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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.
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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
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2.2 A single spring element
Stiffness matrix for the spring elementSpring rate, k
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3
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[K] – Stiffness matrix
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4
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Example 2.1
5 -10 = -fi
-5 +10 = -fj
fi = 5fj = -5
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Let us consider an example with two springs
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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
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FFF
uuu
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[ ]{ } { }FuK =
Or more compactly as
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Number of rows = number of degrees of freedom
6
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17January 23, 2008 Rao, P.N.
⎪⎪⎪
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⎩
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=
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7
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Interchange rows as well as columns in the same sequence as the node element sequence
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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.
8
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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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9
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⎪⎪⎪
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=
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3123
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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
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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.
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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).
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36January 23, 2008 Rao, P.N.
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
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38January 23, 2008 Rao, P.N.
Stress and Strain Review
The basic stress and strain equations:
σ = FA
ε = σE
δ = FL ΑE
δ =0
Lε dx
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39January 23, 2008 Rao, P.N.
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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40January 23, 2008 Rao, P.N.
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)
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41January 23, 2008 Rao, P.N.
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
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FEA Equation Solution
This can be solved with matrix algebra by rearranging the equation as follows:
{d} = [K] {f}-1
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43January 23, 2008 Rao, P.N.
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).
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44January 23, 2008 Rao, P.N.
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
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45January 23, 2008 Rao, P.N.
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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