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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.

A diagram showing a beam loaded with a concentrated force and applied moment
Figure 1 - Beam structure to analyze


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

Equations

Check these reactions by summing the vertical forces.

shear equation

 

 

 

 

The resulting system,

Loaded beam with reactions
Figure 2 - Beam structure with support reactions


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,

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

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

For 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,

For the 21 k support reaction
Figure 5 - Moment diagram due to 21 k support reaction

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

For the 6 k applied load
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.

resultant moment
Figure 7 - Resultant moment diagram


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.

Beam with a unit moment at point A
Figure 8 - Beam with virtual unit load applied


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

Equations

Check these reactions by summing the vertical forces.

shear equations

The resulting system,

Beam diagram with unit moment reactions
Figure 9 - Support reactions due to virtual unit load


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.

Cantilever virtual moment diagram
Figure 10 - Moment diagram due to virtual unit load


 

 

 

 

 

 

 

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.

Area No. Area/EI (ft2-k/EI) Location of centroid
from support (ft)
1. -56x20/EI=-1120/EI X1 = 1/2x20 = 10
2. 1/3x20x-400/EI=-2666.67/EI X2 = 3/4x20 = 15
3. 1/2x20x420/EI=4200/EI X3 = 2/3x20 = 13.33
4. 1/2x6x-36/EI=-108/EI X4 = 1/3x6   = 2

Determine the values - heights (hi) - 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,

 virtual work equation

Heights (hi) and locations by the cantilever method.

Cantilever virtual moment diagram
Figure 11 - Heights on virtual moment diagram


Integrate the equation virtual work 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.

Area No. Area (a)
from M diagram (ft2-k/EI)
Height (h)
from m diagram (ft-k)

Ai*hi (ft3-k2/EI)

1. -1120/EI 1 -1120/EI
1. -1120/EI -1/2 560/EI
2. -2666.67/EI 1 -2666.67/EI
2. -2666.67/EI -3/4 2000/EI
3. 4200/EI 1 4200/EI
3. 4200/EI -2/3 -2800/EI
4. -108/EI 0 0/EI
Total 173.33/EI

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

resulting equation


Contact Dr. Fouad Fanous for more information.