Deflections - Method of Virtual Work
Rotation of a Beam - Cantilever

problem statement

Using the same structure as used in the Beam Deflection examples, determine the rotation at A of the beam shown in the figure below using the Cantilever Method. The modulus of elasticity (E) and the moment of inertia (I) are constant for the entire beam.

Note: The colors of the loads and moments are used to help indicate the contribution of each force to the deflection or rotation being calculated. The moment diagrams show the moments induced by a load using the same color as the load.

Figure 1 - Beam structure to analyze

• calculate the support reactions

Calculate the support reactions (caused by the applied loads) using the following relationships:

Check these reactions by summing the vertical forces.

The resulting system,

Figure 2 - Beam structure with support reactions

• draw moment diagrams (M) for the structure under applied "real" loads

Using the cantilever method, fix the structure at joint B and draw the resulting moment diagram induced by the applied "real" loads.

Plot the moment diagram for each applied load separately, i.e., by parts. The final results can then be obtained by utilizing the method of superposition i.e., by summing the contribution of each individual load to the displacement being calculated. This method is applicable since the structure is assumed to be elastic and the deflections are small.

Note: The centroid of each area is indicated by the numbered arrow and dot.

i) Moment diagram due to the 56 ft-k concentrated moment at A,

Figure 3 - Moment diagram due to 56 ft-k moment

ii) Moment diagram due to the 2 k/ft applied load,

Figure 4 - Moment diagram due to 2 k/ft applied load

iii) Moment diagram due to the 21 k support reaction at A,

Figure 5 - Moment diagram due to 21 k support reaction

iv) Moment diagram due to the 6k applied load at end C

Figure 6 - Moment diagram due to 6 k applied load

Notice that the resultant moment diagram (figure 3 above) is the sum of these four diagrams.

Figure 7 - Resultant moment diagram

• apply virtual load

Apply the virtual load at the point of interest in the desired direction. In this case, we want to know the rotation at point A. Therefore, apply a unit moment at point A in the positive (clockwise) direction.

Figure 8 - Beam with virtual unit load applied

• solve the support reactions due to the virtual load

Following the same procedure used previously, calculate the support reactions (caused by the virtual load) using the following relationships:

Check these reactions by summing the vertical forces.

The resulting system,

Figure 9 - Support reactions due to virtual unit load

• draw virtual moment diagram (m)

Determine the moment diagram due to the virtual load using the same method as used to find the moment diagrams for the applied loads.

Moment diagram due to the virtual load by using the cantilever method and fixing the structure at joint B.

Figure 10 - Moment diagram due to virtual unit load

• calculate areas and centroids

Once the "real" moment diagrams are determined, calculate the area enclosed by each moment diagram and determine the location of the centroid of each of these areas.

• determine heights of virtual moment diagram at centroids

Determine the values - heights (h_i) - on the virtual moment diagram (m) at the centroids of the moments due to the real loads. This is needed to carry out the integration by using the equation given in the introduction,

Heights (h_i) and locations by the cantilever method.

Figure 11 - Heights on virtual moment diagram

• integrate

Integrate the equation by using the visual integration approach.

Multiply the areas of the "real" moment diagram by the heights found in the virtual moment diagram and add them together.

Since EI is constant throughout the structure, the total rotation at A equals +173.33 ft³-k²/EI.

The positive sign indicates that the rotation is in the same direction as the unit moment applied at A - therefore the rotation is in the clockwise direction.

If values of E and I are specified, the vertical deflection at C in inches can be determined.  For example, let E = 29,000 ksi, I = 144 in⁴, and Q = 1 ft-k, then

Contact Dr. Fouad Fanous for more information.