L13: Cylindrical Roof Static Analysis

A reinforced concrete cylindrical roof (barrel vault) is subjected to a uniform load. It has a length of 50 feet, a thickness of 0.25 feet, a radius of 25 feet and an 80 degree arc length. A 90 lb/ft2 uniform downward load is to be applied.

Determine the vertical displacement at the center of the free edge i.e. at joint 49 of the model.

Due to symmetry of the structure and loading, a quarter of the cylindrical shell (as shown above) is used as a model. It consists of 6 x 6 mesh of shell elements.

The edge from joint 1 to 7 is simply supported and from 7 to 49 is free.

The boundary conditions along the edges from joint 1 to 43 and 43 to 49 are prescribed for symmetry. Here are the joint restraints specified for preserving symmetry:

Joints 1 to 7 XT YT ZR

Edge Joints 1 to 43 XT YR ZR

Joints 43 to 49 ZT XR YR

The material property of the shell is:

Modulus of Elasticity = 432,000,000 psf

Poisson's Ratio = 0.00

Shear Modulus = 216,000,000 psf

The uniform downward load of 90 lb/ft2 is equivalent to specifying a gravity loading in Global Y-direction with:

Weight Density = 360 lb/ft3

For a shell thickness of 0.25 ft and weight density of 360 lb/ft3, this combination gives the desired 90 lb/ft2 uniform loading in Global Y-direction.

The input for the model should be generated using a cylindrical coordinate system.

Reference

MacNeal, R.H. and Harder, R.C., "A Proposed Standard Set of Problems to Test Finite Element Accuracy," Finite Elements in Analysis and Design 1 (1985), pp.3-20, North-Holland.
Zienkiewicz, O. C., "The Finite Element Method," McGraw-Hill, 1977.

Result Comparison

Axial Z-Displacement in feet at the Support Joints 1-7	SAP90	Theory	LARSA	OPENBRIM	PERCENT DIFFERENCE (LARSA vs. THEORY)	PERCENT DIFFERENCE (OPENBRIM vs. THEORY)
Location=0.00(deg)	0.0000	0.0004	0.0000	0.0000	-	-
Location=6.67(deg)	0.0005	0.0009	0.0005	0.0005	80%	80%
Location=13.33(deg)	0.0018	0.0020	0.0018	0.0018	11%	11%
Location=20.00(deg)	0.0029	0.0030	0.0029	0.0029	3.45%	3.45%
Location=26.67(deg)	0.0024	0.0021	0.0024	0.0024	12.5%	12.5%
Location=33.33(deg)	-0.0017	-0.0016	-0.0017	-0.0017	5.88%	5.88%
Location=40.00(deg)	-0.0118	-0.0120	-0.0118	-0.0118	1.69%	1.69%

Vertical Y-Displacement in feet at the Center Joints 43-49	SAP90	Theory	LARSA	OPENBRIM	PERCENT DIFFERENCE (LARSA vs. THEORY)	PERCENT DIFFERENCE (OPENBRIM vs. THEORY)
Location=0.00(deg)	0.046	0.045	0.047	0.0463	4.26%	2.81%
Location=6.67(deg)	0.031	0.027	0.031	0.0309	12.9%	12.6%
Location=13.33(deg)	-0.013	-0.018	-0.013	-0.0131	38.5%	37.44%
Location=20.00(deg)	-0.078	-0.082	-0.079	-0.0786	3.8%	4.3%
Location=26.67(deg)	-0.155	-0.155	-0.157	-0.1557	1.3%	0.45%
Location=33.33(deg)	-0.234	-0.241	-0.236	-0.2341	2.1%	2.9%
Location=40.00(deg)	-0.307	-0.309	-0.309	-0.3072	0%	0.6%