The Mechanical APDL Verification Manual includes a variety of nonlinear analysis test cases:
VM7 - Plastic Compression of a Pipe Assembly
VM11 - Residual Stress Problem
VM24 - Plastic Hinge in a Rectangular Beam
VM38 - Plastic Loading of a Thick-Walled Cylinder Under Pressure
VM56 - Hyperelastic Thick Cylinder Under Internal Pressure
VM78 - Transverse Shear Stresses in a Cantilever Beam
VM80 - Plastic Response to a Suddenly Applied Constant Force
VM104 - Liquid-Solid Phase Change
VM124 - Discharge of Water from a Reservoir
VM126 - Heat Transferred to a Flowing Fluid
VM132 - Stress Relaxation of a Bolt Due to Creep
VM133 - Motion of a Rod Due to Irradiation Induced Creep
VM134 - Plastic Bending of a Clamped I-Beam
VM146 - Bending of a Reinforced Concrete Beam
VM185 - Current Carrying Ferromagnetic Conductor
VM198 - Large Strain In-Plane Torsion Test
VM199 - Viscoplastic Analysis of a Body Undergoing Shear Deformation
VM200 - Viscoelastic Sandwich Seal Analysis
VM218 - Hyperelastic Circular Plate
VM220 - Eddy Current Loss in Thick Steel Plate
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Where to Find Other Examples