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)

LarsaOpenBrIM

Percent Difference

(Larsa vs Independent)

Percent Difference

(OpenBrIM vs Independent)

Vertical Displacements (inch)
Station=00.00000.00000.00000%0%
Station=12-0.1693-0.1693-0.16930%0%
Station=24-0.3331-0.3331-0.33310%0%
Station=36-0.4870-0.4870-0.48700%0%
Station=48-0.6270-0.6270-0.62700%0%
Station=60-0.7502-0.7502-0.75020%0%
Station=72-0.8541-0.8541-0.85410%0%
Station=84-0.9367-0.9367-0.93670%0%
Station=96-0.9967-0.9967-0.99670%0%
Station=108-1.0331-1.0331-1.03310%0%
Station=120-1.0453-1.0453-1.04530%0%
Bending Moment (lb-inch)
Station=0
00

Station=12
3420934209

Station=24
6280362803

Station=36
8639786397

Station=48
105570105567

Station=60
120840123839

Station=72
132670132674

Station=84
141460141463

Station=96
147520147515

Station=108
151060151055

Station=120
152220152220