Mixed formulation of nonlinear beam on foundation elements

Mixed formulation of nonlinear beam on foundation elements

Ashraf Ayoub *

Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL 33620, USA

Received 24 April 2002; accepted 24 November 2002

Abstract

This paper presents an inelastic element for the analysis of beams on foundations. The element is derived from a two-

field mixed formulation with independent approximation of forces and displacements. The state determination algo-

rithm for the implementation of the element in a general purpose nonlinear finite element analysis program is presented

and its stability characteristics are discussed. Numerical studies are performed to compare the model with the classical

displacement formulation. The studies confirm the superiority of the proposed model in describing the inelastic be-

havior of beams on foundations.

� 2003 Elsevier Science Ltd. All rights reserved.

Keywords: Beam on foundation; Beam on Winkler support; Beam on lateral support; Mixed formulation; Mixed variational principle;

Mixed finite element

1. Introduction

Beams on foundations are commonly used in mod-

eling civil engineering problems. They are typically used

for modeling beams resting on soils, but are also used in

several other applications (e.g. railroad tracks). The

Winkler foundation model [1] represents the simplest

form of these types of beams. In this model the foun-

dation is treated essentially as an array of closely spaced

but noninteracting springs, each having a spring stiffness

that equals the foundation modulus divided by the

spacing between springs. In most practical applications

the foundation is used to model soil behavior, and is

thus assumed tensionless.

Analysis of beams on foundations have been exten-

sively studied for the linear elastic case. Little attention,

however, has been given to their behavior in the non-

linear range. Hetenyi [2] and Timoshenko [3] presented

an analytical solution for beams on elastic supports

using classical differential equation approach, and

considering several boundary and loading conditions.

Miranda and Nair [4] simplified the process by using the

method of initial condition. Bowles [5] derived a stiffness

matrix for the problem using a conventional beam ele-

ment supported on discrete springs only at its ends. Ting

and Mockry [6] developed a stiffness matrix of an elastic

beam on lateral support suitable for the displacement

method of analysis. The matrix is determined by deriv-

ing the exact solution of the differential equation of the

problem. Lai et al. [7] extended the same work for dy-

namic cases. Yankelevesky and Eisenberg [8,9] used the

same approach and developed an exact stiffness matrix

for beam on elastic foundations including axial effects.

The finite element method using displacement shape

functions have been proposed in recent years by several

researchers [10–12]. Aydogan [13] also used displace-

ment finite element formulations to analyze beams on

lateral supports including shear deformations. Thambi-

ratnam and Zhuge [14] used the same approach to study

the vibration and dynamic behavior of theses types of

structures.

Inelastic analysis of beams on lateral supports have

focused primarily on beams on tensionless foundations.

Weistman [15], Rao [16], and Choros and Adams

[17] studied the effect of moving loads on beams on

tensionless foundations. Kaschiev and Mikhajlov [18]

used the finite element method as a general numerical

technique to solve the problem of elastic beams on

*Tel.: +1-813-974-4746; fax: +1-813-974-2957.

E-mail address: [email protected] (A. Ayoub).

0045-7949/03/$ - see front matter � 2003 Elsevier Science Ltd. All rights reserved.

doi:10.1016/S0045-7949(03)00015-4

Computers and Structures 81 (2003) 411–421

www.elsevier.com/locate/compstruc

mail to: [email protected]

tensionless foundations for different loading conditions.

Yim and Chopra [19] studied the earthquake response of

beams on tensionless lateral supports. To account for

the nonlinear nature of the foundation, several re-

searchers used the concept of beam on nonlinear Win-

kler foundation (BNWF) (e.g. [20,21]). In that simple

model the foundation constitutive law follows a non-

linear p–y relation, while the beam is assumed elastic.

Inelastic models in which the beam moment–curvature

relation, and the foundation load–deformation relation

are both assumed to follow inelastic behavior, are

not commonly used due to their expensive numerical

cost. A displacement-based finite element formulation

represents a limitation in this case, since it usually re-

quires a fine finite element mesh to capture the behavior,

which renders the problem computationally expensive.

The objective of this paper is to develop a new finite

element formulation that is capable of capturing the

nonlinear behavior of both the beam and foundation

elements, without sacrifice of computational efficiency.

Ayoub and Filippou [22,23] proposed a consistent mixed

formulation for inelastic analysis of anchored bars

and composite structures respectively. The new mixed

formulation overcomes most of the limitations of dis-

placement models, and thus represents a superior ap-

proach. This paper extends the newly developed mixed

formulation to analyze inelastic beams on foundations,

where both the beam and foundation are assumed to be

inelastic.

In the next sections, both displacement and mixed

formulations are developed. The models are imple-

mented in the finite element program FEAP, developed

by Taylor, and described in details in Zienkiewicz and

Taylor [24]. Numerical examples that compare the be-

havior of both models are then performed, and conclu-

sions based on these results are derived. The governing

equations of beams on foundations (strong form) are

presented first.

2. Strong form equations for beams on foundations

2.1. Equilibrium

Consider an element of length dx of a beam element

resting on a foundation as shown in Fig. 1. The equa-

tions of equilibrium are:

V;x þ ðw� tfÞ ¼ 0

M;x þ V ¼ 0ð1Þ

V , and M denote the shear force and bending moment,

respectively; tf is the foundation force per unit length, wdenotes the distributed load on the beam, and , and ,,

denote first and second derivatives respectively. All

terms in (1) are functions of x. From the two equations

in (1):

M;xx � wþ tf ¼ 0 ð2Þ

2.2. Compatibility

The compatibility equation of the beam is:

v;xx � v ¼ 0 ð3Þ

where v is the vertical displacement of the beam, and v isthe curvature.

2.3. Constitutive force–deformation relations

The internal moment of the beam MðxÞ is related tothe curvature v by a nonlinear constitutive relation

MðxÞ ¼ ggðvðxÞÞ ð4Þ

In this study the nonlinear relation in (4) is derived

from a fiber discretization of the cross-section of the

beam with nonlinear, uniaxial stress–strain relations for

the constituent materials. This method of discretization

automatically accounts for the spread of yielding along

the section depth. Spacone [25] investigated the effect of

number of fibers on the convergence of the finite element

solution of reinforced concrete columns. He concluded

that sections with very fine fiber discretization do not

improve much the accuracy of the solution, but rather

increase the computational cost considerably.

The foundation force tf is related to the vertical dis-placement v by another nonlinear relation as follows:

tf ¼ ggfðvÞ ð5Þ

In the next sections, the strong form equations (1)–(5)

are solved for using the finite element method. Due to

the nonlinear nature of Eqs. (4) and (5), a Newton–

Raphson iteration strategy is used. The following dis-

cussion refers to a single Newton–Raphson iteration

denoted by subscript i.

V V+dVM M+dM

w

tf

dx

Fig. 1. Infinitesimal segment of a beam on foundation.

412 A. Ayoub / Computers and Structures 81 (2003) 411–421

3. Displacement formulation of beam on foundation

element

In the displacement formulation, the displacements

serve as primary variables. The governing equations of

the problem are cast in the following form:

(1) Assuming predefined continuous displacement

fields along the beam length:

vðxÞ ¼ aðxÞv ð6Þ

where vðxÞ is the vertical displacement at any point x, v isthe vector of end displacements and rotations, and aðxÞis an nd row vector of displacement interpolation func-

tions.

(2) Deriving the weighted integral form of the equi-

librium equations at the current iteration step iZ L

0

dvTðxÞ Mi;xxðxÞ

hþ tifðxÞ � wðxÞ

idx ¼ 0 ð7Þ

where dvðxÞ is a weighting function. For the sake ofsimplicity the effect of element loads wðxÞ is not pursuedfurther in this study. Integrating by parts twice the first

term in (7), we get:Z L

0

dvT;xxðxÞMiðxÞdxþZ L

0

dvTðxÞtifðxÞdx

¼ Boundary terms ðBTÞ ð8Þ

The consistent linearization of the force–deformation

relation of both the beam and foundation yields:

Mi ¼ ki�1Dvi;xx þMi�1

tif ¼ ki�1f Dvi þ ti�1f

ð9Þ

where k and kf are the beam section and foundation

stiffness terms respectively at the end of the last itera-

tion.Substituting (9) into (8) results inZ L

0

dvT;xxðxÞ ki�1Dvi;xxh

þMi�1idx

þZ L

0

dvTðxÞ ki�1f Dvi�

þ ti�1f

�dx ¼ BT ð10Þ

(3) Substituting the predefined displacement shape

functions into the weak form (10), and using a Galerkin

approach for the weighting function, we get

dvTZ L

0

aT;xxðxÞki�1a;xxdx� �

Dvi�

þZ L

0

aTðxÞki�1f adx� �

Dvi

¼ dvT P

��Z L

0

aT;xxðxÞMi�1 dx�Z L

0

aTðxÞti�1f dx�

ð11Þ

From the arbitrariness of dvT, Eq. (11) could be writtenas:,

Ki�1þ Ki�1

f

�DVi ¼ P�Mi�1 �Mi�1

f ð12Þ

where Ki�1 ¼R L0aT;xxk

i�1ðxÞa;xxðxÞdx is the beam ele-

ment stiffness matrix, Ki�1f ¼

R L0aTðxÞki�1f ðxÞaðxÞdx

is the foundation element stiffness matrix, Mi�1 ¼R L0aT;xxM

i�1ðxÞ dx is the beam element resisting load vec-

tor, Mi�1f ¼

R L0aTðxÞti�1f ðxÞ dx is the foundation element

resisting load vector and P is the vector of applied ex-

ternal loads.

Since (12) includes second derivatives of the vertical

displacement interpolation functions aðxÞ, these func-tions have to be C1 continuous. Thus, this formulationdoes not enforce curvature continuity at element

boundaries. Typically, Hermitian cubic functions are

used to approximate the vertical displacements. Higher

fifth order polynomial functions could be also used. In

this case, the middle degrees of freedom are condensed

out at the element level.

4. Mixed formulation of beam on foundation element

In the mixed formulation, the differential equations

are solved based on both a displacement and a force

interpolation functions. Several forms of the mixed

formulation exist: The v�M field, the v� v �M field

and the v� v � tf field, where v is the vector of dis-

placements,M is the vector of element moments, v is the

vector of curvature, and tf is the vector of foundation

force. For the beam on foundation problem, it is known

that the curvature field shows a steep distribution in the

plastic zone, and the foundation force shows a nonuni-

form distribution along the beam length that depends

mainly on the foundation force–deformation constitu-

tive law. The bending moment and displacement fields,

however, show a much smoother distribution along the

beam length. Consequently, it is more advantageous

from a numerical standpoint to approximate the dis-

placement and moment field rather than the curvature

and foundation force fields. Therefore, the v�M field is

adopted in the present study. The v�M model is based

on the following:

(1) Assuming a predefined displacement field vðxÞalong the beam length

vðxÞ ¼ aðxÞv ð13Þ

where vðxÞ is the vertical displacement at any point x, v isthe vector of end displacements and rotations, and aðxÞis an nd row vector of displacement interpolation func-

tions.

(2) Assuming a predefined bending moment inter-

polation function along the beam length

MðxÞ ¼ bðxÞM ð14Þ

where MðxÞ is the internal bending moment at any pointx, M is the vector of beam end moments, and bðxÞ is anns row vector of force interpolation functions.

A. Ayoub / Computers and Structures 81 (2003) 411–421 413

(3) Deriving the weighted integral form of the com-

patibility equation

Z L

0

dMTðxÞ½vi;xxðxÞ � jiðxÞ�dx ¼ 0 ð15Þ

where dMðxÞ is a weighting function.The incremental section constitutive law is then in-

verted in order to ensure symmetry, as discussed in

Zienkiewicz and Taylor [24]:

ji ¼ f i�1DMi þ ji�1 ð16Þ

ji�1 is the beam curvature at the end of the last iteration

and f i�1 the corresponding beam section flexibility co-

efficient. Substituting (16) in (15) yields

Z L

0

dMTðxÞ vi;xxðxÞh

� f i�1DMi � ji�1idx ¼ 0 ð17Þ


functions and force interpolation functions into the

weak form (17), and using a Galerkin approach for the

weighting function, we get

dMT

Z L

0

bTðxÞa;xxðxÞdx� �

vi�

�Z L

0

bTðxÞf i�1ðxÞbðxÞdx� �

DMi

�Z L

0

bTðxÞji�1ðxÞdx

¼ 0 ð18Þ

From the arbitrariness of dM, it follows thatZ L

0

bTðxÞa;xxðxÞdx� �

vi �Z L

0

bTðxÞf i�1ðxÞbðxÞdx� �

DMi

�Z L

0

bTðxÞji�1ðxÞdx ¼ 0 ð19Þ

Substituting vi by vi�1 þ Dvi, (19) becomes

TDvi � Fi�1DMi � vi�1r ¼ 0 ð20Þ

with

T ¼Z L

0

bTðxÞa;xxðxÞdx;

Fi�1 ¼Z L

0

bTðxÞf i�1ðxÞbðxÞdx;

vi�1r ¼Z L

0

bTðxÞji�1ðxÞdx� Tvi�1 ð21Þ

F is the beam flexibility matrix and vi�1r is the vector of

nodal displacement residuals at the end of the previous

Newton–Raphson iteration. These residuals represent

the compatibility error between node displacements vi�1

and the corresponding deformation field ji�1ðxÞ beforereaching convergence.

(5) Deriving the weighted integral form of the equi-

librium equations

Z L

0

dvTðxÞ Mi;xxðxÞ

hþ tifðxÞ � wðxÞ

idx ¼ 0 ð22Þ

where dvðxÞ is a weighting function. As mentioned be-fore, the effect of element loads w is not pursued further

in this study. Integrating by parts twice the first term in

(22), we get:

Z L

0


0

dvTðxÞtifðxÞdx ¼ BT ð23Þ

The incremental force–deformation relation of the

foundation takes the form

tif ¼ ki�1f Dvi þ ti�1f ð24Þ

Substituting (24) into (23) results in

Z L

0


0

dvTðxÞ ki�1f Dvi�

þ ti�1f

�dx ¼ BT

ð25Þ


functions and force interpolation functions into the

weak form (25), and using a Galerkin approach for the

weighting functions, we get

dvTZ L

0

aT;xxðxÞbðxÞdx� �

Mi

�þZ L

0

aTðxÞ ki�1f Dvi�

þ ti�1f

�dx

¼BT ð26Þ

Substituting (13) in the last equation, it becomes

dvTZ L

0

aT;xxðxÞbðxÞdx� �

Mi

�þ

Z L

0

aTðxÞki�1f aðxÞdx� �

Dvi

þZ L

0

aTðxÞti�1f ðxÞdx

¼ BT ð27Þ

Using the first relation from (21) in the last equation

yields

dvT TTMi

þ Ki�1f Dvi þMi�1

f

�¼ BT ð28Þ

where Ki�1f ¼

R L0aTðxÞki�1f ðxÞaðxÞdx is the foundation

element stiffness matrix, Mi�1f ¼

R L0aTðxÞti�1f ðxÞdx is the

foundation element resisting load vector.

Noting that the BT in (28) can be written as dvTP,where P are the applied nodal forces and taking account

of the arbitrary nature of virtual displacements dv sim-plifies (28) to

TTMi þ Ki�1f Dvi þMi�1

f ¼ P ð29Þ

After replacing Mi ¼ Mi�1 þ DMi in (29) and moving

the resisting forces to the right hand side, the equili-

brium equation takes the form

TTDMi þ Ki�1f Dvi ¼ P�Mi�1 �Mi�1

f ð30Þ


Writing Eqs. (30) and (20) in matrix form:

�Fi�1 T

TT Ki�1f

� �DMi

Dvi

� �¼ vi�1r

P�Mi�1 �Mi�1f

� �ð31Þ

Eq. (31) represents the matrix form of the mixed for-

mulation.

It is important to note that at a solution, the residual

deformation vi�1r vanishes independently in each element

satisfying element compatibility.

Since the matrix form given above includes second

derivatives of the vertical shape functions, they have

to be C1 continuous, while the moment interpolationfunction bðxÞ can be discontinuous.

5. State determination of mixed formulation

As discussed by Ayoub and Filippou [22,23], two

algorithms for the mixed formulation exist. The first

requires only a global Newton–Raphson iteration. The

second requires an additional internal element iteration

besides the global Newton–Raphson iteration. The two

algorithms are discussed next.

Algorithm 1. In the first algorithm, the system of equa-

tions in (31) is solved for globally with the displacements

and moments as degrees of freedom. No external loads

are applied to the moment degrees of freedom. At global

convergence, the residual deformation vector vr vanishes

inside each element. This algorithm enforces moment

continuity at element boundaries by solving for the

moments as independent global degrees of freedom.

Algorithm 2. In the second algorithm, the moment de-

grees of freedom are condensed out at the element level

resulting in a generalized displacement stiffness matrix.

Accordingly:

From the first of Eq: ð31Þ :DMi ¼ ðFi�1Þ�1 TDvi

� vi�1r

�ð32Þ

Substituting (32) in the second of Eq. (31), we get

TTðFi�1Þ�1 TDvi

� vi�1r

�þ Ki�1

f Dvi ¼ P�Mi�1 �Mi�1f

ð33Þ

Simplifying (33), we get

TTðFi�1Þ�1Th

þ Ki�1f

iDvi

¼ P�Mi�1 �Mi�1f þ TTðFi�1Þ�1vi�1r ð34Þ

An internal element iteration is required in order to

zero the residual deformation vector vr in every element.

The algorithm used follows the same procedure as the

one discussed in Ayoub [26]. At convergence the residual

vector vr vanishes leading to

Ki�1totalDv

i ¼ P�Mi�1total ð35Þ

where Kitotal ¼ fTT½Fi�1��1Tþ Ki�1

f g, Mi�1total ¼ TTMi�1þ

Mi�1f , Ktotal is the element total generalized global stiff-

ness matrix, and Mtotal is the element total generalized

global resisting load vector.

Since moments are condensed out inside each ele-

ment, this algorithm does not guarantee moment con-

tinuity at element boundaries. However, with the proper

choice of interpolation functions, the jump in moment

distributions at element boundaries could be minimized

as described in the next section.

5.1. Stability of the mixed formulation

As discussed by Zienkiewicz and Taylor [24], it is

recommended that the force continuity be relaxed lo-

cally, since excessive force continuity typically results in

oscillation of results. This can be accomplished only by

using Algorithm 2, where the forces are condensed out

at the element level.

Despite the relaxation of shape function continuity in

mixed principles, a limitation in the choice of individual

shape functions exists. Babuska and Brezzi [27,28] dis-

cussed the reasons for that difficulty. The Babuska–

Brezzi (B–B) condition of stability applied to the beam

on lateral support problem could be derived by simple

reasoning as follow: For stability of the formulation, the

rank of matrix T in the expression TTF�1T in (34) should

not be larger than the rank of the flexibility matrix F for

the limit case where the foundation stiffness matrix is

zero. For this to be the case the number of unknowns n0din vector v after excluding the rigid body modes should

be less or equal to the number of unknowns ns in vectorM. The B–B condition of stability, therefore becomes:

ns P n0d ð36Þ

It is important to note that the number of rigid body

modes for the problem equals 2.

While condition (36) is necessary for stability of the

problem, certain care should be exercised when applying

it. According to De Veubeke�s principle of limitation[29], there is no accuracy gain by increasing the order

of the force field beyond the order of the deformation

field that respects the strain–displacement compatibility

condition. The equality of condition (36), i.e. ns ¼ nd0 istherefore the most efficient choice from a computational

standpoint. While this is only true for linear elastic be-

havior, that selection minimizes the jump in moment

distributions across element boundaries in the nonlinear

range. Conventional cubic Hermitian polynomials for

the vertical displacements are thus used, along with

linear interpolation functions for the bending moment.

It is well known that Hermitian polynomials are C1

continuous. Fifth order polynomials for the vertical

displacements and cubic polynomials for the moment is


another possibility. In this case, the degrees of freedom

of the middle node are condensed out at the element

level. The mixed model with the previously mentioned

choices of interpolation functions also satisfies the inf–

sup condition (Bathe [30,31]), and furthermore repre-

sents the best formulation for the beam on foundation

problem in terms of number of elements needed to reach

convergence. Numerical examples to confirm this state-

ment are presented next.

6. Evaluation of model by numerical studies

In order to evaluate the efficiency of the new formu-

lation, a comparison with standard displacement-based

models is conducted on sample beam on foundation

problems. Two of these examples are described next.

The first beam is simply supported under midspan

loading as shown in Fig. 2. The length of the beam

L ¼ 300 in, and the cross-section is square with dimen-

sion 6.26 in. The beam uniaxial stress–strain relation

is assumed to follow an elastic–plastic behavior. The

modulus of elasticity E ¼ 29,000 ksi, the yield strength is

assumed to be equal to 30 ksi, and a strain hardening

slope of 0.014 is also assumed. The beam section is

subdivided into 16 fibers. The foundation force–defor-

mation relation is also assumed to be elastic–plastic. The

stiffness equals 0.5 kip/in2, the yield force equals 1 k/in,

and the hardening slope equals 0.01. The load is applied

at midspan under displacement control. The analytical

load–deformation response is shown in Figs. 3 and 4 for

both the displacement and mixed models respectively

with different order of interpolation functions using 6

elements. A mesh consisting of 32 elements using dis-

placement-based models with fifth order polynomials, is

used to represent the converged solution. While dis-

placement models showed a large error in the inelastic

zone, mixed models were capable of tracing the behavior

rather well. In fact, the mixed model with cubic moment

functions almost captured the exact behavior. The rea-

son could be explained by considering the distribution of

the local parameters along the length of the beam,

namely the bending moment, curvature, vertical dis-

placement, and foundation force, which are shown in

Figs. 5–12 for the load stage identified by point A in the

global response of Figs. 3 and 4. Figs. 5 and 6 show the

1P

300 in

σ ksi)

ε

30

290001

( p(k/in)

∆(in)

1

0.5

Fig. 2. Simply supported inelastic beam on foundation.

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5

Converged Solution

Displ. Model (5th order displ.)

Displ. Model (Cubic displ.)

A

Fig. 3. Load–deformation response of simply supported beam

(displ. model).

0

20

40

60

80

100

120

140

160

180

200

0 1 2 3 4 5

A

Converged Solution

Mixed Model (Cubic moment)

Mixed Model (linear moment)

Fig. 4. Load–deformation response of simply supported beam

(mixed model).

–2000

–1500

–1000

500

0

500

1000

1500

2000

2500

3000

0 50 100 150 200 250 300

Length along beam (in)

Ben

din

g m

om

ent

(k.in

)

Converged Solution


Displ. Model (cubic displ.)

–

Fig. 5. Moment distribution for simply supported beam at load

point A.


distribution of the moment and curvature using dis-

placement models. From the figures, it is clear that the

bending moment exhibits jumps at element boundaries,

even for the higher fifth order polynomial model. Fur-

thermore, the maximum moment value at midspan

is underestimated. The maximum curvature value at

0.012

0.010

0.008

0.006

0.004

0.002

0.000

-0.0020 50 100 150 200 250 300


Cu

rvat

ure

(1/

in)

Converged Solution



Fig. 6. Curvature distribution for simply supported beam at

load point A.

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.00 50 100 150 200 250 300


Ver

tica

l dis

pla

cem

ent

(in

)

Converged Solution



Fig. 7. Vertical displacement for simply supported beam at

load point A.

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.00 50 100 150 200 250 300


Fo

un

dat

ion

fo

rce

(k/in

)

Converged Solution



Fig. 8. Foundation force distribution for simply supported

beam at load point A.

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

Converged Solution




Ben

din

g m

om

ent

(k.in

)

0 50 100 150 200 250 300

Fig. 9. Bending moment distribution for simply supported


Converged Solution




Cu

rvat

ure

(1/

in)

0 50 100 150 200 250 300

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

-0.002

Fig. 10. Curvature distribution for simply supported beam at

load point A.

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0


Ver

tica

l dis

pla

cem

ent

(in

)

Converged Solution



0 50 100 150 200 250 300

Fig. 11. Vertical displacement distribution for simply sup-

ported beam at load point A.


midspan is also clearly underestimated by a factor

greater than 2. This poor representation of the behavior

is due to the fact that displacement models assume a

specified continuous curvature field, while the curvature

is actually localized in the plastic zone. The vertical

displacements and foundation force shown in Figs. 7

and 8 are well represented though. Figs. 9 and 10 show

the distribution of the moment and curvature using

mixed models. While the moment distribution shows

small jumps for the linear model, the cubic model was

able of capturing the behavior rather accurately. Fur-

thermore, the maximum moment and curvature values

are also accurately represented. This is due to the fact

that mixed models assume a specified continuous field

for the smooth bending moment parameter. The vertical

displacement and foundation force distributions, shown

in Figs. 11 and 12 respectively, are represented even

better than with displacement models. From these re-

sults, it is clear that mixed models represent superior

behavior than displacement models.

The second example represents an inelastic beam

with tensionless elastic support as shown in Fig. 13. The

beam length is 6 m, and the cross-section is square with

60 cm dimension. The beam uniaxial stress–strain rela-

tion is elasto-plastic with Young�s modulus E ¼ 25 GPa,

yield strength equals 17.5 MPa, and hardening slope

equals 1.6%. The beam section is subdivided into 12 fi-

bers. The elastic foundation stiffness equals 100 MPa.

The loading condition consists of an axial force and a

moment acting at midspan, which is typical of founda-

tion structures. The axial force equals 4000 kN, and is

applied under load control, while the moment is applied

incrementally under displacement control. The midspan

moment rotation behavior of the beam is shown in Figs.

14 and 15 for the displacement and mixed models re-

spectively using four finite elements. A mesh consisting

of 32 displacement-based elements with fifth order

polynomials represent the converged solution. While the

lower order models show a large error in the inelastic

region, the higher order models captured the global

behavior rather well. Also, from the plot, it is clear that

mixed models have a higher rate of convergence. Figs.

16–23 show the distribution of the different parameters

of the problem at load stages A and B identified in the

P

6 m

(σ MPa)

e

17.5

250001

D(mm)

p(N/mm)

1100

M

Fig. 13. Inelastic beam on tensionless foundation.

0

200

400

600

800

1000

1200

1400

1600

1800

0.000 0.005 0.010 0.015 0.020

Midspan rotation (rad)

Mid

span

mo

men

t (k

N.m

)

Converged Solution


Displ. Model (Cubic displ.)

A

B

Fig. 14. Moment rotation response of beam on tensionless

foundation (displ. model).

0

200

400

600

800

1000

1200

1400

1600

0.000 0.005 0.010 0.015 0.020Midspan rotation (rad)

Mid

span

mo

men

t (k

N.m

)

Converged Solution

Mixed Model (Cubic moment)

Mixed Model (linear moment)

A

B

Fig. 15. Moment rotation response of beam on tensionless

foundation (mixed model).

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0


Fo

un

dat

ion

fo

rce

(k/in

)

Converged Solution



0 50 100 150 200 250 300

Fig. 12. Foundation force distribution for simply supported



global plot for both the displacement and mixed models.

The displacement model shows big jumps at element

boundaries for the bending moment as shown in Fig. 16,

and clearly underestimates the maximum curvature

shown in Fig. 17 even with higher order polynomials.

The model, however, predicts the vertical displacement

and the corresponding foundation force rather well as

shown in Figs. 18 and 19. The mixed model, on the other

-1500

-1000

-500

0

500

1000

1500

2000

0 1 2 3 4 5 6Length along beam (m)

Ben

din

g m

om

ent

(kN

.m)

A

B

Converged solution

Displ. model (cubic displ.)

Displ. model (5th order displ.)

Fig. 16. Bending moment distribution for beam on tensionless

foundation.

-0.05

0.00

0.05

0.10

0.15

0.20

0.25


Cu

rvat

ure

(1/

m)

A

B

Converged solution



Fig. 17. Curvature distribution for beam on tensionless foun-

dation.

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

0.05


Ver

tica

l dis

pla

cem

ent

(m)

A

B

Converged solution



Fig. 18. Vertical displacement distribution for beam on ten-

sionless foundation.

-3000

-2500

-2000

-1500

-1000

-500

00 1 2 3 4 5 6

Length along beam (m)

Fo

un

dat

ion

fo

rce

(kN

/m)

B

A Converged solution



Fig. 19. Foundation force for beam on tensionless foundation.

-1500

-1000

-500

0

500

1000

1500

2000


Ben

din

g m

om

ent

(kN

.m)

A

B

Converged solution

Mixed model (cubic moment)

Mixed model (linear moment)

Fig. 20. Bending moment distribution for beam on tensionless

foundation.

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6


Cu

rvat

ure

(1/

m)

B

A

Converged solution



Fig. 21. Curvature distribution for beam on tensionless foun-

dation.


hand, successfully describes the bending moment and

curvature, with the higher order polynomial capturing

almost the exact behavior, as shown in Figs. 20 and 21

respectively. The vertical displacement and foundation

force are also accurately modeled as shown in Figs. 22

and 23 respectively, confirming the superiority of mixed

models over displacement models.

7. Conclusions

The paper presents a new inelastic element for the

analysis of beams on foundations. The element is de-

rived from a two-field mixed formulation, where forces

and deformations are approximated with independent

interpolation functions. The state determination algo-

rithm for the implementation of the element in a general

purpose nonlinear analysis program is discussed. Sta-

bility of the algorithm is presented. Numerical examples

that clarify the advantages of the proposed model over

the well-established displacement model are presented.

The studies reveal the superiority of the proposed model

due to the accurate representation of the smooth force

field in the element. This results in reduction of the total

number of degrees of freedom in a structural model and

in appreciable cost savings of the analytical simulation.

Acknowledgements

The author would like to express his deepest thanks

to Prof. F.C. Filippou, his doctoral thesis supervisor at

the University of California, Berkeley, for his constant

advice regarding the formulation and implementation of

general mixed models. The author is also grateful to

Prof. R.L. Taylor for his continuous help regarding the

use of the finite element program FEAP.

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-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

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Ver

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l dis

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cem

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Converged solution



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Mixed formulation of nonlinear beam on foundation elements