static analysis: load factor analysis

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Static Analysis:Load Factor Analysis


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Objectives

The objective of this module is to introduce the equations and numerical methods used to determine if a system is stable.

Buckling is a failure mode, due to compressive components becoming unstable.

The type of analysis presented in this module is frequently called load factor, linear buckling, or eigenbuckling analysis.

Section II - Static Analysis

Module 8 - Load Factor Analysis


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Load Deflection Curve

The figure shows the force versus deflection curve of a system.

As the load is increased, the slope (tangent stiffness) progresses to a point at which it is zero.

When the stiffness is zero, the application of an infinitesimally small load will cause a very large displacement.

This type of behavior is typical of a system subjected to compressive loading.

F

u

0TK




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Governing EquationsSection II - Static Analysis


The governing equations for a static finite element analysis can be written as

.unbT RuK

The tangent stiffness matrix, , has three components

.321 KuKKKT

u

TK

321 and , KuKKwhere are the linear, displacement, and stress dependent contributions.

are the displacement increments.


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Governing Equations

is the unbalanced load array. It is the difference between two arrays.

is an array of external forces acting on the nodes. This array is obtained from the external virtual work term.

is an array of node forces associated with the stresses inside the body. This array is obtained from the internal virtual work term.

intRFR extunb

unbR

extF

intR




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Zero Tangent Stiffness Matrix

When the system is in equilibrium, the unbalanced load vector is equal to zero and the equations reduce to

There is the possibility that the system is unstable and that another equilibrium configuration exists that the system could move to without an increase in the external loads

The stress levels at which unstable configurations exist can be determined by introducing a Load Factor (l) into the above equation

0321 uKuKK

0321 uKuKK l




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Eigenvalue Problem

This equation is a homogeneous set of equations that has a solution only when the determinant of the coefficient matrix is zero.

The determinant is a polynomial that will be equal to zero only for specific values of l. The order of the polynomial is equal to the number of equations.

0det 321 l KuKK

l

det

l1

l2

l3

Characteristic Polynomial




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Load Factor Interpretation

If l is equal to 1, then the system is unstable as loaded.

If l is between 0 and 1, then the system is unstable at a load level less than the applied load.

If l is greater than 1, then the load can be increased by a multiple of l before it becomes unstable.

If l is equal to -1, the system is unstable if the load is reversed.

If l is between 0 and -1 then the system is unstable at a load level less than the applied load if the load is reversed.

If l is greater than, -1 then the system becomes unstable at a multiple of l when the load is reversed.

Only the lowest value of l is of practical interest.




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ExampleSection II - Static Analysis


1 inch wide x 12 inch long x 1/8 inch thick.Material - 6061-T6 aluminum.

Brick elements with mid-side nodes are used to improve bending accuracy through the thin section. 0.0625 inch element size.

Simulation is used to compute the load factor and buckled shape of the cantilevered beam that has an axial load of 5 lbs. located at its free end.

Fixed end boundary condition

5 lbs.


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Example – Analysis Parameters

Use Sparse matrix solver

Use BCS-LIB parallel sparse solver

Compute first five load factors (lowest is only one of practical interest)

Use all threads/cores during the solution (fastest)




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Example - Results

Buckled shape for lowest load factor.

Beam will buckle at a load that is 5.6 times that applied.

The buckling load is 5 lbs. x 5.6 load factor = 28 lbs.



Caution – initial distortions in the shape of the beam could make the buckling load be smaller.


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Summary

A method for determining the stability of a system based on the tangent stiffness matrix was introduced.

A load factor was introduced that acts like a safety factor to tell how far from being unstable the system is at an applied load level.

An example showing results for a simple system were presented.



static analysis: load factor analysis

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load factor analysispage

unbalanced load array

small load

static analysismodule

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unbalanced load vector

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