A reinforced concrete column 400mm by 400mm and 3.5m high is subjected to a load

Question

A reinforced concrete column 400mm by 400mm and 3.5m high is subjected to a load 160kN. The column shortens by 0.07mm.
a) What is the stress in the column?
b) What is the strain in the column?
c) Given that the maximum allowable stress of the concrete is 20MPa, is the concrete column strong enough?
d) Could you reduce the size of the column? What would be the minimum size of the column?
(Please show work)

Explanation / Answer

Partly as a refresher for myself: (a) Stress is defined as force per unit area. The area which is receiving the force is the 400mm square section of the column (A = l*w = [400mm]^2). The force is 160kN. Stress = force/area = 160kN/(160000mm^2) = 160/160 MPa *side note N/mm^2 = MPa* Stress = 1 MPa or 1000kPa (b) Strain is the unitless change in length per unit length. The direction of the strain is the height so we want the change in height (-0.07mm) divided by the height (3.5m = 3500mm). Strain = change/length = -0.07mm/3500mm = -2 E-5 mm/mm *side note, the units mm/mm are not necessary, but sometimes useful to help give a scale* Strain = 2 E-5 mm/mm (c) The maximum allowable stress is 20 MPa and we are only seeing 1 MPa so, YES, I would say that the structure is strong enough. (d) Since the structure is strong enough, we could reduce the size of the column. Now we just set the maximum stress equal to the stress in our equation and solve for the new area we need to maintain: 20 MPa = force/area = 160kN/area --> area = 160kN/20MPa = 8000 mm^2 Area = l*w = side^2 --> set it equal to our new calculated minimum area: side^2 = 8000 mm^2 --> side = SQRT(8000) = 89.4 mm. Therefore, the new minimum size of the column is 89.4 mm by 89.4 mm Cheers!

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