Chapter 17

Let loads \(P_{1}\) and \(P_{2}\) be functions of displacements \(D_{1}\) and \(D_{2}\), that is, \(P_{1}=f_{1}\left(D_{1}, D_{2}\right)\) and \(P_{2}=f_{2}\left(D_{1}, D_{2}\right)\). Let \(D_{A}\) and \(D_{B}\) be exact values of \(D_{1}\) and \(D_{2}\) produced by loads \(P_{A}\) and \(P_{B}\). Let \(D_{\lambda}^{*}\) and \(D_{B}^{*}\) be approximations of \(D_{A}\) and \(D_{B}\). Assume that \(D_{A}=D_{A}^{*}+\Delta D_{A}\) and \(D_{B}=D^{2}+\Delta D_{B}\). Derive the following equations (analogous to Eq. 17.2-9): $$ \left[\begin{array}{ll} \partial P_{1} / \partial D_{1} & \partial P_{1} / \partial D_{2} \\ \partial P_{2} / \partial D_{1} & \partial P_{2} / \partial D_{2} \end{array}\right]_{D_{\lambda}^{\prime}, D_{s}^{\prime}}\left\\{\begin{array}{l} \Delta D_{A} \\ \Delta D_{B} \end{array}\right\\}=\left\\{\begin{array}{l} P_{A}-f_{1}\left(D_{\lambda}^{*}, D_{B}^{*}\right) \\ P_{B}-f_{2}\left(D_{\lambda}^{*}, D_{B}^{\prime}\right) \end{array}\right\\} $$

17,16 The horizontal bar in the sketch is perfectly rigid and is constrained to remain horizontal as load \(P\) and displacement \(v\) increase. The three vertical bars are elastic-perfectly plastic with \(A=1, E=1\) and \(L=2\). These bars have the respective yield point loads \(F_{1}=2, F_{2}=4\), and \(F_{3}=6\). Use the tangent-stiffness method to generate the \(P\) versus \(p\) relation. Use three steps. Scale each step so that one bar begins to yield at the end of the step. Problem \(17.16\)

17\. 18 Assume that members of a truss carry only uniaxial stress, and that members in compression will buckle elastically at their critical loads without yielding. Tensile members are elastic-perfectly plastic. Outline a tangentstiffness algorithm for computation of the displacements produced by applied loads. For simplicity, assume that load reversal does not occur in any mernber

17.19 In both the initial-stiffness algorithm and the tangent-stiffness aigorithm, factor \(m\) of Eq. \(17.3-4\) is used onty to correct a trial solution after it is computed. Can the correction be anticipated instead? Explain. Section \(17.4\)

Section 17.5 17.25 Describe the steps of a tangent-stiffness solution algorithm in which cach load increment causes a single sampling point to be brought to the initiation of yiclding.

17.26 Consider a plane structure modeled by finite elements. The material is isotropic but brittle: it cracks when the tensile stress in any direction exceeds a value \(\sigma_{f}\), Outline a tangent-stiffness algorithm for predicting deformations caused by increasing load. How will the collapse load be detected by this algorithm?

\(17.27\) Imagine that corrective loads \(\left\\{\Delta \mathbf{R}_{c}\right\\}\) are to be computed for a mesh of elements having internal d.o.f. Should internal d.o.f. carry loads that result from \(\\{\sigma\\}\), or should these loads be omitted from internal d.o.f.? If carried, should they be distributed to remaining d.o.f. (that is, condensed) by means of elastic element stiffness equations?

17.28 Imagine that a plane beam of rectangular cross section is modeled by plane finite elements. The material is linearly elastic, but elastic moduli in tension and compression are different. Outline an algorithm that will calculate the stresses produced by a pure bending load. What are the comparative merits of tangent-stiffness and initial-stiffness solutions? Section 17.6

\(17.30\) Starting with the basic definition of the rate of internal work for an element \(e\) as $$ \dot{W}_{e}^{\mathrm{in}}=\int_{v_{f}}\\{\dot{\epsilon}\\}^{T}\\{\boldsymbol{\sigma}\\} d V $$ show that the rate of internal work for the entire structure is \(\dot{W}^{\text {int }}=\\{\mathbf{D}\\}^{T}\left\\{\mathbb{R}^{\text {int }}\right\\}\). behavior, \(W_{n}^{\text {int }}=\frac{1}{2}\\{\mathbf{D}\\}_{n}^{T}[\mathbf{K}]\\{\mathbf{D}\\}_{n}\). Note: \(\frac{d}{d t}\left(\\{\mathbf{D}\\}^{T}[\mathbf{K}]\\{\mathbf{D}\\}\right)=2(\mathbf{D}\\}^{T}[\mathrm{~K}]\\{\mathbf{D}\\}\) if \([\mathbf{K}]\) is symmetric.

\(17.31\) Starting with \(\dot{W}_{n}^{f n}=\\{\mathbf{D}\\}_{n}^{T}\left\\{\mathrm{R}^{\mathrm{im}}\right\\}_{\mathrm{m}}\), show that for linearly elastic material behavior, \(W_{n}^{\text {int }}=\frac{1}{2}\\{\mathbf{D}\\}_{n}^{T}[\mathbf{K}]\\{\mathbf{D}\\}_{n}\). Note: \(\frac{d}{d t}\left(\\{\mathbf{D}]^{T}[\mathbf{K}]\\{\mathbf{D}\\}\right)=2(\mathbf{D}\\}^{\top}[\mathbf{K}]\\{\mathbf{D}\\}\) if [K] is symmetric.