Method of Superposition

Superposition

If we have a beam under a point load and a distributed load, we can analyze them individually and superimpose them to get a linear combination of deformations from the individual loading.

Indeterminate Beams

Using the method of super position is especially useful for indeterminate beams

Here, we essentially superimpose two determinate beams and designate one of the reactions as redundant.

Then you determine the beam deformation without the redundant support and treat the redundant reaction as an unknown load which together with the other loads must produce a deformation compatible with the original supports

Basically, an efficiency coming from using the table (A-9) Basic questions here are asking for slope and deflection at a given point.

General Strategy

1. Split the loading into n cases which correspond to different cases in table A-9 from the MTE 321 textbook
2. The textbook will provide a general equation for deflection (y), from this equation, an appropriate value for x needs to be subbed in for the equation of deflection
3. The equation for the slope of a point (theta) is the derivative of the general deflection equation provided in table A-9, then sub in the appropriate value for x
   1. Note that if this is a cantilever beam question the deflection of the end is less than the deflection somewhere in the middle and can be modelled using the equation $Y_A^{}=Y_B-\left(L-a\right){\theta }_B$ Where ${\theta }_B$(L-a) is the additional deflection of Y_a (this can be derived through y = mx+b where y is the deflection, m is theta_b and x is the horizontal distance (L-a))
4. Once y and theta have been derived, they can be used to find additional y and thetas if required
5. Now the principle of super position can be applied where the total deflection and slope (theta) is equal to the sum of each loading state