Chapter 18

For a straight bar on the \(x\) axis that may stretch and bend in the \(x y\) plane, strain and strain energy for small displacements are given by $$ \varepsilon_{x}=u_{x_{x}}+\frac{1}{2} v_{x}^{2}-y v_{x x} \quad U=\int \frac{1}{2} E \varepsilon_{x}^{2} d V $$ Show how these expressions yield familiar conventional element stiffness matrices and a stress stiffness matrix.

Imagine that a bar element is tapered, so that its cross-sectional area is a continuous function of its axial coordinate. In which element matrices \(\left([\mathbf{k}]\right.\) or \(\left[\mathbf{k}_{\sigma}\right]\) ) does the effect of taper appear, and how is it taken into account?

A massless flexible string of length \(2 L\) is attached to a ceiling. Two particles are attached to the string, each of mass \(m\), one at the middle and one at the lower end. (a) Determine the horizontal deflection at the lower end if a small horizontal force \(Q\) is applied there. (b) Determine the natural frequencies and associated mode shapes.

Model a uniform simply supported beam by a single element. Let \(L=1.0 \mathrm{~m}\), \(A=0.0002 \mathrm{~m}^{2}, E I=300.0 \mathrm{~N} \cdot \mathrm{m}^{2}\), and \(\rho=2100 \mathrm{~kg} / \mathrm{m}^{3}\). Impose symmetry, thus reducing the problem to a single d.o.f., by setting \(\theta_{2}=-\theta_{1}\). Determine (a) the fundamental vibration frequency if there is no axial force. (b) the axial force that makes the vibration frequency \(347 \mathrm{rad} / \mathrm{s}\). (c) the vibration frequency if compressive axial force \(1200 \mathrm{~N}\) is imposed.

Diagonal mass matrices are discussed in Chapter \(11 .\) Why is an analogous diagonal stress stiffness matrix unacceptable if its nonzero terms operate on only translational d.o.f.?