Question

explain briefly relation ship between SF and BM at a section.
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Answer 1

Explanation:

Thus, the rate of change of the bending moment with respect to x is equal to the shearing force, or the slope of the moment diagram at the given point is the shear at that point .

Answer 2

Answer:

The relation between Bending Moment and the Shear Force ( With usual notations) is given by

dMdx=V ------------(1)

From above equation we can also write,

∫21dm=∫21Vdx

M2=M1+∫21Vdx ----------(2)

The above equations are useful while sketching a Bending Moment diagram. Equation (1) gives the slope of the B.M.D, at a point. (Look for the slope of BMD positive/negative, increasing/decreasing in the SFD. ) Equation (2) gives the Bending Moment at a point 2, in terms of Bending Moment at point 1 and the area under the SFD between points 1 and 2.

Example:

(1) Shown below a Cantilever with end load. The Shear Force is constant and is equal to load `P’. The Bending Moment, therefore, will be varying linearly, with positive slope. (The sign convention used for the BMD and SFD, for the following figures, is structural sign convention. In the structural sign convention the Bending Moment acting in the positive `z’ direction on the positive `x’ face is taken as positive. Whereas, the Shear Force, on the positive `x’ face is taken as positive, if it acts in the negative `y’ direction. )

The rea of SFD is P l Bending moment at the free end is zero. The Bending Moment ‘M′ at a fixed end will be

M=0 +Area of SFD between Free end and the fixed end. =P l

(2) Below is shown the cantilever subjected to uniformly distributed load ‘q′ per unit length. The expressions for Shear Force and the Bending Moment are

V=q(l−x)

M=−q(l−x)22

Bending Moment at the fixed end is −q l22 and varies quadratically (as the Shear Force and hence the slope varies linearly) such that it has positive decreasing slope, as we move from fixed end to free end. The bending moment is zero at the free end. The slope of BMD is zero at the free end as V=0 at x=l .

The Bending Moment at the fixed end can be written from Shear Force Diagram as well. Note the area of the SFD is ql22