Journal of Theoretical and Applied Mechanics, Sofia, 2014, vol. 44, No. 2, pp. 57–70

RECURSIVE DIFFERENTIATION METHOD

FOR BOUNDARY VALUE PROBLEMS: APPLICATION

TO ANALYSIS OF A BEAM-COLUMN ON AN ELASTIC

FOUNDATION

Mohamed Taha

Dept. of Eng. Math. and Physics,

Faculty of Engineering, Cairo University, Giza, Egypt,

e-mails:mtahah@eng.cu.edu.eg, mtaha@alfaconsult.org

[Received 19 August 2013. Accepted 10 March 2014]

Abstract. In the present work, the recursive differentiation method(RDM) is introduced and implemented to obtain analytical solutions fordifferential equations governing different types of boundary value prob-lems (BVP). Then, the method is applied to investigate the static behav-iour of a beam-column resting on a two parameter foundation subjected todifferent types of lateral loading. The analytical solutions obtained usingRDM and Adomian decomposition method (ADM) are found similar butthe RDM requires less mathematical effort. It is indicated that the RDMis reliable, straightforward and efficient for investigation of BVP in finitedomains. Several examples are solved to describe the method and theobtained results reveal that the method is convenient for solving linear,nonlinear and higher order ordinary differential equations. However, it isindicated that, in the case of beam-columns resting on foundations, thebeam-column may be buckled in a higher mode rather than a lower one,then the critical load in that case is that accompanies the higher mode.This result is very important to avoid static instability as it is widelycommon that the buckling load of the first buckling mode is always thesmaller one, which is true only in the case of the beam-columns withoutfoundations.Key words: Boundary value problems, recursive differentiation method,differential equations, beams on elastic foundation.

1. Introduction

Most physical and engineering boundary value problems (BVP) can bemodelled as functional equations. However, for most of these equations, exactsolutions are very rare. Several analytical and numerical methods are being de-veloped to obtain approximate solutions for such models. The commonly usedanalytical methods are: Adomian decomposition method (ADM) [1–4], varia-tional iteration method (VIM) [5–6], homotopy perturbation method (HPM)

58 Mohamed Taha

[7–8], deferential transform method (DTM) [9] and perturbation techniques [10–11]. On the other hand, numerical methods such as finite difference method [12],finite element method (FEM) [13], and differential quadrature method (DQM)[14–15], offer tractable alternative solutions for BVP that involve non-uniformcharacteristics or complicated boundaries.

The analytical methods construct a solution for BVP as a polynomialsuch that the coefficients of these polynomials are obtained to satisfy boththe governing differential equation and the boundary conditions. On the otherhand, numerical techniques transform the differential equation into a systemof algebraic equations either on the boundary of the BVP domain or at dis-crete points in the BVP domain and solve the new system. Two main issuesface these methods, the mathematical manipulation in constructing the solu-tion expressions and the accuracy (or convergence) of the results. Indeed, thedegree of method success in overcoming such issues determines its efficiencyand popularity.

The present work introduces the recursive differentiation method (RDM)to solve differential equations governing different types of BVP’s. The methodconstructs an analytical series solution for the differential equations but in away different from the traditional higher order Taylor series method. How-ever, in the RDM, the coefficients of the solution series are obtained throughrecursive differentiations of the governing differential equation. Investigationsof the method reveal that the method yields exact solutions for linear differ-ential equations. For complicated differential equations however, the methodyields, with less effort, the same analytical expressions obtained from othertechniques. The RDM is tested on linear, nonlinear and higher order differen-tial equations which represent practical boundary value problems to illustrateits efficiency. Furthermore, both the ADM and the RDM method are appliedto obtain analytical solutions for the axially loaded beams (beam-columns) ontwo parameter foundation subjected to different types of lateral loading. Exactexpression for the critical loads of a beam-column resting on two parameterfoundation required for stability analysis is obtained. The illustrated examplesshow that the method possesses the same accuracy of DTM and ADM but inrelatively short mathematical manipulations.

2. Recursive differentiation method (RDM)

Consider the nonlinear n-order boundary value problem of the form:

(1) y(n)(x) = F (x, y, y(1), y(2), . . . , y(n−1)), x0 ≤ x ≤ x1,

with the boundary conditions: Bi(y, y(1), . . . , y(n−1)) = bi, i = 1, 2, . . . , n,

Recursive Differentiation Method for Boundary Value Problems . . . 59

where y(n) is the n-th-derivative and bi are constants. In the RDM, the so-lution of Eqn (1) is assumed as a polynomial in the form:

(2) y(x) =

N∑

m=0

Tm

(x − xo)m

m!,

where Tm are coefficients obtained to satisfy both the governing equation andthe boundary conditions, x0 is the boundary coordinates, N is the truncationindex selected to achieve the pre-assigned accuracy. The coefficients Tm arerelated to the governing differential equation on the boundary as:

(3) Tm = y(m)∣

∣

∣

x=x0

.

It is found that to enhance the accuracy in the RDM, the solution do-main is to be transformed to [0, 1]. In addition, to decrease the mathematicaleffort required to obtain the coefficients Tm, a recurrence formula to be de-rived from the recursive differentiations of the governing equation as it will beillustrated in the following numerical examples.

Example 1. Consider the Riccati differential equation which appearsin random process, optimal control and diffusion problems [3];

(4) y(1)(t) = −y2 + 1, 0 ≤ t ≤ 1.

Subject to the initial condition: y(0) = 0.Let the solution of Eqn (4) be in the form:

(5) y(t) =

N∑

m=0

Tm

tm

m!.

Recursive differentiation of Eqn (4) yields the recursive equations:

(6) y(m)(t) = F (y, y(1), y(2), . . . , y(m−1)), m ≥ 2.

Using Eqn (3), the coefficients Tm are obtained as:

T0 = Teven = 0, T1 = 1, T3 = −2, T5 = 16, T7 = −272, T9 = 7936.

Substitution Tm into Eqn (5), the solution for N = 9 is:

(7) y(t) = t −2

3!t3 +

16

5!t5 −

272

7!t7 +

7936

9!t9 + O(t11).

The exact solution to this problem is [3]: y(t) =e2t − 1

e2t + 1.

It is found that the numerical results obtained from the RDM expressionare compatible with those calculated from the exact solution.

60 Mohamed Taha

Example 2. Consider the linear fifth-order boundary value problemwhich arises in the mathematical modelling of viscoelastic flow [4]:

(8) y(5)(x) = y(x) − 15ex− 10xex, 0 ≤ x ≤ 1,

with the boundary conditions:

y(0) = 0, y(1)(0) = 1, y(2)(0) = 0, y(1) = 0 and y(1)(1) = −e.

Let the solution of Eqn (8) be in the form:

(9) y(x) =

N∑

m=0

Tm

xm

m!.

The recursive differentiations of Eqn (8) yield the recurrence relation:

(10) y(5+n)(x) = y(n)(x) − (15 + 10n)ex− 10xex, n = 0, 1, 2, . . .

Using Eqn (3), the coefficients Tm for m≤5 are obtained as:

T0 = 0, T1 = 1, T2 = 0, T5 = −15.

In addition, using Eqn (3) and Eqn (10), a recurrence formula for thecalculations of the coefficients Tm for m > 5 is obtained as:

(11) Tn+5 = Tn − (15 + 10n), n = 1, 2, 3, . . .

Substitution into Eqn (9) yields the solution as:

(12) y(x) = x +1

3!T3x

3 +1

4!T4x

4−

15

5!x5 +

N−5∑

n=1

xn+5

(n + 5)!(Tn − (15 + 10n)) .

The unknown coefficients T3 and T4 are obtained using the boundaryconditions at x = 1 with Eqn (12), for different values of truncation index Nas:

T3 = −2.9653, T4 = −8.17213 for N = 7

T3 = −3, T4 = −8 for N = 12.

Then, the solution of Eqn (8) is:

(13) y(x) = x −3

3!x3

−8

4!x4

−15

5!x5

−24

6!x6

−35

7!x7

−48

8!x8

−63

9!x9 + · · ·

which is the exact solution: y(x) = x(1 − x)ex.

Recursive Differentiation Method for Boundary Value Problems . . . 61

Fig. 1. A beam-column on an elastic foundation

3. Analysis of a beam-column on an elastic foundation

The equation of motion of an infinitesimal element of the axially loadedbeam subjected to lateral load (beam-column) and resting on two-parameterfoundation shown in Fig. 1 may be expressed as:

(14) EId4y

dx4+ (p − k2)

d2y

dx2+ k1y(x) = q(x),

where EI is the flexural stiffness, p is the axial applied load, k1 and k2 are thelinear and shear foundation stiffness per unit length, q(x) is the lateral load,E is the modulus of elasticity, I is the moment of inertia, y(x) is the lateraldisplacement, x is the coordinate along the beam. The applied lateral load isassumed in the nonlinear form:

(15) q(x) = q0 + q1x + q2x2.

Introducing the dimensionless variables; ξ = x/L and w = y/L, whereL is the beam length, Eqn (14) may be rewritten as:

(16)d4w

d ξ4+ (P1 − K2)

d2w

d ξ2+ K1 w(ξ) = Q0 + Q1ξ + Q2ξ

2,

where the following dimensionless parameters are defined:(17)

K1 =k1L

4

EI, K2 =

k2L2

EI, P1 =

pL2

EI, Q0 =

q0L3

EI, Q1 =

q1L4

EIand Q2 =

q2L5

EI.

The boundary conditions at the beam ends for beam pinned at its ends (P-P)in dimensionless forms may be expressed as:

(18) w(0) = w′′(0) = 0 and w(1) = w′′(1) = 0.

In the following sections, the solution of non- homogeneous linear dif-ferential equation (Eqn 16) will be obtained using both the ADM and the RDMto illustrate the advantages of the proposed method.

62 Mohamed Taha

3.1. Adomian Decomposition Method (ADM)

It is an analytical method proposed by Adomian [1] and has been usedby many investigators [2, 3, 4] to obtain approximate analytical solution fornumerous BVP’s.

To apply the ADM, Eqn (16) is rewritten in the operator form as:

(19) ℓw = −(P1 − K2)w′′− K1w(ξ) + Q0 + Q1ξ + Q2ξ

2,

where: ℓ =d4

dξ4.

Appling the inverse operator ℓ−1to both sides of Eqn (19) and substi-tuting of the boundary conditions at ξ=0 yields:

(20) w(ξ) = ξw′(0)+ξ3

3!w′′′(0)−ℓ−1

(

Pw′′− K1w(ξ)

)

+Q0ξ4

4!+Q1

ξ5

5!+Q2

ξ6

6!,

where: P = P1 − K2, w′(0) and w′′′(0) are constants to be obtained. TheADM suggests the solution as a finite series in the form:

(21) w(ξ) =

N∑

n=0

wn(ξ),

where N is the truncation index that achieves the pre-assigned accuracy.

Substitution of Eqn (21) into Eqn (20) and assuming:

(22) w0(ξ) = ξw′(0) +ξ3

3!w′′′(0) + Q0

ξ4

4!+ Q1

ξ5

5!+ Q2

ξ6

6!.

Then, the components wn(ξ), n > 0 may be elegantly derived from therecurrence formula:

(23) wn(ξ) = −P ℓ−1w′′

n−1 − K1 ℓ−1wn−1.

Using Eqn (21), Eqn (22) and Eqn (23), the lateral deflection of thebeam in terms of the two unknowns w′(0) and w′′′(0) may be expressed as:

(24) w(ξ) = w0(ξ) + w1(ξ) + w2(ξ) + · · ·

The substitution of the boundary conditions at ξ = 1 yields the un-knowns w′(0) and w′′′(0).

3.2. Application of Recursive Differentiation Method

To use the RDM, the Eqn (16) is rewritten in the recursive form:

(25) w(4)(ξ) = −Pw(2)(ξ) − K1w(0)(ξ) + Q0 + Q1ξ + Q2ξ2.

Recursive Differentiation Method for Boundary Value Problems . . . 63

Let the solution of Eqn (25) be in the form:

(26) w(ξ) =N

∑

m=0

Tm

ξm

m!.

The recursive differentiations of Eqn (25) yield the recurrence formula:

(27) w(m)(ξ) = −Pw(m−2)(ξ) − K1w(m−4)(ξ) for m ≥ 7.

Using Eqn (3) and the boundary conditions at ξ=0, Tm for m<7 areobtained as:

T0 = 0, T2 = 0, T4 = Q0, T5 = −P T1−K1T3 +Q0 and T6 = −PQ0 +2Q2.

The recurrence formula for coefficients Tm , m ≥ 7 may be expressedas:

(28) Tm = −P Tm−2 − K1Tm−4 for m ≥ 7.

Then, the lateral displacement in terms of the two unknowns T1 and T3

is;

(29) w(ξ) = T1A1(ξ) + T3A2(ξ) + Q0A3(ξ) + Q1A4(ξ) + 2Q2A5(ξ),

where the recursive functions Ai(ξ) , i = 1, 2, . . . , 5 are:

A1(ξ) = ξ − K1ξ5

5!+ P K1

ξ7

7!− (P 2K1 − K12)

ξ9

9!

+(P 3K1 − 2PK1)ξ11

11!+ · · ·

A3(ξ) =ξ4

4!− P

ξ6

6!+ (P 2

− K1)ξ8

8!− (P 3

− 2PK1)ξ10

10!

+(P 4− 3P 2K1 + K12)

ξ12

12!+ · · ·

A5(ξ) =ξ6

6!− P

ξ8

8!+ (P 2

− K1)ξ10

10!− (P 3

− 2PK1)ξ12

12!+ · · ·

A2(ξ) = A′

3(ξ) and A4(ξ) = A′

5(ξ).

The boundary conditions at ξ=1 yield the coefficients T1 and T3 as:(30 − a)

T1 =(A32A21 − A31A22)Q0 + (A42A21 − A41A22)Q1 + (A52A21 − A51A22)Q2

A11A22 − A12A21,

(30 − b)

T3 =(A31A12 − A32A11)Q0 + (A41A12 − A42A11)Q1 + (A51A12 − A52A11)Q2

A11A22 − A12A21,

64 Mohamed Taha

where: Ai1 = Ai(1) and Ai2 = A′′

i(1), i = 1, 2, . . . , 5.

It is clear that the expressions obtained for the lateral displacementusing the RDM are similar to those obtained from the ADM but in a straight-forward procedure and with relatively short mathematical manipulations. Thebending moment distribution M(ξ) and the shearing force distribution V (ξ)along the beam may be obtained as:

(31) M(ξ) = −EI w′′(ξ)/L and V (ξ) = −EI w′′′(ξ)/L2.

Furthermore, inspection of Eqn (30) indicates that the lateral displacementis unbounded as the denominator approaches zero. Actually, this situationrepresents the buckling condition and the axial load in this case is denotedas the buckling (or critical) loads Pcr. Thus, the buckling loads Pcr can becalculated from the condition:

(32) A11A22 − A12A21 = 0.

3.3. VerificationAdditional verification is presented in Table 1 where values of the stabil-

ity parameter λ (λ =√

(PcrL2/EI)) are calculated using RDM and comparedwith those obtained from the FEM [13] and the DQM [14], though the pro-posed RDM is verified against the widely common ADM. It is clear that theRDM results for truncation index N = 12 are in close agreement with othertechniques which validate the accuracy of the proposed method.

Table 1. The stability parameter (λ) for P–P beam on elastic foundation

K2

K1 0 π2 2.5π2 Method

λ

3.1415 4.4428 5.8774 FEM

0 3.1414 4.4425 5.8719 DQM

3.1415 4.4429 5.8773 RDM (N = 12)

4.4723 5.4654 6.6840 FEM

100 4.4642 5.4595 6.6799 DQM

4.4724 5.4654 6.6840 RDM (N = 12)

4. Numerical resultsThe objectives of the present work are: 1) to implement the new pro-

posed RDM; 2) to investigate the distributions of the straining actions alongthe beam (bending moments M(ξ) and shearing force V (ξ)) due to different

Recursive Differentiation Method for Boundary Value Problems . . . 65

types of lateral loading (uniform, linear or nonlinear); and 3) to investigatethe influence of the different system parameters on the critical loads Pcr. Toachieve these objectives, the obtained expressions for the straining actions (Eqn31) and for the critical load (Eqn 32) are to be used.

However, investigation of Eqn (16) reveals that the influence of K2 onthe beam-column behaviour is inversely proportional to the influence of P1.Thus, the effect of K2 can be considered as an axial tension load (−P1) actingon the beam-column.

The obtained expressions may be inserted in a short MATLAB codeor even in an Excel spread sheet to draw the required results. Furthermore,although these expressions are obtained in dimensionless forms to be validfor any specific case, the properties of the beam considered in the presentparametric study are: concrete beam, b = 0.2 m, h = 0.5 m, L = 5 m,E = 2.1E10 Pa, Poisson ratio µ is 0.15 and the total lateral load acting on thebeam QT = 250000 N.

4.1. Identification of the foundation stiffness parameters K1

and K2

There are two models have been widely used to simulate the foundationinfluence, namely, the linear elastic foundation model (Winkler model) and thetwo parameter model (Pasternak model). However, in the case of two parametermodel, both the values of k1 and k2 increase with the increase in the foundationstiffness (weak, medium and stiff foundation etc.). Indeed, values of foundationparameters k1 and k2 depend on both the configurations of the foundation andthe beam. The following expressions may be utilized for the determinationof foundation parameters in the static analysis of rectangular beams on twoparameter elastic foundations [16].

(33) k1 =E0 b

2(1 − µ20)

γ

χand k2 =

E0 b

4(1 − µ)

χ

γ,

where: χ = 3

√

2EI (1 − µ20)

bE0(1 − µ2), E0 =

Es

1 − µ2s

and µ0 =µs

1 − µs

,

E and µ are the elastic modulus and Poisson ratio of the beam respec-tively, Es and µs for the foundation and γ is a parameter that accounts forthe beam-foundation loading configuration (it is a common practice to assumeγ = 1). Typical values of the elastic modulus and Poisson ratio for differenttypes of foundation are given in Table 2.

Investigations of Eqn (17) with Eqn (33) indicate that the values ofK1 and K2 depend on the type of the foundation, the beam material and the

66 Mohamed Taha

Table 2. Typical values for Es and µs for foundation [17]

Type of Soil Es N/m2 µs Type of Soil Es N/m2 µs

Loose sand 5E6 0.42 Soft clay (saturated) 3E6 0.5

Medium sand 1E7 0.38 Medium clay 3E7 0.35

Dense sand 4E7 0.3 Hard clay 1E8 0.25

Sand and gravel 1.2E8 0.25

Fig. 2. Influence of foundation stiffness on the critical load (K2 = 0)

slenderness ratio of the beam L/r (r =√

I/A, A is the beam cross sectionalarea).

4.2. Critical loadUsing Eqn (32), values of the critical loads for different values of foun-

dation linear stiffness parameter K1 are calculated and presented in Fig. 2. It isobserved, that the values of the critical loads parameter Pcr may be correlatedto the foundation stiffness parameters as:

(34) Pcr−n = K2 + n2π2 +K1

n2π2(for P–P beams),

where Pcr is the critical load parameter and n is the buckling mode. Both,the critical load parameter and the buckling mode for a certain configurationsdepend on the foundation stiffness. Actually, the beam buckling mode dependson the minimum potential energy of the beam-foundation system which de-termines the stable equilibrium position in the buckled configuration. As thefoundation stiffness increases, the system potential energy increases, but therate of increase in the higher modes is less than the rate of increase in the lowermodes. In other words, as the foundation stiffness increases to a certain value,the system minimum potential energy due to a buckling mode n approachesthat one for the buckling mode n + 1. However, if the foundation stiffness ex-

Recursive Differentiation Method for Boundary Value Problems . . . 67

ceeds that value, the system minimum potential energy of the buckling moden + 1 becomes smaller than that for buckling mode n and the beam bucklesin the mode n + 1 instead of the mode n. The value of the foundation stiff-ness parameter K1n at which the system minimum potential energy due to abuckling mode n is equal to that for a buckling mode n + 1 is given by:

(35) K1n = n2(n + 1)2π4.

If for a certain configuration of beam-foundation system, K1 exceeds K1n thebeam will buckle in the mode n+1. Further, for 0 < K1 < 4π4, the beam willbuckle in the first mode and for 4π4 < K1 < 36π4 the beam will buckle inthe second mode and so on. This fact reveals that the buckling fundamentalmode not always is the first mode and the critical load may be smaller thanthat calculated one using the first mode. Moreover, the investigations of thehomogeneous solution of Eqn (16) lead to the same conclusions which validatethe accuracy of the RDM.

4.3. Beam straining actionsThe distributions of M∗(ξ) and V ∗(ξ) are shown in Fig. 3 and Fig. 4

respectively, for different loading types and different system parameters. Thefollowing dimensionless parameters are defined:

(36) P ∗ =P1

π2, M∗(ξ) =

M(ξ)

MU (max)and V ∗(1) =

V (ξ)

VU (1),

Fig. 3. Effect of loading type on thedistribution of M∗(ξ)

Fig. 4. Effect of loading type onV ∗(ξ)

where MU (max), VU (1) are the maximum values of the corresponding para-meters for a free beam (beam without foundations) with the same propertiescarrying the same total load distributed uniformly along the beam. The loadingtypes are denoted as U for uniform loading, L for linear and N for nonlinearloading. It is clear that the straining actions increase as the axial load in-creases and as the foundation stiffness decreases. The effects of the type of

68 Mohamed Taha

loading; for the same total load; may be ignored in the calculations of thelateral displacement W , but it is more noticeable for both the bending mo-ment distribution and the shearing force distribution. The maximum values ofthe bending moment for uniform loading and linear loading are the same andslightly greater than the maximum bending moment due to nonlinear loading.It is also observed that the maximum shearing forces for different loading typesare relatively different and should be considered in the shear force calculations.

The effect of the axial load parameter P ∗ on the maximum values of thestraining actions (M∗ and V ∗) for different loading types is shown in Fig. 5. Itis found, that M∗ and V ∗ increase as P∗ increases. However, the influence ofP ∗ on M∗ and V ∗ decreases as the foundation stiffness increases. It should benoted that value of P ∗ for a beam resting on a foundation for a certain bucklingmode is greater than value of P ∗ for the same beam but without foundation.Actually, P ∗ of a beam without foundation is used to produce the given figures.In addition, the effects of the loading type (U , L or N) on both W ∗(max) andM∗(max) may be practically ignored but the effects of the loading type onV ∗(1) should be taken into consideration.

A. Bending moment M∗(max) B. Shearing force V ∗(1)

Fig. 5. Effect of axial load and foundation stiffness on the maximum strainingactions (K2 = 0)

5. Conclusions

The recursive differentiation method (RDM) is introduced and imple-mented to solve numerous examples of boundary value problems (BVP) infinite domain. It is illustrated that the method is simple and straightforwardin constructing analytical solutions for the given differential equations. In addi-tion, it is indicated that the obtained analytical expressions using the RDM forlinear, nonlinear and higher order differential equations are compatible with

Recursive Differentiation Method for Boundary Value Problems . . . 69

those obtained from closed solutions or resulted from other analytical tech-niques. Further, it is found that, the accuracy of the obtained expressionsusing RDM is greatly enhanced when the solution domain is transformed tothe domain [0, 1]. Moreover, a recurrence formula may be derived to decreasethe mathematical effort in constructing the solution expressions.

The RDM is used to investigate the static behaviour of beam-columnsubjected to different types of lateral loading and resting on two parameterfoundation. Exact expressions for critical loads are obtained. However, it isfound that if the foundation stiffness exceeds a certain level, the beam bucklesin the second mode rather than the first mode and the critical load in such casemay be much smaller than the critical load of the first mode. This result is veryimportant for the stability analysis of beams resting on elastic foundation suchas railway tracks, structural elements and piles. Further, the maximum valuesof straining actions affecting the beam cross section are obtained and illustratedagainst different types of loading types and different beam-foundation parame-ters. The results reveal that the method is versatile and efficient in dealingwith beam BVP.

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