Chapter 8

A stiffness matrix and a consistent load vector can be formulated for the three-node bar element of Problem \(8.4\) by use of the displacement field $$ u=\frac{L \div x}{L} u_{1}+\frac{x}{L} u_{2}+x(L-x) a_{1} $$ where \(a_{1}\) is a nodeless d.o.f. (a) Determine the 3 by 3 stiffness matrix \([\mathbf{k}]\) dictated by the given \(u\) field. Let the element be uniform. (b) Under what circumstances do you think the added mode \(x(L-x) a_{1}\) will improve the results given by the basic linear element? In what stage of a finite element stress analysis does \(a_{1}\) have an effect?

Addition to an element of internal d.o.f., such as \(a_{1}\) and \(a_{2}\) in Eq. 8.1-4, can be regarded as a device that permits better approximation of equilibrium. equations within the element, without affecting interelement compatibility. Accordingly, do you think the constant-strain triangle (Section 5.4) would be improved by addition of the bubble function modes \(u=\xi_{1} \xi_{2} \xi_{3} a_{1}\) and \(v=\xi_{1} \xi_{2} \xi_{3} a_{2}\) ? Why or why not?

Imagine that lateral deflection \(w\) of a uniform beam element is defined by three nodal values, as shown. (a) Establish the 3 by 4 transformation matrix \([\mathrm{T}]\) that will convert this element to one that operates on the standard d.o.f. \(w_{1}, \theta_{1}, w_{2}\), and \(\theta_{2}\). (b) Hence, establish the new shape functions \(N_{1}, N_{2}, N_{3}\), and \(N_{4}\). (c) What property does the element have that may pose a difficulty? \(w=\frac{2 x^{2}-3 L x+L^{2}}{L^{2}} w_{1}+\frac{2 x^{2}-L x}{L^{2}} w_{2}+\frac{4 x(L-x)}{L^{2}}_{w_{3}}\)

(a) Write the equation \(\\{\boldsymbol{\sigma}\\}=[\mathbf{P}]\\{\boldsymbol{\beta}\\}\) for a plane elemept if \(\sigma_{x}=\beta_{1}+\beta_{4} x_{1}\) \(\sigma_{y}=\beta_{2}+\beta_{5} y\), and \(\tau_{x y}=\beta_{3}\). Do you think such an element would be a good one? (b) If \(\beta_{4}=\beta_{5}=0\) in part (a), so that \(\left.\\{\beta\\}=\mid \beta_{1} \beta_{2} \quad \beta_{3}\right]^{T}\), what defect would you expect to see in the stiffness matrix of a plane eight-d.o.f. rectangular element?

(a) A bar element of axial stiffness \(k=A E / L\) is permitted only axial nodal displacements \(u_{1}\) and \(u_{2} .\) Write \(\left[\mathrm{k}_{E E}\right]\), and from it determine \([\mathrm{k}]\). (b) Repeat part (a), but let there be four d.o.f. \(\\{d\\}=\left\lfloor\begin{array}{llll}u_{1} & v_{1} & u_{2} & \left.v_{2}\right\\}^{T} \text {, as }\end{array}\right.\) in Eq. \(2.4-3\), so that plane motion is possible.

A long bar, \(100 \mathrm{~mm}\) wide and \(20 \mathrm{~mm}\) thick, is loaded in tension by an axial force \(P\) (a) If the yield strength is \(Y=1150 \mathrm{MPa}\) and \(K_{\mathrm{lC}}=77 \mathrm{MPa} \sqrt{\mathrm{m}}\), and a central crack \(15 \mathrm{~mm}\) long is present, what force \(P\) will fracture the bar? (b) If the yield strength is \(Y=1410 \mathrm{MPa}\) and \(K_{1 C}=50 \mathrm{MPa} \sqrt{\mathrm{m}}\), what is the critical crack length if the force \(P\) determined in part (a) is applied?

Let quarter-point elements be used to solve a certain crack problem (c.g. Fig. 8.9-4). Imagine that the problem is solved again, this time using quarterpoint elements of smaller size. Now the computed results are found to be less accurate than before. Explain how this is possible.

The beam element shown has the usual d.o.f. \(\\{\mathrm{d}\\}=\left[\begin{array}{llll}w_{1} & \theta_{1} & w_{2} & \theta_{2}\end{array}\right]^{T}\). The element has width \(b\) and rests on a Winkler foundation of modulus \(\beta .\) Determine the foundation matrix \(\left[k_{f}\right]\) defined by each of the following approximations. (a) Deflection \(w\) is cubic in \(x\), as in the standard beam element. (b) Deflection \(w\) is quadratic in \(x\) (see Eq. 8.4-2). (c) Deflection \(w\) is linear in \(x\) (and independent of \(\theta_{\mathrm{I}}\) and \(\theta_{2}\) ). (d) Deflection \(w\) is constant.

Imagine that separation is possible between a beam and its Winkler elastic foundation. Outline a solution algorithm for such a problem. In this exercise, do not be concerned with computational efficiency.

If \(\\{\mathbf{R}\\}\) is unchanged and structural alterations are minor, then \(\\{\Delta \mathrm{D}\\} \approx\) \(-[\mathbf{K}]^{-1}([\Delta \mathbf{K}]\\{\mathbf{D}\\})\). Derive this expression for \(\\{\Delta \mathrm{D}\\}\). What are advantages and disadvantages of this method?

Equation 8.13-2 can be cast in the iterative form \([\mathbf{K}]\left[\mathbf{D}^{*}\right\\}_{i+1}=\\{\mathbf{R}\\}-[\Delta \mathbf{K}]\left(\mathbf{D}^{*}\right\\}_{i}\). Consider the application of this equation to single-d.o.f. problems as follows. (a) Let \(K=0.5, K^{*}=0.8\), and \(R=2\). Starting with \(D_{0}^{\prime}=D=4.0\), compute \(D_{3}^{*}\) (i.c., apply five iterative cycles). (b) For what range of values of \(\Delta K / K\) does this iterative method converge?