The reaction at the simple support RA
was solved using two different methods above. answer
Solving for vertical reaction at B
Solving for moment reaction at B
To Draw the Shear Diagram
- The shear at A is equal to RA
- The load at AB is increasing from -wo at A to zero at B, thus, the slope of shear diagram from A to B is also increasing from -wo at A to zero at B.
- The load between AB is 1st degree (linear), thus, the shear diagram between AB is 2nd degree (parabolic) with vertex at B and open upward.
- The magnitude of shear at B is equal to -RB. It is equal to the shear at A minus the triangular load between AB. See the magnitude of RB above.
- The shear diagram from A to B will become zero somewhere along AB, the point is denoted by C in the figure. To locate point C, two solutions are presented below.
Location of C, the point of zero shear
Point C is the location of zero shear which may also be the location of maximum moment.
- By squared property of parabola
- By shear equation
Another method of solving for xC is to pass an exploratory section anywhere on AB and sum up all the vertical forces to the left of the exploratory section. The location of xC is where the sum of all vertical forces equate to zero. Consider the figure shown to the right. Note that this figure is the same figure we used to find the reaction RA by double integration method shown above. The double integration method shows the relationship of x and y which is .
Sum of all vertical forces
At point C, ΣFV = 0 and x = xC
To Draw the Moment Diagram
- The moment at simple support A is zero.
- The shear diagram of AB is 2nd degree curve, thus, the moment diagram between AB is 3rd degree curve.
- The moment at C can be computed in two ways; (a) by solving the area of shear diagram between A and C, and (b) by using the moment equation. For method (a), see the following links for similar situation of solving a partial area of parabolic spandrel.
The solution below is using the approach mentioned in (b). From double integration method of solving RA, the moment equation is given by
For x = xC = 0.3292L, M = MC
- In the same manner of solving for MC, MB can be found by using x = L. Thus,
which confirms the solution above for MB.
- To locate the point of zero moment denoted by D in the figure, we will again use the moment equation; now with M = 0 and x = xD.