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:
and sectional property data as:
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:
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
Independent1 (Timoshenko) | Independent2 (EASE2) | Larsa | OpenBrIM | Percent Difference (Larsa vs Independent1) | Percent Difference (OpenBrIM vs Independent1) | Percent Difference (Larsa vs Independent2) | Percent Difference (OpenBrIM vs Independent2) | |
|---|---|---|---|---|---|---|---|---|
| Vertical Displacements (inch) | ||||||||
| Station=0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0% | 0% | 0% | 0% |
| Station=12 | -0.1693 | -0.169 | -0.1693 | -0.1693 | 0% | 0% | 0.1775% | 0.1775% |
| Station=24 | -0.3331 | -0.332 | -0.3331 | -0.3331 | 0% | 0% | 0.3313% | 0.3313% |
| Station=36 | -0.4870 | -0.486 | -0.4870 | -0.4870 | 0% | 0% | 0.2058% | 0.2058% |
| Station=48 | -0.6270 | -0.625 | -0.6270 | -0.6270 | 0% | 0% | 0.32% | 0.32% |
| Station=60 | -0.7502 | -0.748 | -0.7502 | -0.7502 | 0% | 0% | 0.2941% | 0.2941% |
| Station=72 | -0.8541 | -0.852 | -0.8541 | -0.8541 | 0% | 0% | 0.2465% | 0.2465% |
| Station=84 | -0.9367 | -0.935 | -0.9367 | -0.9367 | 0% | 0% | 0.1818% | 0.1818% |
| Station=96 | -0.9967 | -0.995 | -0.9967 | -0.9967 | 0% | 0% | 0,1709% | 0,1709% |
| Station=108 | -1.0331 | -1.031 | -1.0331 | -1.0331 | 0% | 0% | 0,2037% | 0,2037% |
| Station=120 | -1.0453 | -1.043 | -1.0453 | -1.0453 | 0% | 0% | 0,2205% | 0,2205% |
| 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 | ||||||