Handbook of Civil Engineering Calculations

FIGURE 36. Deflection of simple beam under end moment.

Calculation Procedure:

1. Evaluate the bending moment at a given section
   Make this evaluation in terms of the distance x from the left-hand support to this section.
   Thus RL = N/L; M = NxIL.


2. Write the differential equation of the elastic curve;
   integrate twice
   Thus Elcfy/dx^2 = -M = -Nx/L; Eldyldx = EIS = -Nx^2 /(2L) + C 1 ; Ely = -Nx^3 /(6L) + C 1 X +
   C 2.


3. Evaluate, the constants of integration
   Apply the following boundary conditions: When x = O, y = O; /. C 2 = O; when x = L, y = O;
   /.C 1 =NL/6.


4. Write the slope and deflection equations
   Substitute the constant values found in step 3 in the equations developed in step 2. Thus
   O = [N/(6EIL)](L^2 - 3*^2 ); y = [Nx/(6EIL)](L^2 - x^2 ).


5. Find the slope at the supports
   Substitute the values x = O, x = L in the slope equation to determine the slope at the sup-
   ports. Thus S 1 = NL/(6EI); SR = -NL/(3EI).


6. Solve for the section of maximum deflection
   Set 6 = O and solve for ;c to locate the section of maximum deflection. Thus L^2 - 3x^2 = O;
   x = L/3°^5. Substituting in the deflection equation gives ymsai = M,^2 /(9£/3°^5 ).

MOMENT-AREA METHOD OF DETERMINING

BEAM DEFLECTION

Use the moment-area method to determine the slope of the elastic curve at each support

and the maximum deflection of the beam shown in Fig. 36.

Calculation Procedure:

1. Sketch the elastic curve of the member and draw the
   Mf(EI) diagram
   Let A and B denote two points on the elastic curve of a beam. The moment-area method is
   based on the following theorems:
   The difference between the slope at A and that at B is numerically equal to the area of
   the MI(EI) diagram within the interval AB.