\right) \tag {4} +$$ + +쉘의 차원 기하형상은 다 식으로 표현된다ξ 1 ξ 2 ξ 3 + +$$ +\Omega = \left\{(\xi^ {1}, \xi^ {2}, \xi^ {3}) \in \Re^ {3} | (\xi^ {1}, \xi^ {2}) \in \omega , \xi^ {3} \in \left[ - \frac {t (\xi^ {1} , \xi^ {2})}{2}, \frac {t (\xi^ {1} , \xi^ {2})}{2} \right] \right\} \tag {5} +$$ + +』 』 surface(tensors).2Dmetric여기서 는 쉘의 두께이다 + +$$ +a _ {\alpha \beta} = \vec {a} _ {\alpha} \cdot \vec {a} _ {\beta} \tag {6} +$$ + +서로 공변형 은 다 식과 같다 + +$$ +a ^ {\alpha \beta} = \vec {a} ^ {\alpha} \cdot \vec {a} ^ {\beta} \tag {7} +$$ + +위 식의 반변형 은 다 과 같이 나타난다.(covariant type）斗. + +$$ +b _ {\alpha \beta} = \vec {a} _ {3} \cdot \vec {a} _ {\alpha , \beta} \tag {8} +$$ + +$$ +b _ {\beta} ^ {\alpha} = a ^ {\alpha \lambda} b _ {\lambda \beta} \tag {9} +$$ + +또한 위 식의 합성형 서는 다 식 + +$$ +c _ {\alpha \beta} = b _ {\alpha} ^ {\lambda} b _ {\lambda \beta} \tag {10} +$$ + +세 기본 서는 다 과 같이 정의 된다 w(covariant derivative)E. + +$$ +w _ {\alpha | \beta} = w _ {\alpha , \beta} - \Gamma_ {\alpha \beta} ^ {\lambda} w _ {\lambda} \tag {11} +$$ + +$\Gamma _ { \alpha \beta } ^ { \lambda } \stackrel { \ll } { = }$ Christoffe symbolo)t. + +$$ +\Gamma_ {\alpha \beta} ^ {\lambda} = \vec {a} _ {\alpha , \beta} \cdot \vec {a} ^ {\lambda} \tag {12} +$$ + +여기서 는 면에서의 이다 + +$$ +\vec {g} _ {i} = \frac {\partial \vec {\Phi} (\xi^ {1} , \xi^ {2} , \xi^ {3})}{\partial \xi^ {i}} \tag {13} +$$ + +$$ +\vec {g} _ {\alpha} = \vec {a} _ {\alpha} - \xi^ {3} b _ {\alpha} ^ {\lambda} \vec {a} _ {\lambda} \tag {14} +$$ + +$$ +\vec {g} _ {3} = \vec {a} _ {3} \tag {15} +$$ + +(contravarianto. + +$$ +\vec {g} ^ {i} \cdot \vec {g} _ {j} = \delta_ {j} ^ {i} \tag {16} +$$ + +# 차원 반변 기저 터는 다 + +쉘의 변형거동 + +$$ +\vec {U} \left(\xi^ {1}, \xi^ {2}, \xi^ {3}\right) = \vec {u} \left(\xi^ {1}, \xi^ {2}\right) + \xi^ {3} \theta_ {\lambda} \left(\xi^ {1}, \xi^ {2}\right) \vec {a} ^ {\lambda} \left(\xi^ {1}, \xi^ {2}\right) \tag {17} +$$ + +같이 표 $\vec { u } ( \xi ^ { 1 } , \xi ^ { 2 } )$ 』』(infinitesimaltranslation) E $\theta _ { \lambda } ( \xi ^ { 1 } , \xi ^ { 2 } )$ (infinitesimal rotation) E. $\theta _ { \lambda } { \vec { a } } ^ { \lambda } \equiv$ 여기서 는 $\xi ^ { 3 } \theta _ { \lambda } { \vec { a } } ^ { \lambda } \equiv \ I$ 의 미소변위를 나타내고 는 쉘의 중심면에 수u $\vec { \boldsymbol a } ^ { 1 }$ $\scriptstyle { \vec { \alpha } } ^ { 2 }$ 선 $\vec { a } ^ { 3 } ( = \vec { a } _ { 3 }$ + +전 터 이고 는 그 전 터에 의한 변위를나타낸다 선변위 는 반변 기저 터 + +$$ +e _ {i j} = \frac {1}{2} (\vec {g} _ {i} \cdot \vec {U} _ {, j} + \vec {g} _ {j} \cdot \vec {U} _ {, i}) \tag {18} +$$ + +$$ +\vec {U} _ {, i} = \frac {\partial \vec {U} (\xi^ {1} , \xi^ {2} , \xi^ {3})}{\partial \xi^ {i}}. \tag {19} +$$ + +(14)，(5)(17)』(18)(covariant） + +$$ +e _ {\alpha \beta} = \gamma_ {\alpha \beta} (\vec {u}) + \xi^ {3} \chi_ {\alpha \beta} (\vec {u}, \vec {\theta}) - \left(\xi^ {3}\right) ^ {2} \kappa_ {\alpha \beta} (\vec {\theta}) \tag {20a} +$$ + +$$ +e _ {\alpha 3} = \zeta_ {\alpha} (\vec {u}, \vec {\theta}) \tag {20b} +$$ + +$$ +e _ {3 3} = 0 \tag {20c} +$$ + +$$ +\gamma_ {\alpha \beta} (\vec {u}) = \frac {1}{2} (u _ {\alpha | \beta} + u _ {\beta | \alpha}) - b _ {\alpha \beta} u _ {3} \tag {21a} +$$ + +$$ +\chi_ {\alpha \beta} (\vec {u}, \vec {\theta}) = \frac {1}{2} \left(\theta_ {\alpha | \beta} + \theta_ {\beta | \alpha} - b _ {\beta} ^ {\lambda} u _ {\lambda | \alpha} - b _ {\alpha} ^ {\lambda} u _ {\lambda | \beta}\right) + c _ {\alpha \beta} u _ {3} \tag {21b} +$$ + +$$ +\kappa_ {\alpha \beta} (\vec {\theta}) = \frac {1}{2} (b _ {\beta} ^ {\lambda} \theta_ {\lambda | \alpha} - b _ {\alpha} ^ {\lambda} \theta_ {\lambda | \beta}) \tag {21c} +$$ + +$$ +\zeta_ {\alpha} (\vec {u}, \vec {\theta}) = \frac {1}{2} \left(\theta_ {\alpha} + u _ {3, \alpha} + b _ {\alpha} ^ {\lambda} u _ {\lambda}\right) \tag {21d} +$$ + +(isotropic material)o (plane + + + +을 적용하면 면에 수 인 력은 이며σ 이때 력과 변형률의 상관관계는 다 과 같다 + +$$ +\sigma^ {\alpha \beta} = C ^ {\alpha \beta \lambda \mu} e _ {\lambda \mu} \tag {22a} +$$ + +$$ +\sigma^ {\alpha 3} = \frac {1}{2} D ^ {\alpha \lambda} e _ {\lambda 3} \tag {22b} +$$ + +위의 식 에서 + +$$ +C ^ {\alpha \beta \lambda \mu} = \frac {E}{2 (1 - \nu)} \left(g ^ {\alpha \lambda} g ^ {\beta \mu} + g ^ {\alpha \mu} g ^ {\beta \lambda} + \frac {2 \nu}{1 + \nu} g ^ {\alpha \beta} g ^ {\lambda \mu}\right) \tag {23a} +$$ + +$$ +D ^ {\alpha \lambda} = \frac {2 E}{1 + \nu} g ^ {\alpha \lambda} \tag {23b} +$$ + +여기서 E는 재료의 탄성계수 이고 v는아 비 이며 $g ^ { \alpha \beta } \equiv \ f$ 의 차원 반변기저 터로정의된 서이다 $( g ^ { \alpha \beta } = _ { g } ^ { \alpha } . _ { g } ^ { \beta } )$ + +쉘 구조물에 강체운동 이 일어나지도 적절한 변위경계조건이 주어지면 식 에서 지를 이용하여 기본쉘수학모델의 지배변분식 을 을 수 있다 + +해를 구하는 과정은 모든 임의의 시험함수에 대하여 다 식 와 변위 경계조건을 만 시 는V미지변위 를 찾는 것으로 나타내어질 수 있다U + +$$ +\begin{array}{l} \int_ {\Omega} C ^ {\alpha \beta \lambda \mu} e _ {\alpha \beta} (\vec {U}) e _ {\lambda \mu} (\vec {V}) d V + \int_ {\Omega} D ^ {\alpha \lambda} e _ {\alpha 3} (\vec {U}) e _ {\lambda 3} (\vec {V}) d V \\ = \int_ {\Omega} \vec {F} \cdot \vec {V} d V \tag {24} \\ \end{array} +$$ + +여기서 는 쉘 구조물에 작용하는 외력 F이며 시험함수는 변위경계조건을 만 시켜야 한다 + +$$ +\vec {V} \left(\xi^ {1}, \xi^ {2}, \xi^ {3}\right) = \vec {v} \left(\xi^ {1}, \xi^ {2}\right) + \xi^ {3} \eta_ {\lambda} \left(\xi^ {1}, \xi^ {2}\right) \vec {a} ^ {\lambda} \left(\xi^ {1}, \xi^ {2}\right) \tag {25} +$$ + +# 3. + +휨 막 면외전단작용들은 쉘 구조물이 하중을 지지하는 기본적인 원리이다 그러므로 하중 재하 시 쉘 구조물은 휨막 전단 에 지를 그 내부에 저장하게 된다 쉘의 두께가얇아짐에 따라 쉘의 전단 에 지는 시할 만 작아지므로쉘 구조물의 에 지는 주로 휨 에 지와 막 에 지에 의해구성된다고 할 수 있다 + +쉘은 그 두께가 얇아짐에 따라 특정한 한계거동 휨지배(bending dominated)号，lu(membrane dominated),혼합 지배거동 을 보이게 되며 이를 쉘의 점근거동이라 한다 쉘의 두께가 얇아짐에 따라쉘 구조물이 주로 휨 거동에 의해 하중을 지 할 경우 그쉘 구조물을 휨지배 쉘 구조물이라 부 며 막거동에 의해 하중을 지 할 경우 막지배 쉘구조물이라 한다 또한 두께가얇아짐에 따라 쉘 구조물이 휨과 막거동 두 가지 모두에 의해 외력을 지지할 경우 혼합 지배 쉘 구조물이라 한다 쉘 구조물의 점근거동은 쉘의 형상 경계조건하중 에 따라 라진다and Bathe，2002；Bathe，Chapelle and Lee，2003；Chapelleand Bathe,2003). + +# 3.1 + +식 에서 구해진 선형 쉘 이론의 변분식을 두께 t에 대하여 정리하여 t의 고차항을 제거하면다 과 같이 간 화하여 나타낼 수 있다 + +$\mathrm { F i n d ~ } \ \vec { U } { \in } \ \vec { \Psi } \ \mathrm { \ s u c h ~ \ t h a t }$ + +$$ +\varepsilon^ {3} A _ {b} (\vec {U}, \vec {V}) + \varepsilon A _ {m} (\vec {U}, \vec {V}) = \vec {F} (\vec {V}), \forall \vec {V} \in \vec {\Psi} \tag {26} +$$ + +여기서 ε는 쉘의 두께와 전체 쉘 구조물 기의 비 t Lb m3 A 는 휨 에 지 $\varepsilon A _ { b } ( \cdot , \cdot ) \frac { \circ } { \cdot }$ 막 및 전단 에 지에 대하는 선형식들 이며 는 변위장의 해U는 시험함수 는 공간 1 는 외력V Ψ F( )⋅에 대 하는 선형식 을 나타낸다 일반적으로 쉘의 두께가 얇을 때 전단 에 지는 막 에 지에 비해 매우작으므로 ε A 을 막 에 지에 대 하는 항이라고 부를 수있다 + +ρ 가 해진 외력 을사용한다 + +$$ +\vec {F} (\vec {V}) = \varepsilon^ {\rho} \vec {G} (\vec {V}) \tag {27} +$$ + +여기서 는 의 2 에 속하며 ρ는 실수이G Ψ′ Ψ다 식 의 변의 각 항은 ε 3 과 ε에 비 하므로 ρ는보다 거나 같고 보다 작거나 같은 실수임을 알 수 있다 + +$$ +1 \leq \rho \leq 3 \tag {28} +$$ + +다 의 공간 은 쉘의 점근거동에 중요한 역할을 한다 + +$$ +\overrightarrow {\Psi} _ {0} = \left\{\vec {V} \in \overrightarrow {\Psi} \Big | _ {A _ {m} (\vec {V}, \vec {V}) = 0} \right\} \tag {29} +$$ + +공간은 순수 휨 을 나타내는 변위의 공간Ψ이며 막 및 전단 에 지를 으로 만들 수 있는 모든 변위의 형태들 을 함한다 이 공간 이 단지 모든변위를 으로 하는 변위의 형태들만을 가지고 있을 때 쉘구조물에서 순수 휨거동이 구속되었다고 하며 그러한 쉘을순수 휨이 구속된 쉘 이라 한다 반면에 쉘이모든 변위가 인 형태 가 아 순수 휨 모를 가지고 있을 경우 그 쉘 구조물을 순수 휨이 구속되지은 쉘 구조물이라 한다 쉘의 점근거 + + + +1. + +
경우하중분류
순수 힘이 구속되지않은 셀 구조물, $\vec{\Psi}_{0} \neq \{0\}$ 순수 힘을 유발하는 외력 $\exists \vec{V} \in \vec{\Psi}_{0}$ such that $\vec{G}(\vec{V}) \neq 0$ (i) 훼지배거동
순수 힘을 유발하지 않는 외력 $\vec{G}(\vec{V}) = 0, \forall \vec{V} \in \vec{\Psi}_{0}$ (ii) 불안정한 막지배 또는 혼합지배거동
순수 힘이 구속된 셀 구조물, $\vec{\Psi}_{0} = \{0\}$ Admissible membrane loading $\vec{G} \in \vec{\Psi}_{m}'$ (iii) 막지배거동
Non-admissible membrane loading $\vec{G} \notin \vec{\Psi}_{m}'$ (iv) 혼합지배거동
+ +동은 순수 휨이 구속되어 있는가아 가에 따라 변한다 + +순수 휨이 구속되지 은 경우즉 $\vec { \Psi } _ { 0 } \not = \{ 0 \}$ 는 주로 쉘구조의 휨지배 상태를 이 어낸다 이때 적 한 ρ의 은이며 식 의 막 에 지 항이 사라지면서 이 경우 식의 쉘 문제는 다 과 같이 표현 수 있다 + +Find $\vec { U } ^ { 0 } \in \vec { \Psi } _ { 0 }$ such that + +$$ +A _ {b} (\vec {U} ^ {0}, \vec {V}) = \vec {G} (\vec {V}), \forall \vec {V} \in \vec {\Psi} _ {0} \tag {30} +$$ + +순수 휨이 구속되지 은 경우의 휨지배 상태는 하중이 휨변위를 유발 시켜야만 일어 수 있다 만일 하중이 휨을유발할 수 없다면 이론적인 점근거동은 휨이 구속된 경우와같게 되나 이 경우 거동은 매우 불안정 하다 즉이런 경우 작은 하중의 변화로 쉘 구조물의 점근거동을 막지배거동에서 휨지배거동으로 바 수 있다 + +순수 휨이 구속된 경우즉 $\vec { \Psi } _ { 0 } = \{ 0 \} ) ,$ 적정한 ρ 은 이며 막과 전단 에 지만을 유발시 수 있는 변위공간$\overrightarrow { \Psi } _ { m }$ 에 의해 구조물의 거동이 표현 수 있다 그러므로 이공간 의 기는 $\vec { \Psi }$ 보다 다 막지배거동의 한계문제는 다 과 같이 표현된다 + +Find $\vec { U } ^ { m } \in \vec { \Psi } _ { m }$ such that + +![](images/page-005_353fdda2703e77243cf8511ab96f62613679664877506ed583effba7c69fcb9f.jpg) + +
+text_image + +Z +Y +X +ξ³ -curve +ξ² -curve +u₃ +θ₁ +u₂ +θ₂ +S +a₃ +a₂ +a₁ +a₁ +u₁ +ξ¹ -curve +U̅(ξ¹,ξ²,ξ³)=u̅(ξ¹,ξ²)+ξ³θλ(ξ¹,ξ²) a̅λ(ξ¹,ξ²) +
+ +2. + +$$ +A _ {m} (\vec {U} ^ {m}, \vec {V}) = \vec {G} (\vec {V}), \forall \vec {V} \in \vec {\Psi} _ {m} \tag {31} +$$ + +여기서 외력 가m G $\vec { \Psi } _ { m }$ 의 에 속해야만 막지배거동의 한계문제가 잘 정의 되며 이 조건 $\vec { G } \in \vec { \Psi } _ { m } { } ^ { \prime } ) \frac { \circ } { \textsc { i } }$ 다식과 동일하다 + +$$ +\left| \vec {G} (\vec {V}) \right| \leq c \sqrt {A _ {m} (\vec {V} , \vec {V})}, \forall \vec {V} \in \vec {\Psi} \tag {32} +$$ + +여기서 는 상수이다 위 식은 재하된 외력이 막 력만에의하여 지지 수 있다는 것을 하며 이런 조건을 만 시는 외력을 이라고 부른다}引o] “non-admissible membrane loading $( \vec { G } \in \vec { \Psi } _ { m } ^ { \ \prime } ) ^ { \flat }$ 이라면 이 경우 막지배 문제는 정의 수 없으며 쉘의 점근거동은 막과 휨의 혼합된 형태 를 게 된다 + +표 은 위에서 언 된 쉘 구조물의 점근거동을 정리요하여 보여준다 쉘 구조물의 설계 시 이러한 점근거동에 대한 지식은 매우 유용하며 필수적이다 주어진 하중에 대하여쉘 구조물의 강성은 ερ 에 비 하여 변한다 즉 쉘 구조물의 거동이 휨에 의해 지배 게 경우 구조물의 강성은에 비 하게 되며 막거동에 의해 지배 경우 강성은에 비 하게 된다 그러므로 효과적인 쉘 구조물은 휨거동이 구속되어 막거동에 의해 지배되는 구조물이며 쉘 구조물은 막지배거동을 하도 설계하는 것이 바 하다 주어진 외력에 대하여 쉘의 형상과 경계조건을 적절히 사용하여최대의 기하학적 강성 을 을 수 있도하여야 한다 + +![](images/page-005_4b7a60b2a68ea74e63f7c84a9870ada9b1ae5622c5988abd01bc685583130cdd.jpg) + +
+natural_image + +3D wireframe model of a dome structure with a highlighted top layer (no text or symbols) +
+ +bdry쉘의 + + + +# 3.2 + +쉘의 력/변형률/변위장들은 그 변화가 매 러운 영역들(smooth areas)과 그 지 은 여러 종류의 들(layers)로나누어진다. (layer)은 률(curvature)이나 두께의 변화와같은 쉘의 형상의 변화, 적합하지 은 변위경계조건(incompatible boundary condition), 불규 한 하중(irregularloading) 등에 의하여 유발된다(Lee and Bathe, 2002).(layer)에서는 력/변형률/변위 등이 매우 하게 변하며 변형에 지의 중이 일어난다. 의 특성 이( , characteri-stic length)는 쉘 구조물의 두께에 따라 변하며 쉘의 두께( )와 전체 쉘 구조물 이( )의 함수로 나타나다. + +$$ +L _ {c} = c t ^ {1 - l} L ^ {l} \tag {33} +$$ + +여기서 는 상수이며 은 양의 실수 이다. + +고문 (Lee and Bathe, 2002)는 쉘의 두께가 얇아짐에따라 나타나는 Scodelis-Lo roof shell problem에서의 경계(boundary layer)과 물면(hyperbolic paraboloid) 쉘구조문제의 내부 (inner layer)을 보여준다. 고문 (Bathe,Chapelle and Lee, 2003)에서는 또 다른 형태의 경계 이쉘의 두께가 얇아짐에 따라 변화하는 것을 보여준다. 그3은 쉘의 변형형상(deformed shape)과 유효 력(effectivestress)분 에 나타난 경계 (boundary layer)의 예를 보여주고 있다. + +# 3.3 + +이론적인 방법으로 일반적인 쉘 구조물의 점근거동을 알아내려는 시도는 Lovadina를 비 한 많은 연구자들에 의해 수행되었으나 운 일이 아니었다(Lovadina, 2001). 근 에 수치적인 방법에 의해 점근거동을 알아내는 연구가 진행되었으며 유한요소해석을 통한 여러 가지 방법이 Lee와 Bathe에의해 제안되었다(Lee and Bathe, 2002; Bathe, Chapelle andLee, 2003). 본 논문에서는 가장 운 한가지 방법을 소개한다. + +쉘 구조물의 ρ(load scaling factor) 을 구하게 되면 그구조물의 점근거동을 알아낼 수 있다. 다 은 Lee와 Bathe에 의해 제안된 ρ의 근사 을 구하는 식이다. + +$$ +\rho \cong \frac {\log E (\varepsilon + \Delta \varepsilon) - \log E (\varepsilon)}{\log (\varepsilon + \Delta \varepsilon) - \log \varepsilon} \tag {34} +$$ + +여기서 는 ε(= / )에 대한 유한요소해석으로부터 구해진쉘 구조물의 변형에 지(strain energy) 이다. 주어진 ε에대하여 유한요소해의 정확도가 을수 보다 정확한 ρ 을을 수 있다. 계 된 ρ의 이 1일 경우 구조물은 막지배거동을, 3일 경우 휨지배거동을, 1과 3의 중간 일 경우막과 휨의 혼합지배거동을 한다. + +Lovadina는 보간이론(interpolation theory)을 사용하여 쉘구조물의 점근거동을 연구하 으며 ρ 과 쉘 구조물에 저장되는 에 지들과의 관계를 나타내는 미로운 식을 제안하다(Lovadina, 2001). + +$$ +\lim _ {\varepsilon \rightarrow 0} R (\varepsilon) = \frac {\rho - 1}{2} \tag {35} +$$ + +여기서 (ε)는 쉘 구조물의 전체 변형에 지에 대한 휨 변형에 지의 비이다. + +$$ +R (\varepsilon) = \frac {\varepsilon^ {3} A _ {b} (\vec {U} , \vec {U})}{\varepsilon^ {3} A _ {b} (\vec {U} , \vec {U}) + \varepsilon A _ {m} (\vec {U} , \vec {U})} \tag {36} +$$ + +역으로 쉘 구조물의 (ε)를 계 하면 또한 쉘의 점근거동을찾을 수 있는 ρ 을 알 수 있다. + +Lee와 Bathe는 쉘 유한요소의 성능을 평가(benchmark)하기 위한 쉘 문제로 널리 알려져 있는 Scodelis-Lo roofshell problem을 대상으로 하여 ρ 을 계 하 으며 그 쉘구조물의 점근거동을 보여주었다(Lee and Bathe, 2002). 그후에 Lovadina는 이론적인 방법을 이용해 같은 문제의 ρ을 구하 고 두 결과는 일치하 다. 쉘 구조물의 휨지배거동과 막지배거동에 대한 보다 자세한 예들을 고문 (Leeand Bathe, 2002)에서 수 있으며, 쉘 두께의 변화에 따라 ρ 이 변동(fluctuation)하는 민감한 쉘 구조물의 점근거동에 대한 해석이 고문 (Bathe, Chapelle and Lee,2003)에 제시되어 있다. + +# 4 + +쉘 구조물의 유한요소해석에서 면한 가장 큰 어려 은잠김현상(locking phenomenon)이다. 잠김현상은 쉘 구조물의두께가 얇을 수 유한요소해의 오차(error)를 매우 게증가시 다. 본 장에서는 쉘의 점근거동에 대한 이해를 바탕으로 쉘 유한요소의 잠김현상과 균일최적수렴에 대하여 살펴본다. + +# 4. + +쉘 유한요소의 종류는 게 평면 쉘(flat shell) 유한요소그 과 감절점 쉘(degenerated shell or continuum mechanicsbased shell) 유한요소 그 으로 나 어진다(Ahmad, Ironand Zienkiewicz, 1970; Bathe, 1996; Choi, Lee and Park,1999; 최창근, 2002; Chapelle and Bathe, 2003). 평면 쉘유한요소는 평 유한요소와 평면 력 유한요소의 결합에 의해 만들어진다. 면의 쉘 구조물을 여러 개의 평면으로 나누어 표현한다는 물리적 개 에서 발하 다. 장점은 유한요소 정식화가 고 면내 전자유도(drilling degrees offreedom)를 게 도 하여 절점 6개의 자유도를 가질 수있으며 그로 인해 절점 6개의 자유도를 는 (beam)과 같은 유한요소들과 결합이 리하다는 것이다. 그러나, 기본적으로 요소자체의 형상이 평면이기 때문에 복잡한 면형태의 쉘의 형상과 그 거동을 정밀하게 표현하기 어려면 쉘 구조물의 해석 시 수렴이 리며 정확해로의 수렴에 실 하는 경우가 발생하는 단점을 가지고 있다. 반면 3차원 연속체역학에 근거한 감절점 쉘 요소는 절점 5개의자유도를 가지고 있어 과 같은 유한요소들과의 결합 시특 한 인위적인 방법이 필요하지만 쉘 고유의 복잡한 면형상과 그 거동을 잘 표현해 낼 수 있고 정확해로의 수렴속도가 다. 또한, 2장에서 설명된 바와 같이 가장 일반적인 쉘 수학모델(mathematical shell model)인 기본쉘수학모델에 근거한다. + +쉘 유한요소를 유한요소의 정식화(formulation)방법에 따라 + + + +분류하면 게 변위법 에 의한쉘 유한요소와 혼합법 에 의한 쉘 유한요소로나 수 있다 변위법에 근거한 쉘 유한요소는 막지배 구조물들의 해석에 있어서는 이상적인 거동을 보인다 그러나 요소의 종류와 근사차수 에 상관없이 두께가 얇은 휨지배또는 혼합지배 쉘 구조물들의해석에 있어서 유한요소해석의 해가 매우 리게 수렴하는치명적인 단점을 가지고 있는데 이를 잠김 현상이라고 한다 주어진 요소 에 대하여 유한요소해의 정확성이 쉘의 두께가 얇아짐에 따라 속히 나 지는 현상을하며 최 의 경우 쉘의 두께가 얇아짐에 따라 변위장이에 수렴하게 된다 + +잠김현상은 막잠김 과 전단잠김현상으로 나누어질 수 있으며 잠김현상이 일어나는근본적인 이유는 변위법에 의한 쉘 유한요소가 식 의순수 휨 변위공간 을 분히 근사할 수 없기 때Ψ문이다 즉 잠김현상은 쉘 구조물의 점근거동과 접적인 연관관계를 가지고 있다 쉘 구조물의 점근거동이 쉘의 형상경계조건 하중 등에 따라 변하기 때문에 잠김현상 또한 쉘의 형상 경계조건 하중에 의 적이다 막잠김현상은 률을가진 쉘 구조물에서만 발생하고 평면형상의 쉘 구조물에서는 발생하지 으나 전단잠김현상은 률에 관계없이 발생한다 + +잠김현상은 휨지배 및 혼합지배거동 시 식 와에 있는 변형률항들로부터 발생한다 고문and Hiller，2003) (Lee and Bathe，2005) 召o]유한요소해에 어떻게 나타나는지를 보여주고 잠김현상이 제거되기 어려운 근본적인 이유를 잘 설명하고 있다 + +실제 쉘 유한요소해석에 있어서 잠김현상은 과대한 강성또는 그로 인한 과소한 변위 력 변형률 및 변형에 지로나타난다 그러므로 두께가 얇은 쉘 구조물의 설계 시 분하게 조밀하지 한 유한요소 을 사용하여 해석한 결과를이용할 경우강성이 과대평가되어 과소설계의 오류를 범할수 있다 이와 같이 유한요소해의 오차 또는 정확도가 쉘구조물의 두께에 따라 변하는 현상인 잠김현상은 쉘 구조물의 유한요소해석에 있어서 하게 일어나기 때문에 쉘 구조물의 해석에 앞서 잠김현상에 대한 이해는 필수적이다 + +# 4.2 + +신뢰할 만한 쉘 유한요소의 해는 사용된 요소의 수가 증가함에 따라 장에서 설명된 쉘 유한요소의 수학모델(mathematical shell model) (exact solution)렴하여야 하며 잠김현상을 알아보기 위해서는 쉘 유한요소해의 수렴 선 을 관찰해야 한다 여기서쉘 유한요소의 수렴을 정하기 위하여 적절한 규준을 사용하는 것이 매우 중요하다 그 규준 은 한 점에서 특정 물리 의 수렴이 아니라 쉘구조물 전체 영역에서의 해의 수렴을 고려할 수 있어야 한다 일반적으로 한 점에서의 변위 력변형률 등을 가지고수렴을 정하는 방법이 많이 사용되어 는데 이는 전체영역에서의 수렴을 대표하지 으므로 적절하지 하다 변형에 지의 술적 차이 또한 규준으로 사용할 수 있지만 변위법 정식화에 근거한 유한요소가 아 경우에는 일반적으로 사용 수 없기 때문에 정식화의 방법에 관계없이 일반적으로 쓰일 수 있는 규준이 필요하다 + +와 에 의해 제안된 은 쉘 구조물 ⋅전체의 거동을 반영할 수 있을 만 아니라 물리적인 개으로부터 유도되었기 때문에 정식화 방법에 관계없이 사용(Hiller and Bathe,2003). + +$$ +\left\| \overrightarrow {U} - \overrightarrow {U} h \right\| _ {s} ^ {2} = \int_ {\Omega} \Delta \vec {\mathcal {E}} ^ {T} \Delta \vec {\sigma} d \Omega \tag {37} +$$ + +여기서 는 정확해이며 는 유한요소해이다 와 는ε σ전체 각 표계 에서 정의된 변형률 터와 력 터이다 + +$$ +\vec {\varepsilon} = \left[ \varepsilon_ {x x}, \varepsilon_ {y y}, \varepsilon_ {z z}, 2 \varepsilon_ {x y}, 2 \varepsilon_ {y z}, 2 \varepsilon_ {z x} \right] ^ {T} \tag {38a} +$$ + +$$ +\vec {\sigma} = \left[ \sigma_ {x x}, \sigma_ {y y}, \sigma_ {z z}, \sigma_ {x y}, \sigma_ {y z}, \sigma_ {z x} \right] ^ {T} \tag {38b} +$$ + +식 에서 변형률과 력에 대하여 정확해와 유한요소해의차이 는 다 과 같이 구해질 수 있다 + +$$ +\Delta \vec {\mathcal {E}} = \vec {\mathcal {E}} - \vec {\mathcal {E}} _ {h} = \vec {\mathcal {E}} (\vec {x}) - \mathbf {B} _ {h} (\vec {x} _ {h}) \vec {U} _ {h} \tag {39a} +$$ + +$$ +\Delta \vec {\sigma} = \vec {\sigma} - \vec {\sigma} _ {h} = \vec {\sigma} (\vec {x}) - \mathbf {C} _ {h} (\vec {x} _ {h}) \mathbf {B} _ {h} (\vec {x} _ {h}) \vec {U} _ {h} \tag {39b} +$$ + +여기서 는 재료의 력 변형률 관계 행 이고는 변형률변위 관계 연 자 이다 위치 터 와 는 각각 원 쉘 구조물의 영역과 이 화된 유한요소 모델의 영역에 대 된다 두터의 관계는 일대일 사상 Π에 의하여정의할 수 있다 + +$$ +\vec {x} = \Pi (\vec {x} _ {h}) \tag {40} +$$ + +일반적인 쉘 구조문제에 있어서 이론적인 방법을 이용하여정확해를 찾아내는 것은 거의 불가능하므로 매우 조밀한 유한요소 을 사용하여 계 된 해를 정확해로 고려하여을 계 하는 것이 실용적이다 정확해로 고려된유한요소해를 라고 하면 위식의 은 다 과 같이나타내어질 수 있다 + +$$ +\left\| \vec {U} _ {r e f} - \vec {U} _ {h} \right\| _ {s} ^ {2} = \int_ {\Omega_ {r e f}} \Delta \vec {\varepsilon} ^ {T} \Delta \vec {\sigma} d \Omega_ {r e f}, \tag {41} +$$ + +여기서 + +$$ +\Delta \vec {\varepsilon} = \vec {\varepsilon} _ {r e f} - \vec {\varepsilon} _ {h} = \mathbf {B} _ {r e f} (\vec {x} _ {r e f}) \vec {U} _ {r e f} - \mathbf {B} _ {h} (\vec {x} _ {h}) \vec {U} _ {h}, \tag {42a} +$$ + +$$ +\Delta \vec {\sigma} = \vec {\sigma} _ {r e f} - \vec {\sigma} _ {h} = \mathbf {C} _ {r e f} \left(\vec {x} _ {r e f}\right) \mathbf {B} _ {r e f} \left(\vec {x} _ {r e f}\right) \vec {U} _ {r e f} - \mathbf {C} _ {h} \left(\vec {x} _ {h}\right) \mathbf {B} _ {h} \left(\vec {x} _ {h}\right) \vec {U} _ {h}, \tag {42b} +$$ + +$$ +\vec {x} _ {r e f} = \Pi (\vec {x} _ {h}) \tag {42c} +$$ + +다양한 쉘 구조문제와 여러 경우의 쉘 두께에 대하여 유한요소해의 수렴을 상호 비교 가능하게 하기 위해서는 상대오차 를 사용하여야 한다 상대오차 는 다 과같이 정의된다 + +$$ +E _ {h} = \frac {\left\| \overrightarrow {U} _ {r e f} - \overrightarrow {U} _ {h} \right\| _ {s} ^ {2}}{\left\| \overrightarrow {U} _ {r e f} \right\| _ {s} ^ {2}} \tag {43} +$$ + + + +![](images/page-008_bf153388c62cf5ffb959fea1f220656f3e2d7098e600523e932f659c60a4481d.jpg) + +
+text_image + +Z +q +t +X +
+ +![](images/page-008_482a5acd2a2caffb955b8010addd15a3324d6a27a92279398b29b7ab0aacf39d.jpg) + +
+text_image + +Y +2L +2L +h +X +
+ +![](images/page-008_d5f36e391a2db6312da04bb7410853f21a580803e3d34ef9b823cd8bfcc06080.jpg) + +
+surface_3d + +| X | Y | Z | +|------|------|------| +| -0.5 | 0.0 | 0.0 | +| 0.0 | 0.0 | 0.0 | +| 0.5 | 0.0 | 0.0 | +| 0.5 | 0.5 | 0.0 | +| 0.0 | 0.0 | 0.4 | +| 0.5 | 0.5 | 0.0 | +
+ +(b） + +4.：（a）（=.0）（b）(hyperbolic쉘 구조문제들 네 변이 완 +![](images/page-008_bae77c29be22346584cd73116d7f5d92877cdb935440ffd7b96206b10053cad6.jpg) + +
+line + +| log(h) | t/L=1/10 | t/L=1/100 | t/L=1/1000 | t/L=1/10000 | +| ------ | -------- | --------- | ---------- | ----------- | +| -1.6 | -1.6 | -0.1 | 0.0 | 0.0 | +| -1.2 | -0.8 | -0.1 | 0.0 | 0.0 | +| -0.8 | -0.4 | -0.1 | 0.0 | 0.0 | +| -0.4 | -0.2 | -0.1 | 0.0 | 0.0 | +| 0.0 | 0.0 | -0.1 | 0.0 | 0.0 | +
+ +(a） + +![](images/page-008_9d4f652b3e3d5913766088f8d912b5ae7dcfaa2f377adf6fd5fa6150b5e7eedf.jpg) + +
+line + +| log(h) | log(relative error) for t/L=1/10 | log(relative error) for t/L=1/100 | log(relative error) for t/L=1/1000 | log(relative error) for t/L=1/10000 | +| ------ | ------------------------------- | -------------------------------- | --------------------------------- | ---------------------------------- | +| -1.6 | -2.8 | -2.8 | -2.8 | -2.8 | +| -1.2 | -2.0 | -2.0 | -2.0 | -2.0 | +| -0.8 | -1.2 | -1.2 | -1.2 | -1.2 | +| -0.4 | -0.4 | -0.4 | -0.4 | -0.4 | +
+ +（b） +두께의 변화에 따른 + +유한요소해의 이론적인 수렴은 다 과 같다. + +$$ +E _ {h} \cong c h ^ {2 k} \tag {44} +$$ + +여기서 는 상수, 는 사용된 유한요소의 기, 는 유한요소의 변위 근사함수(displacement interpolation function)차수이다. 예를 들면 선형(linear) 근사함수를 사용하는 3절점 및 4절점 쉘 유한요소에 대하여 =1이며, 2차의(quadratic) 근사함수를 사용하는 6절점 및 9절점 쉘 유한요소에 대하여 = 2이다. S-norm은 수치적인 방법에 의해 계수 있다(Lee, Noh and Bathe, 2007). + +# 4.3 + +다 으로 2개의 휨지배 쉘 구조문제들의 예를 통하여 잠김현상이 s-norm을 이용하여 구해진 수렴 선에 어떻게 나타나는지를 보여주고 균일최적수렴(uniform optimalconvergence)에 대하여 알아본다. + +(plate) 구조는 률이 영인 쉘 구조물로서 쉘 구조의가장 단순한 형태이다. 예로 그 4(a)에 보여진변이 전히 구속된 평 휨(plate bending) 구조문제에대하여 쉘 유한요소해의 수렴 선을 살펴보자. 사용된 탄성계수는 1.7472×10 , 아 비는 0.3, 이 =1.0, 하중은단위 면적 90이 − 방 으로 작용한다. 구조물의 형상,하중, 변위경계조건의 대 성으로 인해 그 4(a)의 된부분만 해석되었다. + +주어진 평 휨 문제를 풀기 위하여 변위법(displacementbased formulation)에 의한 4절점 감절점 쉘 유한요소(QUAD4)와 혼합법(mixed formulation)에 의한 4절점 감절점 쉘 유한요소(MITC43 )를 사용하 다(Ahmad, Irons and + + + +![](images/page-009_4826d01a3dee7593d28fca891b85e8e57136dd881207104f1eb1701ab6e7afd3.jpg) + +
+line + +| log(h) | t/L=1/100 | t/L=1/1000 | t/L=1/10000 | +| ------ | --------- | ---------- | ----------- | +| -1.6 | -1.8 | -0.4 | 0.0 | +| -1.2 | -1.0 | -0.1 | 0.0 | +| -0.8 | -0.4 | 0.0 | 0.0 | +| -0.4 | 0.0 | 0.0 | 0.0 | +
+ +![](images/page-009_694cc196ad8c9d4b86ecbf71ffc2903b3ff0a7bcf55c38766d51a6bf840be433.jpg) + +
+line + +| log(h) | t/L=1/100 | t/L=1/1000 | t/L=1/10000 | +| ------ | --------- | ---------- | ----------- | +| -1.6 | -2.4 | -1.6 | -0.8 | +| -1.2 | -1.6 | -1.2 | -0.4 | +| -0.8 | -0.8 | -0.4 | 0.0 | +| -0.4 | -0.2 | 0.0 | 0.2 | +
+ +두께의 변화에 따른 + +Zienkiewicz, 1970; Bathe and Dvorkin, 1989; Bathe, 1996).각각의 경우 두께 변화에 따른 수렴 선에서 잠김현상이 어떻게 나타나는지를 살펴보자. + +그 5는 두 가지 4절점 쉘 유한요소들에 대하여 쉘 두께의 변화( / =1/10, 1/100, 1/1000, 1/10000)에 따른 수렴선들(convergence curves)을 보여준다. 여기서 는 사용된유한요소의 기를 나타내며 상대오차를 계 하기 위하여 s-norm을 사용하 다. 와 상대오차에 log를 함으로 식(44)의 이론적인 수렴률과 구해진 수렴 선의 수렴률을 비교할 수 있다. 그 5의 각각의 그 에 이론적인 수렴 선의 기 기(2 )가 두께가 은 선으로 그려져 있다. + +그 5(a)는 잠김현상이 일어 때의 전형적인 수렴 선들이다. 쉘의 두께가 얇아짐에 따라 상대오차(relative error)가증하는 것을 알 수 있다. 반면에 그 5(b)의 수렴 선들에서는 상대오차가 쉘 두께( )에 영 을 지 고 오 요소의 기 에만 관계함을 알 수 있다. 또한 그 5(b)에서는 수렴 선들이 이론적인 기 기와 같다. 그 5(b)와 같은형태의 쉘 유한요소해의 수렴형태를 균일최적수렴(uniformoptimal convergence)이라고 한다. + +두 예제로 그 4(b)는 한 변이 구속된 물면(hyperbolic paraboloid) 쉘 구조문제이다. 쉘의 중심면(midsurface)은 다 과 같이 정의된다. + +$$ +\left( \begin{array}{c} X \\ Y \\ Z \end{array} \right) = L \left( \begin{array}{c} \xi^ {1} \\ \xi^ {2} \\ \left(\xi^ {1}\right) ^ {2} - \left(\xi^ {2}\right) ^ {2} \end{array} \right); (\xi^ {1}, \xi^ {2}) \in \left[ - \frac {1}{2}, \frac {1}{2} \right] ^ {2} \tag {45} +$$ + +경계조건은 =−0.5인 변을 따라 전히 구속되며 자중(self-weight)이 − 방 으로 작용한다. 구조물의 형상과 변위 및경계조건이 =0인 면을 따라 대 이므로 그 4(b)의된 부분만 해석되었다. 사용된 탄성계수는 2.0×10 , 아비(Poisson's ratio)는 0.3, 이 =1.0이며 자중은 단위면적80이다. + +주어진 휨지배문제를 풀기 위해 변위법(displacement - MITC6based formulation)에 의한 6절점 각형 쉘 유한요소(QUAD6)와 혼합법(mixed formulation)에 의한 6절점 각형 쉘 유한요소(MITC6)를 사용하 다(Ahmad, Irons andZienkiewicz, 1970; Bathe, 1996; Lee and Bathe, 2004). + +그 6에서 / 이 1/100, 1/1000, 1/10000인 세가지 경우에 대하여 6절점 각형 쉘 유한요소들의 수렴 선들(convergence curves)을 보여준다. 각각의 그 에 이론적인수렴 선의 기 기(2 )가 두께가 은 선으로 그려져 있다.그 6(a)는 QUAD6 유한요소가 쉘의 두께가 얇아짐에 따라 계 된 해의 오차가 어나는 경 , 즉, 잠김현상을 유발한다는 것을 보여준다. 그 6(b)에서는 MITC6 유한요소를사용 을 때 잠김현상이 전히 제거되지는 으나 QUAD6유한요소에 비하여 상 히 화되었 을 나타낸다. + +여기서 우리는 두 가지 쉘 구조문제들을 대상으로 쉘 유한요소해의 수렴 선과 잠김현상에 대하여 알아보 다. 중요한 점은 어 쉘 유한요소가 가지 쉘 구조문제들에 대하여 잠김현상을 일으 지 는다고 해서 다른 쉘 구조문제들에 대하여도 잠김현상을 유발시 지 는다고 할 수 없다는 것이다. 즉, 가지 쉘 구조문제들에 대하여 균일최적수렴을 보이며 잠김현상을 일으 지 는 유한요소도 다른 쉘구조문제들에 대해서 잠김현상을 보일 수 있으며 모든 쉘구조문제들에 대하여 잠김현상을 일으 지 는 쉘 유한요소를 개발하는 것은 극히 어 다. 보다 다양한 쉘 구조문제들에 대한 유한요소해의 수렴 선에 대한 예는 고문 에나와있는 Lee와 Bathe의 논문들에서 찾을 수 있다. + +쉘 유한요소의 잠김현상을 알아보기 위해서는 다양한 형상을 가진 휨 및 혼합지배 쉘 구조문제에 대하여 그 5와 6에 보여진 것과 같이 쉘의 두께를 변화시 며 수렴을 시험하여야 한다. 또한 특정 쉘 요소가 휨지배문제에 은 거동을 보인다고 해서 막지배문제에 대해서도 은 거동을 보이는 것은 아니다. 결론적으로 다양한 형상의 쉘 구조문제들을고려하여 휨과 막지배라는 두 가지 양단의 거동에서 이상적인 수렴을 보이는 쉘 유한요소가 가장 바 하고 이런 쉘유한요소는 혼합지배 쉘 구조문제에 대하여도 은 거동을 + + + +보일 것이다. 즉, 이상적인 쉘 유한요소는 여러 가지 점근적인 거동을 보이는 다양한 형상의 쉘 구조문제들에 대하여균일최적수렴을 보여 주어야 한다. + +# 5. + +나 이 셀 수 없을 만 많은 쉘 유한요소들이 개발되고있지만 쉘 유한요소의 성능평가는 아 도 고전적인 방법을통하여 이루어지고 있다. 본 장에서는 이상적인 쉘 유한요소의 조건과 잠김현상을 화시 는 방법들을 정리하고 쉘 유한요소의 성능평가를 위한 방법론을 제시한다. + +# 5.1 + +유한요소 구조해석 문제는 다 과 같은 변분식(variationalform)으로 나타내어질 수 있다. + +$\mathrm { F i n d ~ } \stackrel { } { U } _ { h } \in \vec { \Psi } _ { h } \mathrm { s u c h ~ t h a t }$ + +$$ +A _ {h} (\vec {U} _ {h}, \vec {V} _ {h}) = \vec {F} (\vec {V} _ {h}), \forall \vec {V} _ {h} \in \vec {\Psi} _ {h}, \tag {46} +$$ + +여기서 $A _ { h } ( \cdot , \cdot )$ 는 유한요소법으로 이 화(discretization)된선형식(bilinear form)이고 는 유한요소 변위장의 ΨSobolev 공간(space)이다. 물론 $\vec { \Psi } _ { h } \subset \vec { \Psi }$ 이 성 한다. + +$$ +A _ {h} (\vec {U} _ {h}, \vec {V} _ {h}) = \vec {V} _ {h} ^ {T} \left(\int_ {\Omega_ {h}} \mathbf {B} _ {h} ^ {T} \mathbf {C} _ {h} \mathbf {B} _ {h} d \boldsymbol {\Omega} _ {h}\right) \vec {U} _ {h} \tag {47} +$$ + +여기서 $\vec { U } _ { h }$ 는 유한요소해(finite element solution), $\vec { \nu } _ { h }$ 는유한요소 시험함수, $\vec { \Psi } _ { h }$ 는 유한요소 변위장의 공간, ( )⋅는 외력을 나타내는 선형식(linear form)이다. 쉘 유한요소해석일 경우 식 (46)은 식 (26)의 형태로 표현 수 있다. + +일반적인 쉘 구조물의 효과적인 유한요소해석에 쓰일 수있는 이상적인 쉘 유한요소의 개발은 매우 어려운 일이다.이상적인 쉘 유한요소의 조건을 다 과 같이 정리할 수 있다. + +● , 쉘 유한요소는 거 영에 지모 (spurious zeroenergy mode)를 지 아야 한다. 변위경계조건이 주어지지 은 임의의 형상을 는 개개의 쉘 유한요소는 물리적인 강체운동에 대 하는 6개의 영에 지모 (zero energymode)만을 가져야 한다. 이 조건을 ellipticity(타원 )라 하며 다 과 같이 정의된다. + +$$ +\exists \alpha > 0 \text { such that } \forall \vec {U} _ {h} \in \vec {\Psi} _ {h}, A _ {h} (\vec {U} _ {h}, \vec {U} _ {h}) \geq \alpha \| \vec {U} _ {h} \| _ {1} ^ {2} \tag {48} +$$ + +여기서 α는 상수이며 는 1차 Sobolev norm4 이다.⋅ + +이 조건은 쉘 유한요소 강성행 (stiffness matrix)의 고유치들(eigenvalues) 중 0인 개수와 그에 대 하는 고유 터들(eigenvectors)을 살펴 으로 시험 수 있다. 탄성체는 강체운동이 아 변위에 대하여 변형에 지를 저장한다. 식A(48)의 조건을 만 시 지 하는 유한요소는 강체운동이 아변위에 대하여 변형에 지를 저장할 수 없으며 이는 물리적으로 적합하지 하다. + +● , 쉘 유한요소는 쉘 수학모델에 근거하 기 때문에 쉘유한요소해석의 해는 주어진 쉘 구조문제에 대하여 사용된요소의 기( )가 어 에 따라 또는 사용된 요소의 수가증가함에 따라 쉘 수학모델의 정확해에 수렴해야 한다. 이조건을 consistency( 모순성 또는 정합성)라 부 며 다 과같이 정의된다. + +$$ +\lim _ {h \rightarrow 0} \overrightarrow {U} _ {h} = \overrightarrow {U} \text { or } \lim _ {h \rightarrow 0} A _ {h} (\overrightarrow {U} _ {h}, \overrightarrow {U} _ {h}) = A (\overrightarrow {U}, \overrightarrow {U}) \tag {49} +$$ + +여기서 (·,·)는 쉘 수학모델의 정확한 선형식(exactbilinear form)이며 는 정확해(exact solution)이다. + +이 조건이 만 되지 을 경우 쉘 유한요소의 해는 이론해에 수렴하지 하므로 신뢰할 만한 결과를 수 없다. + +● , 쉘 유한요소는 모든 종류의 휨 및 혼합지배 쉘 구조문제에 대하여 균일최적수렴(uniform optimal convergence)을 보여야 한다. 이 조건을 만 시 는 쉘 유한요소는 비로소 쉘의 두께와 상관 없이 전단잠김과 막잠김으로부터자유로운 유한요소가 된다. 이는 본 논문의 4장에서 설명된 방법에 의해 시험 수 있다. 혼합법에 의해 정식화(mixed formulation)된 쉘 유한요소에 대하여 이 조건을“inf-sup condition”이라 부른다(Bathe, 1996; Bathe, Iosilevichand Chapelle, 2000b). + +위에서 언 된 세가지 조건을 모두 만 하는 이상적이 쉘유한요소를 개발하는 것은 극히 어 다. 실용적인 쉘 유한요소해석에서는 보다 화된 다 의 조건들을 만 시 는 쉘요소의 사용이 장된다(Lee and Bathe, 2004). + +− 거 영에 지모 (spurious zero energy mode) 없(ellipticity조건 만 ) +− Consistency조건 만 +− 문제의 해석에 있어서 전단잠김 없U U A U U A U U +−막지배거동 쉘 구조물에 대하여 균일최적수렴 +−휨 및 혼합지배거동 쉘 구조물에 대하여 실용적인 / 의범위(1/10\~1/10000)에서 신뢰할 수 있는 결과 +−비선형 해석에 있어서 효 적인 정식화 + +# 5.2 + +지난 수십 년 동안 쉘 유한요소의 잠김현상을 제어하기위해 많은 방법들이 고안되어 다. 그 방법들은 게 세가지로 나누어질 수 있다. + +− 식 (47)의 변형률-변위 관계 연 자 $\mathbf { B } _ { h }$ 를 변형하여 전단 및 막 변형률장의 차수를 여주는 방법, (예)reduced integration, ANS method, MITC method + +− 식 (47)의 $\mathbf { B } _ { h }$ 에 로운 항을 추가하여 전단 및 막 변형률장의 공간(space)을 려주는 방법, (예) EAS method−식 (46)에서 유한요소 변위장 $\vec { U } _ { h }$ 의 공간( )을 려주Ψ는 방법, (예) non-conforming method + +쉘 유한요소의 잠김현상을 화시 기 위한 대부분의 방법들은 위의 세 분류들(categories)에 속하며 세 방법들을 복합적으로 이용한 예들도 있다. + +가장 운 방법은 감차적분(reduced integration)을 사용하 diff --git a/docs/reference-papers/MITC4/쉘구조물의유한요소해석에대하여/쉘구조물의유한요소해석에대하여_002.md b/docs/reference-papers/MITC4/쉘구조물의유한요소해석에대하여/쉘구조물의유한요소해석에대하여_002.md new file mode 100644 index 0000000..1d0a3de --- /dev/null +++ b/docs/reference-papers/MITC4/쉘구조물의유한요소해석에대하여/쉘구조물의유한요소해석에대하여_002.md @@ -0,0 +1,122 @@ + + +는 방법이다(Bathe, 1996). 그러나 이 방법은 거 영에 지모 (spurious zero energy mode)를 유발시 는 치명적인단점을 지니고 있다. 이점을 극복하기 위하여 각종 안정화(stabilization)기법이 사용된다. + +비적합모 (non-conforming or incompatible mode)를 추가함으로 요소의 휨 모 를 복원하여 쉘 유한요소의 잠김현상을 화할 수 있다. 이 방법은 요소간 변위의 적합성(inter-elemental compatibility)을 만 시 지 하는 단점과최종 강성행 을 구하기 위해 정적 (static condensation)을 사용하기 때문에 비선형 해석으로 확장 시 식이 복잡해지는 단점을 가지고 있다(Choi, Lee and Park, 1999; 최창근, 2002). + +변위와 변형률을 각각 따로 근사하는 혼합법(mixedformulation)에 근거한 방법들은 쉘 유한요소의 잠김현상을화시 는 알려진 방법들 중 가장 우수하다고 평가된다. 그중 MITC(Mixed Interpolation of Tensorial Components)방법은 이론적으로 잘 확 되어 있으며 다양한 수치실험으로 잠김현상의 제어에 매우 효과적임이 증되었다(Batheand Dvorkin, 1989; Bathe, 1996; Bathe, Iosilevich andChapelle, 2000a; Hiller and Bathe, 2003). 최근의 연구들은 MITC방법에 의해 만들어진 사각형 쉘 유한요소들이 이상적인 쉘 유한요소에 상 히 접근해 있 을 보여준다(Hillerand Bathe, 2003; Bathe, Lee and Hiller, 2003). + +MITC 방법에서는 변위법에 근거한 쉘 유한요소의 특정한위치들(tying points)에서 공변변형률들(covariant strains)을이용하여 원 공변(covariant)변형률의 근사차수보다 은차수로 변형률장을 근사(interpolation)한다. 일반적으로 근사함수는 은 차수일수 잠김현상 제어에 효과적이지만은 근사차수는 막지배거동을 하는 쉘 구조문제를 풀때 유한요소해가 이론해에 수렴하지 하는(즉, consistency조건을 만 시 지 하는) 현상을 유발시 다. 심한 경우에는 거 영에 지모 를 발생시켜 ellipticity조건 지 만 시수 없게 만든다. 그러므로 MITC방법의 핵심은 휨 및혼합지배거동 시 잠김현상을 여주면서 막지배거동 시 수렴성을 유지하는(즉, consistency를 만 시 는) 균형 잡변형률의 근사장을 찾아내는 것이다. + +대체변형률장(assumed strain field)을 사용하는 ANS(Assumed Natural Strain)방법은 MITC방법과 유사하며,EAS(Extended Assumed Strain)방법은 ANS 또는 MITC방법에 추가적인 변형률장을 도 하여 변형률장이 표현 가능한 형태들(patterns)의 수를 려주는 방법이다. EAS방법은비적합모 를 사용하는 방법과 비 하게 추가된 변형률장의자유도를 제거하기 위해 정적 을 필요로 한다는 단점을가지고 있으며 기 의 ANS나 MITC방법에 비하여 쉘 유한요소의 수렴성을 개선시 수 있지만 그 효과가 지는다고 알려져 있다. + +다른 여러 가지 방법을 사용하여 휨 및 혼합지배거동 쉘구조문제에 대하여 보다 유연(flexible)한 거동을 하는 쉘 유한요소를 개발하는 것은 어 지 . 그러나 그와 동시에consistency와 ellipticity조건들을 모두 만 시 는 것은 지다. 앞으로 여러 종류의 개발된 쉘 유한요소에 대하여 쉘이론에 바탕을 심도 있는 성능시험(benchmark test)과연구가 필요하다. + +# 5.3 + +일반적으로 유한요소법을 사용하여 쉘 구조물을 해석하는대부분의 기술자들(engineers)은 구하여진 해의 오차(error)에대한 평가 없이 해석결과를 아들이기 때문에 쉘 유한요소를 개발하는 연구자들은 개발된 쉘 유한요소를 실제 해석에사용하기에 앞서 성능을 평가하고 그 결과를 보고하여야 한다. 개발된 쉘 유한요소의 오차특성이 명확하게 알려진다면사용자들은 주어진 쉘 유한요소를 어떻게 바 게 사용할수 있을지를 단할 수 있게 된다. 쉘 유한요소의 성능평가는 다 에 거된 사항들을 고려하여 이루어져야 한다. + +# ● 기본시험 + +쉘 유한요소는 표 2에 정리되어있는 기본시험들(basictests)을 통과하여야 한다. 기본시험들을 통과하지 하는 쉘유한요소의 사용은 바 하지 하다. + +# ● 성능평가방법 + +쉘 유한요소의 성능을 평가하기 위해서는 다양한 쉘 구조문제들이 사용되어야 한다. 쉘 유한요소들의 성능을 비교평가하기 위하여 오 전부터 많은 쉘 해석문제들이 제안되어다. 현재 지 가장 널리 쓰이는 성능평가방법은 1985년에MacNeal와 Harder에 의해 정리된 것으로 두께가 정해진가지 쉘 구조문제들을 해석하여 정해진 위치에서 구해진 변위 및 력/변형률 등의 결과치들을 유한요소 을 조밀화 하면서 비교하는 것이다. 이미 언 한 바와 같이 점들에서의 해의 수렴을 정하는 것은 전체 유한요소해의 거동을바 게 반영할 수 없다. 변위형상이나 력/변형률의 분는 유한요소해의 전체적인 수렴정도를 보여 수 있으며 이것들을 비교하는 것은 매우 은 보 방법이다. 그러나 이방법으로 과연 유한요소해가 어 정도 수렴 는지를 정하여 그 정도를 한 개의 으로 보여주기는 대단히 어 다.그러므로, 력과 변형률의 오차분 로부터 구해진 s-norm은전체 유한요소해를 반영하는 은 오차 정의 규준이 된다.또한 4.3절에서 설명한 바와 같이 두께의 변화를 고려한 성 + +
시험대상참고문헌
영에너지 시험 (Zero energy mode test)사각형 셀 유한요소삼각형 셀 유한요소Bathe, 1996
조각 시험 (Patch tests)-Membrane patch test- Bending patch test사각형 셀 유한요소삼각형 셀 유한요소Bathe, 1996Lee and Bathe, 2004
요소 등방성 시험 (Element isotropy test)삼각형 셀 유한요소Lee and Bathe, 2004
+ + + +![](images/page-012_e131c5ba006746699d802f559f97ad57422b70d3abc7e2368eaff4c84c73c457.jpg) + +
+natural_image + +Geometric grid pattern forming a curved, dome-like shape (no text or symbols) +
+ +（a） + +![](images/page-012_e6986c8ee7282f36c8b92d1723a5118b50fb063dbcc6c1b388e94112a84dcc44.jpg) + +
+natural_image + +Curved grid pattern with no text or symbols +
+ +![](images/page-012_a8eaeb95a2d81b756767fd08dfc2c981f1637b77d22e54619033115af6353878.jpg) + +
+natural_image + +Curved wireframe grid pattern with no text or symbols +
+ +（c） +7 . Gaussian : (a) Positive Gaussian curvatu re, (b) Zero Gaussian cu rvatu re, (c) Negative 곡률에 따Gaussian curvature + +3 . ( Bathe , I osi levich and Chapel le , 2000 ; Lee and Bathe , 2002 ; Bathe ,쉘 유한요소의 성능평가를 위한 쉘 구조문제의 예들Chapel le and Lee, 2003 ; Bathe, Lee and H i l ler, 2003 ; Chapel le and Bathe, 2003 ; H i l ler and Bathe, 2003 ; Lee and Bathe,2004 ; Lee and Bathe , 2005 ; Lee , Noh and Bathe , 2007) + +
셀 구조문제 (shell problems)Gaussian 곡률점근거동 ( $\rho$ )
Fully clamped plate problemZerohover지배 ( $\rho = 3.0$ )
Scodelis-Lo roof shell problemZero혼합지배 ( $\rho = 1.75$ )
Modified Scodelis-Lo roof shell problemZero막지배 ( $\rho = 1.0$ )
Free cylindrical shell problemZerohover지배 ( $\rho = 3.0$ )
Fixed cylindrical shell problemZero막지배 ( $\rho = 1.0$ )
Clamped hemispherical cap problemPositive막지배 ( $\rho = 1.0$ )
Monster shell problemPositiveNot well-defined
Partly clamped hyperbolic paraboloid shell problemNegativehover지배 ( $\rho = 3.0$ )
Free hyperboloid shell problemNegativehover지배 ( $\rho = 3.0$ )
Fixed hyperboloid shell problemNegative막지배 ( $\rho = 1.0$ )
+ +능평가방법을 사용하는 것이 바 하다 + +·(layer) + +쉘의 력변형률변위장들이 히 변하는 의즉 특성 이 는 쉘의 두께에 따라 변화한다 일반적으로 특성 이는 쉘의 두께가 얇아짐에 따라 식에 의하여 히 어든다 에서의 에 지 중현상때문에 이런 이 발생하는 쉘 구조문제를 풀 때는 균일한유한요소 을 사용하여 균일최적수렴을 기 들다 각각의 특성 이를 반영한 유한요소 을 사용하여야 한다 즉이 발생하는 영역에서 보다 조밀한 유한요소 의 사용이昱(Bathe，Iosilevich and Chapelle，2000a)．o]弭유한요소 을 라 부른다 + +● 중심면의 률 + +쉘 구조물의 중심면은 률 을 가지고 있다면은 률의 부호에 따라 세가지로 나누어질 수 있다 특히 률이 인 면을 가지는 쉘 구조물은유한요소해석에 있어서 상 한 어려 이 따른다쉘 유한요소의 성능을 평가하기 위한 해석시험문제들 은 다양한 률을 고려하여 구성되어야 한다 이는 어 쉘 유한요소가 특정한 률을 가지는 쉘 구조문제에서 은 수렴성을 보인다고 하여 다른률을 가지는 구조문제에 대하여도 은 수렴을 보이는 것은 아니기 때문이다 그 은 률에 따른 면의예들을 보여주고 있다 + +● 점근거동의 종류 + +장에서 우리는 쉘 구조물의 두께가 얇아짐에 따라 나타나는 가지 점근거동휨지배 막지배 혼합지배거동을 살펴 보다 각각의 점근거동을 모두 시험할 수 있도 해석시험문제들을 구성해 주어야 한다 특히 휨지배 및 혼합지배거동쉘 구조문제들에서는 잠김현상이 일어나는지를 시험하여야하며 막지배거동 쉘 구조문제들에서는 절에서 언 한조건이 만 되는지를 살펴보아야 한다 표 은 쉘유한요소의 성능평가를 위한 쉘 구조문제의 예들을 보여주고 있다 + +● 요소 의 형태 + +유한요소해석의 해는 유한요소 을 어떻게 구성하는지에따라 그 수렴특성이 변화하게 된다 요소형상의 그러짐에민감하지 고 은 수렴성을 유지하는 쉘 유한요소를 개발하는 것은 지 은 일이다 따라서 해석시험문제들은 다양한 유한요소 에 따른 쉘 유한요소의 수렴특성을 반영해 주어야 한다 특히 비등방성 각형 쉘 유한요소의 시험에서는 유한요소 의 형태 만 아니라 요소의 방에 따라 수렴특성이 변하므로 요소의 방 성 또한 고려되(Lee,Noh and Bathe, 2007). + +# 6.号 + +리 에서 언 된 바와 같이 쉘 구조물의 유한요소해석을명확하게 이해하기 위해서는 쉘 구조물의 물리적 거동 수학모델 및 쉘 유한요소해석에 대한 이해가 동시에 체계적이고심도 있게 이루어져야 한다 본 논문에서는 이 세가지 부분에 대한 이해와 이들이 서로 어떻게 유기적으로 관계를 맺 + + + +고 있는지를 최근 주요 연구들을 대로 정리하여 고찰하고 이상적인 쉘 유한요소의 성질과 쉘 유한요소의 성능평가방법을 제시하 다. + +본 논문에서는 대표적인 쉘 수학모델과 휨지배거동, 막지배거동, 혼합지배거동 등으로 나누어지는 쉘 구조물의 점근거동에 대한 기본적인 이론과 그 점근거동을 수치적으로 알아내는 방법을 알아보 다. 또한 휨지배 및 혼합지배거동에서 나타나는 쉘 유한요소의 잠김현상을 두께의 변화에 따른수렴 선을 통하여 고찰하 다. 마지막으로 이상적인 쉘 유한요소의 조건과 잠김현상의 제어하는 방법을 알아보 고 쉘유한요소의 성능평가방법을 제안하 다. + +쉘 구조물의 수학모델과 점근거동은 쉘의 물리적 거동을이해하는데 핵심사항으로 쉘 구조물을 설계하는 기술자나 쉘유한요소해석을 연구하는 연구자들이 명확하게 알아야 할 매우 중요한 부분이다. 유한요소법을 이용하여 쉘 구조물을 해석하기에 앞서 쉘 구조물의 점근거동과 그와 관련된 쉘 유한요소의 감김현상에 대한 이해는 필수적이다. 통합적인 이해의 바탕이 있을 때 신뢰할 만한 쉘 유한요소의 개발이 이루어질 수 있으며 쉘 구조물의 유한요소해석을 통하여 어진 결과를 정확하게 이해할 수 있다. + +# + +글을 맺으며 본 논문에 소개된 기본개 들을 정 하고 정리하는데 많은 가 을 주신 MIT(Massachusetts Instituteof Technology)의 Klaus-Jürgen Bathe 교수 과 KAIST(한국과학기술원)의 최창근 교수 께 은 감사 니다. + +# + +(2002) . . +Ahmad, S., Irons, B.M., and Zienkiewicz, O.C. (1970) Analysis of thick and thin shell structures by curved finite elements. International Journal for Numerical Methods and Engineering, Vol. 2, pp. 419-451. +Bathe, KJ. (1996) Finite Element Procedures. Prentice Hall: New Jersey. +Bathe, K.J., Chapelle, D., and Lee, P.S. (2003) A shell problem ‘highly sensitive’ to thickness changes. International Journal for Numerical Methods and Engineering, Vol. 57, pp. 1039-1052. +Bathe, K.J. and Dvorkin, E.N. (1989) A formulation of general + +shell elements - the use of mixed interpolation of tensorial components. International Journal for Numerical Methods and Engineering, Vol. 22, pp. 697-722. +Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000a) An evaluation of the MITC shell elements. Computers & Structures, Vol. 75, pp. 1-30. +Bathe, K.J., Iosilevich, A., and Chapelle, D. (2000b) An inf-sup test for shell finite elements. Computers & Structures, Vol. 75, pp. 439-456. +Bathe, K.J., Lee, P.S., and Hiller, J.F. (2003) Towards improving the MITC9 shell element. Computers & Structures, Vol. 81, pp. 477-489. +Chapelle, D. and Bathe, K.J. (1998) Fundamental considerations for the finite element analysis of shell structures. Computers & Structures, Vol. 66, pp. 19-36, pp. 711-712. +Chapelle, D. and Bathe, K.J. (2003) The finite element analysis of shells? fundamentals. Berlin:Springer-Verlag. +Choi, C.K., Lee, P.S., and Park, Y.M. (1999) Defect-free 4-node flat shell element: NMS-4F element. Structural Engineering and Mechanics, Vol. 8, pp. 207-231. +Hiller, J.F. and Bathe, K.J. (2003) Measuring convergence of mixed finite element discretizations: an application to shell structures. Computers & Structures, Vol. 81, pp. 639-654. +Lee, P.S. and Bathe, K.J. (2002) On the asymptotic behavior of shell structures and the evaluation in finite element solutions. Computers & Structures, Vol. 80, pp. 235-255. +Lee, P.S. and Bathe, K.J. (2004) Development of MITC isotropic triangular shell finite elements. 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Engineering Structures, Vol. 28, pp. 399-410. + +( : 2006.7.18/ : 2007.1.16/ : 2007.3.27) diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/images/page-001_12127e76479b6d859567b289820a6ef3314514c3cef8b7213c330f618fea48bd.jpg b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/images/page-001_12127e76479b6d859567b289820a6ef3314514c3cef8b7213c330f618fea48bd.jpg new file mode 100644 index 0000000..3724d66 Binary files /dev/null and b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/images/page-001_12127e76479b6d859567b289820a6ef3314514c3cef8b7213c330f618fea48bd.jpg differ diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/images/page-001_292faa5016d98cd4f99523a1c780f9b1f90e27f380b3e9f1ffecf6b955fd6a16.jpg b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/images/page-001_292faa5016d98cd4f99523a1c780f9b1f90e27f380b3e9f1ffecf6b955fd6a16.jpg new file mode 100644 index 0000000..5c9e0d1 Binary files /dev/null and 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배포의 경우, 이 저작물에 적용된 이용허락조건을 명확하게 나타내어야 합니다. +l 저작권자로부터 별도의 허가를 받으면 이러한 조건들은 적용되지 않습니다. + +저작권법에 따른 이용자의 권리는 위의 내용에 의하여 영향을 받지 않습니다. + +이것은 이용허락규약(Legal Code)을 이해하기 쉽게 요약한 것입니다. + +Disclaimer + + + +공학석사학위 청구논문 + +# 유한요소해석법을 이용한 쉘 구조물의 동적 좌굴 해석 + +# Dynamic Buckling Analysis of Shell Structures using Finite Element Method + +2012년 2월 + +인하대학교 대학원 + +항공우주공학과 + +이 희 준 + + + +공학석사학위 청구논문 + +# 유한요소해석법을 이용한 쉘 구조물의 동적 좌굴 해석 + +# Dynamic Buckling Analysis of Shell Structures using Finite Element Method + +2012년 2월 + +지도교수 조 진 연 + +이 논문을 석사학위 논문으로 제출함 + +인하대학교 대학원 + +항공우주공학과 + +이 희 준 + + + +이 논문을 이희준의 석사학위논문으로 인정함 + +2012년 2월 + +![](images/page-004_aa26d7161fd983c6face85227f5b85c668e5c074a5e71892fade4c0d8d0b28f6.jpg) + +
+text_image + +인하 대학 +INHA UNIVERSITY +1954 +
+ +주심 김기욱 + +부심 조 진 연 + +위원 이승수 + + + +# 요 약 + +매개 변수 공진으로도 알려진 동적 좌굴 현상은 구조물이 축 방향의동적 압축 하중을 받을 때 발생하는 동적 불안정 현상으로서 구조물에심각한 파손을 유발할 수 있다. 특히 초음속으로 운동하는 항공기나 탄도미사일, 발사용 로켓과 지구 대기권 재돌입체 그리고 초공동 수중운동체와 같이 동적 압축 하중을 받는 구조물을 설계할 때 구조물의 동적 좌굴거동을 예측하여 설계하는 것이 중요하다. 이에 본 논문에서는 축 방향의동적 압축 하중을 받는 쉘 구조물에 대해 동적 좌굴 해석을 하기 위한유한요소해석 프로그램을 개발하였으며, 선형/비선형 정적 해석과 진동및 정적 좌굴 해석을 통해 프로그램에 사용된 쉘 요소의 신뢰성을 확인하였다. 또한 다양한 모델에 대한 동적 좌굴 해석 결과를 이론적인 해나실험을 통해 나온 결과와 비교함으로써 본 프로그램의 타당성을 검증하였다. + + + +# ABSTRACT + +Dynamic buckling, also known as parametric resonance, is one of the dynamic instability phenomena which may lead to serious failure of structure. It occurs when compressive dynamic loading of axial direction is applied to the structures. Therefore it is essential to consider the dynamic buckling behaviors of structures, especially when the structures is designed to be utilized in compressive dynamic loading of axial direction such as faster supersonic aircrafts, ballistic missiles, launcher, re-entry vehicles and supercavitating underwater vehicles. In this study, the finite element program is developed for dynamic buckling analysis. Linear and nonlinear static analyses, dynamic analysis and static buckling analysis are performed to demonstrate the accuracy of the developed program. Also the dynamic buckling analyses are carried out for various models and the computational results are verified by comparing with analytical and experimental solutions. + + + +# 목 차 + +요약 + +ABSTRACT ⅱ + +목차 ⅲ + +그림 목차 ⅴ + +표 목차 ⅵ + +1. 서 론 1 + +2. 이 론 3 + +2.1. MITC4 Shell Element 3 + +2.2. Geometric Nonlinear Formulation 7 + +2.2.1. Finite Rotation Formulation 21 + +2.2.2. Constitutive Matrix 22 + +2.2.3. Mass Matrix 25 + +2.2.4. 6-DOF Shell Element 26 + +2.3. Buckling Theory ········· 30 + +2.3.1. Static Buckling ······· 31 + +2.3.2. Dynamic Buckling 32 + +2.3.3. Dynamic Buckling Theory of Beam 34 + +3. Numerical Example 39 + +3.1. Linear Static Analysis 39 + +3.1.1. Patch Test 39 + +3.1.1.1. Constant Curvature Patch Test 40 + +3.1.1.2. Constant Shear Patch Test 41 + +3.1.1.3. Constant Twist Patch Test 43 + +3.1.2. Pinched Cylinder 44 + +3.1.3. Hemispherical Shell 46 + +3.2. Geometric Nonlinear Analysis 48 + +3.3. Static Buckling Analysis 50 + + + +3.3.1. Rectangular Plate Shell 50 +3.3.2. Cylindrical Shell 52 +3.3.3. Stiffened Square Plate Shell 55 + +3.4. Dynamic Buckling Analysis 57 + +3.4.1. Dynamic Buckling Analysis of Beam 57 +3.4.2. Dynamic Buckling Analysis of Plate 59 +3.4.3. Dynamic Buckling Analysis of Stiffened Plate 61 + +4. 결 론 63 + +참고문헌 64 + +부 록 65 + +![](images/page-008_fff1eb14d5df4d865edacd55e71045b09245bdbb21cd614ebf92712cc2e1d129.jpg) + +
+text_image + +인하대학년 +1954 +INHA UNIVERSITY +
+ + + +# 그림 목차 + +Fig. 1 Four-node shell element ····· 4 +Fig. 2 Interpolation function for the transverse shear strains 6 +Fig. 3 An arbitrary surface with global Cartesian coordinate system, natural coordinate system and local covariant coordinate system spanned by ig ······· 10 +Fig. 4 Local Cartesian coordinate system .... ·········· 23 +Fig. 5 Global Cartesian coordinate system and local coordinate system ····· ····· 26 +Fig. 6 Dynamic Buckling Model of Beam · 34 +Fig. 7 Patch Test Mesh ········· ·········· 39 +Fig. 8 Constant Curvature Patch Test Model ······ 40 +Fig. 9 Constant Shear Patch Test Model ·········· 41 +Fig. 10 Constant Twist Patch Test Model · 43 +Fig. 11 Pinched Cylinder Model ·· 44 +Fig. 12 Pinched Cylinder 1/8 Model ································· ·· 44 +Fig. 13 Comparison of Convergence for Pinched Cylinder with ABAQUS · 45 +Fig. 14 Comparison of Linear Static Analysis for Pinched Cylinder with ABAQUS ········ 46 +Fig. 15 Hemispherical Shell Model ··········· 46 +Fig. 16 Comparison of Convergence for Hemispherical Shell with ABAQUS ················· 47 +Fig. 17 Comparison of Linear Static Analysis for Hemisphrical Shell with ABAQUS ····· 48 +Fig. 18 Beam Model for Geometric Nonlinear Analysis ············ · 48 +Fig. 19 Comparison of Geometric Nonlinear Analysis for Beam with ABAQUS ······· ····· 49 +Fig. 20 Geometry Change of Beam According to Loads Increase 49 +Fig. 21 Rectangular Plate Shell Model ········· 50 +Fig. 22 Cylindrical Shell Model · 52 +Fig. 23 Stiffened Square Plate Shell Model · 55 +Fig. 24 Dynamic Buckling Analysis Model for Beam · 57 +Fig. 25 Dynamic Instability Region of Beam ······ ······· 58 +Fig. 26 Dynamic Buckling Model for Plate ··59 +Fig. 27 Dynamic Instability Region of Plate · 60 +Fig. 28 Dynamic Buckling Analysis Model for Stiffened Plate 61 +Fig. 29 Comparison of Dynamic Instability Region for Stiffened Plate with Plate ··········· 62 + + + +# 표 목차 + +Table. 1 Constant Curvature Patch Test Results ···· 41 + +Table. 2 Constant Shear Patch Test Results ····· 42 + +Table. 3 Constant Twist Patch Test Results ····· 43 + +Table. 4 Comparison of Linear Static Analysis for Pinched Cylinder with Exact Solution ·45 + +Table. 5 Comparison of Linear Static Analysis for Hemispherical Shell with Exact Solution ·47 + +Table. 6 Comparison of Eigenvalue for Rectangular Plate Shell with ABAQUS ····· 51 + +Table. 7 Comparison of Mode Shape for Rectangular Plate Shell with ABAQUS····· ····· 51 + +Table. 8 Comparison of Eigenvalue for Cylindrical Shell with ABAQUS ······ 53 + +Table. 9 Comparison of Critical Buckling Pressure for Cylindrical Shell with Analytic Solution · 53 + +Table. 10 Comparison of Mode Shape for Cylindrical Shell with ABAQUS ·· 54 + +Table. 11 Comparison of Eigenvalue for Stiffened Square Plate Shell with ABAQUS ····· 56 + +Table. 12 Comparison of Mode Shape for Stiffened Square Plate Shell with ABAQUS ·· 56 diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_002.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_002.md new file mode 100644 index 0000000..acee649 --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_002.md @@ -0,0 +1,242 @@ + + +# 1. 서 론 + +좌굴(buckling, 挫屈)이란 주로 길이가 그 횡단면의 치수에 비해 큰 구조물의 양단에 압축하중이 가해졌을 경우, 하중이 어느 크기에 이르면 기둥이 갑자기 휘는 현상을 말한다. 특히 긴 기둥이나 쉘을 많이 사용하는항공기, 차량, 선박, 건축물 등의 설계에서 좌굴 문제가 중요하며, 원통형쉘의 경우 가스와 같은 액체를 저장하는 압력용기로 사용될 뿐만 아니라잠수함이나 항공기와 같이 외압을 받는 구조물에 사용된다. 이와 같이 다양한 분야에서 사용되는 쉘 구조물은 그 두께가 길이에 비해 얇기 때문에 진동이나 좌굴에 취약한 특성을 보인다. 이러한 취약점을 해결하기 위해서는 우선 구조물에 작용하는 하중의 특성을 파악하고 문제가 발생하는 부분에 대해 구조물을 어떻게 보강할지 결정해야 한다. + +일반적으로 좌굴 문제를 다룰 때 정적 하중만을 고려하지만 특수한경우에 대해서는 동적 하중도 함께 고려해야만 한다. 예를 들어 초음속으로 운동하는 항공기나 탄도 미사일, 발사용 로켓과 지구 대기권 재돌입체그리고 초공동 수중운동체의 경우 빠른 속도로 인해 축 방향으로 매우큰 동적 압축하중이 작용하게 된다. 이러한 동적 압축하중으로 인하여 발생하는 좌굴을 동적 좌굴(dynamic buckling) 또는 매개변수 공진(parametricresonance)이라고 한다. 동적 좌굴이 발생하게 되면 구조물의 횡 방향 운동이 커지게 되고 이로 인해 구조물의 불안정성이 증가하여 치명적인 손상을 유발할 수 있다. 그러므로 동적 좌굴 현상을 방지하기 위해서는 이러한 하중이 작용하는 구조물에 대한 해석을 통해 구조물이 불안정해지는 영역을 파악하고 이를 고려하여 설계하는 것이 중요하다. + +하지만 기존 상용유한요소해석 프로그램의 경우 정적 하중에 대한 정적 좌굴 해석만 가능하며, 동적 하중 또는 정적 하중과 동적 하중이 동시에 가해지는 경우에 대한 좌굴 해석이 불가능하다. 이에 본 논문에서는축 방향의 동적 압축 하중을 받는 쉘 구조물에 대해 동적 좌굴 해석을하기 위한 유한요소해석 프로그램을 개발하였으며, 선형/비선형 정적 해석과 진동 및 정적 좌굴 해석을 통해 프로그램에 사용된 쉘 요소의 신뢰 + + + +성을 확인하였다. 또한 다양한 모델에 대한 동적 좌굴 해석 결과를 이론적인 해나 실험을 통해 나온 결과와 비교함으로써 본 프로그램의 타당성을 검증하였다. + +![](images/page-012_a65bfb59e631405e50af874426c098c14260ed2a657986fb1f2693086e9be335.jpg) + +
+text_image + +인하 대학 +INHA UNIVERSITY +1954 +
+ + + +# 2. 이론 + +# 2.1 MITC4 Shell Element + +셀 구조물을 유한요소 모델로 만들기 위해서 다양한 셀 요소 가운데 Bathe와 Dvorkin에 의해 개발된 MITC4라는 셀 요소를 선정하였다. MITC4 셀 요소는 3차원 솔리드 형상으로부터 셀 형상을 표현하므로 지배방정식의 유한요소 정식화가 다른 셀 요소에 비해 간단하다. 또한 셀 이론을 사용하지 않고 3차원 응력, 변형률을 사용하여 표현되며, 임의의 형상에 대한 두꺼운 셀과 얇은 셀 모두 적용 가능하다는 장점이 있다. 그리고 대 변형/회전(작은 변형률)과 재료 비선형에 모두 적용 가능하다. [1] + +MITC4 셀 요소 내부의 임의의 점은 고유 좌표계(natural coordinate system)에 대해 정의 할 수 있으며, 위치 벡터는 식(2.1)과 (2.2)같이 나타낼 수 있다. + +초기 형상(t=0)에서 + +$$ +\mathbf {X} = \sum_ {I = 1} ^ {4} N _ {I} (\xi , \eta) ^ {0} \mathbf {X} _ {I} + \frac {\zeta}{2} \sum_ {I = 1} ^ {4} t _ {I} N _ {I} (\xi , \eta) ^ {0} \mathbf {V} _ {I} ^ {n} \tag {2.1} +$$ + +임의의 시간 t에서 + +$$ +\mathbf {x} = \sum_ {I = 1} ^ {4} N _ {I} (\xi , \eta) ^ {t} \mathbf {X} _ {I} + \frac {\zeta}{2} \sum_ {I = 1} ^ {4} t _ {I} N _ {I} (\xi , \eta) ^ {t} \mathbf {V} _ {I} ^ {n} \tag {2.2} +$$ + +이 때 $N_{I}(\xi,\eta)$ 는 형상함수이고, $^{t}X_{I}$ 는 시간 t일 때 노드 I의 좌표를 나타내며, 시간 t=0이면 초기 형상에서 노드 I의 좌표를 나타낸다. 그리고 $t_{I}$ 는 노드 I의 두께이고 $^{t}V_{I}^{n}$ 은 시간 t일 때 노드 I의 두께방향의 법선벡터(normal vector)를 나타내며, 시간 t=0이면 초기 형상에서 노드 I의 + + + +두께방향의 법선 벡터를 나타낸다. + +임의의 시간 t에서의 MITC4 셀 요소의 변위는 식(2.3)과 같이 나타낼 수 있다. + +$$ +\begin{array}{l} { } ^ { t } \mathbf { u } = \mathbf { x } - \mathbf { X } = \sum _ { I = 1 } ^ { 4 } N _ { I } ( \xi , \eta ) \left( { } ^ { t } \mathbf { X } _ { I } - { } ^ { 0 } \mathbf { X } _ { I } \right) + \frac { \zeta } { 2 } \sum _ { I = 1 } ^ { 4 } t _ { I } N _ { I } ( \xi , \eta ) \left( { } ^ { t } \mathbf { V } _ { I } ^ { n } - { } ^ { 0 } \mathbf { V } _ { I } ^ { n } \right) \tag {2.3} \\ = \sum_ {I = 1} ^ {4} N _ {I} (\xi , \eta) ^ {t} \mathbf {u} _ {I} + \frac {\zeta}{2} \sum_ {I = 1} ^ {4} t _ {I} N _ {I} (\xi , \eta) \left(^ {t} \mathbf {V} _ {I} ^ {n} - ^ {0} \mathbf {V} _ {I} ^ {n}\right) \\ \end{array} +$$ + +임의의 시간 t에서의 변위는 시간 t일 때의 형상과 초기 형상의 차로부터 구할 수 있으며, 이 때 $^{t}u_{I}$ 는 시간 t일 때 노드 I의 변위를 나타낸다. 그리고 이로부터 변위의 중분은 식(2.4)와 같이 나타낼 수 있다. + +![](images/page-014_6bad3d258bf1884089055f2a5f01b90d096b0efdeb1ed4a80d27f8ef6066034c.jpg) + +
+text_image + +- \alpha_I +\beta_I +0 V_I^n +t V_I^n +z +x +\alpha_I +0 V_I^1 +y +0 V_I^2 +\beta_I +node I +1954 +
+ +Fig. 1 Four-node shell element + +$$ +\Delta \mathbf {u} = \sum_ {I = 1} ^ {4} N _ {I} (\xi , \eta) \Delta \mathbf {u} _ {I} + \frac {\zeta}{2} \sum_ {I = 1} ^ {4} t _ {I} N _ {I} (\xi , \eta) \left(- \alpha_ {I} ^ {t} \mathbf {V} _ {I} ^ {2} + \beta_ {I} ^ {t} \mathbf {V} _ {I} ^ {1}\right) \tag {2.4} +$$ + + + +여기서 와 는 각각 $\mathbf { V } ^ { 1 }$ 과 $\mathbf { V } ^ { 2 }$ 방향 벡터의 회전각이고, 이 때 1 V과 $\mathbf { V } ^ { 2 }$ 는 n V 으로부터 식(2.5)와 같이 구할 수 있다. + +$$ +\mathbf {V} ^ {1} = \frac {\mathbf {e} _ {2} \times \mathbf {V} _ {n}}{\left\| \mathbf {e} _ {2} \times \mathbf {V} _ {n} \right\|}, \quad \mathbf {V} ^ {2} = \mathbf {V} _ {n} \times \mathbf {V} ^ {1} \tag {2.5} +$$ + +이 때 $\mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } , \mathbf { e } _ { 3 }$ 는 전역 직교 좌표계(global Cartesian coordinatesystem)의 기저(basis)이다. 그리고 만약 $\mathbf { e } _ { 2 } \times \mathbf { V } _ { n } \approx 0 \ { \textdegree } ]$ 라면, $\mathbf { V } ^ { 1 }$ 과 $\mathbf { V } ^ { 2 } \triangleq$ 식(2.6)과 같이 나타낼 수 있다. + +$$ +\mathbf {V} ^ {1} = \mathbf {e} _ {3}, \quad \mathbf {V} ^ {2} = \mathbf {e} _ {1} \tag {2.6} +$$ + +하지만 위와 같이 변위를 정의할 경우 일정한 굽힘 모멘트가 가해질때 요소의 모든 점에서 횡 전단 변형률(transverse shear strain)이 영(零)이될 수 없고, 이로 인해 얇은 형상에 대해 요소의 „잠김현상(lockingphenomenon)‟이 발생하게 된다. 그러므로 연속체 역학의 가정이 Kirchhoff쉘의 가정을 포함할지라도 유한요소이산화를 통해 이러한 가정을 표현할수가 없다. 이러한 결점을 해결하기 위해 MITC4 쉘 요소에서는 식(2.7)과같은 횡 전단 변형률에 대한 보간법을 적용하였다. [1] + + + +![](images/page-016_155c1e5def3164c6ccf9c34fbbba090268cfdbb56cc16fe007ca76c27a746f3e.jpg) +Fig. 2 Interpolation function for the transverse shear strains + +$$ +\widetilde {\varepsilon} _ {\xi \zeta} = \frac {1}{2} (1 + \eta) \widetilde {\varepsilon} _ {\xi \zeta} ^ {A} + \frac {1}{2} (1 - \eta) \widetilde {\varepsilon} _ {\xi \zeta} ^ {C} \tag {2.7a} +$$ + +$$ +\widetilde {\varepsilon} _ {\eta \zeta} = \frac {1}{2} (1 + \xi) \widetilde {\varepsilon} _ {\eta \zeta} ^ {D} + \frac {1}{2} (1 - \xi) \widetilde {\varepsilon} _ {\eta \zeta} ^ {B} \tag {2.7b} +$$ + + + +# 2.2 Geometric Nonlinear Formulation + +좌굴 해석을 하기 위해서는 기하 강성 행렬(geometric stiffness matrix)이 필요하며, 이를 구하기 위해 MITC4 셀 요소에 대한 비선형 유한요소 정식화 과정을 수행하였다. + +현재 형상에 대한 평형방정식은 식(2.8)과 같다. + +$$ +\nabla_ {\mathbf {X}} \cdot \boldsymbol {\sigma} + \rho \mathbf {f} = \rho \ddot {\mathbf {u}} \tag {2.8} +$$ + +식(2.8)에 가상 변위에 대한 현재 형상에서의 가상 일 정리를 사용하면, 식 (2.9)와 같이 쓸 수 있다. + +$$ +\int_ {V} \delta \mathbf {u} \cdot \left(\nabla_ {\mathbf {X}} \cdot \boldsymbol {\sigma} + \rho \mathbf {f}\right) d V = \int_ {V} \delta \mathbf {u} \cdot \rho \ddot {\mathbf {u}} d V \tag {2.9} +$$ + +식(2.9)를 인덱스를 사용하여 표현하면 식(2.10)과 같다. + +$$ +\int_ {V} \delta u _ {i} \cdot \left(\frac {\partial \sigma_ {i j}}{\partial x _ {j}} + \rho f _ {i}\right) d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \tag {2.10} +$$ + +이 때 식(2.11)을 식(2.10)에 대입하면 식(2.12)와 같은 결과를 얻을 수 있다. + +$$ +\delta u _ {i} \frac {\partial \sigma_ {i j}}{\partial x _ {j}} = \frac {\partial \left(\delta u _ {i} \sigma_ {i j}\right)}{\partial x _ {j}} - \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} \tag {2.11} +$$ + +$$ +\int_ {V} \frac {\partial \left(\delta u _ {i} \sigma_ {i j}\right)}{\partial x _ {j}} - \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \tag {2.12} +$$ + +식(2.12)에 가우스의 발산 정리(Gauss' divergence theorem)를 사용하면 식(2.13)과 같이 나타낼 수 있다. + + + +$$ +\int_ {\partial V} \delta u _ {i} \sigma_ {i j} n _ {j} d A + \int_ {V} \left(- \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i}\right) d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \tag {2.13} +$$ + +식(2.13)에서 좌측 첫 번째 항의 면 적분 부분은 식(2.14)와 같이 기하학적 경계조건(geometric boundary condition)과 자연적 경계조건(natural boundary condition)으로 나눌 수 있으며, 이 때 식(2.14)의 좌측 첫 번째 항은 기하학적 경계조건에서 가상변위가 영(零)이므로 생략할 수 있다. + +$$ +\int_ {\partial V _ {g}} \delta u _ {i} \sigma_ {i j} n _ {j} d A + \int_ {\partial V _ {m}} \delta u _ {i} \bar {t} _ {i} d A + \int_ {V} \left(- \frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} + \delta u _ {i} \rho f _ {i}\right) d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V \tag {2.14} +$$ + +$$ +\frac {\partial \delta u _ {i}}{\partial x _ {j}} \sigma_ {i j} = \frac {1}{2} \left(\frac {\partial \delta u _ {i}}{\partial x _ {j}} + \frac {\partial \delta u _ {j}}{\partial x _ {i}}\right) \sigma_ {i j} = \delta \varepsilon_ {i j} \sigma_ {i j} \tag {2.15} +$$ + +그리고 식(2.14)에 식(2.15)를 대입하여 정리하면 현재 형상에서의 가상일에 대한 식을 식(2.16)과 같이 얻을 수 있다. + +$$ +\int_ {\partial V _ {m}} \delta u _ {i} \bar {t} _ {i} d A + \int_ {V} \delta u _ {i} \rho f _ {i} d V = \int_ {V} \delta u _ {i} \cdot \rho \ddot {u} _ {i} d V + \int_ {V} \delta \varepsilon_ {i j} \sigma_ {i j} d V \tag {2.16} +$$ + +식(2.16)에서 좌변은 외력에 의한 가상 일이고, 우변은 내력에 의한 가상 일이다. + +비선형 해석을 수행하기 위해서는 Total Lagrangian 기법과 Updated Lagrangian 기법이 있으며, 전자의 방법은 문제를 초기 형상에 대해 정의하는 방법이고, 후자의 방법은 현재 형상에 대해 정의하는 방법이다. 본 논문에서는 전자의 방법을 사용하였으며, 식(2.17)을 사용하여 변형된 현재 형상의 응력과 변형률 등을 초기 형상에 대해 정의하였다. + +$$ +\int_ {V} (A) d V = \int_ {V _ {0}} (A) \det (\mathbf {F}) d V _ {0} = \int_ {V _ {0}} (A) J d V _ {0} \tag {2.17a} +$$ + + + +$$ +\int_ {\partial V} (A) \mathbf {n} d A = \int_ {\partial V _ {0}} (A) J \mathbf {F} ^ {- T} \widetilde {\mathbf {n}} d A _ {0} \quad (\text { Nanson's formular }) \tag {2.17b} +$$ + +$$ +\int_ {V} \delta \mathbf {e}: \mathbf {o} d V = \int_ {V _ {0}} \delta \mathbf {F}: \mathbf {P} d V _ {0} = \int_ {V _ {0}} \delta \mathbf {E}: \mathbf {S} d V _ {0} \tag {2.17c} +$$ + +이 때 (A)는 임의의 값을 의미하고, F는 변형 구배(deformation gradient), J는 자코비안(Jacobian)으로 체적변화율을 나타낸다. 그리고 ε과 σ는 현재 형상에서의 미소 변형률(infinitesimal strain)과 Cauchy 응력을 의미하고, E는 Green-Lagrange 변형률로써 초기 형상에서 정의된다. P는 1차 Piola-Kirchhoff 응력을 나타내고, S는 2차 Piola-Kirchhoff 응력을 나타내며, 두 응력 모두 초기 형상에서 정의된다. + +식(2.17)을 사용하여 식(2.16)을 초기 형상에 대해 정의해주면 식(2.18)과 같이 나타낼 수 있다. + +$$ +\int_ {V _ {0}} \delta \mathbf {u} \cdot \rho_ {0} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta \mathbf {E}: \mathbf {S} d V _ {0} = \int_ {\partial V _ {0 _ {m}}} \delta \mathbf {u} \cdot \mathbf {F} \mathbf {S} \widetilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta \mathbf {u} \rho_ {0} \mathbf {f} d V _ {0} \tag {2.18a} +$$ + +$$ +\Leftrightarrow \int_ {V _ {0}} \delta \mathbf {u} \cdot \rho_ {0} \ddot {\mathbf {u}} d V _ {0} + \int_ {V _ {0}} \delta \mathbf {F}: \mathbf {P} d V _ {0} = \int_ {\partial V _ {0 _ {m}}} \delta \mathbf {u} \cdot \mathbf {P} \widetilde {\mathbf {n}} d A _ {0} + \int_ {V _ {0}} \delta \mathbf {u} \rho_ {0} \mathbf {f} d V _ {0} \tag {2.18b} +$$ + +식(2.18b)는 1차 Piola-Kirchhoff 응력으로 표현한 식이고, 식(2.18a)는 2차 Piola-Kirchhoff 응력으로 표현한 식이다. 본 논문에서는 2차 Piola-Kirchhoff 응력을 사용한 식(2.18a)를 사용하였다. + + + +![](images/page-020_65b2c3dfb5e6b1bcd01a1c99307eddea412691fa93db34cbdf3157059e2e0814.jpg) + +
+text_image + +g₃ +g₁ +g₂ +η +V +ξ +z +y +x +
+ +Fig. 3 An arbitrary surface with global Cartesian coordinate system(x, y, z), natural coordinate system $(\xi, \eta, \zeta)$ and local covariant coordinate system spanned by $\mathbf{g}_i$ + +고유 좌표계(natural coordinate system)에서 공변 기저(covariant basis)는 식(2.19)와 같이 나타낼 수 있고, 변위에 대한 기울기(gradient)를 고유 좌표계와 반공변 기저(contravariant basis)를 사용하여 표현하면 식(2.20)과 같이 표현할 수 있다. + +$$ +\mathbf {g} _ {i} = \frac {\partial \mathbf {x}}{\partial \xi^ {i}} \tag {2.19} +$$ + +$$ +\nabla \otimes \mathbf {u} = \frac {\partial \mathbf {u}}{\partial \xi^ {i}} \otimes \mathbf {g} ^ {i} \tag {2.20} +$$ + +변형률은 식(2.20)을 사용하여 식(2.21)과 같이 나타낼 수 있다. + +$$ +\boldsymbol {\varepsilon} = \frac {1}{2} \left[ \nabla \otimes \mathbf {u} + (\nabla \otimes \mathbf {u}) ^ {T} \right] = \frac {1}{2} \left[ \frac {\partial \mathbf {u}}{\partial \xi^ {i}} \otimes \mathbf {g} ^ {i} + \mathbf {g} ^ {i} \otimes \frac {\partial \mathbf {u}}{\partial \xi^ {i}} \right] \tag {2.21} +$$ + +그리고 변형률 텐서(strain tensor)에서 공변 성분(covariant component)을 diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_003.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_003.md new file mode 100644 index 0000000..54ed79f --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_003.md @@ -0,0 +1,187 @@ + + +얻기 위해 변형률 텐서 좌우에 공변 기저(covariant basis)를 내적하면, 식(2.22)와 같은 결과를 얻을 수 있다. + +$$ +\begin{array}{l} \boldsymbol {\varepsilon} _ {p q} = \mathbf {g} _ {p} \cdot \boldsymbol {\varepsilon} \cdot \mathbf {g} _ {q} \\ = \frac {1}{2} \mathbf {g} _ {p} \cdot \left[ \frac {\partial \mathbf {u}}{\partial \xi^ {i}} \otimes \mathbf {g} ^ {i} + \mathbf {g} ^ {i} \otimes \frac {\partial \mathbf {u}}{\partial \xi^ {i}} \right] \cdot \mathbf {g} _ {q} \tag {2.22} \\ = \frac {1}{2} \left[ \mathbf {g} _ {p} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}} + \frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \mathbf {g} _ {q} \right] \\ \end{array} +$$ + +그리고 식(2.22)를 식(2.19)를 사용하여 표현하면 변형률 텐서의 공변성분(covariant component)은 식(2.23)과 같이 나타낼 수 있다. + +$$ +\varepsilon_ {p q} = \frac {1}{2} \left[ \frac {\partial \mathbf {x}}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}} + \frac {\partial \mathbf {x}}{\partial \xi^ {q}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {p}} \right] \tag {2.23} +$$ + +이와 같은 방법으로 Green-Lagrange 변형률을 표현하기 위해서 우선 Green-Lagrange 변형률은 식(2.24)와 같이 정의 할 수 있다. + +$$ +\begin{array}{l} \mathbf {E} = \frac {1}{2} \left(\mathbf {F} ^ {T} \mathbf {F} - \mathbf {I}\right) = \frac {1}{2} \left(\frac {\partial (\mathbf {X} + \mathbf {u}) ^ {T}}{\partial \mathbf {X}} \frac {\partial (\mathbf {X} + \mathbf {u})}{\partial \mathbf {X}} - \mathbf {I}\right) \\ = \frac {1}{2} \left(\left[ \mathbf {I} + \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] ^ {T} \left[ \mathbf {I} + \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] - \mathbf {I}\right) = \frac {1}{2} \left(\left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] + \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] ^ {T} + \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] ^ {T} \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right]\right) \tag {2.24} \\ = \frac {1}{2} \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \otimes \mathbf {G} ^ {i} + \mathbf {G} ^ {i} \otimes \frac {\partial \mathbf {u}}{\partial \xi^ {i}} + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j}\right) \\ \end{array} +$$ + +그리고 Green-Lagrange 변형률 텐서에서 공변 성분(covariant component)을 얻기 위해 Green-Lagrange 변형률 텐서 좌우에 공변 기저(covariant basis)를 내적하면, 식(2.25)와 같은 결과를 얻을 수 있다. + + + +$$ +\begin{array}{l} E _ {p q} = \mathbf {G} _ {p} \cdot \mathbf {E} \cdot \mathbf {G} _ {q} \\ = \frac {1}{2} \mathbf {G} _ {p} \cdot \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \otimes \mathbf {G} ^ {i} + \mathbf {G} ^ {i} \otimes \frac {\partial \mathbf {u}}{\partial \xi^ {i}} + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j}\right) \cdot \mathbf {G} _ {q} \\ = \frac {1}{2} \left(\left(\mathbf {G} _ {p} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \mathbf {G} _ {q}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right)\right) \tag {2.25} \\ = \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right)\right) \\ \end{array} +$$ + +식(2.25)를 변위에 대한 증분형태(incremental form)로 나타내기 위해 $u = u_t + \Delta u$ 을 대입하여 정리하면, 식(2.26)와 같은 결과를 얻을 수 있다. 이 때 식(2.26)의 첫 번째 항은 $\Delta u$ 에 대한 상수 항이고, 두 번째 항은 $\Delta u$ 에 대한 선형 항 그리고 세 번째 항은 $\Delta u$ 에 대한 비선형 항을 나타내며, 이를 간략하게 기호로 표시하면 다음과 같이 쓸 수 있다. + +$$ +\begin{array}{l} E _ {p q} = \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) \\ = \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial \left(\mathbf {u} _ {t}\right)}{\partial \xi^ {q}}\right) + \left(\frac {\partial \left(\mathbf {u} _ {t}\right)}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \left(\mathbf {u} _ {t}\right)}{\partial \xi^ {p}} \cdot \frac {\partial \left(\mathbf {u} _ {t}\right)}{\partial \xi^ {q}}\right)\right) \\ + \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {p}} \cdot \frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {q}}\right)\right) \\ + \frac {1}{2} \left(\frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {q}}\right) \\ = E _ {0 _ {p q}} + \Delta E _ {p q} + \Delta^ {2} E _ {p q} \tag {2.26} \\ \end{array} +$$ + +가상 일에 대한 변형률을 구하기 위해 Green-Lagrange 변형률에 변분(variation)을 취하면 식(2.27)과 같이 나타낼 수 있다. + + + +$$ +\begin{array}{l} \delta \mathbf {E} = \frac {1}{2} \left(\left[ \frac {\partial (\delta \mathbf {u})}{\partial \mathbf {X}} \right] + \left[ \frac {\partial (\delta \mathbf {u})}{\partial \mathbf {X}} \right] ^ {T} + \left[ \frac {\partial (\delta \mathbf {u})}{\partial \mathbf {X}} \right] ^ {T} \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] + \left[ \frac {\partial \mathbf {u}}{\partial \mathbf {X}} \right] ^ {T} \left[ \frac {\partial (\delta \mathbf {u})}{\partial \mathbf {X}} \right]\right) \\ = \frac {1}{2} \left( \begin{array}{l} \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \otimes \mathbf {G} ^ {i} + \mathbf {G} ^ {i} \otimes \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \\ + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j} + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j} \end{array} \right) \tag {2.27} \\ \end{array} +$$ + +위와 마찬가지로 Green-Lagrange 변형률 텐서에서 공변 성분(covariant component)을 얻기 위해 식(2.27)의 좌우에 공변 기저(covariant basis)를 내적하면, 식(2.28)과 같은 결과를 얻을 수 있다. + +$$ +\begin{array}{l} \delta E _ {p q} = \mathbf {G} _ {p} \cdot \delta \mathbf {E} \cdot \mathbf {G} _ {q} \\ = \frac {1}{2} \mathbf {G} _ {p} \cdot \left( \begin{array}{l} \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \otimes \mathbf {G} ^ {i} + \mathbf {G} ^ {i} \otimes \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \\ + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {i}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j} + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {i}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {j}}\right) \mathbf {G} ^ {i} \otimes \mathbf {G} ^ {j} \end{array} \right) \cdot \mathbf {G} _ {q} \tag {2.28} \\ = \frac {1}{2} \left(\mathbf {G} _ {p} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}} + \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \mathbf {G} _ {q} + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) \\ = \frac {1}{2} \left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}} + \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}} + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) \\ \end{array} +$$ + +식(2.28)을 변위에 대한 증분형태로 나타내기 위해 $u = u_{t} + \Delta u$ 을 대입하여 정리하면, 식(2.29)와 같은 결과를 얻을 수 있다. 이 때 식(2.29)의 첫 번째 항은 $\Delta u$ 에 대한 상수 항이고, 두 번째 항은 $\Delta u$ 에 대한 선형 항을 나타내며, 이를 기호로 간략하게 표현하면 다음과 같이 쓸 수 있다. 여기서 $u_{t}$ 는 고정된 값이므로 $u_{t}$ 에 대한 변분(variation)은 영(雰)이 된다. + + + +$$ +\begin{array}{l} \delta E _ {p q} = \frac {1}{2} \left( \begin{array}{l} \frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}} + \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}} \\ + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t} + \Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right) \end{array} \right) \\ = \frac {1}{2} \left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}} + \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}} + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u} _ {t}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u} _ {t}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) \tag {2.29} \\ + \frac {1}{2} \left(\left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \Delta \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \Delta \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) \\ = \delta \left(E _ {0 _ {p q}} + \Delta E _ {p q} + \Delta^ {2} E _ {p q}\right) \\ = \delta \Delta E _ {p q} + \delta \Delta^ {2} E _ {p q} \\ = \delta \mathcal {E} _ {0 _ {p q}} + \delta \Delta \mathcal {E} _ {p q} \\ \end{array} +$$ + +식(2.28)과 식(2.29)의 결과를 식(2.18a)의 $\int_{V_{0}}\delta E:SdV_{0}$ 항에 대입하면 식(2.30)과 같이 정리할 수 있다. 이 때 식(2.30)의 첫 번째 항은 상수 항이고, 두 번째 항은 선형 항, 세 번째 항은 고차 항이며, 네 번째 항은 선형 항, 다섯 번째와 여섯 번째 항은 고차 항을 나타낸다. + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta \mathbf {E}: \mathbf {S} d V _ {0} = \int_ {V _ {0}} \delta E _ {i j} S ^ {i j} d V _ {0} = \int_ {V _ {0}} \delta E _ {i j} C ^ {i j k l} E _ {k l} d V _ {0} \\ = \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}} + \delta \Delta \mathcal {E} _ {i j}\right) C ^ {i j k l} \left(E _ {0 _ {k l}} + \Delta E _ {k l} + \Delta^ {2} E _ {k l}\right) d V _ {0} \\ = \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}}\right) C ^ {i j k l} \left(E _ {0 _ {k l}}\right) d V _ {0} + \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}}\right) C ^ {i j k l} \left(\Delta E _ {k l}\right) d V _ {0} \tag {2.30} \\ + \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}}\right) C ^ {i j k l} \left(\Delta^ {2} E _ {k l}\right) d V _ {0} + \int_ {V _ {0}} \left(\delta \Delta \mathcal {E} _ {i j}\right) C ^ {i j k l} \left(E _ {0 _ {k l}}\right) d V _ {0} \\ + \int_ {V _ {0}} \left(\delta \Delta \mathcal {E} _ {i j}\right) C ^ {i j k l} \left(\Delta E _ {k l}\right) d V _ {0} + \int_ {V _ {0}} \left(\delta \Delta \mathcal {E} _ {i j}\right) C ^ {i j k l} \left(\Delta^ {2} E _ {k l}\right) d V _ {0} \\ \end{array} +$$ + +따라서 식(2.30)에서 고차 항을 제외하면 식(2.31)과 같은 결과를 얻을 수 있으며, 첫 번째 항은 상수 항이고, 두 번째와 세 번째 항은 선형 + + + +항을 나타낸다. + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta E _ {i j} S ^ {i j} d V _ {0} \approx \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}}\right) C ^ {i j k l} \left(E _ {0 _ {k l}}\right) d V _ {0} + \int_ {V _ {0}} \left(\delta \mathcal {E} _ {0 _ {i j}}\right) C ^ {i j k l} \left(\Delta E _ {k l}\right) d V _ {0} \tag {2.31} \\ + \int_ {V _ {0}} \left(\delta \Delta \mathcal {E} _ {i j}\right) C ^ {i j k l} \left(E _ {0 _ {k l}}\right) d V _ {0} \\ \end{array} +$$ + +앞서 정의한 MITC4 셀 요소의 위치 벡터 식(2.1), (2.2)를 행렬 형태로 표현하면 식(2.32), (2.33)과 같이 나타낼 수 있고, 이를 간단하게 기호로 표현하면 다음과 같이 표현할 수 있다. 이 때 1 $_{3}$ 은 3행 3열의 단위 행렬(identity matrix)을 나타낸다. + +$$ +\begin{array}{l} \mathbf {X} = \left[ N _ {1} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {1} N _ {1} \mathbf {1} _ {3} N _ {2} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {2} N _ {2} \mathbf {1} _ {3} N _ {3} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {3} N _ {3} \mathbf {1} _ {3} N _ {4} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {4} N _ {4} \mathbf {1} _ {3} \right] \left\{ \begin{array}{l} ^ {0} \mathbf {X} _ {1} \\ ^ {0} \mathbf {V} _ {1} ^ {n} \\ ^ {0} \mathbf {X} _ {2} \\ ^ {0} \mathbf {V} _ {2} ^ {n} \\ ^ {0} \mathbf {X} _ {3} \\ ^ {0} \mathbf {V} _ {3} ^ {n} \\ ^ {0} \mathbf {X} _ {4} \\ ^ {0} \mathbf {V} _ {4} ^ {n} \end{array} \right\} \\ = \mathbf {S X} _ {0} \tag {2.32} \\ \end{array} +$$ + +$$ +\begin{array}{l} \mathbf {x} = \left[ N _ {1} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {1} N _ {1} \mathbf {1} _ {3} N _ {2} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {2} N _ {2} \mathbf {1} _ {3} N _ {3} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {3} N _ {3} \mathbf {1} _ {3} N _ {4} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {4} N _ {4} \mathbf {1} _ {3} \right] \left\{ \begin{array}{l} ^ {t} \mathbf {X} _ {1} \\ ^ {t} \mathbf {V} _ {1} ^ {n} \\ ^ {t} \mathbf {X} _ {2} \\ ^ {t} \mathbf {V} _ {2} ^ {n} \\ ^ {t} \mathbf {X} _ {3} \\ ^ {t} \mathbf {V} _ {3} ^ {n} \\ ^ {t} \mathbf {X} _ {4} \\ ^ {t} \mathbf {V} _ {4} ^ {n} \end{array} \right\} \\ = \mathbf {S} \mathbf {X} _ {t} \tag {2.33} \\ \end{array} +$$ + + + +또한 임의의 시간 t에서의 변위 식(2.3)과 변위의 중분 식(2.4)를 행렬 형태로 표현하면 식(2.34), (2.35)로 각각 나타낼 수 있고, 이를 간략하게 기호로 표현하면 다음과 같이 표현할 수 있다. + +$$ +\begin{array}{l} \mathbf {u} _ {t} = \mathbf {x} - \mathbf {X} \\ = \left[ N _ {1} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {1} N _ {1} \mathbf {1} _ {3} N _ {2} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {2} N _ {2} \mathbf {1} _ {3} N _ {3} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {3} N _ {3} \mathbf {1} _ {3} N _ {4} \mathbf {1} _ {3} \frac {\zeta}{2} t _ {4} N _ {4} \mathbf {1} _ {3} \right] \left\{ \begin{array}{l} ^ {t} \mathbf {X} _ {1} - ^ {0} \mathbf {X} _ {1} \\ ^ {t} \mathbf {V} _ {1} ^ {n} - ^ {0} \mathbf {V} _ {1} ^ {n} \\ ^ {t} \mathbf {X} _ {2} - ^ {0} \mathbf {X} _ {2} \\ ^ {t} \mathbf {V} _ {2} ^ {n} - ^ {0} \mathbf {V} _ {2} ^ {n} \\ ^ {t} \mathbf {X} _ {3} - ^ {0} \mathbf {X} _ {3} \\ ^ {t} \mathbf {V} _ {3} ^ {n} - ^ {0} \mathbf {V} _ {3} ^ {n} \\ ^ {t} \mathbf {X} _ {4} - ^ {0} \mathbf {X} _ {4} \\ ^ {t} \mathbf {V} _ {4} ^ {n} - ^ {0} \mathbf {V} _ {4} ^ {n} \end{array} \right\} \\ = \mathbf {S} \left(\mathbf {X} _ {t} - \mathbf {X} _ {0}\right) \tag {2.34} \\ \end{array} +$$ + +$$ +\begin{array}{l} \Delta \mathbf {u} = \left[ N _ {1} \mathbf {1} _ {3} - \frac {\zeta}{2} t _ {1} N _ {1} ^ {t} \mathbf {V} _ {1} ^ {2} \frac {\zeta}{2} t _ {1} N _ {1} ^ {t} \mathbf {V} _ {1} ^ {1} \dots N _ {4} \mathbf {1} _ {3} - \frac {\zeta}{2} t _ {4} N _ {4} ^ {t} \mathbf {V} _ {4} ^ {2} \frac {\zeta}{2} t _ {4} N _ {4} ^ {t} \mathbf {V} _ {4} ^ {1} \right] \left\{ \begin{array}{l} \Delta \mathbf {u} _ {1} \\ \alpha_ {1} \\ \beta_ {1} \\ \Delta \mathbf {u} _ {2} \\ \alpha_ {2} \\ \beta_ {2} \\ \Delta \mathbf {u} _ {3} \\ \alpha_ {3} \\ \beta_ {3} \\ \Delta \mathbf {u} _ {4} \\ \alpha_ {4} \\ \beta_ {4} \end{array} \right\} \\ = \mathbf {N} (\Delta \mathbf {U}) \tag {2.35} \\ \end{array} +$$ + +이 때 $V^{n}$ 이 현재 형상의 변화에 따라 변하므로 $V^{1}$ 과 $V^{2}$ 도 계속 + + + +변하게 되고, 행렬 N 도 현재 형상에 따라 변하게 된다. 즉 $V^{1}$ 과 $V^{2}$ 는 회전각 $\alpha$ , $\beta$ 에 따라 회전하게 되고, 두 축의 회전에 의해 $V^{n}$ 이 시간에 따라 변하게 된다. 따라서 시간에 따른 회전각 $\alpha$ , $\beta$ 의 변화에 따라 $V^{n}$ 을 계산해 주어야 하고, 또한 그에 따라 $V^{1}$ 과 $V^{2}$ 도 다시 계산해 주어야 한다. 이에 대한 자세한 내용은 2.2.1 절에서 다루도록 하겠다. + +식(2.32), (2.33), (2.34), (2.35)의 행렬식을 식(2.31)에서 $\delta E_{0_{pq}}$ , $\delta \Delta E_{pq}$ , $E_{0_{pq}}$ , $\Delta E_{pq}$ 의 각각의 항에 대입하면 식(2.36), (2.37), (2.38), (2.39)와 같이 나타낼 수 있다. + +$$ +\begin{array}{l} E _ {0 _ {p q}} = \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {q}}\right)\right) \\ = \frac {1}{2} \left\{\mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {S}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {S}}{\partial \xi^ {p}} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \\ + \frac {1}{2} \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {S}}{\partial \xi^ {q}} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \\ = \frac {1}{2} \left\{\mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {S}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {S}}{\partial \xi^ {p}} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \tag {2.36} \\ + \frac {1}{2} \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} ^ {T} \frac {1}{2} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {S}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {S}}{\partial \xi^ {p}} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \\ = \frac {1}{2} \left\{\mathbf {X} _ {t} + \mathbf {X} _ {0} \right\} ^ {T} \frac {1}{2} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {S}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {S}}{\partial \xi^ {p}} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \\ = \frac {1}{2} \left\{\mathbf {X} _ {t} + \mathbf {X} _ {0} \right\} ^ {T} \left[ \mathbf {e} _ {p q} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} \\ \end{array} +$$ + + + +$$ +\begin{array}{l} \Delta E _ {p q} = \frac {1}{2} \left(\left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {p}} \cdot \frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {q}}\right) + \left(\frac {\partial (\Delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial (\mathbf {u} _ {t})}{\partial \xi^ {q}}\right)\right) \\ = \frac {1}{2} \left\{\mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\Delta \mathbf {U} \right\} \\ + \frac {1}{2} \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\Delta \mathbf {U} \right\} \\ = \frac {1}{2} \left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\Delta \mathbf {U} \right\} \\ = \left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {p q} \right] \left\{\Delta \mathbf {U} \right\} \tag {2.37} \\ \end{array} +$$ + +$$ +\delta \Delta E _ {p q} = \frac {1}{2} \left(\frac {\partial \mathbf {X}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}} + \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {X}}{\partial \xi^ {q}} + \left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \mathbf {u} _ {t}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \mathbf {u} _ {t}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) +$$ + +$$ += \frac {1}{2} \left\{\mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\delta \mathbf {U} \right\} +$$ + +$$ ++ \frac {1}{2} \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\partial \mathbf {U} \right\} +$$ + +$$ += \frac {1}{2} \left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {S} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\delta \mathbf {U} \right\} +$$ + +$$ += \left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {p q} \right] \left\{\boldsymbol {\partial} \mathbf {U} \right\} \tag {2.38} +$$ + +$$ +\delta \Delta \mathcal {E} = \frac {1}{2} \left(\left(\frac {\partial (\delta \mathbf {u})}{\partial \xi^ {p}} \cdot \frac {\partial \Delta \mathbf {u}}{\partial \xi^ {q}}\right) + \left(\frac {\partial \Delta \mathbf {u}}{\partial \xi^ {p}} \cdot \frac {\partial (\delta \mathbf {u})}{\partial \xi^ {q}}\right)\right) +$$ + +$$ += \frac {1}{2} \left\{\boldsymbol {\partial} \mathbf {U} \right\} ^ {T} \left[ \frac {\partial \mathbf {N} ^ {T}}{\partial \xi^ {p}} \frac {\partial \mathbf {N}}{\partial \xi^ {q}} + \frac {\partial \mathbf {N} ^ {T}}{\partial \xi^ {q}} \frac {\partial \mathbf {N}}{\partial \xi^ {p}} \right] \left\{\Delta \mathbf {U} \right\} \tag {2.39} +$$ + +$$ += \left\{\boldsymbol {\delta} \mathbf {U} \right\} ^ {T} \left[ \mathbf {c} _ {p q} \right] \left\{\Delta \mathbf {U} \right\} +$$ + +식(2.36), (2.37), (2.38), (2.39)을 식(2.31)에 대입하면 식(2.40)과 같이 정리할 수 있다. + + + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta E _ {i j} S ^ {i j} d V _ {0} \approx \int_ {V _ {0}} \left(\left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) C ^ {i j k l} \left(\frac {1}{2} \left\{\mathbf {X} _ {t} + \mathbf {X} _ {0} \right\} ^ {T} \left[ \mathbf {e} _ {k l} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\}\right) d V _ {0} \\ + \int_ {V _ {0}} \left(\left\{\boldsymbol {\partial} \mathbf {U} \right\} ^ {T} \left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) C ^ {i j k l} \left(\left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {k l} \right] \left\{\Delta \mathbf {U} \right\}\right) d V _ {0} \\ + \int_ {V _ {0}} \left(\left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {c} _ {i j} \right] \left\{\Delta \mathbf {U} \right\}\right) C ^ {i j k l} \left(\frac {1}{2} \left\{\mathbf {X} _ {t} + \mathbf {X} _ {0} \right\} ^ {T} \left[ \mathbf {e} _ {k l} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\}\right) d V _ {0} \tag {2.40} \\ \end{array} +$$ + +그리고 현재 형상(t=0)일 때의 2차 Piola-Kirchhoff 응력을 식(2.41)과 같이 정의해주면, 식(2.40)은 식(2.42)와 같이 정리할 수 있다. + +$$ +S _ {0} ^ {i j} = C ^ {i j k l} \left(\frac {1}{2} \left\{\mathbf {X} _ {t} + \mathbf {X} _ {0} \right\} ^ {T} \left[ \mathbf {e} _ {k l} \right] \left\{\mathbf {X} _ {t} - \mathbf {X} _ {0} \right\}\right) \tag {2.41} +$$ + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta E _ {i j} S ^ {i j} d V _ {0} \approx \int_ {V _ {0}} \left(\left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) S _ {0} ^ {i j} d V _ {0} \\ + \int_ {V _ {0}} \left(\left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) C ^ {i j k l} \left(\left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {k l} \right] \left\{\Delta \mathbf {U} \right\}\right) d V _ {0} \\ + \int_ {V _ {0}} \left(\left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {c} _ {i j} \right] \left\{\Delta \mathbf {U} \right\}\right) S _ {0} ^ {i j} d V _ {0} \\ \end{array} +$$ + +$$ +\begin{array}{l} = \left\{\boldsymbol {\partial} \mathbf {U} \right\} ^ {T} \left[ \int_ {V _ {0}} \left(S _ {0} ^ {i j} \left[ \mathbf {a} _ {i j} \right] ^ {T}\right) d V _ {0} \right] \left\{\mathbf {X} _ {t} \right\} \\ + \left\{\boldsymbol {\delta} \mathbf {U} \right\} ^ {T} \left[ \int_ {V _ {0}} \left(\left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) C ^ {i j k l} \left(\left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {k l} \right]\right) d V _ {0} \right] \left\{\Delta \mathbf {U} \right\} \\ + \left\{\boldsymbol {\delta} \mathbf {U} \right\} ^ {T} \left[ \int_ {V _ {0}} \left(S _ {0} ^ {i j} \left[ \mathbf {c} _ {i j} \right]\right) d V _ {0} \right] \left\{\Delta \mathbf {U} \right\} \tag {2.42} \\ \end{array} +$$ + +식(2.42)를 고유 좌표계(natural coordinate system)에서 2×2×2 가우스 적분(Gauss integration)해주면, 식(2.43)과 같이 쓸 수 있다. + + + +$$ +\begin{array}{l} \int_ {V _ {0}} \delta E _ {i j} S ^ {i j} d V _ {0} \\ \approx \left\{\delta \mathbf {U} \right\} ^ {T} \left[ \iiint_ {\xi} \left(S _ {0} ^ {i j} \left[ \mathbf {a} _ {i j} \right] ^ {T}\right) | J | d \xi d \eta d \zeta \right] \left\{\mathbf {X} _ {t} \right\} \\ + \left\{\boldsymbol {\delta} \mathbf {U} \right\} ^ {T} \left[ \iiint_ {\xi} \left(\left[ \mathbf {a} _ {i j} \right] ^ {T} \left\{\mathbf {X} _ {t} \right\}\right) [ T ] ^ {T} \left[ \widetilde {D} ^ {i j k l} \right] [ T ] \left(\left\{\mathbf {X} _ {t} \right\} ^ {T} \left[ \mathbf {a} _ {k l} \right]\right) | J | d \xi d \eta d \zeta \right] \left\{\Delta \mathbf {U} \right\} \tag {2.43} \\ + \left\{\delta \mathbf {U} \right\} ^ {T} \left[ \iiint_ {\xi} \left(S _ {0} ^ {i j} \left[ \mathbf {c} _ {i j} \right]\right) | J | d \xi d \eta d \zeta \right] \left\{\Delta \mathbf {U} \right\} \\ = \left\{\delta \mathbf {U} \right\} ^ {T} \left\{\mathbf {F} \right\} \left\{\mathbf {X} _ {t} \right\} + \left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {K} _ {N L} \right] \left\{\Delta \mathbf {U} \right\} + \left\{\delta \mathbf {U} \right\} ^ {T} \left[ \mathbf {K} _ {L} \right] \left\{\Delta \mathbf {U} \right\} \\ \end{array} +$$ + +여기서 $K_{L}$ 은 초기 변위와 관련된 강성 행렬(initial displacement stiffness matrix), $K_{NL}$ 은 초기 응력과 관련된 강성 행렬(initial stress stiffness matrix)또는 기하 강성 행렬(geometric stiffness matrix)이고, F는 변형으로 인한 힘 벡터를 나타내며, $|J|$ 는 자코비안 행렬(Jacobian matrix)에 대한 determinant를 의미한다. 그리고 $\widetilde{D}^{ijkl}$ 는 지역 직교 좌표계(local Cartesian coordinate system)에서 정의되어있는 구성 행렬(constitutive matrix)로 이를 고유 좌표계(natural coordinate system)로 변환해주기 위해 변환 행렬(transformation matrix) $[T]$ 를 사용하였다. 이에 대한 자세한 내용은 2.2.2 절에서 다루도록 하겠다. + +식(2.18a) 우변은 표면력(surface force)과 체적력(body force)을 나타내며, 식(2.44), (2.45)와 같이 이산화하여 나타낼 수 있다. + +$$ +\int_ {\partial V _ {0 _ {m}}} \delta \mathbf {u} \cdot \mathbf {F} \mathbf {S} \widetilde {\mathbf {n}} d A _ {0} = \left\{\delta \mathbf {U} \right\} ^ {T} \int_ {\partial V _ {0 _ {m}}} \left[ \mathbf {N} \right] ^ {T} \left\{\overline {{\mathbf {T}}} \right\} d A _ {0} = \left\{\delta \mathbf {U} \right\} ^ {T} \left\{\mathbf {Q} _ {T} \right\} \tag {2.44} +$$ + +$$ +\int_ {V _ {0}} \delta \mathbf {u} \rho_ {0} \mathbf {f} d V _ {0} = \left\{\delta \mathbf {U} \right\} ^ {T} \int_ {V _ {0}} \rho_ {0} [ \mathbf {N} ] ^ {T} \left\{\mathbf {f} \right\} d V _ {0} = \left\{\delta \mathbf {U} \right\} ^ {T} \left\{\mathbf {Q} _ {B} \right\} \tag {2.45} +$$ + +식(2.43), (2.44), (2.45)을 식(2.18a)에 대입하고, 정적인 문제이므로 시간에 대한 항을 무시하면, 식(2.18a)는 식(2.46)과 같이 표현할 수 있다. diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_004.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_004.md new file mode 100644 index 0000000..f8a46d2 --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_004.md @@ -0,0 +1,279 @@ + + +$$ +0 = \left\{\delta \mathbf {U} \right\} ^ {T} \left(\left[ \mathbf {K} _ {N L} \right] + \left[ \mathbf {K} _ {L} \right]\right) \left\{\Delta \mathbf {U} \right\} - \left\{\delta \mathbf {U} \right\} ^ {T} \left(\left\{\mathbf {Q} _ {T} \right\} + \left\{\mathbf {Q} _ {B} \right\} - \left\{\mathbf {F} \right\}\right) \tag {2.46} +$$ + +이때 δU 는 임의의 값이므로 식(2.46)을 만족하기 위해서는 식(2.47)과 같이 쓸 수 있고, 식(2.47)은 Newton-Raphson 형식으로 ΔU 에 대한 반복 계산을 통해 수렴값을 찾아 나감으로써 비선형 해석을 수행할 수 있다. + +$$ +\left(\left[ \mathbf {K} _ {N L} \right] + \left[ \mathbf {K} _ {L} \right]\right) \left\{\Delta \mathbf {U} \right\} = \left\{\mathbf {Q} _ {T} \right\} + \left\{\mathbf {Q} _ {B} \right\} - \left\{\mathbf {F} \right\} \tag {2.47} +$$ + +여기서 $K_{NL}$ 과 $K_{L}$ 의 합이 기울기 강성 행렬(tangent stiffness matrix)이 된다. + +# 2.2.1 Finite Rotation Formulation + +셀의 형상이 변함에 따라 각각의 노드에서 정의 된 벡터 $\left(\mathbf{V}^{1}, \mathbf{V}^{2}, \mathbf{V}^{n}\right)$ 역시 시간에 따라 변하게 된다. 즉 시간에 따른 회전각 $\alpha, \beta$ 의 변화에 따라 $\mathbf{V}^{n}$ 이 변하고, 또한 그에 따라 $\mathbf{V}^{1}$ 과 $\mathbf{V}^{2}$ 도 변하게 된다. 이를 관계식으로 표현하면, 식(2.48)과 같이 쓸 수 있다. + +$$ +{ } ^ { t + \Delta t } \mathbf { V } _ { I } ^ { n } = { } _ { t } ^ { t + \Delta t } \mathbf { R } _ { I } \cdot { } ^ { t } \mathbf { V } _ { I } ^ { n } \tag {2.48} +$$ + +식(2.48)은 시간이 t부터 $t+\Delta t$ 까지 변할 때 노드 I에서의 법선 벡터의 변화를 보여주는 식으로 ${}^{t+\Delta t}_{t}R_{I}$ 는 회전 텐서(rotation tensor)를 의미하며, $\left\|^{t}V_{I}^{n}\right\|=\left\|^{t+\Delta t}V_{I}^{n}\right\|=1$ 이다. [9] + +그리고 회전 텐서 $^{t+\Delta t}_{t}R_{I}$ 은 $V^{1}, V^{2}, V^{n}$ 을 정규직교 기저(orthonormal basis)로 하는 좌표계에서 식(2.49)와 같이 행렬 형태로 나타낼 수 있다. [10] + + + +$$ +\left[ \begin{array}{l} t + \Delta t \\ t \end{array} R _ {I} \right] = \left[ \mathbf {1} _ {3} \right] + \frac {\sin \left(\widetilde {\theta} _ {I}\right)}{\widetilde {\theta} _ {I}} \left[ \Theta_ {I} \right] + \frac {1}{2} \left[ \frac {\sin \left(\frac {\widetilde {\theta} _ {I}}{2}\right)}{\left(\frac {\widetilde {\theta} _ {I}}{2}\right)} \right] ^ {2} \left[ \Theta_ {I} \right] ^ {2} \tag {2.49} +$$ + +이 때 $\left[1_{3}\right]$ 은 3행 3열의 단위 행렬을 의미하며, $\widetilde{\theta}_{I}$ 와 $\left[\Theta_{I}\right]$ 는 각각식(2.50), (2.51)과 같이 나타낼 수 있다. + +$$ +\widetilde {\theta} _ {I} = \left[ \left(\alpha_ {I}\right) ^ {2} + \left(\beta_ {I}\right) ^ {2} \right] ^ {\frac {1}{2}} \tag {2.50} +$$ + +$$ +\left[ \Theta_ {I} \right] = \left[ \begin{array}{c c c} 0 & 0 & \beta_ {I} \\ 0 & 0 & - \alpha_ {I} \\ - \beta_ {I} & \alpha_ {I} & 0 \end{array} \right] \tag {2.51} +$$ + +여기서 $\alpha_{I}$ 와 $\beta_{I}$ 가 미소 증분 회전(infinitesimal incremental rotation)이면, 각각 $V^{1}$ , $V^{2}$ 에 대한 독립적인 미소 회전(independent infinitesimal rotation)을 의미하고, $\alpha_{I}$ 와 $\beta_{I}$ 가 유한 증분 회전(finite incremental rotation)이면, $\alpha_{I}$ 와 $\beta_{I}$ 는 서로 독립적이지 않으며 회전 텐서를 정의하는 변수가 된다. [9] + +# 2.2.2 Constitutive Matrix + +구성 행렬(constitutive matrix)의 경우 평면응력(plane stress) 가정을 사용하였으며, 식(2.52)와 같다. 이 때 $\kappa$ 는 전단 보정 계수(shear correction factor)로 5/6를 사용하였다. + + + +$$ +\left[ \widetilde {D} \right] _ {x y z} = \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c c c c} 1 & \nu & 0 & 0 & 0 & 0 \\ \nu & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \kappa \frac {1 - \nu}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \kappa \frac {1 - \nu}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] \tag {2.52} +$$ + +하지만 식(2.52)의 경우 지역 직교 좌표계(local Cartesian coordinate system)에서 정의되므로 이를 고유 좌표계(natural coordinate system)에서 정의하기 위해서는 변환 행렬(transformation matrix)을 사용해야 한다. + +지역 직교 좌표계와 고유 좌표계 사이의 관계는 Fig. 4와 같으며, 여기서 $G_{1}, G_{2}, G_{3}$ 는 고유 좌표계의 콩변 기저(covariant basis)이고, $\hat{e}_{1}, \hat{e}_{2}, \hat{e}_{3}$ 는 지역 직교 좌표계의 기저를 의미한다. + +![](images/page-033_a55842b4e3905a4b0b3cf04b2cbdeb74411bdf717a6ed7808bb011a550afdb67.jpg) + +
+text_image + +η +G₃ +ê₂ +ê₁ +ê₃ +G₂ +ξ +G₁ +
+ +Fig. 4 Local Cartesian coordinate system + +변형률과 응력에 대한 좌표변환은 식(2.53), (2.54)와 같으며, 이로부터 + + + +구성 행렬의 좌표변환은 식(2.55)와 같이 나타낼 수 있다. 여기서 $\widetilde{E}$ , $\widetilde{S}$ , $\widetilde{D}$ 와 E, S, D는 각각 지역 직교 좌표계와 고유 좌표계에서의 변형률, 응력, 구성 행렬을 의미한다. 그리고 $\left[T\right]_{\xi \rightarrow x}$ 는 고유 좌표계에서 정의된 값을 지역 직교 좌표계로 변환해주는 변환 행렬이다. + +$$ +\left\{\widetilde {E} \right\} _ {x y z} = [ T ] _ {\xi \rightarrow x} \left\{E \right\} _ {\xi \eta \zeta} \tag {2.53} +$$ + +$$ +\begin{array}{l} \left\{S \right\} _ {\xi \eta \zeta} = \left[ T \right] _ {x \rightarrow \xi} \left\{\widetilde {S} \right\} _ {x y z} = \left[ T \right] _ {\xi \rightarrow x} ^ {T} \left[ \widetilde {D} \right] _ {x y z} \left\{\widetilde {E} \right\} _ {x y z} \tag {2.54} \\ = \left[ T \right] _ {\xi \rightarrow x} ^ {T} \left[ \widetilde {D} \right] _ {x y z} \left[ T \right] _ {\xi \rightarrow x} \left\{E \right\} _ {\xi \eta \zeta} = \left[ D \right] _ {\xi \eta \zeta} \left\{E \right\} _ {\xi \eta \zeta} \\ \end{array} +$$ + +$$ +\left[ D \right] _ {\xi \eta \zeta} = \left[ T \right] _ {\xi \rightarrow x} ^ {T} \left[ \widetilde {D} \right] _ {x y z} \left[ T \right] _ {\xi \rightarrow x} \tag {2.55} +$$ + +이 때 $[T]_{\xi \rightarrow x}$ 는 식(2.56)과 같이 나타낼 수 있고, 각각의 성분은 식(2.57)과 같다. + +$$ +\left[ T \right] _ {\xi \rightarrow x} = \left[\begin{array}{c c c c c c c}a _ {1} a _ {1}&b _ {1} b _ {1}&c _ {1} c _ {1}&b _ {1} c _ {1}&a _ {1} c _ {1}&a _ {1} b _ {1}\\a _ {2} a _ {2}&b _ {2} b _ {2}&c _ {2} c _ {2}&b _ {2} c _ {2}&a _ {2} c _ {2}&a _ {2} b _ {2}\\a _ {3} a _ {3}&b _ {3} b _ {3}&c _ {3} c _ {3}&b _ {3} c _ {3}&a _ {3} c _ {3}&a _ {3} b _ {3}\\2 a _ {2} a _ {3}&2 b _ {2} b _ {3}&2 c _ {2} c _ {3}&b _ {2} c _ {3} + c _ {2} b _ {3}&a _ {2} c _ {3} + c _ {2} a _ {3}&a _ {2} b _ {3} + b _ {2} a _ {3}\\2 a _ {1} a _ {3}&2 b _ {1} b _ {3}&2 c _ {1} c _ {3}&b _ {1} c _ {3} + c _ {1} b _ {3}&a _ {1} c _ {3} + c _ {1} a _ {3}&a _ {1} b _ {3} + b _ {1} a _ {3}\\2 a _ {1} a _ {2}&2 b _ {1} b _ {2}&2 c _ {1} c _ {2}&b _ {1} c _ {2} + c _ {1} b _ {2}&a _ {1} c _ {2} + c _ {1} a _ {2}&a _ {1} b _ {2} + b _ {1} a _ {2}\end{array}\right] \tag {2.56} +$$ + +$$ +a _ {1} = \mathbf {r} _ {1} \cdot \mathbf {G} ^ {1} \quad b _ {1} = \mathbf {r} _ {1} \cdot \mathbf {G} ^ {2} \quad c _ {1} = \mathbf {r} _ {1} \cdot \mathbf {G} ^ {3} +$$ + +$$ +a _ {2} = \mathbf {r} _ {2} \cdot \mathbf {G} ^ {1} \quad b _ {2} = \mathbf {r} _ {2} \cdot \mathbf {G} ^ {2} \quad c _ {2} = \mathbf {r} _ {2} \cdot \mathbf {G} ^ {3} \tag {2.57} +$$ + +$$ +a _ {3} = \mathbf {r} _ {3} \cdot \mathbf {G} ^ {1} \quad b _ {3} = \mathbf {r} _ {3} \cdot \mathbf {G} ^ {2} \quad c _ {3} = \mathbf {r} _ {3} \cdot \mathbf {G} ^ {3} +$$ + +그리고 지역 직교 좌표계의 기저 $\hat{\mathbf{e}}_{1}, \hat{\mathbf{e}}_{2}, \hat{\mathbf{e}}_{3}$ 는 식(2.58)와 같이 고유 좌표계의 공변 기저(covariant basis) $\mathbf{G}_{1}, \mathbf{G}_{2}, \mathbf{G}_{3}$ 로부터 구할 수 있다. + +$$ +\hat {\mathbf {e}} _ {3} = \frac {\mathbf {G} _ {3}}{\left\| \mathbf {G} _ {3} \right\|}, \quad \hat {\mathbf {e}} _ {1} = \frac {\mathbf {G} _ {2} \times \hat {\mathbf {e}} _ {3}}{\left\| \mathbf {G} _ {2} \times \hat {\mathbf {e}} _ {3} \right\|}, \quad \hat {\mathbf {e}} _ {2} = \hat {\mathbf {e}} _ {3} \times \hat {\mathbf {e}} _ {1} \tag {2.58} +$$ + + + +# 2.2.3 Mass Matrix + +질량 행렬(mass matrix)은 물체 내에 연속적으로 분포되어 있는 물체의질량을 요소망 내 각 절점(node)에 집중 질량(lumped mass) 형식으로이산화시켜 놓은 것으로, 이 질량 행렬 내 각 행렬요소를 합하면, 물체의전체 질량과 같게 되며, 물체의 자중, 운동량, 관성력을 표현한다. + +진동 및 동적 좌굴 해석을 하기 위해서는 이러한 질량 행렬이필요하며, 일반적으로 일관 질량 행렬(consistent mass matrix)과 집중 질량행렬(lumped mass matrix)있다. 일관 질량 행렬은 식(2.59)와 같이 나타낼수 있다.[2] 여기서 는 밀도, N은 형상 함수 행렬을 나타낸다. + +$$ +\left[ \widetilde {\mathbf {M}} \right] = \int_ {V} \rho [ \mathbf {N} ] ^ {T} [ \mathbf {N} ] d V \tag {2.59} +$$ + +집중 질량 행렬의 경우 일관 질량 행렬을 대각화(diagonalization)함으로써 연산에 필요한 용량과 연산 시간을 줄일 수 있다는 장점이있지만[2] 본 논문에서는 물체의 강성을 줄임으로써 유연한 결과를 얻기위해 집중 질량 행렬을 사용하였다. 일관 질량 행렬을 대각화하는방법에는 다양한 방법들이 존재하며, 본 논문에서는 각각의 행을 합하는방법(row-sum technique)을 사용하였다.[3] 식(2.60)으로부터 각각의 요소에대한 일관 질량 행렬 M\~ 의 행의 합을 구한다. 그리고 이를 식(2.61)과같이 새로운 집중 질량 행렬 M 의 대각항(diagonal entries)에 대입하고,비대각항(off-diagonal entries)은 영(零)을 대입한다. 이 때 n은 요소의절점의 개수와 자유도의 곱을 나타낸다. + +$$ +S (i) = \sum_ {j = 1} ^ {n} \widetilde {\mathbf {M}} (i, j) \quad \text { for } i = 1, n \tag {2.60} +$$ + +$$ +\mathbf {M} (i, j) = S (i) \quad \text {for} i = 1, n \tag {2.61} +$$ + +$$ +\mathbf {M} (i, j) = 0 \quad \text { for } i \neq j +$$ + + + +# 2.2.4 6-DOF Shell Element + +일반적으로 셀 요소는 5개의 자유도(degree of freedom)를 사용하며, 법선 벡터(normal vector)는 각 절점당 하나의 법선 벡터를 갖는다. 하지만 셀 요소를 보강재(stiffener)로 사용하는 경우, 셀 요소와 셀 요소의 결합을 위해서는 6개의 자유도가 필요하며, 이와 더불어 각 절점에서 정의되는 법선 벡터에 대한 구속 조건이 필요하다. 이에 본 논문에서는 각 절점에서 정의되는 지역 좌표계(local coordinate system)를 전역 직교 좌표계(global Cartesian coordinate system)로 변환해 줌으로써 셀 요소와 셀 요소의 결합을 구성하였다. 이 때 지역 좌표계는 $V^{1}$ , $V^{2}$ , $V^{n}$ 을 기저로 하는 좌표계로 표현된다. + +전역 직교 좌표계에서 정의되는 회전 자유도 $\theta_{1}, \theta_{2}, \theta_{3}$ 와 지역 좌표계에 의해 정의되는 회전 자유도 $\alpha, \beta, \gamma$ 는 식(2.62)와 같은 관계식을 만족해야 한다. + +![](images/page-036_c0798e8e1b5c65528553a1952a1b69e5aa456e12a10541cd4829660e7ddef352.jpg) + +
+text_image + +1954 +INIA UNIVERSITY +η +ξ +γ +Vⁿ +V² +β +α +θ₁ +θ₂ +e₁ +e₂ +e₃ +θ₃ +e₃ +
+ +Fig. 5 Global Cartesian coordinate system and local coordinate system + +$$ +\theta_ {1} \mathbf {e} _ {1} + \theta_ {2} \mathbf {e} _ {2} + \theta_ {3} \mathbf {e} _ {3} = \alpha \mathbf {V} ^ {1} + \beta \mathbf {V} ^ {2} + \gamma \mathbf {V} ^ {n} \tag {2.62} +$$ + + + +이 때 식(2.62)의 좌변과 우변에 각각 벡터 $V^{1}$ 을 곱해주면, 식(2.63)과 같이 나타낼 수 있고, 마찬가지 방법으로 $V^{2}$ , $V^{n}$ 을 곱해주면, 각각 식(2.64), (2.65)와 같이 나타낼 수 있다. + +$$ +\alpha = \theta_ {1} \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {1}\right) + \theta_ {2} \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {1}\right) + \theta_ {3} \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {1}\right) \tag {2.63} +$$ + +$$ +\beta = \theta_ {1} \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {2}\right) + \theta_ {2} \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {2}\right) + \theta_ {3} \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {2}\right) \tag {2.64} +$$ + +$$ +\gamma = \theta_ {1} \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {n}\right) + \theta_ {2} \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {n}\right) + \theta_ {3} \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {n}\right) \tag {2.65} +$$ + +이를 행렬 형태로 표현해 주면 식(2.66)과 같이 쓸 수 있다. + +$$ +\left\{ \begin{array}{l} \alpha \\ \beta \\ \gamma \end{array} \right\} = \left[ \begin{array}{l} \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {1}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {1}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {1}\right) \\ \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {2}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {2}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {2}\right) \\ \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {n}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {n}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {n}\right) \end{array} \right] \left\{ \begin{array}{l} \theta_ {1} \\ \theta_ {2} \\ \theta_ {3} \end{array} \right\} \tag {2.66} +$$ + +위 식으로부터 전역 직교 좌표계에서 지역 좌표계로 변환해 주는 행렬은 식(2.67)과 같이 정의 할 수 있으며, 이 때 L 은 식(2.68)과 같다. [5] + +$$ +\left[ \widetilde {T} \right] = \left[ \begin{array}{c c c c c c} \mathbf {1} _ {3} & 0 & 0 & \dots & 0 & 0 \\ & \mathbf {L} & 0 & & & 0 \\ & & \ddots & \ddots & & \vdots \\ & & & \ddots & 0 & 0 \\ & \text {sym.} & & & \mathbf {1} _ {3} & 0 \\ & & & & & \mathbf {L} \end{array} \right] \tag {2.67} +$$ + +$$ +[ \mathbf {L} ] = \left[ \begin{array}{l} \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {1}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {1}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {1}\right) \\ \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {2}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {2}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {2}\right) \\ \left(\mathbf {e} _ {1} \cdot \mathbf {V} ^ {n}\right) \left(\mathbf {e} _ {2} \cdot \mathbf {V} ^ {n}\right) \left(\mathbf {e} _ {3} \cdot \mathbf {V} ^ {n}\right) \end{array} \right] \tag {2.68} +$$ + +만약 변위 벡터와 힘 벡터를 전역 좌표계에서 지역 좌표계로 변환해주면 각각 식(2.69), (2.70)과 같이 나타낼 수 있다. 이 때 위 첨자 + + + +e는 요소에 대해 정의 되었음을 의미한다.[4] + +$$ +\{\widetilde {\mathbf {u}} \} ^ {e} = \left[ \widetilde {T} \right] \{\mathbf {u} \} ^ {e} \tag {2.69} +$$ + +$$ +\left\{\widetilde {\mathbf {f}} \right\} ^ {e} = \left[ \widetilde {T} \right] \left\{\mathbf {f} \right\} ^ {e} \tag {2.70} +$$ + +그리고 강성 행렬(stiffness matrix)은 지역 좌표계에서 식(2.71)과 같은 선형 관계식으로 나타낼 수 있다.[4] + +$$ +\left\{\widetilde {\mathbf {f}} \right\} ^ {e} = \left[ \widetilde {\mathbf {K}} \right] ^ {e} \left\{\widetilde {\mathbf {u}} \right\} ^ {e} \tag {2.71} +$$ + +여기서 식(2.69)와 (2.70)을 식(2.71)에 대입해주면, 식(2.72)와 같이 나타낼 수 있고, 따라서 강성 행렬을 지역 좌표계에서 전역 좌표계로 변환해주는 관계식은 식(2.73)과 같다. [4] + +$$ +\left\{\mathbf {f} \right\} ^ {e} = \left[ \widetilde {T} \right] ^ {T} \left[ \widetilde {\mathbf {K}} \right] ^ {e} \left[ \widetilde {T} \right] \left\{\mathbf {u} \right\} ^ {e} \tag {2.72} +$$ + +$$ +\left[ \mathbf {K} \right] ^ {e} = \left[ \widetilde {T} \right] ^ {T} \left[ \widetilde {\mathbf {K}} \right] ^ {e} \left[ \widetilde {T} \right] \tag {2.73} +$$ + +하지만 6-자유도의 도입으로 강성 행렬 $\tilde{\mathbf{K}}^{e}$ 의 경우 여섯 번째 자유도에 해당하는 행과 열이 모두 영(雫)이 되고, 이로 인해 특이점(singularity)이 발생하게 된다. 이에 본 논문에서는 이를 방지하기 위해서 여섯 번째 자유도의 대각항(diagonal entries)에 식(2.75)와 같이 대각항 최소값의 $10^{-3}$ 비율을 갖는 값을 대입하였다. + + + +$$ +\left[ \widetilde {\mathbf {K}} \right] ^ {e} = \left[ \begin{array}{c c c c c c c c c c c c c c c c} & & & & & 0 & & & & & & & & & 0 \\ & & & & & 0 & & & & & & & & & 0 \\ & & \mathbf {K} _ {1} & & & 0 & & & & \mathbf {K} _ {2} & & & & 0 \\ & & & & & 0 & & & & & & & & 0 \\ & & & & & 0 & & & & & & & & 0 \\ 0 & 0 & 0 & 0 & 0 & d & \dots & \dots & 0 & 0 & 0 & 0 & 0 & 0 \\ & & & & & \vdots & \ddots & & & & & & & \vdots \\ & & & & & \vdots & & \ddots & & & & & & \vdots \\ & & & & & 0 & & & & & & & & 0 \\ & & & & & 0 & & & & & & & & 0 \\ & & \mathbf {K} _ {3} & & & 0 & & & & \mathbf {K} _ {4} & & & 0 \\ & & & & & 0 & & & & & & & & 0 \\ & & & & & 0 & & & & & & & & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \dots & \dots & 0 & 0 & 0 & 0 & 0 & d \end{array} \right] \tag {2.74} +$$ + +$$ +d = \min \left(\widetilde {\mathbf {K}} ^ {e} (i, i)\right) \times 1 0 ^ {- 3} \quad \text { for } i = 1, n \tag {2.75} +$$ + +그리고 6-자유도의 도입으로 회전 텐서를 구하는데 필요한 식(2.50)과 (2.51)은 식(2.76), (2.77)과 같은 형태가 된다. + +$$ +\widetilde {\theta} _ {I} = \left[ \left(\alpha_ {I}\right) ^ {2} + \left(\beta_ {I}\right) ^ {2} + \left(\gamma_ {I}\right) ^ {2} \right] ^ {\frac {1}{2}} \tag {2.76} +$$ + +$$ +\left[ \Theta_ {I} \right] = \left[ \begin{array}{c c c} 0 & - \gamma_ {I} & \beta_ {I} \\ \gamma_ {I} & 0 & - \alpha_ {I} \\ - \beta_ {I} & \alpha_ {I} & 0 \end{array} \right] \tag {2.77} +$$ + + + +# 2.3 Buckling Theory + +가느다란 기둥을 축 방향으로 누르거나 얇은 판을 판과 평행한방향으로 압축하면, 하중이 어느 크기에 도달하는 순간 갑자기 판이 횡방향으로 과도하게 휘어지는 축 방향 변위(lateral displacement)가 발생한다.물체의 이러한 거동을 좌굴 혹은 붕괴라고 정의하며 구조물의 안전성에치명적인 문제점을 야기시킨다. + +좌굴이 발생하기 전까지 물체는 정적인 평형상태를 유지하지만, 일단좌굴이 발생하면 평형상태가 깨어지고 횡 방향으로 큰 변형이 발생하여외부 하중을 더 이상 지탱할 수 없게 된다. 이러한 좌굴은 비단 가느다란기둥이나 얇은 판의 휨 좌굴(flexural buckling)에만 국한되는 것이 아니며,물체의 국부 영역에 지역적으로 발생하는 국부 좌굴(local buckling),전단력에 의하여 야기되는 전단 좌굴(shear buckling) 그리고 비틀림에의해 발생하는 비틀림 좌굴(torsion buckling) 등이 있다. + +한편 좌굴에 의한 물체의 변형이 구조물이 이루는 평면 내에 있느냐아니면 바깥에 있느냐에 따라 면내 좌굴(in-plane buckling) 그리고 면외좌굴(out of plane buckling)로 구분하기도 한다. 좌굴은 거의 대부분 물체의형상이나 하중 조건의 불완전성(imperfection)에 기인한다. 예를 들어,기둥의 단면 중심에 정확히 축 방향으로 집중 압축력을 가한다고 했을때, 이론적으로는 횡 방향으로 휨을 발생시킬 하중이나 모멘트 성분이전혀 없기 때문에 좌굴이 발생해서는 안 된다. + +하지만 실제 기둥은 정확히 원형 단면이 아닐 뿐만 아니라 압축력이작용하는 지점도 정확히 축의 중심에 위치하지 않는다. 따라서기하학적인 불완전성과 축 중심에서 어느 정도 편심된 위치에 압축력이작용함에 따른 불완전함에 따라 횡 방향으로의 변위가 발생하게 된다. + +좌굴은 물체의 가느다란 정도를 나타내는 형상 종횡비(aspect ratio)가클수록 보다 쉽게 발생한다. 다시 말해 길이가 긴 기둥이 짧은 기둥에비해 좌굴이 보다 쉽게 발생한다. 그리고 좌굴은 동일한 재질, 형상 및하중조건에서도 물체를 구속하는 경계조건(boundary condition)에 크게영향을 받는다. + +또한 좌굴은 작용하는 힘이 정적 하중이냐 동적 하중이냐에 따라 정적 diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_005.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_005.md new file mode 100644 index 0000000..16803b2 --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_005.md @@ -0,0 +1,306 @@ + + +좌굴(static buckling)과 동적 좌굴(dynamic buckling)로 나눌 수 있으며, 이절에서는 정적 좌굴과 동적 좌굴에 대해 알아보도록 하겠다. + +# 2.3.1 Static Buckling + +이처럼 좌굴은 다양한 힘과 형상, 조건 등에서 발생하며 이러한 좌굴 문제를 해석하기 위한 지배방정식은 가상일에 기반을 둔 운동 에너지(total potential energy) 식(2.78)과 위치 에너지(kinetic energy)에 Hamilton's principle을 적용하여 구할 수 있다. + +$$ +\Pi = U + V = \frac {1}{2} \mathbf {q} ^ {T} \mathbf {K} \mathbf {q} + \frac {1}{2} \mathbf {q} ^ {T} \mathbf {K} _ {g} \mathbf {q} \tag {2.78} +$$ + +$$ +T = \frac {1}{2} \dot {\mathbf {q}} ^ {T} \mathbf {M} \dot {\mathbf {q}} \tag {2.79} +$$ + +식(2.78)과 식(2.79)에 Euler-Lagrange 방정식 식(2.80)을 적용하면, 좌굴해석을 위한 지배 방정식 식(2.81)을 구할 수 있다. + +$$ +\frac {\partial L}{\partial \mathbf {q}} - \frac {d}{d t} \left(\frac {\partial L}{\partial \dot {\mathbf {q}}}\right) = 0, \quad \text { where } L = U + V - T \tag {2.80} +$$ + +$$ +\mathbf {M} \{\ddot {\mathbf {q}} \} + \left(\mathbf {K} + \mathbf {K} _ {g}\right) \{\mathbf {q} \} = \{\mathbf {0} \} \tag {2.81} +$$ + +여기서 M은 질량 행렬(mass matrix), K는 강성 행렬(stiffness matrix), $K_{g}$ 는 기하 강성 행렬(geometric stiffness matrix)를 의미한다. 지배 방정식으로부터 정적 좌굴 해석은 식(2.82)의 형태로 나타난다. + +$$ +\left(\mathbf {K} + \lambda \mathbf {K} _ {g}\right) \{\mathbf {q} \} = \{\mathbf {0} \} \tag {2.82} +$$ + +이 식은 고유치 문제(eigenvalue problem)와 유사한 형태를 가지고 있으므로 고유치 해석 솔버를 사용하여 고유치 $\lambda_{i}$ 와 고유치 + + + +벡터(eigenvector)를 구할 수 있다. 이 때 고유치 해석을 위해서 BlockLanczos 방법을 사용한 “BLZPACK”이라는 오픈 소스(open source)를활용하였다. 일반적인 고유치 해석의 경우 기하 강성 행렬 K 의 자리에질량 행렬 M을 사용하며, 질량 행렬의 특성 상 집중 행렬(lumped matrix)또는 일관 행렬(consistent matrix)을 이용하여 고유치를 계산한다. 하지만정적 좌굴 해석의 경우 기하 강성 행렬 K 는 집중 행렬 형태를 구현하기어려우므로 일관 행렬 형태로 계산을 수행해야 한다. 따라서 정적 좌굴해석의 경우 식(2.82)에 대한 고유치 해석을 통해 식(2.83)과 같이 임계좌굴 압력(critical buckling pressure) 또는 임계 좌굴 하중(critical bucklingload)를 계산하게 된다. [6] + +$$ +P _ {c r _ {i}} = \lambda_ {i} P \quad \text {or} \quad F _ {c r _ {i}} = \lambda_ {i} F \tag {2.83} +$$ + +# 2.3.2 Dynamic Buckling + +동적 좌굴 해석의 경우 정적 좌굴 해석과는 달리 식(2.84)와 같이시간에 따라 변화하는 하중이 가해진다. 여기서 P 는 정적 압축하중(static compressive loading), P 는 축 방향의 동적 하중(dynamic axialloading), 는 축 방향의 동적 하중에 대한 가진 주파수(excitationfrequency), t는 시간을 의미한다. + +$$ +P (t) = P _ {0} + P _ {t} \cos (\theta t) \tag {2.84} +$$ + +식(2.84)와 같이 시간에 따라 변화하는 하중이 가해질 때 운동방정식(equation of motion)은 지배 방정식 식(2.81)로부터 식(2.85)와 같이구할 수 있다.[6] + +$$ +\mathbf {M} \{\ddot {\mathbf {q}} \} + \left(\mathbf {K} + \mathbf {K} _ {g} ^ {(s)} + \beta \cos (\theta t) \mathbf {K} _ {g} ^ {(d)}\right) \{\mathbf {q} \} = \{\mathbf {0} \} \tag {2.85} +$$ + +여기에서 K 는 하중이 가해지지 않은 상태에서의 강성 행렬(stiffnessmatrix)이고, (s) K 는 정적 압축 하중에 대한 기하 강성 행렬(geometric + + + +stiffness matrix)이며, $\mathbf{K}_{g}^{(d)}$ 는 축 방향의 동적 하중에 대한 기하 강성 행렬이다. M 은 질량 행렬(mass matrix)이고, $\beta$ 는 동적 하중의 척도 인자(dynamic load scale factor)를 의미한다. 이 때 주요 동적 불안정 경계 영역(principal region of dynamic instability)을 찾기 위해서 식(2.86)과 같이 주기가 $2T(=2\times2\pi/\theta)$ 인 해를 가정하였다. 여기서 a 와 b 는 임의의 벡터를 의미한다.[6] + +$$ +\mathbf {q} (t) = \mathbf {a} \sin \frac {\theta t}{2} + \mathbf {b} \cos \frac {\theta t}{2} \tag {2.86} +$$ + +식(2.86)을 식(2.85)에 대입하여 푸리에 급수 전개(Fourier series expansion)를 하면 식(2.87)과 같이 나타낼 수 있다. + +$$ +\begin{array}{l} \mathbf {a} \sin \frac {\theta t}{2} \left[ - \frac {\theta^ {2}}{4} \mathbf {M} + \mathbf {K} + \mathbf {K} _ {g} ^ {(s)} + \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)} \right] \\ + \mathbf {b} \cos \frac {\theta t}{2} \left[ - \frac {\theta^ {2}}{4} \mathbf {M} + \mathbf {K} + \mathbf {K} _ {g} ^ {(s)} - \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)} \right] + \underbrace {\dots \dots} _ {\text { high order term }} = 0 \tag {2.87} \\ \end{array} +$$ + +식(2.87)에서 고차항을 무시하면 식(2.88)과 같이 쓸 수 있다. + +$$ +\mathbf {a} \sin \frac {\theta t}{2} \left[ - \frac {\theta^ {2}}{4} \mathbf {M} + \mathbf {K} + \mathbf {K} _ {g} ^ {(s)} + \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)} \right] + \mathbf {b} \cos \frac {\theta t}{2} \left[ - \frac {\theta^ {2}}{4} \mathbf {M} + \mathbf {K} + \mathbf {K} _ {g} ^ {(s)} - \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)} \right] \approx 0 \tag {2.88} +$$ + +모든 시간에 대해 식(2.88)를 만족하는 자명하지 않은 해(non-trivial solution)를 얻기 위해서는 식(2.89)와 같은 고유치 문제(eigenvalue problem)형태를 갖는다.[6] + +$$ +\left| \mathbf {K} + \left(\mathbf {K} _ {g} ^ {(s)} \pm \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)}\right) - \frac {\theta^ {2}}{4} \mathbf {M} \right| = 0 \tag {2.89} +$$ + + + +동적 좌굴 해석의 경우 정적 좌굴 해석 문제와는 달리 질량 행렬을사용하며, 본 논문에서는 집중 질량 행렬(lumped mass matrix)을 사용하여고유치를 구했으며, 이 때 고유치 $\lambda _ { i } = \theta _ { i } ^ { 2 } / 4 \ \textdegree ]$ 된다. 그리고 이로부터구조물이 불안정해지는 가진 주파수(excitation frequency) $\theta _ { i }$ 를 구할 수있으며, 동적 하중 변화에 따른 불안정 경계 영역(instability region)을 구할수 있다. + +# 2.3.3 Dynamic Buckling Theory of Beam + +![](images/page-044_fcfedf3c59a8d6fb74523b8ebe985160d32e497787348b98ce12f9c632f772eb.jpg) + +
+text_image + +P₀ + Pₜ cos(θt) +L +v(x,t) +1954 +y +x +
+ +Fig. 6 Dynamic Buckling Model of Beam + +Fig. 6과 같이 보(beam) 구조물에 축 방향으로 정적 압축하중(compressive static load)과 동적 압축 하중(compressive dynamic load)이동시에 가해질 경우 보의 동적 좌굴 현상을 이론적으로 전개할 수 있다. + +우선 보의 정적 휨(static bending)에 관한 식은 식(2.90)과 같다. [7] + +$$ +E I \frac {d ^ {2} v}{d x ^ {2}} + P v = 0 \tag {2.90} +$$ + +식(2.90)을 두 번 미분하면 식(2.91)과 같이 쓸 수 있다. + + + +$$ +E I \frac {d ^ {4} v}{d x ^ {4}} + P \frac {d ^ {2} v}{d x ^ {2}} = 0 \tag {2.91} +$$ + +이 때 E는 탄성 계수(Young's Modulus)이고, I는 면적 관성 모멘트(area moment of inertia)이며, P는 축 방향의 힘을 의미한다. 그리고 Fig. 6과 같이 정적 압축 하중과 동적 압축 하중이 동시에 가해지는 경우 힘 P는 식(2.92)와 같이 나타낼 수 있다. + +$$ +P (t) = P _ {0} + P _ {t} \cos (\theta t) \tag {2.92} +$$ + +여기서 $P_{0}$ 는 보에 가해지는 정적 압축 하중이고, $P_{t}$ 는 축 방향의 동적 하중에 대한 진폭(amplitude)를 의미하며, $\theta$ 는 축 방향의 동적 하중에 대한 가진 주파수(excitation frequency)를 의미한다. 이 때 동적 하중의 주기 T는 $2\pi/\theta$ 이다. + +횡 방향 관성력만을 고려하여 운동 방정식을 구성하면, 식(2.91)은 식(2.93)과 같이 표현할 수 있다. + +$$ +E I \frac {\partial^ {4} v (x , t)}{\partial x ^ {4}} + \left(P _ {0} + P _ {t} \cos (\theta t)\right) \frac {\partial^ {2} v (x , t)}{\partial x ^ {2}} + \rho \frac {\partial^ {2} v (x , t)}{\partial t ^ {2}} = 0 \tag {2.93} +$$ + +식(2.93)에서 $v(x,t)$ 는 시간에 따른 처짐을 의미하며, $\rho$ 는 단위 길이당 밀도를 의미한다. 위의 식(2.93)이 식(2.94)와 같은 형태의 해를 갖는다면 변수 분리(separation of variable)를 통해 식(2.93)은 식(2.95)와 같이 표현할 수 있다. 이 때 식(2.94)는 경계 조건(boundary condition)을 만족한다. + +$$ +v (x, t) = \sum_ {k = 1} ^ {\infty} f _ {k} (t) \sin \frac {k \pi x}{L} \tag {2.94} +$$ + +$$ +\sum_ {k = 1} ^ {\infty} \left[ E I \frac {k ^ {4} \pi^ {4}}{L ^ {4}} f _ {k} (t) - \left(P _ {0} + P _ {t} \cos (\theta t)\right) \frac {k ^ {2} \pi^ {2}}{L ^ {2}} f _ {k} (t) + \rho \frac {d ^ {2} f _ {k} (t)}{d t ^ {2}} \right] \sin \frac {k \pi x}{L} = 0 \tag {2.95} +$$ + + + +식(2.95)를 만족하기 위해서는 모든 k에 대해 sin 함수의 계수가 영(雫)이 되어야 하며, 따라서 모든 k에 대해 식(2.96)이 만족해야 한다. + +$$ +E I \frac {k ^ {4} \pi^ {4}}{L ^ {4}} f _ {k} (t) - \left(P _ {0} + P _ {t} \cos (\theta t)\right) \frac {k ^ {2} \pi^ {2}}{L ^ {2}} f _ {k} (t) + \rho \frac {d ^ {2} f _ {k} (t)}{d t ^ {2}} = 0 \quad (k = 1, 2, 3, \dots) \tag {2.96} +$$ + +그리고 이를 자유진동 상태에서의 고유 주파수(natural frequency) $\omega_{k}$ 와 오일러 보(Euler beam)의 임계 좌굴 하중(critical buckling load) $P_{k}^{cr}$ 을 각각 식(2.98), (2.99)와 같이 정의하고 이를 사용하여 식(2.96)를 다시 쓰면 식(2.97)과 같이 나타낼 수 있다. + +$$ +\frac {d ^ {2} f _ {k} (t)}{d t ^ {2}} + \omega_ {k} ^ {2} \left(1 - \frac {P _ {0} + P _ {t} \cos (\theta t)}{P _ {k} ^ {c r}}\right) f _ {k} (t) = 0 \quad (k = 1, 2, 3, \dots) \tag {2.97} +$$ + +$$ +\omega_ {k} = \frac {k ^ {2} \pi^ {2}}{L ^ {2}} \sqrt {\frac {E I}{\rho}} \tag {2.98} +$$ + +$$ +P _ {k} ^ {c r} = \frac {k ^ {2} \pi^ {2}}{L ^ {2}} E I \tag {2.99} +$$ + +또한 일정한 압축 하중 $P_{0}$ 가 가해지는 보의 고유 주파수 $\Omega_{k}$ 와 가진 매개변수(excitation parameter) $\mu_{k}$ 를 식(2.101), (2.102)와 같이 정의하고, 이를 식(2.97)에 적용하면 식(2.100)과 같이 쓸 수 있다. + +$$ +\frac {d ^ {2} f _ {k} (t)}{d t ^ {2}} + \Omega_ {k} ^ {2} \left(1 - 2 \mu_ {k} \cos (\theta t)\right) f _ {k} (t) = 0 \quad (k = 1, 2, 3, \dots) \tag {2.100} +$$ + +$$ +\Omega_ {k} ^ {2} \equiv \omega_ {k} ^ {2} \frac {\left(P _ {k} ^ {c r} - P _ {0}\right)}{P _ {k} ^ {c r}} \tag {2.101} +$$ + +$$ +\mu_ {k} \equiv \frac {P _ {t}}{2 \left(P _ {k} ^ {c r} - P _ {0}\right)} \tag {2.102} +$$ + + + +그리고 여기서 k=1일 때의 기본모드(fundamental mode)만 고려하면 식(2.100)은 식(2.103)과 같이 나타낼 수 있고, 마찬가지로 식(2.101)과 식(2.102)도 식(2.104)와 식(2.105)로 나타낼 수 있다. + +$$ +\frac {d ^ {2} f (t)}{d t ^ {2}} + \Omega^ {2} \big (1 - 2 \mu \cos (\theta t) \big) f (t) = 0 \tag {2.103} +$$ + +$$ +\Omega^ {2} = \left(\frac {\pi^ {2}}{L ^ {2}} \sqrt {\frac {E I}{\rho}}\right) ^ {2} \left[ 1 - P _ {0} \frac {L ^ {2}}{\pi^ {2} E I} \right] \tag {2.104} +$$ + +$$ +\mu = \frac {P _ {t}}{2 \left[ \left(\pi^ {2} E I\right) / L ^ {2} - P _ {0} \right]} \tag {2.105} +$$ + +이 때 식(2.103)은 식(2.106)과 같은 Mathieu-Hill 방정식의 형태를 갖게 된다. [8] + +$$ +f ^ {\prime \prime} + \Omega^ {2} [ 1 - 2 \mu \Phi (t) ] f = 0 \tag {2.106} +$$ + +여기서 $\Phi(t)$ 는 가진 주기가 $T=2\pi/\theta$ 인 임의의 주기 함수이고, 다음과 같이 매개변수의 변화에 따라 해가 안정하거나 불안정해질 수 있다. + +⇒ 해의 주기가 T(또는 2T)에서 주기 2T(또는 T)로 바 Parks 경우 : 안정 +⇒ 해의 주기가 T(또는 2T)에서 주기 T(또는 2T)로 바 Parks 경우 : 불안정 + +즉 주기 2T, T인 해가 존재하므로 이를 푸리에 급수(Fourier series)로 가정하고, 이를 통해 매개변수에 따른 해의 안정성을 판별할 수 있다. + +불안정 영역(instability region)을 구하기 위해 주기가 $2T(=2\times2\pi/\theta)$ 인해를 식(2.107)과 같이 정의하면 다음과 같다. + +$$ +f (t) = \sum_ {k = 1, 3, 5, \dots} ^ {\infty} \left(a _ {k} \sin \frac {k \theta t}{2} + b _ {k} \cos \frac {k \theta t}{2}\right) \tag {2.107} +$$ + + + +이 중에서 k=1인 경우만 고려하면, 식(2.108)과 같다. + +$$ +f (t) = a \sin \frac {\theta t}{2} + b \cos \frac {\theta t}{2} \tag {2.108} +$$ + +이를 식(2.103)에 대입하면 식(2.109)와 같으며, 이 때 주기 2T에 대해 푸리에 급수 전개를 하면 식(2.110)과 같이 나타낼 수 있다. + +$$ +\begin{array}{l} - \frac {\theta^ {2}}{4} a \sin \frac {\theta t}{2} - \frac {\theta^ {2}}{4} b \cos \frac {\theta t}{2} \tag {2.109} \\ + \Omega^ {2} (1 - 2 \mu \cos (\theta t)) a \sin \frac {\theta t}{2} + \Omega^ {2} (1 - 2 \mu \cos (\theta t)) b \cos \frac {\theta t}{2} = 0 \\ \end{array} +$$ + +$$ +a \sin \frac {\theta t}{2} \left(- \frac {\theta^ {2}}{4} + \Omega^ {2} + \mu \Omega^ {2}\right) + b \cos \frac {\theta t}{2} \left(- \frac {\theta^ {2}}{4} + \Omega^ {2} - \mu \Omega^ {2}\right) + \underbrace {\dots \dots} _ {\text { high order term }} = 0 \tag {2.110} +$$ + +그리고 식(2.110)에서 주기가 2T인 항을 취하고 나머지 고차항을 무시하면, 식(2.111)과 같이 나타낼 수 있고, 이 때 모든 시간에 대해 식(2.111)을 만족하는 자명하지 않은 해(non-trivial solution)을 얻기 위해서는 식(2.112)를 만족해야 한다. + +$$ +a \sin \frac {\theta t}{2} \left(- \frac {\theta^ {2}}{4} + \Omega^ {2} + \mu \Omega^ {2}\right) + b \cos \frac {\theta t}{2} \left(- \frac {\theta^ {2}}{4} + \Omega^ {2} - \mu \Omega^ {2}\right) \approx 0 \tag {2.111} +$$ + +$$ +\Omega^ {2} \left[ 1 \pm \mu - \frac {1}{4} \left(\frac {\theta}{\Omega}\right) ^ {2} \right] = 0 \tag {2.112} +$$ + +이는 불안정 영역을 결정짓는 매개변수와 가진 진동수의 근사화된식으로 식(2.113)과 같이 무차원 가진 진동수 $\theta/(2\Omega)$ 를 매개변수 $\mu$ 에대한 양함수의 형태로 표현할 수 있으며, 이를 도시하여 가진 매개 변수 $\mu$ 에 따른 불안정 영역의 변화를 살펴 볼 수 있다. + +$$ +\frac {\theta}{2 \Omega} = \sqrt {1 \pm \mu} \tag {2.113} +$$ + + + +# 3. Numerical Example + +본 연구에서 개발한 동적 좌굴 유한요소해석 프로그램을 이용하여동적 좌굴 해석을 하기 위한 기본적인 해석을 수행하였다. 기본적으로선형 정적 해석, 비선형 정적 해석 그리고 정적 좌굴 해석을수행하였으며, 최종적으로 본 연구의 목적인 동적 좌굴 해석을 수행하여이론값 또는 실험값과의 비교를 통해 동적 좌굴 유한요소해석프로그램의 타당성과 신뢰성을 검증하였다. + +# 3.1 Linear Static Analysis + +# 3.1.1 Patch Test + +![](images/page-049_c8c3db0a29650c52d571b8844dc5f45ba045cac37affa5f0165668ab6b54b8d8.jpg) + +
+scatter + +| Point | X1 | X2 | Label | +|---|---|---|---| +| 1 | 0,0 | 0 | 1 (0,0) | +| 2 | 2,2 | 0 | 2 (2,2) | +| 3 | 10,0 | 0 | 3 (10,0) | +| 4 | 8,3 | 0 | 4 (8,3) | +| 6 | 4,7 | 0 | 6 (4,7) | +| 7 | 10,10 | 0 | 7 (10,10) | +| 8 | 8,7 | 0 | 8 (8,7) | +The label 'INHE UNIVERSITY' is not present in the image. The data points are explicitly labeled with numbers and coordinates. +
+ +Fig. 7 Patch Test Mesh + +패치 테스트 모델(patch test model)은 Fig. 1과 같다. 이 모델에 대해서constant curvature, constant shear, constant twist 패치 테스트를 수행하였으며,재료의 탄성계수(Young‟s modulus)는 2.1E+06이고, 푸아송의 비(Poisson‟sratio)는 0.3이다. 그리고 모델의 두께는 1.0과 0.001 두 모델에 대해 패치테스트를 수행하였다. + + + +# 3.1.1.1 Constant Curvature Patch Test + +![](images/page-050_c08b014c3f927023e595e9c9500d7e043903c2a949d1e6daf8b2586c46c6e47b.jpg) + +
+text_image + +U_{1-2-3} = 0 +β = 0 +BENDING +U_{1-3} = 0 +β = 0 +M +M +
+ +Fig. 8 Constant Curvature Patch Test Model + +Constant curvature 패치 테스트 모델에 대한 경계조건은 우선 1번절점에 대해 1,3 방향의 변위와 2 방향의 회전을 구속하였고, 5번 절점에대해 1,2,3 방향의 변위와 2 방향의 회전을 구속하였다. 또한 외력은 3번,7번 절점에 2 방향의 모멘트를 가하였으며, 이 때 힘의 크기는 두께1.0일 때 M=1000, 두께 0.001일 때 M=0.0001을 가하였다. 이러한 경계조건과 외력은 Fig. 8로부터 확인할 수 있으며, constant curvature 패치테스트 해석 결과 각 절점의 곡률는 식(3.1)과 같이 구할 수 있다. + +$$ +\frac {1}{\rho} = \frac {d \theta}{d x} = \frac {d ^ {2} w}{d x ^ {2}} \tag {3.1} +$$ diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_006.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_006.md new file mode 100644 index 0000000..79cef1a --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_006.md @@ -0,0 +1,268 @@ + + +
두께NODEx-coord. $\beta$ $\rho$
1.022.00.00114291750
310.00.00571431750
48.00.00457141750
64.00.00228571750
710.00.00571431750
88.00.00457141750
0.00122.00.11428617.50
310.00.57142917.50
48.00.45714317.50
64.00.22857117.50
710.00.57142917.50
88.00.45714317.50
+ +Table. 1 Constant Curvature Patch Test Results + +Table. 1은 constant curvature 패치 테스트 결과로 두 가지 두께에 대해구속조건이 적용된 절점을 제외한 나머지 절점에서 모두 일정한 곡률이나오는 것을 볼 수 있다. + +# 3.1.1.2 Constant Shear Patch Test + +![](images/page-051_7437aa69d3ca380949c5c6675030071df961bd6b511226d8ab2c4c76f6bd67c0.jpg) + +
+text_image + +U₃ = 0 +SHEAR +U₁₋₂ = 0 +β = 0 +U₃ = 0 +Q +
+ +Fig. 9 Constant Shear Patch Test Model + + + +Constant shear 패치 테스트 모델에 대한 경계조건은 우선 1번과 5번절점에 대해 3 방향의 변위를 구속하였고, 모든 절점에 대해 1,2 방향의변위와 2 방향의 회전을 구속하였다. 또한 외력은 3번, 7번 절점에 3방향의 집중 하중을 가하였으며, 이 때 힘의 크기는 두께 1.0일 때Q=2000, 두께 0.001일 때 Q=2.0을 가하였다. 이러한 경계 조건과 외력은Fig. 9로부터 확인할 수 있으며, constant shear 패치 테스트 해석 결과는Table. 2와 같다. + +
두께NODEx-coord.z-disp. (w)dw/dx
1.022.00.0004952382.47619E-04
310.00.002476192.47619E-04
48.00.001980952.47619E-04
64.00.0009904762.47619E-04
710.00.002476192.47619E-04
88.00.001980952.47619E-04
0.00122.00.0004952382.47619E-04
310.00.002476192.47619E-04
48.00.001980952.47619E-04
54.00.0009904762.47619E-04
610.00.002476192.47619E-04
710.00.002476192.47619E-04
88.00.001980952.47619E-04
+ +Table. 2 Constant Shear Patch Test Results + +Constant shear 패치 테스트 결과로 두 가지 두께에 대해 구속조건이적용된 절점을 제외한 나머지 절점에서 모두 일정한 기울기가 나타나는것을 확인할 수 있으며, 실제로 3 방향 변위는 식(3.2)를 사용하여 계산할수 있다. + +$$ +\tau = \frac {V}{A} = G \gamma , \quad \gamma = \frac {w}{L}, \quad G = \frac {E}{2 (1 + \nu)} \quad \Rightarrow \quad w = \frac {2 (1 + \nu) V L}{E A} \tag {3.2} +$$ + + + +# 3.1.1.3 Constant Twist Patch Test + +![](images/page-053_a67d8b522896c8b34a1424a628f080f8afa42b9ab5ecb2f46d336c9a45f1941b.jpg) + +
+text_image + +U₃ = 0 +P +TWISTING +U₁₋₂ = 0 +U₃ = 0 +U₃ = 0 +
+ +Fig. 10 Constant Twist Patch Test Model + +Constant twist 패치 테스트 모델에 대한 경계조건은 우선 1, 3, 5번절점에 대해 3 방향의 변위를 구속하였고, 모든 절점에 대해 1,2 방향의변위를 구속하였다. 또한 외력은 7번 절점에 3 방향의 집중 하중을가하였으며, 이 때 힘의 크기는 두께 1.0일 때 P=1000, 두께 0.001일 때P=1.0E-06을 가하였다. 이러한 경계 조건과 외력은 Fig. 10으로부터확인할 수 있으며, constant twist 패치 테스트 해석 결과는 Table. 3과 같다. + +
두께NODEx-coord. $\alpha$ $\rho_1$ y-coord. $\beta$ $\rho_2$
1.022.00.00751761266.042.00.0075345265.45
48.00.03103950257.743.00.0114079262.98
64.00.01518240263.467.00.0272324257.05
710.00.03767130265.4510.00.0379748263.33
88.00.03104290257.717.00.0272190257.17
0.00122.00.00742857269.232.00.00742857269.23
48.00.02971430269.233.00.0111429269.23
64.00.01485710269.237.00.0260000269.23
710.00.03714290269.2310.00.0371429269.23
88.00.02971430269.237.00.0260000269.23
+ +Table. 3 Constant Twist Patch Test Results + + + +Constant twisting 테스트 결과를 보면 두께가 0.001일 때 x와 y방향에대해 일정한 곡률이 나오는 것을 볼 수 있지만 두께가 1.0일 때는 그렇지않음을 알 수 있다. 이 문제는 두께가 두꺼울 때 횡 전단 변형(transverseshear deformation)이 지배적으로 나타나기 때문에 발생하는 것으로 전단보정 계수(shear correction factor)를 사용하여 전단 변형(shear deformation)을억제하거나 사각형 요소를 사용함으로써 이 문제를 해결할 수 있다.[1] + +# 3.1.2 Pinched Cylinder + +![](images/page-054_c96037aeca2cdff42c52ed06354ae6fb9cb2290b53cbd2d037e28e0795343bc5.jpg) + +
+text_image + +P +D +C +R +A +B +End +diaphragm +End +diaphragm +P +
+ +Fig. 11 Pinched Cylinder Model + +![](images/page-054_fa626a0d336adc88792ee5552a20af6c950d37f0824274263bbbc09fdb2948fd.jpg) + +
+text_image + +P/4 +Sym. +Sym. +BC(12) +Sym. +y +z +x +
+ +Fig. 12 Pinched Cylinder 1/8 Model + +Fig. 11과 같이 양 끝에 막이 있는 pinched cylinder 쉘은 이론적인 해를비교할 수 있기 때문에 쉘 검증 문제에 적합하다. 이 때 하중은중간면에서 서로 반대 방향으로 집중 하중 P가 가해지고 있으며, 형상 및하중이 대칭이기 때문에 Fig. 12와 같이 1/8 모델을 사용하였다. + +Pinched cylinder 쉘의 형상은 길이(L) 600, 반경(R) 300, 두께(t) 3이고,재료 특성은 탄성 계수(Young‟s modulus) 3.0E+06, 푸아송 비(Poisson‟s ratio)0.3이며, 외력 P의 크기는 1이다. 격자(mesh)는 20x20, 30x30, 40x40 격자를사용하였으며, 경계 조건은 절단면은 각각 대칭 경계 조건을적용하였으며, pinched cylinder 쉘의 end diaphragm 부분은 X와 Y 방향의변위를 구속하였다. + +해석 결과 Table. 4 로부터 ABAQUS와 현재 쉘 모두 이론적인 해와유사하게 나옴을 확인할 수 있으며, ABAQUS결과가 보다 이론적인 해에 + + + +가까움을 확인할 수 있다. 이는 쉘 요소가 다르기 때문에 발생하는차이로 ABAQUS 쉘 요소가 현재 쉘 요소에 비해 보다 유연함을 알 수있다. 그리고 Fig. 13으로부터 요소의 수가 증가함에 따라 ABAQUS와현재 쉘의 결과가 모두 이론적인 해에 수렴함을 알 수 있다. 또한 Fig.14는 Y방향의 변위에 대해 해석 결과를 도시한 그림으로 변형 형상이유사하게 나옴을 확인할 수 있다. + +
MeshExactPresent $W_{present}$ / $W_{exact}$ ABAQUS $W_{abaqus}$ / $W_{exact}$
20x201.8248E-051.74362E-050.95551.77866E-050.9747
30x301.79593E-050.98421.81676E-050.9956
40x401.81878E-050.99671.82150E-050.9982
+ +Table. 4 Comparison of Linear Static Analysis for Pinched Cylinder with Exact Solution + +![](images/page-055_7a7972ecc99093be92b2351247263cc4f518f1ae1b6313d57820e26c69c9e3fe.jpg) + +
+line + +| Number of elements per side | Present | ABAQUS | +| --------------------------- | ------- | ------ | +| 20 | 0.955 | 0.975 | +| 30 | 0.985 | 0.995 | +| 40 | 0.998 | 0.999 | +
+ +Fig. 13 Comparison of Convergence for Pinched Cylinder with ABAQUS + + + +![](images/page-056_4d3e19a9e3285506c0fff929f8e03b5a403f1b6d2c9df512530e9b5a2a71b54f.jpg) +Fig. 14 Comparison of Linear Static Analysis for Pinched Cylinder with ABAQUS + +# 3.1.3 Hemispherical Shell + +![](images/page-056_df796ac2e35030758d83f8ec769d60e13478da3be1305dc7b7ccd299cddd12d9.jpg) + +
+text_image + +Sym. +Sym. +P +P +x +y +z +
+ +Fig. 15 Hemispherical Shell Model + +Fig. 15와 같은 반구형(hemispherical) 쉘의 1/4 모델을 사용하여이론적인 해와 ABAQUS 결과값에 대해 비교해 보았다. 먼저 반경은 10m,두께는 0.04m이고, 요소는 한 면당 9개, 17개 격자(mesh)를 사용하였다.탄성계수는 68.25MPa이고, 푸아송 비는 0.3의 물성치를 주었다.경계조건은 좌우 면에 대해 대칭 경계조건을 적용하였고, 하중은 Fig.15와 같은 지점에 (+)Z, (-)X방향으로 각각 1씩 가하였다. 해석 결과 Table. + + + +5로부터 ABAQUS는 이론값보다 큰 값이 나오고, 현재 코드 결과는이론값보다 작은 값이 나옴을 알 수 있는데 이는 사용한 쉘 요소가다르므로 요소의 특성에 따른 차이로 볼 수 있다. 하지만 Fig. 16로부터두 결과값 모두 요소 수가 증가함에 따라 이론값에 수렴함을 확인할 수있으며, Fig. 17은 Y방향 변위에 대해 ABAQUS와 현재 코드의 해석결과를 직접 도시한 그림으로 서로 유사한 결과가 나옴을 알 수 있다. + +
Node/sideExactPresent $W_{present}$ / $W_{exact}$ ABAQUS $W_{abaqus}$ / $W_{exact}$
90.09240.08884390.96150.09411531.0186
170.09190910.99470.09330831.0098
250.09213860.99720.09287991.0052
+ +Table. 5 Comparison of Linear Static Analysis for Hemispherical Shell with Exact Solution + +![](images/page-057_e01f38ae80ca9e3efac4c4342ef66ba8fade9195c30f63268506c341a2f29cf4.jpg) + +
+line + +| Number of nodes per side | Present | ABAQUS | +| ------------------------ | ------- | ------ | +| 9 | 0.96 | 1.02 | +| 17 | 0.995 | 1.01 | +| 25 | 0.998 | 1.005 | +
+ +Fig. 16 Comparison of Convergence for Hemispherical Shell with ABAQUS + + + +![](images/page-058_d94a9a81dc807b91bb65c57b1b5b57fce656f369bfbbb0c11b04f19c86c4a5c4.jpg) +Fig. 17 Comparison of Linear Static Analysis for Hemisphrical Shell with ABAQUS + +# 3.2 Geometric Nonlinear Analysis + +![](images/page-058_121dd25446c976f330becfc0905b54b3ea6b25cad4f75d4d3791fc39e6c34945.jpg) + +
+text_image + +BC(Fixed) +y +x +Moment +
+ +Fig. 18 Beam Model for Geometric Nonlinear Analysis + +Fig. 18과 같은 보(beam) 형상에 대해 기하비선형 정적 해석(geometricnonlinear static analysis)을 수행하여 ABAQUS와 그 결과를 비교해 보았다.먼저 가로 12m, 세로 1m, 두께 0.01m 이고, 요소는 총 12개를사용하였다. 탄성계수는 1.0MPa이고, 푸아송 비는 0.0의 물성치를 주었다.경계조건은 한 면은 XYZ방향의 변위와 회전을 모두 구속하였고, 다른 한면은 Y방향으로 모멘트를 가하였다. 그리고 물체가 Y방향에 대해 회전할때 X방향의 회전으로 인한 형상의 뒤틀림을 방지하기 위해 모든 절점에 + + + +대해 X방향 회전을 구속하였다. 해석 결과 Fig. 19로부터 ABAQUS 해석결과와 유사한 결과가 나옴을 확인 할 수 있으며, 이 때 힘의 증가에따른 형상 변화는 Fig. 20과 같다. + +![](images/page-059_3df9745334afc5e1729f42e8dac6b1abdee78b0d767ed5d6877e162da4f871fe.jpg) + +
+line + +| Displacement & Rotation | Present_Disp.X | Present_Disp.Z | Present_Rota.Y | Abaqus_Disp.X | Abaqus_Disp.Z | Abaqus_Rota.Y | +| ----------------------- | -------------- | -------------- | -------------- | ------------- | ------------- | ------------- | +| 0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | +| 2 | 0.0070 | 0.0020 | 0.0100 | 0.0080 | 0.0040 | 0.0060 | +| 4 | 0.0100 | 0.0050 | 0.0200 | 0.0120 | 0.0080 | 0.0140 | +| 6 | 0.0150 | 0.0080 | 0.0300 | 0.0160 | 0.0120 | 0.0220 | +| 8 | 0.0180 | 0.0120 | 0.0350 | 0.0200 | 0.0160 | 0.0280 | +| 10 | 0.0200 | 0.0160 | 0.0380 | 0.0240 | 0.0200 | 0.0320 | +| 12 | 0.0220 | 0.0200 | 0.0400 | 0.0280 | 0.0240 | 0.0360 | +| 14 | 0.0250 | 0.0240 | 0.0420 | 0.0320 | 0.0280 | 0.0400 | +| 16 | 0.0280 | 0.0280 | 0.0440 | 0.0360 | 0.0320 | 0.0440 | +
+ +Fig. 19 Comparison of Geometric Nonlinear Analysis for Beam with ABAQUS + +![](images/page-059_4b7590bcc293d5f6eb41f747a43988e7a1b5d0087b90d0a8972ec210e6158ab2.jpg) + +
+line + +| M | Value | +|-------|-------| +| 0.04 | 1.0 | +| 0.03 | 0.8 | +| 0.02 | 0.6 | +| 0.01 | 0.4 | +
+ +Fig. 20 Geometry Change of Beam According to Loads Increase + + + +# 3.3 Static Buckling Analysis + +# 3.3.1 Rectangular Plate Shell + +정적 좌굴 해석 프로그램의 검증을 위해 먼저 Fig. 21과 같은 직사각형평판 쉘(rectangular plate shell) 형상에 대해 정적 좌굴 해석을 수행하였다.크기는 가로 20m, 세로 8m이고, 두께는 0.01m로 4노드 쉘 40x16 격자로모델링 하였으며, 재료의 물성치는 탄성계수 29MPa, 푸아송 비는 0.3을적용하였다. 경계 조건은 한 면은 XYZ방향의 변위를 구속하였고, 반대쪽면은 YZ방향의 변위를 구속하였다. 그리고 나머지 두 면은 Z방향의변위를 구속하였다. 마지막으로 하중은 YZ방향의 변위를 구속한 면에X방향으로 총 8N의 압축력을 가하였다. + +![](images/page-060_15245d58568e3fbbd37222b6a830741c514df6b83cabeb987fcb107945a81103.jpg) + +
+text_image + +BC(23) +F +BC(3) +BC(3) +BC(123) +Z +X Y +
+ +Fig. 21 Rectangular Plate Shell Model + +해석 결과 나오는 고유치(eigenvalue) $\lambda _ { i }$ 는 Table. 6과 같고, ABAQUS와유사한 결과가 나옴을 확인할 수 있다. 또한 고유치 벡터(eigenvector)로부터 확인할 수 있는 좌굴 형상 역시 ABAQUS와 유사함을 확인할 수있다.(Table. 7) diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_007.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_007.md new file mode 100644 index 0000000..bbd3a30 --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_007.md @@ -0,0 +1,178 @@ + + +
MODEPresentABAQUS
11.671891.6736
21.699821.7033
32.018932.0198
42.570522.5706
53.300943.3003
63.464713.4751
74.20564.2041
85.290425.2879
96.478776.5154
106.568396.5648
+ +Table. 6 Comparison of Eigenvalue for Rectangular Plate Shell with ABAQUS + +
MODEPresentABAQUS
1
2
6
+ +Table. 7 Comparison of Mode Shape for Rectangular Plate Shell with ABAQUS + + + +# 3.3.2 Cylindrical Shell + +정적 좌굴 해석 프로그램의 검증을 위해 Fig. 22과 같은 원통형쉘(cylindrical shell) 형상에 대해 정적 좌굴 해석을 수행하였다. 크기는직경 0.3m, 길이 2m이고, 두께는 0.005m로 4노드 쉘 40x80 격자로모델링 하였으며, 재료의 물성치는 탄성계수 71GPa, 푸아송 비는 0.3을적용하였다. 경계 조건은 한 면은 XYZ방향의 변위를 구속하였고, 반대쪽면은 XY방향의 변위를 구속하였으며, 하중은 YZ방향의 변위를 구속한면에 X방향으로 총 40000N의 압축력을 가하였다. + +![](images/page-062_f418696d943b46f77842d561a9092117be9b0e90f14d1ed581f2cf7b7cab2d24.jpg) + +
+text_image + +BC(12) +BC(123) +P +Y +z +X +UNIVERSITY +
+ +Fig. 22 Cylindrical Shell Model + +해석 결과 고유치는 Table. 8과 같으며, ABAQUS 해석 결과와 유사하게나옴을 확인할 수 있다. 또한 고유치 벡터(eigenvector)로부터 확인할 수있는 좌굴 형상 역시 ABAQUS와 유사함을 확인할 수 있다.(Table. 10) + +이 때 정적 좌굴 해석으로부터 나온 고유치를 식(2.83)에 대입하면,Table. 9와 같은 임계 좌굴 압력(critical buckling pressure)을 구할 수 있으며,이론값[6]과 비교했을 때 유사한 결과가 나옴을 알 수 있고 더불어ABAQUS 해석 결과보다 오차가 적음을 알 수 있다. + + + +
MODEPresentABAQUS
11204.051242.9
31418.571442.7
51502.831534.4
71564.761626.3
91616.481641.0
+ +Table. 8 Comparison of Eigenvalue for Cylindrical Shell with ABAQUS + +
MODE $P_{cr} \times 10^{9} [N/m^{2}]$ (Present) $P_{cr} \times 10^{9} [N/m^{2}]$ (Analytical) $P_{cr} \times 10^{9} [N/m^{2}]$ (ABAQUS)
11.02200.99261.0550
31.20411.16411.2246
51.27561.17221.3024
71.32821.26021.3804
91.37211.29931.3929
+ +Table. 9 Comparison of Critical Buckling Pressure for Cylindrical Shell with Analytic Solution + + + +
MODEPresentABAQUS
1
3
5
+ +Table. 10 Comparison of Mode Shape for Cylindrical Shell with ABAQUS + + + +# 3.3.3 Stiffened Square Plate Shell + +정적 좌굴 해석 프로그램의 검증을 위해 Fig. 23과 같은 보강된 정사각형평판 쉘(stiffened square plate shell) 형상에 대해 정적 좌굴 해석을수행하였다. 평판은 가로 10m, 세로 10m, 두께 0.01m이고,보강재(stiffener)는 가로 10m, 세로 1m, 두께 0.01m이며, 4노드 쉘 4000개요소로 모델링 하였다. 재료의 물성치는 탄성계수 71GPa, 푸아송 비0.3을 적용하였다. 경계 조건은 한 면은 XYZ방향의 변위를 구속하였고,반대쪽 면은 XZ방향의 변위를 구속하였으며, 하중은 XZ방향의 변위를구속한 면에 Y방향으로 총 80N의 압축력을 가하였다. + +![](images/page-065_c204f528b88a134059878e7a786ee71cf74cf9f886c027f70386bfbd7e123f9c.jpg) + +
+text_image + +BC(123) +BC(13) +F +z y +x +
+ +Fig. 23 Stiffened Square Plate Shell Model + +해석 결과 고유치는 Table. 11과 같으며, ABAQUS 해석 결과와유사하게 나옴을 확인할 수 있다. 또한 고유치 벡터(eigenvector)로부터확인할 수 있는 좌굴 형상 역시 ABAQUS와 유사함을 확인할 수 있다.(Table. 12) + + + +
MODEPresentABAQUS
19640.019694.7
29908.39965.8
310106.510167.
41042410483.
51047610531.
610894.610970.
711054.211118.
81125011307.
911357.811400.
1011615.111657.
+ +Table. 11 Comparison of Eigenvalue for Stiffened Square Plate Shell with ABAQUS + +
MODEPresentABAQUS
1
2
4
+ +Table. 12 Comparison of Mode Shape for Stiffened Square Plate Shell with ABAQUS + + + +# 3.4 Dynamic Buckling Analysis + +# 3.4.1 Dynamic Buckling Analysis of Beam + +![](images/page-067_309b61bf87ec2ca57d19c19ad6e0ca799da003615208708f893792f0128cc533.jpg) + +
+text_image + +BC(123) +BC(23) +P(t) +Y +Z +Y +X +
+ +Fig. 24 Dynamic Buckling Analysis Model for Beam + +동적 좌굴 해석 프로그램의 검증을 위해 먼저 Fig. 24와 같은 보(beam)형상에 대한 동적 좌굴 이론값과 유한요소해석 결과를 비교해 보았다.보의 크기는 먼저 가로 800mm, 세로 30mm, 두께 3mm 이고, 요소는4노드 쉘 요소 160x6격자를 사용하였다. 탄성계수(Young‟s modulus)는68.9GPa이고, 프아송 비(Poisson‟s ration)는 0.33, 밀도는 2700kg/m3의물성치를 주었다. 경계조건은 왼쪽 면은 XYZ방향의 변위를 구속하였고,오른쪽 면은 YZ방향의 변위를 구속하였다. 하중은 식 (3.3)와 같은시간에 따른 하중을 가하였으며, 이 때 동적 좌굴 해석을 위한 방정식은식 (3.4)와 같다. 이 때 $P _ { 0 } \frac { 2 } { 2 }$ 정적 하중, P는 동적 하중의 크기, 는 가진주파수(excitation frequency), 는 동적 매개변수(Dynamic parameter), K는강성 행렬(stiffness matrix), ${ \bf K } _ { g } ^ { ( s ) }$ 는 정적 하중에 대한 기하 강성행렬(geometric stiffness matrix), $\mathbf { K } _ { g } ^ { ( d ) }$ 는 동적 하중에 대한 기하 강성 행렬,M은 질량 행렬(Mass matrix)을 의미하며, 고유치 해석(eigenvalue analysis)을통해 가진 주파수를 구할 수 있다. + +$$ +P (t) = P _ {0} + P _ {t} \beta \cos (\theta t) \tag {3.3} +$$ + +$$ +\left| \mathbf {K} + \mathbf {K} _ {g} ^ {(s)} \pm \frac {\beta}{2} \mathbf {K} _ {g} ^ {(d)} - \frac {\theta^ {2}}{4} \mathbf {M} \right| = 0 \tag {3.4} +$$ + + + +Fig. 24와 같은 보 모델에 대해 유한요소해석을 이용한 고유치 해석결과 고유진동수(natural frequency)는 10.7Hz이고, 임계좌굴하중(criticalbuckling loads) 71.73N으로 계산되었다. 그리고 동적 좌굴 해석을 수행하여불안정 경계 영역(instability region)을 구하고, 이를 이론값과 비교한 결과Fig. 25과 같이 나타낼 수 있으며, 유사한 결과가 나옴을 확인할 수 있다.이 때 $P _ { 0 } { = } 0 .$ , 인 경우 이다. 그리고 Fig. 25에서 하중이 0일 때구조물이 불안정 해지는 가진 주파수는 고유진동수의 두 배임을 확인 할수 있으며, 동적 하중의 크기가 증가할수록 불안정 경계 영역이 넓어짐을확인할 수 있다. + +![](images/page-068_b6fe05303e1d039ffa0e45254c119c5ef19fbbb8eecf465f17720ce5f2537333.jpg) + +
+line + +| Pt(N) | Lower_Present (Hz) | Upper_Present (Hz) | Lower_Analytic (Hz) | Upper_Analytic (Hz) | +|-------|---------------------|---------------------|----------------------|----------------------| +| 0 | 21.5 | 21.5 | 21.5 | 21.5 | +| 10 | 21.0 | 22.0 | 21.0 | 22.0 | +| 20 | 20.5 | 22.5 | 20.5 | 22.5 | +| 30 | 20.0 | 23.0 | 20.0 | 23.0 | +| 40 | 19.5 | 23.5 | 19.5 | 23.5 | +
+ +Fig. 25 Dynamic Instability Region of Beam + + + +# 3.4.2 Dynamic Buckling Analysis of Plate + +![](images/page-069_0a6df941f6428fb9863912d1331f6827b792593efe82c42fd7e245c179189eb8.jpg) + +
+text_image + +BC(Fixed) +x +y +BC(Fixed) +P(t) +z +BC(23) +
+ +Fig. 26 Dynamic Buckling Model for Plate + +동적 좌굴 해석 프로그램의 검증을 위해 Fig. 26과 같은 판(plate)형상에 대한 동적 좌굴 실험값과 유한요소해석 결과를 비교해 보았다.판의 크기는 먼저 가로 1000mm, 세로 250mm, 두께 1mm 이고, 요소는4노드 쉘 요소 100x25격자(mesh)를 사용하였다. 탄성계수(Young‟smodulus)는 63.3GPa이고, 프아송 비(Poisson‟s ration)는 0.33, 밀도는2678.4kg/m3의 물성치를 주었다. 경계조건은 실험 모델을 바탕으로 한쪽면은 양 끝 단의 50x50mm2 영역을 고정 구속하였고, 반대쪽 면은 한쪽끝 단의 50x50mm2 영역을 XZ방향의 변위에 대해 구속하였다. 하중은 식(3.3)와 같은 시간에 따른 하중을 가하였으며, 이 때 동적 좌굴 해석을위한 방정식은 식 (3.4)와 같다. + +Fig. 26과 같은 판(plate) 형상에 대해 유한요소해석을 이용한 고유치해석 결과 고유진동수(natural frequency)는 5.17Hz이고, 임계 좌굴하중(critical buckling loads)는 49.6N으로 계산되었다. 그리고 동적 하중에대한 동적 좌굴 해석을 수행하여 불안정 경계 영역(instability region)을구한 결과 Fig. 27과 같은 결과를 얻을 수 있으며, 실험값과 유한요소해석결과가 유사한 경향성을 보이는 것을 확인 할 수 있다. 이 때 실선이유한요소해석 결과이고 사각형 모양의 점들이 실험결과를 나타낸다.[11] + + + +![](images/page-070_ffdee295d77a54d40cf90e831b53b5117b06759314d80a4261cd8bc7d900ce20.jpg) + +
+line + +| Pt [N] | FE Analysis (rpm) | Experiment (rpm) | +| ------ | ----------------- | ---------------- | +| 10 | 750 | - | +| 20 | 780 | 780 | +| 30 | 800 | 800 | +| 40 | 820 | 820 | +| 50 | 840 | - | +| 60 | 860 | - | +
+ +Fig. 27 Dynamic Instability Region of Plate diff --git a/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_008.md b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_008.md new file mode 100644 index 0000000..23e2459 --- /dev/null +++ b/docs/reference-papers/MITC4/유한요소해석법을이용한쉘구조물의동적좌굴해석/유한요소해석법을이용한쉘구조물의동적좌굴해석_008.md @@ -0,0 +1,115 @@ + + +# 3.4.3 Dynamic Buckling Analysis of Stiffened Plate + +![](images/page-071_98adff82a3b4a72ea29eacce054f85a4d83d8ba1d01885d634769801f1565dac.jpg) + +
+text_image + +BC(Fixed) +BC(Fixed) +P(t) +BC(12) +y +z +x +하 대 밤 +
+ +Fig. 28 Dynamic Buckling Analysis Model for Stiffened Plate + +동적 좌굴 해석 프로그램의 검증을 위해 Fig. 28과 같은 보강된판(stiffened plate) 형상에 대한 동적 좌굴 실험값과 유한요소해석 결과를비교해 보았다. 판의 크기는 먼저 가로 1000mm, 세로 250mm, 두께 1mm이고, 요소는 4노드 쉘 요소 100x25개를 사용하였다. 보강재는 가로1000mm, 세로 10mm, 두께 1mm이고, 요소는 4노드 쉘 요소 100x1개를사용하였다. 탄성계수(Young‟s modulus)는 62.4GPa이고, 프아송 비(Poisson‟sration)는 0.33, 밀도는 2696.7kg/m3의 물성치를 주었다. 경계조건은 실험모델을 바탕으로 한쪽 면은 양 끝 단의 50x50mm2 영역을 고정구속하였고, 반대쪽 면은 한쪽 끝 단의 50x50mm2 영역을 XY방향의변위에 대해 구속하였다. 하중은 식 (3.3)와 같은 시간에 따른 하중을가하였으며, 이 때 동적 좌굴 해석을 위한 방정식은 식 (3.4)와 같다. + +동적 하중에 대한 동적 좌굴 해석을 수행하여 불안정 경계영역(instability region)을 구한 결과 실험값과 유한요소해석 결과가 평판에비해 다소 차이가 남을 볼 수 있다.(Fig. 29) 그 이유는 유한요소해석의경우 모델이 “균질하다(homogeneous)”는 가정하에 해석이 이루어지지만 + + + +실제 구조물의 경우 다양한 결함(imperfection)이 존재하므로 이러한차이가 발생할 수 있다. 특히 보강된 판의 경우 일반 평판에 비해보강재를 만드는 과정에서 구조물에 결함(imperfection)이 발생할 가능성이높기 때문에 이러한 결과가 나타날 수 있다. 하지만 두 결과값에 대한전체적인 경향성은 유사함을 알 수 있다. 또한 앞서 해석한 평판과보강된 평판을 비교해 보면 보강된 평판의 임계 좌굴 하중이 증가하고,불안정 경계 영역이 전체적으로 상승함을 알 수 있다.[11] + +![](images/page-072_9cc0d6cbe9c6ce32d9c934ae42af1934fd020ad5a5899692093942b736fe250f.jpg) +Fig. 29 Comparison of Dynamic Instability Region for Stiffened Plate with Plate + + + +# 4. 결 론 + +동적 좌굴 현상은 구조물이 동적 압축 하중을 받을 때 발생하는 동적불안정 현상으로서, 특히 큰 동적 압축 하중 환경하에 노출되어 있는초음속 항공기나 탄도 미사일, 발사용 로켓과 지구 대기권 재돌입체그리고 고속의 수중운동체 등을 설계할 때 반드시 고려되어야 하는현상이다. 이에 본 연구에서는 동적 압축 하중을 받는 구조물의 동적좌굴 해석을 위한 프로그램을 개발하였다. + +본 연구를 위해 MITC4 쉘 요소를 사용하여 3차원 쉘 구조물을모델링 하였으며, 고유치 해석 시 필요한 기하 강성 행렬(Geometricstiffness matrix)을 구하기 위해 Total Lagrangian 방법을 사용하여 기하비선형 해석을 구현하였다. 그리고 질량 행렬은 구조물을 유연하게 하기위해서 집중 질량 행렬(Lumped mass matrix)을 사용하였다. 또한 진동 및정적/동적 좌굴 해석 시 필요한 고유치 해석 솔버(Eigenvalue analysissolver)는 Block Laczos 방법을 사용한 “BLZPACK”이라는 오픈 소스(Opensource)를 사용하여 해석 프로그램을 구현하였다. + +본 연구에서 개발한 해석 프로그램을 사용하여 선형/비선형 정적해석과 진동 해석 그리고 정적 좌굴 해석을 수행하였고, 이를 이론값 및상용유한요소해석 프로그램 해석 결과와 비교하여 프로그램의 타당성과정확성을 검증하였다. 또한 보의 동적 좌굴 이론으로부터 구한 이론값과평판 그리고 보강된 평판의 동적 좌굴 실험을 통해 구한 실험값을유한요소해석 프로그램 결과와 비교하여 본 연구에서 개발한 동적 좌굴해석 프로그램의 타당성을 검증하였다. + +본 연구에서 개발한 동적 좌굴 해석을 위한 유한요소해석 프로그램을사용하여 구조물 설계 시 동적 좌굴 해석이 필요한 구조물에 대해해석을 수행 할 수 있으며, 이를 통해 불안정 경계 영역을 예측하고구조물의 안정화 방안을 모색할 수 있을 것으로 기대된다. + + + +# 참고문헌 + +[1] Eduardo N. Dvorkin and Klaus-Jurgen Bathe, “A continuum mechanics based four-node shell element for general nonlinear analysis”, Eng. Comput., Vol. 1, No. 1, pp.77-88, 1984. +[2] Robert D. Cook, David S. Malkus, Michael E. Plesha, Robert J. Witt, Concepts and application of finite element analysis., John Wiley & Sons, 2001. +[3] Thomas J.R. Hughes, The Finite Element Method : Linear Static and Dynamic Finite Element Analysis., Prentice-Hall, 1987. +[4] O.C. Zienkiewicz, R. L. Taylor, The Finite Element Method : Basic Formulation and Linear Problems., McGraw-Hill, 1989. +[5] O.C. Zienkiewicz, R. L. Taylor, The Finite Element Method : Solid and Fluid Mechanics Dynamics and Non-linearity., McGraw-Hill, 1991. +[6] M. Ruzzene, “Dynamic buckling of periodically stiffened shells : application to supercavitating vehicles”, International Journal of Solids and Structures, Vol. 41, No.3-4, pp.1039-1059, 2004. +[7] V. V. Bolotin, The Dynamic Stability of Elastic System., Holden-Day, 1964. +[8] Leonard Meirovitch, Method of Analytical Dynamics., McGraw-Hill, 1985. +[9] Eduardo N. Dvorkin, “Nonlinear Analysis of Shells Using the MITC Formulation”, Archives of Computational Methods in Engineering, Vol. 2, No. 2, pp.1-50, 1995. +[10] J. Argyris, “An excursion into large rotations”, Comput. Methods Appl. Mech. Engrg., Vol. 32, No. 1-3, pp.85-155, 1982. +[11] Minho Chung, Hee Jun Lee, Woo-Bin Lim, Jin Yeon Cho, Wanil Byun, Seung Jo Kim and Sung-Han Park, “Experimental study on dynamic buckling phenomena for supercavitating underwater vehicle”, International Journal of Naval Architecture and Ocean Engineering, submitted. + + + +# 부록 + +# 1. Strain/Stress Vector + +$$ +\text {Strain} \quad \{\mathbf {E} \} = \left\{ \begin{array}{l} E _ {\xi \xi} \\ E _ {\eta \eta} \\ E _ {\varsigma \varsigma} \\ 2 E _ {\eta \varsigma} \\ 2 E _ {\xi \varsigma} \\ 2 E _ {\xi \eta} \end{array} \right\}, \qquad \text {Stress} \quad \{\mathbf {S} \} = \left\{ \begin{array}{l} S _ {\xi \xi} \\ S _ {\eta \eta} \\ S _ {\varsigma \varsigma} \\ 2 S _ {\eta \varsigma} \\ 2 S _ {\xi \varsigma} \\ 2 S _ {\xi \eta} \end{array} \right\} +$$ + +# 2. Shape Function + +$$ +N _ {1} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 - \eta), \quad N _ {2} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 - \eta) +$$ + +$$ +N _ {3} (\xi , \eta) = \frac {1}{4} (1 + \xi) (1 + \eta), \quad N _ {4} (\xi , \eta) = \frac {1}{4} (1 - \xi) (1 + \eta) +$$ + +# 3. Jacobian Matrix + +$$ +[ J ] = \left[ \begin{array}{c c c} X, _ {\xi} & Y, _ {\xi} & Z, _ {\xi} \\ X, _ {\eta} & Y, _ {\eta} & Z, _ {\eta} \\ X, _ {\varsigma} & Y, _ {\varsigma} & Z, _ {\varsigma} \end{array} \right] = \left[ \begin{array}{c c c} \frac {\partial X}{\partial \xi} & \frac {\partial Y}{\partial \xi} & \frac {\partial Z}{\partial \xi} \\ \frac {\partial X}{\partial \eta} & \frac {\partial Y}{\partial \eta} & \frac {\partial Z}{\partial \eta} \\ \frac {\partial X}{\partial \varsigma} & \frac {\partial Y}{\partial \varsigma} & \frac {\partial Z}{\partial \varsigma} \end{array} \right] +$$ + +$$ += \left[ \begin{array}{c c c c} \frac {\partial N _ {1}}{\partial \xi} & \frac {\partial N _ {2}}{\partial \xi} & \frac {\partial N _ {3}}{\partial \xi} & \frac {\partial N _ {4}}{\partial \xi} \\ \frac {\partial N _ {1}}{\partial \eta} & \frac {\partial N _ {2}}{\partial \eta} & \frac {\partial N _ {3}}{\partial \eta} & \frac {\partial N _ {4}}{\partial \eta} \\ 0 & 0 & 0 & 0 \end{array} \right] \left[ \begin{array}{c c c} X _ {1} & Y _ {1} & Z _ {1} \\ X _ {2} & Y _ {2} & Z _ {2} \\ X _ {3} & Y _ {3} & Z _ {3} \\ X _ {4} & Y _ {4} & Z _ {4} \end{array} \right] + \frac {1}{2} \left[ \begin{array}{c c c c} \varsigma \frac {\partial N _ {1}}{\partial \xi} t _ {1} & \varsigma \frac {\partial N _ {2}}{\partial \xi} t _ {2} & \varsigma \frac {\partial N _ {3}}{\partial \xi} t _ {3} & \varsigma \frac {\partial N _ {4}}{\partial \xi} t _ {4} \\ \varsigma \frac {\partial N _ {1}}{\partial \eta} t _ {1} & \varsigma \frac {\partial N _ {2}}{\partial \eta} t _ {2} & \varsigma \frac {\partial N _ {3}}{\partial \eta} t _ {3} & \varsigma \frac {\partial N _ {4}}{\partial \eta} t _ {4} \\ N _ {1} t _ {1} & N _ {2} t _ {2} & N _ {3} t _ {3} & N _ {4} t _ {4} \end{array} \right] \left[ \begin{array}{c} \mathbf {V} _ {1} ^ {n} \\ \mathbf {V} _ {2} ^ {n} \\ \mathbf {V} _ {3} ^ {n} \\ \mathbf {V} _ {4} ^ {n} \end{array} \right] +$$ + +$$ += \left[ \begin{array}{c} \mathbf {G} _ {1} \\ \mathbf {G} _ {2} \\ \mathbf {G} _ {3} \end{array} \right] +$$ + + + +# 감사의 글 + +이렇게 밤늦게 실험실에서 책상 앞에 앉아 있는 날도 얼마 남지 않았습니다.이제는 졸업을 앞두고 논문의 마지막 장을 써 내려간다는 것이 실감나지 않고아쉬운 마음이 크지만 다사다난했던 지난 2년의 시간들은 아마도 제 인생 중에큰 전환점이었으며 귀중한 발판으로 삼으려 합니다. + +아직은 부족하지만 제가 이렇게 많은 발전과 함께 자그마한 결실을 맺고졸업을 앞두게 되어 이 자리까지 도움을 주신 고마운 분들에게 감사의 글로써졸업논문을 맺으려 합니다. + +먼저, 너무도 부족했던 저를 대학원 짧은 기간 동안에 이렇게 발전하도록일일이 깨우쳐주시고 함께 고민해 주시며 아낌없이 지도해주신 조진연 교수님께심심한 감사를 드립니다. 그리고 학부시절 공학도의 길을 제시해주시고 저에게큰 힘이 되어주신 김기욱 교수님, 항상 항공과의 발전을 위해 헌신적으로학생들을 지도해 주시는 김범수 교수님, 최동환 교수님, 최기영 교수님, 노태성교수님, 이승수 교수님, 유창경 교수님께 감사와 존경을 전합니다. + +실험실에서 동고동락했던 민환이형, 민호, 순신이, 연철이, 재연이, 이제대학원 생활을 시작하게 될 소영이, 우빈이 모두에게 고마운 마음을 전하며모두들에게 앞으로도 좋은 일이 가득하길 바랍니다. 그리고 대학원에 입학하여도움을 주신 장훈이형, 형수형, 영민이형, 규원이형에게도 감사의 뜻을 전합니다. + +초등학교 때부터 지금까지도 변치 않는 우정을 나누고 있는 동현이, 용덕이,항상 자기자리에서 최선을 다하는 대학 친구들 상형이형, 재필이, 광규,영민이에게도 감사의 뜻을 전합니다. + +그리고 저의 소중한 가족들에게 감사합니다. 어려운 여건에서도 항상큰아들을 믿어주시고 노심초사 걱정해 주시며 뒷바라지 해주신 부모님의 은혜에깊은 감사를 드립니다. 또한 멀리 있고 잘해주지도 못하는 오빠를 믿고따라주는 동생 진아, 민희에게도 고마움을 전하며 항상 사랑으로 지켜봐 주고힘이 되어준 저의 가장 소중한 친구이자 연인인 미연이에게 고마움을 전합니다. + +마지막으로 저를 믿어주시고 사랑해주신 많은 분들의 기대에 보답하도록사회에 꼭 필요한 일꾼이 되며 항상 최선을 다하는 모습을 보여드림을다짐하겠습니다. + +2011년 12월 + +이 희 준