1. Beam Design for Geometric Nonlinearities
Jordan Radas
Kantaphat Sirison
Wendy Zhao

2. Premise Large deflection Linear assumptions no longer apply
Is necessary form many real life applications

3. Design Overview
Linear
Nonlinear

4. Geometric Nonlinearity Assumptions
Large deformation
Plane cross section remains plane
Linear elastic material
Constant cross section

5. Kinematics
Exactly the same as what we showed in class but without the small angle approximation
Location of particle at deformed configuration relative to displacement and original configuration

6. Kinematics Green Lagrange Strain Tensor
Characterize axial strain, shear strain and curvature in terms of the derivatives of the displacement

7. Strain Displacement Matrix
[B]
The components of the strain displacement matrix can be determined explicitly by differentiation.
Converting u’x, u’y to theta and converting theta to ux1, ux2, uy1, uy2
where

8. Tangent Stiffness Matrix
[K][d] R
Through discretization and linearization of the weak form

10. Newton-Raphson Method
Load
Displacement

11. Newton-Raphson Method

12. Restoring load
Corresponds to element internal loads of current stress state.
Definition of deformation gradient
[Bi] is the strain-displacement matrix in terms of the current geometry {Xn} and [Di] is the current stress- strain matrix. The deformation gradient can be separated into a rotation and a shape change using the right polar decomposition theorem:
From right polar decomposition theorem
Spatial Decomposition

13. Incremental Approximation
With
From
With
With
Evaluated at midpoint geometry

14. Nonlinear solution levels
Load steps:
Adjusting the number of load steps account for:
abrupt changes in loading on a structure
specific point in time of response desired
Substeps:
Application of load in incremental substeps to obtain a solution within each load step
Equilibrium Iterations:
Set maximum number of iterations desired

15. Substeps Equilibrium iterations performed until convergence
Opportunity cost of accuracy versus time
Automatic time stepping feature
Chooses the size and number of substeps to optimize
Bisections method
Activates to restart solution from last converged step if a solution does not converge within a substep

16. Modified Newton-Raphson
Incremental Newton-Raphson Initial-Stiffness Newton- Raphson

17. Displacement iteration
As opposed to residual iteration

18. Ansys Features
Predictor
Line Search Option

19. Ansys Features
Adaptive Descent

20. Design challenge: Olympic diving board
L = 96in
b = in
h = 1.625in
P = -2500lbs
Al 2024 – T6 (aircraft alloy)
E = 10500ksi
v = .33
Yield Strength = 50ksi

21. Solid Beam: Linear/nonlinear
Mesh Size
Linear
Nonlinear
.125in
8.1043in
2.1671
2.2383
8.1082in
2.1757
2.2442
.25in
8.0255in
2.2170
2.2675
7.9951in
2.2254
2.2749
.5in
8.0566in
2.3051
2.3144
8.0282in
2.3151
2.3195

22. Optimization problem
ANSYS Goal Driven Optimization is used to create a geometry where hole diameter is the design variable.
Goals include minimizing volume and satisfying yield strength criterion.

23. Optimization and element technology
Optimization samples points in the user specified design space.
The number of sampling points is minimized using statistical methods and an FEA calculation is made for each sample.
Samples are chosen based on goals set for output variables, such as volume and safety factor.

24. Optimization results Problem Method Volume Deflection Max Tensile
Von Mises
Solid Beam
Linear
3061.5in3
8.0566in
2.3051
2.3144
Nonlinear
8.0282in
2.3151
2.3195
Optimized Beam
2719.5in3
8.7993in
1.2754
1.2555
8.7576in
1.2756
1.2757

25. Conclusion
Analysis serves as a proof of concept that real-world situations involving large structural displacements benefit from nonlinear modeling considerations
Extra computing power and time is worth it
Recommendations/suggestions

26. Questions?