Description
A cantilever made of the material with different tensile and compressive plastic strength is fully fixed on the left end and loaded by a bending moment according to the following sketch. The problem is described by the following set of parameters. Small deformations are considered and the self-weight is neglected in this example. Determine the maximum deflection uz,max.
| Material | Elastic-Plastic | Modulus of Elasticity | E | 210000.000 | MPa |
| Poisson's Ratio | ν | 0.000 | - | ||
| Shear Modulus | G | 105000.000 | MPa | ||
| Tensile Plastic Strength | ft | 200.000 | MPa | ||
| Comprerssive Plastic Strength | fc | 280.000 | MPa | ||
| Geometry | Cantilever | Length | L | 2.000 | m |
| Width | w | 0.005 | m | ||
| Thickness | t | 0.005 | m | ||
| Load | Bending Moment | M | 6.000 | Nm | |
Analytical Solution
The cantilever is loaded by the bending moment M. Due to the different plastic strength in the tension and compression the neutral axis is not necessary coincident with the axis of the symmetry according to the following figure. The parameter z0 is introduced and it is defined so that σx(x,z0)=0, note that it changes during loading as well as parameters zt and zc. The bending stress is defined by the following formula:
To obtain the maximum deflection uz,max the curvature κ has to be solved. The elastic-plastic moment Mep (internal force) has to equal to the bending moment M (external force).
because of the unknown parameters zt, zc and z0 it is necessary to write further equations. The stresses in the interface between the elastic and plastic zones are defined as follows:
The last condition is defined by the equilibrium of the axial forces.
Solving these equations numerically, the curvature κ and the maximum deflection uz,max can be calculated. The result can be found in the following table.
RFEM Settings
- Modeled in RFEM 5.16 and RRFEM 6.06
- The element size is lFE= 0.020 m
- Geometrically linear analysis is considered
- The number of increments is 5
- Shear stiffness of the members is neglected
Results
| Material Model | Analytical Solution | RFEM 6 | RFEM 5 | ||
| uz,max [m] | uz,max [m] | Ratio [-] | uz,max [m] | Ratio [-] | |
| Orthotropic Plastic 2D | 1.272 | 1.277 | 1.004 | 1.277 | 1.004 |
| Isotropic Nonlinear Elastic 1D | 1.272 | 1.000 | 1.272 | 1.000 | |
| Nonlinear Elastic 2D/3D,Mohr - Coulomb, Plate | 1.283 | 1.009 | 1.283 | 1.009 | |
| Nonlinear Elastic 2D/3D,Drucker - Prager, Plate | 1.283 | 1.009 | 1.283 | 1.009 | |
| Isotropic Plastic 2D/3D,Mohr - Coulomb, Plate | 1.284 | 1.009 | 1.284 | 1.009 | |
| Isotropic Plastic 2D/3D,Drucker - Prager, Plate | 1.272 | 1.000 | 1.272 | 1.000 | |
| Nonlinear Elastic 2D/3D,Mohr - Coulomb, Solid | 1.308 | 1.028 | 1.307 | 1.028 | |
| Nonlinear Elastic 2D/3D,Drucker - Prager, Solid | 1.313 | 1.032 | 1.312 | 1.031 | |
| Isotropic Plastic 2D/3D,Mohr - Coulomb, Solid | 1.302 | 1.024 | 1.293 | 1.017 | |
| Isotropic Plastic 2D/3D,Drucker - Prager, Solid | 1.283 | 1.009 | 1.283 | 1.009 | |
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