Chapter 15

An isotropic thin rectangular plate has dimension \(a\) parallel to the \(x\) axis and dimension \(b\) parallel to the \(y\) axis. The \(x\)-parallel edges are simply supported; the \(y-\) parallel edges are free. Uniform downward pressure \(p\) is applied to the upper surface. What are the principal stresses at the middle of the lower surface? What is the lateral deflection at the center of the plate?

Consider an isotropic thin square plate, with edges parallel to \(x\) and \(y\) axes, loaded only along its edges. Describe loads applied to the edges if the lateral deflection is (a) \(w=c_{1}\left(x^{2}+y^{2}\right)\), and (b) \(w=c_{2}\left(y^{2}-x^{2}\right)\), where \(c_{1}\) and \(c_{2}\) are constants.

Establish coordinates \(n s\), rotated by angle \(\beta\) with respect to \(x y\) coordinates (as in Fig. 15.5-1, for example). Let \(n\) and \(s\) be principal axes of an orthotropic material. Express transverse shear coefficients in Eq. 15.1-5 in terms of \(\beta\) and principal shear moduli \(G_{n}\) and \(G_{S}\). A procedure is suggested in the text.

Let a slender beam of length \(L\) and rectangular cross section (width \(b\) and depth \(t\) ) carry uniformly distributed lateral load \(q .\) Supports at either end allow rotation but prevent beam ends from moving laterally or axially. (a) Assume that lateral deflection w has the paraholic distribution \(w=4 w_{c} x(L-x) / L^{2}\), where \(w_{c}\) is the midspan deflection. Show that the fraction of \(q\) supported by axial (membrane) stress is \(q_{m}=(64 / 3)\left(E b t^{4} / L^{4}\right)\left(w_{c} / t\right)^{3} .\) Suggestions: Assume that tensile force is independent of \(x\), and recall that for \(w_{c} \ll