Beam on Elastic Foundation Analysis

For Soil Supported Beam, Combined Footing, Slab Strip or Mat Strip of Assumed Finite Length with Both Ends Free

Input Data

Beam Data:
Length, L = ft.  
Width, B = ft.  
Thickness, T = ft.  
Modulus, E = ksi  
Subgrade, K = pci  
Beam Loadings:
Full Uniform: w = kips/ft.
Distributed: b (ft.) wb (kips/ft.) e (ft.) we (kips/ft.)
1#:
2#:
3#:
4#:
5#:
6#:
Point Loads: a (ft.) P (kips)  
1#  
2#:  
3#:  
4#:  
5#:  
6#:  
7#:  
8#:  
9#:  
10#:  
11#:  
12#:  
Moments: c (ft.) M (ft-kips)  
1#:  
2#:  
3#:  
4#:  

Result :

Beam Flexiblity Criteria:
Inertia, I = ft.^4 I = B*T^3/12
b = b = ((K*B)/(4*E*I))^(1/4)
b*L = b*L = Flexibility Factor
   
Max. Shears and Locations:
+V(max) = k @x= ft.
-V(max) = k @x= ft.
Max. Moments and Locations:
+M(max) = ft-k @x= ft.
-M(max) = ft-k @x= ft.
Max. Deflection and Location:
D(max) = in. @x= ft.
Max. Soil Pressure and Location:
Q(max) = ksf @x= ft.
Min. Thickness for Theoretically "Semi-Rigid" Beam:
T(min) = ft. (12*K/(4*E*(p/L)^4))^(1/3)
Min. Thickness for Theoretically "Rigid" Beam:
T(min) = ft.(12*K/(4*E*(p/(4*L))^4))^(1/3)
Estimated Effective Strip Width for Slab or Mat:
Lr = ft.(E*T^3/(12*(1-m^2)*K))^(1/4) (where: m = 0.15)
%B = %B = user assumed percentage
Lr = ft. B(est.) = (%B)*(2*Lr)
User Designated Points:
x1= ft.  
x2= ft.  
x3= ft.  
x4= ft.  
     
Results for User Designated Points:
  For x1: For x2:
Vx1 = kips Vx2 = kips
Mx1 = ft-kips Mx2 = ft-kips
Dx1 = in. Dx2 = in.
Qx1 = ksf Qx2 = ksf
 For x3:For x4:
Vx3 = kips Vx4 = kips
Mx3 = ft-kips Mx4 = ft-kips
Dx3 = in. Dx4 = in.
Qx3 = ksf Qx4 = ksf
c
e
b
a
+P
+wb +we +M +w
T
     
E, I L Subgrade
x

Nomenclature



'E' is the modulus of elasticity for the beam material.
for Concrete:
f'cE
3000 psi3120 ksi
4000 psi 3600 ksi
5000 psi4030 ksi
Note: values of 'E' listed above are determined by:
        E = 57*SQRT(f'c) in units of ksi
Modulus of Subgrade Reaction, K
Sandy Soils:
Loose Sand 30-100 kcf
Medium Sand 60-500 kcf
Dense Sand 400-800 kcf
Sand w/Clay (mix) 200-500 kcf
Sand w/Silt (mix) 150-300 kcf


Clayey Soils:
qu < 4 ksf 75-150 kcf
4 ksf < qu < 8 ksf 150-300 kcf
qu > 8 ksf > 300 kcf
where: qu = unconfined compression strength

Note: if 'K' is known/given in units of kcf,
        E = multiply by 1000/1728 = 0.5787 to
         convert kcf to pci for input.
        This program takes 'K' input in pci and
         multiplies it by 1728/1000 = 1.728 to
        E = convert to kcf for use throughout.
Moment of inertia of beam cross-section:
   I = B*T^3/12
Note: in formula for 'b',
'E' is multiplied by 144 to convert from units of
ksi to units of ksf for consistency.
'+V(max)' is the maximum positive shear in beam.
This value represents total shear, +V(max),
on assumed beam/strip width = B.
'-V(max)' is the maximum negative shear in beam.
This value represents total shear, -V(max),
on assumed beam/strip width = B.
'x' is the location of the maximum positive
shear from left end of beam.
'x' is the location of the maximum negative
shear from left end of beam.
'+M(max)' is the maximum positive moment in beam.
Positive (+) moment = tension in bottom of beam.
This value represents total moment, +M(max) on
assumed beam/strip width = B
'-M(max)' is the maximum negative moment in beam.
Negative (-) moment = tension in top of beam.
This value represents total moment, -M(max) on
assumed beam/strip width = B
'x' is the location of the maximum positive
moment from left end of beam.
'x' is the location of the maximum negative
moment from left end of beam.
'D(max)' is the maximum deflection in beam.
Negative deflection is in downward direction.
'x' is the location of the maximum
deflection from left end of beam.
'Q(max)' is the maximum bearing pressure.
'x' is the location of the maximum
soil pressure from left end of beam.
'Lr' is the radius relative of stiffness for the
slab/mat and subgrade, from H.M.
Westergaard's theory for slabs on grade
subjected to concentrated loads (1926).

Lr = ((E*T^3/(12*(1-m^2)*K))^(1/4)
where: m = Poisson's Ratio
         (assumed =0.15 for concrete)
The user assumed percentage, '%B', may vary between
50% up to a limiting value of 88.62% for determining the
estimated effective strip width for slab or mat.

The maximum value may be determined by equating the
area of a circle of diameter = 2*Lr to an equivalent square:
%B(max) = SQRT(p*(2*Lr)^2/4)/(2*Lr)*100
         = SQRT (p)/2*100 = 88.62%

A square inscribed inside a circle of diameter = 2*Lr
would have a side dimension = SQRT(2)*Lr, thus:
%B = SQRT(2)*Lr/(2*Lr)*100
         = SQRT(2)/2*100 = 70.71%
Estimated effective width of strip of slab/mat,
'B(est)', is assumed
to be an assumed percentage of 2 times the radius of slab/mat
relative stiffness, 'Lr'.
Shear at distance = x1
from left end of beam.
Sign convention:
positive (+) = upward.
Moment at distance = x1
from left end of beam.
Sign convention: positive
(+) = tension in bottom of beam.
Deflection at distance = x1
from left end of beam.
Sign convention: positive
(+) = upward.
Bearing pressure at distance = x1
from left end of beam.
Sign convention: positive
(+) = compression.
Shear at distance = x2
from left end of beam.
Sign convention:
positive (+) = upward.
Moment at distance = x2
from left end of beam.
Sign convention: positive (+)
= tension in bottom of beam.
Deflection at distance = x2
from left end of beam.
Sign convention: positive (+)
= upward.
Bearing pressure at distance = x2
from left end of beam.
Sign convention: positive (+)
= compression.
Shear at distance = x3
from left end of beam.
Sign convention:
positive (+) = upward.
Moment at distance = x3
from left end of beam.
Sign convention: positive (+)
= tension in bottom of beam.
Deflection at distance = x3
from left end of beam.
Sign convention:
positive (+) = upward.
Bearing pressure at distance = x3
from left end of beam.
Sign convention:)
positive (+ = compression.
Shear at distance = x4
from left end of beam.
Sign convention:
positive (+) = upward.
Moment at distance = x4
from left end of beam.
Sign convention: positive (+)
= tension in bottom of beam.
Deflection at distance = x4
from left end of beam. Sign convention:
positive (+) = upward.
Bearing pressure at distance = x4
from left end of beam.
Sign convention: positive (+)
= compression.
Beam Flexiblity Criteria:
for b*L <= p/4 beam is rigid
for p/4 < b*L < p beam is semi-rigid
for b*L >= p beam is flexible
for b*L >= 6 beam is semi-infinite long


Disclaimer: This calculator are not intended to be used for the design of actual structures, but only for schematic (preliminary) understanding of structural design principals. For the design of an actual structure, a competent professional should be consulted.

‘Calculations courtesy of Alex Tomanovich, PE