Problem 1: A 12 ft long steel W8 x 58 beam carries a 6 kip load at its center and is simply supported at the ends. Determine the slope at A and the deflection at B.

Solution: It is clear that the reactions are 3 kips. The shear and bending moment diagrams are:

The shear can be written in this way:

Integrating, I get

Applying the flexural rigidity formula, and integrating,

But the slope y' is zero at x = 6, so 0 = 3(6)²/2 + C₁, and C₁ = -54 kip-ft². Substituting the constant and integrating again,

Since it is obvious that y = 0 at x = 0, C₂ = 0. y = 0 also at x = 12; this provides a check.

The deflection at x = 6 ft is given by (I was taken from Appendix C):

The slope at x = 0 is