Chapter 8

The member \(A B C\) consists of two straight legs \(A B\) and \(B C\) of round bar stock of radius \(r\) welded together at \(B\). It is built-in at \(A\) and lies in a horizontal plane when it is unloaded. Find the deflection at \(C\) under the vertical load \(P\) in terms of the dimensions given, and the elastic constants \(E\) and \(v\) of the bar stock.

The singularity functions introduced in Sec. \(3.6\) are useful for writing the bending moment at a section \(x\) based on a free body which extends to the lefi end of the beam. Investigate the family of reversed singularity functions $$ g_{n}(x)=\langle a-x\rangle^{n} $$ and show that they are useful for writing the bending moment at a section \(x\) based on a free body which extends to the right end of the beam. Show that when \(n \geqq 0\) $$ \int_{x}^{\infty}\langle a-x\rangle^{n} d x=\frac{\langle a-x\rangle^{n+1}}{n+1} $$ and $$ \int\langle a-x\rangle^{n} d x=-\frac{\langle a-x\rangle^{n+1}}{n+1}+c $$

A simply supported floor beam carries a uniformly distributed loading of \(15 \mathrm{kN} / \mathrm{m} .\) In order to avoid possible cracking of the plaster on the ceiling beneath the beam, it is desired that the deflection should not exceed \(1 / 360\) of the span length \(L\). If \(L=3.6 \mathrm{~m}\) and \(E=7 \mathrm{GN} / \mathrm{m}^{2}\), what is the minimum allowable value of the section moment of inertia \(T\) ?

The beams \(A B\) and \(C D\) are each of length \(L\) and both have the same bending stiffness \(E I .\) The beam \(C D\) is simply supported, while \(A B\) is built-in at \(A\). Before the load \(P\) is applied the two beams make light contact at their midpoints. Find the deflection under the load \(P\).

A girder bridge is simply supported at both ends and is supported by a pontoon in the center of the bridge. How much will the pontoon sink into the water if the bridge is loaded by a uniformly distributed load \(w\) per unit of length? Let the length of the bridge be \(L\), the weight per unit volume of water be \(\gamma\), and the water-line area of the pontoon be \(A\).

A more-or-less flexible uniform box of weight \(W\) and length \(L / 3\) rests in the middle of a simply supported beam of length \(L\). Estimate the deflection at midspan and compare it with an approximate solution obtained by replacing the box by a concentrated load \(W\) at the center.

In the text the deflection of a beam was considered to be due only to the bending, i.e., the effect of shear was neglected. An estimate of the order of magnitude of the shear deflection can be obtained as follows. Let the total deflection \(v(x)\) be considered as the sum of \(v_{b}(x)\) due to bending and \(v_{s}(x)\) due to shear. The bending deflection is obtained, as in the text, by neglecting the presence of the shear forces. The shear deflection, as indicated in the figure, may be estimated by neglecting the presence of the bending moments. The deflection model shown in (b) is based on the simplifying assumption that the originally vertical plane cross sections remain plane and vertical, and that the element is subjected to a uniform shear strain \(\gamma_{x y}\). This is an oversimplification because, as indicated in Fig. 7.16, the shear strain varies across the section, and plane sections do not remain plane. This model will, however, overestimate the shear deflection if we take the uniform shear strain \(\gamma_{x y}\), to be equal to the maximum shear strain in the actual distribution. If \(\tau_{x y}(x)\) represents the maximum shear stress in the cross section as obtained by the equilibrium analysis of Chapter 7, the shear deflection of the model pictured here then satisfies the following differential equation: $$ \frac{d v_{s}}{d x}=\frac{\tau_{x y}}{G} $$ For a rectangular cantilever beam of the dimensions given in (a), show that the shear and bending deflections at \(x=L\) have the following ratio: $$ \left(\frac{v_{s}}{v_{b}}\right)_{x=L}=\frac{3(1+v)}{4} \frac{h^{2}}{L} $$ and thus that the shear deflection is less than 1 percent of the bending deflection if the beam length is more than 10 times the beam cross-sectional height.

The beam \(A D\) is built-in at both ends. The magnitude of its fully plastic bending moment is \(M_{L}\). Find the limiting value of the load \(P\) on the swing seat which corresponds to plastic collapse, as a function of the dimension \(a\), where \(0

In Example \(8.5\) it was shown that when a flexible rod overhangs a table by a length \(a\), it does not come in continuous contact with the table until a distance \(\sqrt{2} a\) back from the edge. The length of the rod, \(L\), must clearly be greater than \((1+\sqrt{2}) a\) for this solution to be valid. If the length of the rod is less than \(2 a\), equilibrium is impossible: the rod falls off the table. Analyze the intermediate situation where \(2 a

Consider a very long, uniform beam of bending modulus \(E I\) which rests on a uniformly distributed elastic foundation. Assume that the foundation exerts a distributed reaction of intensity \(-k v\) per unit length, where the displacement of the beam is \(v(x)\). Verify that when an external loading \(q(x)\) is applied, the displacement satisfies the following differential equation: $$ E I \frac{d^{2} v}{d x^{4}}+k v=q $$ Show that when \(q=0\) the general solution of this equation is $$ \begin{aligned} v=& e^{-\beta x}\left(C_{1} \cos \beta x+C_{2} \sin \beta x\right) \\ &+e^{\beta_{1}}\left(C_{3} \cos \beta x+C_{4} \sin \beta x\right) \end{aligned} $$ where \(\beta^{4}=k / 4 E I\). Verify that the deflection under a single concentrated load \(P\) at the middle of the very long beam is $$ \delta=\frac{P}{\left(64 E I k^{3}\right)^{1 / 4}} $$

In one of his discourses, Galileo tells a story about some Roman engineers who had to transport a large stone column to the construction site of a new temple. The standard method at that time was to place the column on two large logs and draw it forward with oxen. However, previous experience with columns of this size showed that they would break over one of the rollers. In order to avoid this problem, the Romans added a third support. Galileo reports that the column "broke upon the middle support." Can you explain why?

A rectangular beam of minimum weight is to be designed to satisfy the following requirements: 1\. The span \(L\) and the depth \(h\) of the beam are fixed. 2\. The beam is to carry a given total load \(W\), uniformly distributed along the length of the beam. The weight of the beam itself \(W_{B}\) is negligible in comparison with \(W\). 3\. Several materials having various values of maximum allowable stress \(\sigma_{\max }\), modulus \(E\), and weight density \(\gamma\) are to be considered. (a) In the first design, deflection is not considered to be important and the beam is designed strictly on the basis of strength. Show that the maximum value of \(W / W_{B}\) is obtained by using the material with the largest value of \(\sigma_{\max } / \gamma\). (b) In the second design the beam is deflection-limited; i.e., the maximum beam deflection cannot be greater than a specified value \(\delta_{\max }\), where for every material to be considered the deflection limit is reached before the stress limit. In this case show that \(W / W_{B}\) is maximized by the material with the largest value of \(E / \gamma\).