Influence Line Diagram For Shearing Force:

Next, we shall study how to draw the influence line diagram for shear force in a simply supported beam. For description we take the same beam and the same point P as in Example, for which we now proceed to draw the shear force influence line diagram.

Example

Draw the influence line diagram for shear force at point P in the simply supported beam AB of span 10 m. P is 4 m from the hold up A.

Solution

Let us assume that the moving unit load is at a distance x from A. the reactions R_A = 10 - x /10, and RB=  x/10  as determined earlier.

Case (a) : If the unit load is between the points A and P, then considering forces to left of P,

Shear force at P = RA - 1 = (10 - x)/10 - 1 = -  x/10. By our sign convention, a shear force is negative when the left hand portion of the beam tends to move downward. Therefore, the shear force in this case will be negative and will depend upon the value of x (i.e. its distance from A). The ordinate of the diagram will be zero then the load is at A (x = 0) and it will be- 4 /10= - 0.4 when the rolling load is at P (i.e. x = 4). The diagram will be a straight line A?P?.

Case (b) : When the unit load crosses to the right of the point P and lies between P and B, then considering forces to left of P the shear force at P = (10 - x)/10 which is also a straight line P? B such that the ordinate at P (when the unit load has just crossed to right) is equal to 10 - 4 /10= + 0.6 and it is zero when the unit load is at B, ordinate = 10 - 10/10 = 0 .

Thus, we see that the shear force influence line consists of two parts : the part between A and P has negative ordinates and the part between P and B has positive ordinates, showing that the SF changes sign as the unit rolling load crosses the point P.