Problem Description

In this example a straight cantilever beam, modeled with shell elements, is subjected to unit forces at the tip in the three orthogonal directions and unit moments at the tip about the three orthogonal directions, each in a different load case. The tip displacements in the direction of the load are compared with hand calculated results.

It is important to note that this example is an extreme case presented for testing and verification of the shell element. Shell elements are not in general intended for use in modeling a beam with a 2 to 1 depth-to-width ratio.

The basic geometry, properties and loading are as described in MacNeal and Harder 1985. The cantilever beam is 6 inches long, 0.2 inch wide parallel to the Z direction and 0.1 inch wide parallel to the Y direction. Five different models are created, each with a different mesh. Models A, B and C use a 6x1 mesh with rectangular-, trapezoidal- and parallelogram-shaped elements, respectively, as suggested in MacNeal and Harder 1985. Model D starts with the 6x1 rectangular mesh and then divides each rectangle into two triangles. Model E starts with the 6x1 rectangular mesh and then divides each rectangle into four triangles. The meshes used in models D and E are not included in MacNeal and Harder 1985.

Six load cases are created for each model. The six load cases apply a unit axial force, a unit in-plane force, a unit out-of-plane force, a unit twisting moment, a unit in-plane moment and a unit out-of-plane moment at the tip of the cantilever, respectively. The twisting moment is applied as a couple of Y direction forces. The in-plane moment is applied as a couple of X direction forces. The out-ofplane moment is applied as moments.

The independent solution is derived using elementary beam theory that assumes no local Poisson’s effect occurs at the support. The beam is modeled in this model to match this assumption. At the fixed end, joint 1 is restrained in the Ux, Uy, Uz and Rz degrees of freedom and joint 8 is restrained in the Ux, Uy and Rz degrees of freedom. Joint 8 is not restrained in the Uz degree of freedom to avoid imposing the unwanted local Poisson’s effect into the model. Also, when the beam is loaded with in-plane shear, an in-plane force equal to half the applied tip load is applied to joint 8 in the opposite direction of the tip load. This special load at joint 8 is applied to model the reaction without the Poisson’s effect.

Tested Features

Membrane analysis using shell elements
Plate bending analysis using shell elements
Effect of shell element aspect ratio
Effect of geometrical distortion of shell element from rectangular
Joint force loading

Result Comparision

App

Object Link: https://openbrim.org/objid8n4c4jwt55je6wvd7hulp.project

Library

Object Link: https://openbrim.org/objidqy5k37frzawgdwjugvidp.libobj

The independent results are hand calculated using the unit load method described on page 244 in Cook and Young 1985. In addition, the torsional stiffness of the section, J, is calculated using item 4 in Table 20 on page 290 in Roark and Young 1975. Independent results are also published in MacNeal and Harder 1985.

COMPARISION

Load Case And Type	Model And Element Shape	Output Parameter	OpenBrIM	SAP2000	Independent	Percent Difference (SAP2000 vs Independent)	Percent Difference (OpenBrIM vs Independent)
Load Case 1 Axial Extension	A- Rectangle	Ux Average of jts 7 and 14 in	0.00003	0.00003	0.00003	0.00%	0.00%
	B - Trapezoid		0.00003	0.00003		0.00%	0.00%
	C - Parallelogram		0.00003	0.00003		0.00%	0.00%
	D - Triangle 2		0.00003	0.00003		0.00%	0.00%
	E – Triangle 4		0.00003	0.00003		0.00%	0.00%
Load case 2 In-plane shear and bending	A- Rectangle	Uz Average of jts 7 and 14 in	0.09768	0.1072	0.1081	-1%	-9.64%
	B - Trapezoid		0.08817	0.0227		-79%	-18.44%
	C - Parallelogram		0.09755	0.0804		-26%	-9.76%
	D - Triangle 2		0.11573	0.0032		-97%	7.06%
	E – Triangle 4		0.00865	0.0066		-94%	-92.00%
Load case 3 Out-of-plane shear and bending	A- Rectangle	Uy Average of jts 7 and 14 in	0.43200	0.4320	0.43210	0%	0%
	B - Trapezoid		0.43227	0.4322		0%	0%
	C - Parallelogram		0.43219	0.4322		0%	0%
	D - Triangle 2		0.42844	0.4296		-1%	-1%
	E – Triangle 4		0.42900	0.4314		0%	-1%

Load Case And Type	Model And Element Shape	Output Parameter	OpenBrIM	SAP2000	Independent	Percent Difference (SAP2000 vs Independent)	Percent Difference (OpenBrIM vs Independent)
Load case 4 Twist	A- Rectangle	Uy Average of absolute values at jts 7 and 14 in	0.00233	0.00233	0.00341	-32%	-32%
	B - Trapezoid		0.00233	0.00233		-32%	-32%
	C - Parallelogram		0.00233	0.00233		-32%	-32%
	D - Triangle 2		0.00231	0.00231		-32%	-32%
	E – Triangle 4		0.00201	0.00230		-33%	-41.03%
Load case 5 In-Plane Moment	A- Rectangle	Ux Average of absolute values at jts 7 and 14 in	0.00082	8.990E-04	0.0009	0%	-8.89%
	B - Trapezoid		0.00074	1.395E-04		-85%	-17.78%
	C - Parallelogram		0.00082	7.225E-04		-20%	-8.89%
	D - Triangle 2		0.00097	0.265E-04		-97%	7.78%
	E – Triangle 4		0.00007	0.550E-04		-94%	-92.22%
Load case 6 Out-of-Plane Moment	A- Rectangle	Rz Average of values at jts 7 and 14 radians	0.036	0.036	0.036	0.00%	0.00%
	B - Trapezoid		0.036	0.036		0.00%	0.00%
	C - Parallelogram		0.036	0.036		0.00%	0.00%
	D - Triangle 2		0.036	0.036		0.00%	0.00%
	E – Triangle 4		0.036	0.036		0.00%	0.00%