Displacement of Trusses Using the Principle of Virtual Work

The principle of virtual work sometimes known as the “unit-load” method was developed by John Bernoulli in 1717. It is a direct application of the principle of conservation of energy which states that the work by all external forces acting on a structure, We, is transformed into internal work or strain energy, Wi developed as a result of the structure’s deformation under these forces. Supposing these forces are removed. the strain energy stored will return the structure to it undeformed state-provided the elastic limit has not been exceeded. Thus, energy is conserved.

The principle can be mathematically expressed as :

{ W }_{ e }={ W }_{ i }

Where:

  • We is the external work done by the external forces
  • Wi is the internal work done or strain energy stored in the structure
descibes principle of virtual work
Figure 1: Principle of Virtual Work

The concept of virtual work can be extended further by considering a structure subjected to a series of external forces P1, P2, P3……Pn. Supposing it is required to determine the displacement of the structure at a point A (figure 1) due to the applied external forces. Since there is no external force applied at point A, the displacement of point A can be determined by applying a “virtual force” at point A acting in the same direction of the displacement. Its value is often taken to have a magnitude of “unity”. The term virtual load is chosen to depict an imaginary load that doesn’t exist as part of the real loads. However, the unit load does cause internal forces in the member as shown in figure 1. We can, therefore, apply the principle of conservation of energy described above to determine this displacement.

Once the unit virtual force P’ is applied, the structure is subjected to real loads and point A will displaced an amount Δ causing the structure to also displace internally by dL. Thus the displacement caused by the unit virtual force P’ is Δ and that caused by the internal forces u is dL. Therefore applying the principle of conservation of energy, we have:

P’.\Delta =u.dL

Where:

  • P’ = 1=Unit Virtual Force acting in the direction of Δ
  • u = virtual internal forces acting in the direction of dL
  • Δ = external deflection caused by the real loads
  • dL = internal deformation of the element caused by the real loads

By choosing P’ as “unity” we can see that the solution of Δ is straight forward.

Application of Virtual Work to Trusses

The principle of virtual work can be applied to determine the displacement of trusses subjected to external forces. Supposing we intend to determine the displacement of a joint in a truss, the virtual work equation for a truss can be expressed as:

1\cdot \Delta =\sum { \frac { nNL }{ AE } }

Where :

  • 1= external virtual unit load acting on the truss joint in the stated direction of Δ
  • n = internal virtual normal force in a truss member caused by the external virtual unit load
  • Δ = external joint displacement caused by the real loads on the truss
  • N = internal normal force in a truss member caused by the real loads
  • L = length of a member
  • A = cross-sectional area of a member
  • E = modulus of elasticity of a member.
Procedure for Analysis

The following steps may be applied to obtain the displacement at any joint in the truss using the virtual work method.

1. Analyze the truss using the method of sections or joint to determine the internal force N in each member. These forces are caused as a result of the application of the real loads only. Assume tensile forces as positive and compressive forces as negatives.

2. Place the unit load on the truss at the joint where the desired displacement is to be determined. The load should be in the same direction as the specified displacement, e.g., horizontal or vertical.

3. With the unit load so placed, and all the real loads removed from the truss, use the method of joints or the method of sections and calculate the internal n force in each truss member. Again assume the tensile forces as positive and the compressive forces as negatives.

4. Apply the equation of virtual work, to determine the desired displacement. It is important to retain the algebraic sign for each of the corresponding n and N forces when substituting these terms into the equation.

Worked Example

The simple truss shown is hinged at point A and on roller support at joint B. Determine the vertical and horizontal displacement of joint C. Assuming the cross-sectional area and modulus of elasticity of each member is constant.

The worked example for this text
Figure 2: Simple Truss

The truss is first analyzed for the internal forces using the method of joints, and since the truss is statically determinate the equations of statics can be employed in determining these internal forces.

Support Reactions

Summation of all the horizontal forces in the structure must be equal to zero.

\sum { { F }_{ x } } =0
10-{ H }_{ A }=0—-1
{ H }_{ A }=10kN\leftarrow

Summation of all the vertical forces in the structure must be equal to zero.

\sum { { F }_{ y } } =0
{ V }_{ A }+{ V }_{ B }=25kN—-2

Summation of moment about point any point must be equal to zero. Therefore take moment about point A.

\sum { { M }_{ A } } =0
{ 16V }_{ B }-\left( 10\times 6 \right) -\left( 25\times 8 \right) =0
{ V }_{ B }=\frac { 260 }{ 16 } =16.25kN\uparrow

Substitute the vertical reaction at point B into equation 2

{ V }_{ A }=25-{ V }_{ B }=25-16.25=8.75kN\uparrow
Real Forces

Applying equations of statics to joint A and B we have

{ N }_{ AB }=21.64kN\quad ;\quad { N }_{ BC }=-27.08kN;\quad { N }_{ AC }=-14.58kN
analysis of the members for real forces
Figure 3: Analysis of Member for Real Forces
Vertical Virtual-Force

To determine the vertical deflection of support C we apply a virtual unit load as shown in figure 4 and carry out the analysis for the member forces.

analysis of the members for vertical virtual forces
Figure 4: Analysis of Members for Vertical Virtual Force

Applying the equations of equilibrium, we have :

{ n }_{ 1,ab }=0.667kN;\quad { n }_{ 1,bc }={ n }_{ 1,ac }=-0.833kN
Horizontal Virtual Force

To determine the horizontal deflection of support C we apply a virtual unit load as shown in figure 4 and carry out the analysis for the member forces.

analysis of the members for horizontal virtual forces
Figure 5: Analysis of Members for Horizontal Virtual Force

Applying the equations of equilibrium, we have :

{ n }_{ 2,ab }=0.50kN;\quad { n }_{ 2,bc }=-0.625kN;\quad { n }_{ 2,ac }=0.625kN
Virtual-Force Equation

Since AE is constant, each of the terms nNL can be arranged in tabular form and computed. Here positive numbers indicate tensile forces and negative numbers indicate compressive forces.

Thus applying the virtual work equation for trusses, we have:

1.\Delta =\sum { \frac { nNL }{ AE } }

Vertical deflection of point C :

{ \Delta }_{ V }=\sum { \frac { { n }_{ 1 }NL }{ AE } = } \frac { 578 }{ AE }\downarrow

Horizontal deflection of point C :

{ \Delta }_{ H }=\sum { \frac { { n }_{ 2 }NL }{ AE } = } \frac { 251.24 }{ AE } \rightarrow

In this example, we used a simple truss to illustrate the principle of virtual work. However, its application encompasses beams, frames and even complex trusses with numerous members. Calculation by hand will no doubt be tedious however, a computer program can be written for the entire structure and then subsequently used to compute the displacements at the desired points. Most software packages of today are either based on this principle or an extension of this principle.

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