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Principle of Virtual Work

The document discusses the principle of virtual work and its application in the finite element method. It introduces: 1) The principle of virtual work, which states that the change in strain energy from a virtual displacement equals the work done by forces. 2) Derivation of the element stiffness matrix and load vector using virtual displacements and constitutive relationships. 3) Examples of plane stress elements - the constant strain triangle (CST) and linear strain triangle (LST) - including their displacement fields, strain-displacement relationships, and stiffness matrix formulations.

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218 views11 pages

Principle of Virtual Work

The document discusses the principle of virtual work and its application in the finite element method. It introduces: 1) The principle of virtual work, which states that the change in strain energy from a virtual displacement equals the work done by forces. 2) Derivation of the element stiffness matrix and load vector using virtual displacements and constitutive relationships. 3) Examples of plane stress elements - the constant strain triangle (CST) and linear strain triangle (LST) - including their displacement fields, strain-displacement relationships, and stiffness matrix formulations.

Uploaded by

NITBEHL77
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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1

Principle of Virtual Work


• The principle of virtual work states that at
equilibrium the strain energy change due to a
small virtual displacement is equal to the work
done by the forces in moving through the virtual
displacement.
• A virtual displacement is a small imaginary
change in configuration that is also a admissible
displacement
• An admissible displacement satisfies kinematic
boundary conditions
• Note: Neither loads nor stresses are altered by
the virtual displacement.

R.T. Haftka EML5526 Finite Element Analysis University of Florida


2

Principle of Virtual Work


• The principle of virtual work can be written as follows
∫ {δε }T
{σ }dV = ∫ {δu }T
{F }dV + ∫ {δu }T
{Φ}dS
• The same can be obtained by the Principle of Stationary
Potential Energy

• The total potential energy of a system Π is given by

δΠ = δU − δW = δ U + δV = 0
– U is strain energy, W is work done, or V is potential of the forces

Π = U −W

R.T. Haftka EML5526 Finite Element Analysis University of Florida


3

Element and load derivation

• Interpolation {u} = [ N ]{d} {u} = u v w


• Strain displacement {ε } = [ B ]{d} [ B ] = [∂ ][ N ]
• Virtual {δu} = {δd} [ N ] and {δε } = {δd} [ B ]
T T T T T T

• Constitutive law {σ } = [ E ]{ε }


• Altogether
{δd} ( ∫ [ B ] [ E ][ B ] dV {d} − ∫ [ B ] [ E ]{ε 0 } dV + ∫ [ B ] {σ 0 } dV
T T T T

− ∫ [ N ] {F} dV − ∫ [ N ] {φ } dS ) = 0
T T

R.T. Haftka EML5526 Finite Element Analysis University of Florida


4

Stiffness matrix and load vector


• Equations of equilibrium
[ k ]{d} = {re }
• Element stiffness matrix
[ k ] = ∫ [ B ] [ E ][ B ] dV
T

• Element load vector


{re} =∫[ B] [ E]{ε0} dV−∫[ B] {σ0} dV+∫ [ N] {F} dV+∫ [ N] {φ} dS )
T T T T

• Loads due to initial strain, initial stress,


body forces and surface tractions

R.T. Haftka EML5526 Finite Element Analysis University of Florida


5

Plane Problems: Constitutive Equations


• Constitutive equations for a linearly elastic and
isotropic material in plane stress (i.e., σz=τxz=τyz=0):

Initial thermal strains ε x 0 = ε y 0 = α∆T , γ xy0 = 0


• In matrix form,

where

R.T. Haftka EML5526 Finite Element Analysis University of Florida


6

Plane Problems: Strain-Displacement


Relations
•.

R.T. Haftka EML5526 Finite Element Analysis University of Florida


7

Plane Problems: Displacement


Field Interpolated
•

• From the previous two equations,

where B is the strain-displacement matrix.

R.T. Haftka EML5526 Finite Element Analysis University of Florida


8
Constant Strain Triangle (CST)

• The node numbers sequence must go counter


clockwise
• Linear displacement field so strains are constant!

R.T. Haftka EML5526 Finite Element Analysis University of Florida


9

Constant Strain Triangle (CST):


Stiffness Matrix
•Strain-displacement relation, ε=Bd

A is the area of the triangle and xij=xi- xj. (textbook has results for a
coordinate system with x aligned with side 1-2
T
• From the general formula k = B EBtA
where t: element thickness (constant)

R.T. Haftka EML5526 Finite Element Analysis University of Florida


10

Linear Strain Triangle (LST)

• The element has six nodes and 12 dof. Not available


in Nastran-Genesis!
R.T. Haftka EML5526 Finite Element Analysis University of Florida
11

Linear Strain Triangle (LST)


• The quadratic displacement field in terms of
generalized coordinates:

• The linear strain field:

R.T. Haftka EML5526 Finite Element Analysis University of Florida

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