Slope & Deflection of Beams Using Virtual Work Method, Structural Analysis for Beams Example 3

Structural Analysis for Beam Deflection and Slope Using Virtual Work Method - Example 3
Find the slope at support A and the deflection at support C for the simply supported beam using the virtual work method .
EI is constant.
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The method of virtual work can be used to determine the displacement or slope of a point on a beam which is subjected to an external loading, temperature change, or fabrication errors.
The first step is to establish an appropriate coordinate system. There may be a need for more than one coordinate system depending on the loading scenario. A coordinate system is valid only within a region of the beam where there is no discontinuity of real or virtual loads.
The second step is to calculate the support reactions of the beam and evaluate the real internal bending moment as a function of the chosen coordinate system(s). The real bending moment functions can be generated using basic statics.
The third step is to apply a virtual unit load or virtual unit couple on the beam at the point where the displacement or slope is to be found and remove all real loads. It is important to keep in mind that the virtual unit load or unit couple should be in the same direction as the specified displacement or rotation. After placing the virtual unit load, determine the virtual bending moment as a function of the same coordinate systems as that of the real bending moment functions. The virtual bending moment functions can be generated using basic statics.
The last step is to use the equation of virtual work to calculate the desired displacement or rotation. If the algebraic sum of all the integrals is positive, the real displacement or rotation is in the same direction as the virtual unit load or virtual unit couple respectively. If the final summation is negative, the real displacement or rotation is in the opposite direction as the virtual unit load or virtual unit couple respectively. Lastly, divide the summation by the external virtual unit load or virtual unit couple to get the final desired displacement or rotation.
1 = virtual unit load or couple
Delta = Unknown displacement
Theta = Unknown rotation
m = virtual moment function
M = Real moment Function
E = Young’s Modulus, Modulus of Elasticity
I = Moment of inertia of cross-sectional area, computed about the neutral axis
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[@KhabisangLekena]

Thanks, that was highly helpful

[@NicholleWillisLoves]

When you cut the beam to find the virtual moment function for the displacement at C, why do you cut it before the virtual load?/ why don't you include the virtual load in the cut?
Nevermind, I see why you can do that now, haha. When you solved for the displacement using the function, you just did the bounds to L/2, and multiplied by 2, since it's symmetrical, which is way easier than solving for the virtual moment function including the virtual load, and then making the boundary include the entire length of the beam when solving for the displacement. That only works when it's symmetrical though...