Beam Design for Geometric Nonlinearities Jordan Radas Kantaphat Sirison Wendy Zhao

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

Design Overview Linear Nonlinear

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

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

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

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

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

Newton-Raphson Method Load Displacement

Newton-Raphson Method

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

Incremental Approximation With From With With Evaluated at midpoint geometry

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

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

Modified Newton-Raphson Incremental Newton-Raphson Initial-Stiffness Newton- Raphson

Displacement iteration As opposed to residual iteration

Ansys Features Predictor Line Search Option

Ansys Features Adaptive Descent

Design challenge: Olympic diving board L = 96in b = 19.625in h = 1.625in P = -2500lbs Al 2024 – T6 (aircraft alloy) E = 10500ksi v = .33 Yield Strength = 50ksi

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

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.

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.

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

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

Questions?