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김경종
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Figure 3.13-2. Shape functions of a cubic curve fitted to ordinates and slopes at x = 0 and at x = L.
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different interpolation schemes are possible. All are lengthy to write out, and their comparative merits are intimately connected with plate theory (Chapter 11).
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# PROBLEMS
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# Section 3.2
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3.1 The system shown consists of a rigid half-cylinder that can roll without friction on a horizontal surface, a linear nondissipative spring, and a force P of constant magnitude that can move around the cylinder, but is always directed toward point C on the cylinder. Show that this system is not conservative.
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<details>
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<summary>text_image</summary>
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P
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C
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k
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</details>
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Problem 3.1
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3.2 Redefine $\Omega$ to be $\Omega = P(D_{\mathrm{eq}} - D)$ , as suggested near the end of Section 3.2. Redraw Fig. 3.2-4 as required.
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3.3 Reverse the direction of load P in Fig. 3.2-3. Solve for the equilibrium value of D by use of $\Pi_{p}$ . Revise Fig. 3.2-4 as required.
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3.4 Imagine that the spring in Fig. 3.2-3 is not linear but exerts a force proportional to the square of its stretch. Write an expression for $\Pi_p$ , and from it determine the equilibrium value of $D$ .
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# Section 3.3
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3.5 Redefine $D_{2}$ and $D_{3}$ in Fig. 3.3-1 so that $D_{2}$ is an axial displacement relative to $D_{1}$ and $D_{3}$ is an axial displacement relative to $D_{2}$ . Write an expression for $\Pi_{p}$ , analogous to Eq. 3.3-3. For the special case $k_{1} = k_{2} = k_{3} = k$ and $P_{1} = P_{2} = P_{3} = P$ , solve for the $D_{i}$ and show that they give the same absolute axial displacements as Eq. 3.3-3.
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<details>
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<summary>text_image</summary>
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y
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L
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d4
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d3
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d2
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A,E
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β
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d1
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x
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</details>
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Problem 3.7
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3.6 Show that Eqs. 3.3-6 and 3.3-7 yield the rows of Eq. 2.4-3 from the stationary condition $d\Pi_{p} = 0$ .
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3.7 Displacement d.o.f. $d_{i}$ at ends of a uniform bar element have the directions shown. Write an expression for $\Pi_p$ in terms of these d.o.f. From the stationary condition $d\Pi_p = 0$ , obtain the element stiffness matrix that operates on these four d.o.f.
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3.8 Verify that Eqs. 2.3-8 are produced when the stationary condition $d\Pi_{p} = 0$ is applied to Eq. 3.3-8.
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3.9 Use the method of stationary potential energy to derive the 4 by 4 stiffness matrix for the structure described in (a) Problem 1.13, and (b) Problem 2.2.
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3.10 Verify that Eqs. 3.3-5 and 3.3-9 yield Eq. 3.3-3.
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# Section 3.4
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Section 3.4
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3.11 Write the term $\frac{1}{2}\{\epsilon\}^{T}[E]\{\epsilon\}$ in Eq. 3.4-1 for an isotropic material in a condition of plane stress in the xy plane (for which [E] is 3 by 3). Specialize this expression for the case of uniaxial stress $\sigma_{x}$ . Check your result against Eq. 3.4-6.
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3.12 In the beam of Fig. 3.4-1b, let initial strain and initial stress be given by $\epsilon_0 = -z\kappa_0$ and $\sigma_0 = -m_0z/I$ , respectively. Here $\kappa_0$ and $m_0$ are regarded as initial curvature and initial moment, both considered positive when associated with a concave-up condition of the beam. Determine the contributions of $\kappa_0$ and $m_0$ to Eq. 3.4-8.
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3.13 Repeat the example of Eqs. 3.4-9 to 3.4-11, but account for heating by use of $\sigma_0$ rather than $\epsilon_0$ .
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# Section 3.5
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Section 3.5
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3.14 Verify that the first of Eqs. 3.5-6a is indeed given by the conditions $\partial\Pi_{p}/\partial a_{1}=\partial\Pi_{p}/\partial a_{2}=0.$
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3.15 Verify that the $a_{i}$ of Eq. 3.5-9 result from the use of the three-term polynomial $u = a_{1}x + a_{2}x^{2} + a_{3}x^{3}$ in a Rayleigh-Ritz solution.
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3.16 Consider the two approximate solutions and the one exact solution in Section 3.5 (Eqs. 3.5-5, 3.5-6, and 3.5-8). At what point or points is the differential equation of equilibrium satisfied by each solution?
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3.17 Obtain one-term and two-term solutions for a uniform axially loaded bar, analogous to Eqs. 3.5-5 and 3.5-6, if load $q$ is replaced by concentrated forces at $x = L_T / 3$ , $x = 2L_T / 3$ , and $x = L_T$ . Each of the three forces is directed to the right and is of magnitude $P$ .
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<!-- source-page: 123 -->
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3.18 Consider a cantilever beam of length L, fixed at end x = 0 and carrying a moment load $M_{L}$ at x = L. Write an admissible series for lateral displacement w based on either sine or cosine functions.
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3.19 Consider a uniform cantilever beam of length $L$ , fixed at end $x = 0$ and carrying a transverse force $F$ at $x = L$ .
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(a) Let the lateral displacement field be $w = a_{1}x^{3}$ , where $a_{1}$ is a constant. Is this field admissible? Explain.
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(b) Write a polynomial field for $w$ that is better than that of part (a). Let the field contain three terms, each of the form $a_{i}x^{j}$ , where $i = 1,2,3$ and $j$ is an integer such that the term is admissible.
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(c) Without calculation, can you predict the quality of the answers obtainable from the field of part (b) and the numerical value of any of the $a_{i}$ ?
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(d) Use the field of part (a) to find the deflection of force $F$ .
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3.20 A uniformly loaded beam of constant flexural stiffness $EI$ is simply supported at its ends $x = 0$ and $x = L$ . In parts (a) and (b), determine the deflection and bending moment predicted at $x = L / 2$ by a Rayleigh-Ritz solution that has a single d.o.f. Compare exact and approximate results.
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(a) Use a single-d.o.f. algebraic expression—that is, a d.o.f. $a_1$ times a function that contains $x$ and $x^2$ .
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(b) Use one term of a sine series.
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(c) Why should you anticipate that part (b) will be better than part (a) if part (a) is the simplest admissible function?
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3.21 The uniform cantilever beam shown carries uniformly distributed load of intensity q, tip force $P_{L}$ , and tip moment $M_{L}$ . In parts (a) and (b), compute Rayleigh–Ritz approximations for displacement and rotation at the tip. Compare these results with formulas from beam theory, and explain why the Rayleigh–Ritz result is or is not exact.
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(a) Use one term of a polynomial series.
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(b) Use two terms of a polynomial series.
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(c) In part (a), for which of the given loadings is $w = w(x)$ exact away from the tip—that is, for $0 < x < L$ ? Why?
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<details>
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<summary>text_image</summary>
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y
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q
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P_L
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x
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L
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M_L
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</details>
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Problem 3.21
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<details>
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<summary>text_image</summary>
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z₁w
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P
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x
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L/2
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L/2
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</details>
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Problem 3.23
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3.22 If loads $P_{L}$ and $M_{L}$ in Problem 3.21 were applied at x = L/2 rather than at the tip, how many series terms would be needed in order to obtain the exact displacement at x = L? Explain.
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3.23 The uniform beam shown is simply supported and carries a force P at its center. Use the infinite series
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$$
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w = \sum_ {i = 1} ^ {n} a _ {i} \sin \frac {i \pi x}{L}
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$$
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<!-- source-page: 124 -->
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in a Rayleigh–Ritz solution for deflection and bending moment at the center. Compare exact and approximate results for n = 1, 2, 3, and 4.
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3.24 Repeat Problem 3.23 with load P replaced by a uniformly distributed downward load of intensity q.
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# Section 3.6
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3.25 A complete polynomial of degree n in x, y, and z contains $(n + 3)!/6n!$ terms. With this in mind, construct a “Pascal tetrahedron” analogous to the Pascal triangle in Fig. 3.6-1.
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3.26 (a) Compute the work done by applied load $q = cx$ in going through the exact displacement $u$ of Eq. 3.5-8.
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(b) Similarly, compute the work done by load $q$ in each of the two approximate solutions (Eqs. 3.5-5 and 3.5-6). What conclusion can you draw?
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3.27 (a) Compute the strain energy associated with the exact solution given in Eq. 3.5-8. How is this energy related to the work computed in Problem 3.26a, and why?
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(b) Similarly, compute the strain energy associated with the two approximate solutions (Eqs. 3.5-5 and 3.5-6), and compare answers with work values computed in Problem 3.26b.
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# Section 3.7
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3.28 A certain functional is $\Pi = \int F dx$ , in which $F = c_{1}\phi_{,xx}^{2} + c_{2}\phi_{,x}^{2} + c_{3}\phi^{2} + c_{4}\phi + c_{5}$ and the five $c_{i}$ are constants. What is the Euler equation?
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3.29 A certain physical problem has the functional
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$$
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\Pi = \int_ {0} ^ {L} \left(\frac {1}{2} \phi_ {, x} ^ {2} - 5 0 \phi\right) d x
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$$
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Essential boundary conditions are $\phi = 0$ at $x = 0$ and $\phi = 20$ at $x = L$ . What is $\phi$ as a function of $x$ and $L$ ?
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3.30 Discard terms that contain $\{\mathbf{F}\}$ and $\{\mathbf{M}\}$ from Eq. 3.4-8, and find the Euler equation of a uniform beam under lateral load $q$ . What is the Euler equation if $EI$ is not constant?
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3.31 The potential energy of an isotropic plate that carries lateral pressure q is
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$$
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\Pi_ {p} = \frac {D}{2} \int \int \left\{(w _ {, x x} + w _ {, y y}) ^ {2} - 2 (1 - \nu) [ w _ {, x x} w _ {, y y} - w _ {, x y} ^ {2} ] - \frac {2 q}{D} \right\} d x d y
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$$
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where $D$ is a constant called flexural rigidity. Show that the Euler equation is $\nabla^{4}w = q / D$ , where $\nabla^{4}$ is the biharmonic operator.
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3.32 Consider the functional
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$$
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\Pi = \int \left(\frac {M ^ {2}}{2 E I} + M _ {, x} w _ {, x} + q w\right) d x
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$$
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for a uniform beam that carries distributed lateral load q. Bending moment M and lateral deflection w are each regarded as dependent variables. Derive the two Euler equations. Do they have the form expected from beam theory?
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<!-- source-page: 125 -->
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<details>
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<summary>text_image</summary>
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y
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i j
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x,u
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x_i
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x_j
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</details>
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Problem 3.33
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<details>
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<summary>text_image</summary>
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y
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l
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l
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i
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k
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j
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x,u
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</details>
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Problem 3.34
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# Section 3.8
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3.33 The bar element shown is to have an axial displacement field $u$ that is linear in $x$ and depends on nodal d.o.f. $u_{i}$ and $u_{j}$ . The shape function matrix $[\mathbf{N}]$ is a function of $x_{i}, x_{j}$ , and $x$ . Write $[\mathbf{N}]$ by inspection if you can. Then derive $[\mathbf{N}]$ by use of the [A]-matrix method described in Section 3.8.
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3.34 The three-node bar element shown is to have an axial displacement field $u$ that is quadratic in $x$ and depends on nodal d.o.f. $u_i, u_j$ , and $u_k$ . Determine the shape function matrix $\lfloor \mathbf{N} \rfloor$ in terms of $x$ and $\ell$ .
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# Section 3.9
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3.35 Use $\{\mathbf{D}\}$ as given by Eq. 3.9-10, and $[\mathbf{K}]$ and $\{\mathbf{R}\}$ as given in Eq. 3.9-9, to compute $U = \frac{1}{2}\{\mathbf{D}\}^T [\mathbf{K}]\{\mathbf{D}\}$ and $\Omega = -\{\mathbf{D}\}^T \{\mathbf{R}\}$ . How is $U$ related to $\Omega$ ? What are the precentage errors of $U$ and $\Omega$ ? (Exact values of $U$ and $\Omega$ are computed in Problems 3.26a and 3.27a.)
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3.36 Consider a uniform bar under axial load. Displacements and stresses may be obtained by either the classical or the finite element form of the Rayleigh-Ritz method. Which form (if either) gives exact results, and how many d.o.f. are required for exactness, if the axial loading is (a) distributed in the form $q = c(L_T - x)$ , and (b) concentrated, with equal forces $P$ at $x = L$ , $x = 2L$ , and $x = 3L$ ? The total length of the bar is $L_T = 3L$ .
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3.37 Consider a bar of length $L_{T}$ that is fixed at $x = 0$ and free at $x = L_{T}$ . The bar carries a distributed axial loading of intensity $q = c(L_{T} - x)$ , where $c$ is a constant. Generate a finite element solution like that in Section 3.9, using
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(a) a single element of length $L_{T}$ .
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(b) two elements, each of length $L = L_{T} / 2$ .
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(c) three elements, each of length $L = L_{T} / 3$ .
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3.38 Repeat the finite element solution of Section 3.9 but let the respective elements have lengths $L_{T} / 6$ , $L_{T} / 3$ , and $L_{T} / 2$ (reading left to right).
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3.39 (a) Apply Eq. 3.9-3, with $\lfloor \mathbf{B} \rfloor$ taken from Eq. 3.9-1, to determine the stiffness matrix of a bar element that is tapered: its cross-sectional area is $A = A_0(3L - 2x)/L$ , where $A_0$ is the cross-sectional area at the right end ( $x = L$ ).
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(b) What is the exact stiffness matrix of this tapered bar? Find out by displacing d.o.f. one at a time and using an elementary mechanics of materials analysis to compute the axial force required.
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3.40 The rigid bar shown rests on an elastic foundation. When displaced laterally an amount w, the foundation applies a force kw dx to a length dx of the bar, where k is a constant. Determine the 2 by 2 stiffness matrix that operates
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<!-- source-page: 126 -->
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<details>
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<summary>text_image</summary>
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w_i
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L
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w_j
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i
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j
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</details>
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Problem 3.40
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on $w_{i}$ and $w_{j}$ . Suggestion: Express strain energy U in terms of k, L, $w_{i}$ , and $w_{j}$ , then write U in the form $\{d\}^{T}[k]\{d\}/2$ , and identify [k].
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# Section 3.10
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3.41 As suggested below Eq. 3.10-8, indicate assembly of elements when $\Pi$ is written, and see whether Eq. 3.10-8 again results.
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3.42 For each of the following problems, obtain finite element formulations in a form analogous to Eqs. 3.10-6 and 3.10-7. Details such as specific element shape functions are not required.
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(a) Plane beam (use the integrals in Eq. 3.4-8).
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(a) Plane beam (see Problem 3.32). Use two interpolating fields, one for $M$ that depends on nodal moments $\{\mathbf{M}_e\}$ and one for $w$ that depends on nodal lateral deflections $\{\mathbf{w}_e\}$ .
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(c) Acoustic modes in a cavity with rigid walls. The functional is
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$$
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\Pi = \int \left(p _ {, x} ^ {2} + p _ {, y} ^ {2} + p _ {, z} ^ {2} - \frac {\omega^ {2}}{c ^ {2}} p ^ {2}\right) d V
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$$
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where $p = p(x,y,z)$ is the amplitude of gas pressure that varies with time, $\omega$ is the circular frequency, and $c$ is the speed of sound. Let $p = \lfloor \mathbf{N} \rfloor \{\mathbf{p}_e\}$ , where $\{\mathbf{p}_e\}$ represents nodal pressures.
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3.43 The uniform bar shown is to act as a heat conduction element, with $T =$ temperature, $k =$ thermal conductivity, $q =$ axial heat flux in the bar per unit of cross-sectional area $A$ , and $H =$ lateral heat flux per unit length. From the functional
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$$
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\Pi = A q _ {j} T _ {j} - A q _ {i} T _ {i} + \frac {1}{2} \int_ {0} ^ {L} k T _ {, x} ^ {2} A d x - \int_ {0} ^ {L} H T d x
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$$
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determine expressions for [k] and $\{\mathbf{r}_Q\}$ in the element equations $[\mathbf{k}]\{\mathbf{T}_e\} = \{\mathbf{r}_O\}$ , where $\{\mathbf{T}_e\} = \lfloor T_l - T_j \rfloor^T$ . Let $T = \lfloor \mathbf{N} \rfloor \{\mathbf{T}_e\}$ and $H = \lfloor \mathbf{N} \rfloor \left\lfloor H_i - H_j \right\rfloor^T$ , where
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$$
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[ \mathbf {N} ] = \left\lfloor \frac {L - x}{L} \frac {x}{L} \right\rfloor .
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$$
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<details>
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<summary>text_image</summary>
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H = H(x)
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y
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q_i A i j q_j A x
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L
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</details>
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Problem 3.43
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<!-- source-page: 127 -->
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# Section 3.12
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3.44 In Fig. 3.12-1b, let $x_{1} = 0$ , $x_{2} = 2$ , and $x_{3} = 3$ . Then do as follows.
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(a) Verify numerically that $\Sigma N_{i} = 1$ .
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(b) According to Eq. 3.12-4, $\Sigma N_{i,x} = 0$ . Verify this property numerically.
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3.45 For the four points in Fig. 3.12-2, let the respective $x_{i}$ be 1, 3, 5, and 8, and let the respective $\phi_{i}$ be 2, 2, 2, and 5.
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(a) Use Lagrange's formula to obtain the interpolating curve.
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(b) What values of $\phi$ does Lagrange's formula predict at $x = 2$ , $x = 4$ , and $x = 7$ ?
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3.46 Sketch the four $N_{i}$ of Eqs. 3.12-10 in a fashion analogous to Fig. 3.12-1. That is, sketch the element in isometric view, with $\phi = N_{i}\phi_{i}$ shown normal to the $xy$ plane, as in Fig. 1.1-3.
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3.47 (a) Determine shape functions $N_{i}$ for the nine-node Lagrange element shown.
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(b) What is $\phi = \phi(x)$ for this element when written in the form of Eq. 3.12-7?
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(c) Show how $N_{1}$ varies over the element by making an isometric sketch and showing $\phi = N_{1}\phi_{1}$ normal to the $xy$ plane (similar to Fig. 1.1-3). Do the same for $N_{8}$ and $N_{9}$ .
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<details>
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||||
<summary>text_image</summary>
|
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|
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y
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a a
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4 7 3
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8 9 6
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1 5 2
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x
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b
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b
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</details>
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Problem 3.47
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3.48 The element shown has three linear edges and one quadratic edge. Determine the five shape functions $N_{i}$ . Suggestion: after interpolating along edges $y = -b$ and $y = +b$ , interpolate linearly in the $y$ direction.
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3.49 Determine shape functions $N_{i}$ for the eight-node rectangular parallelepiped shown. Overall side lengths are 2a, 2b, and 2c. Suggestion: Infer answers from the pattern seen in Eqs. 3.12-10, then test the answers.
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<details>
|
||||
<summary>text_image</summary>
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|
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y
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a a
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4 3
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b
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x
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1 5 2
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||||
b
|
||||
</details>
|
||||
|
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Problem 3.48
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<details>
|
||||
<summary>text_image</summary>
|
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|
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y
|
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3
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4
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8
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7
|
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2c
|
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2b
|
||||
x
|
||||
z
|
||||
2
|
||||
1
|
||||
2a
|
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5
|
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6
|
||||
</details>
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|
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Problem 3.49
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|
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<!-- source-page: 128 -->
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|
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# Section 3.13
|
||||
|
||||
3.50 Determine matrix $[A]^{-1}$ of Eq. 3.13-4 and verify the shape functions given in Fig. 3.13-2. Suggestion: Regard Eqs. 3.13-3 as four equations to be solved for the four $a_{i}$ . Arrange the results in matrix format and identify a 4 by 4 matrix as $[A]^{-1}$ .
|
||||
3.51 Imagine that a curve $\phi = \phi(x)$ is to be fitted to three data values: $\phi_1$ and $\theta_1$ at $x = 0$ and $\phi_2$ at $x = L$ (analogous to Fig. 3.13-1, but with $\theta_2$ unspecified). Determine the shape functions. Also sketch them and check their behavior at $x = 0$ and at $x = L$ (in the fashion of Fig. 3.13-2).
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3.52 Show that cubic shape functions (Fig. 3.13-2) do not provide $C^2$ continuity. Suggestion: Examine the node shared by two adjacent elements, only one of which has nonzero d.o.f.
|
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3.53 Imagine that at points $A, B$ , and $C$ in the sketch one knows both ordinate and slope data. Slope is indicated by a short line through a data point. Without calculation, sketch
|
||||
|
||||
(a) a Lagrange interpolation curve through all three points.
|
||||
|
||||
(b) piecewise interpolation of $C^0$ continuity.
|
||||
|
||||
(c) piecewise interpolation of $C^1$ continuity.
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
φ
|
||||
A
|
||||
B
|
||||
C
|
||||
Problem 3.53
|
||||
x
|
||||
</details>
|
||||
|
||||
3.54 Shape functions $N_{i}$ of $C^0$ elements satisfy the relation $\Sigma N_{i} = 1$ . Such is not the case for the $N_{i}$ of Fig. 3.13-2. Why?
|
||||
|
||||
<!-- source-page: 129 -->
|
||||
|
||||
# DISPLACEMENT-BASED ELEMENTS FOR STRUCTURAL MECHANICS
|
||||
|
||||
General expressions for the element stiffness matrix [k] and the element load vector $\{r_{e}\}$ are derived, then used to formulate simple elements. Also discussed are how equilibrium and compatibility are approximated by the solution, requirements for convergence with mesh refinement, and procedures for stress computation.
|
||||
|
||||
# 4.1 FORMULAS FOR ELEMENT
|
||||
|
||||
MATRICES [k] AND $\{\mathbf{r}_e\}$
|
||||
|
||||
Our discussion is restricted to elements based on displacement fields. Other elements, often equally good but less popular, are based on stress fields. Stress field elements are not discussed in this chapter.
|
||||
|
||||
The derivation of finite element formulas is a straightforward procedure, which can be verbally summarized as follows [4.1]. Displacements are taken as the dependent variables. Therefore, the appropriate functional for a Rayleigh-Ritz solution is $\Pi_p$ , the expression for potential energy. We select an admissible displacement field, defined in piecewise fashion so that displacements within any element are interpolated from nodal d.o.f. of that element, then evaluate $\Pi_p$ in terms of nodal d.o.f. Using the principle of stationary potential energy, we write $d\Pi_p = 0$ , from which we obtain algebraic equations to be solved for the nodal d.o.f. In the course of this argument we identify certain expressions as the element stiffness matrix [k] and the element load vector $\{\mathbf{r}_e\}$ . Details of this derivation are now described.
|
||||
|
||||
The starting point is the expression for potential energy in a linearly elastic body, Eq. 3.4-1, which is repeated here as Eq. 4.1-1,
|
||||
|
||||
$$
|
||||
\Pi_ {p} = \int_ {V} \left(\frac {1}{2} \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} \} - \{\boldsymbol {\epsilon} \} ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} + \{\boldsymbol {\epsilon} \} ^ {T} \{\boldsymbol {\sigma} _ {0} \}\right) d V \tag {4.1-1}
|
||||
$$
|
||||
|
||||
$$
|
||||
- \int_ {V} \{\mathbf {u} \} ^ {T} \{\mathbf {F} \} d V - \int_ {S} \{\mathbf {u} \} ^ {T} \{\boldsymbol {\Phi} \} d S - \{\mathbf {D} \} ^ {T} \{\dot {\mathbf {P}} \}
|
||||
$$
|
||||
|
||||
in which $\{\mathbf{u}\} = \lfloor u \quad v \quad w \rfloor^T$ , the displacement field
|
||||
|
||||
$\{\epsilon\} = \left\lfloor \epsilon_x \quad \epsilon_y \quad \epsilon_z \quad \gamma_{xy} \quad \gamma_{yz} \quad \gamma_{zx} \right\rfloor^T$ , the strain field
|
||||
|
||||
[E] = the material property matrix (e.g., Eq. 1.7-3)
|
||||
|
||||
$\{\pmb{\epsilon}_0\} ,\{\pmb{\sigma}_0\} =$ initial strains and initial stresses (Eq. 1.7-8)
|
||||
|
||||
$\{\mathbf{F}\} = \left\lfloor F_x\quad F_y\quad F_z\right\rfloor^T$ , body forces (Eq. 1.6-4)
|
||||
|
||||
$\{\Phi\} = \left\lfloor \Phi_x \quad \Phi_y \quad \Phi_z \right\rfloor^T$ , surface tractions (Eq. 1.6-4)
|
||||
|
||||
<!-- source-page: 130 -->
|
||||
|
||||
$\{D\} = \text{nodal d.o.f. of the structure}$
|
||||
|
||||
$\{\mathbf{P}\} =$ loads applied to d.o.f. by external agencies
|
||||
|
||||
S, V = surface area and volume of the structure
|
||||
|
||||
Shorter expressions may be used for problems that are two- or one-dimensional, as will be seen subsequently.
|
||||
|
||||
s will be seen subsequently.
|
||||
Displacements within an element are interpolated from element nodal d.o.f. {d},
|
||||
|
||||
$$
|
||||
\{\mathbf {u} \} = [ \mathbf {N} ] \{\mathbf {d} \} \tag {4.1-2}
|
||||
$$
|
||||
|
||||
where [N] is the shape function matrix. The particular form of [N] need not be specified yet, but a form must eventually be written. The form selected has much to do with the quality of the approximate solution.
|
||||
|
||||
From here onward, the derivation requires only straightforward manipulation. Strains are obtained from displacements by differentiation. Thus
|
||||
|
||||
$$
|
||||
\{\epsilon \} = [ \partial ] \{\mathbf {u} \} \quad \text { yields } \quad \{\epsilon \} = [ \mathbf {B} ] \{\mathbf {d} \}, \quad \text { where } \quad [ \mathbf {B} ] = [ \partial ] [ \mathbf {N} ] \tag {4.1-3}
|
||||
$$
|
||||
|
||||
The differential operator matrix $[\partial]$ is given by Eq. 1.5-6; its size is 6 by 3 for three-dimensional problems and 3 by 2 for two-dimensional problems. Substitution of the expressions for $\{\mathbf{u}\}$ and $\{\epsilon\}$ into Eq. 4.1-1 yields
|
||||
|
||||
$$
|
||||
\Pi_ {p} = \frac {1}{2} \sum_ {n = 1} ^ {\text { numel }} \{\mathbf {d} \} _ {n} ^ {T} [ \mathbf {k} ] _ {n} \{\mathbf {d} \} _ {n} - \sum_ {n = 1} ^ {\text { numel }} \{\mathbf {d} \} _ {n} ^ {T} \{\mathbf {r} _ {e} \} _ {n} - \{\mathbf {D} \} ^ {T} \{\mathbf {P} \} \tag {4.1-4}
|
||||
$$
|
||||
|
||||
where summation symbols indicate that we include contributions from all numel elements of the structure, and we have defined
|
||||
|
||||
$$
|
||||
\text { element stiffness matrix } \quad \boxed {[ \mathbf {k} ] = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] [ \mathbf {B} ] d V} \tag {4.1-5}
|
||||
$$
|
||||
|
||||
element load vector
|
||||
|
||||
$$
|
||||
\begin{array}{r l} \{\mathbf {r} _ {e} \} & = \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} [ \mathbf {E} ] \{\boldsymbol {\epsilon} _ {0} \} d V - \int_ {V _ {e}} [ \mathbf {B} ] ^ {T} \{\boldsymbol {\sigma} _ {0} \} d V \\ & + \int_ {V _ {e}} [ \mathbf {N} ] ^ {T} \{\mathbf {F} \} d V + \int_ {S _ {e}} [ \mathbf {N} ] ^ {T} \{\boldsymbol {\Phi} \} d S \end{array} \tag {4.1-6}
|
||||
$$
|
||||
|
||||
where $V_{e}$ denotes the volume of an element and $S_{e}$ its surface. In the surface integral, [N] is evaluated on $S_{e}$ .
|
||||
|
||||
Despite its formidable appearance, the expression for $\{\mathbf{r}_e\}$ is less important than the expression for [k]. In essence, the $\{\mathbf{r}_e\}$ expression states how certain loads can be dealt with to best advantage. Use of an ad hoc method instead may provide acceptable accuracy. Examples follow (see below Eq. 4.3-14).
|
||||
|
||||
To complete the derivation we must determine the algebraic equations to be solved for nodal d.o.f., as follows. Every d.o.f. in an element vector $\{\mathbf{d}\}$ also appears in the vector of global (i.e., structural) d.o.f. $\{\mathbf{D}\}$ . Therefore, as argued in Sections 2.6 and 2.7, $\{\mathbf{D}\}$ can replace $\{\mathbf{d}\}$ in Eq. 4.1-4 if $[\mathbf{k}]$ and $\{\mathbf{r}_e\}$ of every element are conceptually expanded to structure size. Thus Eq. 4.1-4 becomes
|
||||
|
||||
$$
|
||||
\Pi_ {p} = \frac {1}{2} \{\mathbf {D} \} ^ {T} [ \mathbf {K} ] \{\mathbf {D} \} - \{\mathbf {D} \} ^ {T} \{\mathbf {R} \} \tag {4.1-7}
|
||||
$$
|
||||
Reference in New Issue
Block a user