Analysis of beam resting on elastic foundation

3.2 Analysis of beam resting on elastic foundation

A beam is a one-dimensional member. A schematic diagram of a beam (of unit width) resting on numerous springs with spring constant k (known as Winkler’s model, discussed later in

48 Chapter 3. Load stress in concrete pavement

Figure 3.1: A beam resting on numerous springs is acted on by a concentrated load Q.

Section 3.2.1) subjected to a pointed loading Q at x = 0 is pre- sented as Figure 3.1. The free-body diagram of a portion of the beam (other than the load application point) is shown in Fig- ure 3.2. The moment (M ) and shear force (V ) are shown in the free-body diagram. The upward force of the spring on the element of length dx is ωkdx, where ω is the displacement of the beam in the Z direction. From Figure 3.2, using the force equilibrium one can write,

V − (V + dV ) + kωdx = 0 (3.1) Taking the moment equilibrium one obtains,

dM V=

(3.2) dx

Putting Equation 3.2 in Equation 3.1 and considering that

d 2 EI ω dx 2 = −M , (that is, for the Euler−Bernoulli beam) one ob- tains,

− kω = 0 (3.3) where, EI is the flexural rigidity of the beam. This is a widely

EI dx 4

used basic expression for a beam resting on an elastic foundation

3.2. Analysis of beam resting on elastic foundation

dx

V V+dV

M+dM

kwdx Z

Figure 3.2: Free-body diagram of dx elemental length of the beam presented in Figure 3.1.

applied to foundation engineering [109, 136, 144, 247]. The general solution of Equation 3.3 is given as,

ω=e λx (c

1 cos λx + c 2 sin λx)+e −λx (c 3 cos λx + c 4 sin λx) (3.4)

where, λ =

, and c 1 ,c 2 ,c 3 and c 4 are constants. These con- stants can be determined from the boundary conditions pertaining to the specific geometry of the problem. Considering one side of the beam (for instance, the right side) with respect to the load application point of the infinite beam (as shown in Figure 3.1), one can write the following boundary conditions and subsequently derive the following results [109].

4EI

• lim x→∞ ω = 0. This condition leads to c 1 =c 2 = 0. Thus, the equation reduces to ω = e −λx (c 3 cos λx + c 4 sin λx)

• Due to symmetry, dω dx | x→0 = 0 . This leads to c 3 =c 4 =c (say)

• Total upward force generated by the springs must be equal

50 Chapter 3. Load stress in concrete pavement R to the downward force Q applied, that is, 2 ∞

0 kω dx = Q. This leads to c = Qλ

2k

Hence, the expression for deflection of an infinite beam resting on an elastic foundation is obtained as,

Qλ ω=

e −λx (cos λx + sin λx) (3.5) 2k

It may be noted that the developed equation (Equation 3.5) is valid only for the right side (i.e., x ≥ 0) of the infinite beam. In a similar manner, an expression can be developed for the left side of the beam. Then, the first boundary condition changes as, lim x→−∞ = 0; the other two conditions remain the same. From

these boundary conditions, the constants are obtained as c 3 =

c Qλ

4 = 0, and c 1 = −c 2 = 2k . The equation (for the left side of the infinite beam from x = 0) takes the form of,

Qλ λx ω=

e (cos λx − sin λx) (3.6) 2k

Equations 3.5 or 3.6 can be utilized (by successive differentiation) to obtain the rotation, bending moment, and shear profile [109]. The maximum deflection is under the load and is obtained as,

Qλ ωmax =

(3.7) 2k

In case there is a uniformly distributed loading (instead of point loading) of q per unit length, the deflection can be obtained by integration.

For the loading diagram shown as in Figure 3.3, the deflection at AA ′ (ω AA ′ ) can be obtained by using superposition of deflection calculated from Equations 3.5 and 3.6, and can be expressed as follows:

Z n ω AA ′

qλ =

e −λx (cos λx + sin λx) dx

0 2k Z m qλ

+ λx e (cos λx − sin λx) dx

0 2k

3.2. Analysis of beam resting on elastic foundation

dx x

A' Z

Figure 3.3: Analysis of an infinite beam with distributed loading.

If the section AA ′ is outside the loaded area (of length n + m), Equation 3.5 or 3.6 needs be used with appropriate integration limits. Further, the beam can be assumed as semi-infinite (that is, the beam has a definite ending at one side, and the other side is infinite), or finite (that, is the beam has a finite length). In such cases, an appropriate boundary condition can be used. Alterna- tively, one can solve it as a superposition of two infinite beams. One can refer to [109, 247], for example, for the details of the various approaches.

Such one dimensional analysis is useful, for example, for analysis of the problem of the dowel bar. Figure 3.4 illustrates how a single dowel bar can be idealized as a finite beam resting on an elastic foundation. However, additional considerations are involved in the dowel bar analysis problem (refer to Figure 3.4), for example, (i) there is a discontinuity of support in the middle portion, (ii) one side of the dowel bar is embedded in concrete but the other side is free to move horizontally, (iii) the wheel load does not directly act on the dowel bar, etc. Interested readers can refer to past works by Friberg [87, 88] and Bradbury [27] and a relatively recent study by Porter [223] on dowel bar analysis and the assumptions involved.

In line with the development of Equation 3.3, an equilibrium con- dition of a beam (refer to Equation 3.1) with an arbitrary loading

52 Chapter 3. Load stress in concrete pavement

Wheel

Dowel bar

Concrete slab

Gap = z s

Concrete slab

(a) Schematic diagram of a dowel bar in a concrete slab

(b) Idealized representation of dowel bar

Figure 3.4: Idealization of a dowel bar for analysis. (of q per unit length, which may include self-weight) and arbi-

trary foundation support (of p per unit length) condition (refer to Figure 3.5) can be written as:

dV =p−q

=q−p=q ∗ (3.10)

Figure 3.5: A beam with an arbitrary loading.

53 where, q ∗ is the net loading per unit length in the downward di-

3.2. Analysis of beam resting on elastic foundation

rection. Depending on the foundation support, the expression for p may become different. It may be noted that if p = 0, it becomes equivalent to the beam bending equation, without any spring sup- port. Winkler, Pasternak, and Kerr are the examples of different types of supports, and are briefly discussed in the following.

3.2.1 Beam resting on a Winkler foundation

Unconnected (linear) springs are known as Winkler’s springs [124, 142, 144, 182]. Formulation for a beam resting on a Winkler spring for a pointed loading was already discussed in the beginning of this section (Section 3.2). That is, for a Winkler spring, p = kω. Thus, a beam resting on a Winkler spring subjected to loading q (following Equation 3.3) can be represented as,

d 4 ω EI = q − kω

(3.11) dx 4

The Winkler spring constant (k) used here in the formulation in- dicates the pressure needed on the spring system to cause unit displacement. 1 Its unit is therefore MPa/mm. As discussed in Sec-

tion 2.2, the modulus of subgrade reaction (k) is also the pressure needed to cause unit deformation to the medium (that is, subgrade or sub-base or base layer). Thus, the spring constant used in the present formulation is conceptually equivalent to the modulus of subgrade reaction of the supporting layer.

One can refer to the paper written by Terzaghi [270] for a de- tailed discussion on the evaluation of the k value. Non-uniqueness of the modulus of the subgrade reaction in terms of the prediction of the (i) deflected shape (especially at the edges and corners of the slab) or (ii) stresses, is an issue raised by past researchers.

1 The Winkler model is also known as the dense liquid model, and k repre- sents the pressure needed to cause unit vertical displacement to a hypothetical

floating body against buoyancy.

54 Chapter 3. Load stress in concrete pavement

dx

Shear layer X

(a) A beam resting on a Pasternak pdx foundation

V'

V'+dV'

kwdx (b) Free body diagram of an element of the shear layer

Figure 3.6: A beam resting on a Pasternak foundation and a free- body-diagram of the shear layer.

One can, for example, refer to [62, 117, 233] for discussions on the issues involved. This has prompted researchers in the past to develop multi-parameter models to capture the response of struc- tures resting on soil. Some of these are discussed in the following.

Beam resting on a Pasternak foundation In a Pasternak foundation, it is assumed that there is a hypothet-

ical shear layer placed at the top of the spring system (refer to Figure 3.6(a)). Thus, considering the equilibrium of an element of length dx of shear layer (refer to Figure 3.6(b)), one can write,

pdx − V ′ + (V + dV ) − kωdx = 0

dV ′

p = kω −

dx

3.2. Analysis of beam resting on elastic foundation

If the shear force developed within the shear layer is assumed to

be proportional to the slope, then it can be written,

dω

V =G s (3.13) dx

where G s is the shear modulus of the foundation. Putting Equa- tion 3.13 in Equation 3.12, one obtains,

(3.14) Putting, Equation 3.14 in Equation 3.10 (i.e., the general equation

p = kω − G s dx 2

for a beam resting on an elastic foundation) the equation for a one- dimensional beam resting on a Pasternak foundation becomes,

(3.15) dx

EI 4 −G s 2 + kω = q

dx One can refer to, for example, [37] for the solutions for various

problem geometries on a Pasternak foundation.

3.2.2 Beam resting on a Kerr foundation

The Kerr foundation model [144, 145] consists of two layers of Winkler springs (with spring constants k 1 and k 2 , say) with a shear layer in between (refer to Figure 3.7(a)). The free-body diagram of an element of length dx of the shear layer in the Kerr foundation is shown in Figure 3.7(b). The pressures transmitted on the top

and the bottom of the shear layer are shown as p 1 and p 2 , and the displacements that the top and the bottom set of springs undergo are ω 1 and ω 2 respectively.

Considering the equilibrium of the shear layer, dV ′

p 1 −p 2 =− dx

dω 2

= −G s (Similar to Equation 3.13) (3.16)

dx

56 Chapter 3. Load stress in concrete pavement

dx

Shear layer X

(a) A beam resting on a Kerr p 1 dx foundation

V'

V'+dV' p 2 dx

(b) Free body diagram of an element of the shear layer

Figure 3.7: A beam resting on a Kerr foundation and a free-body diagram of the shear layer.

Further, ω=ω 1 +ω 2

(3.17) where, ω is the deflection of the beam. The spring conditions can

be written as, p 1 =k 1 ω 1

p 2 =k 2 ω 2

Putting Equations 3.17 and 3.18 in Equation 3.16, one can write

Putting Equation 3.18 in Equation 3.10 (i.e., the general equation for a beam resting on an elastic foundation and considering that

3.2. Analysis of beam resting on elastic foundation

p=p 1 in the present case) it can be written as,

− 1+ q (3.19) k 1

k 1 dx 2

3.2.3 Various other models

There is a large number of models of beams on elastic founda- tions (for example, the Filonenko–Borodich model, the Vlasov model, the Rhines model, the Reissner model, the Het´eni model, etc., some of which are continuum models) and varieties of solu- tion techniques proposed and studied by various researchers [73, 137, 237, 299]. Interested readers may refer to, for example, pa- pers/reports such as [124, 142, 144, 153] for a review of various types of foundation models, or refer to the book by Selvadurai [247] for a detailed discussion.

If a beam resting on a Winkler spring is subjected to tensile axial force N , it can be shown (from the free-body diagram, in a similar manner the other equations are developed) [109],

(3.20) dx

EI 4 −N 2 + kω = q

dx It can be seen that the form of Equation 3.20 is similar to Equa-

tion 3.15 (hence the solution approach will be similar), even though these have been derived for two different types of problems.

Starting from the above approach, models can be further devel- oped to represent a geotextile/geogrid placed within the shear layer [179, 181]. Other than the spring models (generally classified as “lumped parameter models”), continuum models are also used to represent subgrade support. This will be discussed in Chapter 5.

58 Chapter 3. Load stress in concrete pavement