김경종
393 lines
23 KiB
Markdown
393 lines
23 KiB
Markdown
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TABLE 4.3 Generalized stress-strain matrices for isotropic materials and the problems in Table 4.2
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<table><tr><td>Problem</td><td>Material matrix C</td></tr><tr><td>Bar</td><td>E</td></tr><tr><td>Beam</td><td>EI</td></tr><tr><td>Plane stress</td><td><img src="images/60c55558278de8313b88d9ebadf95cc8dbd46c8e31d98f6022d525de791af319.jpg"/></td></tr><tr><td>Plane strain</td><td><img src="images/060d6855da4cf83b022c25b9731ade1ec878adb7cedea38944926ad41e455aad.jpg"/></td></tr><tr><td>Axisymmetric</td><td><img src="images/1c5927576a323bbb2575d585ed9b6de698a5749130da54dd4562b6c15d0df3db.jpg"/></td></tr><tr><td>Three-dimensional</td><td><img src="images/f451e6381798e656bedfa0a64855ce4d8589b50ea259f7be68f290383c28d42d.jpg"/></td></tr><tr><td>Plate bending</td><td><img src="images/b2ef1eb80cb848e4c09acc3f5295b7c32bf292696f7801c2c763f921ea2f758c.jpg"/></td></tr></table>
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unknown coefficients in the polynomials as generalized coordinates. The number of unknown coefficients in the polynomials was equal to the number of element nodal point displacements. Expressing the generalized coordinates in terms of the element nodal point displacements, we found that, in general, each polynomial coefficient is not an actual physical displacement but is equal to a linear combination of the element nodal point displacements.
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Finite element matrices that are formulated by assuming that the displacements vary in the form of a function whose unknown coefficients are treated as generalized coordinates are referred to as generalized coordinate finite element models. A rather natural class of functions to use for approximating element displacements are polynomials because they are commonly employed to approximate unknown functions, and the higher the degree of the polynomial, the better the approximation that we can expect. In addition, polynomials are easy to differentiate; i.e., if the polynomials approximate the displacements of the structure, we can evaluate the strains with relative ease.
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Using polynomial displacement assumptions, a very large number of finite elements for practically all problems in structural mechanics have been developed.
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The objective in this section is to describe the formulation of a variety of generalized coordinate finite element models that use polynomials to approximate the displacement fields. Other functions could in principle be used in the same way, and their use can be effective in specific applications (see Example 4.20). In the presentation, emphasis is given to the general formulation rather than to numerically effective finite elements. Therefore, this section serves primarily to enhance our general understanding of the finite element method. More effective finite elements for general application are the isoparametric and related elements described in Chapter 5.
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In the following derivations the displacements of the finite elements are always described in the local coordinate systems shown in Fig. 4.5. Also, since we consider one specific element, we shall leave out the superscript (m) used in Section 4.2.1 [see (4.31)].
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For one-dimensional bar elements (truss elements) we have
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$$
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u (x) = \alpha_ {1} + \alpha_ {2} x + \alpha_ {3} x ^ {2} + \dots \tag {4.51}
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$$
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where x varies over the length of the element, u is the local element displacement, and $\alpha_{1}$ , $\alpha_{2}$ , $\ldots$ , are the generalized coordinates. The displacement expansion in (4.51) can also be used for the transverse and longitudinal displacements of a beam.
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For two-dimensional elements (i.e., plane stress, plane strain, and axisymmetric elements), we have for the u and v displacements as a function of the element x and y coordinates,
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$$
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u (x, y) = \alpha_ {1} + \alpha_ {2} x + \alpha_ {3} y + \alpha_ {4} x y + \alpha_ {5} x ^ {2} + \dots \tag {4.52}
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$$
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$$
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v (x, y) = \beta_ {1} + \beta_ {2} x + \beta_ {3} y + \beta_ {4} x y + \beta_ {5} x ^ {2} + \dots
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$$
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where $\alpha_{1},\alpha_{2},\ldots$ , and $\beta_{1},\beta_{2},\ldots$ , are the generalized coordinates.
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In the case of a plate bending element, the transverse deflection w is assumed as a function of the element coordinates x and y; i.e.,
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$$
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w (x, y) = \gamma_ {1} + \gamma_ {2} x + \gamma_ {3} y + \gamma_ {4} x y + \gamma_ {5} x ^ {2} + \dots \tag {4.53}
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$$
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where $\gamma_{1},\gamma_{2},\ldots$ , are the generalized coordinates.
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<details>
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<summary>text_image</summary>
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Across section AA:
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τxx is uniform
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All other stress components
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are zero
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</details>
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(a) Uniaxial stress condition: frame under concentrated loads
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<details>
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<summary>text_image</summary>
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Hole
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Differential
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element
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τyy
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τxy
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τxx
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τxx, τyy, τxy are uniform
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across the thickness
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All other stress components
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are zero
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</details>
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(b) Plane stress conditions: membrane and beam under in-plane actions
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<details>
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<summary>text_image</summary>
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1
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y, v
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x, u
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z, w
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1
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τyy
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τxy
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τxx
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τzz
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u(x, y), v(x, y) are nonzero
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w = 0, εzz = γyz = γzx = 0
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</details>
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(c) Plane strain condition: long dam subjected to water pressure
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Figure 4.5 Various stress and strain conditions with illustrative examples
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Finally, for elements in which the u, v, and w displacements are measured as a function of the element x, y, and z coordinates, we have, in general,
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$$
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u (x, y, z) = \alpha_ {1} + \alpha_ {2} x + \alpha_ {3} y + \alpha_ {4} z + \alpha_ {5} x y + \dots
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$$
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$$
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v (x, y, z) = \beta_ {1} + \beta_ {2} x + \beta_ {3} y + \beta_ {4} z + \beta_ {5} x y + \dots \tag {4.54}
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$$
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$$
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w (x, y, z) = \gamma_ {1} + \gamma_ {2} x + \gamma_ {3} y + \gamma_ {4} z + \gamma_ {5} x y + \dots
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$$
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where $\alpha_{1},\alpha_{2},\ldots ,\beta_{1},\beta_{2},\ldots$ , and $\gamma_{1},\gamma_{2},\ldots$ are now the generalized coordinates.
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(d) Axisymmetric condition: cylinder under internal pressure
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$\tau_{zz} = 0$ All other stress components are nonzero
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(e) Plate and shell structures
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Figure 4.5 (continued)
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As in the discussion of the plane stress element in Example 4.6, the relations (4.51) to (4.54) can be written in matrix form,
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$$
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\mathbf {u} = \boldsymbol {\Phi} \boldsymbol {\alpha} \tag {4.55}
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$$
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where the vector u corresponds to the displacements used in (4.51) to (4.54), the elements of $\Phi$ are the corresponding polynomial terms, and $\alpha$ is a vector of the generalized coordinates arranged in the appropriate order.
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To evaluate the generalized coordinates in terms of the element nodal point displacements, we need to have as many nodal point displacements as assumed generalized coordinates. Then, evaluating (4.55) specifically for the nodal point displacements $\hat{u}$ of the element, we obtain
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$$
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\hat {\mathbf {u}} = \mathbf {A} \alpha \tag {4.56}
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$$
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Assuming that the inverse of $\mathbf{A}$ exists, we have
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$$
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\boldsymbol {\alpha} = \mathbf {A} ^ {- 1} \hat {\mathbf {u}} \tag {4.57}
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$$
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The element strains to be considered depend on the specific problem to be solved. Denoting by € a generalized strain vector, whose components are given for specific problems in Table 4.2, we have
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$$
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\epsilon = \mathbf {E} \alpha \tag {4.58}
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$$
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where the matrix E is established using the displacement assumptions in (4.55). A vector of generalized stresses $\tau$ is obtained using the relation
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$$
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\boldsymbol {\tau} = \mathbf {C} \boldsymbol {\epsilon} \tag {4.59}
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$$
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where C is a generalized elasticity matrix. The quantities $\tau$ and C are defined for some problems in Tables 4.2 and 4.3. We may note that except in bending problems, the generalized $\tau$ , $\epsilon$ , and C matrices are those that are used in the theory of elasticity. The word “generalized” is employed merely to include curvatures and moments as strains and stresses, respectively. The advantage of using curvatures and moments in bending analysis is that in the stiffness evaluation an integration over the thickness of the corresponding element is not required because this stress and strain variation has already been taken into account (see Example 4.15).
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Referring to Table 4.3, it should be noted that all stress-strain matrices can be derived from the general three-dimensional stress-strain relationship. The plane strain and axisymmetric stress-strain matrices are obtained simply by deleting in the three-dimensional stress-strain matrix the rows and columns that correspond to the zero strain components. The stress-strain matrix for plane stress analysis is then obtained from the axisymmetric stress-strain matrix by using the condition that $\tau_{zz}$ is zero (see the program QUADS in Section 5.6). To calculate the generalized stress-strain matrix for plate bending analysis, the stress-strain matrix corresponding to plane stress conditions is used, as shown in the following example.
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EXAMPLE 4.15: Derive the stress-strain matrix C used for plate bending analysis (see Table 4.3).
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The strains at a distance z measured upward from the midsurface of the plate are
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$$
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\left[ - z \frac {\partial^ {2} w}{\partial x ^ {2}} - z \frac {\partial^ {2} w}{\partial y ^ {2}} - z \frac {2 \partial^ {2} w}{\partial x \partial y} \right]
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$$
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In plate bending analysis it is assumed that each layer of the plate acts in plane stress condition and positive curvatures correspond to positive moments (see Section 5.4.2). Hence, integrating the normal stresses in the plate to obtain moments per unit length, the generalized stress-strain matrix is
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$$
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\mathbf {C} = \int_ {- h / 2} ^ {+ h / 2} z ^ {2} \frac {E}{1 - \nu^ {2}} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right] d z
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$$
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$$
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\mathbf {C} = \frac {E h ^ {3}}{1 2 (1 - \nu^ {2})} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {1 - \nu}{2} \end{array} \right]
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$$
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Considering (4.55) to (4.59), we recognize that, in general terms, all relationships for evaluation of the finite element matrices corresponding to the local finite element nodal point displacements have been defined, and using the notation of Section 4.2.1, we have
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$$
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\mathbf {H} = \boldsymbol {\Phi} \mathbf {A} ^ {- 1} \tag {4.60}
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$$
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$$
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\mathbf {B} = \mathbf {E} \mathbf {A} ^ {- 1} \tag {4.61}
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$$
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Let us now consider briefly various types of finite elements encountered, which are subject to certain static or kinematic assumptions.
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Truss and beam elements. Truss and beam elements are very widely used in structural engineering to model, for example, building frames and bridges [see Fig. 4.5(a) for an assemblage of truss elements].
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As discussed in Section 4.2.1, the stiffness matrices of these elements can in many cases be calculated by solving the differential equations of equilibrium (see Example 4.8), and much literature has been published on such derivations. The results of these derivations have been employed in the displacement method of analysis and the corresponding approximate solution techniques, such as the method of moment distribution. However, it can be effective to evaluate the stiffness matrices using the finite element formulation, i.e., the virtual work principle, particularly when considering complex beam geometries and geometric nonlinear analysis (see Section 5.4.1).
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Plane stress and plane strain elements. Plane stress elements are employed to model membranes, the in-plane action of beams and plates as shown in Fig. 4.5(b), and so on. In each of these cases a two-dimensional stress situation exists in an xy plane with the stresses $\tau_{zz}$ , $\tau_{yz}$ , and $\tau_{zx}$ equal to zero. Plane strain elements are used to represent a slice (of unit thickness) of a structure in which the strain components $\epsilon_{zz}$ , $\gamma_{yz}$ , and $\gamma_{zx}$ are zero. This situation arises in the analysis of a long dam as illustrated in Fig. 4.5(c).
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Axisymmetric elements. Axisymmetric elements are used to model structural components that are rotationally symmetric about an axis. Examples of application are pressure vessels and solid rings. If these structures are also subjected to axisymmetric loads, a two-dimensional analysis of a unit radian of the structure yields the complete stress and strain distributions as illustrated in Fig. 4.5(d).
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On the other hand, if the axisymmetric structure is loaded nonaxisymmetrically, the choice lies between a fully three-dimensional analysis, in which substructuring (see Section 8.2.4) or cyclic symmetry (see Example 4.14) is used, and a Fourier decomposition of the loads for a superposition of harmonic solutions (see Example 4.20).
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Plate bending and shell elements. The basic proposition in plate bending and shell analyses is that the structure is thin in one dimension, and therefore the following assumptions can be made [see Fig. 4.5(e)]:
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1. The stress through the thickness (i.e., perpendicular to the midsurface) of the plate/shell is zero.
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2. Material particles that are originally on a straight line perpendicular to the midsurface of the plate/shell remain on a straight line during deformations. In the Kirchhoff theory, shear deformations are neglected and the straight line remains perpendicular to the midsurface during deformations. In the Reissner/Mindlin theory, shear deformations are included, and therefore the line originally normal to the midsurface in general does not remain perpendicular to the midsurface during the deformations (see Section 5.4.2).
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The first finite elements developed to model thin plates in bending and shells were based on the Kirchhoff plate theory (see R. H. Gallagher [A]). The difficulties in these approaches are that the elements must satisfy the convergence requirements and be relatively effective in their applications. Much research effort was spent on the development of such elements; however, it was recognized that more effective elements can frequently be formulated using the Reissner/Mindlin plate theory (see Section 5.4.2).
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To obtain a shell element a simple approach is to superimpose a plate bending stiffness and a plane stress membrane stiffness. In this way flat shell elements are obtained that can be used to model flat components of shells (e.g., folded plates) and that can also be employed to model general curved shells as an assemblage of flat elements. We demonstrate the development of a plate bending element based on the Kirchhoff plate theory and the construction of an associated flat shell element in Examples 4.18 and 4.19.
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EXAMPLE 4.16: Discuss the derivation of the displacement and strain-displacement interpolation matrices of the beam shown in Fig. E4.16.
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The exact stiffness matrix (within beam theory) of this beam could be evaluated by solving the beam differential equations of equilibrium, which are for the bending behavior
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$$
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\frac {d ^ {2}}{d \xi^ {2}} \left(E I \frac {d ^ {2} w}{d \xi^ {2}}\right) = 0; \quad E I = E \frac {b h ^ {3}}{1 2} \tag {a}
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$$
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and for the axial behavior
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$$
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\frac {d}{d \xi} \left(E A \frac {d u}{d \xi}\right) = 0; \quad A = b h \tag {b}
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$$
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where E is Young's modulus. The procedure is to impose a unit end displacement, with all other end displacements equal to zero, and solve the appropriate differential equation of equilibrium of the beam subject to these boundary conditions. Once the element internal displacements for these boundary conditions have been calculated, appropriate derivatives give the element end
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<details>
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<summary>text_image</summary>
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Section A-A
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b
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h
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η
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w1
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h1
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θ1
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u1
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ξ
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A
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w2
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h2
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θ2
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u2
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A
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z
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x
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L
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</details>
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Figure E4.16 Beam element with varying section
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forces that together constitute the column of the stiffness matrix corresponding to the imposed end displacement. It should be noted that this stiffness matrix is only “exact” for static analysis because in dynamic analysis the stiffness coefficients are frequency-dependent.
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Alternatively, the formulation given in (4.8) to (4.17) can be used. The same stiffness matrix as would be evaluated by the above procedure is obtained if the exact element internal displacements [that satisfy (a) and (b)] are employed to construct the strain-displacement matrix. However, in practice it is frequently expedient to use the displacement interpolations that correspond to a uniform cross-section beam, and this yields an approximate stiffness matrix. The approximation is generally adequate when $h_{2}$ is not very much larger than $h_{1}$ (hence when a sufficiently large number of beam elements is employed to model the complete structure). The errors encountered in the analysis are those discussed in Section 4.3, because this formulation corresponds to displacement-based finite element analysis.
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Using the variables defined in Fig. E4.16 and the “exact” displacements (Hermitian functions) corresponding to a prismatic beam, we have
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$$
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u = \left(1 - \frac {\xi}{L}\right) u _ {1} + \frac {6 \eta}{L} \left(\frac {\xi}{L} - \frac {\xi^ {2}}{L ^ {2}}\right) w _ {1} - \eta \left(1 - 4 \frac {\xi}{L} + 3 \frac {\xi^ {2}}{L ^ {2}}\right) \theta_ {1}
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$$
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$$
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+ \frac {\xi}{L} u _ {2} - \frac {6 \eta}{L} \left(\frac {\xi}{L} - \frac {\xi^ {2}}{L ^ {2}}\right) w _ {2} + \eta \left(2 \frac {\xi}{L} - 3 \frac {\xi^ {2}}{L ^ {2}}\right) \theta_ {2}
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$$
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Hence,
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$$
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\begin{array}{l} \mathbf {H} = \left[ \left(1 - \frac {\xi}{L}\right) \mid \frac {6 \eta}{L} \left(\frac {\xi}{L} - \frac {\xi^ {2}}{L ^ {2}}\right) \mid - \eta \left(1 - \frac {4 \xi}{L} + 3 \frac {\xi^ {2}}{L ^ {2}}\right) \mid \frac {\xi}{L} \right| \\ \left. - \frac {6 \eta}{L} \left(\frac {\xi}{L} - \frac {\xi^ {2}}{L ^ {2}}\right) \mid \eta \left(\frac {2 \xi}{L} - 3 \frac {\xi^ {2}}{L ^ {2}}\right) \right] \tag {c} \\ \end{array}
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$$
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For (c) we ordered the nodal point displacements as follows
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$$
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\hat {\mathbf {u}} ^ {T} = \left[ \begin{array}{l l} u _ {1} w _ {1} \theta_ {1} & u _ {2} w _ {2} \theta_ {2} \end{array} \right]
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$$
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Considering only normal strains and stresses in the beam, i.e., neglecting shearing deformations, we have as the only strain and stress components
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$$
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\epsilon_ {\xi \xi} = \frac {d u}{d \xi}; \quad \tau_ {\xi \xi} = E \epsilon_ {\xi \xi}
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$$
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and hence
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$$
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\mathbf {B} = \left[ - \frac {1}{L} \left| \frac {6 \eta}{L} \left(\frac {1}{L} - \frac {2 \xi}{L ^ {2}}\right) \right| - \eta \left(\frac {- 4}{L} + \frac {6 \xi}{L ^ {2}}\right) \right| \left| \frac {1}{L} \right| - \frac {6 \eta}{L} \left(\frac {1}{L} - \frac {2 \xi}{L ^ {2}}\right) \left| \eta \left(\frac {2}{L} - \frac {6 \xi}{L ^ {2}}\right) \right] \tag {d}
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$$
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The relations in (c) and (d) can be used directly to evaluate the element matrices defined in (4.33) to (4.37); e.g.,
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$$
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\mathbf {K} = E b \int_ {0} ^ {L} \int_ {- h / 2} ^ {h / 2} \mathbf {B} ^ {T} \mathbf {B} d \eta d \xi
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$$
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where $h = h_{1} + (h_{2} - h_{1})\frac{\xi}{L}$
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This formulation can be directly extended to develop the element matrices corresponding to the three-dimensional action of the beam element and to include shear deformations (see K. J. Bathe and S. Bolourchi [A]).
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EXAMPLE 4.17: Discuss the derivation of the stiffness, mass, and load matrices of the axisymmetric three-node finite element in Fig. E4.17.
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This element was one of the first finite elements developed. For most practical applications, much more effective finite elements are presently available (see Chapter 5), but the element is conveniently used for instructional purposes because the equations to be dealt with are relatively simple.
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The displacement assumption used is
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$$
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u (x, y) = \alpha_ {1} + \alpha_ {2} x + \alpha_ {3} y
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$$
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$$
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v (x, y) = \beta_ {1} + \beta_ {2} x + \beta_ {3} y
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$$
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Therefore, a linear displacement variation is assumed, just as for the derivation of the four-node plane stress element considered in Example 4.6 where the fourth node required that the term xy be included in the displacement assumption. Referring to the derivations carried out in Example 4.6, we can directly establish the following relationships:
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$$
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\left[ \begin{array}{l} u (x, y) \\ v (x, y) \end{array} \right] = \mathbf {H} \left[ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ v _ {1} \\ v _ {2} \\ v _ {3} \end{array} \right]
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$$
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where $\mathbf{H} = \begin{bmatrix} 1 & x & y & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & x & y \end{bmatrix} \mathbf{A}^{-1}$
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$$
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\mathbf {A} ^ {- 1} = \left[ \begin{array}{l l} \mathbf {A} _ {1} ^ {- 1} & \mathbf {0} \\ \mathbf {0} & \mathbf {A} _ {1} ^ {- 1} \end{array} \right]; \quad \mathbf {A} _ {1} = \left[ \begin{array}{l l l} 1 & x _ {1} & y _ {1} \\ 1 & x _ {2} & y _ {2} \\ 1 & x _ {3} & y _ {3} \end{array} \right]
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$$
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<!-- source-page: 220 -->
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<details>
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<summary>text_image</summary>
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y, v
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||
3 (x₃, y₃)
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||
v₁
|
||
u₁
|
||
Node 1 (x₁, y₁)
|
||
2 (x₂, y₂)
|
||
z, w
|
||
x, u
|
||
</details>
|
||
|
||
(a) Nodal points
|
||
|
||
|
||
|
||
<details>
|
||
<summary>text_image</summary>
|
||
|
||
y
|
||
3
|
||
f_y^S
|
||
f_x^S
|
||
s
|
||
2
|
||
x
|
||
</details>
|
||
|
||
(b) Surface loading
|
||
Figure E4.17 Axisymmetric three-node element
|
||
|
||
Hence $\mathbf{A}_{1}^{-1} = \frac{1}{\det\mathbf{A}_{1}}\left[ \begin{array}{cccc}x_{2}y_{3} - x_{3}y_{2} & x_{3}y_{1} - x_{1}y_{3} & x_{1}y_{2} - x_{2}y_{1}\\ y_{2} - y_{3} & y_{3} - y_{1} & y_{1} - y_{2}\\ x_{3} - x_{2} & x_{1} - x_{3} & x_{2} - x_{1} \end{array} \right]$
|
||
|
||
where $\operatorname{det} \mathbf{A}_1 = x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)$
|
||
|
||
We may note that $\det \mathbf{A}_1$ is zero only if the three element nodal points lie on a straight line. The strains are given in Table 4.2 and are
|
||
|
||
$$
|
||
\epsilon_ {x x} = \frac {\partial u}{\partial x}; \quad \epsilon_ {y y} = \frac {\partial v}{\partial y}; \quad \gamma_ {x y} = \frac {\partial u}{\partial y} + \frac {\partial v}{\partial x}; \quad \epsilon_ {z z} = \frac {\partial w}{\partial z} = \frac {u}{x}
|
||
$$
|
||
|
||
Using the assumed displacement polynomials, we obtain
|
||
|
||
$$
|
||
\left[ \begin{array}{l} \boldsymbol {\epsilon} _ {x x} \\ \boldsymbol {\epsilon} _ {y y} \\ \gamma_ {x y} \\ \boldsymbol {\epsilon} _ {z z} \end{array} \right] = \mathbf {B} \left[ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \\ v _ {1} \\ v _ {2} \\ v _ {3} \end{array} \right]; \quad \mathbf {B} = \left[ \begin{array}{l l l l l l} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ \frac {1}{x} & 1 & \frac {y}{x} & 0 & 0 & 0 \end{array} \right] \mathbf {A} ^ {- 1} = \mathbf {E} \mathbf {A} ^ {- 1}
|
||
$$
|