Slope and Deflection by Moment Area Theorems

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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/EI) diagram is shown in figure 7-6(b)-(c) and the elastic curve (deflected shape of beam) 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 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.

It is clear from the elastic curve in Figure 7-6(d) that the deflection at point C is equal the tangential deviation of point C with respect to tangent at A .

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Hence deflection at C is calculated as given below.

The negative sign of deviation indicates that it is below the tangent. Hence the deflection is in the downward direction.

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