Instructional Objectives
After reading this lesson, the reader will be able to:
1. State and prove CrottiEngesser theorem.
2. Derive simple expressions for calculating deflections in trusses subjected to mechanical loading using unitload method.
3. Derive equations for calculating deflections in trusses subjected to temperature loads.
4. Compute deflections in trusses using unitload method due to fabrication errors.
Introduction
In the previous lesson, we discussed the principle of virtual work and principle of virtual displacement. Also, we derived unit – load method from the principle of virtual work and unit displacement method from the principle of virtual displacement. In this lesson, the unit load method is employed to calculate displacements of trusses due to external loading. Initially the Engesser’s theorem, which is more general than the Castigliano’s theorem, is discussed. In the end, few examples are solved to demonstrate the power of virtual work.
CrottiEngesser Theorem
The CrottiEngesser theorem states that the first partial derivative of the complementary strain energy (U*) expressed in terms of applied forces F_{j} is equal to the corresponding displacement.
(6.1)
For the case of indeterminate structures this may be stated as,
(6.2)
Note that Engesser’s theorem is valid for both linear and nonlinear structures. When the complementary strain energy is equal to the strain energy (i.e. in case of linear structures) the equation (6.1) is nothing but the statement of Castigliano’s first theorem in terms of complementary strain energy.
In the above figure the strain energy (area OACO) is not equal to complementary strain energy (area OABO)
(6.3)
Differentiating strain energy with respect to displacement,
dU/du = F (6.4)
This is the statement of Castigliano’s second theorem. Now the complementary energy is equal to the area enclosed by OABO.
(6.5)
Differentiating complementary strain energy with respect to force F,
(6.6)
This gives deflection in the direction of load. When the load displacement relationship is linear, the above equation coincides with the Castigliano’s first theorem given in equation (3.8).
Unit Load Method as applied to Trusses
External Loading
In case of a plane or a space truss, the only internal forces present are axial as the external loads are applied at joints. Hence, equation (5.7) may be written as,
(6.7)
wherein, δF_{j} is the external virtual load, u_{j} are the actual deflections of the truss, δP_{v} is the virtual stress resultant in the frame due to the virtual load and is the actual internal deformation of the frame due to real forces. In the above equation L, , E A respectively represent length of the member, crosssectional area of a member and modulus of elasticity of a member. In the unit load method, δF_{j} = 1 and all other components of virtual forces are zero. Also, if the cross sectional area A of truss remains constant throughout, then integration may be replaced by summation and hence equation (6.7) may be written as,
(6.8)
where m is the number of members, (δP_{v}) is the internal virtual axial force in member i due to unit virtual load at j and is the total deformation of memberi due to real loads. If we represent total deformation by Δ_{i} ,then
(6.9)
where, Δ_{i} is the true change in length of member i due to real loads.
Temperature Loading
Due to change in the environmental temperature, the truss members either expand or shrink. This in turn produces joint deflections in the truss. This may be calculated by equation (6.9). In this case, the change in length of member Δ_{i} is calculated from the relation,
Δ_{i} = αTL_{i} (6.10)
where α is the coefficient of thermal expansion member, L_{i} is the length of member and T is the temperature change.
Fabrication Errors and Camber
Sometimes, there will be errors in fabricating truss members. In some cases, the truss members are fabricated slightly longer or shorter in order to provide camber to the truss. Usually camber is provided in bridge truss so that its bottom chord is curved upward by an equal to its downward deflection of the chord when subjected to dead. In such instances, also, the truss joint deflection is calculated by equation (6.9). Here,
Δ_{i} = e_{i} (6.11)
where, e_{i} is the fabrication error in the length of the member. e_{i} is taken as positive when the member lengths are fabricated slightly more than the actual length otherwise it is taken as negative.
Procedure for calculating truss deflection
1. First, calculate the real forces in the member of the truss either by method of joints or by method of sections due to the externally applied forces. From this determine the actual deformation (Δ_{i}) in each member from the equation
Assume tensile forces as positive and compressive forces as negative.
2. Now, consider the virtual load system such that only a unit load is considered at the joint either in the horizontal or in the vertical direction, where the deflection is sought. Calculate virtual forces (δP_{v})_{ij} in each member due to the applied unit load at the jth joint.
3. Now, using equation (6.9), evaluate the jth joint deflection u_{j}.
4. If deflection of a joint needs to be calculated due to temperature change, then determine the actual deformation (Δ_{i}) in each member from the equation Δ_{i} = αTL_{i} .
The application of equation (6.8) is shown with the help of few problems.
Example 6.1 Find horizontal and vertical deflection of joint C of truss ABCD loaded as shown in Fig. 6.2a. Assume that, all members have the same axial rigidity.
The given truss is statically determinate one. The reactions are as shown in Fig 6.2b along with member forces which are determined by equations of static equilibrium. To evaluate horizontal deflection at ‘C’, apply a unit load as shown in Fig 6.2c and evaluate the virtual forces δP_{v} in each member. The magnitudes of internal forces are also shown in the respective figures. The tensile forces are shown as +ve and compressive forces are shown as –ve. At each end of the bar, arrows have been drawn indicating the direction in which the force in the member acts on the joint.
Horizontal deflection at joint C is calculated with the help of unit load method. This may be stated as,
(1)
For calculating horizontal deflection at C, u_{c}, apply a unit load at the joint C as shown in Fig.6.2c. The whole calculations are shown in table 6.1. The calculations are self explanatory.
Table 6.1 Computational details for horizontal deflection at C
Member  Length  L / AE  P  (P ),  (PPA EA 
units  m  m/kN  kN  kN  kN.m 
AB  4  4/AE  0  0  0 
BC  4  4/AE  0  0  0 
CD  4  4/AE  15  1  60/AE 
DA  4  4/AE  0  0  0 
AC  /AE  /AE  





(2)
Vertical deflection at joint C
(3)
In this case, a unit vertical load is applied at joint C of the truss as shown in Fig. 6.2d.
Table 6.2 Computational details for vertical deflection at C
Member  Length  L / AE  P  (P^{v}),  (P )_{t}PL_{t} EA 
units  m  m/kN  kN  kN  kN.m 
AB  4  4/AE  0  0  0 
BC  4  4/AE  0  0  0 
CD  4  4/AE  15  1  60/AE 
DA  4  4/AE  0  0  0 
AC  /AE  0  0  



 60/AE 
(4)