Beam on Elastic Foundation Analysis

7/29/2019 Beam on Elastic Foundation Analysis

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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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Formulas Used to Determine Shear, Moment, Slope, Deflection, and Pressure in Beam on Elastic Foundation

General Constants and Functions:

I = W*T^3/12

b = ((K*W)/(4*E*144*I))^(1/4) (Note: units of 'K' are "kcf".)F1 = COSH(b*x)*COS(b*x)

F2 = COSH(b*x)*SIN(b*x) + SINH(b*x)*COS(b*x)

F3 = SINH(b*x)*SIN(b*x)

F4 = COSH(b*x)*SIN(b*x) - SINH(b*x)*COS(b*x)

C1 = COSH(b*L)*COS(b*L)

C2 = COSH(b*L)*SIN(b*L) + SINH(b*L)*COS(b*L)

C3 = SINH(b*L)*SIN(b*L)

C4 = COSH(b*L)*SIN(b*L) - SINH(b*L)*COS(b*L)

C11 = SINH(b*L)^2 - SIN(b*L)^2

For Full Uniform or Distributed Loads:

Specific Constants and Functions:

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

Ca3 = SINH(b*(L-b))*SIN(b*(L-b))

Cb2 = COSH(b*(L-e))*SIN(b*(L-e)) + SINH(b*(L-e))*COS(b*(L-e))

Cb3 = SINH(b*(L-e))*SIN(b*(L-e))

If x > b:

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

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

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

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

Fa5 = 1- Fa1 else: Fa5 = -Fa1

If x > e:

Fb1 = COSH(b*(x-e))*COS(b*(x-e)) else: Fb1 = 0

Fb2 = COSH(b*(x-e))*SIN(b*(x-e)) + SINH(b*(x-e))*COS(b*(x-e)) else: Fb2 = 0

Fb3 = SINH(b*(x-e))*SIN(b*(x-e)) else: Fb3 = 0

Fb4 = COSH(b*(x-e))*SIN(b*(x-e)) - SINH(b*(x-e))*COS(b*(x-e)) else: Fb4 = 0

Fb5 = 1- Fb1 else: Fb5 = -Fb1

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

Va = 0

Ma = 0

qa = w/(2*E*144*I*b^3)*(C2*Ca3-C3*Ca2)/C11

Da = w/(4*E*144*I*b^4)*(C4*Ca2-2*C3*Ca3)/C11

Loading functions for each uniform or distributed 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 - w/(2*b)*Fa2

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

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

FDx = Da*F1 + qa/(2*b)*F2 + Ma/(2*E*144*I*b^2)*F3 + Va/(4*E*144*I*b^3)*F4 - w/(4*E*144*I*b^4)*Fa5


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For Point Loads:


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:


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: Subject:

Job Number: Originator: Checker:

Input Data:c

Beam Data: e

b

Length, L = 25.0000 ft. a

Width, W = 4.0000 ft. +P

Thickness, T = 0.7500 ft. +wb +we +M +w

Modulus, E = 3600 ksi

Subgrade, K = 100 pci T

Inertia, I = 0.141 ft.^4E,I L

Beam Loadings: x

Nomenclature

Full Uniform:w = 0.6000 kips/ft. Results:

Start End Beam Flexiblity Criteria:

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

#3: forb*L >= 6 beam is semi-infinite long

#4:

#5: b = 0.221 b = ((K*W)/(4*E*144*I))^(1/4)

#6: b*L = 5.52 b*L = Flexibility Factor

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

#1: 5.0000 8.00#2: 20.0000 12.00

#3: Max. Shears and Locations:

#4: +V(max) = 6.27 k @ x = 20.00 ft.

#5: -V(max) = -5.73 k @ x = 20.00 ft.

#6:

#7: Max. Moments and Locations:

#8: +M(max) = 13.00 ft-k @ x = 20.00 ft.

#9: -M(max) = -4.95 ft-k @ x = 12.75 ft.

#10:

#11: Max. Deflection and Location:

#12: D(max) = -0.034 in. @ x = 20.25 ft.

Moments: c (ft.) M (ft-kips) Max. Soil Pressure and Location:#1: Q(max) = 0.494 ksf @ x = 20.25 ft.

#2:

#3:

#4:

Subgrade

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"BOEF.xls" Program

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Beam on Elastic Foundation Analysis

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