segments 1 and 2, x = 10 and w1 = w2 and dw1/dx = dw2/dx. These boundary conditions give us Therefore, Step 9: Solve for Mc and Ra Because w2 = 0 at x = 25, we can solve for Mc in terms of Ra to get Also, since w1 = 0 at x = 10, expressing the deflection in terms of Ra (after eliminating Mc) and solving for Ra, gives Step 10: Plot bending moment and shear force diagrams We can now calculate the reactions Rb and Rc, the bending moments M1, M2, M3, M4, and the shear forces V1, V2, V3, V4. These expressions can then be plotted as a function of length for each segment. Relationship between shear force and bending moment It is important to note the relationship between the two diagrams. The moment diagram is a visual representation of the area under the shear force diagram. That is, the moment is the integral of the shear force. If the shear force is constant over an interval, the moment equation will be in terms of x (linear). If the shear force is linear over an interval, the moment equation will be quadratic (parabolic). Another note on the shear force diagrams is that they show where external force and moments are applied. With no external forces, the piecewise functions should attach and show no discontinuity. The discontinuities on the graphs are the exact magnitude of either the external force or external moments that are applied. For example, at x = 10 on the shear force diagram, there is a gap between the two equations. This gap goes from -10 to 15.3. The length of this gap is 25.3, the exact magnitude of the external force at that point. At section 3 on the moment diagram, there is a discontinuity of 50. This is from the applied moment of 50 on the structure. The maximum and minimum values on the graphs represent the max forces and moments that this beam will have under these circumstances. Relationships among load, shear, and moment diagrams Since this method can easily become unnecessarily complicated with relatively simple problems, it can be quite helpful to understand different relations between the loading, shear, and moment diagram. The first of these is the relationship between a distributed load on the loading diagram and the shear diagram. Since a distributed load varies the shear load according