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27.06.2020

VE 0017 | Flexion plastique - Charge continue

Description du projet

Une plaque mince est entièrement fixée à l'extrémité gauche et soumise à une pression uniforme. 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.

Matériau	Elastic-Plastic	Module d’élasticité	E	210000.000	MPa
		coefficient de Poisson	ν	0.000
		Module de cisaillement	G	105000.000	MPa
		Limite d’élasticité	f_y	40,000	MPa
Géométrie	Plate	Périmètre	L	1.000	m
		Largeur	w	0.050	m
		Épaisseur	t	0.005	m
Load		Uniform Pressure	p	2.750	kPa

Flexion plastique - Charge continue

Analytical Solution

The quantities of the load are discussed at first. The moment M_e when the first yield is occurred and the ultimate moment M_p when the structure becomes plastic hinge are calculated as follows:

M_{e} = 2 \int_{0}^{t / 2} σ (z) zw dz = 2 \int_{0}^{t / 2} \frac{2 f_{y}}{t} z^{2} w dz = \frac{f_{y} {wt}^{2}}{6}

Moment élastique

M_{p} = 2 \int_{0}^{t / 2} σ (z) zw dz = 2 \int_{0}^{t / 2} f_{y} zw dz = \frac{f_{y} {wt}^{2}}{4}

Moment plastique ultime

The plate is brought into the elastic-plastic state by the pressure p. The bending stress is defined according to the following formula:

σ_{x} (x, z) = - κ (x) Ez

Contrainte de flexion

where κ is the curvature. The elastic-plastic zone length is described by the parameter x_p. The bending stress quantity on the surface is equal to the plastic strength f_y at the point x_p, see the following schema.

Distribution des contraintes de flexion

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{1}{E} \sqrt{\frac{\frac{f_{y}^{3} w}{3}}{- \frac{q (L - x)^{2}}{2} + \frac{f_{y} t^{2} w}{4}}}

Courbure dans la zone élastique-plastique

u_{z, \max} = \int_{0}^{x_{p}} κ_{p} (L - x) dx + \int_{x_{p}}^{L} κ_{e} (L - x) dx = 83.117 + 83.117 = 166.234 mm

Flèche maximale du porte-à-faux

Paramètres RFEM

Modeled in RFEM 5.26 and RFEM 6.01
The element size is l_FE=0.020 m
In case of solid models mesh refinement across the thickness is used (6 elements per thickness)
Geometrically linear analysis is considered
The number of increments is 5
Shear stiffness of the members is neglected

Résultats

Model	Solution analytique	RFEM5		RFEM6
Model	u_z,max [mm]	u_z,max [mm]	Ratio [-]	u_z,max [mm]	Ratio [-]
Isotrope plastique 1D	166.234	166.214	1.000	166.018	0.999
Isotropic Plastic 2D/3D, Plate		162.987	0.980	162.960	0.980
Isotropic Nonlinear Elastic 2D/3D, Plate, von Mises		165.730	0.997	165.700	0.997
Isotropic Nonlinear Elastic 2D/3D, Plate, Tresca		166.998	1.005	166.969	1.004
Isotropic Plastic 2D/3D, Solid		160.601	0.966	162.429	0.977
Isotropic Nonlinear Elastic 2D/3D, Solid, von Mises		163.003	0.981	165.593	0.996
Isotropic Nonlinear Elastic 2D/3D, Solid, Tresca		168.725	1.015	169.691	1.021
Isotrope élastique non-linéaire 1D		166.214	1.000	166.018	0.999

Remarque : The results deviation is also caused by the difference between the analytical and numerically calculated torsional constant.

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Exemple de vérification 0017 | 1

Références

Licence, J. (1990). Théorie de la plasticité. New York : Macmillan, 1993

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