Chapter 3, Problem 32P

Using our experience with concentrated loading on a simple beam. Prob. 3-31, consider a uniformly loaded simple beam (Table A-9-7).

1. (a)   Show that the stress-to-load equation for a rectangular-cross-section beam is given by

   $W=\frac{4}{3}\frac{\sigma b{h}^2}{l}$

   where W = wl.

2. (b)   Subscript every- parameter with m (for model) and divide the model equation into the prototype equation. Introduce the scale factor s as in Prob. 3-31, setting σ_m/σ = 1. Express W_m and w_m in terms of the scale factor, and comment on what you have learned.

3-31 The Roman method for addressing uncertainty in design was to build a copy of a design that was satisfactory and had proven durable. Although the early Romans did not have the intellectual tools to deal with scaling size up or down, you do. Consider a simply supported, rectangular-cross-section beam with a concentrated load F, as depicted in the figure.

1. (a)   Show that the stress-to-load equation is

   $F=\frac{\sigma b{h}^{2}l}{6ac}$

2. (b)   Subscript every parameter with m (for model) and divide into the above equation. Introduce a scale factor, s = a_m/a = b_m/b = c_m/c etc. Since the Roman method was to not “lean on” the material any more than the proven design, set σ_m/σ = 1. Express F_m in terms of the scale factors and F, and comment on what you have learned.

Problem 3-31