김경종
24 KiB
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1 2 3 4 3 1 2 3 4 1
(a)
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3 3 3 3 2 2 4 1
(b)
(t)
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3 y 3 4 3 1 2 2 2 y
{d}
Problem 6.13
Section 6.4
6.14 Derive the locations and weights of an order 2 Gauss rule by requiring that it integrate exactly the polynomial \phi = a_{1} + a_{2}\xi + a_{3}\xi^{2} + a_{4}\xi^{3} in the range -1 \leqslant \xi \leqslant 1 . Assume that weights and points are symmetric with respect to the axis \xi = 0 .
6.15 Use one-, two-, and three-point Gauss quadrature to integrate each of the following functions. Compare these answers with the exact answers.
(a) \phi = \cos \pi x / 2 between x = -1 and x = 1 .
(b) \phi = (2 - x) / (2 + x) between x = -1 and x = 1 .
(c) \phi = 1 / (x^2 - 3x + 4) between x = -1 and x = 1 .
(d) \phi = 1 / x between x = 1 and x = 7 .
6.16 Write an expression for I , analogous to Eq. 6.4-8, for (a) a 2 by 3 quadrature rule, and (b) a 3 by 4 quadrature rule.
6.17 For any of the quadrature rules in Table 6.4-1, weights W_{i} sum to 2 in one dimension, weight products W_{i}W_{j} sum to 4 in two dimensions, and weight products W_{i}W_{j}W_{k} sum to 8 in three dimensions. Why? Check this behavior in Eq. 6.4-8.
6.18 Use a 2 by 2 Gauss rule to approximate I over the rectangular region shown.
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y ← 6 → I = ∏(3+x²)/2+y² dx dy 4 x
Problem 6.18
6.19 In Problem 6.8, what are the r and s coordinates of the Gauss points of an order 2 rule? And what are the corresponding weights W_{i} ? (Integration is from 0 to 1 for both r and s .)
6.20 (a) Determine the element stiffness matrix [k] of a two-node uniform bar element of length L by use of an order 2 Gauss quadrature rule. Check your result against Eq. 2.4-5.
(b) Repeat part (a), but let the cross-sectional area vary linearly from A_{1} at node 1 to A_{2} at node 2.
(c) Repeat part (b), but use one-point Gauss quadrature.
6.21 Determine the 3 by 3 element stiffness matrix [k] if node 3 in Fig. 6.2-1a is at the middle of the bar and AE is constant. Use an order 2 Gauss quadrature rule.
Section 6.5
6.22 Subroutine QUAD4 (Fig. 6.5-2) would be more efficient if DV were removed from statement 60 and placed elsewhere. Where? And why?
6.23 Using Figs. 6.5-1 and 6.5-2 as a guide, write Fortran statements that will generate the stiffness matrix of the bar element in Fig. 6.2-1 by three-point Gauss quadrature. Let cross-sectional area A be linearly interpolated from known values at node 1 and node 2. Node 3 is not necessarily at the midpoint.
6.24 Using Figs. 6.5-1 and 6.5-2 as a guide, write Fortran statements that will generate the stiffness matrix of a uniform beam element (Fig. 4.2-2 and Eq. 4.2-5) by two-point Gauss quadrature.
Section 6.6
6.25 For the element described by Eqs. 6.6-1, write the eight-term displacement function \phi = a_{1} + a_{2}\xi + \cdots + a_{8}\xi\eta^{2} . Then write the 8 by 8 matrix [A] that arises in exchanging the a_{i} for nodal d.o.f. (in the manner of Eq. 3.8-6).
6.26. The element of Fig. 6.6-1a can be called subparametric. Why?
6.27 (a) Sketch a rectangular eight-node element for which J is a function of \eta but not of \xi .
(b) Sketch a nonrectangular eight-node element with midside nodes for which J is a function of \eta but not of \xi .
6.28 What changes would be needed in Figs. 6.5-1 and 6.5-2 to convert these subroutines so that they apply to the eight-node element described by Eqs. 6.6-1? Do not code the shape functions and their derivatives, but otherwise describe the changes precisely.
6.29 Identify the defects associated with connecting four-node and eight-node elements in the pattern shown.
flowchart
graph TD
A["①"] --> B["②"]
B --> C["③"]
C --> D["④"]
D --> E["End"]
A --> C
B --> C
C --> D
Problem 6.29
6.30 For the quadratic Lagrange element (nine nodes), sketch shape function N_{1} in the manner of Fig. 6.6-2d. Decide whether N_{1} is positive or negative in each quadrant by evaluating N_{1} at \xi = \pm \frac{1}{2} and \eta = \pm \frac{1}{2} .
6.31 Consider the nine-node element whose shape functions are given by Table 6.6-1. Any of nodes 5 through 9 can be omitted. In similar fashion, could node 1 be omitted, as shown, so as to produce a valid element with two straight edges, whose displacements are governed by nodes 2, 4, 5, and 8? Suggestion: Consider the horizontal (or the vertical) displacement at the lower left corner.
Problem 6.31
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Geometric line drawing of a polyhedron with interconnected vertices and edges (no text or symbols)
Problem 6.32
6.32 As an alternative to the nine-node element whose shape functions appear in Table 6.6-1, a nine-node element can be formed by combining eight linear triangles, as shown. What are comparative advantages and disadvantages of these two alternatives?
Section 6.7
6.33 What changes would be needed in Figs. 6.5-1 and 6.5-2 to convert these subroutines so that they apply to the eight-node solid of Fig. 6.7-1? Describe the changes precisely, including the coding of new shape functions and their derivatives. Assume that a subroutine can be called to invert [J] and compute J.
Section 6.8
6.34 Show that the bilinear element (Fig. 6.3-1a) becomes a constant-strain triangle if nodes 1 and 4 coalesce. For simplicity, use the particular geometry shown in the sketch.
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y a 3 b 1,4 2 x
Problem 6.34
6.35 Use the arguments of Eqs. 6.8-1 to 6.8-5 to evaluate [B] for a linear (three-node) triangle. Compare your result with Eqs. 5.2-5 and 5.4-3.
6.36 If \phi is constant, the product [\mathbf{B}]\{\phi_e\} must be zero. Use Eq. 6.8-5 to show that this is so.
6.37 As an alternative to Eqs. 6.8-2 through 6.8-4, one can eliminate (say) \xi_{3} from shape functions N_{i} by use of the constraint relation \xi_{1} + \xi_{2} + \xi_{3} = 1 , then take the derivatives \partial N_{i} / \partial \xi_{1} and \partial N_{i} / \partial \xi_{2} . Verify that this procedure also yields Eq. 6.8-5.
6.38 Consider a six-noded triangle. Imagine that we wish to move the side nodes to the positions shown, where a and b are dimensionless fractions of edge length (so that a + b = 1 ). We can accomplish this positioning in isoparametric fashion, using Eqs. 5.3-5, but the element displacement field is then not a complete quadratic in Cartesian coordinates. Show that we can accomplish the positioning in subparametric fashion, thus retaining the quad-
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3 6 a b b a b 1 4 2 5
Problem 6.38
ratic field. Suggestion: Abandon uniform side-node spacing on the reference element, and apply Eq. 5.3-2 to element geometry. Obtain shape functions N_{i} , and show that they reduce to Eqs. 5.3-5 for a = b = \frac{1}{2} .
6.39 (a) If \phi = 1 in Eq. 6.8-7, one concludes that \Sigma W_{i} = 1 . Verify this property in Table 6.8-1.
(b) All points listed in Table 6.8-1 should satisfy the constraint relation \xi_{1} + \xi_{2} + \xi_{3} = 1 . Verify that this is so for the 13-point formula.
6.40 (a) Integrate the function \phi = (1 + \xi_1\xi_2)^{-1} , using each of the first four integration formulas in Table 6.8-1. Let A = 1 and J = 2 .
(b) Integrate the function \phi = \xi_1\xi_2\xi_3 by use of the appropriate rule in Table 6.8-1. Verify your result by use of Eq. 5.2-8. Let A = 1 and J = 2 .
6.41 Evaluate Eq. 5.2-10 for the triangle shown. Evaluate this same integral by use of the first three-point formula in Table 6.8-1, and compare results.
6.42 In the wedge-shaped elements shown, let \zeta be a coordinate that has values +1 and -1 , respectively, on the top and bottom triangular faces. Write shape functions N_{i} for (a) the 6-node element, and (b) the 15-node element.
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(-1, 5) (-3, 0) 1 3 2 (4, -5) x y
Problem 6.41
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1 2 3 4 5 6
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1 9 3 7 8 2 10 15 11 4 6 13 5 14
Problem 6.42
Section 6.9
6.43 Verify the results given in Eq. 6.9-1.
6.44 Let the centroid of the triangle of Fig. 6.8-1a lie at x = y = 0 . Imagine that pressure p = cx , where c is a constant, acts normal to area A of the triangle. Evaluate the consistent nodal loads produced by p , in terms of c, A, x_1, x_2, and x_3 .
6.45 Let a uniform traction act on the surface of a nine-node quadratic element of rectangular shape. In the consistent load vector \{\mathbf{r}_e\} , what fraction of the total force appears at each node? (See Fig. 4.3-5 for the corresponding eight-node case.)
Section 6.10
6.46 Use shape functions of Table 6.6-1 to demonstrate the interelement compatibility argument made in Section 6.10 with reference to Fig. 6.10-1.
6.47 Verify that \Sigma N_{i} = 1 for the N_{i} of (a) Eqs. 5.1-5, (b) Eqs. 5.1-6, (c) Eqs. 5.3-5, (d) Eqs. 6.3-2, and (e) Eqs. 6.6-1.
6.48 Verify that Eqs. 6.10-7 are satisfied for the N_{i} of (a) Eqs. 5.1-5, (b) Eqs. 5.1-6, (c) Eqs. 5.4-1, and (d) Eqs. 6.3-2.
6.49 Let u_{1} = 0 and \dot{u}_{2} > 0 in the three-node bar element shown. For 0 < x < L , interpolate axial displacement u linearly, so that the element is superparametric. Calculate axial strain \epsilon_{x} = \lfloor \mathbf{B} \rfloor \{\mathbf{u}\} , where \{\mathbf{u}\} = \lfloor u_{1} - u_{2} \rfloor^{T} and J (the denominator of \lfloor \mathbf{B} \rfloor ) is given by Eq. 6.2-5. Hence, show that the element fails unless x_{3} = L / 2 .
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y l 1 3 2 x,u x₃
Problem 6.49
6.50 Write out the formula x = \sum N_i x_i for the element of Fig. 6.6-1a. Hence, show the correctness of the completeness argument in Section 6.10 for this subparametric element.
Section 6.11
6.51 Let u_{1}, u_{2} , and u_{3} be prescribed in the three-node bar of Fig. 6.2-1. Let AE be constant. What order of Gauss rule is needed to calculate strain energy in the element if (a) x_{3} = L / 2 , and (b) x_{3} \neq L / 2 ?
6.52 If element thickness t can vary and is computed as t = \sum N_{i}t_{i} from nodal values t_{i} , what order of Gauss quadrature is needed to compute the exact volume of (a) a bilinear element (four nodes), and (b) a quadratic element (eight nodes)?
6.53 Show that the volume of a trilinear solid element is correctly computed by an order 2 Gauss rule.
6.54 Let the following elements be rectangular in geometry, with side nodes evenly spaced and thicknesses constant. What order of Gauss quadrature is needed to obtain the exact stiffness matrix—that is, to integrate each k_{ij} exactly?
(a) Plane bilinear element (four nodes).
(b) Plane quadratic element (eight nodes).
(c) Solid trilinear element (eight nodes).
(d) Plane quadratic triangle (six nodes).
6.55 Repeat Problem 6.54 if element thickness t is variable and interpolated from nodal values, t = \Sigma N_{i}t_{i} .
6.56 A 2 by 2 Gauss rule is used to form [k] for each element in Table 6.14-1. What will the qualitative change in deflection v_{A} in each of the three cases be if the Gauss rule is changed to 3 by 3? Why?
Section 6.12
6.57 The plane structure shown is built of four bilinear elements, each integrated by one-point Gauss quadrature.
(a) Sketch the possible mechanisms of the structure.
(b) If you ask a computer program to solve for the displacement of load P , what do you think will happen?
(c) Add one roller support that will prevent an instability.
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P
Problem 6.57
6.58 Verify that Eqs. 6.12-3 yield zero strains at Gauss points of an order 2 rule.
6.59 For each of the following elements, write (if possible) a vector \{\mathbf{d}\} of nodal d.o.f. that represents an instability mode under one-point Gauss quadrature.
(a) The three-node bar element of Fig. 6.2-1.
(b) The standard four-d.o.f. beam element, Fig. 4.2-2.
6.60 Imagine that the bilinear element of Fig. 6.12-1 is integrated with a 2 by 1 Gauss rule. What is the rank of the element stiffness matrix?
6.61 (a) Consider the quadratic serendipity solid element (a hexahedron having eight corner nodes and twelve side nodes). If integrated by 'an order 2 Gauss rule, what do you anticipate will be the rank of its stiffness matrix? State your reason.
(b) Can these elements be put together so that the mesh has a mechanism?
6.62 There exists a six-point quadrature rule for hexahedra that uses a sampling point at the middle of each face [6.9]. What mechanisms are possible for a rectangular eight-node element whose stiffness matrix is formed by this rule? Can a mesh of elements also display these mechanisms?
6.63 Consider the upper right-hand element in Fig. 6.12-2d. Let the undeformed element be square, two units on a side.
(a) Show that strains are zero at the center of the element. Let all nonzero d_{i} in \{\mathbf{d}\} have magnitude c .
(b) Show that \{\mathbf{d}\} of the deformed element can be obtained by combining modes 7 and 8 of Fig. 6.12-1 with a rigid-body rotation.
6.64 (a) Let a vector \{\mathbf{d}\} contain nodal d.o.f. of an arbitrary plane element, with all u_{i} in the upper half and all v_{i} in the lower half. Consider the following similarly-partitioned vectors of 0's, 1's, and nodal coordinates x_{i} and y_{i} :
\left\{ \begin{array}{l} \mathbf {1} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {x} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {y} \\ \mathbf {0} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {1} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {x} \end{array} \right\}, \left\{ \begin{array}{l} \mathbf {0} \\ \mathbf {y} \end{array} \right\}
Identify these vectors, singly or in combination, with three rigid-body modes and three constant-strain modes.
(b) Let the element be square, two units on a side and have nine nodes. Write \{\mathbf{d}\} for the mode of Eq. 6.12-3b, and show that it is orthogonal to the modes of part (a).
6.65 Let the trilinear solid element of Fig. 6.7-1 be rectangular and two units on a side, so that \xi = x , \eta = y and \zeta = \dot{z} .
(a) What is the rank of the element stiffness matrix if it is integrated by use of a single Gauss point?
(b) Consider only x -direction displacements u . Let \{\mathbf{d}_x\} represent the u_i of the eight nodes. Write a \{\mathbf{d}_x\} of arbitrary magnitude for each zero-energy mode that involves only the u_i .
(c) Similarly, write a \{\mathbf{d}_x\} for a total of four independent rigid-body and constant-strain modes that involve only the u_{i} .
(d) Show that the modes of part (b) are orthogonal to those of part (c).
6.66 Determine a_{7} in Eq. 6.12-5 so that a rectangular element of uniform thickness has the exact strain energy in mode 7 (a pure bending mode). Express your answer in terms of the elastic modulus and element dimensions.
Section 6.13
6.67 Use the bending-deformation mode of Fig. 6.13-1b and the N_{i} of Eq. 6.3-2 to show that \gamma_{xy} = -c\xi on \eta = 0 , where c is a positive constant. For simplicity, assume that elements are square.
6.68 (a) Verify the numerical factors in Eq. 6.13-5.
(b) Apply Eq. 6.13-3 to nodes B , C , and D in Fig. 6.13-3. (Obtain numerical factors, as in Eq. 6.13-5.)
(c) Apply Eq. 6.13-3 to nodes E, F, G , and H in Fig. 6.13-3. (Obtain numerical factors, as in Eq. 6.13-5.)
6.69 Write a formula analogous to Eq. 6.13-3 that uses \sigma_{i} at the eight Gauss points of an order 2 rule in a solid element. Use it to write expressions for \sigma_{x} , analogous to Eq. 6.13-5, at (a) node 8 in Fig. 6.7-1, and (b) the point where axis \xi pierces the right-hand face in Fig. 6.7-1.
6.70 In the bilinear element (four nodes), stresses calculated directly at nodes agree exactly with stress extrapolated to nodes from four Gauss points, if the element is a parallelogram. Results disagree if the element is an arbitrary quadrilateral. Why?
6.71 Imagine that the bilinear element is not of constant thickness. What role does the thickness variation play in stress calculation according to Eq. 6.13-1? Suggest an ad hoc adjustment for thickness variation that might improve the accuracy of computed stresses.
Section 6.14
6.72 In Fig. 6.14-1b, locate the points described by the following coordinates.
(a) \xi = -1 and \xi = -1 / \sqrt{3} .
(b) \eta = 0 and \eta = -1 .
6.73 Consider an isosceles triangle, created by moving nodes 3 and 4 of a rectangular bilinear element (Fig. 4.2-4) so that they coincide on the \eta axis.
(a) Sketch the element and the Gauss points of a 2 by 2 rule, in the manner of Fig. 6.14-1.
(b) If J is computed at each Gauss point, what is the ratio J_{\max} / J_{\min} ?
Summary Questions
6.74 Each of the structures shown may be analyzed as two-dimensional. Greatest stresses and greatest deflections are desired. Imagine that an initial (coarse mesh) analysis is to be undertaken, so that errors will be roughly 10% or less. For each structure, sketch a suitable mesh, first using linear elements and then using quadratic elements. State your assumptions and approximations regarding how loads and supports are specified, the use of symmetry, the quadrature rule needed, treatment of stress concentrations, and so on.
COORDINATE TRANSFORMATION
Uses of coordinate transformations in structural mechanics are described, with emphasis on transformation of stiffness properties.
7.1 INTRODUCTION
Coordinate transformation permits the calculation of elastic property matrix [E'] and stiffness matrix [k'] in one coordinate system with subsequent transformation to matrices [E] and [k] in another coordinate system. Other uses of coordinate transformation include condensation techniques in structural dynamics and imposition of constraints. Constraints are discussed in detail in Chapter 9.
The form [Q] = [T]^{7}[Q'][T] appears repeatedly. Here [Q'] is the matrix to be transformed and [T] is the transformation matrix. The transformed matrix [Q] is symmetric if [Q'] is symmetric. Matrix [T] may be rectangular or square. If square it may not be orthogonal. The specific form of [T] depends on the problem at hand.
One often has the option of taking [Q'] as either an element matrix or the corresponding structure matrix. Computer programming is usually easiest when transformations are done before elements are assembled, even though we must then transform several small matrices instead of one large one.
Formal matrix multiplication to produce [T]^{T}[Q^{\prime}][T] is often wasteful because [T] is often sparse. Sparsity should be exploited, or terms in the product should be hand-calculated and then coded.
Caution. Transformations modify stiffness matrices. Errors and inconsistencies in stiffness matrices can lead to numerical difficulties and seriously degrade accuracy. It matters little if errors in [T] produce only a slightly different geometry than intended. But damage is done if errors in [T] act to falsify equilibrium equations. To avoid damage, we should state and manipulate transformation matrices and constraint equations with as much precision as is granted to stiffness coefficients K_{ij} .
7.2 TRANSFORMATION OF VECTORS
Consider a vector V whose scalar components in the x, y, and z directions are u, v, and w (Fig. 7.2-1). Components of V in the x' , y' and z' directions are u' , v' , and w' . We wish to express u' , v' , and w' in terms of u, v, w, and the cosines of
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y' y v' v V x' x u y w' w z z'
Direction cosines of axes:
| x | y | z | |
| $x'$ | $\ell_1$ | $m_1$ | $n_1$ |
| $y'$ | $\ell_2$ | $m_2$ | $n_2$ |
| $z'$ | $\ell_3$ | $m_3$ | $n_3$ |
Figure 7.2-1. Coordinate systems xyz and x'y'z' , with table of direction cosines of angles between axes; for example, \ell_{1} is the cosine of the angle between axes x and x' . Components of a vector V can be expressed in either coordinate system.
angles between axes x'y'z' and xyz. Vector V can be regarded as a position vector, a nodal force or moment vector, or a nodal displacement or rotation vector. ^{1}
Component u' can be regarded as the sum of components of displacements u , v , and w parallel to the x' axis. That is, u' = \ell_1 u + m_1 v + n_1 w . Components v' and w' can be written similarly. In matrix format, these relations are
\left\{ \begin{array}{l} u ^ {\prime} \\ v ^ {\prime} \\ w ^ {\prime} \end{array} \right\} = [ \Lambda ] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\}, \quad \text { where } \quad [ \Lambda ] = \left[ \begin{array}{l l l} \ell_ {1} & m _ {1} & n _ {1} \\ \ell_ {2} & m _ {2} & n _ {2} \\ \ell_ {3} & m _ {3} & n _ {3} \end{array} \right] \tag {7.2-1}
Matrix [\Lambda] is called the rotation matrix. Equation 7.2-1 relates vectorial components in two systems. Matrix [\Lambda] is orthogonal (i.e., its inverse is equal to its transpose). Therefore, the inverse of the transformation in Eq. 7.2-1 is
\left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} = [ \Lambda ] ^ {T} \left\{ \begin{array}{l} u ^ {\prime} \\ v ^ {\prime} \\ w ^ {\prime} \end{array} \right\} \tag {7.2-2}
Vector V in Fig. 7.2-1 may represent a displacement whose components can be expressed as \{d\} = \left[u \quad v \quad w\right]^{T} or as \{d'\} = \left[u' \quad v' \quad w'\right]^{T} . Or, V may represent a force whose components can be expressed as \{r\} = \left[f_{x} \quad f_{y} \quad f_{z}\right]^{T} or as \{r'\} = \left[f_{x}' \quad f_{y}' \quad f_{z}'\right]^{T} . Accordingly, components of displacements and forces obey the transformation rules
\{\mathbf {d} ^ {\prime} \} = [ \Lambda ] \{\mathbf {d} \} \quad \text { and } \quad \{\mathbf {d} \} = [ \Lambda ] ^ {T} \{\mathbf {d} ^ {\prime} \} \tag {7.2-3}
\{\mathbf {r} ^ {\prime} \} = [ \Lambda ] \{\mathbf {r} \} \quad \text { and } \quad \{\mathbf {r} \} = [ \Lambda ] ^ {T} \{\mathbf {r} ^ {\prime} \} \tag {7.2-4}
^{1} Provided that rotation is small, as is usual. Finite (large) rotations do not combine vectorially.