Question

Let \(D = \left( {\begin{aligned}{*{20}{c}}{.{\bf{005}}}&{.{\bf{002}}}&{.{\bf{001}}}\\{.{\bf{002}}}&{.{\bf{004}}}&{.{\bf{002}}}\\{.{\bf{001}}}&{.{\bf{002}}}&{.{\bf{005}}}\end{aligned}} \right)\) be a flexibility matrix, with flexibility measured in inches per pound. Suppose that forces of 30, 50, and 20 lb are applied at points 1, 2, and, 3, respectively, in figure 1 of Example 3. Find the corresponding deflections.

Short answer

0.27 in, 0.30 in, and 0.23 in

Step-by-step solution

Find the force matrix

The force matrix is \(f = \left( {\begin{aligned}{*{20}{c}}{30}\\{50}\\{20}\end{aligned}} \right)\).

Find the corresponding deflection

Solve the equation \(y = Df\) to find the corresponding deflection.

\(\begin{aligned}{c}y = \left( {\begin{aligned}{*{20}{c}}{.005}&{.002}&{.001}\\{.002}&{.004}&{.002}\\{.001}&{.002}&{.005}\end{aligned}} \right)\left( {\begin{aligned}{*{20}{c}}{30}\\{50}\\{20}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{0.15 + 0.1 + 0.02}\\{0.06 + 0.2 + 0.04}\\{0.03 + 0.1 + 0.1}\end{aligned}} \right)\\ = \left( {\begin{aligned}{*{20}{c}}{0.27}\\{0.30}\\{0.23}\end{aligned}} \right)\end{aligned}\)

So, the deflections are 0.27 in, 0.30 in, and 0.23 in at points 1, 2, and 3, respectively.