Elastic Beam

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ANALYSIS OF BEAMS ON ELASTIC FOUNDATION: THE FINITEDEFFERENCES APPROACH

Teodoru I. Bogdan1

Abstract

In the solution of beams on elastic foundation problem, it is usual to use Winkler’s assumption or the concept ofmodulus of subgrade reaction. The general solution for the Winkler foundation is of limited use, since practical problems(may) involve beams of finite length, or changes in moment of inertia. Moreover, the solution cannot easily adapt to achange of soil’s modulus of subgrade reaction. Because of these shortcomings, the discrete elements methods, amongFinite Differences Method (FDM) and Finite Element Method (FEM), are most powerful and popular, are preferred for use,since all kind of contingencies may be accounted for.

This paper examines the use of the Finite Differences Method (FDM) for the analysis of elastic beams resting onWinkler medium. To estimate the deformation, soil reaction distribution and internal forces of a continuous footingsubjected to external loads, a computer program, based on Matlab code, has been developed. A comparison between FDM,FEM and analytical solutions is also presented.

Keywords

beams on elastic foundation, finite differences method, numerical analysis, Winkler’s assumption, continuous beam

1 INTRODUCTION

The acceptance of numerical analysis in engineering problems is growing. In particular, the development of numericalanalysis and its application to geotechnical problems over the last time have provided geotechnical engineers with anextremely powerful analysis tool. Moreover, the new codes of practice (e.g. Eurocode 7), are not as prescriptive as the oldercodes and allow the designer to choose an appropriate method of analysis, [3]. Nevertheless, beams on elastic foundationare most usually analysed based on Winkler’s concept in which the soil is treated as a bed of springs. To obtain thetheoretical solution of this approach is laborious and classical solutions are not general in their application. Several distinctdisadvantages of the classical solution are presented below [1]:

• Difficult to remove soil effect when footing tends to separate from soil

• Difficult to apply multiple types of loads to a footing

• Difficult to change cross section of the footing

• Difficult to allow for change in subgrade reaction along footing

For these reasons numerical analyses of a beam resting on an elastic foundation are shown in full detail and modellingaspects will be discussed (e.g. discretization dependency). Finally, the results of the numerical analyses are compared withthe result of the general solution.

2 BASICS OF FINITE DIFFERENCES FORMULATION

Finite difference schemes provide an alternative route to the conversion of continuum field equations into

relationships between discrete numerical values. This method discretizes the domain into a regular grid defined by a certainnumber of nodes which are separated in the coordinates direction by a certain spatial interval. When applying the FDMover the domain, we will be able to approximate the objective function (e.g. displacements) at each one of the nodes.

The FDM consists of transforming the partial derivatives in difference equations over a small interval, using thedevelopment in Taylor's series of the objective function at each node, leading to an equation that relates the value of thefunction at a particular node with its value at the neighbouring nodes. This procedure is repeated at each node composingthe grid and the assembly of the obtained equations yields a system of equations which may be numerically solved.

The method is best illustrated by a physical example, from structural mechanics. Taking the case of a beam inbending, the average slope of the elastic curve at distance xi is given by:

1 Teodoru I. Bogdan, Ph.D. student, “Gh. Asachi” Technical Univesity of Ia ş i, Faculty of Civil Engineering, Department of

Transportation Infrastructure and Foundations, Bd. Dimitrie Mangeron nr. 67, 700050 Iasi IS, Romania, e-mail:[email protected]



where y is deflection.

In terms of finite differences, assumed ∆ x = λ as finite values, Equation (1) can be evaluated by three approximations:

1) Central difference

2) Forward difference

3) Backward difference

The last equations are called a finite-difference equation . Solving these equations gives an approximate solution to thedifferential equation.

In an analogous way, we can obtain finite difference approximations to higher order derivatives and differentialoperators. For example, by using the above central difference formula with step λ /2 for ( x + λ /2) and ( x − λ /2) and thenapplying a central difference formula for the derivative at xi, we obtain the central difference approximation of the secondderivative:

By differentiating Equation (5) one, obtains the central difference expression for the third derivative as

These last two expressions are sufficient to solve the beam resting on an elastic foundations problem by FDM.

3 SOLUTION OF A BEAM ON AN ELASTIC FOUNDATION USING FDM

The differential equation of the deflection curve for a bending beam is given by:

where:

- E is Young’s modulus;

- I is the moment of inertia;

- M is bending moment.

By differentiating the above equation, with respect to x and taking into account relationship between bending momentand shearing force, we obtain expression of the shearing force:

By substituting Equations (5) and (6) into Equations (7) and (8) we have:

For a foundation beam, by considering Winkler’s concept, one can replace the foundation with a series ofconcentrated springs on the base of the footing, as shown in Fig.1. The soil pressure at any point on the beam is directly

( )ii

dytan

dx

θ = (1)

i 1 i 1

i

y ydydx 2

+ −− ≈

λ (2)

i 1 i

i

y ydydx

+ − ≈

λ (3)

i i 1

i

y ydydx

−− ≈

λ (4)

2i 1 i i 1

2 2i

y 2y yd ydx

+ − − +

≈ λ

(5)

3i 2 i 1 i 1 i 2

3 3i

y 2y 2y yd ydx 2

+ + − − − + +

≈ λ

(6)

2

2

d yEI M

dx= − (7)

3

3

d yEI V

dx= − (8)

( )i 1 i i 1 i2

EIy 2y y M+ −− + = −

λ (9)

( )i 2 i 1 i 1 i 2 i3

EIy 2y 2y y V

2 + + − −− + + = −λ

(10)



proportional with beam deflection ( y) and modulus of subgrade reaction ( k ). For instance at the point i, soil pressure isrelated by:

Fig. 1 Winkler’s concept

By discretizing the beam domain into n elements of ∆ x = λ = constant, and considering pressure distribution of soilgiven by known function p = f(x) , one can compute the soil reaction against the beam, at each node i = 1 to n + 1 . One mayuse any type of pressure distribution of soil to footing (for details see [4]), but for computational simplicity a stepped

pressure distribution is use. Thus, the reactions against the beam of Fig. 2 become:

where B is width of footing.

Fig. 2 Mathematical model for the FD solution for a beam on elastic foundation

By transposing Equation (7) in finite differences, for each node composing the grid, except extremity points (to avoidlimit condition), the bending moment at the point k = 2 to i-1 is given by:

where:

- M P is the bending moment of axial loads;

-

M is concentrated moment.

( ) ( )i ip k y= (11)

( ) ( ) ( )k k k R p B k y B= λ = λ for k = 2 to i-1 (12)

( ) ( ) ( )1 1 1

1 1R p B k y B

2 2= λ = λ (13)

( ) ( ) ( )i i i

1 1R p B k y B

2 2= λ = λ (14)

( ) ( ) j i 1

k 1 k k 1 j P2 j 1

EIy 2y y R k j M ( ) M

= −

− +=

− + = − − λ − + −

λ ∑ ∑ ∑ (15)



Making notation EI / λ 2 = C and arranging, we have:

Thus we have obtained i-2 equations in i unknown values of displacements y. The number of equations with finite

differences being less than unknown’s number, we must complete with two equilibrium equations.The sum of the moments with respect to right end, for instance:

and the sum of forces in the vertical direction:

We have now obtained i simultaneous linear equations in i unknown values of displacements y.

A numerical scheme for solving a system of equations is matrix method. By arranging system of equations (19) for acomputer solution by the matrix method, we have:

• Coefficient matrix:

where:

- C R =k B λ

- m is the row’s indices.

The foregoing matrix can be easily carried out by hand, for a low number of division points (in literature [4] it isrecommended to use 10 divisions); increasing number of division points leading to a more accurate solution, but in thesame time yields difficulty in writing coefficient matrix.

• Free terms vector: is formed by right hand side terms from Equations (19) (moments of all external loads withrespect to the each station point i =2 to i-1 )

Once the system of equations (19) is solved, one can compute bending moment and shearing force, by backsubstitution of the beam deflection ( yi) into Equations (9) and (10) at each division point form the beam domain. Soilreaction against the beam at a point i is given by Equation (11).

( ) j i 1

k 1 k k 1 j P j 1

Cy 2Cy Cy R k j M ( ) M= −

− +=

− + + − λ = + −∑ ∑ ∑ (16)

j i 1 m 1

P j 1 m i 1

R m M= − =

= = −

λ =∑ ∑∏ (17)

i

i1

R P=∑ ∑ (18)

( ) ( )

( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

1 2 3 1 P 1 1

2 3 4 1 2 P 2 2

3 4 5 1 2 3 P 3 3

i 2 i 1 i 1 2 i 1 P i 1 i 1

1 2 3 i 1

Cy 2Cy Cy R M ( ) M

Cy 2Cy Cy R 2 R M ( ) MCy 2Cy Cy R 3 R 2 R M ( ) M

Cy 2Cy Cy R i 1 R i 2 ... R M ( ) M

R i 1 R i 2 R i 3 ... R

− − − − −

−

− + + λ = + −

− + + λ + λ = + −− + + λ + λ + λ = + −

− + + − λ + − λ + + λ = + −

− λ + − λ + − λ + + λ =

∑ ∑

∑ ∑∑ ∑

∑ ∑M

( ) ( )P i i

1 2 3 i

M ( ) M

R R R ... R P

+ −

+ + + + =

∑ ∑∑

(19)

( )( ) ( )( ) ( ) ( )( ) ( ) ( )

R

R R

R R R

R R R R

R R R R R

R R R R R R

R R R R

C 0,5C 2C C 0 0 0 0 0C C C 2C C 0 0 0 0

1,5C 2C C C 2C C 0 0 0 0

0,5 m C C 2C C C 2C C 0 0m 10,5 m C C C 2C C C 2C C 0m 1 m 20,5 m C C C C 2C C C 2C Cm 3m 1 m 20,5 m C C C Cm 3m 1 m 2

+ λ −λ + λ −

λ λ + λ −

λ λ λ + λ −− λ λ λ λ + λ −− − λ λ λ λ λ + λ −−− − λ λ λ λ−− −

L L

L LL

ML L

L LL L

( ) R R R

R R R R R R R R R

C 2C C 0m 40,5C C C C C C C C 0,5C

λ λ λ−

L LL

(20)



4 COMPUTATIONAL EXAMPLE AND COMPARATIVE ANALYSIS

In order to estimate the capabilities and the advantages of the solution described foregoing, let us consider in somedetail a continuous footing as a beam on elastic foundation with loading condition shown in Fig. 4. The solution can beadapted to continuous footings with any number of column loads which may include both axial loads as well as moments.

Fig. 4 Assumed loading for Winkler foundation

For the analysis of the longitudinal bending behaviour of the beam, three different methods (FEM, FDM and generalmethod) were performed.

In the analytical solution and FE analysis, the beam is divided into 13 elements by 0,4 m (left span), 0,5 m (middlespans) and 0,35 m for the right span. In FD method the beam is also divided into the same number of elements, but ofconstant length.

For the assumed discretization of the given beam, FDM requires the solution of 14 simultaneous equations, thusnecessitating the use of a computer. In order to generate the matrix (20) for FD analysis, a computer code listing inAppendix, based on Matlab language, has been developed. This can be used for any number of division points along beam.

Considering for reference the solution obtaining by general method, comparative results obtained by FD and FEanalysis are plotted in Fig. 5. From the plot it is evident that either of the FEM or FDM solutions tends, with a goodaccuracy, to analytical result. The FDM solution yields more appropriate results, regarding negative bending moments,whereas FEM solution gives a good approach for positive moments.

For this discretization step, FDM doesn’t give a very accurate solution regarding the positive bending moment, but itsaccuracy can be easily controlled by changing the grid size. For instance, by increasing five times the number of divisionsyields the bending behaviour of the beam plot in Fig. 6.

-80

-60

-40

-20

0

20

4060

0 1 2 3 4 5 6Distance along beam [m]

B e n d i n g

M o m e n t

[ k N m

] F DM

F EMAna lytical So lution

Fig. 5 Graphical comparison of bending behaviour of the beam

FDM computes displacements strictly at predetermined grid points only (unlike FEM, it doesn’t computedisplacement functions that can be used to interpolate displacements at the points that are not located at the grid); to obtainan appropriate solution in singularity points (e.g. points of application of external forces), those must be in coincidencewith a point from the grid domain.



Referring to beam with loading condition given in Fig. 4, filling its domain with a discretization with step chosenaccording with foregoing, so with λ = 10, 5, 2, 1, 0,5 (cm) and so on, the values of bending moments are in goodagreement with the resuls obtained by FEM for 65…80 division points (see Table 1).

Tab. 1 Comparison of bending moments values (in kNm) obtained by FD and FE analyses

-80

-60

-40

-20

0

2040

600 1 2 3 4 5 6

Distance along beam [m]

B e n d i n g

M o m e n t

[ k N m

] FDM - 65 elements

Ana lytica l So lutio n

Fig. 6 Discretization step dependency for bending behaviour of the beam

For another discretization steps, different then λ = 10, 5, 2, 1, 0,5 (cm)..., the solution regarding positive bendingmoments is plotted in Fig. 7. From this, one can see that for a coarse grid the solution is very unstable and the convergenceis obtained for a nodes number by 3000 ( λ = 0,1 cm).

The discretization with such a huge number of division points (which can increase computing time) can be avoid ifone chose different discretization steps, but not in the same time. For instance, if we want to find the value of the leftpositive bending moment we can perform an analysis with λ = 40 cm (so 15 elements); a discretization step by λ = 30 cm(so 20 elements) will give the value of the middle positive bending moment.

30

35

40

45

50

55

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Number of divisi on points [x 10 2]

B e n d i n g

M o m e n t s

[ k N m

]

leftmiddleright

Fig. 7 Discretization step dependency for positive bending moments

The soil pressure distribution evaluated from the numerically computed using FD and FE and pressure distributionobtained by analytical method are compared in Fig. 8. Good agreement between the foregoing is seen, the diagrams ofpressure being almost identically for any discretization step.

M+left M -

left M +middle M -

right M +right

FDM 52,5 72,6 45,6 39,0 45,1

FEM 52,5 72,7 45,6 39,1 45,0



0

50

100150

200

250

300

350

0 1 2 3 4 5 6Distance al ong beam [m]

S o

i l p r e s s u r e

[ k P

a ]

FD MFE M

Analytica l So lution

Fig. 8 Soil pressure distribution along beam

5 CONCLUSIONThe results from FD analysis are generally in good agreement with both analytical and FE results. One limitation of

the FDM, however, is that it computes displacements at predetermined grid points only and thus accuracy of the solutionobtained is affected somehow by grid size. To obtain an appropriate solution in singularity points (e.g. points of applicationof external forces), those must be in coincidence with a point from the grid domain; if this can’t be reached one can chosedifferent discretization steps, but not in the same time.

Anyway, the FD approach can provide satisfactory prediction of the structural behaviour of beams resting on elasticfoundation and thus of use in professional engineering work.

ACKNOWLEDGEMENTS

The author is grateful to Professor Mu ş at Vasile for his helpful comments, guidance and support in the research work

presented in this paper.

Appendix - Computer code to generate coefficient matrix for FD analysis

%F.D.M. FOR BEAMS ON ELASTIC FOUNDATION%Coeffs matrix%========================================================% INPUT DATAL=input ('Beam length -- L = ');n=input('Number of elements -- n = ');l=L/n; % << -------- Discretization step <<lambda>>k=input('Modulus of subgrade reaction -- k = ');B=input('The width of the footing -- B = ');I=input('Beam''s moment of inertia -- I = ');E=input('Young''s modulus for footing -- E = ');C=E*I/(l^2); CR=k*B*l;%========================================================

% Nodesi=n+1;%Domain gridVl=1:(i-1); VL=Vl*l;%Eq 1 to i-2

for li=1:i-2,for c=1:i,

for m=1:i-3,if li-c == -2,MCRl(li,c)=C;elseif li-c == -1,MCRl(li,c)=-2*C;elseif li == c,MCRl(li,c)=C+CR*l;elseif li-c == m,

MCRl(li,c)=(m+1).*CR*l;end



endend

endMc=MCRl;%M_end=0

for j=i-1:-1:1VM0(j)=j*CR*l;VMc=[fliplr(VM0),0];

end%sumFv=0

for j=i:-1:1VMf(j)=CR;

end%Assembly

M=[Mc;VMc;VMf];M(:,1)=M(:,1)*0.5;M(1,1)=C+0.5*CR*l;M(i,i)=0.5*CR;M %< --- Coefficient matrix

Literature

[1] Bowles J. E., - Foundation Analysis and Design, 3rd Ed. , New York: MCGRAW-HILL EDUCATION, 1982, 816pages, ISBN 19820070661928.

[2] Muir Wood D., - Geotechnical Modelling , Abingdon: TAYLOR & FRANCIS, 2004, 480 pages, ISBN 0415343046.

[3] Potts D. M., - Numerical analysis: a virtual dream or practical reality? , GEOTECHNIQUE, Vol. 53, No. 6, pp. 535-572, 2003, ISSN 0016-8505.

[4] Winterkorn H.F., Fang H.Y. - Foundation Engineering Handbook , New York: VAN NOSTRAND REINHOLD CO.,1975, 751 pages, ISBN 0442295642.

Reviewer

Mu ş at Vasile, Professor, Ph.D., “Gh. Asachi” Technical Univesity of Ia ş i, Faculty of Civil Engineering, Department ofTransportation Infrastructure and Foundations, Bd. Dimitrie Mangeron nr. 67, 700050 Iasi IS, Romania, Phone: +40 745574 061, e-mail: [email protected].

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