Example 7⁷

Problem: A concrete beam is reinforced by three steel rods placed as shown. The modulus of elasticity is 20 GPa for concrete and 200 GPa for steel. The allowable stresses are 10 MPa for the concrete (that's in compression; it's zero in tension) and 150 MPa for the steel. Determine the largest bending moment which may be applied to the beam.

Solution: To compute the stresses, the moment of inertia will be necessary. For that, the centroid must be located. I begin by drawing the steel as its equivalent in concrete:

The dashed rectangle indicates that part of the concrete below the neutral axis (N.A.); since it is tension and concrete cannot support tension, it is not included in the calculations.

 

section
1
2		50	6.7858x105
sums

Multiplying this out, collecting terms, and solving the quadratic equation, I get (I have ignored the other root, 801 mm, as inapplicable to the problem.) Now I compute the moment of inertia:

For the steel, I have ignored the centroidal contribution, since it is less than 0.01 of the parallel axis contribution. Now I have all of the information necessary to compute the moments:

The 10⁹ factor takes care of the units (mm to m). The maximum moment that can be applied is 79.1 kNm.

⁷ This is Problem 4.46 in Beer & Johnston's Mechanics of Materials, 2^nd Edition, McGraw-Hill.