# V. Beam on Elastic Foundation

To find deflection and stress at the center due to a uniform, static load on a simply supported beam on elastic foundation.

## Reference

Hand calculation using the following reference:

Peterson, F.E., Elastic Analysis for Structural Engineering (EASE2), Example Problem Manual, Engineering Analysis Corporation, Berkeley, CA, 1981.

## Problem

Find the vertical deflection and bending stress at the center of the beam.

• beam width, b = 1.0 in.
• beam depth, h = 7.114 in.

 E = 30×(10)6 psi

 Note: Due to symmetry, only half of the beam is modeled.

 L = 240 in.

 wu = 43.3 lb/in.

## Comparison

Table 1. Comparison of results
Bending stress, σ (psi) 18,052 18,053.29 none
Vertical deflection (in.) 1.0453 1.04549 none

The file C:\Users\Public\Public Documents\STAAD.Pro CONNECT Edition\Samples\ Verification Models\01 Beams\Beam on Elastic Foundation.STDis typically installed with the program.

STAAD SPACE :A SIMPLY SUPPORTED BEAM ON ELASTIC FOUNDATION
START JOB INFORMATION
ENGINEER DATE 18-Sep-18
END JOB INFORMATION
*
* REFERENCE: PETERSON, EASE2, EXAMPLE PROBLEM MANUAL
*
INPUT WIDTH 72
UNIT INCHES POUND
JOINT COORDINATES
1 0 0 0; 2 0 7.114 0; 3 0 0 6; 4 0 7.114 6; 5 0 0 12; 6 0 7.114 12;
7 0 0 18; 8 0 7.114 18; 9 0 0 24; 10 0 7.114 24; 11 0 0 30;
12 0 7.114 30; 13 0 0 36; 14 0 7.114 36; 15 0 0 42; 16 0 7.114 42;
17 0 0 48; 18 0 7.114 48; 19 0 0 54; 20 0 7.114 54; 21 0 0 60;
22 0 7.114 60; 23 0 0 66; 24 0 7.114 66; 25 0 0 72; 26 0 7.114 72;
27 0 0 78; 28 0 7.114 78; 29 0 0 84; 30 0 7.114 84; 31 0 0 90;
32 0 7.114 90; 33 0 0 96; 34 0 7.114 96; 35 0 0 102; 36 0 7.114 102;
37 0 0 108; 38 0 7.114 108; 39 0 0 114; 40 0 7.114 114; 41 0 0 120;
42 0 7.114 120;
ELEMENT INCIDENCES SHELL
1 1 2 4 3; 2 3 4 6 5; 3 5 6 8 7; 4 7 8 10 9; 5 9 10 12 11;
6 11 12 14 13; 7 13 14 16 15; 8 15 16 18 17; 9 17 18 20 19;
10 19 20 22 21; 11 21 22 24 23; 12 23 24 26 25; 13 25 26 28 27;
14 27 28 30 29; 15 29 30 32 31; 16 31 32 34 33; 17 33 34 36 35;
18 35 36 38 37; 19 37 38 40 39; 20 39 40 42 41;
ELEMENT PROPERTY
1 TO 20 THICKNESS 1
DEFINE MATERIAL START
ISOTROPIC MATERIAL1
E 3e+07
POISSON 0.290909
END DEFINE MATERIAL
CONSTANTS
MATERIAL MATERIAL1 ALL
SUPPORTS
1 FIXED BUT KFY 78.125
41 FIXED BUT FZ MX
2 FIXED BUT FY
3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 -
39 FIXED BUT FZ MX KFY 156.25
4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 FY 260.4
2 42 FY 130.2
PERFORM ANALYSIS
PRINT JOINT DISPLACEMENTS LIST 1 2
PRINT ELEMENT JOINT STRESSES LIST 1 2
FINISH

JOINT DISPLACEMENT (INCH RADIANS)    STRUCTURE TYPE = SPACE
------------------
JOINT  LOAD   X-TRANS   Y-TRANS   Z-TRANS   X-ROTAN   Y-ROTAN   Z-ROTAN
1    1    0.00000   1.04548   0.00000   0.00000   0.00000   0.00000
2    1    0.00000   1.04549   0.00000   0.00000   0.00000   0.00000
************** END OF LATEST ANALYSIS RESULT **************
47. PRINT ELEMENT JOINT STRESSES LIST 1 2
ELEMENT  JOINT    STRESSES LIST
:A SIMPLY SUPPORTED BEAM ON ELASTIC FOUNDATION           -- PAGE NO.    4
ELEMENT STRESSES    FORCE,LENGTH UNITS= POUN INCH
----------------
STRESS = FORCE/UNIT WIDTH/THICK, MOMENT = FORCE-LENGTH/UNIT WIDTH
ELEMENT  LOAD       SQX        SQY          MX          MY          MXY
VONT       VONB         SX          SY          SXY
TRESCAT    TRESCAB
1      1         0.00        0.00        0.00        0.00        0.00
37.23       37.23       35.30       -0.00       -6.82
37.85       37.85
TOP : SMAX=      36.57 SMIN=      -1.27 TMAX=      18.92 ANGLE=-10.6
BOTT: SMAX=      36.57 SMIN=      -1.27 TMAX=      18.92 ANGLE=-10.6
JOINT         0.00        0.00        0.00        0.00        0.00
1      18071.09    18071.09       35.32   -18053.40       -6.80
TOP : SMAX=      35.32 SMIN=  -18053.41 TMAX=    9044.36 ANGLE= -0.0
BOTT: SMAX=      35.32 SMIN=  -18053.41 TMAX=    9044.36 ANGLE= -0.0
JOINT         0.00        0.00        0.00        0.00        0.00
2      18035.75    18035.75       35.32    18053.38       -6.84
TOP : SMAX=   18053.38 SMIN=      35.31 TMAX=    9009.03 ANGLE=-90.0
BOTT: SMAX=   18053.38 SMIN=      35.31 TMAX=    9009.03 ANGLE=-90.0
JOINT         0.00        0.00        0.00        0.00        0.00
4      18035.79    18035.79       35.29    18053.40       -6.84
TOP : SMAX=   18053.41 SMIN=      35.28 TMAX=    9009.06 ANGLE=-90.0
BOTT: SMAX=   18053.41 SMIN=      35.28 TMAX=    9009.06 ANGLE=-90.0
JOINT         0.00        0.00        0.00        0.00        0.00
3      18071.05    18071.05       35.29   -18053.38       -6.80
TOP : SMAX=      35.29 SMIN=  -18053.38 TMAX=    9044.33 ANGLE= -0.0
BOTT: SMAX=      35.29 SMIN=  -18053.38 TMAX=    9044.33 ANGLE= -0.0
2      1         0.00        0.00        0.00        0.00        0.00
50.05       50.05       35.22       -0.00      -20.53
54.10       54.10
TOP : SMAX=      44.66 SMIN=      -9.44 TMAX=      27.05 ANGLE=-24.7
BOTT: SMAX=      44.66 SMIN=      -9.44 TMAX=      27.05 ANGLE=-24.7
JOINT         0.00        0.00        0.00        0.00        0.00
3      18001.93    18001.93       35.27   -17984.23      -20.48
TOP : SMAX=      35.29 SMIN=  -17984.25 TMAX=    9009.77 ANGLE= -0.1
BOTT: SMAX=      35.29 SMIN=  -17984.25 TMAX=    9009.77 ANGLE= -0.1
JOINT         0.00        0.00        0.00        0.00        0.00
4      17966.57    17966.57       35.27    17984.15      -20.58
TOP : SMAX=   17984.17 SMIN=      35.25 TMAX=    8974.46 ANGLE=-89.9
BOTT: SMAX=   17984.17 SMIN=      35.25 TMAX=    8974.46 ANGLE=-89.9
JOINT         0.00        0.00        0.00        0.00        0.00
6      17966.71    17966.71       35.17    17984.23      -20.58
TOP : SMAX=   17984.25 SMIN=      35.15 TMAX=    8974.55 ANGLE=-89.9
BOTT: SMAX=   17984.25 SMIN=      35.15 TMAX=    8974.55 ANGLE=-89.9
JOINT         0.00        0.00        0.00        0.00        0.00
5      18001.79    18001.79       35.17   -17984.15      -20.48
TOP : SMAX=      35.20 SMIN=  -17984.17 TMAX=    9009.68 ANGLE= -0.1
BOTT: SMAX=      35.20 SMIN=  -17984.17 TMAX=    9009.68 ANGLE= -0.1
:A SIMPLY SUPPORTED BEAM ON ELASTIC FOUNDATION           -- PAGE NO.    5
**** MAXIMUM STRESSES AMONG SELECTED PLATES AND CASES ****
MAXIMUM       MINIMUM       MAXIMUM       MAXIMUM       MAXIMUM
PRINCIPAL     PRINCIPAL       SHEAR       VONMISES       TRESCA
STRESS        STRESS        STRESS        STRESS        STRESS
1.805341E+04 -1.805341E+04  9.044362E+03  5.004878E+01  5.409544E+01
PLATE NO.       1             1             1             2             2
CASE  NO.       1             1             1             1             1