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Deflections - Method of Virtual Work
Deflection of a Truss


The virtual work method can be used to determine the deflection of trusses.  We know from the principle of virtual work for trusses that the deflection can be calculated by the equation eq with n equal to the virtual force in the member and delta equal to the change in length of the member.  Therefore, the deflection of a truss due to any condition that causes a change in length of the members can be calculated.  This change in length can be caused by the applied loads acting on each member, temperature changes, and by fabrication errors.

Axial Deformation:

From statics we know how to determine member forces in a truss by using either the method of joints or the method of sections. Once these forces are known we can determine the axial deformation of each member by using the equation:

eq

The equation for the deflection can be modified with this value for delta.

eq

where m is equal to the number of members, n is the force in the member due to the virtual load, N is the force in the member due to the applied load, L is the length, A is the area, and E represents Young's Modulus of Elasticity.

Temperature Changes:

The axial deformation of a truss member of length L due to a change in temperature of delta T is given by:

axial deformation equation

where alpha is the coefficient of thermal expansion.

The equation for the deflection is then modified with this value for delta.

eq

where j is the number of members experiencing temperature change and n is the force in the member due to the virtual load.

Fabrication Errors:

In the case of fabrication errors, the deformation of each member is known. Therefore, the original equation for deflection of a truss can be modified.

eq

where k is the number of members undergoing fabrication errors and n is the force in the member due to the virtual load and delta is the change in length of the member due to fabrication errors.

The total deflection of a truss is made up of the sum of all of these cases.

This equation is now used to find the deflection of a truss. Please refer to an introductory text book on structural analysis for a complete description of this approach.


problem statement

Using the method of virtual work, determine the vertical deflection at joint G in the truss below, under the loading conditions show in figures i), ii), and iii).

The member properties are A=2 in2 and E=29x103 ksi.

The truss is subjected to the following applied loads:

i)    The truss with applied loads
Figure 1 - Truss structure to analyze

And the following fabrication errors are present:

ii)    The truss showing fabrication errors
Figure 2 - Fabrication errors present

NOTE: TO PREVENT ERRORS, CALCULATE THE INFLUENCE OF EACH CASE INDEPENDENTLY AND THEN ADD THE RESULTS AT THE END


simplified truss diagram
Figure 3 - Frame structure with applied loads

Calculate the support reactions (caused by the applied loads) by summing the moments about A and E: (answers in Kips)

Equations

Check these reactions by summing vertical and horizontal forces:

equations

The resulting system,

Loaded truss with reactions
Figure 4 - Support reactions due to applied loads


For equilibrium at joint A,

Joint A
Figure 5 - Joint equilibrium at A

Sum vertical and horizontal forces to determine the force in each member, (Kips)

Remember that in the method of joints, a joint reaction is in the opposite direction to how the force acts on the member.  Therefore, member AB is in compression.

Continue this method for each joint in the structure.

Truss diagram with internal forces due to applied loads,

Truss member reactions
Figure 6 - Truss member reactions


Apply the virtual load at the point of interest in the desired direction. In this case, we want to know the deflection at point G. Therefore, apply a unit load at point G.

Truss with unit load at joint G
Figure 7 - Truss with virtual unit load applied


Following the same procedure used previously, calculate the support reactions (caused by the virtual load).

The resulting system,

Truss with unit load reactions
Figure 8 - Support reactions due to virtual unit load


Use the method of joints as illustrated in Step 2 to determine the member results due to the unit virtual load.  Add the results to your existing table:

Truss diagram with internal forces due to virtual load,

Truss member reactions due to virtual load at G
Figure 9 - Internal forces due to virtual unit load


The deflection of the truss can now be determined by completing the equation:

For the case of Axial Deformation

Member n(k) N(k) L(in) AE (in2-ksi) nNL/AE (in-k)
AB -0.67 -33.33 48 58000 0.0184
BC -0.67 -33.33 48 58000 0.0184
CD -0.67 -46.66 48 58000 0.0257
DE -0.67 -46.66 48 58000 0.0257
AF 0.83 41.67 60 58000 0.0359
BF 0 -10 36 58000 0
CF -0.83 -25 60 58000 0.0216
FG 1.33 53.33 48 58000 0.0589
CG 1 0 36 58000 0
CH -0.83 -8.33 60 58000 0.0072
GH 1.33 53.33 48 58000 0.0589
DH 0 -30 36 58000 0
HE 0.83 58.33 60 58000 0.0503
Total 0.3209

For the case of Fabrication Error

Member n(k) Change in Length (Delta)(in) n(delta)(k-in)
AB -0.67 + 0.4 -0.268
FG 1.33 + 0.6 0.798
HE 0.83 - 0.3 -0.249
    Sum 0.281

Since there were no temperature effects included in this example, the total deflection at point G is the sum of these two results.

(1 k)() = 0.281 in-k + 0.321 in-k = 0.602 in-k
= 0.602 in-k / 1 k = 0.602 in

The positive answer of 0.602 in indicates that the structure will deflect down in the direction of the virtual load.


Contact Dr. Fouad Fanous for more information.