Stiffness Method Chapter 2

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Stiffness MethodChapter 2
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Definition
For an element, a stiffness matrix is a
matrix such that where relates local
coordinates nodal displacements to local
forces of a single element.
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Spring Element
k
1
2
L
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Definitions
k - spring constant
node
node
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Examples of Stiffness

• Uniaxial Bar k AE/L
• Circular Torsion k GJ/L
• One-dimensional heat conduction
  k AKxx/L
• One-dimensional fluid flow (porous medium)
  k AKxx/L

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Stiffness Relationship for a Spring
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Steps in Process

• Discretize and Select Element Type
• Select a Displacement Function
• Define Strain/Displacement and Stress/Strain
  Relationships
• Derive Element Stiffness Matrix Eqs.
• Assemble Equations and Introduce B.C.s
• Solve for the Unknown Degrees of Freedom
• Solve for Element Stresses and Strains
• Interpret the Results

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General Steps

• Outlined on Previous Slide
• Derive Stiffness Matrix
• Illustrate Usage for Spring Assemblies

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Step 1 - Select the Element Type
k
1
2
T
T
L
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Step 2 - Select a Displacement Function

• Assume a displacement function
• Assume a linear function.
• Number of coefficients number of d-o-f
• Write in matrix form.

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Express as function of and
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Substituting back into
Yields
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In matrix form
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Shape Functions
N1 and N2 are called Shape Functions or
Interpolation Functions. They express the shape
of the assumed displacements. N1 1 N2 0 at
node 1 N1 0 N2 1 at node 2 N1 N2 1
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N1
1
2
L
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N2
1
2
L
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N1
N2
1
2
L
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Step 3 - Define Strain/Displacement and
Stress/Strain Relationships
T - tensile force ? - total elongation
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Deformed Linear Spring Element
k
1
2
L
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Step 4 - Derive the Element Stiffness Matrix and
Equations
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Step 5 - Assemble the Element Equations to Obtain
the Global Equations and Introduce the B.C.
Note not simple addition!
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Step 6 - Solve for Nodal Displacements
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Step 7 - Solve for Element Forces
Once displacements at each node are known, then
substitute back into element stiffness
equations to obtain element nodal forces.
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Two Spring Assembly
2
1
3
x
F3x
F2x
k1
k2
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Elements 1 and 2 remain connected at node 3. This
is called the continuity or compatibility
requirement.
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Nodal forces consistent with element force sign
convention.
2
3
1
F1x
F2x
F3x
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Assembly of K - An Alternative Look.
2
1
3
x
F3x
F2x
k1
k2
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Assembly of K
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Expand Local k matrices to Global Size
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Force Equilibrium
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Compatibility
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Boundary Conditions

• Must Specify B.C.s to prohibit rigid body
  motion.
• Two type of B.C.s
• Homogeneous - displacements 0
• Nonhomogeneous - displacements nonzero value

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Partitioning
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2
1
3
x
F3x
F2x
k1
k2
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Homogeneous B.C.s

• Delete row and column corresponding to B.C.
• Solve for unknown displacements.
• Compute unknown forces (reactions) from original
  (unmodified) stiffness matrix.

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Nonhomogeneous B.C.s
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Nonhomogeneous B.C.s
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Nonhomogeneous B.C.s

• Transfer terms associated with known d-o-f to
  RHS.
• Solve for unknown displacements.
• Compute unknown forces (reactions) from original
  (unmodified) stiffness matrix.

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Properties of K Matrix

• Symmetric - both element k and global K
• K is singular. Must apply B.C. to prohibit
  rigid body motion.
• Terms on main diagonal are positive Kii and kii

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EXAMPLE Three Spring Assembly
k22000 lb/in
k11000 lb/in
k33000 lb/in
2
4
1
3
x
2
5000 lb
1
3
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Element 1
1
909.1 lb
3
909.1 lb
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Element 2
2
909.1 lb
4
3
909.1 lb
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Element 3
3
4090.9 lb
2
4090.9 lb
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EXAMPLENonhomogeneous B.C.
k
k
k
k
4
2
3
x
1
5
1
2
3
4
d
k200 kN/m
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Element 1
1
1.0 kN
2
1.0 kN
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Three Spring Assembly
k2
3
k1
2
2
P
x
k3
2
1
4
1
2
3
Rigid Bar
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Free Body Diagram
1
2
1
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Free Body Diagram
P
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Free Body Diagram
2
3
3
2
2
4
4
3
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Matrix Form of Stiffness Equations
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Applying B.C.
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Solving for Global Forces
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Potential Energy Approach

• Equilibrium at minimum potential energy.
• Total potential energy defined as the sum of
  internal strain energy U and potential energy of
  external forces W.
• ?p U W

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System
F
x
k
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Force-Deformation Curve
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Stationary Value
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Stationary Values
G
maximum
neutral
minimum
x
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Stationary Value
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Principle of Minimum Potential Energy
Equilibrium occurs when the qi define a state
such that ??p 0 for arbitrary admissible
variations in ?q1 from the equilibrium state
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Admissible Variations in Displacements
An admissible variation is one in which the
displacement field satisfies the
boundary conditions and inter-element continuity.
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Admissible Displacements
u
Admissible Displacement Function u ?u
?u
Actual Displacement Function
x
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For Admissible Variations in Displacements
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For Admissible Variations in Displacements
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F
F
x
k
k 500 lb/in
x
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EXAMPLE
k22000 lb/in
k11000 lb/in
k33000 lb/in
2
4
1
3
x
2
5000 lb
1
3
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