Page 1 Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows that for the statically determinate structure; the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F - F - system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,......., , 2 1 n ,....., 2 , 1 n u u u ,......, , 2 1 n F F F ,......., , 2 1 ij s and real internal strains ij e inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F d d d ,......, , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement Page 2 Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows that for the statically determinate structure; the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F - F - system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,......., , 2 1 n ,....., 2 , 1 n u u u ,......, , 2 1 n F F F ,......., , 2 1 ij s and real internal strains ij e inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F d d d ,......, , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement configuration n u u u d d d , ,......... , 2 1 . The virtual loading system produces virtual internal stresses ij d s and virtual internal strains ij d e inside the beam. Now, apply the second system of forces on the beam which has been deformed by first system of forces. Then, the external loads and internal stresses i F ij s do virtual work by moving along i u d and ij d e . The product i i u F d ? is known as the external virtual work. It may be noted that the above product does not represent the conventional work since each component is caused due to different source i.e. i u d is not due to . Similarly the product i F ij ij s de ? is the internal virtual work. In the case of deformable body, both external and internal forces do work. Since, the beam is in equilibrium, the external virtual work must be equal to the internal virtual work. Hence, one needs to consider both internal and external virtual work to establish equations of equilibrium. 5.3 Principle of Virtual Displacement A deformable body is in equilibrium if the total external virtual work done by the system of true forces moving through the corresponding virtual displacements of the system i.e. is equal to the total internal virtual work for every kinematically admissible (consistent with the constraints) virtual displacements. i i u F d ? Page 3 Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows that for the statically determinate structure; the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F - F - system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,......., , 2 1 n ,....., 2 , 1 n u u u ,......, , 2 1 n F F F ,......., , 2 1 ij s and real internal strains ij e inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F d d d ,......, , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement configuration n u u u d d d , ,......... , 2 1 . The virtual loading system produces virtual internal stresses ij d s and virtual internal strains ij d e inside the beam. Now, apply the second system of forces on the beam which has been deformed by first system of forces. Then, the external loads and internal stresses i F ij s do virtual work by moving along i u d and ij d e . The product i i u F d ? is known as the external virtual work. It may be noted that the above product does not represent the conventional work since each component is caused due to different source i.e. i u d is not due to . Similarly the product i F ij ij s de ? is the internal virtual work. In the case of deformable body, both external and internal forces do work. Since, the beam is in equilibrium, the external virtual work must be equal to the internal virtual work. Hence, one needs to consider both internal and external virtual work to establish equations of equilibrium. 5.3 Principle of Virtual Displacement A deformable body is in equilibrium if the total external virtual work done by the system of true forces moving through the corresponding virtual displacements of the system i.e. is equal to the total internal virtual work for every kinematically admissible (consistent with the constraints) virtual displacements. i i u F d ? That is virtual displacements should be continuous within the structure and also it must satisfy boundary conditions. dv u F ij ij i i de s d ? ? = (5.1) where ij s are the true stresses due to true forces and i F ij de are the virtual strains due to virtual displacements i u d . 5.4 Principle of Virtual Forces For a deformable body, the total external complementary work is equal to the total internal complementary work for every system of virtual forces and stresses that satisfy the equations of equilibrium. dv u F ij ij i i e ds d ? ? = (5.2) where ij d s are the virtual stresses due to virtual forces i F d and ij e are the true strains due to the true displacements . i u As stated earlier, the principle of virtual work may be advantageously used to calculate displacements of structures. In the next section let us see how this can be used to calculate displacements in a beams and frames. In the next lesson, the truss deflections are calculated by the method of virtual work. 5.5 Unit Load Method The principle of virtual force leads to unit load method. It is assumed throughout our discussion that the method of superposition holds good. For the derivation of unit load method, we consider two systems of loads. In this section, the principle of virtual forces and unit load method are discussed in the context of framed structures. Consider a cantilever beam, which is in equilibrium under the action of a first system of forces causing displacements as shown in Fig. 5.2a. The first system of forces refers to the actual forces acting on the structure. Let the stress resultants at any section of the beam due to first system of forces be axial force ( n F F F ,....., , 2 1 n u u u ,....., , 2 1 P ), bending moment (M ) and shearing force (V ). Also the corresponding incremental deformations are axial deformation ( ), flexural deformation ( ? d ? d ) and shearing deformation ( ? d ) respectively. For a conservative system the external work done by the applied forces is equal to the internal strain energy stored. Hence, Page 4 Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows that for the statically determinate structure; the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F - F - system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,......., , 2 1 n ,....., 2 , 1 n u u u ,......, , 2 1 n F F F ,......., , 2 1 ij s and real internal strains ij e inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F d d d ,......, , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement configuration n u u u d d d , ,......... , 2 1 . The virtual loading system produces virtual internal stresses ij d s and virtual internal strains ij d e inside the beam. Now, apply the second system of forces on the beam which has been deformed by first system of forces. Then, the external loads and internal stresses i F ij s do virtual work by moving along i u d and ij d e . The product i i u F d ? is known as the external virtual work. It may be noted that the above product does not represent the conventional work since each component is caused due to different source i.e. i u d is not due to . Similarly the product i F ij ij s de ? is the internal virtual work. In the case of deformable body, both external and internal forces do work. Since, the beam is in equilibrium, the external virtual work must be equal to the internal virtual work. Hence, one needs to consider both internal and external virtual work to establish equations of equilibrium. 5.3 Principle of Virtual Displacement A deformable body is in equilibrium if the total external virtual work done by the system of true forces moving through the corresponding virtual displacements of the system i.e. is equal to the total internal virtual work for every kinematically admissible (consistent with the constraints) virtual displacements. i i u F d ? That is virtual displacements should be continuous within the structure and also it must satisfy boundary conditions. dv u F ij ij i i de s d ? ? = (5.1) where ij s are the true stresses due to true forces and i F ij de are the virtual strains due to virtual displacements i u d . 5.4 Principle of Virtual Forces For a deformable body, the total external complementary work is equal to the total internal complementary work for every system of virtual forces and stresses that satisfy the equations of equilibrium. dv u F ij ij i i e ds d ? ? = (5.2) where ij d s are the virtual stresses due to virtual forces i F d and ij e are the true strains due to the true displacements . i u As stated earlier, the principle of virtual work may be advantageously used to calculate displacements of structures. In the next section let us see how this can be used to calculate displacements in a beams and frames. In the next lesson, the truss deflections are calculated by the method of virtual work. 5.5 Unit Load Method The principle of virtual force leads to unit load method. It is assumed throughout our discussion that the method of superposition holds good. For the derivation of unit load method, we consider two systems of loads. In this section, the principle of virtual forces and unit load method are discussed in the context of framed structures. Consider a cantilever beam, which is in equilibrium under the action of a first system of forces causing displacements as shown in Fig. 5.2a. The first system of forces refers to the actual forces acting on the structure. Let the stress resultants at any section of the beam due to first system of forces be axial force ( n F F F ,....., , 2 1 n u u u ,....., , 2 1 P ), bending moment (M ) and shearing force (V ). Also the corresponding incremental deformations are axial deformation ( ), flexural deformation ( ? d ? d ) and shearing deformation ( ? d ) respectively. For a conservative system the external work done by the applied forces is equal to the internal strain energy stored. Hence, 1 11 1 1 d ? d ? d ? 22 2 2 n ii i Fu P M V = =+ + ? ? ?? ? ? ? + + = L L L AG ds V EI ds M EA ds P 0 2 0 2 0 2 2 2 2 (5.3) Now, consider a second system of forces n F F F d d d ,....., , 2 1 , which are virtual and causing virtual displacements n u u u d d d ,....., , 2 1 respectively (see Fig. 5.2b). Let the virtual stress resultants caused by virtual forces be v v M P d d , and v V d at any cross section of the beam. For this system of forces, we could write ? ? ? ? + + = = L v L v L v n i i i AG ds V EI ds M EA ds P u F 0 2 0 2 0 2 1 2 2 2 2 1 d d d d d (5.4) where v v M P d d , and v V d are the virtual axial force, bending moment and shear force respectively. In the third case, apply the first system of forces on the beam, which has been deformed, by second system of forces n F F F d d d ,....., , 2 1 as shown in Fig 5.2c. From the principle of superposition, now the deflections will be ()( ) ( n n u u u u u u ) d d d + + + ,......, , 2 2 1 1 respectively Page 5 Instructional Objectives After studying this lesson, the student will be able to: 1. Define Virtual Work. 2. Differentiate between external and internal virtual work. 3. Sate principle of virtual displacement and principle of virtual forces. 4. Drive an expression of calculating deflections of structure using unit load method. 5. Calculate deflections of a statically determinate structure using unit load method. 6. State unit displacement method. 7. Calculate stiffness coefficients using unit-displacement method. 5.1 Introduction In the previous chapters the concept of strain energy and Castigliano’s theorems were discussed. From Castigliano’s theorem it follows that for the statically determinate structure; the partial derivative of strain energy with respect to external force is equal to the displacement in the direction of that load. In this lesson, the principle of virtual work is discussed. As compared to other methods, virtual work methods are the most direct methods for calculating deflections in statically determinate and indeterminate structures. This principle can be applied to both linear and nonlinear structures. The principle of virtual work as applied to deformable structure is an extension of the virtual work for rigid bodies. This may be stated as: if a rigid body is in equilibrium under the action of a system of forces and if it continues to remain in equilibrium if the body is given a small (virtual) displacement, then the virtual work done by the F - F - system of forces as ‘it rides’ along these virtual displacements is zero. 5.2 Principle of Virtual Work Many problems in structural analysis can be solved by the principle of virtual work. Consider a simply supported beam as shown in Fig.5.1a, which is in equilibrium under the action of real forces at co-ordinates respectively. Let be the corresponding displacements due to the action of forces . Also, it produces real internal stresses n F F F ,......., , 2 1 n ,....., 2 , 1 n u u u ,......, , 2 1 n F F F ,......., , 2 1 ij s and real internal strains ij e inside the beam. Now, let the beam be subjected to second system of forces (which are virtual not real) n F F F d d d ,......, , 2 1 in equilibrium as shown in Fig.5.1b. The second system of forces is called virtual as they are imaginary and they are not part of the real loading. This produces a displacement configuration n u u u d d d , ,......... , 2 1 . The virtual loading system produces virtual internal stresses ij d s and virtual internal strains ij d e inside the beam. Now, apply the second system of forces on the beam which has been deformed by first system of forces. Then, the external loads and internal stresses i F ij s do virtual work by moving along i u d and ij d e . The product i i u F d ? is known as the external virtual work. It may be noted that the above product does not represent the conventional work since each component is caused due to different source i.e. i u d is not due to . Similarly the product i F ij ij s de ? is the internal virtual work. In the case of deformable body, both external and internal forces do work. Since, the beam is in equilibrium, the external virtual work must be equal to the internal virtual work. Hence, one needs to consider both internal and external virtual work to establish equations of equilibrium. 5.3 Principle of Virtual Displacement A deformable body is in equilibrium if the total external virtual work done by the system of true forces moving through the corresponding virtual displacements of the system i.e. is equal to the total internal virtual work for every kinematically admissible (consistent with the constraints) virtual displacements. i i u F d ? That is virtual displacements should be continuous within the structure and also it must satisfy boundary conditions. dv u F ij ij i i de s d ? ? = (5.1) where ij s are the true stresses due to true forces and i F ij de are the virtual strains due to virtual displacements i u d . 5.4 Principle of Virtual Forces For a deformable body, the total external complementary work is equal to the total internal complementary work for every system of virtual forces and stresses that satisfy the equations of equilibrium. dv u F ij ij i i e ds d ? ? = (5.2) where ij d s are the virtual stresses due to virtual forces i F d and ij e are the true strains due to the true displacements . i u As stated earlier, the principle of virtual work may be advantageously used to calculate displacements of structures. In the next section let us see how this can be used to calculate displacements in a beams and frames. In the next lesson, the truss deflections are calculated by the method of virtual work. 5.5 Unit Load Method The principle of virtual force leads to unit load method. It is assumed throughout our discussion that the method of superposition holds good. For the derivation of unit load method, we consider two systems of loads. In this section, the principle of virtual forces and unit load method are discussed in the context of framed structures. Consider a cantilever beam, which is in equilibrium under the action of a first system of forces causing displacements as shown in Fig. 5.2a. The first system of forces refers to the actual forces acting on the structure. Let the stress resultants at any section of the beam due to first system of forces be axial force ( n F F F ,....., , 2 1 n u u u ,....., , 2 1 P ), bending moment (M ) and shearing force (V ). Also the corresponding incremental deformations are axial deformation ( ), flexural deformation ( ? d ? d ) and shearing deformation ( ? d ) respectively. For a conservative system the external work done by the applied forces is equal to the internal strain energy stored. Hence, 1 11 1 1 d ? d ? d ? 22 2 2 n ii i Fu P M V = =+ + ? ? ?? ? ? ? + + = L L L AG ds V EI ds M EA ds P 0 2 0 2 0 2 2 2 2 (5.3) Now, consider a second system of forces n F F F d d d ,....., , 2 1 , which are virtual and causing virtual displacements n u u u d d d ,....., , 2 1 respectively (see Fig. 5.2b). Let the virtual stress resultants caused by virtual forces be v v M P d d , and v V d at any cross section of the beam. For this system of forces, we could write ? ? ? ? + + = = L v L v L v n i i i AG ds V EI ds M EA ds P u F 0 2 0 2 0 2 1 2 2 2 2 1 d d d d d (5.4) where v v M P d d , and v V d are the virtual axial force, bending moment and shear force respectively. In the third case, apply the first system of forces on the beam, which has been deformed, by second system of forces n F F F d d d ,....., , 2 1 as shown in Fig 5.2c. From the principle of superposition, now the deflections will be ()( ) ( n n u u u u u u ) d d d + + + ,......, , 2 2 1 1 respectively Since the energy is conserved we could write, 22 2 2 11 1 00 00 22 00 0 0 0 11 22 2 2 2 2 22 LL LL nn n vv v jj j j jj jj j LL L L L vv Pds M ds V ds Pds Fu F u Fu v EAEI AG E Mds Vds Pd M d Vd EI AG dd d dd d A d d? d == = ++ = + + + ++ ?+ + ?? ? ?? ?? ?? ? ? ? ? + (5.5) In equation (5.5), the term on the left hand side ( ) ? j j u F d , represents the work done by virtual forces moving through real displacements. Since virtual forces actRead More

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