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2020-07-26

VE 0018 | Plastic Bending – Tapered Cantilever

Description

A tapered cantilever is fully fixed on the left end and subjected to a continuous load q. Small deformations are considered and the self-weight is neglected in this example. The problem is described by the following set of parameters. Determine the maximum deflection u_z,max.

Material	Elastic-Plastic	Modulus of Elasticity	E	210000.000	MPa
		Poisson's Ratio	ν	0.000	-
		Shear Modulus	G	105000.000	MPa
		Yield Strength	f_y	40.000	MPa
Geometry	Cantilever	Length	L	4.000	m
		Width	w	0.005	m
		Left Side Height	h₁	0.250	m
		Right Side Height	h₂	0.150	m
Load		Continuous Load	q	2300.000	N/m

01 Plastic Bending – Tapered Cantilever

Analytical Solution

This is more complex variant of the verification example 17. The tapered cantilever is considered in this case. The continuous load q causes the elastic-plastic state of the plate. The calculation procedure is similar to verification example 17.

02 Bending Stress Distribution

The elastic-plastic moment M_ep (internal force) has to equal to the bending moment M (external force). The curvature κ_p in the elastic-plastic zone results from this equality.

κ_{p} = \frac{2 f_{y}}{E \sqrt{3}} \frac{1}{\sqrt{{h (x)}^{2} - \frac{2 q (L - x)^{2}}{{wf}_{y}}}}

Curvature in Elastic-Plastic Zone

The total deflection of the structure is defined as a superposition of the elastic-plastic and the elastic contribution using the Mohr's integral.

u_{z, \max} = \int_{0}^{x_{p}} κ_{p} (L - x) dx + \int_{x_{p}}^{L} κ_{e} (L - x) dx = 27.908 + 58.091 = 85.999 mm

Maximum Deflection of Cantilever

RFEM Settings

Modeled in RFEM 5.26 and RFEM 6.02
The element size is l_FE=0.020 m for files 0018.01-0018.03 and l_FE=0.005 m for files 0018.04-0018.05
Geometrically linear analysis is considered
The number of increments is 10
Shear stiffness of the members is neglected

Results

Model	Analytical Solution	RFEM 5		RFEM 6
Model	u_z,max [mm]	u_z,max [mm]	Ratio [-]	u_z,max [mm]	Ratio [-]
Isotropic Plastic 1D	85.999	86.215	1.003	86.139	1.002
Isotropic Nonlinear Elastic 2D, Plate		86.566	1.007	86.431	1.005
Isotropic Plastic 2D/3D, Plate		84.142	0.978	84.142	0.978
Isotropic Nonlinear Elastic 2D, Plate, Variable Thickness		83.728	0.974	83.121	0.967
Isotropic Plastic 2D/3D, Plate, Variable Thickness		83.088	0.966	83.088	0.966
Isotropic Nonlinear Elastic 1D		86.215	1.003	86.136	1.002

References

Lubliner, J. (1990). Plasticity Theory. New York: Macmillan.

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