Chapter 3

(a) Imagine that, at each end node, a uniform bar element is to have not only axial displacement d.o.f., but axial strain do.f. as well, so that \(\\{\mathbf{d}\\}=\left\lfloor u_{1} \quad \varepsilon_{x 1} \quad u_{2} \quad \varepsilon_{x 2}\right\rfloor\). Derive the resulting 4 by 4 element stiffness matrix. (b) How can this element be used to model a bar that carries concentrated axial loads, or has abrupt changes in elastic modulus or in cross-sectional area?

Shape functions of \(C^{0}\) elements satisfy the relation \(\sum N_{i}=0\), but such is not the case for shape functions of a \(C^{1}\) element such as a plane beam. Why?

Imagine that stresses in the \(x y\) plane are reported to be \(\sigma_{x}=-6 a_{1} x^{2}, \sigma_{y}=12 a_{1} x^{2}\), and \(\tau_{x y}=12 a_{1} y^{2}\), where \(a_{1}\) is a constant.(a) Consider the square region \(0 \leq x \leq b, 0 \leq y \leq b\). Write expressions for tractions \(\Phi_{x}\) and \(\Phi_{y}\) on each side of this square, in terms of \(x, y, b\), and \(a_{1}\) * (b) If body forces are zero, is this state of stress in fact possible? Explain.

Determine if the following stress field is a valid solution of a plane elasticity problem: \(\sigma_{x}=3 a_{1} x^{2} y, \sigma_{y}=a_{1} y^{3}\), and \(\tau_{x y}=-3 a_{1} x y^{2}\), where \(a_{1}\) is a constant. The body is isotropic and linearly elastic, and body forces are zero.

(a) Consider volume \(V\) of a differential element and its change \(\Delta V\) under stress. Show that \(\Delta V / V=\varepsilon_{x}+\varepsilon_{y}+\varepsilon_{z}\) if strains are small. (b) Let hydrostatic pressure \(p\) be applied. Obtain an expression for \((\Delta V / V) / p\). (c) Hence, show that a rubberlike material is almost incompressible.

Use the virtual work argument to determine nodal moment \(q L^{2} / 12\) associated with uniformly distributed transverse load \(q\) on a beam element (as shown in Eq. 2.9-2 and Fig. 2.9-2a). That is, calculate work done as \(q\) moves through the displacement created by virtual nodal rotation \(\delta \theta_{z 1}\) and beam shape function \(N_{2}\), and equate it to work done by the nodal moment in acting through rotation \(8 \theta_{21}\).

For the following loads on a beam element, determine the consistent nodal load vector. Also show that this load vector is statically equivalent to the given load. (a) Uniformly distributed transverse load \(q\) acts on the left half of the element only. (b) Concentrated moment \(M_{c}\) is applied at midspan.

Show that initial stresses \(\left\\{\sigma_{0}\right]\) on a \(C^{0}\) element lead 'to nodal loads \(\left\\{\mathbf{r}_{e}\right\\}\) that are self- equilibrating. That is, show that nodal forces in \(\left(\mathbf{r}_{e}\right\\}\) provide a zero resultant.

Show that initial stresses \(\left\\{\sigma_{0}\right]\) on a \(C^{0}\) element lead 'to nodal loads \(\left\\{\mathbf{r}_{e}\right\\}\) that are self- equilibrating. That is, show that nodal forces in \(\left(\mathbf{r}_{e}\right\\}\) provide a zero resultant.

Show that initial stresses \(\left\\{\sigma_{0}\right]\) on a \(C^{0}\) element lead to nodal loads \(\left\\{\mathbf{r}_{e}\right\\}\) that a self- equilibrating. That is, show that nodal forces in \(\\{\mathbf{r}\). \\} provide a zero resulta