Chapter 4

If element d.o.f. are given virtual (i.e., small imaginary) displacements \(\\{\delta \mathbf{d}\\}\), strains are changed in the amount \(\\{\delta \epsilon\\}=[\mathrm{B}]\\{\delta \mathrm{d}\\} .\) Loads acting on the structure do work that is stored as the strain energy \(\delta U=\int\\{\delta \epsilon\\}^{T}\\{\boldsymbol{\sigma}\\} d V .\) Complete this virtual work argument to obtain the structure equations \([\mathrm{K}]\\{\mathbf{D}\\}=\\{\mathbf{R}\\}\). In the process, identify formulas for \([k]\) and \(\left\\{\mathbf{r}_{e}\right\\}\).

Imagine that a pin-jointed plane truss is modeled by plane frame elements. If rotational d.o.f. \(\theta\) are suppressed at all nodes, is the truss cortectly modeled? Explain.

The best approximation of a state of pure bending that a plane bilinear element can display is \(u=\bar{u} x y / a b\) and \(v=0\), where \(\bar{u}\) is the magnitude of a corner displacement. (a) Sketch the deformed element. Show nodal forces \(F\) that produce the deformation state. (b) If Poisson's ratio is taken as zero, strain energy per unit volume is \(\left(E \epsilon_{x}^{2}+E \epsilon_{y}^{2}+G \gamma_{x y}^{2}\right) / 2 .\) Use this information to determine \(F\) as a function of \(a, b, E, G, \bar{u}\), and element thickness \(t .\) (c) What is the correct value of \(F\), according to elementary beam theory? (d) In parts (b) and (c), one can define a stiffness measure \(S\) as \(S=F / \bar{u}\). Give a physical explanation as to why \(S_{(b)}>S_{(c)}\). (e) Show that the ratio \(S_{(b)} / S_{(c)}\) approaches unity only as \(a / b\) approaches zero. What happens as \(a / b\) becomes large?

The block of material shown is loaded by axial force \(P=\sigma_{c} b t\), which produces axial deflection \(D\). Axial stiffness is \(k=P / D\). (a) Show that \(k\) is inversely proportional to \(L\) if cross-sectional area \(A:\) \(b t\) remains constant. (b) Show that \(k\) is independent of \(b\) and \(L\) if \(t\) remains constant and the aspect ratio \(b / L\) is not changed. (c) Show that \(k\) is directly proportional to a linear dimension if the shape of the element is not changed. (These behaviors are in fact observed in axial, plane, and solid elements, respectively.)

Consider the uniform, three-node bar element of Problem \(4.3 .\) Let the element be fixed at the left end and loaded by a uniformly distributed axial load of intensity \(q .\) The respective rows of \([k]\) are \(\lfloor 7,-8,1\rfloor,\lfloor-8,16,-8\rfloor\), and \(11,-8,7]\), each times \(A E / 3 L\). Calculate \(u_{2}\) and \(u_{3}\) if nodal loads are (a) calculated in consistent fashion. (b) of magnitude \(q L / 3\) at each of the three nodes.

Let a uniformly distributed load of intensity \(q\) act over only the left half of a beam element. Compute the 4 by 1 load vector \(\left\\{r_{e}\right\\}\). Check that loads in \(\left\\{\mathbf{r}_{e}\right\\}\) are statically equivalent to the original load \(q\).

The uniform cantilever beam shown carries a concentrated lateral force \(P .\) Model the beam by a single beam element. (a) Calculate \(w_{2}\) (the lateral deflection at the right end). Use the consistent load vector at node 2. Express your answer in terms of \(P, L ; E, I\), and \(x\). (b) Again calculate \(w_{2}\), but use load lumping: let the lateral force at node 2 be \(P x / L\) and ignore the moment load at node 2 . (c) Compute the exact \(w_{2}\) according to elementary beam theory. (d) Compute the ratio of the finite element \(w_{2}\) to the exact \(w_{2}\). Do this for part (a) and for part (b). Plot these ratios versus \(x / L\). (e) Compute the bending moment at the left end, first as given by the consistent loads, then as given by lumped loads. Compute the ratio of each moment to the exact value, and plot the two ratios versus \(x\) for \(0<\) \(x

A uniform simply supported beam is loaded by moment \(M_{c}\) at midspan, as shown. If the entire beam is modeled by one element, what rotation at midspan is computed? (The exact answer is \(\theta=-M L / 12 E I .\) )

If each of the two coefficients \(x y\) in Eq. 4.2-12 is replaced by \(x^{2}+y^{2}\), the element becomes incompatible. Why? Suggestion: Let two adjacent elements have two corner nodes in common. Along the boundary between elements, d.o.f. of these corner nodes must produce the same boundary displacement in each element if the elements are to be compatible. But how many d.o.f. are needed to define a quadratic curve? And what does this imply?

\(7 \mathrm{It}\) is proposed that a beam element be based on a cubic polynomial but that d.o.f. are to be only lateral displacements \(w_{i}\), where \(i=1,2,3,4\). Nodes are to be at either end and at the third points. What convergence criterion is violated by this element?

In three dimensions, which of the cubic terms would you add to a complete quadratic field (10 terms), if there are to be no preferred directions and the total number of terms in the expansion is to be (a) \(11,(b) 13,(c) 14,(d) 16\), (e) \(17,(\mathrm{f}) 19 ?\)

Sketch an assembly of hexahedral solid elements, suitable for use as a patch test mesh. Let the elements have corner nodes only and some corner angles other than \(90^{\circ} .\) Show supports and nodal loads appropriate to a test for uniform tensile stress \(\sigma_{z}\)

Nodal d.o.f. of (say) plane elements need not be restricted to displacements \(u\) and \(v .\) One might also use all four first derivatives of \(u\) and \(v\), for a total of six d.o.f. per node. Such an approach has both advantages and disadvantages. What do you think they are?