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Consistent Deformations - Force Method
Indeterminate Beam with Moment Reaction as Redundant

problem statement

Using the method of consistent deformations, determine the reactions, moments and shears under the loading conditions shown. The modulus of elasticity (E) and the moment of inertia (I) are constant for the entire beam.

Beam with applied loads
Figure 1 - Beam structure to analyze

The structure is statically indeterminate to the first degree (r = 4, e = 3, n = r-e = 4-3 = 1).

To solve for this single degree of indeterminacy, the structure has to be reduced to a statically determinate and stable structure. This is done by removing a redundant support.

In this example, the moment reaction at support A is selected as a redundant to remove in order to obtain a primary determinate structure.

beam with redundant support removed
Figure 2 - Primary structure

Calculate the support reactions of the primary structure.

The resulting system, (A0 indicates the resulting deflection or deformation at the location of the removed redundant for the primary structure).

Support reactions
Figure 3 - Support reactions

Determine the moment diagram M0 due to the applied loads on the primary structure.

In this example, the cantilever method is used to develop the moment diagram. (See a Virtual Work Cantilever Example for a complete description of this step)

Moment diagram due to applied loads
Figure 4 - M0 - Moment diagram due to applied loads

Using the virtual work method, calculate the rotational translation of support A that correponds to the redundant MA. Remove all loads an apply a unit moment in the direction of the redundant, draw the moment diagram, ma, and sketch the deflected shape due to the unit moment.

The resulting system, (faa is the deformation caused by the unit load).

member forces due to unit load
Figure 5 - Primary structure with unit load applied and resulting deflected shape

Moment diagram due to unit load
Figure 6 - Moment diagram ma with MA = 1 ft-k

Calculate the rotational translation corresponding to the redundant MA at support A using the following equation:

Calculate the deformation at the redundant, A0. Use the method of virtual work, calculate the areas on the M0 diagram and multiply each area by the corresponding heights, hi, measured at the centroid of this area on the ma diagram:

Area No. Area/EI (A) (k-ft2)/EI Height (h) on ma diagram (k-ft) Ai*hi (k2-ft3)/EI
A01 2/3 x 20 x 100/EI = 1333.33/EI -1/2 -666.67/EI
A02 1/2 x 20 x -36/EI = -360/EI -1/3 120/EI
A03 1/2 x 6 x -36/EI = -108/EI 0 0
Total = Q(A0) = -546.67/EI

Therefore, with Q = 1 ft-k;
Q(A0) = -546.67 (k2-ft3)/EI
A0 = -546.67 (k-ft2)/EI

Calculate the flexibility coefficient, faa, by determining the deformation of the primary structure when subjected to the redundant load, MA = 1 ft-k.

Again, using the method of virtual work, calculate the areas on the ma diagram and multiply by the corresponding heights, hi, measured at the centroid of each area:

Area No. Area/EI (A) (k-ft2)/EI Height (h) on ma diagram (k-ft) Ai*hi (k2-ft3)/EI
A11 1/2 x 20 x -1/EI = -10/EI -2/3 6.67/EI
Total = Q(faa) = 6.67/EI

Therefore, with Q = 1 ft-k;
Q(faa) = 6.67 (k2-ft3)/EI
faa = 6.67 (k-ft2)/EI

The consistent deformation equation that corresponds to the redundant MA (the moment reaction at support A) is:

A0 + faa * MA = 0   (1)

This equation is set equal to zero since the fixed support at A does not allow any rotation.

Using equation (1), solve for MA:

-546.67 (k-ft2)/EI + 6.67 (k-ft2)/EI * MA = 0
MA = 82

Multiply the unit moment, Q, at MA by 82 to get the final reaction. The positive answer indicates that the reaction is in the direction of the applied unit moment.

Impose the value of the calculated MA along with the other applied loads on the original structure. Calculate the remaining reactions using the three static equilibrium equations, (Fx = 0, Fy = 0 and M = 0).

Resulting beam reactions
Figure 7 - Beam with support reactions


Figure 8 - Final shear diagram


Figure 9 - Final moment diagram

Deflected Shape:

Figure 10 - Deflected shape

Contact Dr. Fouad Fanous for more information.