Handbook of Civil Engineering Calculations

Diagram A Full load on entire span

Diagram B Dead load on overhangs; full load

between supports

(b) Bending-moment diagrams

FIGURE 27

Calculation Procedure:

1. Place full load on the overhangs, and compute the
   negative moment
   Refer to the moment diagrams. For every position of the supports, there is a correspon-
   ding maximum bending stress. The position for which this stress has the smallest value
   must be identified.
   As the supports are moved toward the interior of the beam, the bending moments be-
   tween the supports diminish in algebraic value. The optimal position of the supports is
   that for which the maximum potential negative moment M 1 is numerically equal to the
   maximum potential positive moment M 2. Thus, M 1 = -1.2w(x^2 /2) = — 0.6w;t^2.


2. Place only the dead load on the overhangs and the full load
   between the supports. Compute the positive moment.
   Sum the areas under the shear diagram to compute M 2. Thus, M 2 =^1 A[1.2w(L/2 - x)^2 -
   0.2w.r^2 ] = w(0.15L^2 - 0.6Lx + 0.5*^2 ).


3. Equate the absolute values of M 1 and M 2 and solve for x
   Substituting gives 0.6^2 = 0.15L^2 - 0.6Lx + 0.5^2 ; x = 110.5°^5 - 3) = 0.240L.

FLEXURAL CAPACITY OFA COMPOUND BEAM

A Wl6 x 45 steel beam in an existing structure was reinforced by welding an WT6 x 20
to the bottom flange, as in Fig. 28. If the allowable bending stress is 20,000 lb/in^2
(137,900 kPa), determine the flexural capacity of the built-up member.

(a) Loads carried by overhanging beam