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2020-11-29

VE0033 | Plate with Hole

Description

The wide plate with a hole is loaded in one direction by means of the tensile stress σ according to the following sketch. The plate width is large with respect to the hole radius and it is very thin, considering the state of the plane stress.

Material	Steel	Modulus of Elasticity	E	210000.000	MPa
Material	Steel	Poisson's Ratio	ν	0.270	-
Geometry		Plate Width	2L	800.000	mm
		Plate Thickness	t	1.000	mm
		Hole Radius	R	20.000	mm
Load		Tension	σ	100.000	MPa

Determine the radial stress σ_r, tangential stress σ_θ, and shear stress τ_rθ in test points A and B according to Figure 1. The plate is considering infinite wide. Thus the plate modeled in RFEM 5 and RFEM 6 is relatively large.

01 Plate with Hole

Analytical Solution

The analytical solution of the stress state can be determined by means of the Airy stress function Φ, which is defined in the state of plane stress. The Airy stress function Φ in polar coordinates is defined as follows:

σ_{r} = \frac{1}{r} \frac{\partial ϕ}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} ϕ}{\partial θ^{2}}

r	Radial coordinate
θ	Tangential coordinate
Φ	Airy stress function

Radial Stress

σ_{θ} = \frac{\partial^{2} ϕ}{\partial θ^{2}}

Tangential Stress

τ_{r θ} = - \frac{\partial}{\partial r} (\frac{1}{r} \frac{\partial ϕ}{\partial θ})

Shear Stress

The radial stress σ_r and tangential stress σ_θ in the hole proximity result as follows:

σ_{r} = \frac{σ}{2} (1 - \frac{r^{2}}{R^{2}}) + \frac{σ}{2} (1 + \frac{3 r^{4}}{R^{4}} - \frac{4 r^{2}}{R^{2}}) \cos (2 θ)

Radial Stress

σ_{θ} = \frac{σ}{2} (1 + \frac{r^{2}}{R^{2}}) - \frac{σ}{2} (1 + \frac{3 r^{4}}{R^{4}}) \cos (2 θ)

Tangential Stress

τ_{r θ} = - \frac{σ}{2} (1 + \frac{3 r^{4}}{R^{4}} + \frac{2 r^{2}}{R^{2}}) \sin (2 θ)

Shear Stress

The specific results for test points A and B are listed in the result table below.

RFEM Settings

Modeled in RFEM 5.05 and RFEM 6.01
The global element size is l_FE = 0.005 m
The mesh refinement (circular) is used, l_FE = 0.001 m
Geometrically linear analysis is considered
The number of increments is 5
Plate entity is used

Results

Test Point A	Analytical Solution	RFEM 5	Ratio	RFEM 6	Ratio
σ_r [MPa]	0.000	2.449	-	2.632	-
σ_θ [MPa]	300.000	300.529	1.002	300.753	1.003
τ_rθ [MPa]	0.000	-0.002	-	-0.001	-

Test Point B	Analytical Solution	RFEM 5	Ratio	RFEM 6	Ratio
σ_r [MPa]	0.000	-1.753	-	-1.828	-
σ_θ [MPa]	-100.000	-100.216	1.002	-100.398	1.004
τ_rθ [MPa]	0.000	0.000	-	0.000	-

The resultant stress fields around the hole are demonstrated in the following figures. In RFEM 5 and RFEM 6, the values of stresses σ_r,σ_θ, and τ_rθ are read from the values of σ_x, σ_y, and τ_xy in the appropriate point and directions.

02 Plate with Hole – σ_x

03 Plate with Hole – σ_y

04 Plate with Hole – τ_xy

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