Using non-linear analysis solver in GSA 1Advanced Features of GSA

P-delta & Buckling Analysis

Thomas Li

October 2009

Using non-linear analysis solver in GSA 2Advanced Features of GSA

P-delta and buckling analysis P-delta analysis

The fact: element stiffness is affected by the forces/moments within the elements Stabilising forces (e.g. tensile force in beam element)

strengthen the element stiffness

Destabilising forces (e.g. compressive force in beam element) weaken the element stiffness

The real stiffness of an element is the sum of linear and geometric stiffness

Using non-linear analysis solver in GSA 3Advanced Features of GSA

P-delta and buckling analysis P-delta analysis

Linear static analysis ignores geometric stiffness The element stiffness matrix is constructed assuming that

all the forces and moments of the elements are zero (linear stiffness matrix)

Linear and geometric stiffness matrixes Linear stiffness matrix – depends on element geometry &

material properties Geometric stiffness matrix – depends on the element

geometry & element internal forces

Using non-linear analysis solver in GSA 4Advanced Features of GSA

P-delta and buckling analysis P-delta analysis

Linear, p-delta & non-linear analyses Linear analysis

– only uses linear stiffness matrix– equilibrium condition is satisfied at non-deformed

geometry P-delta analysis

– uses both linear and geometric stiffness matrixes– No guarantee the equilibrium conditions are satisfied at

both deformed and non-deformed geometries, but the results are more closer to non-linear analysis compared with linear analysis

Non-linear analysis– uses both linear and geometric stiffness matrixes– Equilibrium condition is satisfied at deformed geometry

(the true equilibrium)

Using non-linear analysis solver in GSA 5Advanced Features of GSA

P-delta and buckling analysis P-delta analysis

Procedures of doing p-delta analysis Construct linear stiffness matrix Do linear analysis to obtain the internal forces/moments Construct geometric stiffness matrix according to the

element forces/moments obtained Combine the linear and geometric stiffness matrixes to

have the general stiffness matrix Do another linear analysis using the combined stiffness

matrix (as shown below) to obtain displacements and forces etc

FKK ge

Using non-linear analysis solver in GSA 6Advanced Features of GSA

P-delta and buckling analysis P-delta analysis

Results of p-delta analysis Better than linear analysis in general Not as good as true non-linear analysis Results are reliable only if displacements are relatively

small

Limitation of p-delta analysis The final displacements should relatively small compared

with the structure sizes Results are not combinable

Using non-linear analysis solver in GSA 7Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

For a single degree of freedom problem, the general stiffness becomes zero when the critical (buckling) load applies

For multiple degrees of freedom problem, the formula will be in matrix form, i.e.

Where: l is the buckling load factor

FFKK ge

Using non-linear analysis solver in GSA 8Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

Similar to SDF problem, the critical loads for MDF will make the determinant of the stiffness matrix be zero. Solving for l from following the following equation to obtain the buckling load factors {l}

0det ge FKK

Using non-linear analysis solver in GSA 9Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

Procedures of buckling analysisConstruct linear stiffness matrix

Do linear analysis to obtain element forces etc

Construct geometric stiffness matrix according to the forces obtained

Solve for factor l and mode shapes according to the equation shown above (eigen value analysis)

Using non-linear analysis solver in GSA 10Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

Particular data requirementsOnly bar, beam and 2D shell elements are

subjected to buckling in GSA If bar elements can be used (e.g. small section

elements and both ends are pinned), use bar elements rather than beam elements when global bucking modes are interested

Using non-linear analysis solver in GSA 11Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

Doing buckling analysis in GSA Through analysis wizard

Recognising an unsuccessful analysis Error norm not too large All modes converged, otherwise increase the number of

iterations or adjust advanced settings

Using non-linear analysis solver in GSA 12Advanced Features of GSA

P-delta and buckling analysis Buckling analysis

Interpreting results Load factors

Model stiffness

Model geometric stiffness

If there are negative load factors, ignore them for current applied loads. It may indicate that if the loads are reversed, that will be the buckling loads for this mode.