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2025-04-22

VE0035 | Mixed Dimensional Coupling

Description

Prove that coupling of different dimensional elements doesn't affect the results. A cantilever with a rectangular cross-section is fixed at one end and loaded at the other with forces F_x and F_z. Neglecting its self-weight and assuming only small deformations, determine cantilever's maximum axial deflection u_x, transversal deflection u_z, and total deflection u.

Material	Linear Elastic	Modulus of Elasticity	E	200000.000	MPa
Material	Linear Elastic	Poisson's Ratio	ν	0.000	-
Geometry	Cantilever	Length	L	1.000	m
		Width	b	100.000	mm
		Height	h	10.000	mm
Load		Axial Force	F_x	1000.000	kN
Load		Transversal Force	F_z	0.100	kN

Mixed Dimensional Coupling

Analytical Solution

Deflection of the cantilever with a rectangular cross-section in the axial (X) direction can be calculated according to the following formula:

u_{x} = \frac{F_{x} L}{E b h} = 5.000 m m

Axial Deflection

Deflection in the transversal direction (Z), including shear effect, can be calculated according to the following formula:

u_{z} = \frac{4 F_{z} L^{3}}{E b h^{3}} + β \frac{2 F_{z} L}{E b h} = 20.001 mm

β	Parameter dependent on the shape of the cross-section

Transversal Deflection

Parameter β is for the rectangular cross-section equal to 1.2. Finally, the total deflection of the cantilever can be calculated.

u = \sqrt{u_{x}^{2} + u_{z}^{2}} = 20.617 m m

Total Deflection of the Cantilever

RFEM Settings

Modeled in version RFEM 5.26 and RFEM 6.06
The element size is l_FE= 0.010 m
Geometrically linear analysis is considered
The Mindlin plate theory is used
Isotropic linear elastic material model is used
Shear stiffness of members is activated

Results

Quantity	Analytical Solution	RFEM 6 (Plate-Member)		RFEM 6 (Solid-Member)		RFEM 6 (Solid-Plate)
Quantity	[mm]	[mm]	Ratio [-]	[mm]	Ratio [-]	[mm]	Ratio [-]
u_x	5.000	5.000	1.000	5.018	1.004	5.000	1.000
u_z	20.001	20.001	1.000	20.002	1.000	20.002	1.000
u_z	20.617	20.616	1.000	20.622	1.000	20.617	1.000

Quantity	Analytical Solution	RFEM 5 (Plate-Member)		RFEM 5 (Solid-Member)		RFEM 5 (Solid-Plate)
Quantity	[mm]	[mm]	Ratio [-]	[mm]	Ratio [-]	[mm]	Ratio [-]
u_x	5.000	5.000	1.000	5.018	1.004	5.000	1.000
u_z	20.001	20.001	1.000	20.002	1.000	20.002	1.000
u_z	20.617	20.617	1.000	20.621	1.000	20.617	1.000

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Verification Example 0035 | 1

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