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"BOEF" --- BEAM ON ELASTIC FOUNDATION ANALYSIS

Program Description:

"BOEF" is a spreadsheet program written in MS-Excel for the purpose of analysis a finite length beam with free ends

supported continuously on an elastic foundation. This program is ideally suited for analyzing a soil supported beam,

a combined footing, or a strip of a slab or a mat. Specifically, the beam shear, moment, deflection, and soil bearingpressure are calculated for 100 equal beam segments, as well as the maximum values. Plots of both the shear,

moment, and soil bearing pressure diagrams are produced, as well as a tabulation of the shear, moment,

deflection, and bearing pressure for the beam.

This program is a workbook consisting of two (2) worksheets, described as follows:

Worksheet Name Description

Doc This documentation sheet

Beam on Elastic Foundation Beam on elastic foundation analysis

Program Assumptions and Limitations:

1. The following reference was used in the development of this program (see below): "Formulas for Stress and Strain" - Fifth Edition

by Raymond R. Roark and Warren C. Young, McGraw-Hill Book Company (1975), pages 128 to 146.

2. This program uses the equations for a "finite-length" beam in the analysis. This usually gives very similar to

exact results for a "semi-infinite" beam which has had end-corrections applied to "force" the moment and shear

values to be equal to zero at the ends. (Note: a "semi-infinite" beam is defined as one that has a b*L value > 6.)

3. This program uses the five (5) additional following assumptions as a basis for analysis:

a. Beam must be of constant cross section (E and I are constant for entire length, L).

b. Beam must have both ends "free". ("Pinned" or "fixed" ends are not permitted.)

c. Elastic support medium (soil) has a constant modulus of subgrade, K, along entire length of beam.

d. Applied loads are located in the center of the width, W, of the beam and act along a centroidal line of the

beam-soil contact area.

e. Bearing pressure is linearly proportional to the deflection, and varies as a function of subgrade modulus, K.

4. This program can handle up to twelve (12) concentrated (point) loads, a full uniformly distributed load with up tosix (6) additional full or partial uniformly distributed loads, and up to four (4) externally applied moments.

5. Beam self-weight is NOT automatically included in the program analysis, but may be accounted for as a full

uniformly distributed applied load. Beam self-weight will only affect the deflection and bearing pressure, and not

the moment or shear.

6. This program will calculate the maximum positive and negative shears, the maximum positive and negative

moments, the maximum negative deflection, and the maximum soil bearing pressure. The calculated values

for the maximum shears, maximum moments, deflection, and bearing pressure are determined from dividing

the beam into 100 equal segments with 101 points, and including all of the point load and applied moment

locations as well.

7. The user is given the ability to input four (4) specific locations from the left end of the beam to calculate the

shear, moment, deflection, and bearing pressure.

8. The plots of the shear, moment, and bearing pressure diagrams as well as the displayed tabulation of shear,

moment, deflection, and bearing pressure are based on the beam being divided up into 100 equal segments

with 101 points.

9. This program contains numerous comment boxes which contain a wide variety of information including

explanations of input or output items, equations used, data tables, etc. (Note: presence of a comment box

is denoted by a red triangle in the upper right-hand corner of a cell. Merely move the mouse pointer to the

desired cell to view the contents of that particular "comment box".)

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For Point Loads:

Specific Constants and Functions:

Ca1 = COSH(b*(L-a))*COS(b*(L-a))

Ca2 = COSH(b*(L-a))*SIN(b*(L-a)) + SINH(b*(L-a))*COS(b*(L-a))

If x > a:Fa1 = COSH(b*(x-a))*COS(b*(x-a)) else: Fa1 = 0

Fa2 = COSH(b*(x-a))*SIN(b*(x-a)) + SINH(b*(x-a))*COS(b*(x-a)) else: Fa2 = 0

Fa3 = SINH(b*(x-a))*SIN(b*(x-a)) else: Fa3 = 0

Fa4 = COSH(b*(x-a))*SIN(b*(x-a)) - SINH(b*(x-a))*COS(b*(x-a)) else: Fa4 = 0

Loading functions for each point load evaluated at distance x = L from left end of beam:

Va = 0

Ma = 0

qa = P/(2*E*144*I*b^2)*(C2*Ca2-2*C3*Ca1)/C11

Da = P/(2*E*144*I*b^3)*(C4*Ca1-C3*Ca2)/C11

Loading functions for each point load evaluated at distance = x from left end of beam:

Fvx = Va*F1 - Da*2*(E*144)*I*b^3*F2 - qa*2*(E*144)*I*b^2*F3 - Ma*b*F4 - P*Fa1

Fmx = Ma*F1 + Va/(2*b)*F2 - Da*2*(E*144)*I*b^2*F3 - qa*(E*144)*I*b*F4 - P/(2*b)*Fa2

Fqx = qa*F1 + Ma/(2*E*144*I*b)*F2 + Va/(2*E*144*I*b^2)*F3 - Da*b*F4 - P/(2*E*144*I*b^2)*Fa3

FDx = Da*F1 + qa/(2*b)*F2 + Ma/(2*E*144*I*b^2)*F3 + Va/(4*E*144*I*b^3)*F4 - P/(4*E*144*I*b^3)*Fa4

For Applied Moments:

Specific Constants and Functions:

Ca1 = COSH(b*(L-c))*COS(b*(L-c))

Ca4 = COSH(b*(L-c))*SIN(b*(L-c)) - SINH(b*(L-c))*COS(b*(L-c))

If x > c:

Fa1 = COSH(b*(x-c))*COS(b*(x-c)) else: Fa1 = 0

Fa2 = COSH(b*(x-c))*SIN(b*(x-c)) + SINH(b*(x-c))*COS(b*(x-c)) else: Fa2 = 0

Fa3 = SINH(b*(x-c))*SIN(b*(x-c)) else: Fa3 = 0

Fa4 = COSH(b*(x-c))*SIN(b*(x-c)) - SINH(b*(x-c))*COS(b*(x-c)) else: Fa4 = 0

Loading functions for each applied moment evaluated at distance x = L from left end of beam:

Va = 0

Ma = 0

qa = -M/(E*144*I*b)*(C3*Ca4+C2*Ca1)/C11

Da = M/(2*E*144*I*b^2)*(2*C3*Ca1+C4*Ca4)/C11

Loading functions for each applied moment evaluated at distance = x from left end of beam:

Fvx = Va*F1 - Da*2*(E*144)*I*b^3*F2 - qa*2*(E*144)*I*b^2*F3 - Ma*b*F4 - M/(2*b)*Fa2

Fmx = Ma*F1 + Va/(2*b)*F2 - Da*2*(E*144)*I*b^2*F3 - qa*(E*144)*I*b*F4 - M*Fa1

Fqx = qa*F1 + Ma/(2*E*144*I*b)*F2 + Va/(2*E*144*I*b^2)*F3 - Da*b*F4 - M/(2*E*144*I*b)*Fa2

FDx = Da*F1 + qa/(2*b)*F2 + Ma/(2*E*144*I*b^2)*F3 + Va/(4*E*144*I*b^3)*F4 - M/(2*E*144*I*b^2)*Fa3

Summations of shear, moment, slope, and deflection at distance = x from left end of beam:

Shear: Vx = S(Fvx)

Moment: Mx = S(Fmx)

Slope: qx = S(Fqx)

Deflection: Dx = S(FDx)

Pressure: Qx = Dx*K

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"BOEF.xls" Program

Version 1.2

BEAM ON ELASTIC FOUNDATION ANALYSISFor Soil Supported Beam, Combined Footing, Slab Strip or Mat Strip

of Assumed Finite Length with Both Ends Free

Job Name: Koenders MFG - Pipe Sleeper Subject: Sleeper on Grade

Job Number: 401EN12 Originator: JMT Checker:

Input Data: c

Beam Data: e

b

Length, L = 20.6690 ft. a

Width, W = 0.7500 ft. +P

Thickness, T = 0.5000 ft. +wb +we +M +w

Modulus, E = 130 ksi

Subgrade, K = 55 pci T

Inertia, I= 0.003 ft.^4E,I L

Beam Loadings: x

Nomenclature

Full Uniform:w = 0.0000 kips/ft. Results:

Start End Beam Flexiblity Criteria:

Distributed: b (ft.) wb(kips/ft.) e (ft.) we(kips/ft.) for b*L = p beam is flexible

#3: 15.3270 8.0920 16.5773 8.0920 for b*L >= 6 beam is semi-infinite long

#4:

#5: b= 0.760 b= ((K*W)/(4*E*144*I))^(1/4)

#6: b*L = 15.70 b*L = Flexibility Factor

Point Loads: a (ft.) P (kips) Beam is f lexible

#1:#2:

#3: Max. Shears and Locations:

#4: +V(max)= 2.91 k @x = 15.30 ft.

#5: -V(max)= -2.77 k @x = 10.95 ft.

#6:

#7: Max. Moments and Locations:

#8: +M(max)= 2.02 ft-k @ x = 10.33 ft.

#9: -M(max)= -1.07 ft-k @ x = 13.23 ft.

#10:

#11: Max. Deflection and Location:

#12: D(max)= -0.597 in. @x = 4.75 ft.

Moments: c (ft.) M (ft-kips) Max. Soil Pressure and Location:#1: Q(max)= 4.726 ksf @x = 4.75 ft.

#2:

#3:

#4:

Subgrade

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Compatibility Report for BOEF Full Beam Length.xls

Run on 9/18/2012 10:53

The following features in this workbook are not supported by

earlier versions of Excel. These features may be lost or

degraded when opening this workbook in an earlier version of

Excel or if you save this workbook in an earlier file format.

Significant loss of functionality # of

occurrences

Version

One or more cells in this workbook contain data validation

rules which refer to values on other worksheets. These data

validation rules will not be saved.

25

Beam on

Elastic

Foundation'!E

25:E30

Excel 97-2003

Beam on

ElasticFoundation'!L

21:L22

Beam on

Elastic

Foundation'!B

47:B50

Beam on

Elastic

Foundation'!B

25:B30

Beam on

ElasticFoundation'!B

33:B44

Beam on

Elastic

Foundation'!D

29:D30

Beam on

Elastic

Foundation'!D

25:D28

Beam on

Elastic

Foundation'!L

23:L24