Chapter 4

It is proposed that a beam element be based on a cubic polynomial but that d.o.f. are to be lateral displacements \(v_{p}\), where \(i=1,2,3,4\). Nodes are to be at either end and at the third-points (that is, at \(x=0, L 3,2 L / 3, L)\). What convergence criterion is violated by this element?

A certain physical problem has the functional $$ \Pi_{p}=\int_{0}^{L}\left(\frac{1}{2} \phi_{x}^{2}-50 \phi\right) d x $$ Determine the Euler equation for this functional. Then solve the Euler equation with the essential boundary conditions \(\phi=0\) at \(x=0\) and \(\phi=20\) at \(x=L\) to determine \(\phi\) as a function of \(x\) and \(L\).

The block shown has weight \(W\) and rests on a rough horizontal surface having friction coefficient \(\mu . \mathbf{A}\) horizontal force \(T\) is applied to produce sliding displacement D. Show that the system is not conservative.

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

Load \(P\) applied to the tip of the column shown is called a follower load: it always has the same orientation \(\theta\) as the tip. By considering work done by \(P\) in undergoing small displacements \(v\) and \(\theta\) from initial position to final position, show that the system is not conservative. Neglect work associated with displacement of \(P\) in the \(x\) direction.

Consider a uniform straight bar of leagth \(L\) that is supported against rigid body, motion, and is loaded by a smoothly varying distributed axial force \(q\). The bar is to be modeled using two meshes. The first uses \(n\) linear- displacement bar elements, where \(n\) is an even number. The second mesh uses \(n / 2\) quadratic-displacement bar clements. The number and locations of nodes for each mesh are the same. (a) Which mesh will likely provide the more accurate displacement and stress. results? Explain. (b) If each mesh is refined by subdividing elements into two, by what factors will errors in displacements and stresses likely be reduced for each of the finer meshes?

Consider a uniform cantilever beam of length \(L\), fixed at end \(x=0\) and carrying a transverse force \(F\) at \(x=L\) (a) Let the lateral displacement field be \(v=a_{1} x^{3}\), where \(a_{1}\) is a generalized d.o.f. Is this field admissible? Explain. (b) Write a polynomial field for \(v\) that is better than that of part (a). Let the field contain three terms, each of the form \(a_{i} x^{j}\), where \(i=1,2,3\) and \(j\) is an integer such that the term is admissible. (c) Without calculation, can you predict the quality of the answers obtainable from the field of part (b) and the numerical value of any of the \(a_{i} ?\) (d) Use the field of part (a) to calculate the deflection of force \(F\).

A uniformly loaded beam of uniform flexural stiffness \(E I\) is simply supported at its ends \(x=0\) and \(x=L\). In parts (a) and (b), determine the deflection and bending moment predicted at \(x=L / 2\) by a Rayleigh-Ritz solution that has a single d.o.f. Compare exact and approximate results. (a) Use the single-d.o.f. algebraic expression \(v=a_{1} x(L-x)\). (b) Use one term of a sine series. (c) Why should you anticipate that part (b) will provide a more accurate result than part (a)?

A beam with uniform \(E I\) and length \(L\) has a pin support at \(x=0\), a roller support at \(x=L / 2\), and is loaded by a transverse tip force \(P\) at \(x=L\). Starting with a complete quadratic polynomial trial function, determine the tip displacement using the Rayleigh-Ritz method. What is the percentage error of this result, and of the associated bending moment at \(x=L / 2 ?\)

The uniform bar shown has modulus \(E\), cross sectional area \(A\), and coefficient of thermal expansion \(\alpha\). At a reference temperature of zero, the bar is stress-free and exactly fits between the two rigid walls. The bar is then subjected to a temperature field \(T(x)\) that varies linearly from zero at \(x=0\) to \(T_{0}\) at \(x=L\) where \(T_{0}>0\). Starting with a complete quadratic polynomial trial function, use the Rayleigh-Ritz method to predict the axial displacement field \(u(x)\) and the axial stress field.

Consider a straight column with uniform \(A\) and \(E I\) that supports an axial load \(P\) (positive in tension) as shown. Potential energy for small deflections is $$ \Pi_{p}=\int_{0}^{L} \frac{A E}{2} u_{x}^{2} d x+\int_{0}^{L} \frac{P}{2} v_{t_{x}}^{2} d x+\int_{0}^{L} \frac{E I}{2} v,_{x x}^{2} d x-P \delta $$ Using the Rayleigh-Ritz method, with separate trial functions for \(u\) and \(t\), estimate the buckling load for the column. Use the simplest possible kinematically admissible polynomial trial functions for \(u\) and \(v\). Express the result in terms of constants such as \(A, E I, L\), etc. One of the resulting equations will allow one of the generalized d.o.f. to be zero. Rather than accept this solution (physically, it means the column remains straight and does not buckle), let the d.o.f. be nonzero; hence obtain an equation for the buckling load.

A uniform beam is simply supported and carries a force \(P\) at its center. Use the Rayleigh-Ritz method with the infinite series $$ v=\sum a_{i} \sin \frac{i \pi x}{L} \text { with } i=1,3,5, \ldots $$ to compute deflection and bending moment at the center. Compare exact and approximate results for \(1,2,3\), and 4 terms retained.