Problem
In order to support the required loads and maximum deflection limits, a rectangular timber floor beam needs to be selected such that it has the following criteria: \[ \begin{array}{l} I_{x} \geq 100 i n^{4} \\ S_{x} \geq 30 i n^{3} \end{array} \] Which of the following sections (width \( x \) depth) is the most economical (i.e. has the)smallest area) section while meeting all criteria? a. \( 2^{\prime \prime} \) wide \( x 8^{\prime \prime} \) deep b. \( 4^{\prime \prime} \) wide \( x 8^{\prime \prime} \) deep c. 2" wide \( \times 10^{\prime \prime} \) deep d. 6" wide \( x \) 6" deep e. 2" wide \( x 12^{\prime \prime} \) deep
Solution
Step 1 ：\(I_x = \frac{1}{12}bh^3\)
Step 2 ：\(S_x = \frac{1}{6}bh^2\)
Step 3 ：\begin{array}{|c|c|c|} \hline \text{Section} & I_x & S_x \\ \hline a & 2 \times (\frac{1}{12})(2)(8^3) = 853.3333 & 2 \times (\frac{1}{6})(2)(8^2) = 170.6667 \\ \hline b & 4 \times (\frac{1}{12})(4)(8^3) = 6826.6667 & 4 \times (\frac{1}{6})(4)(8^2) = 682.6667 \\ \hline c & 2 \times (\frac{1}{12})(2)(10^3) = 1666.6667 & 2 \times (\frac{1}{6})(2)(10^2) = 666.6667 \\ \hline d & 6 \times (\frac{1}{12})(6)(6^3) = 6480 & 6 \times (\frac{1}{6})(6)(6^2) = 432 \\ \hline e & 2 \times (\frac{1}{12})(2)(12^3) = 2880 & 2 \times (\frac{1}{6})(2)(12^2) = 576 \\ \hline \end{array}\)
Step 4 ：\text{The most economical section is } a
