김경종
26 KiB
original configuration with respect to shape, material properties, and support conditions. In the plate of Fig. 8.15-1a, each dashed line is an axis of reflective symmetry. An axis normal to the plate through its center is an axis of rotational symmetry, because successive 90^{\circ} rotations bring the structure into coincidence with itself. Other examples of rotational symmetry include solids of revolution (Chapter 10) and cyclic symmetry (Section 8.16).
A symmetric structure may carry symmetric or antisymmetric loads. Antisymmetry of loads exists if a single reflection of the structure, with its loads, followed by reversal of all loads, results in self-coincidence. An example appears in Fig. 8.15-1b, in which the plane of reflection is normal to the beam and intersects its center.
Symmetric loads, when acting on symmetric structures, produce symmetric effects in linear static analysis [8.33] . To exploit this rule, we analyze a part of the structure with appropriate boundary conditions. At a plane of geometric symmetry, displacement boundary conditions for symmetric loading are
- No translational motion is perpendicular to a plane of geometric symmetry.
- Rotation vectors have no component in a plane of geometric symmetry.
At a plane of geometric symmetry, displacement boundary conditions for anti-symmetric loading are
- Translational motion has no component in a plane of geometric symmetry.
- Rotation vectors have no component perpendicular to a plane of geometric symmetry.
For symmetric and antisymmetric loads acting on symmetric structures, the resulting displacements and stresses are respectively symmetric and antisymmetric.
Consider, for example, the plate of Fig. 8.15-1a. Let finite elements (not shown) have as d.o.f. lateral translation w and rotations \theta_{y} and \theta_{x} about x and y axes, respectively (e.g., \theta_{y} = w_{,y} ). Under uniform load, the loading has symmetry with respect to x and y axes. Quadrant ABCD can be analyzed with boundary conditions w = 0 along AB and AD, \theta_{x} = 0 along AB and BC, and \theta_{y} = 0 along AD and CD. If the load is uniform but alternately up and down over the four quadrants in checkerboard fashion, the loading has antisymmetry with respect to x and y axes. Quadrant ABCD can be analyzed with boundary conditions w = 0 along AB, BC, CD and DA, \dot{\theta}_{y} = 0 along AD and BC, and \theta_{x} = 0 along AB and CD.
Remarks. In vibration analysis, symmetry must be exploited with caution, as symmetry of geometry does not imply symmetry of all vibration modes. Similarly, caution is needed if buckling or other nonlinear behavior arises, as symmetries present in an initial linear analysis may subsequently disappear.
It may be expedient to express a load as the sum of symmetric and antisymmetric components (Fig. 8.15-1b). Thus, rather than analyzing the entire structure once, one analyses half the structure twice [8.34].
Additional types of symmetries, known as skew-symmetric and skew-antisymmetric, can be identified and exploited [8.35].
Before a large model is analyzed, a model with few d.o.f. can be studied to check that anticipated symmetries indeed exist, and perhaps discover unanticipated symmetries.
8.16 CYCLIC SYMMETRY
A structure such as an impeller in a centrifugal pump is not a solid of revolution and cannot be analyzed as such. Nor is there a plane of reflective symmetry. Yet one can recognize a repetition of geometry and loading (Fig. 8.16-1). This circumstance is called cyclic symmetry, sectorial symmetry, or rotational periodicity. It is possible to analyze a representative substructure rather than the entire structure. The procedure is as follows [8.36].
For a typical substructure AABB, Fig. 8.16-1b, let [K]\{D\} = \{R\} represent the substructure equations, where \{D\} includes all substructure d.o.f. In partitioned form, these equations are
\left[ \begin{array}{l l l} \mathbf {K} _ {I I} & \mathbf {K} _ {I A} & \mathbf {K} _ {I B} \\ \mathbf {K} _ {I A} ^ {T} & \mathbf {K} _ {A A} & \mathbf {K} _ {A B} \\ \mathbf {K} _ {I B} ^ {T} & \mathbf {K} _ {A B} ^ {T} & \mathbf {K} _ {B B} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \\ \mathbf {D} _ {B} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {I} \\ \mathbf {R} _ {A} \\ \mathbf {0} \end{array} \right\} + \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {F} _ {A} \\ \mathbf {F} _ {B} \end{array} \right\} \tag {8.16-1}
where \{D_{A}\} and \{D_{B}\} contain d.o.f. on interface boundaries AA and BB, respectively, and \{D_{I}\} contains all remaining d.o.f. (from nodes within the substructure and noninterface nodes on boundaries r = a and r = b). Loads \{F_{A}\} and \{F_{B}\} result from elastic deformations and are applied along AA and BB by neighboring substructures. Loads \{R_{I}\} and \{R_{A}\} represent imposed loads, typically caused by rotation and uneven (but cyclically symmetric) heating. Loads \{R_{B}\} are absent because imposed loads on interface boundaries must appear on only one interface boundary; if mistakenly placed on both, the substructure receives twice the load intended.
All repeating substructures are identical. Therefore, subject to a subsequent caution,
\{\mathbf {D} _ {B} \} = \{\mathbf {D} _ {A} \} \quad \text { and } \quad \{\mathbf {F} _ {B} \} = - \{\mathbf {F} _ {A} \} \tag {8.16-2}
(a)
Figure 8.16-1. (a) A hypothetical pump impeller, viewed along its axis of rotation. Vanes such as DD, seen here in edge view, are mounted on a circular disk. (b) A typical repeating substructure.
Using transformation procedures explained in Chapter 7, we write
\left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \\ \mathbf {D} _ {B} \end{array} \right\} = [ \mathbf {T} ] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \end{array} \right\}, \quad \text { where } \quad [ \mathbf {T} ] = \left[ \begin{array}{l l} \mathbf {I} & \mathbf {0} \\ \mathbf {0} & \mathbf {I} \\ \mathbf {0} & \mathbf {I} \end{array} \right] \tag {8.16-3}
and [I] is a unit matrix. The transformations [T]^{T}[K][T] and [T]^{T}\{R\} , applied to [K] and \{R\} of Eq. 8.16-1, yield
\left[ \begin{array}{c c} \mathbf {K} _ {I I} & \mathbf {K} _ {I A} + \mathbf {K} _ {I B} \\ \mathbf {K} _ {I A} ^ {T} + \mathbf {K} _ {I B} ^ {T} & \mathbf {K} _ {A A} + \mathbf {K} _ {A B} + \mathbf {K} _ {A B} ^ {T} + \mathbf {K} _ {B B} \end{array} \right] \left\{ \begin{array}{l} \mathbf {D} _ {I} \\ \mathbf {D} _ {A} \end{array} \right\} = \left\{ \begin{array}{l} \mathbf {R} _ {I} \\ \mathbf {R} _ {A} \end{array} \right\} \tag {8.16-4}
in which \{F_{A}\} and \{F_{B}\} do not appear because of Eq. 8.16-2. Solution for nodal d.o.f. and stresses now proceeds in the usual way.
Caution. The number and location of nodes along AA and BB must correspond exactly, and d.o.f. at corresponding nodes must have the same orientation with respect to the interface boundary. For example, in Fig. 8.16-1b, if i and j are corresponding nodes (e.g., both the kth node on their respective boundaries), then we must have r_{i} = r_{j} , and d.o.f. at i and j must have directions such as those shown, where u and v are respectively tangent and normal to each interface boundary. Directions u (radial) and v (tangential) in polar coordinates with point C as pole are also acceptable, but directions u and v in Cartesian coordinates are not acceptable.
Equation 8.16-4 can be produced automatically by the assembly process, thus avoiding the transformation defined by Eq. 8.16-3. The trick is to assign the same node number to corresponding nodes along AA and BB; for example, nodes i and j cited in the preceding paragraph would both be given the number (say) 125. Thus, the additions seen in Eq. 8.16-4 are produced automatically when elements are assembled. One must of course use actual node point coordinates in the formulation of element matrices.
PROBLEMS
Section 8.1
8.1 Apply Eq. 8.1-3 to the problem in Fig. 2.11-1. Specifically, eliminate u_{2} and u_{3} in Fig. 2.11-1b, and obtain the condensed equation 2u_{4} = 8 seen in Fig. 2.11-1d.
8.2 What values of the 18 d.o.f. in Eq. 8.1-4 are associated with rigid-body translation of magnitude \bar{u} in the +x direction?
8.3 A 9 by 9 transformation matrix [T] can be applied to the shape function matrix [N] associated with Table 6.6-1 to yield the shape function matrix associated with Eq. 8.1-4. Write this matrix [T] and the equation that relates the two shape function matrices.
8.4 A three-node bar element with a central node 3 is shown, along with its stiffness matrix, which operates on d.o.f. \lfloor u_1 \quad u_2 \quad u_3 \rfloor .
(a) Determine the 2 by 2 matrix [k] produced by condensation of u_{3} .
text_image
1 3 2 x,u L/2 L/2
[ \mathbf {k} ] = \frac {A E}{3 L} \left[ \begin{array}{r r r} 7 & 1 & - 8 \\ 1 & 7 & - 8 \\ - 8 & - 8 & 1 6 \end{array} \right]
Problem 8.4
(b) Apply a uniformly distributed axial load of intensity q . From the consistent load vector \{\mathbf{r}_e\} , obtain a condensed load vector associated with u_{1} and u_{2} .
8.5 A stiffness matrix and a consistent load vector can be formulated for the three-node bar element of Problem 8.4 by use of the displacement field
u = \frac {L - x}{L} u _ {1} + \frac {x}{L} u _ {2} + x (L - x) a _ {1}
where a_1 is a nodeless d.o.f.
(a) Determine the 3 by 3 stiffness matrix [k] dictated by the given u field. Let the element be uniform.
(b) Under what circumstances do you think the added mode x(L - x)a_1 will improve the results given by the basic linear element? In what stage of a finite element stress analysis does a_1 have an effect?
8.6 Imagine that d.o.f. \theta_{1} and \theta_{2} of the standard four-d.o.f. beam element (Fig. 4.2-2 and Eq. 4.2-5) are to be eliminated. What do you think the 2 by 2 condensed matrix [k] will be? Verify your prediction.
8.7 Let a plane frame element be joined to a rotational spring at each end, with respective spring stiffnesses k_{1} and k_{2} (moment per radian). Let \beta_{1} and \beta_{2} be structure node rotations. Rotational d.o.f. \theta_{1} and \theta_{2} of the frame element are to be connected to structure nodes through the rotational springs, so that in Fig. 4.2-2 \theta_{1} \neq \beta_{1} and \theta_{2} \neq \beta_{2} unless k_{1} and k_{2} approach infinity. Translational d.o.f. are to be connected directly, as usual. Beginning with an 8 by 8 stiffness matrix that operates on d.o.f. \lfloor u_{1} \quad w_{1} \quad \theta_{1} \quad u_{2} \quad w_{2} \quad \theta_{2} \quad \beta_{1} \quad \beta_{2} \rfloor^{T} , describe how to determine a 6 by 6 matrix [k] that operates on d.o.f. \lfloor u_{1} \quad w_{1} \quad \beta_{1} \quad u_{2} \quad w_{2} \quad \beta_{2} \rfloor^{T} and is a function of A, E, I, L, k_{1} , and k_{2} .
8.8 Addition to an element of internal d.o.f., such as a_1 and a_2 in Eq. 8.1-4, can be regarded as a device that permits better approximation of equilibrium equations within the element, without affecting interelement compatibility. Accordingly, do you think the constant-strain triangle (Section 5.4) would be improved by addition of the bubble function modes u = \xi_1\xi_2\xi_3a_1 and v = \xi_1\xi_2\xi_3a_2 ? Why or why not?
8.9 Consider the frame of Fig. 8.1-2. Imagine that, before assembly, rotation \theta_{A} is condensed in all four elements that meet at node A . What do you think will be the effect of these condensations, both physically and in the numerical process?
8.10 Cantilever beams AB and BC are identical and are connected by a hinge at B, as shown. Use condensation, as described in connection with Fig. 8.1-2, to evaluate the rotation in both beams at B. Verify your result by elementary beam theory.
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L L P A ① B ② C
Problem 8.10
8.11 In Problem 8.10, how would you determine the value of P needed to produce a prescribed amount of relative rotation between the beams at B?
Section 8.2
8.12 Modify Figs. 8.2-1 and 8.2-2 to allow for NL load cases rather than only one.
Section 8.3
8.13 (a) For the elements shown in Figs. 8.3-1b and 8.3-1c, compute the ratio of element strain energies, U_{1} / U_{2} . (b) Use this result to verify the correctness of Eq. 8.3-5.
8.14 Consider the two beams built of rectangular elements in Table 6.14-1 (one-element case and the first five-element case). If one assumes that Eq. 8.3-5 is approximately true for these beams, what end deflections would be expected? Compare these results with those in Table 6.14-1.
8.15 (a) Do the a_i of Eqs. 8.3-6 represent relative or absolute motions? (b) If, after computation of nodal d.o.f. in a mesh of QM6 elements, the nodeless d.o.f. a_i are omitted from stress computation, what consequences do you expect? Consider, for example, the rectangular-element test cases in Table 6.14-1.
8.16 The sketch shows a cantilever beam modeled by QM6 elements. For the loading shown, will exact values of stresses \sigma_{x} be computed? Why or why not?
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P/2' P/2
Problem 8.16
flowchart
graph TD
A["1"] --> B["2"]
B --> C["3"]
C --> A
style A fill:#fff,stroke:#000
style B fill:#fff,stroke:#000
style C fill:#fff,stroke:#000
Problem 8.17
8.17 All three elements in the beam shown are plane QM6 elements. Examine displacements along sides of element 2 under the moment loading shown. Hence, show that pure bending is not modeled exactly by nonrectangular QM6 elements.
8.18 For the nodal d.o.f. \overline{u} applied in Fig. 8.3-1, show that Eq. 8.3-7 yields Eqs. 8.3-3.
8.19 Imagine that Figs. 6.5-1 and 6.5-2 are to be modified so that they will apply to the QM6 element. Clearly state what changes and additions are required, and supply new coding where needed in Fig. 6.5-1.
Section 8.4
8.20 Show that \delta in Eq. 8.4-1 can be regarded as a beam midspan deflection, as noted below Eq. 8.4-1.
8.21 Following the procedure suggested below Eq. 8.4-2, write shape functions for the nine-d.o.f. triangle of Fig. 8.4-1b.
8.22 (a) Establish the contents of matrix [T] in Eq. 8.4-4.
(b) Use this [T] to determine shape functions N_{i} of a nine-d.o.f. triangle from shape functions N_{i}^{\prime} of a linear-strain triangle.
8.23 Imagine that lateral deflection w of a uniform beam element is defined by three nodal values, as shown.
(a) Establish the 3 by 4 transformation matrix [T] that will convert this element to one that operates on the standard d.o.f. w_{1} , \theta_{1} , w_{2} , and \theta_{2} .
(b) Hence, establish the new shape functions N_{1}, N_{2}, N_{3} , and N_{4} .
(c) What property does the element have that may pose a difficulty?
text_image
w₁ w₃ w₂ 1 3 2 L/2 L/2 x
w = \frac {2 x ^ {2} - 3 L x + L ^ {2}}{L ^ {2}} w _ {1} + \frac {2 x ^ {2} - L x}{L ^ {2}} w _ {2} + \frac {4 x (L - x)}{L ^ {2}} w _ {3}
Problem 8.23
Section 8.5
8.24 Complete the steps of generating [k] for the element of Fig. 8.5-1b, as follows. Use [P] and [R] from Eq. 8.5-12.
(a) Complete matrix [L], begun in Eqs. 8.5-14.
(b) Generate matrix [G], Eq. 8.5-7.
(c) Generate matrix [H], Eq. 8.5-4. For simplicity, let \nu = 0 , so that [\mathbf{E}] = \mathrm{E}[1, 1, \frac{1}{2}] .
(d) Generate [k], Eq. 8.5-10, again for \nu = 0 .
8.25 Use the assumed-stress hybrid method to evaluate [k] for the six-d.o.f. plane triangle shown. Use \{\pmb{\beta}\} = [\beta_1 \beta_2 \beta_3]^T . For simplicity, take Poisson's ratio as zero. (This [k] should agree with the [k] obtained in Problem 4.11c.)
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y,v a 3 b 1 2 x,u
Problem 8.25
8.26 (a) Write the equation \{\sigma\} = [\mathbf{P}]\{\boldsymbol{\beta}\} for a plane element if \sigma_x = \beta_1 + \beta_4x , \sigma_y = \beta_2 + \beta_5y , and \tau_{xy} = \beta_3 . Do you think such an element would be a good one?
(b) If \beta_{4} = \beta_{5} = 0 in part (a), so that \{\pmb{\beta}\} = [\beta_{1}, \beta_{2}, \beta_{3}]^{T} , what defect would you expect to see in the stiffness matrix of a plane eight-d.o.f. rectangular element?
8.27 The beam element shown is to include the effects of transverse shear deformation. If bending moment M is taken as M = \beta_1 + \beta_2x , then the shear force V = \beta_2 satisfies the equilibrium equation dM/dx = V . With \{\sigma\} = [M, V]^T , strain energy in the element is U = \frac{1}{2} \int \{\sigma\}^T \left[ \frac{1}{EI} \cdot \frac{f}{AG} \right] \{\sigma\} dx , where f is a “form factor” ( f = 1.2 for a rectangular cross section). [R] relates nodal moments and shear forces to \{\beta\} , [L] is a unit matrix, and [R] ^T [L] requires no integration. Derive [k] and show that it reduces to Eq. 4.2-5 as shear modulus G becomes large [4.11].
text_image
M₁ → x V₁ L M₂ V₂
text_image
w₁ θ₁ 1 w₂ θ₂ 2 L
Problem 8.27
Section 8.7
8.28 (a) Following the example of Eqs. 8.7-8 and 8.7-9, determine [k] for a standard four-d.o.f. beam element. However, use \{\mathbf{d}_R\} = \lfloor w_1 - w_2 \rfloor^T .
(b) Imagine that the leading diagonal coefficient of [\mathbf{k}_{EE}] in part (a) is in error by an amount e . Show that [\mathbf{k}] still represents rigid-body motion correctly.
8.29 The two-spring structure shown is allowed axial nodal displacements u_{1}, u_{2} , and u_{3} . If \{\mathbf{d}_R\} = u_1 , write [\mathbf{k}_{EE}] , and from it determine [\mathbf{k}] .
8.30 (a) A bar element of axial stiffness k = AE / L is permitted only axial nodal displacements u_{1} and u_{2} . Write [\mathbf{k}_{EE}] , and from it determine [\mathbf{k}] .
(b) Repeat part (a), but let there be four d.o.f. \{\mathbf{d}\} = \lfloor u_1 \quad v_1 \quad u_2 \quad v_2 \rfloor^T , as in Eq. 2.4-3, so that plane motion is possible.
8.31 A flat elastic disk has inside and outside radii r_1 and r_2 , as shown. Nodal d.o.f. are circumferential displacements v_1 and v_2 . When the disk is fixed at r = r_1 , the ratio of torque T_2 on edge r = r_2 to the resulting angle of twist \theta_2 is a number C . Determine the stiffness matrix [k] that operates on d.o.f. v_1 and v_2 , in terms of C, r_1 , and r_2 . Verify that [k]{d} = {0} if {d} represents rigid-body motion.
Problem 8.29
text_image
r₂ r₁ v₁ v₂ T₂,θ₂
Problem 8.31
Section 8.8
8.32 The structure shown is built of plane elements. D.o.f. (at each corner node) consist of u, v, u_{,x}, v_{,x}, u_{,y} , and v_{,y} . Pressure p acts along edge AB . Edge BC is fixed. What boundary conditions should be imposed on nodal d.o.f. along edges AB, BC, CD , and DA ? Assume that the material is isotropic. What is different if the material is anisotropic?
text_image
t y A p B C x s α D
Problem 8.32
Section 8.9
8.33 A long bar, 100 mm wide and 20 mm thick, is loaded in tension by an axial force P.
(a) If the yield strength is Y = 1150 \mathrm{MPa} and K_{\mathrm{IC}} = 77 \mathrm{MPa} \sqrt{\mathrm{m}} , and a central crack 15 \mathrm{~mm} long is present, what force P will fracture the bar?
(b) If the yield strength is Y = 1410 MPa and K_{IC} = 50 MPa \sqrt{m} , what is the critical crack length if the force P determined in part (a) is applied?
8.34 Rather than use Eq. 8.9-8a to determine K_{\mathrm{I}} , one can determine K_{\mathrm{I}} from displacements of points B1 and B2 alone in Fig. 8.9-4. Derive the appropriate formula.
8.35 Consider the bar element of Fig. 8.9-3, but place node 3 at the third point rather than at the quarter point. At what value of x / L is a stress singularity indicated?
8.36 Let quarter-point elements be used to solve a certain crack problem (e.g., Fig. 8.9-4). Imagine that the problem is solved again, this time using quarter-point elements of smaller size. Now the computed results are found to be less accurate than before. Explain how this is possible.
Section 8.10
8.37 The beam element shown has the usual d.o.f. \{\mathbf{d}\} = \left[w_1 \quad \theta_1 \quad w_2 \quad \theta_2\right]^T . The element has width b and rests on a Winkler foundation of modulus \beta . Determine the foundation matrix [\mathbf{k}_f] defined by each of the following approximations.
(a) Deflection w is cubic in x , as in the standard beam element.
(b) Deflection w is quadratic in x (see Eq. 8.4-2).
(c) Deflection w is linear in x (and independent of \theta_1 and \theta_2 ).
(d) Deflection w is constant.
text_image
z,w L 1 2 x
Problem 8.37
8.38 The sketches represent top views of triangular elements that rest on a Winkler foundation of modulus \beta . Assume that vertical deflection w depends only on nodal values of w. Determine an expression for [k_{f}] of
(a) the three-node element (Eqs. 5.3-4).
(b) the six-node element (Eqs. 5.3-5), if sides are straight and side nodes are at midsides.
(a)
(b)
Problem 8.38
8.39 Imagine that separation is possible between a beam and its Winkler elastic foundation. Outline a solution algorithm for such a problem. In this exercise, do not be concerned with computational efficiency.
8.40 Imagine that a Winkler elastic foundation, which has translational modulus \beta , is augmented by a rotational modulus \alpha (whose units are force divided by length). For an element on such a composite foundation, what formula for [k_{f}] replaces Eq. 8.10-2? (A symbolic result is desired, with terms defined, rather than specifics of a particular element.)
Section 8.11
8.41 Show that Eqs. 8.11-5 and 8.11-6 indeed result from the manipulations described.
8.42 Use \phi from Eq. 8.11-7 and the mapping of Eq. 8.11-1 to determine the following:
(a) \phi as a function of r (analogous to Eq. 8.11-6).
(b) J as a function of \xi and a .
(c) Element matrix [k]. Use exact integration.
(d) Element matrix [k]. Use a two-point Gauss rule.
8.43 (a) For the infinite element shown, let field variable \phi depend on nodal values \phi_{1}, \phi_{2}, \phi_{5} , and \phi_{6} only (not on \phi_{3} and \phi_{4} ). Write mapping functions and shape functions, in the manner of Fig. 8.11-3.
text_image
a x y 2b 1 2 3 η 4 ξ 5 6
Problem 8.43
text_image
1 2 3 4 5 η 6 ξ
Problem 8.44
(b) Let sides 1–3–5 and 2–4–6 be parallel. Evaluate [J] and J.
(c) If \phi_5 = \phi_6 = 0 , what 2 by 2 element characteristic matrix [k] operates on \phi_1 and \phi_2 ? Again let sides 1-3-5 and 2-4-6 be parallel. (See Eq. 6.3-5, and let t = element thickness and k = material characteristic, both uniform over the element.)
8.44 Write mapping functions for the infinite plane element shown.
Section 8.13
8.45 If \{\mathbf{R}\} is unchanged and structural alterations are minor, then \{\Delta \mathbf{D}\} \approx -[\mathbf{K}]^{-1}([\Delta \mathbf{K}]\{\mathbf{D}\}) . Derive this expression for \{\Delta \mathbf{D}\} . What are advantages and disadvantages of this method?
8.46 Equation 8.13-2 can be cast in the iterative form \{\mathbf{K}\} \{\mathbf{D}^{*}\}_{i+1} = \{\mathbf{R}\} - [\Delta \mathbf{K}\} \{\mathbf{D}^{*}\}_{i} . Consider the application of this equation to single-d.o.f. problems as follows.
(a) Let K = 0.5, K^{*} = 0.8 , and R = 2 . Starting with D_0^* = D = 4.0 , compute D_0^* (i.e., apply five iterative cycles).
(b) For what range of values of \Delta K / K does this iterative method converge?
Section 8.14
8.47 In Fig. 8.14-2b, imagine that the reduced [k] for substructure AEFGH is known. How can one transform this [k] so that it pertains to substructure HGIJD, ready for assembly with substructure AEFGH? For brevity, consider only translational d.o.f. u_{i} and v_{i} at the lettered corners.
Section 8.15
8.48 Let the plate of Fig. 8.15-1a be uniformly loaded. Imagine that octant ABC is modeled by square elements, as shown in the sketch for this problem, so that some elements straddle the symmetry axis AC. What boundary conditions should be applied to these elements, for example, to typical element 1–2–3–4? State these conditions with reference to (a) st axes, and (b) xy axes.
text_image
A 1 2 3 4 C B
Problem 8.48
text_image
P y, v x, u
Problem 8.49