A simply supported beam is uniformly loaded and is on an elastic foundation. The beam cross section is rectangular with moment of inertia of 30 in^4. The elastic modulus of the beam is 30x10^6 psi. The span length is 240 inches. The uniform load is 43.4 lb/in. The foundation modulus (i.e. distributed reaction for a deflection of unity) is 26.041667 psi (lb/in^2).

Determine the transverse deflections and bending moments along the beam.

The model consists of 21 joints, 20 beam elements and 19 grounded springs using a 2D model in the global XZ plane. The degrees of freedom for Y translation, X and Z rotations can be deleted from the analysis. Joint 1 is a pin support and joint 21 is a roller support.

The material property data is given:

Modulus of Elasticity = 30,000,000 psi

and sectional property data as:

A = 7.11 in^2

Izz = 30 in^4

The member orientation angle is 90 degrees.

A distributed load of 43.4 lb/inch is applied in the vertical (Z)- direction as uniform beam load on all beam elements. Linear static analysis with one (1) load case is performed.

The foundation support is modeled using a set of foundation springs with stiffness of 312.50 lb/in in the vertical (Z) direction. The spring constant is computed as the product of the foundation modulus and tributary area for each joint. At all joints the tributary area is 12 inches (width of the beam) by 1 inch (unit length). Therefore, the spring constant for a grounded spring element is:

k = (Foundation Modulus) x (Tributary Area)

k = (26.041667) x (12)

k = 312.50 lb/in

In this example, linear grounded spring elements are used and linear static analysis is performed. Since we know the solution will yield the vertical displacement as downward with soil resistance active at all joints, we can use linear static analysis. However, the foundations can be subject to loads causing uplifts and loss of soil contact. It is more appropriate to model the soil as compression-only foundation elements and perform nonlinear static analysis.

Tested Features

App

Object Link: https://openbrim.org/objidlnk1epipj4i6z9lr2gi2sj.project

Library

Object Link: https://openbrim.org/objidnx78iybf4yel5euxk69xl.libobj

Sources

Timoshenko, S. and Wionowsky-Krieger, S.,"Theory of Plates and Shells", McGraw-Hill Book Co., N.Y., 1959.
"EASE2 - Elastic Analysis for Structural Engineering - Example Problem Manual," Engineering Analysis Corporation, 1981, pp. 1.05.
LARSA 4D

Result Comparison

	Independent (Timoshenko)	Larsa	OpenBrIM	Percent Difference (Larsa vs Independent)	Percent Difference (OpenBrIM vs Independent)
Vertical Displacements (inch)
Station=0	0.0000	0.0000	0.0000	0%	0%
Station=12	-0.1693	-0.1693	-0.1693	0%	0%
Station=24	-0.3331	-0.3331	-0.3331	0%	0%
Station=36	-0.4870	-0.4870	-0.4870	0%	0%
Station=48	-0.6270	-0.6270	-0.6270	0%	0%
Station=60	-0.7502	-0.7502	-0.7502	0%	0%
Station=72	-0.8541	-0.8541	-0.8541	0%	0%
Station=84	-0.9367	-0.9367	-0.9367	0%	0%
Station=96	-0.9967	-0.9967	-0.9967	0%	0%
Station=108	-1.0331	-1.0331	-1.0331	0%	0%
Station=120	-1.0453	-1.0453	-1.0453	0%	0%
Bending Moment (lb-inch)
Station=0		0	0
Station=12		34209	34209
Station=24		62803	62803
Station=36		86397	86397
Station=48		105570	105567
Station=60		120840	123839
Station=72		132670	132674
Station=84		141460	141463
Station=96		147520	147515
Station=108		151060	151055
Station=120		152220	152220

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