Question 4.8: Finding Reactions of a Statically Indeterminate Beam Using C...

See Figure 4-22e.

1    Consider the reaction force at A to be redundant and remove it temporarily. The beam will then be statically determinate and will deflect at A. Now consider the reaction force R_{A} to be an unknown applied load that will force the deflection to be zero (as it must be if point A is supported). If we write an equation for the deflection at A in terms of the force R_{A} and then solve it for R_{A} with the deflection set to zero, we will determine the necessary reaction force R_{A}.

2     Write equation 4.21 for the deflection y_{A} at the unknown applied load R_{A} in terms of the strain energy in the beam at that point.

\Delta=\frac{\partial U}{\partial Q}       (4.21)

y_A=\frac{\partial U}{\partial R_A}     (a)

3    Substitute equation 4.22d and differentiate:

U=\frac{1}{2} \int_0^1 \frac{M^2}{E I} d x      (4.22d)

y_A=\frac{\partial\left(\frac{1}{2} \int_0^l \frac{M^2}{E I} d x\right)}{\partial R_A}=\int_0^l \frac{M}{E I} \frac{\partial M}{\partial R_A} d x      (b)

4    Write an expression for the bending moment at a distance x from A:

M=R_A x-\frac{1}{2} w x^2     (c)

5    Its derivative with respect to R_{A} is:

\frac{\partial M}{\partial R_A}=x    (d)

6    Substitute (c) and (d) in (b) to get:

y_A=\frac{1}{E I} \int_0^l\left\lgroup R_A x^2-\frac{1}{2} w x^3\right\rgroup d x=\frac{1}{E I}\left\lgroup \frac{R_A l^3}{3}-\frac{w l^4}{8}\right\rgroup     (e)

7    Set y_{A} = 0 and solve for R_{A} to get:

R_A=\frac{3}{8}  w l      (f)

8    From sum of forces and sum of moments we get:

R_R=\frac{5}{8} w l \quad M_1=\frac{1}{8} w l^2    (g)