532
9 Variational Principles in Mechanics and Their Applications
state of equilibrium produces the virtual displacement, the external forces have acted
on the elastic body, and in the virtual displacement, the external forces and stress all
retain unchanged, it is the work done by the constant forces. Thus the virtual work
is simply expressed as the product of external forces and virtual displacements, no
the factor 1/2. This is the important characteristics of the virtual work different from
the real work.
According to Eq. (9.2.32), the principle of virtual displacement of elastic body
can be expressed as: The necessary and sufficient in the state of equilibrium for
an elastic body is: For any small virtual displacement, the virtual work done by
the external forces (volume force and surface force) acting on the elastic body
in the process of any virtual displacement is equal to the virtual strain energy
of the elastic body.
The principle of virtual displacement the expression (9.2.28), expression (9.2.29)
or expression (9.2.32) is the fundamental of variational principle in elasticity, it also
has important application value in finite element method.
It is worth pointing out that although from the angle of the elastic equilibrium
of object the expression of the principle of virtual displacement is derived above,
generally speaking, the principle of virtual displacement has universal significance.
It can be applied to all structures, no matter the material is linear or nonlinear, elastic
or plastic, static load or dynamic load, etc. it all can applied.
As a matter of fact, the principle of virtual displacement derived above can be
achieved in an easy way. Using the principle of generalized virtual displacement
(9.2.20), take σ
p
i j as the true stress σ i j , and note that δu i and δε i j are also the possible
displacement and possible strain. So σ i j , δu i and δε i j can be used to replace σ
p
i j , u
p
i
and ε
p
i j in the expression (9.2.20), and the expression (9.2.29) can be obtained.
Example 9.2.1 As shown in Fig. 9.3, a structure composed of n hinged bars hinges
at the node A, the length of every bar is L i , the section area is A i , the modulus of
elasticity is E i , the various bars are at angle of α i (i = 1, 2, …, n) with the horizontal
plane respectively. The load P acts on the node A and is at an angle β from horizontal
plane. Assume that the bars are linear elastic material, omit the act of volume force
(bar’s own weight), find the internal force of each bar.
Solution The deformations of various bars in the structure can be calculated through
the displacements of the node A, and the displacements of the node A can be
represented by trigonometric function.
Let the node A have a virtual displacement δx in the x direction, then the length
of each bar will change and turn around each fixed endpoint B i (i = 1, 2, …, n), this
does not violate the constraint conditions given by this problem.
The change quantity of each bar in length is
δL i = δx cos α i
(1)
The virtual strain of each bar is
δε i =
δL i
L i
=
δx cos α i
L i
(2)
9 Variational Principles in Mechanics and Their Applications
state of equilibrium produces the virtual displacement, the external forces have acted
on the elastic body, and in the virtual displacement, the external forces and stress all
retain unchanged, it is the work done by the constant forces. Thus the virtual work
is simply expressed as the product of external forces and virtual displacements, no
the factor 1/2. This is the important characteristics of the virtual work different from
the real work.
According to Eq. (9.2.32), the principle of virtual displacement of elastic body
can be expressed as: The necessary and sufficient in the state of equilibrium for
an elastic body is: For any small virtual displacement, the virtual work done by
the external forces (volume force and surface force) acting on the elastic body
in the process of any virtual displacement is equal to the virtual strain energy
of the elastic body.
The principle of virtual displacement the expression (9.2.28), expression (9.2.29)
or expression (9.2.32) is the fundamental of variational principle in elasticity, it also
has important application value in finite element method.
It is worth pointing out that although from the angle of the elastic equilibrium
of object the expression of the principle of virtual displacement is derived above,
generally speaking, the principle of virtual displacement has universal significance.
It can be applied to all structures, no matter the material is linear or nonlinear, elastic
or plastic, static load or dynamic load, etc. it all can applied.
As a matter of fact, the principle of virtual displacement derived above can be
achieved in an easy way. Using the principle of generalized virtual displacement
(9.2.20), take σ
p
i j as the true stress σ i j , and note that δu i and δε i j are also the possible
displacement and possible strain. So σ i j , δu i and δε i j can be used to replace σ
p
i j , u
p
i
and ε
p
i j in the expression (9.2.20), and the expression (9.2.29) can be obtained.
Example 9.2.1 As shown in Fig. 9.3, a structure composed of n hinged bars hinges
at the node A, the length of every bar is L i , the section area is A i , the modulus of
elasticity is E i , the various bars are at angle of α i (i = 1, 2, …, n) with the horizontal
plane respectively. The load P acts on the node A and is at an angle β from horizontal
plane. Assume that the bars are linear elastic material, omit the act of volume force
(bar’s own weight), find the internal force of each bar.
Solution The deformations of various bars in the structure can be calculated through
the displacements of the node A, and the displacements of the node A can be
represented by trigonometric function.
Let the node A have a virtual displacement δx in the x direction, then the length
of each bar will change and turn around each fixed endpoint B i (i = 1, 2, …, n), this
does not violate the constraint conditions given by this problem.
The change quantity of each bar in length is
δL i = δx cos α i
(1)
The virtual strain of each bar is
δε i =
δL i
L i
=
δx cos α i
L i
(2)