Question

A steel beam is placed vertically in the basement of a building to keep the floor above from sagging. The load on the beam is $5.8 \times 10^{4} \mathrm{N},\( the length of the beam is \)2.5 \mathrm{m},$ and the cross- sectional area of the beam is \(7.5 \times 10^{-3} \mathrm{m}^{2}\) Find the vertical compression of the beam.

Short answer

Answer: The vertical compression of the steel beam is approximately \(0.000096625\) meters.

Step-by-step solution

Write down the given information

We know the following values: Load on the beam (F) = \(5.8 \times 10^{4} \mathrm{N}\) Length of the beam (L) = \(2.5 \mathrm{m}\) Cross-sectional area of the beam (A) = \(7.5 \times 10^{-3} \mathrm{m}^{2}\)

Find the stress

Using the formula for stress, we have: Stress (σ) = \(\frac{F}{A}\) Plug in the values: σ = \(\frac{5.8 \times 10^{4} \mathrm{N}}{7.5 \times 10^{-3} \mathrm{m^2}}\) σ = \(7.73 \times 10^6 \mathrm{Pa}\)

Find the strain

Using Hooke's Law, we have: Strain (ε) = \(\frac{σ}{E}\) The modulus of elasticity of steel (E) is approximately \(2 \times 10^{11} \mathrm{Pa}\). Plug in the values: ε = \(\frac{7.73 \times 10^6 \mathrm{Pa}}{2 \times 10^{11} \mathrm{Pa}}\) ε = \(3.865 \times 10^{-5}\)

Calculate the vertical compression

Using the formula for strain, we have: ε = \(\frac{ΔL}{L}\) We need to find ΔL (the vertical compression), so we rearrange the formula: ΔL = ε × L Plug in the values: ΔL = \(3.865 \times 10^{-5} \times 2.5 \mathrm{m}\) ΔL = \(9.6625 \times 10^{-5} \mathrm{m}\) Therefore, the vertical compression of the steel beam is approximately \(9.6625 \times 10^{-5}\) meters or \(0.000096625\) meters.