General information 
Description 
The purpose of this calculation is to obtain information about shear, bending moment, and deflection distribution over the length of a beam, which is under various transverse loads: couples, concentrated and linearly distributed loads. The result of calculation is represented by shear force, bending moment and deflection diagrams. Extreme values of these functions as well as their values at the ends of each segment of the beam are also calculated. In addition all main steps of the solution are represented by formulas. 
Assumptions 




Sign conventions 
Coordinate system has an origin on the left side of the beam. In Figures 13 all components are shown as positive. Clockwise external moments, downward forces, upward deflections are considered positive. Reaction is positive if it acts upward. Positive bending moment compresses the upper part of the beam; positive shear force acts clockwise against the material (Fig. 1). 
Figure 1. Sign conventions for shear force V and bending moment M. 
Figure 2. Simply supported beam under load of different types. 
Methodology 
At the first step of calculation we find the reactions using two equilibrium equations of the entire beam. From the equation of moment equilibrium about left end point of the beam we have the reaction force R_{2} : 
In this formula the linearly distributed load is divided into two parts. First one is the uniformly distributed load with intensity q_{Li} and the second part corresponds to the distributed load whose left side intensity equals to 0 and the right one is equal to q_{Ri} q_{Li}. We will use this representation in all following formulas.
From the equation of moment equilibrium about right pin support we obtain the formula for the reaction force R_{1} : 
Shear force and bending moment at a cross section located at distance x from the left end of the beam are obtained from equilibrium equations of freebody diagram of the lefthand part of the beam. 
Figure 3. Freebody diagram of the lefthand part of the beam. 
From the equation of vertical equilibrium we have the formula for shear force: 
From the equation of moment equilibrium about right end point of the freebody diagram we obtain the formula for bending moment: 
If the bending moment function M(x) is known, the deflection function v(x) could be found from the basic differential equation of the deflection curve of a prismatic beam: 
After integration of this equation we obtain the formula for deflection function: 
where v_{0} and θ_{0} are deflection and slope of the deflection curve at x=0 respectively. In the case of beam with the pin support at the left end v_{0} is equal to 0. The value of EIθ_{0} we obtain from the condition v(l)=0. 
Note that in the above formulas only those members that correspond to the load components located to the left side of the considered cross section with coordinate x should be taken into account: values of a_{i}, b_{i}, c_{i}, d_{i}, l must be less than x. Formulas can be simplified in the case of uniformly distributed load discarding members with factors (q_{Ri} q_{Li}), which are equal to 0. 
References 
James M. Gere, Stephen P. Timoshenko. Mechanics of Materials, PWS Publishing Co., 4th edition, 1997, 832 p. 