김경종
425 lines
26 KiB
Markdown
425 lines
26 KiB
Markdown
<!-- source-page: 241 -->
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shown. Establish explicitly all matrices you need but do not perform any multiplications and integrations.
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(b) Explain (by physical reasoning) that your assumptions on u, v, w make sense.
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4.21. An inviscid fluid element (for acoustic motions) can be obtained by considering only volumetric strain energy (since inviscid fluids provide no resistance to shear). Formulate the finite element fluid stiffness matrix for the four-node plane element shown and write out all matrices required. Do not actually perform any integrations or matrix multiplications. Hint: Remember that $p = -\beta \Delta V / V$ and $\tau^T = [\tau_{xx} \quad \tau_{yy} \quad \tau_{xy} \quad \tau_{zz}] = [-p - p - 0 - p]$ and $\Delta V / V = \epsilon_{xx} + \epsilon_{yy}$ .
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<details>
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<summary>text_image</summary>
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Thickness t
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Bulk modulus β
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b
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2
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1
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3
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4
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a
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y
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x
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</details>
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4.22. Consider the element assemblages in Exercises 4.18 and 4.19. For each case, evaluate a lumped mass matrix (using a uniform mass density $\rho$ ) and a lumped load vector.
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4.23. Use a finite element program to solve the model shown of the problem in Example 4.6.
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(a) Print out the element stresses and element nodal point forces and draw the “exploded element views” for the stresses and nodal point forces as in Example 4.9.
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(b) Show that the element nodal point forces of element 5 are in equilibrium and that the element nodal point forces of elements 5 and 6 equilibrate the applied load.
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(c) Print out the reactions and show that the element nodal point forces equilibrate these reactions.
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(d) Calculate the strain energy of the finite element model.
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<details>
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<summary>text_image</summary>
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P = 100
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①
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②
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③
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④
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⑤
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⑥
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⑦
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⑧
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</details>
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Eight constant-strain triangles
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4.24. Use a finite element program to solve the model shown of the problem in Example 4.6. Print out the element stresses and reactions and calculate the strain energy of the model. Draw the "exploded element views" for the stresses and nodal point forces. Compare your results with those for Exercise 4.23 and discuss why we should not be surprised to have obtained different results (although the same kind and same number of elements are used in both idealizations).
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<details>
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<summary>text_image</summary>
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P = 100
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①
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②
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③
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④
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⑤
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⑥
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⑦
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⑧
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</details>
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Eight constant-strain triangles
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# 4.3 CONVERGENCE OF ANALYSIS RESULTS
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Since the finite element method is a numerical procedure for solving complex engineering problems, important considerations pertain to the accuracy of the analysis results and the convergence of the numerical solution. The objective in this section is to address these issues. We start by defining in Section 4.3.1 what we mean by convergence. Then we consider in a rather physical manner the criteria for monotonic convergence and relate these criteria to the conditions in a Ritz analysis (introduced in Section 3.3.3). Next, some important properties of the finite element solution are summarized (and proven) and the rate of convergence is discussed. Finally, we consider the calculation of stresses and the evaluation of error measures that indicate the magnitude of the error in stresses at the completion of an analysis.
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We consider in this section displacement-based finite elements leading to monotonically convergent solutions. Formulations that lead to a nonmonotonic convergence are considered in Sections 4.4 and 4.5.
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# 4.3.1 The Model Problem and a Definition of Convergence
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Based on the preceding discussions, we can now say that, in general, a finite element analysis requires the idealization of an actual physical problem into a mathematical model and then the finite element solution of that model (see Section 1.2). Figure 4.8 summarizes these concepts. The distinction given in the figure is frequently not recognized in practical analysis because the differential equations of motion of the mathematical model are not dealt with, and indeed the equations may be unknown in the analysis of a complex problem,
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<!-- source-page: 243 -->
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<details>
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<summary>flowchart</summary>
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```mermaid
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graph TD
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A["Actual Physical Problem"] --> B["Geometric domain"]
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B --> C["Material"]
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B --> D["Loading"]
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B --> E["Boundary conditions"]
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E --> F["Mathematical Model (which corresponds to a mechanical idealization)"]
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F --> G["Kinematics, e.g., truss"]
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G --> H["plane stress"]
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G --> I["three-dimensional"]
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G --> J["Kirchhoff plate"]
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G --> K["etc."]
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F --> L["Material, e.g., isotropic linear"]
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L --> M["elastic"]
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L --> N["Mooney-Rivlin rubber"]
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L --> O["etc."]
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F --> P["Loading, e.g., concentrated"]
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P --> Q["centrifugal"]
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P --> R["etc."]
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F --> S["Boundary"]
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S --> T["prescribed"]
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S --> U["displacements"]
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S --> V["etc."]
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F --> W["Conditions, e.g.,"]
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W --> X["finite Element Solution"]
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W --> Y["Choice of elements and solution procedures"]
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F --> Z["Yields: Governing differential equation(s) of motion e.g.,"]
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Z --> AA["\frac{\partial}{\partial x} \left( EA \frac{\partial u}{\partial x} \right) = -p(x) \"]
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Z --> AB["and principle of virtual work equation (see Example 4.2)"]
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Z --> AC["Yields: Approximate solution of the mathematical model (that is, approximate response of mechanical idealization)"]
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```
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</details>
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Figure 4.8 Finite element solution process
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such as the response prediction of a three-dimensional shell. Instead, in a practical analysis, a finite element idealization of the physical problem is established directly. However, to study the convergence of the finite element solution as the number of elements increases, it is valuable to recognize that a mathematical model is actually implied in the finite element representation of the physical problem. That is, a proper finite element solution should converge (as the number of elements is increased) to the analytical (exact) solution of the differential equations that govern the response of the mathematical model. Furthermore, the convergence behavior displays all the characteristics of the finite element scheme because the differential equations of motion of the mathematical model express in a very precise and compact manner all basic conditions that the solution variables (stress, displacement, strain, and so on) must satisfy. If the differential equations of motion are not known, as in a complex shell analysis, and/or analytical solutions cannot be obtained, the convergence of the finite element solutions can be measured only on the fact that all basic kinematic, static, and constitutive conditions contained in the mathematical model must ultimately (at convergence) be satisfied. Therefore, in all discussions of the convergence of finite element solutions we imply that the convergence to the exact solution of a mathematical model is meant.
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Here it is important to recognize that in linear elastic analysis there is a unique exact solution to the mathematical model. Hence if we have $a$ solution that satisfies the governing
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mathematical equations exactly, then this is the exact solution to the problem (see Section 4.3.4).
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In considering the approximate finite element solution to the exact response of the mathematical model, we need to recognize that different sources of errors affect the finite element solution results. Table 4.4 summarizes various general sources of errors. Round-off errors are a result of the finite precision arithmetic of the computer used; solution errors in the constitutive modeling are due to the linearization and integration of the constitutive relations; solution errors in the calculation of the dynamic response arise in the numerical integration of the equations of motion or because only a few modes are used in a mode superposition analysis; and solution errors arise when an iterative solution is obtained because convergence is measured on increments in the solution variables that are small but not zero. In this section, we will discuss only the finite element discretization errors, which are due to interpolation of the solution variables. Thus, in essence, we consider in this section a model problem in which the other solution errors referred to above do not arise: a linear elastic static problem with the geometry represented exactly with the exact calculation of the element matrices and solution of equations, i.e., also negligible round-off errors. For ease of presentation, we assume that the prescribed displacements are zero. Nonzero displacement boundary conditions would be imposed as discussed in Section 4.2.2, and such boundary conditions do not change the properties of the finite element solution.
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For this model problem, let us restate for purposes of our discussion the basic equation of the principle of virtual work governing the exact solution of the mathematical model
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$$
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\int_ {V} \overline {{{\boldsymbol {\epsilon}}}} ^ {T} \boldsymbol {\tau} d V = \int_ {S _ {f}} \overline {{{\mathbf {u}}}} ^ {S _ {f} ^ {T}} \mathbf {f} ^ {S _ {f}} d S + \int_ {V} \overline {{{\mathbf {u}}}} ^ {T} \mathbf {f} ^ {B} d V \tag {4.62}
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$$
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TABLE 4.4 Finite element solution errors
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<table><tr><td>Error</td><td>Error occurrence in</td><td>See section</td></tr><tr><td>Discretization</td><td>Use of finite element interpolations for geometry and solution variables</td><td>4.2.14.2.3, 5.3</td></tr><tr><td>Numerical integration in space</td><td>Evaluation of finite element matrices using numerical integration</td><td>5.56.8.4</td></tr><tr><td>Evaluation of constitutive relations</td><td>Use of nonlinear material models</td><td>6.6.36.6.4</td></tr><tr><td>Solution of dynamic equilibrium equations</td><td>Direct time integration, mode superposition</td><td>9.2–9.4</td></tr><tr><td>Solution of finite element equations by iteration</td><td>Gauss-Seidel, conjugate gradient, Newton-Raphson, quasi-Newton methods, eigensolutions</td><td>8.3, 8.49.510.4</td></tr><tr><td>Round-off</td><td>Setting up equations and their solution</td><td>8.2.6</td></tr></table>
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We recall that for $\tau$ to be the exact solution of the mathematical model, (4.62) must hold for arbitrary virtual displacements $\overline{\mathbf{u}}$ (and corresponding virtual strains $\overline{\epsilon}$ ), with $\overline{\mathbf{u}}$ zero at and corresponding to the prescribed displacements. A short notation for (4.62) is
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Find the displacements $\mathbf{u}$ (and corresponding stresses $\tau$ ) such that
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$$
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a (\mathbf {u}, \mathbf {v}) = (\mathbf {f}, \mathbf {v}) \quad \text { for all admissible } \mathbf {v} \tag {4.63}
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$$
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Here $a(\cdot, \cdot)$ is a bilinear form and $(\mathbf{f}, \cdot)$ is a linear form $^9$ —these forms depend on the mathematical model considered— $\mathbf{u}$ is the exact displacement solution, $\mathbf{v}$ is any admissible virtual displacement [“admissible” because the functions $\mathbf{v}$ must be continuous and zero at and corresponding to actually prescribed displacements (see (4.7)], and $\mathbf{f}$ represents the forcing functions (loads $\mathbf{f}^{\mathrm{S}f}$ and $\mathbf{f}^{\mathrm{B}}$ ). Note that the notation in (4.63) implies an integration process. The bilinear forms $a(\cdot, \cdot)$ that we consider in this section are symmetric in the sense that $a(\mathbf{u}, \mathbf{v}) = a(\mathbf{v}, \mathbf{u})$ .
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From (4.63) we have that the strain energy corresponding to the exact solution u is $1/2 a(u, u)$ . We assume that the material properties and boundary conditions of our model problem are such that this strain energy is finite. This is not a serious restriction in practice but requires the proper choice of a mathematical model. In particular, the material properties must be physically realistic and the load distributions (externally applied or due to displacement constraints) must be sufficiently smooth. We have discussed the need of modeling the applied loads properly already in Section 1.2 and will comment further on it in Section 4.3.4.
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Assume that the finite element solution is $u_{h}$ : this solution lies of course in the finite element space given by the displacement interpolation functions (h denoting here the size of the generic element and hence denoting a specific mesh). Then we define “convergence” to mean that
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$$
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a (\mathbf {u} - \mathbf {u} _ {h}, \mathbf {u} - \mathbf {u} _ {h}) \rightarrow 0 \quad \text { as } h \rightarrow 0 \tag {4.64}
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$$
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or, equivalently [see (4.90)], that
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$$
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a (\mathbf {u} _ {h}, \mathbf {u} _ {h}) \rightarrow a (\mathbf {u}, \mathbf {u}) \quad \text { as } h \rightarrow 0
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$$
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Physically, this statement means that the strain energy calculated by the finite element solution converges to the exact strain energy of the mathematical model as the finite element mesh is refined. Let us consider a simple example to show what we mean by the bilinear form $a(\cdot,\cdot)$ .
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$^{9}$ The bilinearity of $a(\cdot,\cdot)$ refers to the fact that for any constants $\gamma_{1}$ and $\gamma_{2}$ ,
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$$
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a \left(\gamma_ {1} \mathbf {u} _ {1} + \gamma_ {2} \mathbf {u} _ {2}, \mathbf {v}\right) = \gamma_ {1} a \left(\mathbf {u} _ {1}, \mathbf {v}\right) + \gamma_ {2} a \left(\mathbf {u} _ {2}, \mathbf {v}\right)
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$$
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$$
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a (\mathbf {u}, \gamma_ {1} \mathbf {v} _ {1} + \gamma_ {2} \mathbf {v} _ {2}) = \gamma_ {1} a (\mathbf {u}, \mathbf {v} _ {1}) + \gamma_ {2} a (\mathbf {u}, \mathbf {v} _ {2})
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$$
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and the linearity of $(\mathbf{f},\cdot)$ refers to the fact that for any constants $\gamma_{1}$ and $\gamma_{2}$ ,
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$$
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(\mathbf {f}, \gamma_ {1} \mathbf {v} _ {1} + \gamma_ {2} \mathbf {v} _ {2}) = \gamma_ {1} (\mathbf {f}, \mathbf {v} _ {1}) + \gamma_ {2} (\mathbf {f}, \mathbf {v} _ {2}).
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$$
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<!-- source-page: 246 -->
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EXAMPLE 4.22: Assume that a simply supported prestressed membrane, with (constant) prestress tension T, subjected to transverse loading p is to be analyzed (see Fig. E4.22). Establish for this problem the form (4.63) of the principle of virtual work.
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<details>
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<summary>text_image</summary>
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Hinged on all edges
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p
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T > 0
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T = 0
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z
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w
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y
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x
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Tension T
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</details>
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Figure E4.22 Prestressed membrane
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The principle of virtual work gives for this problem
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$$
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\int_ {A} \left[ \begin{array}{l} \frac {\partial \overline {{w}}}{\partial x} \\ \frac {\partial \overline {{w}}}{\partial y} \end{array} \right] ^ {T} T \left[ \begin{array}{l} \frac {\partial w}{\partial x} \\ \frac {\partial w}{\partial y} \end{array} \right] d x d y = \int_ {A} p \overline {{w}} d x d y
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$$
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where $w(x, y)$ is the transverse displacement. The left-hand side of this equation gives the bilinear form $a(v, u)$ , with $v = \overline{w}$ , u = w, and the integration on the right-hand side gives $(f, v)$ .
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Depending on the specific (properly formulated) displacement-based finite elements used in the analysis of the model problem defined above, we may converge monotonically or nonmonotonically to the exact solution as the number of finite elements is increased. In the following discussion we consider the criteria for the monotonic convergence of solutions. Finite element analysis conditions that lead to nonmonotonic convergence are discussed in Section 4.4.
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# 4.3.2 Criteria for Monotonic Convergence
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For monotonic convergence, the elements must be complete and the elements and mesh must be compatible. If these conditions are fulfilled, the accuracy of the solution results will increase continuously as we continue to refine the finite element mesh. This mesh refinement should be performed by subdividing a previously used element into two or more elements; thus, the old mesh will be “embedded” in the new mesh. This means mathematically that the new space of finite element interpolation functions will contain the previously used space, and as the mesh is refined, the dimension of the finite element solution space will be continuously increased to contain ultimately the exact solution.
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The requirement of completeness of an element means that the displacement functions of the element must be able to represent the rigid body displacements and the constant strain states.
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The rigid body displacements are those displacement modes that the element must be able to undergo as a rigid body without stresses being developed in it. As an example, a two-dimensional plane stress element must be able to translate uniformly in either direction of its plane and to rotate without straining. The reason that the element must be able to undergo these displacements without developing stresses is illustrated in the analysis of the cantilever shown in Fig. 4.9: the element at the tip of the beam—for any element size—must translate and rotate stress-free because by simple statics the cantilever is not subjected to stresses beyond the point of load application.
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The number of rigid body modes that an element must be able to undergo can usually be identified without difficulty by inspection, but it is instructive to note that the number of element rigid body modes is equal to the number of element degrees of freedom minus the number of element straining modes (or natural modes). As an example, a two-noded truss has one straining mode (constant strain state), and thus one, three, and five rigid body modes in one-, two-, and three-dimensional conditions, respectively. For more complex finite
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(a) Rigid body modes of a plane stress element
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<details>
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<summary>text_image</summary>
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p
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Distributed
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load p
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</details>
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<details>
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<summary>text_image</summary>
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Rigid body translation
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and rotation;
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element must be
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stress-free for any
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element size
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</details>
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(b) Analysis to illustrate the rigid body mode condition
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Figure 4.9 Use of plane stress element in analysis of cantilever
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elements the individual straining modes and rigid body modes are displayed effectively by representing the stiffness matrix in the basis of eigenvectors. Thus, solving the eigenproblem
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$$
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\mathbf {K} \boldsymbol {\phi} = \lambda \boldsymbol {\phi} \tag {4.65}
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$$
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we have (see Section 2.5)
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$$
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\mathbf {K} \boldsymbol {\Phi} = \boldsymbol {\Phi} \boldsymbol {\Lambda} \tag {4.66}
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$$
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where $\Phi$ is a matrix storing the eigenvectors $\phi_1, \ldots, \phi_n$ and $\Lambda$ is a diagonal matrix storing the corresponding eigenvalues, $\Lambda = \text{diag}(\lambda_i)$ . Using the eigenvector orthonormality property we thus have
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$$
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\Phi^ {T} \mathbf {K} \Phi = \Lambda \tag {4.67}
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$$
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We may look at $\Lambda$ as being the stiffness matrix of the element corresponding to the eigenvector displacement modes. The stiffness coefficients $\lambda_{1},\ldots,\lambda_{n}$ display directly how stiff the element is in the corresponding displacement mode. Thus, the transformation in (4.67) shows clearly whether the rigid body modes and what additional straining modes are present. $^{10}$ As an example, the eight eigenvectors and corresponding eigenvalues of a four-node element are shown in Fig. 4.10.
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The necessity for the constant strain states can be physically understood if we imagine that more and more elements are used in the assemblage to represent the structure. Then in the limit as each element approaches a very small size, the strain in each element approaches a constant value, and any complex variation of strain within the structure can be approximated. As an example, the plane stress element used in Fig. 4.9 must be able to represent two constant normal stress conditions and one constant shearing stress condition. Figure 4.10 shows that the element can represent these constant stress conditions and, in addition, contains two flexural straining modes.
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The rigid body modes and constant strain states that an element can represent can also be directly identified by studying the element strain-displacement matrix (see Example 4.23).
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The requirement of compatibility means that the displacements within the elements and across the element boundaries must be continuous. Physically, compatibility ensures that no gaps occur between elements when the assemblage is loaded. When only translational degrees of freedom are defined at the element nodes, only continuity in the displacements u, v, or w, whichever are applicable, must be preserved. However, when rotational degrees of freedom are also defined that are obtained by differentiation of the transverse displacement (such as in the formulation of the plate bending element in Example 4.18), it is also necessary to satisfy element continuity in the corresponding first displacement derivatives. This is a consequence of the kinematic assumption on the displacements over the depth of the plate bending element; that is, the continuity in the displacement w and the derivatives $\partial w/\partial x$ and/or $\partial w/\partial y$ along the respective element edges ensures continuity of displacements over the thickness of adjoining elements.
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Compatibility is automatically ensured between truss and beam elements because they join only at the nodal points, and compatibility is relatively easy to maintain in
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<!-- source-page: 249 -->
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<details>
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<summary>text_image</summary>
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1.0
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1.0
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</details>
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Rigid body mode $\lambda_{1} = 0$
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Thickness = 1.0
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Young's
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modulus = 1.0
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Poisson's ratio = 0.30
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram with a rectangle and dashed lines, no text or symbols present
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</details>
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Rigid body mode $\lambda_{2} = 0$
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram of a square with dashed lines indicating hidden edges (no text or symbols)
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</details>
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Rigid body mode $\lambda_{3} = 0$
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<details>
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<summary>natural_image</summary>
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Geometric diagram of a square with internal dashed and solid lines (no text or symbols)
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</details>
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Flexural mode $\lambda_{4} = 0.495$
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<details>
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<summary>natural_image</summary>
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Geometric diagram of a square with internal dashed and solid lines (no text or symbols)
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</details>
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Flexural mode $\lambda_{5} = 0.495$
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<details>
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<summary>natural_image</summary>
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Geometric diagram of a square with dashed and solid lines forming an inner circle (no text or symbols)
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</details>
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Shear mode $\lambda_6 = 0.769$
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram of a square with dashed border (no text or symbols)
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</details>
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Stretching mode $\lambda_{7} = 0.769$
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<details>
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<summary>natural_image</summary>
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Simple geometric diagram of a square with dashed border (no text or symbols)
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</details>
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Uniform extension mode $\lambda_8 = 1.43$
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Figure 4.10 Eigenvalues and eigenvectors of four-node plane stress element
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two-dimensional plane strain, plane stress, and axisymmetric analysis and in three-dimensional analysis, when only u, v, and w degrees of freedom are used as nodal point variables. However, the requirements of compatibility are difficult to satisfy in plate bending analysis, and particularly in thin shell analysis if the rotations are derived from the transverse displacements. For this reason, much emphasis has been directed toward the development of plate and shell elements, in which the displacements and rotations are
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variables (see Section 5.4). With such elements the compatibility requirements are just as easy to fulfill as in the case of dealing only with translational degrees of freedom.
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Whether a specific element is complete and compatible depends on the formulation used, and each formulation need be analyzed individually. Consider the following simple example.
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EXAMPLE 4.23: Investigate if the plane stress element used in Example 4.6 is compatible and complete.
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We have for the displacements of the element,
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$$
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u (x, y) = \alpha_ {1} + \alpha_ {2} x + \alpha_ {3} y + \alpha_ {4} x y
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$$
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$$
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v (x, y) = \beta_ {1} + \beta_ {2} x + \beta_ {3} y + \beta_ {4} x y
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$$
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Observing that the displacements within an element are continuous, in order to show that the element is compatible, we need only investigate if interelement continuity is also preserved when an element assemblage is loaded. Consider two elements interconnected at two node points (Fig. E4.23) on which we impose two arbitrary displacements. It follows from the displacement assumptions that the points (i.e., the material particles) on the adjoining element edges displace linearly, and therefore continuity between the elements is preserved. Hence the element is compatible.
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<details>
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<summary>text_image</summary>
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Node 3
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2, 3
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u2 = u3
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V2 = V3
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2
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Particles on element edges
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remain together
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V1 = V4
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1, 4
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u1 = u4
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4
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1
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y, v
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x, u
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</details>
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Figure E4.23 Compatibility of plane stress element
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Considering completeness, the displacement functions show that a rigid body translation in the x direction is achieved if only $\alpha_{1}$ is nonzero. Similarly, a rigid body displacement in the y direction is imposed by having only $\beta_{1}$ nonzero, and for a rigid body rotation $\alpha_{3}$ and $\beta_{2}$ must be nonzero only with $\beta_{2} = -\alpha_{3}$ . The same conclusion can also be arrived at using the matrix E that relates the strains to the generalized coordinates (see Example 4.6). This matrix also shows that the constant strain states are possible. Therefore the element is complete.
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