Handbook of Civil Engineering Calculations

FIGURE 39

section; wx = load intensity at the given section; M and m = bending moment at the given

section induced by the actual load and by the unit moment, respectively.

2. Evaluate the moments in step 1
   Evaluate M and m. By proportion, Wx = w(L - x)/L; M = -(x^2 /6)(2w + wj = -(wx^2 /6)[2 +
   (L - x)IL\ = -wx^2 (3L - x)/(6L); m = -.


3. Apply a suitable slope equation

Use the equation (^0) A = /§ [MmI(EI)] dx. Then ElBA = So [^x^2 3L - x)l(6L)] dx = [w/(6L)]
x/£ (ILx^2 - jc^3 ) dx = [wl'(6L)](SLx^3 /3 - *^4 /4)]£ = [w/(6L)](L^4 - L^4 /4); thus, ^ - %wL^3 /
(EI) counterclockwise. This is the slope at A.

4. Apply a unit load to the beam
   Apply a unit downward load at A as shown in Fig. 39c. Let m' denote the bending mo-
   ment at a given section induced by the unit load.


5. Evaluate the bending moment induced by the unit load;
   find the deflection
   Apply yA = So [Mm'I(EI)] dx. Then m' =-x; EIyA = /§ [wx\3L - x)l(6L)] dx = [w/(6L)]
   x/J x\3L -x) dx;yA = (1 l/120)wZ^4 /(£7).
   The first equation in step 3 is a statement of the work performed by the unit moment at
   A as the beam deflects under the applied load. The left-hand side of this equation express-
   es the external work, and the right-hand side expresses the internal work. These work
   equations constitute a simple proof of Maxwell's theorem of reciprocal deflections, which
   is presented in a later calculation procedure.

DEFLECTION OFA CANTILEVER FRAME

The prismatic rigid frame ABCD (Fig. 4Oa) carries a vertical load P at the free end. Deter-

mine the horizontal displacement of A by means of both the unit-load method and the mo-

ment-area method.

( b) Superimposed moment to find (^0) A
( Q) Actual load on beam