| Applies To | |||
| Product(s): | STAAD.Pro | ||
| Version(s): | All | ||
| Environment: | N/A | ||
| Area: | Analysis Solutions | ||
| Subarea: | Buckling Analysis | ||
| Original Author: | Biswatosh Purkayastha | ||
BUCKLING ANALYSIS
In science, the buckling is a mathematical instability, leading to a failure mode before reaching the material strength.
Mathematically, this point is also defined as a point of Bifurcation to the solution of the Static equilibrium.
Buckling occurs lateral to the direction of load transfer in an element.
Common types of Buckling in Structural Engineering
Types of Buckling:
(3) Lateral Torsional Buckling
(4) Flexural Torsional Buckling
(3) Local Buckling
Common Buckling Modes
Perform Buckling Analysis in STAAD
Use the command PERFORM BUCKLING ANALYSIS in STAAD (See below)
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Both solvers in STAAD are solving this expression.
The Advanced solver uses the eigen-solution method,
The Basic solver uses a starting trial value for Buckling Factor
and refines it using the iterative method.
Calculation of Buckling Factors and Buckling Modes
The buckling refers to the loss of stability of a component and is usually independent of material strength. This loss of stability usually occurs within the elastic range of the material.
Stiffness matrix can be decomposed into two part
(1) First Order or primary stiffness matrix , K
(2) Second Order or Geometric Stiffness Matrix, Kg
(3) Resultant matrix, Ke
Ke = K +Kg
Kg matrix gets changed with the axial load applied P on along the element.
The more the Compressive load, the softer the material becomes to bear any lateral load.
The more the Tensile force , stiffer the material becomes against flexure.
So a stage will come when after certain incremental compressive force, the element has almost lost its strength or capacity to bear any slightest magnitude of lateral force and the element is susceptible to buckle due to elastic instability.
Mathematically,
“Ke” becomes "0"
|[K+Kg]| = 0
Please note that “Kg” is the function of axial load applied P on along the member.
If the Applied load P is Compressive in nature then P is taken negative
If the Applied load P is Tensile in nature then P is taken positive
More the Compressive load, the softer the material becomes to bear any lateral load.
More the Tensile force , stiffer the material becomes against flexure.
Rearranging the internal parameters of the matrix in the previous expression, considering “P” as compressive in nature
we get
|(K-lambda*KG')*Q| = 0
Lambda is a coefficient also known as BF (Buckling Factor) is the elastic critical buckling load factor is Pcr*L^2/EI
Q is the Buckling mode shape
The smallest BF is the critical one.
Considering the Buckling equation as shown above
if the Lambda <1, the element has already buckled
if the Lambda >1, the element has not yet buckled
if the Lambda <0, the element will buckle in the direction opposite to
the direction of the applied load
Buckling Modes
•SET BUCKLING MODES n ----(For n number of modes. This command is to be inserted before
FEW POINTS TO BE CONSIDERED
•The program can obtain only the load value that causes the overall structure to buckle.
Though this may be due to the buckling of the weakest member, but it doesn't tell you which member buckles.
Example of a Finite Element Model