Stiffness Matrices

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330:155g Finite Element Analysis

Nageswara Rao Posinasetti

2January 23, 2008 Rao, P.N.

2 Stiffness Matrices

Review Matrix Algebra given in App A.Direct stiffness method is used which simple to understandThis can be used for spring, bar and beam elements.


2.1 Spring element (1-dim)

Parts are 3DSome times 1D yields results that can be applied to 3D under certain circumstancesUse one dimensional spring elementObeys Hooke’s law

Deflection is linearly proportional to the force within the spring divided by the spring ratef = k u

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u – Displacement

F - ForceF = k u

Nodal point


2.2 A single spring element

Stiffness matrix for the spring elementSpring rate, k


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[K] – Stiffness matrix


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Example 2.1

5 -10 = -fi

-5 +10 = -fj

fi = 5fj = -5


Let us consider an example with two springs

⎥⎦

⎤⎢⎣

⎡−

−

11

11

kkkk and

⎥⎦

⎤⎢⎣

⎡−

−

22

22

kkkk

5


2.3 Assembling Total Structure’s stiffness matrix

Sum of the internal forces should be equal to the external forces applied at each nodek1 u1 – k1 u2 = F1

-k1 u1 + k1 u2 + k2 u2 – k2 u3 = F2

-k2u2 + k2 u3 = F3


⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

−−+−

−

3

2

1

3

2

1

22

2211

11

0

0

FFF

uuu

kkkkkk

kk

[ ]{ } { }FuK =

Or more compactly as


Number of rows = number of degrees of freedom

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⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−+−

−+−−

4

3

2

1

4

3

2

1

33

3322

2211

11

000

000

FFFF

uuuu

kkkkkk

kkkkkk


⎪⎪⎪

⎭

⎪⎪⎪

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⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

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⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−+−

−+−−+−

−

5

4

3

2

1

5

4

3

2

1

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2211

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kk


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Interchange rows as well as columns in the same sequence as the node element sequence


Bandwidth

Such that the stiffness values (non-zero elements) concentrated closer to the diagonalBandwidth refers to the number of terms we must move away from the main diagonal before we encounter all zeroes.

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2.4 Boundary conditions

These are the restrained movements of the nodal points

Homogeneous typeFixed

Non-homogeneous typeSpecified displacement


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⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

+−−−+−−

−

4

3

2

1

4

3

2

1

3123

2211

33

11

00

0000

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uuuu

kkkkkkkkkk

kk


10




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What is a DOF?

The unknowns in a finite element problem are referred to as degrees of freedom (DOF).Degrees of freedom vary by element and analysis type.

ThermalHeat Flow Rate

TemperatureStructuralForceDisplacementApplicationActionDOF Type

Courtesy Algor Inc, Pittsburgh


What is a DOF?

Node

Uy

Rot x

Rot y

UzRot z

Ux


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Node

A node is a coordinate location in space where the DOF are defined. The DOF of this point represent the possible response at this point due to the loading of the structure.



Element

An element is a mathematical relation that defines how the DOF of a node relate to the next. These elements can be lines (beams), areas (2-D or 3-D plates) or solids (bricks and tetrahedrals).



Nodes and ElementsA node has a given set of DOF, which characterize the response. For structural analyses, these DOF include translations and rotations in the three global directions.The type of element being used will also characterize which type of DOF a node will have.Some analysis types have only one DOF at a node. Examples of these analysis types are temperature in a heat transfer analysis and velocity in a fluid flow analysis.


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Element ConnectivityElements can only transfer loads to one another via common nodes.

No CommunicationBetween the Elements

CommunicationBetween the Elements



Stress and Strain Review

The basic stress and strain equations:

σ = FA

ε = σE

δ = FL ΑE

δ =0

Lε dx



StressBasic equations do not require the use of a computer to solve.Computer-based analysis is needed when complexity is added as follows:

Geometric complexity makes the elasticity equation difficult or impossible to solve.Variations in material properties exist throughout the part.Multiple load cases and complex or combined loading exists.Dynamics are of interest.


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General Case

The DOF components of each element combine to form a matrix equation:

[K] {d} = {A}

[K] = element stiffness components{d} = DOF results (unknown){A} = action value (e.g., force, temperature)



Structural FEA Equation

To determine the displacement of a simple linear spring under load, the relevant equation is:

{f} = [K] {d}

Known Unknownwhere {f} = force vector

[K] = stiffness matrix{d} = displacement vector



FEA Equation Solution

This can be solved with matrix algebra by rearranging the equation as follows:

{d} = [K] {f}-1


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Calculation of σ and ε

Strains are computed based on the classical differential equations previously discussed.Stress can then be obtained from the strains using Hooke’s law (F = kx).



Dynamic EquationFor a more complex analysis, more terms are needed. This is true indynamic analysis, which is defined bythe following equation:

{f} = [K] {d} + [c] {v} + [m] {a}where {f} = force vector

[K] = stiffness matrix{d} = displacement vector[c] = damping matrix{v} = velocity vector[m] = mass matrix{a} = acceleration vector



Other Applications

FEA can be applied to a wide variety of applications such as:

DynamicsNonlinear MaterialsHeat TransferFluid FlowElectrostaticsPiping Design and Analysis


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Questions/ Comments?

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Stiffness Matrices