Strain Energy Method Notes | EduRev

: Strain Energy Method Notes | EduRev

 Page 1


Module 4 : Deflection of Structures
Lecture 4 : Strain Energy Method
 Objectives
   In this course you will learn the following
Deflection by strain energy method.
Evaluation of strain energy in member under different loading.
Application of strain energy method for different types of structure.
4.5 Deflection by Strain Energy Method
 
The concepts of strain, strain-displacement relationships are very useful in computing energy-related
quantities such as work and strain energy. These can then be used in the computation of deflections. In the
special case, when the structure is linear elastic and the deformations are caused by external forces only,
(the complementary energy U * is equal to the strain energy U ) the displacement of structure in the
direction of force  is expressed by   
 
   
(4.16)
 
This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the
calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is
equivalent to the virtual work transformation by the unit-load theorem.
4.5.1 Calculation of Strain Energy
 
When external loads are applied on an elastic body they deform. The work done is transformed into elastic
strain energy U that is stored in the body. We will develop expressions for the strain energy for different
types of loads.
 
Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied
gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external
work done by the axial force i.e
 
(4.17)
 
 
Figure 4.24 Member subjected to axial force
 
Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as
shown in Figure 4.25. Consider a small differential element of length, dx . The energy stored in the small
element is given by
Page 2


Module 4 : Deflection of Structures
Lecture 4 : Strain Energy Method
 Objectives
   In this course you will learn the following
Deflection by strain energy method.
Evaluation of strain energy in member under different loading.
Application of strain energy method for different types of structure.
4.5 Deflection by Strain Energy Method
 
The concepts of strain, strain-displacement relationships are very useful in computing energy-related
quantities such as work and strain energy. These can then be used in the computation of deflections. In the
special case, when the structure is linear elastic and the deformations are caused by external forces only,
(the complementary energy U * is equal to the strain energy U ) the displacement of structure in the
direction of force  is expressed by   
 
   
(4.16)
 
This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the
calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is
equivalent to the virtual work transformation by the unit-load theorem.
4.5.1 Calculation of Strain Energy
 
When external loads are applied on an elastic body they deform. The work done is transformed into elastic
strain energy U that is stored in the body. We will develop expressions for the strain energy for different
types of loads.
 
Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied
gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external
work done by the axial force i.e
 
(4.17)
 
 
Figure 4.24 Member subjected to axial force
 
Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as
shown in Figure 4.25. Consider a small differential element of length, dx . The energy stored in the small
element is given by
                                
(4.18)
 
The total strain energy in the entire beam will be
 
    
(4.19)
 
  
 
Figure 4.25 Member under bending
 Shear Force: The strain energy stored in the member due to shearing force is expressed by
 
 
(4.20)
 
where V is the shearing force; and  is the shearing rigidity of the member.
 
Twisting Moment: The strain energy stored in the member due to twisting moment is expressed by
 
(4.21)
 
where T is the twisting moment; and GJ is the torsional rigidity of the member.
 
Example 4.18 Find the horizontal deflection at joint C of the pin-jointed frame as shown in Figure 4.26(a). AE
is constant for all members.
 
Page 3


Module 4 : Deflection of Structures
Lecture 4 : Strain Energy Method
 Objectives
   In this course you will learn the following
Deflection by strain energy method.
Evaluation of strain energy in member under different loading.
Application of strain energy method for different types of structure.
4.5 Deflection by Strain Energy Method
 
The concepts of strain, strain-displacement relationships are very useful in computing energy-related
quantities such as work and strain energy. These can then be used in the computation of deflections. In the
special case, when the structure is linear elastic and the deformations are caused by external forces only,
(the complementary energy U * is equal to the strain energy U ) the displacement of structure in the
direction of force  is expressed by   
 
   
(4.16)
 
This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the
calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is
equivalent to the virtual work transformation by the unit-load theorem.
4.5.1 Calculation of Strain Energy
 
When external loads are applied on an elastic body they deform. The work done is transformed into elastic
strain energy U that is stored in the body. We will develop expressions for the strain energy for different
types of loads.
 
Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied
gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external
work done by the axial force i.e
 
(4.17)
 
 
Figure 4.24 Member subjected to axial force
 
Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as
shown in Figure 4.25. Consider a small differential element of length, dx . The energy stored in the small
element is given by
                                
(4.18)
 
The total strain energy in the entire beam will be
 
    
(4.19)
 
  
 
Figure 4.25 Member under bending
 Shear Force: The strain energy stored in the member due to shearing force is expressed by
 
 
(4.20)
 
where V is the shearing force; and  is the shearing rigidity of the member.
 
Twisting Moment: The strain energy stored in the member due to twisting moment is expressed by
 
(4.21)
 
where T is the twisting moment; and GJ is the torsional rigidity of the member.
 
Example 4.18 Find the horizontal deflection at joint C of the pin-jointed frame as shown in Figure 4.26(a). AE
is constant for all members.
 
   
 
Solution: The force in various members of the frame is shown in Figure 4.26(b). Calculation of strain energy
of the frame is shown in Table 4.4.   
 
 Table 4.4
 
Member Length ( L ) Force ( P )
AB L P
BC L P
BD
 
CD L 0 0
 
                                                                
 
Horizontal displacement of joint C , 
                                                               
 
Example 4.19 A bar of uniform cross-section is bent into a quadrant of circle of radius R . One end of the
bent is fixed and other is free. At the free end it carries a vertical load W . Determine the vertical and
horizontal deflection at A .
 Solution:
 
   
Page 4


Module 4 : Deflection of Structures
Lecture 4 : Strain Energy Method
 Objectives
   In this course you will learn the following
Deflection by strain energy method.
Evaluation of strain energy in member under different loading.
Application of strain energy method for different types of structure.
4.5 Deflection by Strain Energy Method
 
The concepts of strain, strain-displacement relationships are very useful in computing energy-related
quantities such as work and strain energy. These can then be used in the computation of deflections. In the
special case, when the structure is linear elastic and the deformations are caused by external forces only,
(the complementary energy U * is equal to the strain energy U ) the displacement of structure in the
direction of force  is expressed by   
 
   
(4.16)
 
This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the
calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is
equivalent to the virtual work transformation by the unit-load theorem.
4.5.1 Calculation of Strain Energy
 
When external loads are applied on an elastic body they deform. The work done is transformed into elastic
strain energy U that is stored in the body. We will develop expressions for the strain energy for different
types of loads.
 
Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied
gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external
work done by the axial force i.e
 
(4.17)
 
 
Figure 4.24 Member subjected to axial force
 
Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as
shown in Figure 4.25. Consider a small differential element of length, dx . The energy stored in the small
element is given by
                                
(4.18)
 
The total strain energy in the entire beam will be
 
    
(4.19)
 
  
 
Figure 4.25 Member under bending
 Shear Force: The strain energy stored in the member due to shearing force is expressed by
 
 
(4.20)
 
where V is the shearing force; and  is the shearing rigidity of the member.
 
Twisting Moment: The strain energy stored in the member due to twisting moment is expressed by
 
(4.21)
 
where T is the twisting moment; and GJ is the torsional rigidity of the member.
 
Example 4.18 Find the horizontal deflection at joint C of the pin-jointed frame as shown in Figure 4.26(a). AE
is constant for all members.
 
   
 
Solution: The force in various members of the frame is shown in Figure 4.26(b). Calculation of strain energy
of the frame is shown in Table 4.4.   
 
 Table 4.4
 
Member Length ( L ) Force ( P )
AB L P
BC L P
BD
 
CD L 0 0
 
                                                                
 
Horizontal displacement of joint C , 
                                                               
 
Example 4.19 A bar of uniform cross-section is bent into a quadrant of circle of radius R . One end of the
bent is fixed and other is free. At the free end it carries a vertical load W . Determine the vertical and
horizontal deflection at A .
 Solution:
 
   
 Vertical displacement of A : The vertical displacement of A is given by
 
 
For evaluation of the total strain energy in the system, consider a small element  as shown in the Figure.
The bending moment at this element,  . Thus, 
  
 
 
 
   
 
Since there is no horizontal force acting at point A , apply a horizontal force, F at A as shown in Figure
4.27(b). From the Castigliano's theorem, the horizontal displacement of A due to applied external load W is
given by
 
 
   
 
The bending moment at the small element  is  . Thus, the horizontal
displacement of A
 
 
 
   
Page 5


Module 4 : Deflection of Structures
Lecture 4 : Strain Energy Method
 Objectives
   In this course you will learn the following
Deflection by strain energy method.
Evaluation of strain energy in member under different loading.
Application of strain energy method for different types of structure.
4.5 Deflection by Strain Energy Method
 
The concepts of strain, strain-displacement relationships are very useful in computing energy-related
quantities such as work and strain energy. These can then be used in the computation of deflections. In the
special case, when the structure is linear elastic and the deformations are caused by external forces only,
(the complementary energy U * is equal to the strain energy U ) the displacement of structure in the
direction of force  is expressed by   
 
   
(4.16)
 
This equation is known as Castigliano's theorem. It must be remembered that its use is limited to the
calculation of displacement in linear elastic structures caused by applied loads. The use of this theorem is
equivalent to the virtual work transformation by the unit-load theorem.
4.5.1 Calculation of Strain Energy
 
When external loads are applied on an elastic body they deform. The work done is transformed into elastic
strain energy U that is stored in the body. We will develop expressions for the strain energy for different
types of loads.
 
Axial Force : Consider a member of length L and axial rigidity AE subjected to an axial force P applied
gradually as shown in the Figure 4.24. The strain energy stored in the member will be equal to the external
work done by the axial force i.e
 
(4.17)
 
 
Figure 4.24 Member subjected to axial force
 
Bending Moment: Consider a beam of length L and flexural rigidity EI subjected to a general loading as
shown in Figure 4.25. Consider a small differential element of length, dx . The energy stored in the small
element is given by
                                
(4.18)
 
The total strain energy in the entire beam will be
 
    
(4.19)
 
  
 
Figure 4.25 Member under bending
 Shear Force: The strain energy stored in the member due to shearing force is expressed by
 
 
(4.20)
 
where V is the shearing force; and  is the shearing rigidity of the member.
 
Twisting Moment: The strain energy stored in the member due to twisting moment is expressed by
 
(4.21)
 
where T is the twisting moment; and GJ is the torsional rigidity of the member.
 
Example 4.18 Find the horizontal deflection at joint C of the pin-jointed frame as shown in Figure 4.26(a). AE
is constant for all members.
 
   
 
Solution: The force in various members of the frame is shown in Figure 4.26(b). Calculation of strain energy
of the frame is shown in Table 4.4.   
 
 Table 4.4
 
Member Length ( L ) Force ( P )
AB L P
BC L P
BD
 
CD L 0 0
 
                                                                
 
Horizontal displacement of joint C , 
                                                               
 
Example 4.19 A bar of uniform cross-section is bent into a quadrant of circle of radius R . One end of the
bent is fixed and other is free. At the free end it carries a vertical load W . Determine the vertical and
horizontal deflection at A .
 Solution:
 
   
 Vertical displacement of A : The vertical displacement of A is given by
 
 
For evaluation of the total strain energy in the system, consider a small element  as shown in the Figure.
The bending moment at this element,  . Thus, 
  
 
 
 
   
 
Since there is no horizontal force acting at point A , apply a horizontal force, F at A as shown in Figure
4.27(b). From the Castigliano's theorem, the horizontal displacement of A due to applied external load W is
given by
 
 
   
 
The bending moment at the small element  is  . Thus, the horizontal
displacement of A
 
 
 
   
  
  
 
 
 
 (i.e. deflection is in direction)
 
Example 4.20 Determine the deflection of the end A of the beam as shown in Figure 4.28. The flexibility of
the spring is .
 
   
 
Solution: Reactions at support B and C are  
 
 (upward) and  (downward)
 
Force in the spring = Reaction, 
 
Deflection under the load is given by 
 
 
where  is the total strain energy stored in the system;  is the energy stored in the
member AB ;  is the energy stored in the member BC ; and  = strain energy in the spring.
 Strain energy in the spring is given by
                                                 
 
Consider member AB : ( x measured from A )
  
                                               
 
Consider member BC : ( x measured from C )
Read More
Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!