# Slope and Deflection by Moment Area Theorems

Problem 7-6: Use moment-area theorems to determine the slope and deflection at point C of the cantilever beam given above in Figure 7-6(a)

Solution:

Step 1: The M/EI (bending moment divided by EI) diagram is shown in figures 7-6(b) due to point load and 7-6(c) due to uniformly distributed load (udl). We will use principle of superposition to find resultant M/EI. The elastic curve (deflected shape of beam) is shown in figure 7-6(d).  As the point A is fixed it will have zero slope hence the tangent to the elastic curve will be horizontal at A as shown in the elastic curve in figure 7-6(d). Step 2: According to first moment-area theorem, the change in slope of two points on the elastic curve is equal to the area of M/EI diagram between these two points.

Hence change in slope between A and C is equal to the area of M/EI diagram between A and C. The M/EI diagram due to point load shown in Figure 7-6(b) is a triangle (area =bh/2) whereas the M/EI diagram due to udl shown in Figure 7-6(c) is parabolic (area=bh/3) . Hence we can write as follows, It is to be noted that the negative sign of M/EI is used because the bending moment is hogging due to point load and uniform load. The negative sign of slope indicates that it is clock-wise.

Step 3: The deflection at C can be calculated by using second moment area theorem which says that tangential deviation of a point C with respect to tangent at A is equal to the moment of area of M/EI diagram between A and C taken about C. 