4.
ME 5040 FEM 1
Section 2
Stiffness (Displacement) Method
Truss Elements
1 (NKM)
Outline
Finite Element Formulation Direct Method
Nodal equilibriums
Element and global stiffness matrices
Superposition of stiffness matrix
System equation and boundary conditions
Displacements, reaction forces, stresses, and strains
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Introduction
3 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Spring Element
Consider the following linear spring element:
Points 1 and 2 are reference points called nodes
f1x and f2x are the local nodal forces on the x-axis
1 and 2 are the local nodal displacements
k is the spring constant or stiffness of the spring
L is the distance between the nodes
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A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Spring Element
We have selected our element type and now need to define
the deformation relationships
For the spring subject to tensile forces at each node:
= 2 - 1 & T = k
Where is the total deformation and T is the tensile force
Combine to obtain: T = k(2 - 1 )
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A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Spring Element
Performing a basic force balance yields:
Combining these force eqs with the previous eqs:
Express in matrix form:
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A First Course in the Finite Element Method, 6th Edition
Logan
Establishing the Global Stiffness
Matrix for a Spring Assemblage
Consider the two-spring assemblage:
Node 1 is fixed and axial forces are applied at
nodes 3 and 2.
The x-axis is the global axis of the assemblage.
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A First Course in the Finite Element Method, 6th Edition
Logan
Establishing the Global Stiffness
Matrix for a Spring Assemblage
For element 1:
For element 2:
Elements 1 and 2 must remain connected at common
node 3. The is called the continuity or compatibility
requirement given by:
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A First Course in the Finite Element Method, 6th Edition
Logan
Establishing the Global Stiffness
Matrix for a Spring Assemblage
From the Free-body diagram of the assemblage:
We can write the equilibrium nodal equations:
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A First Course in the Finite Element Method, 6th Edition
Logan
Establishing the Global Stiffness
Matrix for a Spring Assemblage
Combining the nodal equilibrium equations with the
elemental force/displacement/stiffness relations we obtain
the global relationship:
Which takes the form: {F} = [K]{d}
{F} is the global nodal force matrix
{d} is the global nodal displacement matrix
[K] is the total or global or system stiffness matrix
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A First Course in the Finite Element Method, 6th Edition
Logan
Homogenous Boundary
Conditions
Where is the homogenous boundary condition for
the spring assemblage?
It is at the location which is fixed, Node 1
Because Node 1 is fixed 1 = 0
The system relation can be written as:
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A First Course in the Finite Element Method, 6th Edition
Logan
Nonhomogeneous Boundary
Conditions
Consider the case where there is a known
displacement, , at Node 1
Let 1 =
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A First Course in the Finite Element Method, 6th Edition
Logan
Minimum Potential Energy
Approach
Alternative method often used to derive the
element equations and stiffness matrix.
More adaptable to the determination of element
equations for complicated elements such as:
Plane stress/strain element
Axisymmetric stress element
Plate bending element
Three-dimensional solid stress element
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Logan
Total Potential Energy
Defined as the sum of the internal strain energy,
U, and the potential energy of the external forces,
Strain energy is the capacity of internal forces to
do work through deformations in the structure.
The potential energy of external forces is the
capacity of forces such as body forces, surface
traction forces, or applied nodal forces to do work
through deformation of the structure.
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A First Course in the Finite Element Method, 6th Edition
Logan
Total Potential Energy of Spring
The strain energy can be expressed as:
The potential energy of the external force can be
expressed as:
Therefore, the total potential energy of a spring is:
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A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Spring Element Eqs.
Consider the linear spring subject to nodal forces:
The total potential energy is:
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A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Spring Element Eqs.
To minimize the total potential energy the partial
derivatives of p with respect to each nodal
displacement must be taken:
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A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Spring Element Eqs.
Simplify to:
In matrix form:
The results are identical to the direct method
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A First Course in the Finite Element Method, 6th Edition
Logan
Development of Truss Equations
Lift bridge truss over the Illinois River (By Daryl L. Logan)
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A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Bar Element
Consider the following bar subjected to tensile forces:
For this derivation, assume that:
The bar element has a constant cross-sectional area A,
modulus of elasticity E, and initial length L.
The bar cannot sustain shear force or bending moment.
Any effect of transverse displacement is ignored.
Hookes law applies, axial stress is related to axial strain.
No intermediate applied loads
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A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Bar Element
By combining the strain/displacement and stress/strain
relationships with the equation for force in terms of stress
and area we get:
From the force equilibrium equations:
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A First Course in the Finite Element Method, 6th Edition
Logan
Derivation of the Stiffness Matrix
for a Bar Element
Express the two force equations in matrix form:
We now have the local matrix form we have seen in
past chapters:
Thus the local stiffness matrix for a linear-elastic bar
element is:
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Nodal Equilibriums
Element Stiffness Matrices
1 1
L1 1 1
[K1 ] = E1 A1
1 1
1 1
[K 2 ] = E2 A2
L2
Nodal Equilibriums at
E1 A1 1 1 u1 f1
=
L1 1 1 u2 f 2
E2 A2
L2
1 1 u2 f 3
1 1 u = f
3 4
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Introduction
Global Stiffness Matrix
System Equation
? ? ? u1 F1 R1 Nodal Displacements
? ? ? u = F = 0
u1 k1=E1A1/L1 u2
2 2
? ? ? u3 F3 P
1
Superposition
? ? ? u1 F1
?K1? ? u = F
2 2
? ? ? u3 F3
? ? ? u1 F1
? ? ? u = F
2 2
K
? ? 2? u3 F3
u2 k2=E2A2/L2 u3
2
Nodal Forces
f1
1
k1=E1A1/L1 f
2
1
f3
2
k2=E2A2/L2 f
4
2
F1 f1
F2 = f 2 + f 3
F f
3 4
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Introduction
Global Stiffness Matrix (contd)
Global Stiffness Matrix
E1 A1
?
? L1
?
EA
K1
[K ] = ?
?
? = 1 1
L1
K2
?
?
?
0
E1 A1
L1
E1 A1 E2 A2
+
L1
L2
E2 A2
L2
E2 A2
L2
E2 A2
L2
0
Recall
k1
[K ] = k1
0
k1
k1 + k 2
k2
0
k 2
k 2
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Introduction
Example: Forming Stiffness Matrix
E1 = E2 = E
P
A1
A2
L1
EA2/L2
EA1/L1
x
1
L2
Element Stiffness Matrices
E1 A1 1 1 u1
[K1 ] =
L1 1 1 u3
1 1 u3 E2 A2
u = L
1
1
2
2
[K 2 ] = E2 A2
L2
1 1 u2
1 1 u
3
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Introduction
Example: Forming Stiffness Matrix
Global Stiffness Matrix
0
[K ] = 0
0
0
0
0 u1
0 u 2
0 u3
Add [K2]
Add [K1]
EA1 L1
[K ] = 0
EA1 L1
EA1 L1
[K ] = 0
EA1 L1
0
EA2 L2
EA2 L2
0
0
0
EA1 L1 u1
u2
0
EA1 L1 u3
EA1 L1
u1
u2
EA2 L2
EA1 L1 + EA2 L2 u3
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Introduction
Alternative Approach
Expanding element stiffness matrices
1
L1 1 1
[K1 ] = E1 A1
1 1
[K 2 ] = E2 A2
L2 1 1
1
u1 E1 A1
0
=
u3
L1
1
0
u3 E2 A2
0
=
u2
L2
0
0 1 u1
0 0 u2
0 1 u3
0 0 u1
1 1 u2
1 1 u3
Obtaining global stiffness matrix by superposition
EA1 L1
[K ] = [K1 ] + [K 2 ] = 0
EA1 L1
0
EA2 L2
EA2 L2
EA1 L1
u1
u2
EA2 L2
EA1 L1 + EA2 L2 u3
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Introduction
Node 1
Node 2
Node 3
Node 4
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Node 1
Node 2
Node 3
Node 4
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A First Course in the Finite Element Method, 6th Edition
Logan
Guidelines for Selecting a
Displacement Function
It is difficult, if not impossible at time, to obtain a closed form
or exact solution, so we assume a solution shape or
distribution of displacement.
The approximation function should be:
Continuous within the bar element
Should provide interelement continuity for all degrees of
freedom
Should allow for rigid-body displacement and for a state
of constant strain within the element
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Interpolations over Elements
1-D Element
u1
u( x ) = a0 + a1 x
u(0) = a0 = u1
u( L) = a0 + a1L = u2
u2
2
u( x ) = u1 +
u2 u1
x
L
u( x ) = (1 x L)u1 + ( x L)u2
x
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Interpolations over Elements
1-D Element
x x
N1 = 2
x2 x1
x x
N2 = 1
x1 x2
2
u( x ) = N i ui
i =1
2
x
49 (NKM)
Introduction
Interpolations over Elements
1-D Element
( x x )( x3 x )
N1 = 2
( x2 x1 )( x3 x1 )
( x x )( x3 x )
N2 = 1
( x1 x2 )( x3 x2 )
( x x )( x2 x )
N3 = 1
( x1 x3 )( x2 x3 )
3
u( x ) = N i ui
i =1
x
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Introduction
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FE Formulation by MTPE
Principle of Minimum Total Potential Energy (MTPE)
Of all the geometrically possible shapes that a body can
assume, the true one, corresponding to the satisfaction of
stable equilibrium of the body, is identified by a minimum
value of the total potential energy
MTPE is applicable only to elastic behavior
MTPE is the sum of internal strain energy and
potential energy of external work (done by external
forces)
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Introduction
Internal Strain Energy
Strain
energy
L
L
d 0
Element Strain Energy
V0
Ue =
0 d dV = L A 0 Ed dy = L A 1 EA 2 dy
0 0
0 2 0
0
1 L
1 EA 2
= EA
L
L=
2 L
2 L
2
Total Internal Strain Energy: U = U i
i =1
Introduction
64 (NKM)
Potential Energy of External Work
External Work
n
W = Fju j
j =1
Potential Energy of External Work
= W = F j u j
j =1
65 (NKM)
Introduction
Minimum Total Potential Energy
Total Potential Energy
m
i =1
j =1
= U + = U W = U i Fju j
Total potential energy is minimum when
n
U
F
u
=
i
j j u1 = 0
j =1
u1
i =1
n
= U i F j u j u2 = 0
u2
j =1
i =1
...
m
n
= U F u u = 0
n
i j j
un
i
j
=
1
=
1
Minimum
at
/x=0
x
66 (NKM)
Introduction
Stiffness Matrix by MTPE
Example: a two-bar system
Internal strain energy
1E A
1E A
2
2
U = U1 + U 2 = 1 1 ( L1 ) + 2 2 ( L2 )
2 L1
2 L2
1
1
2
2
= k1 (u 2 u1 ) + k 2 (u3 u2 )
2
2
Potential energy of external work
u1
k1
u2
k2
u3
= W = F j u j = ( R1u1 + 0 u2 + Pu3 ) = R1u1 Pu3
j =1
Total potential energy
=U + =
1
1
2
2
k1 (u2 u1 ) + k 2 (u3 u2 ) + (R1u1 Pu3 )
2
2
67 (NKM)
Introduction
Stiffness Matrix by MTPE (contd)
Minimum total potential energy
= k1 (u2 u1 ) + R1 = 0
u1
k1u1 k1u2 + 0 u3 = R1
= k1 (u2 u1 ) k 2 (u3 u2 ) = 0
u2
k1u1 + (k1 + k 2 )u2 k 2u3 = 0
= k 2 (u3 u2 ) P = 0
u3
0 u1 k 2u2 + k 2u3 = P
Stiffness matrix and system equation
k1
k
1
0
k1
k1 + k 2
k2
0 u1 R1
k 2 u2 = 0
k 2 u3 P
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Introduction
Element Stiffness Matrix by MTPE
Internal strain energy
1 EA
1
2
2
U=
( L1 ) = k (u2 u1 )
2 L
2
Potential energy of external work
= W = ( f1u1 + f 2u2 )
u1
u1
k=EA/L
u2
f2
Nodal Forces
k=EA/L
f1
Minimum total potential energy
(U + )
=
= k (u u ) f = 0
u1
Nodal Displacements
(U + )
=
= k (u2 u1 ) f 2 = 0
u2
u2
1 1 u1 f1
k
u = f
1
1
2 2
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Introduction
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A First Course in the Finite Element Method, 6th Edition
Logan
Global Stiffness Matrix for
Arbitrarily Oriented Bar
Consider the bar element arbitrarily oriented in the global
x y plane
We require a stiffness matrix which solves this
configuration for any angle
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A First Course in the Finite Element Method, 6th Edition
Logan
Global Stiffness Matrix for
Arbitrarily Oriented Bar
Performing the derivation yields the explicit stiffness matrix for a bar
arbitrarily oriented in the x y plane
C and S represent cos() and sin(), respectively.
The matrix can be summed using the direct stiffness method since it is
piece-wise continuous from element to element.
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Section 3.4 & Example 3.3
(Pages 84 - 89)
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5.11
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A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Bar Element Equations
Recall that the total potential energy is defined as:
To evaluate the strain energy for a bar, we consider only the work
done by the internal forces during deformation.
The differential internal work (or strain energy) is:
86 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Bar Element Equations
Letting the volume of the element approach zero and then integrating
for the whole bar we get:
For a linear-elastic material (Hookes law) and further simplifying for
a uniform-cross sectional area A with stress/strain dependent on the
x coordinate we get:
We can observe from this integral that the strain energy is the area
under the stress/strain curve.
87 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Bar Element Equations
The potential energy of the external forces is given by:
Where the first term is body forces, the second term is surface
loading or traction, and the third term is nodal concentrated forces
moving through nodal displacements
Combining both potential energy factors we get:
Expressed in matrix form where {P} is the concentrated nodal loads:
88 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Bar Element Equations
Combining the axial displacement function in terms of the shape
function and nodal displacements gives us:
The minimization of the potential energy requires that:
Further rearranging and substitution gives:
89 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Potential Energy Approach to
Derive Bar Element Equations
Writing these two equations in matrix form gives:
It can be seen that we have derived the identical bar stiffness matrix
as earlier:
Finally, we can use the matrix differentiation performed in Chapter 2
and apply it directly:
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A First Course in the Finite Element Method, 6th Edition
Logan
Comparison of Finite Element
Solution to Exact Solution For
Bar
Comparison of exact and finite element solutions for axial
displacement (along length of bar)
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A First Course in the Finite Element Method, 6th Edition
Logan
Comparison of Finite Element
Solution to Exact Solution For Bar
Comparison of exact and finite element solutions for axial stress
(along length of bar)
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A First Course in the Finite Element Method, 6th Edition
Logan
Weighted Residual Methods
The methods of weighted residuals applied directly to a differential
equation can be used to develop the finite element equations.
In weighted residual methods, a trial or approximate function is
chosen to approximate the independent variable, such as
displacement or temperature, in a problem defined by a differential
equation.
It is required that the weighted value of the residual be a minimum
over the whole region. The general form of the weighted residual
integral is:
Where W is the weighting function
93 (NKM)
A First Course in the Finite Element Method, 6th Edition
Logan
Types of Residual Methods
Collocation Method
Requires that the error or residual function, R, be forced to zero
at as many points as there are unknown coefficients
Subdomain Method
Requires that the integral of the error or residual function over
some selected subintervals to be set to zero
Least Squares Method
Requires the integral of the error function squared to be
minimized with respect to each of the unknown coefficients
Galerkins Method
Requires the error to be orthogonal to some weighting functions
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A First Course in the Finite Element Method, 6th Edition
Logan
Flowchart for
Solution of
ThreeDimensional
Truss Problems
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