**Introduction**

Fig 1: Schematic representation of beam on elastic foundation |

For instance, the beam shown in Fig. 1 will deflect due to the externally applied load, and produce continuously distributed reaction forces in the supporting medium. The intensity of these reaction forces at any point is proportional to the deflection of the beam *y(x)* at that point via the constant *k _{s}*:

*R(x) =*

*k*

_{s}*.y(x)*(1)

Where *k _{s }*is the soil’s modulus of subgrade reaction which is the pressure per unit settlement of the soil (unit in kN/m

^{2}/m).

*R(x)*act vertically and opposing the deflection of the beam. Hence, where the deflection is acting downward there will be a compression in the supporting medium. Where the deflection happens to be upward in the supporting medium tension will be produced which is not possible (for soils).

If we assume that the beam under consideration has a constant cross section with constant width *b* which is supported by the foundation. A unit deflection of this beam will cause reaction equal to *k _{s}.b* in the foundation, therefore the intensity of distributed reaction (per unit length of the beam) will be:

*R(x) = b.**k _{s}*

*.y(x) = k.y(x)*(2)

where k = *k _{0}*.b is the constant of the foundation, known as Winkler’s constant, which includes the effect of the width of the beam, and has unit of kN/m/m.

The general 4th order differential equation for beam on elastic foundation is given by equation (3);

*EI(d*(3)

^{4}y)/dx^{4}+ k.y = qThe homogenous equation is given by;

*EI(d ^{4}y)/dx^{4} + k.y = 0*

*(d ^{4}y)/dx^{4} + 4β^{4}y = 0*

Where *β* = ∜(*k*/4EI) = (*k*/EI)^{(1/4)}

The general solution for the equation is available, which is given by;

y = e^{βx} (C_{1}*sinβx* + C_{2}*cosβx*) + e^{-βx}(C_{3}*sinβx* + C_{4}*cosβx*)

Warren and Richard (2002) published tables containing equations for analysing beams on elastic foundation subjected to different loads. The method described in the book has been employed to analyse a beam on elastic foundation and compare the results with Staad Pro software.

**Solved Example**

A 600mm x 400mm rectangular beam is resting on a homogenous soil of modulus of subgrade reaction *ks* = 10000 kN/m^{2}/m. The beam is 10m long, and carrying a concentrated load of W = 300 kN from the left hand side. Neglecting the self weight of the beam, and assuming freely supported ends, obtain the bending moment at the point of the concentrated load. Take modulus of elasticity of concrete Ec = 21.7 x 10^{6} kN/m^{2}.

**Solution **

Second moment of area of concrete beam I_{B} = (bh^{3})/12 = (0.4 × 0.6^{3})/12 = 7.2 x 10^{-3} m^{4}

Flexural rigidity of the beam E_{c}I_{B} = 21.7 × 10^{6} × 7.2 × 10^{-3} = 156240 kN.m^{2}

*β* = (bk_{s}/4EI)^{(1/4)} = [(0.4 × 10000)/(4 × 156240)]^{(1/4)} = 0.2828

*βl* = 0.2828 × 10 = 2.828 m; *β(l – a)* = 0.282(10 – 3) = 1.979

Where a is the distance of the concentrated load from the left end of the beam.

Since *βl* < 6.0, we can use Table 8.5 of Roark’s Table for Stress and Strain (Warren and Richard, 2002).

For a beam with both ends free;

R_{A} = 0; M_{A} = 0

The equations for bending moment and shear force along the beam is as given below;

M*x* = M_{A}F_{1} + R_{A}/2*β*F_{2} – y_{A}2EI*β*^{2}F_{3} – θ_{A}EI*β*F_{4} – *W*/2*β*F_{a2}

V*x* = R_{A}F_{1} – y_{A}2EIβ^{3}F_{2} – θ_{A}EIβ^{2}F_{3} – M_{A}*β*F_{4} – *W*F_{a1}

Where;

θ_{A} (rotation at point A) = [*W*/(2EI*β*^{2})] × [(C_{2}C_{a2} – 2C_{3}C_{a1})/C_{11}]

y_{A} (vertical deflection at point A) = [*W*/(2EI*β*^{3})] × [(C_{4}C_{a1} – C_{3}C_{a2})/C_{11}]

We can therefore compute the constants as follows;

C_{2} = *coshβl.sinβl* + *sinhβl.cosβl* = [*cosh*(2.828) × *sin*(2.828)] + [*sinh*(2.828) × *cos*(2.828)] = (8.485 × 0.3084) + (8.426 × –0.9512 ) = –5.398

C_{3} = *sinhβl.sinβl* = (8.426 × 0.3084) = 2.5985

C_{4} = *coshβl.sinβl* – *sinhβl.cosβl* = [*cosh*(2.828) × *sin*(2.828)] + [*sinh*(2.828) × *cos*(2.828)] = (8.485 × 0.3084) – (8.426 × –0.9512) = 10.631

C_{a1} = *coshβ(l – a).cosβ(l – a)* = cosh(1.974) × cos(1.974) = 3.669 × -0.392 = -1.438

C_{a2} = *coshβ(l – a).sinβ(l – a)* + *sinhβ(l – a).cosβ(l – a)* = [*cosh*(1.974) ×* sin*(1.974)] + [*sinh*(1.974) × *cos*(1.974)] = (3.669 × 0.9198) + (3.530 × -0.392) = 1.9909

C_{a3} = *sinhβ(l – a).sinβ(l – a)* = (3.530 × 0.9198) = 3.247

C_{11} = *sinh ^{2}βl – sin^{2}βl* = 8.4262 – 0.30842 = 70.902

θ_{A} = *W*/(2EI*β*^{2}) × [(C_{2}C_{a2} – 2C_{3}C_{a1})/C_{11}]

θ_{A} = [300/(2 × 156240 × 0.2828^{2} ) × [(–5.398 × 1.9909) – (2 × 2.5985 × –1.438)/70.902] = (0.012 × –0.0462) = -0.0005544 radians

y_{A} (vertical deformation at point A) = [*W*/(2EI*β*^{3})] × [(C_{4}C_{a1} – C_{3}C_{a2})/C_{11}]

y = [300/(2 × 156240 × 0.2828^{3})] × [(10.631 × –1.438) –2.5985 × 1.9909)/70.902] = (0.0424 × –0.2885) = –0.01223 m = -12.23 mm

*Bending moment at point C*

Substituting the values of deflection and slope into the equation for bending moment (note that the first and second terms of the equation goes to zero since R_{A} = M_{A} = 0);

Mx = – y_{A}EI*β*^{2}F_{3} – θ_{A}EI*β*F_{4} – *W*/2*β*F_{a2}

Mx = – (–0.01223 × 2 × 156240 × 0.28282)F_{3} – (– 0.0005544 × 156240 × 0.2828)F_{4} – 300/(2 × 0.2828)F_{a2}

_{
}M*x* = 305.638F_{3} + 24.495F_{4} – 530.410F_{a2}

Substituting F_{3}, F_{4}, and F_{a2} (see Table 8.5, Warren and Richard, 2002) into the equation;

Mx = 305.638(*sinhβx.sinβx*) + 24.495(*coshβx.sinβx – sinhβx.cosβx*) – 530.410[*coshβ(x –a).sin β(x –a) + sinh β(x –a).cos β(x –a)*]

The bending moment under the concentrated load (point C);

*x *= 3m; *(x – a)* = 3 – 3 = 0

*βx* = 0.8484

M*x* = 305.638[*sinh*(0.8484) × *sin*(0.8484)] + 24.495[*cosh*(0.8484) × *sin*(0.8484) – *sinh*(0.8484) × cos(0.8484)]

Mc = 305.638 (0.953 × 0.750) + 24.495(1.382 × 0.750 – 0.9539 × 0.6611) = 218.45 + 9.942 = 228.392 kN.m

**Verification**

This manual calculation has been verified using Staad Pro software. The steps adopted were as follows:

*(1) Modelling*

The 10m beam was modelled as a one dimensional line element connected by nodes at 1m length interval. This was to represent/attach soil springs at 1m interval.

**(2) Section Properties**

^{6}kN/m

^{2}, with dimensions of 600mm x 400mm.

**(3) Support**

**FIXED BUT**’ support option. The support must be released for moment since this is the ideal scenario for the structure we are trying to model.

*kb*= 10000 kN/m

^{2}/m x 0.4m = 4000 kN/m

^{2}

**(4) Analysis and Results**

*(a) Soil deformation*

*(b) Bending Moment*

**References**