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LINEAR BUCKLING ANALYSIS

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Buckling refers to sudden large displacements due to compressive loads. Slender structures subject to axial loads

can fail due to buckling at load levels lower than those required to cause material failure.

Buckling can occur in different modes under the effect of different load levels. In most cases, only the lowest

buckling load is of interest.

To grasp the concept of buckling, note that any structural load affects structural stiffness. Tensile loads induce a

positive stress stiffness which gets added to the elastic stiffness of the structure (also called shape stiffness).

A compressive load induces a negative stress stiffness which gets subtracted from the elastic stiffness of the

structure.

Buckling takes place when, as a result of subtracting the stress stiffness induced by compressive load from elastic

stiffness, the resultant structures stiffness drops to zero.

This is analogous to modal analysis where the inertial stiffness is subtracted from the elastic stiffness also

producing a zero resultant stiffness.

MECHANISM OF BUCKLING

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MECHANISM OF BUCKLING

The cancellation of resultant stiffness can be described by equation:

Eigenvalue multiplied by the applied load gives the critical loading

The first mode and its associated magnitude of buckling force is the most important because buckling most often

causes catastrophic failure or renders the structure unusable even if the structure can still withstand the load in its

buckled shape.

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Buckling can be thought of as a situation where a very small increase in the load causes very large displacements.

Buckling analysis, which is more precisely called linear buckling analysis, calculates that load, called buckling load, and the

shape assumed under the buckling load. However, linear buckling analysis, does not offer any quantitative information on

the deformed post-buckling shape.

Linear buckling analysis just finds the eigenvalues of structure for given loads and restraints disregarding any imperfections

and nonlinear effects which always exist in real structures. Those imperfections and non-linear effects very significantly

lower the buckling loads as compared to those predicted by linear buckling analysis.

For this reason, the results of linear buckling analysis must be interpreted with caution remembering that real buckling load

may be very significantly lower than that predicted by linear buckling analysis.

Nonlinear buckling analysis must be used to find accurate values of buckling load as well as to study post-bucking effects.

Some buckling problems that always require nonlinear buckling analysis and can not even be approximated by linear

buckling analysis include: inelastic or nonlinear material behavior prior to instability, re-alignment of applied pressure during

displacement or finite displacements prior to buckling.

Buckling should always be considered as potential mode of failure in structure consists of slender members in

compression. In fact many structural disasters are initiated by buckling and only the final destruction is caused by excessive

stresses in post buckling stage.

LINEAR VS. NONLINEAR BUCKLING

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BUCKLING LOAD FACTOR

cr

app

cr

app

PBLF =

P

P - critical load

P - applied load

The buckling load safety factor BLF is expressed by a number by which the applied load must be multiplied

in order to obtain the buckling load magnitude.

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COLUMN

Model file COLUMN.sldprt

Model solid

Material 1060 alloy

Restraints edge support

Load 1000 N compressive load

Objective

• calculate buckling load and buckling load factor

• analyze several modes of buckling

Split line restrained in all directions

1,000 N compressive load to split line

Split line restrained in y direction

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FEA results

Load factor 1.576 FBUCKLING 1.576 * 1,000 N = 1,576 N

COLUMN

Analytical results

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2BUCKLING

E IF

l

E = 6.9*105 MPa

I = 208.33 mm4

L = 300mm

FBUCKLING = 1,576 N

First buckling mode

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support

2500N

I BEAM

Model file I BEAM.sldprt

Model solid

Material Alloy steel

Restraints as shown

Load as shown

Objective

• calculate safety factor to yield

• calculate safety factor to buckling

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100 N vertical load

All legs can slide

PLASTIC TABLE

Model file PLASTIC TABLE.sldprt

Model shell

Thickness 2mm

Material ABS

Restraints as shown

Load 100N vertical load

Objective

• meshing on faces of solid geometry

• analysis of buckling load

• calculate static load safety factor

• exercise proper support definition

• soft springs solution option

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PLASTIC TABLE

Surface geometry Buckling analysis resultsSolid geometry Shell element mesh