LINEAR BUCKLING ANALYSIS
1
MECHANISM OF BUCKLING
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.
2
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.
3
LINEAR VS. NONLINEAR BUCKLING
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.
4
BUCKLING LOAD FACTOR
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.
Pcr
BLF =
Papp
Pcr - critical load
Papp - applied load
5
COLUMN
Model file
COLUMN.sldprt
Model
solid
Material
1060 alloy
Restraints
edge support
Load
1000 N compressive load
Split line restrained in all
directions
Objective
• calculate buckling load and buckling load factor
• analyze several modes of buckling
Split line restrained
in y direction
1,000 N compressive
load to split line
6
COLUMN
First buckling mode
Analytical results
FBUCKLING 
FEA results
2 EI
l2
Load factor 1.576
FBUCKLING
1.576 * 1,000 N = 1,576 N
E = 6.9*105 MPa
I = 208.33 mm4
L = 300mm
FBUCKLING = 1,576 N
7
I BEAM
support
Model file
I BEAM.sldprt
Model
solid
Material
Alloy steel
Restraints
as shown
Load
as shown
2500N
Objective
• calculate safety factor to yield
• calculate safety factor to buckling
8
PLASTIC TABLE
Model file
PLASTIC TABLE.sldprt
Model
shell
Thickness
2mm
Material
ABS
Restraints
as shown
Load
100N vertical load
100 N vertical load
Objective
• meshing on faces of solid geometry
• analysis of buckling load
• calculate static load safety factor
• exercise proper support definition
All legs can slide
• soft springs solution option
9
PLASTIC TABLE
Solid geometry
Surface geometry
Shell element mesh
Buckling analysis results
10