김경종
515 lines
26 KiB
Markdown
515 lines
26 KiB
Markdown
<!-- source-page: 281 -->
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<details>
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<summary>text_image</summary>
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(0, 10)
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(10, 10)
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(4, 7)
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(8, 7)
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(2, 2)
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(8, 3)
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(10, 0)
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y
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z
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x
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</details>
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(a) Patch of elements, two-dimensional elements, plate bending elements, or side view of three-dimensional elements. Each quadrilateral domain represents an element; for triangular and tetrahedral elements, each quadrilateral domain is further subdivided
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<details>
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<summary>text_image</summary>
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τyy
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τxy
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τxx
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y
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z
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x
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</details>
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Plane stress and plane strain: $\tau_{xx}$ , $\tau_{yy}$ , $\tau_{xy}$ constant; in three-dimensional analysis the additional three stress conditions $\tau_{zz}$ , $\tau_{zx}$ , $\tau_{yz}$ constant are tested
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<details>
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<summary>text_image</summary>
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τyy
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τxy
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τxx
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R
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Φ
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</details>
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Axisymmetric; here perform the test with $R \to \infty$
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<details>
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<summary>text_image</summary>
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M_y
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M_x M_xy
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y
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z x
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Plate bending
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</details>
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<details>
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<summary>text_image</summary>
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τyz
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τxz
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(This test also produces bending)
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</details>
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Plate bending (see Section 5.4.2)
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(b) Stress conditions to be tested
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Figure 4.17 Patch tests
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<!-- source-page: 282 -->
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<details>
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<summary>text_image</summary>
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Constant normal traction t_y
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Constant shear
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traction t_xy
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(2)
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(1)
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2
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5
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8
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3
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7
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1
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4
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6
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y, v
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t_xy
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t_y
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x, u
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</details>
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(a) Patch test of compatible mesh of 8-node elements (discussed in Section 5.3.1). The patch test is passed; that is, all calculated element stresses are equal to the applied tractions
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<details>
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<summary>other</summary>
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| Component | Value |
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|-----------|-------|
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| τ_xx | 2.74 |
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| τ_yy | 0.19 |
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| τ_xy | 0.08 |
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| τ_xx | 0.44 |
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| τ_yy | -0.11 |
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| τ_xy | -0.10 |
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| τ_xx | 0.08 |
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| τ_yy | -0.58 |
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| τ_xy | 0.02 |
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| τ_xx | 3.44 |
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| τ_yy | 0.04 |
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| τ_xy | -0.18 |
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| τ_xx | 2.0 |
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| τ_yy | 0.0 |
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| τ_xy | 0.0 |
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</details>
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(b) Patch test of incompatible mesh of 8-node elements. All element midside nodes are now element individual nodes with degrees of freedom not coupled to the adjacent element. Hence, two nodes are located where in Fig. 4.18(a) only one node was located. Patch test results are shown at center of elements for external traction applied in the x-direction. (Note that only the corner nodes of the complete patch are subjected to externally applied loads)
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Figure 4.18 Patch test results using the patch and element geometries of Fig. 4.17
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<!-- source-page: 283 -->
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and then as a general quadrilateral element. We also present a remedy to correct the element so that it will always pass the patch test (see E. L. Wilson and A. Ibrahimbegovic [A]).
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EXAMPLE 4.28: Consider the four-node square element with incompatible modes in Fig. E4.28(a) and determine whether the patch test is passed. Then consider the general quadrilateral element in Fig. E4.28(b) and repeat the investigation.
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We notice that the square element is really a special case of the general quadrilateral element. In fact, the quadrilateral element is formulated using the square element as a basis and using the natural coordinates $(r, s)$ in the interpolations as discussed in Section 5.2.
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<details>
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<summary>text_image</summary>
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2
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2
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Node 1
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y, v
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x, u
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2
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3
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4
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</details>
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$$
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h _ {1} = \frac {1}{4} (1 + x) (1 + y)
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$$
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$$
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h _ {2} = \frac {1}{4} (1 - x) (1 + y)
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$$
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$$
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h _ {3} = \frac {1}{4} (1 - x) (1 - y)
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$$
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$$
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h _ {4} = \frac {1}{4} (1 + x) (1 - y)
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$$
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Displacement interpolation functions
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$$
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u = \sum_ {i = 1} ^ {4} h _ {i} u _ {i} + \alpha_ {1} \phi_ {1} + \alpha_ {2} \phi_ {2}
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$$
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$$
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V = \sum_ {i = 1} ^ {4} h _ {i} v _ {i} + \alpha_ {3} \phi_ {1} + \alpha_ {4} \phi_ {2}
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$$
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$$
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\phi_ {1} = (1 - x ^ {2}); \phi_ {2} = (1 - y ^ {2})
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$$
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(a) Square element
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<details>
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<summary>text_image</summary>
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y, v
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x, u
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s
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r
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</details>
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(b) General quadrilateral element (here $h_{i}$ and $\phi_{i}$ are used with r, s coordinates; see Section 5.2)
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Figure E4.28 Four-node plane stress element with incompatible modes, constant thickness
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<!-- source-page: 284 -->
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For this element formulation we can analytically investigate whether, or under which conditions, the patch test is passed. First, we recall that the patch test is passed for the four-node compatible element (i.e., when the $\phi_{1}$ , $\phi_{2}$ displacement interpolations are not used).
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Next, let us consider that the element is placed in a condition of constant stresses $\tau^{c}$ . Then the requirement for passing the patch test is that, in these constant stress conditions, the element should behave in the same way as the four-node compatible element.
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The formal mathematical condition can be derived by considering the stiffness matrix of the element with incompatible modes.
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Let
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$$
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\hat {\mathbf {u}} ^ {*} = \left[ \begin{array}{c} \hat {\mathbf {u}} \\ \alpha \end{array} \right]
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$$
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with $\hat{\mathbf{u}}^T = [u_1\ldots u_4:v_1\ldots v_4]$
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and $\alpha^T = [\alpha_1\ldots \alpha_4]$
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Then $\pmb{\epsilon} = [\mathbf{B}:\mathbf{B}_{\mathrm{IC}}]\left[ \begin{array}{c}\hat{\mathbf{u}}\\ \cdot \cdot \cdot \\ \alpha \end{array} \right]$
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where B is the usual strain-displacement matrix of the four-node element and $B_{IC}$ is the contribution due to the incompatible modes.
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Hence, with our usual notation, we have
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$$
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\left[ \begin{array}{c c} \int_ {V} \mathbf {B} ^ {T} \mathbf {C B} d V & \int_ {V} \mathbf {B} ^ {T} \mathbf {C B} _ {\mathrm{IC}} d V \\ \hdashline \int_ {V} \mathbf {B} _ {\mathrm{IC}} ^ {T} \mathbf {C B} d V & \int_ {V} \mathbf {B} _ {\mathrm{IC}} ^ {T} \mathbf {C B} _ {\mathrm{IC}} d V \end{array} \right] \left[ \begin{array}{c} \hat {\mathbf {u}} \\ \boldsymbol {\alpha} \end{array} \right] = \left[ \begin{array}{c} \mathbf {R} \\ \mathbf {0} \end{array} \right] \tag {a}
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$$
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In practice, the incompatible displacement parameters $\alpha$ would now be statically condensed out to obtain the element stiffness matrix corresponding to only the $\hat{u}$ degrees of freedom.
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If the nodal point displacements are the physically correct values $\hat{\mathbf{u}}^c$ for the constant stresses $\tau^c$ , we have
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$$
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\int_ {V} \mathbf {B} _ {\mathrm{IC}} ^ {T} \mathbf {C B} d V \hat {\mathbf {u}} ^ {c} = \int_ {V} \mathbf {B} _ {\mathrm{IC}} ^ {T} \boldsymbol {\tau} ^ {c} d V \tag {b}
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$$
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To now force the element to behave under constant stress conditions in the same way as the four-node compatible element, we require that (since the entries $\pi^c$ are independent of each other)
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$$
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\int_ {V} \mathbf {B} _ {\mathrm{IC}} ^ {T} d V = \mathbf {0} \tag {c}
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$$
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Namely, when (c) is satisfied, we find from (a):
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If the nodal point forces of the element are those of the compatible four-node element, the solution is $\hat{u} = \hat{u}^{c}$ and $\alpha = 0$ . Also, of course, if we set $\hat{u} = \hat{u}^{c}$ and $\alpha = 0$ , we obtain from (a) the nodal point forces of the compatible four-node element and no forces corresponding to the incompatible modes.
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Hence, under constant stress conditions the element behaves as if the incompatible modes were not present.
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<!-- source-page: 285 -->
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We can now easily check that the condition in (c) is satisfied for the square element:
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$$
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\int_ {V} \left[ \begin{array}{c c c c} - 2 x & 0 & 0 & 0 \\ 0 & 0 & 0 & - 2 y \\ 0 & - 2 y & - 2 x & 0 \end{array} \right] d V = \mathbf {0}
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$$
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However, we can also check that the condition is not satisfied for the general quadrilateral element (here the Jacobian transformation of Section 5.2 is used to evaluate $B_{IC}$ ). In order to satisfy (c) we therefore modify the $B_{IC}$ matrix by a correction $B_{IC}^{C}$ and use
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$$
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\mathbf {B} _ {\mathrm{IC}} ^ {\text { new }} = \mathbf {B} _ {\mathrm{IC}} + \mathbf {B} _ {\mathrm{IC}} ^ {\mathrm{C}}
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$$
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The condition (c) on $\mathbf{B}_{\mathrm{IC}}^{\mathrm{new}}$ gives
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$$
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\mathbf {B} _ {\mathrm{IC}} ^ {\mathrm{C}} = - \frac {1}{V} \int_ {V} \mathbf {B} _ {\mathrm{IC}} d V
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$$
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The element stiffness matrix is then obtained by using $B_{IC}^{new}$ in (a) instead of $B_{IC}$ . In practice, the element stiffness matrix is evaluated by numerical integration (see Chapter 5), and $B_{IC}^{C}$ is calculated by numerical integration prior to the evaluation of (a).
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With the above patch test we test only for the constant stress conditions. Any patch of elements with incompatibilities must be able to represent these conditions if convergence is to be ensured.
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In essence, this patch test is a boundary value problem in which the external forces are prescribed (the forces $f^{B}$ are zero and the tractions $f^{S}$ are constant) and the deformations and internal stresses are calculated (the rigid body modes are merely suppressed to render the solution possible). If the deformations and constant stresses are correctly predicted, the patch test is passed, and (because at least constant stresses can be correctly predicted) convergence in stresses will be at least $o(h)$ .
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This interpretation of the patch test suggests that we may in an analogous manner also test for the order of convergence of a discretization. Namely, using the same concept, we may instead apply the external forces that correspond to higher-order variations of internal stresses and test whether these stresses are correctly predicted. For example, in order to test whether a discretization will give a quadratic order of stress convergence, that is, whether the stresses converge $o(h^{2})$ , a linear stress variation needs to be correctly represented. We infer from the basic differential equations of equilibrium that the corresponding patch test is to apply a constant value of internal forces and the corresponding boundary tractions. While numerical results are again of interest and are valuable as in the test for constant stress conditions, only analytical results can ensure that for all geometric element distortions in the patch the correct stresses and deformations are obtained (see Section 5.3.3 for further discussion and results).
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Of course, in practice, when testing element formulations, this formal procedure of evaluating the order of convergence frequently is not followed, and instead a sequence of simple test problems is used to identify the predictive capability of an element.
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# 4.4.2 Mixed Formulations
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To formulate the displacement-based finite elements we have used the principle of virtual displacements, which is equivalent to invoking the stationarity of the total potential energy $\Pi$ (see Example 4.4). The essential theory used can be summarized briefly as follows.
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<!-- source-page: 286 -->
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1. We use $^{15}$
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$$
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\Pi (\mathbf {u}) = \frac {1}{2} \int_ {V} \boldsymbol {\epsilon} ^ {T} \mathbf {C} \boldsymbol {\epsilon} d V - \int_ {V} \mathbf {u} ^ {T} \mathbf {f} ^ {B} d V - \int_ {S _ {f}} \mathbf {u} ^ {S _ {f} ^ {T}} \mathbf {f} ^ {S _ {f}} d S \tag {4.109}
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$$
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$$
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= \text { stationary }
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$$
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with the conditions
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$$
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\epsilon = \partial_ {\epsilon} \mathbf {u} \tag {4.110}
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$$
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$$
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\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p} = \mathbf {0} \tag {4.111}
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$$
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where $\partial_{\epsilon}$ represents the differential operator on u to obtain the strain components, the vector $u_{p}$ contains the prescribed displacements, and the vector $u^{S_{u}}$ lists the corresponding displacement components of u.
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If the strain components are ordered as in (4.3), we have
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$$
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\mathbf {u} = \left[ \begin{array}{l} u (x, y, z) \\ v (x, y, z) \\ w (x, y, z) \end{array} \right]; \quad \boldsymbol {\partial} _ {\epsilon} = \left[ \begin{array}{c c c} \frac {\partial}{\partial x} & 0 & 0 \\ 0 & \frac {\partial}{\partial y} & 0 \\ 0 & 0 & \frac {\partial}{\partial z} \\ \frac {\partial}{\partial y} & \frac {\partial}{\partial x} & 0 \\ 0 & \frac {\partial}{\partial z} & \frac {\partial}{\partial y} \\ \frac {\partial}{\partial z} & 0 & \frac {\partial}{\partial x} \end{array} \right]
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$$
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2. The equilibrium equations are obtained by invoking the stationarity of $\Pi$ (with respect to the displacements which appear in the strains),
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$$
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\int_ {V} \delta \boldsymbol {\epsilon} ^ {T} \mathbf {C} \boldsymbol {\epsilon} d V = \int_ {V} \delta \mathbf {u} ^ {T} \mathbf {f} ^ {B} d V + \int_ {S _ {f}} \delta \mathbf {u} ^ {S _ {f} ^ {T}} \mathbf {f} ^ {S _ {f}} d S \tag {4.112}
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$$
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The variations on u must be zero at and corresponding to the prescribed displacements on the surface area $S_{u}$ . We recall that to obtain from (4.112) the differential equations of equilibrium and the stress (natural) boundary conditions we substitute $C\epsilon = \tau$ and reverse the process of transformation employed in Example 4.2 (see Sections 3.3.2 and 3.3.4). Therefore, the stress-strain relationship, the strain-displacement conditions [in (4.110)], and the displacement boundary conditions [in (4.111)] are directly fulfilled, and the condition of differential equilibrium (in the interior and on the boundary) is a consequence of the stationarity condition of $\Pi$ .
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3. In the displacement-based finite element solution the stress-strain relationship, the strain-displacement conditions [in (4.110)], and the displacement boundary conditions [in (4.111)] are satisfied exactly, but the differential equations of equilibrium in the interior and the stress (natural) boundary conditions are satisfied only in the limit as the number of elements increases.
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<!-- source-page: 287 -->
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The important point to note concerning the use of $(4.109)$ to $(4.112)$ for a finite element solution is that the only solution variables are the displacements which must satisfy the displacement boundary conditions in $(4.111)$ and appropriate interelement conditions. Once we have calculated the displacements, other variables of interest such as strains and stresses can be directly obtained.
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In practice, the displacement-based finite element formulation is used most frequently; however, other techniques have also been employed successfully and in some cases are much more effective (see Section 4.4.3).
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Some very general finite element formulations are obtained by using variational principles that can be regarded as extensions of the principle of stationarity of total potential. These extended variational principles use not only the displacements but also the strains and/or stresses as primary variables. In the finite element solutions, the unknown variables are therefore then also displacements and strains and/or stresses. These finite element formulations are referred to as mixed finite element formulations.
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Various extended variational principles can be used as the basis of a finite element formulation, and the use of many different finite element interpolations can be pursued. While a large number of mixed finite element formulations has consequently been proposed (see, for example, H. Kardestuncer and D. H. Norrie (eds.) [A] and F. Brezzi and M. Fortin [A]), our objective here is only to present briefly some of the basic ideas, which we shall then use to formulate some efficient solution schemes (see Sections 4.4.3 and 5.4).
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To arrive at a very general and powerful variational principle we rewrite (4.109) in the form
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$$
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\begin{array}{l} \Pi^ {*} = \Pi - \int_ {V} \boldsymbol {\lambda} _ {\epsilon} ^ {T} (\boldsymbol {\epsilon} - \boldsymbol {\partial} _ {\epsilon} \mathbf {u}) d V - \int_ {S _ {u}} \boldsymbol {\lambda} _ {u} ^ {T} \left(\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p}\right) d S \tag {4.113} \\ = \text { stationary } \\ \end{array}
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$$
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where $\lambda_{\epsilon}$ and $\lambda_{u}$ are Lagrange multipliers and $S_{u}$ is the surface on which displacements are prescribed. The Lagrange multipliers are used here to enforce the conditions (4.110) and (4.111) (see Section 3.4). The variables in (4.113) are $\mathbf{u}$ , $\epsilon$ , $\lambda_{\epsilon}$ , and $\lambda_{u}$ . By invoking $\delta\Pi^{*}=0$ the Lagrange multipliers $\lambda_{\epsilon}$ and $\lambda_{u}$ are identified, respectively, as the stresses $\tau$ and tractions over $S_{u}$ , $\mathbf{f}^{S_{u}}$ , so that the variational indicator in (4.113) can be written as
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$$
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\Pi_ {\mathrm{HW}} = \Pi - \int_ {V} \boldsymbol {\tau} ^ {T} (\boldsymbol {\epsilon} - \boldsymbol {\partial} _ {\boldsymbol {\epsilon}} \mathbf {u}) d V - \int_ {S _ {u}} \mathbf {f} ^ {S _ {u} T} \left(\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p}\right) d S \tag {4.114}
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$$
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This functional is referred to as the Hu-Washizu functional (see H. C. Hu [A] and K. Washizu [A, B]). The independent variables in this functional are the displacements u, strains $\epsilon$ , stresses $\tau$ , and surface tractions $f^{S_{u}}$ . The functional can be used to derive a number of other functionals, such as the Hellinger-Reissner functionals (see E. Hellinger [A] and E. Reissner [A], Examples 4.30 and 4.31, and Exercise 4.36) and the minimum complementary energy functional, and can be regarded as the foundation of many finite element methods (see H. Kardestuncer and D. H. Norrie (eds.) [A], T. H. H. Pian and P. Tong [A], and W. Wunderlich [A]).
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Invoking the stationarity of $\Pi_{\mathsf{HW}}$ with respect to $\mathbf{u},\epsilon ,\tau$ , and $\mathbf{f}^{S_u}$ , we obtain
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$$
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\int_ {V} \delta \boldsymbol {\epsilon} ^ {T} \mathbf {C} \boldsymbol {\epsilon} d V - \int_ {V} \delta \mathbf {u} ^ {T} \mathbf {f} ^ {B} d V - \int_ {S _ {f}} \delta \mathbf {u} ^ {S _ {f} T} \mathbf {f} ^ {S _ {f}} d S - \int_ {V} \delta \boldsymbol {\tau} ^ {T} (\boldsymbol {\epsilon} - \boldsymbol {\partial} _ {\boldsymbol {\epsilon}} \mathbf {u}) d V
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$$
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<!-- source-page: 288 -->
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$$
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- \int_ {V} \boldsymbol {\tau} ^ {T} (\delta \boldsymbol {\epsilon} - \boldsymbol {\partial} _ {\epsilon} \delta \mathbf {u}) d V - \int_ {S _ {u}} \delta \mathbf {f} ^ {S _ {u} T} \left(\mathbf {u} ^ {S _ {u}} - \mathbf {u} _ {p}\right) d S - \int_ {S _ {u}} \mathbf {f} ^ {S _ {u} T} \delta \mathbf {u} ^ {S _ {u}} d S = 0 \tag {4.115}
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$$
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where $S_{f}$ is the surface on which known tractions are prescribed.
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The above discussion shows that the Hu-Washizu variational formulation may be regarded as a generalization of the principle of virtual displacements, in which the displacement boundary conditions and strain compatibility conditions have been relaxed but then imposed by Lagrange multipliers, and variations are performed on all unknown displacements, strains, stresses, and unknown surface tractions. That this principle is indeed a valid and most general description of the static and kinematic conditions of the body under consideration follows because $(4.115)$ yields, since $(4.115)$ must hold for the individual variations used, the following.
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For the volume of the body:
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The stress-strain condition,
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$$
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\tau = \mathbf {C} \epsilon \tag {4.116}
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$$
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The compatibility condition,
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$$
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\epsilon = \partial_ {\epsilon} \mathbf {u} \tag {4.117}
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$$
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The equilibrium conditions,
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$$
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\frac {\partial \tau_ {x x}}{\partial x} + \frac {\partial \tau_ {x y}}{\partial y} + \frac {\partial \tau_ {x z}}{\partial z} + f _ {x} ^ {B} = 0
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$$
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$$
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\frac {\partial \tau_ {y x}}{\partial x} + \frac {\partial \tau_ {y y}}{\partial y} + \frac {\partial \tau_ {y z}}{\partial z} + f _ {y} ^ {B} = 0 \tag {4.118}
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$$
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$$
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\frac {\partial \tau_ {z x}}{\partial x} + \frac {\partial \tau_ {z y}}{\partial y} + \frac {\partial \tau_ {z z}}{\partial z} + f _ {z} ^ {B} = 0
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$$
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For the surface of the body:
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The applied tractions are equilibrated by the stresses,
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$$
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\mathbf {f} ^ {S _ {f}} = \overline {{{\boldsymbol {\tau}}}} \mathbf {n} \quad \text { on } S _ {f} \tag {4.119}
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$$
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The reactions are equilibrated by the stresses,
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$$
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\mathbf {f} ^ {S _ {u}} = \overline {{{\boldsymbol {\tau}}}} \mathbf {n} \quad \text { on } S _ {u} \tag {4.120}
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$$
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where n represents the unit normal vector to the surface and $\overline{\tau}$ contains in matrix form the components of the vector $\tau$ .
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The displacements on $S_{u}$ are equal to the prescribed displacements,
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$$
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\mathbf {u} ^ {S _ {u}} = \mathbf {u} _ {p} \quad \text { on } S _ {u} \tag {4.121}
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$$
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The variational formulation in (4.115) represents a very general continuum mechanics formulation of the problems in elasticity.
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<!-- source-page: 289 -->
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Considering now the possibilities for finite element solution procedures, the Hu-Washizu variational principle and principles derived therefrom can be directly employed to derive various finite element discretizations. In these finite element solution procedures the applicable continuity requirements of the finite element variables between elements and on the boundaries need to be satisfied either directly or to be imposed by Lagrange multipliers. It now becomes apparent that with this added flexibility in formulating finite element methods a large number of different finite element discretizations can be devised, depending on which variational principle is used as the basis of the formulation, which finite element interpolations are employed, and how the continuity requirements are enforced. The various different discretization procedures have been classified as hybrid and mixed finite element formulations (see H. Kardestuncer and D. H. Norrie (eds.) [A] and T. H. H. Pian and P. Tong [A]).
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We demonstrate the use of the Hu-Washizu principle in the following examples.
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EXAMPLE 4.29: Consider the three-node truss element shown in Fig. E4.29. Assume a parabolic variation for the displacement and a linear variation in strain and stress. Also, let the stress and strain variables correspond to internal element degrees of freedom so that only the displacements at nodes 1 and 2 connect to the adjacent elements. Use the Hu-Washizu variational principle to calculate the element stiffness matrix.
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<details>
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<summary>text_image</summary>
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Young's modulus E
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Area A
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2 3 1
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x,u
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1 1
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</details>
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Figure E4.29 Three-node truss element
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We can start directly with (4.115) to obtain
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$$
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\underbrace {\int_ {V} \delta \epsilon^ {T} (C \epsilon - \tau) d V} _ {①} - \underbrace {\int_ {V} \delta \tau^ {T} (\epsilon - \partial_ {\epsilon} u) d V} _ {②} \tag {a}
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$$
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$$
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+ \underbrace {\int_ {V} \left(\partial_ {\epsilon} \delta u\right) ^ {T} \tau d V} _ {③} - \int_ {V} \delta u ^ {T} f ^ {B} d V + \text { boundary terms } = 0
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$$
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where $\epsilon = \epsilon_{xx};\quad \partial_{\epsilon} = \frac{\partial}{\partial x};\quad \tau = \tau_{xx};\quad C = E;\quad f^{B} = f_{x}^{B}$
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and the boundary terms correspond to expressions for $S_{f}$ and $S_{u}$ and are not needed to evaluate the element stiffness matrix.
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We now use the following interpolations:
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$$
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u = \mathbf {H} \hat {\mathbf {u}}; \quad \mathbf {H} = \left[ \frac {(1 + x) x}{2} - \frac {(1 - x) x}{2} 1 - x ^ {2} \right]
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$$
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$$
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\hat {\mathbf {u}} ^ {T} = \left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \end{array} \right]
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$$
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<!-- source-page: 290 -->
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$$
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\tau = \mathbf {E} \hat {\mathbf {r}}; \quad \mathbf {E} = \left[ \frac {1 + x}{2} \quad \frac {1 - x}{2} \right]
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$$
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$$
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\epsilon = \mathbf {E} \hat {\epsilon}
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$$
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$$
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\hat {\pmb {\tau}} ^ {T} = \left[ \begin{array}{l l} \tau_ {1} & \tau_ {2} \end{array} \right]; \quad \hat {\pmb {\epsilon}} ^ {T} = \left[ \begin{array}{l l} \epsilon_ {1} & \epsilon_ {2} \end{array} \right]
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$$
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Substituting the interpolations into (a), we obtain corresponding to term 1:
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$$
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\delta \hat {\boldsymbol {\epsilon}} ^ {T} \left[ \left(\int_ {V} \mathbf {E} ^ {T} C \mathbf {E} d V\right) \hat {\boldsymbol {\epsilon}} - \left(\int_ {V} \mathbf {E} ^ {T} \mathbf {E} d V\right) \hat {\boldsymbol {\tau}} \right]
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$$
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corresponding to term 2:
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$$
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\delta \hat {\mathbf {r}} ^ {T} \left[ - \left(\int_ {V} \mathbf {E} ^ {T} \mathbf {E} d V\right) \hat {\mathbf {e}} + \left(\int_ {V} \mathbf {E} ^ {T} \mathbf {B} d V\right) \hat {\mathbf {u}} \right]
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$$
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corresponding to term 3: $\delta \hat{\mathbf{u}}^T\left(\int_V\mathbf{B}^T\mathbf{E}dV\right)\hat{\boldsymbol{\tau}}$
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where $\mathbf{B} = \left[(\frac{1}{2} + x) - (-\frac{1}{2} + x) - 2x\right]$
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Hence, we obtain
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$$
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\left[ \begin{array}{c c c} \mathbf {0} & \mathbf {0} & \mathbf {K} _ {u \tau} \\ \mathbf {0} & \mathbf {K} _ {\epsilon \epsilon} & \mathbf {K} _ {\epsilon \tau} \\ \mathbf {K} _ {u \tau} ^ {T} & \mathbf {K} _ {\epsilon \tau} ^ {T} & \mathbf {0} \end{array} \right] \left[ \begin{array}{c} \hat {\mathbf {u}} \\ \hat {\boldsymbol {\epsilon}} \\ \hat {\boldsymbol {\tau}} \end{array} \right] = \dots \tag {b}
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$$
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where $\mathbf{K}_{\epsilon \epsilon} = \int_{V}\mathbf{E}^{T}\mathbf{C}\mathbf{E}dV$
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$$
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\mathbf {K} _ {u \tau} = \int_ {V} \mathbf {B} ^ {T} \mathbf {E} d V
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$$
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and $\mathbf{K}_{\epsilon \tau} = -\int_{V}\mathbf{E}^{T}\mathbf{E}dV$
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If we now substitute the expressions for B and E and eliminate the $\epsilon_{i}$ and $\tau_{i}$ degrees of freedom (because they are assumed to pertain only to this element, thus allowing jumps in stresses and strains between adjacent elements), we obtain from (b)
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$$
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\frac {E A}{6} \left[ \begin{array}{r r r} 7 & 1 & - 8 \\ 1 & 7 & - 8 \\ - 8 & - 8 & 1 6 \end{array} \right] \left[ \begin{array}{l} u _ {1} \\ u _ {2} \\ u _ {3} \end{array} \right] = \dots
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$$
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This stiffness matrix is identical to the matrix of a three-node displacement-based truss element—as should be expected using a linear strain and parabolic displacement assumption.
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However, we should note that if the element stress and strain variables are not eliminated on the element level and instead are used to impose continuity in stress and strain between elements, then clearly with the element stiffness matrix in (b) the stiffness matrix of the complete element assemblage is not positive definite.
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This derivation could of course be extended to obtain the stiffness matrices of truss elements with various displacement, stress, and strain assumptions. However, a useful element is obtained only if the interpolations are “judiciously” chosen and actually fulfill specific requirements (see Section 4.5).
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