Chapter 5

The uniform elastic bar shown is loaded by axial force \(P\) at its left end. The bar is embedded in an elastic medium that applies end load \(F\) and distributed axial load \(q\), both of which are directly proportional to axial displacement \(u\) of the bar. Use the Galerkin method to establish matrices of a one-element FE formulation based on a linear axial displacement field.

Verify that despite collocation at \(x=L_{T} / 3\), the \(\bar{u}\) field of Eqs. 5.2-8 and the \(a_{i}\) of Eqs. 5.2-9 do not yield \(\bar{u}=u\) at \(x=L_{T} / 3\). Does this indicate that something is wrong? Explain.

Consider a uniform, simply supported beam of length \(L\). For each of the following loadings, use the Galerkin method to determine gereralized d.o.f. \(a\) in the approximating lateral deflection field \(\tilde{v}=a x(L-x)\). At midspan, determine the percentage errors of deflection \(\tilde{v}\) and bending moment \(\bar{M}=E I \tilde{v}_{1}\), . (a) Sinusoidal distributed lateral load \(\boldsymbol{q}=q_{o} \sin (\pi x / L)\) over \(0

The equation of motion of a string is \(T w_{v x}-\rho_{L} \ddot{w}=0\), where \(T=\) constant axial tension, \(w=\) small lateral displacement, \(x=\) axial coordinate, \(\rho_{L}=\) mass per unit length, and \(\tilde{w}=d^{2} w / d t^{2}\). (a) Use the Galerkin method to establish matrices of a two-d.o.f. element. (b) Consider a simply supported uniform string of length \(2 L\). Model it by two elements, each of length \(L\) and of the type formulated in part (a). Solve for the fundamental frequency of vibration. (The exact answer is \(\left.\omega^{2}=\pi^{2} T / 4 \rho_{L} L^{2}\right)\)

Consider the differential equation \(u_{, x x}+4 u=12\) in the range \(0

The differential equation for wind-driven circulation in a shallow lake is $$ \psi_{x x}+\psi_{r y y}+A \psi_{\eta x}+B \psi_{y}+C=0 $$ where \(\psi\) is the stream function and \(A, B\), and \(C\) are functions of \(x\) and \(y\). Coordinates \(x\) and \(y\) are horizontal, tangent to the lake surface. With \(h=\) depth, depthwise average velocities are \(u=\phi_{1 y} / h\) and \(v=-\psi_{x} / h\). The boundary condition is \(\psi_{n}=0\), where \(n\) is a direction normal to the shoreline. Use the Galerkin method to derive formulas for element matrices in terms of shape functions and constants [5.9].

Consider the differential equation \(u_{x x}+2 u-16 x=0\) in the range \(0

An isotropic flat disk is used as a flywheel. The disk has unit thickiness and spins about a central axis normal to its plane at constant angular velocity \(a\). The differential equation of equilibrium is $$ \frac{1}{r} \frac{d}{d r}\left(r \sigma_{r}\right)-\frac{\sigma_{6}}{r}+p \omega^{2} r=0 $$ where \(\sigma_{r}=\) radial stress, \(\sigma_{6}=\) circemferential stress, and \(\rho=\) mass density. An FE formulation can be based on annular elements of inger radius \(r_{i}\) and outer radius \(r_{o^{+}}\)Nodal d.o.f. are radial displacements, one at \(r=r_{i}\) and another at \(r=r_{o}\). Use the Galerkin method to derive formulas for element matrices in terms of shape functions and constants.