**Geometrically nonlinear problems** in three-dimensional structural analysis variational formulation and step-by-step approach of their solution. Unconditionally-stable subtended difference scheme is applied. **Test problems** are presented.

**GEOMETRICALLY NONLINEAR PROBLEMS AFTER BUCKLING FAILURE**

Evzerov I.D., Doctor of Technical Sciences

**Annotation**

Geometrically nonlinear problems in three-dimensional structural analysis variational formulation and incremental method of their solution are being considered. Transition to corresponding bar and plate problems has been carried out. Geometrically nonlinear dynamic problems and difference scheme have been considered also. The following algorithm of solution of geometrically nonlinear problems after buckling failure has been suggested. Step-by-step approach is applied at first. If after particular step buckling failure has been determined, then corresponding dynamic problem with right-hand side is equal to zero are being solved. Initial conditions are defined accordingly to computed the first buckling mode. Unconditionally-stable subtended difference scheme is being applied. Using this method, equilibrium under buckling load is determined. Next, step-by-step approach is applied again. Test problems of hinged-rod system, circular arch and axially loaded cantilever are presented, which confirm algorithm effectiveness.

**Author's notes**

In the work presented here contains mathematical justification of solution methods of geometrically nonlinear problems, which are applied in SP LIRA 10. Bar's stability theory has been discussed with professor Slivker V.I., Doctor of Technical Sciences, and matches results obtained via other methods in monograph Perelmuter A.V., Slivker V.I. Structural equilibrium stability and related problems. - M.: SCAD SOFT, 2009.

**Contents list**

Symbols

Static problem and step-by-step approach

Dynamic problem and difference scheme

Post-buckling analysis

Test problems solving

Hinged-rod system 1

Hinged-rod system 2

Hinged-rod system 3

Hinged-rod system 4

Large displacements and buckling failure of pinched circular arch

Cantilever bending after buckling failure

Conclusions

Bibliography

**Introduction**

Equilibrium equations and step-by-step approach equations are presented for three-dimensional geometrically nonlinear problem, difference scheme for dynamic problem. Transition to bars and plates has been carried out. New algorithm of solution of geometrically nonlinear problems has been suggested. After buckling failure the corresponding dynamic problem with right-hand side is equal to zero are being solved, which gives opportunity to determinate equilibrium under the same load, that caused buckling failure of the structure. Initial conditions are defined accordingly to computed the first buckling mode. Unconditionally-stable subtended difference scheme is being applied. Presented algorithm has been implemented in SP LIRA 10. Test problems, which confirm algorithm effectiveness, are presented.