Question

If the left vertical support column in Example 9–5 is made of steel, what is its cross-sectional area? Assume that a safety factor of 3 was used in its design to avoid fracture.

Short answer

The cross-sectional area of the steel support is \(2.6 \times {10^{ - 4}}\;{{\rm{m}}^2}\).

Step-by-step solution

Identification of the given data

The compressive strength of steel is\(\sigma = 500 \times {10^6}\;{\rm{N/}}{{\rm{m}}^2}\).

The force in the left vertical support column is\({F_{{\rm{left}}}} = 44100\;{\rm{N}}\).

Definition of stress

Stress is defined as the force experienced by a body per unit area. The formula for stress is

\(\sigma = \frac{{{\rm{Force}}}}{{{\rm{Area}}}} = \frac{F}{A}\). … (i)

Evaluation of the cross-sectional area of the support column

If the force in the left vertical support is increased by a factor of 3, then

\(\begin{array}{c}F' = 3{F_{{\rm{left}}}}\\ = 3\left( {44100\;{\rm{N}}} \right)\\ = 132300\;{\rm{N}}\end{array}\)

Using equation (i),

\(\begin{array}{c}A = \frac{{F'}}{\sigma }\\ = \frac{{132300\;{\rm{N}}}}{{500 \times {{10}^6}\;{\rm{N/}}{{\rm{m}}^2}}}\\ = 2.646 \times {10^{ - 4}}\;{{\rm{m}}^2}\\ \approx 2.6 \times {10^{ - 4}}\;{{\rm{m}}^2}\end{array}\)

Thus, the cross-sectional area of the steel support is \(2.6 \times {10^{ - 4}}\;{{\rm{m}}^2}\).