International Journal of Mechanical Sciences 101-102 (2015) 429–436

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International Journal of Mechanical Sciences

http://d0020-74

n CorrE-m

journal homepage: www.elsevier.com/locate/ijmecsci

Static analysis of a thick ring on a unilateral elastic foundation

Chunjian Wang a,n, Beshah Ayalew a, Timothy Rhyne b, Steve Cron b, Benoit Dailliez c

a Clemson University International Center for Automotive Research, No. 4 Research Dr., Greenville, SC 29607, USAb Michelin America Research Corporation, 515 Michelin Rd, Greenville, SC 29605, USAc Centre de Technologie Europe, Michelin Corporation Clermont Ferrand, France

a r t i c l e i n f o

Article history:Received 17 April 2015Received in revised form12 August 2015Accepted 20 August 2015Available online 29 August 2015

Keywords:Ring on elastic foundationUnilateral elastic foundationTimoshenko beamDeformation of thick ringIterative compensation method

x.doi.org/10.1016/j.ijmecsci.2015.08.01503/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 864 905 3259.ail address: chunjiw@g.clemson.edu (C. Wang

a b s t r a c t

A thick ring on a unilateral elastic foundation can be used to model important applications such as non-pneumatic tires or bushing bearings. This paper presents a reduced-order compensation scheme forcomputing the static deformation response of a thick ring supported by a unilateral elastic foundation toan arbitrary applied force. The ring considered is an orthotropic and extensible ring that can be treated asa Timoshenko beam. The elastic foundation is a two-parameter foundation with a linear torsionalstiffness but a unilateral radial stiffness whose value vanishes when compressed or tensioned. The paperfirst derives the deformation response for the linear foundation case for which Fourier expansiontechniques can be applied to obtain an analytical solution. Then, the nonlinear unilateral foundationproblem is solved via an iterative compensation scheme that identifies regions with vanishing radialstiffness and applies a compensation force to the linear foundation model to counteract the excessivefoundation forces that would not be there with the unilateral foundation. This scheme avoids the needfor solving the complex set of nonlinear differential equations and gives a computationally efficient toolfor rapidly analyzing and designing such systems. Representative results are compared with Finite Ele-ment Analysis (FEA) results to illustrate the validity of the proposed approach.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The flexible Ring on Elastic Foundation (REF) model [1,2] is aclassical one that has been studied for decades. This is because ofits broad and important applications such as automotive tires[3–5], railway wheels and gears [6], and others [7].

Different criteria can be used to classify the focus of existingworks that analyze REFs. The simplest categories are the treatmentof the static deformation problem [7,8] vs. the dynamic problem asfree vibration [6] or forced vibration [2,9]. Considering the ringmechanics, the ring has been treated as a tensioned string that hasdirect tensile strain but no bending stiffness [10]; as an Euler–Bernoulli beam or thin ring whose plane section remains planeand always normal to the neutral axis of the ring after deformation[7,5,9]; or as a Timoshenko beam or thick ring [11] which takes theshear deformation into account by assuming that the normal of aplane section is subject to rotation. Further distinctions existbetween extensional and inextensional rings. [12,1] studied thevibration problem for both a rotating thin ring and a thick ring,and pointed out that the inextensional assumptions in thick ring

).

theories are improper because extensional coupling effects are asimportant as shearing effects especially for a rotating ring.

As an important component of the REF model, the treatment ofthe elastic foundation can be used as another criterion to classifythe existing research. Numerous works, including all of the onescited above, assume a linear and uniformly distributed stiffness forthe whole elastic foundation, independent of location and defor-mation state. The distributed stiffness can be modeled with oneparameter [7,10] where the foundation has a stiffness only in theradial direction; or with two parameters [2,9] involving both radialand torsional stiffness values; or even more parameters [6], wherein addition to the radial and torsional stiffness, a stiffness asso-ciated with the distortion of the foundation due to in-plane rota-tion of a cross-section of the ring is included.

Although much has been gained from linear and uniformelastic foundation assumptions, not all REF problems have a per-fect linear elastic foundation with uniformly distributed stiffness.Examples for nonuniform distribution include planetary gearingwhere tooth meshes for the ring and planets are not equallyspaced [13] and tires with non-uniformity [14]. For these type ofproblems, [13] studied the free vibration of rings on a generalelastic foundation, whose stiffness distribution can be variablecircumferentially in the radial, tangential, or inclined orientation,and gave the closed-form expression for natural frequencies andvibration modes. [14] studied the natural frequencies and mode

Fig. 1. Thick ring on a two-parameter elastic foundation.

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436430

shapes of rings supported by a number of radial spring elementswith a constant radial stiffness using modal expansion andreceptance method. However, in both works, the distributions ofthe stiffnesses for the elastic foundations still did not change withthe deformation of the ring resting on them.

A more complicated problem was invoked by consideringdeformation-dependent elastic foundations, such as those withunilateral stiffness whose values vanish when compressed ortensioned. The difficulty in solving this group of problems is in thefact that the compressed or tensioned region is not known inadvance. It depends on the loading and consequently on thedeformation. An example for the application is the non-pneumatictire presented in [15], whose structure is a deformable shear ringsupported by collapsible spokes which offer stiffness only in ten-sion [16]. Another example is a bushing bearing, whose externalsleeve can be modeled as a ring on a tensionless foundation andthe internal sleeve as a ring on a collapsible foundation. Not toomany works exist that deal with such rings on unilateral founda-tions. [17] worked on the forced response of a thin and inexten-sible ring on a tensionless two-parameter foundation under a timevarying in-plane load. To solve the unilateral problem, an auxiliaryfunction is used in the coefficients of the equations to track andreflect the status of the foundation. This auxiliary function makesthe differential equations of the system nonlinear and difficult tosolve. Furthermore, the tangential displacement of the ring isobtained from the inextensible assumption, which cannot beadopted in a more general case, such as that with extensibleTimoshenko ring. [18] studied the static deformation and thecontact pressure of a non-pneumatic tire resting on collapsiblespokes, when it contacts against a rigid plane ground. The gov-erning differential equations were derived only for the thick ringmodeled via Timoshenko beam theory by treating the supportingforce by collapsible spokes as radial distributed force which van-ishes in collapsed spoke regions. The ring was divided into threeregions according to the post-deformation status of the spokes(tensioned or buckled spokes) and contact status with the ground(contact region or free region). Closed-form expressions for thedeformation and contact stress are given in terms of angularbounds of these three regions, which then need to be solvednumerically. However, the method has limitations in two aspects:(1) For more complex loading cases, such as that with multipleforces applied at multiple locations, the number of regions intowhich the ring must be divided grows with the number of the loadregions. Multiple unknown angular bounds would then need to besolved numerically. (2) It is difficult to extend this method topractical dynamic cases.

The present paper studies the deformation of a thick ring on anelastic two-parameter foundation where the radial stiffness isunilateral. The deformation response to an arbitrary in-plane forceis considered. The ring is modeled as an orthotropic and extensiblecircular Timoshenko beam. As the first step, the linear foundationproblem is solved analytically using Fourier expansion techniquesfor both the radial and tangential directions. It is then shown thatthe linear foundation case includes an excessive foundation forcecompared with that of the unilateral foundation. An iterativecompensation scheme is then set up to both find the region ofvanishing radial stiffness with the unilateral foundation and thatof the required compensation force to counteract the excessiveforce predicted via the linear foundation. The method is anintuitive and efficient alternative to numerically solving the cou-pled and complex system of nonlinear differential equations for aflexible ring on a unilateral foundation. In addition, compared withdiscretization-based numerical methods such as nonlinear FEA,the proposed scheme avoids the time-consuming modeling andmeshing work, which makes it attractive specially for rapidparameter studies at the design stage. Compared with the method

in [18], the method proposed in this paper is capable of handlingarbitrary force distributions and directions, without increasingcomplexity. Furthermore, since the proposed scheme is Fourierexpansion-based, it can be easily extended to the dynamic cases(both forced response and dynamic contact) as we illustrate in ourother work [19].

The rest of the paper is organized as follows. Section 2 restatesthe problem and gives the governing equations. Section 3 gives theanalytical solutions for the linear foundation problem and extendsthem to the unilateral case. Then, in Section 4, discussions of someexample results are given and compared with FEA results. Con-clusions and future work are given in Section 5.

2. Statement of problem and governing equations

Fig. 1 shows a schematic of the model for a thick ring on a two-parameter elastic foundation. The ring with thickness h is assumedto have a radius R at its centroid. The width perpendicular to theplane of the ring is b. The uniformly distributed radial and tan-gential stiffnesses are assumed to be Kr and Kθ, respectively. Theyhave units of stiffness per radian. For a linear elastic foundation,the distributed radial stiffness Kr is constant. However, for theunilateral elastic foundation, the radial stiffness vanishes when theelastic foundation is tensioned or compressed. A polar coordinatesystemwith origin located at the ring center is adopted. The centerof the ring is fixed and friction is neglected.

The radial and tangential displacements at the centroid areassumed to be u R,r θ( ) and u R, θ( )θ , respectively. Following [20],the cross-section of the ring is assumed to have a rotation R,ϕ θ( )at the centroid at circumferential position θ and keeps its planeafter deformation. Then, the radial and tangential displacements atan arbitrary point on the ring with radius r and circumferentialposition θ, u r,r θ( ) and u r, θ( )θ , can be represented by

u r u R

u r u R r R R

, ,

, , , 1

r r( ) ( )( ) ( ) ( ) ( )

θ θ

θ θ ϕ θ

=

= + − ( )θ θ

The strain–displacement relationships in polar coordinates are[21]

rr

u r

rr

u rr

u r

rr

u rr

u rr

u r

, ,

,1

,1

,

,1

, ,1

, 2

rr r

r

r r

( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

θ θ

θθ

θ θ

γ θθ

θ θ θ

ϵ = ∂∂

ϵ = ∂∂

+

= ∂∂

+ ∂∂

− ( )

θθ θ

θ θ θ

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436 431

where r,rr θϵ ( ), r, θϵ ( )θθ , r,rγ θ( )θ are the radial, tangential and shearstrains, respectively.

The ring is assumed to be orthotropic and homogeneous. Weconsider that the ring material axes coincide with the polarcoordinate system adopted. The stress–strain relationships are[21]

rE r r

rE r r

r G r

,, ,

1

,, ,

1

, , 3

rrr r rr

r r

r rr

r r

r r

( )

( )( ) ( ) ( )

( ) ( ) ( )

( ) ( )

σ θν θ θ

ν ν

σ θν θ θ

ν ν

τ θ γ θ

=ϵ + ϵ− +

=ϵ + ϵ− +

= ( )

θ θθ

θ θ

θθθ θ θθ

θ θ

θ θ

where r,rrσ θ( ), r,σ θ( )θθ and r,rτ θ( )θ are radial, tangential and shearstresses, respectively. Er, Eθ and G are the elastic moduli in theradial and tangential directions and the shear modulus, respec-tively. ν is the Poisson's ratio.

The strain energy stored in the ring is given by

Ub

r r2

d d4R h

R h

rr rr r r1/2

/2∫ ∫ σ σ τ γ θ= ( ϵ + ϵ + )

( )π

πθθ θθ θ θ

− −

+

The strain energy stored in the elastic foundation is given by

⎛⎝⎜⎜

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎞⎠⎟⎟U K u R

hK u R

h12 2

,2

, d5

r r2

2 2

∫ θ θ θ= − + −( )π

πθ θ

−

Note here that the radial and tangential displacements for internaledge of the ring u R h/2,r θ( − ) and u R h/2, θ( − )θ couple the ringand the elastic foundation.

The work by the applied forces is obtained from

⎛⎝⎜

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟W b q u R

hq u R

hR

h2

,2

,2

d6r r∫ θ θ θ= + + + +

( )π

π

θ θ−

where q q R h/2,r r θ= ( + ) and q q R h/2, θ= ( + )θ θ are distributedforces applied to the external edge of the ring (at the radial loca-tion R h/2+ ) in the radial and tangential directions, respectively.The units of qr and qθ are in Force Area/ .

Invoking the principle of virtual work [22]:

U U W 71 2( )δ δ+ = ( )

After substitution of (1)–(6) into (7) and some manipulationsaccording to Euler–Lagrange equation, the governing equations forthe present thick ring on an elastic foundation problem can bederived. Here, for brevity, rν θ and rνθ are set to zero for the gov-erning equations given as (8). However, the proposed approachwill hold for the general case where these are not zero:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

GAb

GARb

uEARb

GARb

u

EARb

Kb

u Rh

q

GAb

K hb

EARb

uEARb

GARb

u

GARb

Kb

u Rh

q

GA Rb

K hb

EIRb

GAb

u

GAb

K hb

u Rh

hq

2

12

2

14

12

12 2 8

r

rr r

r

r

2

2

2

2

2 2

2

θϕ

θ θ

ϕθ θ

ϕθ

ϕθ

− ∂∂

− ∂∂

+ + ∂∂

+ + = +

− − − ∂∂

+ − − ∂∂

+ + = +

+ − ∂∂

+ ∂∂

+ − − = +( )

θθ

θ

θ θθ

θ

θθ θ

θ θ

θθ θ

where the following notations are used:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

u u R

u u R

R

q q Rh

q q Rh

,

,

,

2,

2,

9

r r

r r

( )( )

( )

θ

θ

ϕ ϕ θ

θ

θ

=

=

=

= +

= +( )

θ θ

θ θ

and

EA E A

GA G A

EI E I 10

= ×

= ×= × ( )

θ θ

θ θ

where A b h= × is the cross-sectional area of the ring andI bh1/12 3= ( ) is the area moment of inertia of the cross-section.The following approximations are used in the manipulations toobtain the governing equations above, considering that R h:⪢

rr

Rr

hR

r R

rr

Rr R r

r R

rr

Rr R r

hR

1d

1d

d1

d 0

d1

d1

12 11

R h

R h

R h

R h

R h

R h

R h

R h

R h

R h

R h

R h

/2

/2

/2

/2

/2

/2

/2

/2

/2

/2 2

/2

/22

3

∫ ∫

∫ ∫

∫ ∫

( ) ( )

( ) ( )

≐ =

−≐ − =

−≐ − =

( )

−

+

−

+

−

+

−

+

−

+

−

+

3. Method of solution

In this section, the governing equations (8) are first solvedanalytically by considering a linear foundation with constant cir-cumferential distribution of the foundation stiffnesses. Theapproach we use is to first get Fourier series expansion of theapplied arbitrary force, and then the governing equations aresolved harmonic-wise in both the radial and tangential directions.The solution for the total displacements will then be the super-position of the harmonic contributions in both directions. Then thesolutions are extended to the unilateral foundation case via theiterative compensation scheme proposed in this work.

3.1. Solution for linear elastic foundation

An arbitrary circumferentially distributed (with respect to θ) orconcentrated force F θ( ) can be decomposed into its radial andtangential components as

F F r F 12r ( )θ θ θ θ→

( ) = ( )→ +→

( )θ

where Fr θ( ) and F θ( )θ are force components in the radial andtangential directions, respectively. Using Fourier expansion on

,π π[ − ], these components can be approximated by

F Q n Q n

F Q n Q n

cos sin

cos sin13

rn N

N

n r cn N

N

n r s

n N

N

n cn N

N

n s

, , 0 , , 0

, , 0 , , 0

( ) ( )

( ) ( )

∑ ∑

∑ ∑

θ θ θ θ θ

θ θ θ θ θ

( ) = ( − ) + ( − )

( ) = ( − ) + ( − )( )

θ θ θ

=− =−

=− =−

where N is the cut-off harmonic number, Q n r c, , , Q n r s, , , Q n c, ,θ andQ n s, ,θ are corresponding coefficients of the nth harmonic force forradial, tangential and cosine, sine components. The subscript r or θindicates that the coefficient is for radial or tangential direction,respectively, while c or s represents cosine or sine component,

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436432

respectively. θ0 is used to define the rotation of the local cylind-rical coordinate system with respect to the global Cartesian coor-dinate system. In this paper, θ0 is always set to 0, which indicates

0θ = corresponds to the bottom point of the ring.Considering only the nth harmonic force, the force per unit area

applied on the external edge of the ring (where the radial locationis R h/2+ ) can be written for the radial and tangential directionsindependently, where each direction consists of cosine and sinecomponents:

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

q Rh

q Rh

q Rh

q Rh

q Rh

q Rh

2,

2,

2,

2,

2,

2,

14

nr n r c n r s

n n c n s

, , , ,

, , , ,

θ θ θ

θ θ θ

+ = + + +

+ = + + +( )θ θ θ

The cosine or sine component in each direction can be expressedin terms of the width of the ring, the radial coordinate and thecorresponding force component:

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

q Rh

b Rh

Q n

q Rh

b Rh

Q n

q Rh

b Rh

Q n

q Rh

b Rh

Q n

2,

1

2

cos

2,

1

2

sin

2,

1

2

cos

2,

1

2

sin

15

n r c n r c

n r s n r s

n c n c

n s n s

, , , , 0

, , , , 0

, , , , 0

, , , , 0

( ( ))

( ( ))

( ( ))

( ( ))

θ θ θ

θ θ θ

θ θ θ

θ θ θ

+ =+

−

+ =+

−

+ =+

−

+ =+

−

( )

θ θ

θ θ

The four components of the nth harmonic force in (15) can besolved separately and then superposed to get the deformation forthe nth harmonic. In order to solve for the response to the firstforce component in (15), which is the cosine part in the radialdirection, qr and qθ in (8) can be replaced with qnr and qnθ:

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

q q Rh

b Rh

Q n

q

2,

1

2

cos

0 16

nr n r c n r c

n

, , , , 0( ( ))θ θ θ= + =+

−

= ( )θ

Noting that the right hand side of the first equation in (8) has acosine function, combined with the order of differentiation to thevariables, it is easy to obtain the assumed ansatz:

u R Cur n n

u R Cu n n

R C n n

, cos

, sin

, sin 17

r n r c r c

n r c r c

n r c r c

, , , , 0

, , , , 0

, , , 0

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

( ) ( ) ( ( ))

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

= −

= −

= − ( )

θ

where u R, ,r n r c, , , θ( ) u R,n r c, , , θ( )θ are displacement solutions in radialand tangential direction for the centroid and R,n r c, ,ϕ θ( ) is thecross-section rotation. The subscripts n r c, , indicate that they arethe solution components due to the cosine part of nth harmonicforce in the radial direction. Curr c, , Cu r c,θ and C r c,ϕ are modalcoefficients to be solved for. They are indexed by the harmonic(circumferential mode) number n. Substituting (16) and (17) into(8) and noting that the coefficients of ncos 0θ θ( ( − )) or

nsin 0θ θ( ( − )) in all three equations in (8) should be equal, oneobtains 3 equations with 3 unknown coefficients Curr c, , Cu r c,θ andC r c,ϕ . These can be solved explicitly and the results are given in theAppendix. The responses to the other three components of the nthharmonic force in (15) can be solved for in a similar way.

The solutions for the sine component in the radial direction aregiven by

u R Cur n n

u R Cu n n

R C n n

, sin

, cos

, cos 18

r n r s r s

n r s r s

n r s r s

, , , , 0

, , , , 0

, , , 0

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

( ) ( ) ( ( ))

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

= −

= −

= − ( )

θ

And the solutions for the cosine and sine components in thetangential direction are given by

u R Cur n n

u R Cu n n

R C n n

u R Cur n n

u R Cu n n

R C n n

, sin

, cos

, cos

, cos

, sin

, sin 19

r n c c

n c c

n c c

r n s s

n s s

n s s

, , , , 0

, , , , 0

, , , 0

, , , , 0

, , , , 0

, , , 0

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

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

( ) ( ) ( ( ))

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

= −

= −

= −

= −

= −

= − ( )

θ θ

θ θ θ

θ θ

θ θ

θ θ θ

θ θ

The corresponding modal coefficients are given in the Appendix.The final displacement solutions for the governing equations in

(8) are then written as follows:

⎡⎣ ⎤⎦20

u R u R u R u R u R

u R u R u R u R u R

R R R R R

, , , , ,

, , , , ,

, , , , ,

rn N

N

r n r c r n r s r n c r n s

n N

N

n r c n r s n c n s

n N

N

n r c n r s n c n s

, , , , , , , , , , , ,

, , , , , , , , , , , ,

, , , , , , , ,( ) ( ) ( ) ( )

[ ]

[ ]

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

∑

∑

∑( )

θ θ θ θ θ

θ θ θ θ θ

ϕ θ ϕ θ ϕ θ ϕ θ ϕ θ

= + + +

= + + +

= + + +

θ θ

θ θ θ θ θ θ θ

θ θ

=−

=−

=−

3.2. Solution for unilateral elastic foundation

The radial displacement solution u R,r θ( ) obtained from (20)considers support by the foundation whether the displacement ispositive and negative. With a collapsible foundation, there is nosupport in regions where u R,r θ( ) is negative. In this region, anexcessive force is included by the linear foundation assumption.The amount of this excessive force Fe θ( ) is proportional to theextent of negative deformation:

⎧⎨⎩Fu R Kr u R

u R

, where , 00, where , 0 21

er r

r( )θ

θ θθ

=( ) ( ) <

( ) ≥ ( )

For tensionless foundation, a similar excessive force exists inregions where u R,r θ( ) is positive. But the remainder of the analysiswill be the same as the case of the collapsible foundation.

In order to counteract the excessive force and obtain thedeformation for the unilateral foundation case, a compensationforce Fcp θ( ) which has the same magnitude as the excessive forcecan be applied to the linear elastic foundation:

F F 22cp e( ) ( )θ θ= ( )

Fcp can be expanded into Fourier series on ,π π[ − ] using the sameharmonic numbers as F θ( ):

⎡⎣ ⎤⎦

F F

H n H ncos sin23

cpn N

N

cp n

n N

N

n r c n r s

,

, , 0 , , 0( ) ( )

( ) ( )

( ) ( )

∑

∑

θ θ

θ θ θ θ

=

= − + −( )

=−

=−

Because the unilateral property of the foundation only exists in theradial direction, this compensation force will only have radialcomponents.

The compensation force is then applied to the external edge ofthe foundation, at internal ring radius R h/2− . The distributed

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436 433

compensation force per unit area in both the radial and tangentialdirections is given by

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟ ⎛

⎝⎜⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

⎛⎝⎜

⎞⎠⎟

q Rh

b Rh

H n

q Rh

b Rh

H n

q Rh

q Rh

2,

1

2

cos

2,

1

2

sin

2, 0

2, 0

24

cp n r c n r c

cp n r s n r s

cp n c

cp n s

, , , , , 0

, , , , , 0

, , ,

, , ,

( ( ))

( ( ))

θ θ θ

θ θ θ

θ

θ

− =−

−

− =−

−

− =

− =( )

θ

θ

Substitution of (24) into (8) for qnr and qnθ leads to the governingequations for the compensation displacements driven by thecompensation force. Note that the external ring radius R h/2+ inthe right hand sides of (8) needs to be replaced by the internal ringradius R h/2− . The governing equations can then be solved usingthe same approach as above as they are still linear differentialequations. In so doing, the nonlinear unilateral foundation pro-blem is approximated by applying a compensation force to thelinear foundation model instead of directly solving the system ofcoupled nonlinear differential equations for the problem.

However, instead of solving the linear differential equationsagain with the compensation force, the following observations canbe used to get the compensation displacements to further reducethe computational load. Since the radius on the right hand sides of(8) (updated) will be canceled out by those in the denominators of(24), the displacement solutions are only affected by the magni-tude of the forces regardless of how they are applied on theexternal edge of the foundations or the outside of the ring. So,based on (17) and (18), the cosine and sine components of thecompensation displacements caused by the compensation forcecan be obtained directly by

u RHQ

Cur n n

u RHQ

Cu n n

RHQ

C n n

u RHQ

Cur n n

u RHQ

Cu n n

RHQ

C n n

, cos

, sin

, sin

, sin

, cos

, cos25

r cp n r cn r c

n r cr c

cp n r cn r c

n r cr c

cp n r cn r c

n r cr c

r cp n r sn r s

n r sr s

cp n r sn r s

n r sr s

cp n r sn r s

n r sr s

, , , ,, ,

, ,, 0

, , , ,, ,

, ,, 0

, , ,, ,

, ,, 0

, , , ,, ,

, ,, 0

, , , ,, ,

, ,, 0

, , ,, ,

, ,, 0

( ) ( ) ( ( ))

( ) ( ) ( ( ))

( ) ( ) ( ( ))

( ) ( ) ( ( ))

( ) ( ) ( ( ))

( ) ( ) ( ( ))

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

θ θ θ

θ θ θ θ

ϕ θ ϕ θ θ

= −

= −

= −

= −

= −

= −( )

θ

θ

By using these simple algebraic equations, even the linear differ-ential equations need to be solved only once.

Correspondingly, the compensation displacements are obtainedfrom

⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

u R u R u R

u R u R u R

R R R

, , ,

, , ,

, , ,26

r cpn N

N

r cp n r c r cp n r s

cpn N

N

cp n r c cp n r s

cpn N

N

cp n r c cp n r s

, , , , , , , , ,

, , , , , , , , ,

, , , , , ,( ) ( )

( ) ( )

( ) ( )

( )

( )

( )

∑

∑

∑

θ θ θ

θ θ θ

ϕ θ ϕ θ ϕ θ

= +

= +

= +( )

θ θ θ

=−

=−

=−

The new total displacements are the summation of the originalsolutions (20) and the compensation displacements (26):

u R u R u R

u R u R u R

R R R

, , ,

, , ,

, , , 27

r r r cp

cp cp

cp cp

,

, ,

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

( ) ( ) ( )

θ θ θ

θ θ θ

ϕ θ ϕ θ ϕ θ

^ = +^ = +^ = + ( )

θ θ θ

Due to the change of radial displacement from u R,r θ( ) tou R,r θ^ ( ), the excessive force in (21) needs to be updated as well.Thus, the procedure from (21) to (27) needs to be repeated,thereby setting up an iterative scheme. This scheme is consideredto converge when values of the compensated displacements in(27) do not change anymore with further iteration steps. Theconverged displacements in (27) can then be taken as the solutionfor the unilateral foundation case.

4. Results and discussion

In this section, two examples are included to demonstrate theutility of the proposed method. In the first example, a radial pointforce with both positive and negative magnitudes is consideredand the results obtained using the iterative compensationapproach introduced above are compared with those obtained viaFEA. In the second one, a more general distributed force, with bothradial, tangential and sine, cosine components are applied and thesolutions are shown.

4.1. Response to a concentrated force

As a basic comparison, we consider a radial concentrated forceFc applied at the bottom of the ring. A Gaussian function repre-sentation allows one to model different degrees of “concentration”of this force:

f Q1

e28c 2

/02 2( ) ( )θ

π σ=

( )θ θ σ− −

where Q 1000 N= ± is the magnitude of the force, s is a para-meter that determines how concentrated the applied force is. Thesmaller s is, the more concentrated the force is. Fig. 2 shows theeffect of s to the distribution density of a unit force. It can be seenthat with 0σ → , fc 0θ( ) → ∞, and the force becomes an idealconcentrated force with 0 distribution width. The integration ofthe distribution density function around the ring always equals tothe concentrated force:

f Fd 29c c∫ ( )θ θ = ( )π

π

−

fc θ( ) can be expanded into Fourier series in ,π π[ − ]:

f Q n

Q n

cos

12

e cos30

cn N

N

n

n N

Nn

0

1/40

2 2

( ) ( ( ))

( ( ))

∑

∑

θ θ θ

πθ θ

= −

= −( )

σ

=−

=−

−( )

that is

Q Q1

2e 31n

n1/4 2 2

π= ( )

σ−( )

The displacement responses due to this concentrated force arethen solved by using the iterative scheme described above. Acollapsible unilateral foundation is considered and compared witha linear foundation. Parameter values used for the results givenbelow are listed in Table 1. In this case, negligible tangentialstiffness is considered for the comparison with FEA results.

A 2-D FE model of the ring on a unilateral foundation is builtusing the commercial finite element software Abaqus/Standard.

Fig. 2. Effect of s to the distribution density of a unit force.

Table 1Value of parameters.

Parameters Definition Value Unit

R Radius of ring centroid 0.2 mh Ring thickness 0.02 mb Ring width 0.06 mEθ Extensional modulus of the ring 1e10 PaG Shear modulus of the ring 4e6 PaKr Radial stiffness of the foundation 1e5 N/m per radianKθ Radial stiffness of the foundation 1 N/m per radianks Stiffness of discretized spring element 5.236e3 N/mFc Magnitude of concentrated force 1000± Ns Distribution factor of Gaussian

function0.02 NA

Fig. 3. Comparison of deformed centroid: collapsible foundation vs. linearfoundation.

Fig. 4. Comparison of deformed centroid: response by positive and negative force.

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436434

The thick ring is meshed using the second-order quadrilateralplane stress element with reduced integration, CPS R8 , with 6 lay-ers of elements in the radial direction and 120 divisions in thecircumferential direction. In the examples, only radial stiffness istaken into account for the foundation, which is modeled using 120evenly distributed and discretized spring elements SpringA. Theproperty of these spring elements can be set to linear or nonlinearto represent the linear or unilateral elastic foundation. The stiff-ness of every linear spring element (or the effective non-vanishedstiffness for the nonlinear spring element representing unilateralfoundation) is calculated by

kK

N2

32sr

s

π=( )

where Ns¼120 is the number of the spring elements.Fig. 3 shows the deformation of the ring centroid for both

collapsible and linear foundation with F 1000 Nc = − (pointing tothe ring center). The deformations are amplified by 10 times forclarity. It can be seen that the solutions obtained by the proposediterative approach match the results by the FEA very well, for boththe linear and unilateral foundation cases.

For the unilateral foundation case, it is expected that themagnitude of displacements will be different under a positive andnegative force with same magnitude. Fig. 4 shows the deformationresponses under F 1000 Nc = ± for the ring on collapsible foun-dation. In this figure, the displacements by positive force areshown with flipped sign so that they can be easily compared withthose by negative force. It can be seen that for the collapsiblefoundation case, a negative force will lead to a larger magnitude of

ur at the position where the force is applied because negative forcewill directly compress the local area of the foundation to make itcollapse while the positive force will stretch it so the effectivestiffness there is higher than that for a negative force.

4.2. Response to a more complex distributed force

A more complex distributed force is also considered as shownin Fig. 5. This arbitrarily assigned distributed force consists of bothtangential and radial components, and the corresponding dis-tribution density is plotted separately. Due to the asymmetry ofthe distribution, the Fourier expansion of this general distributedforce will include both cosine and sine components. The sameparameters as in Table 1 are used for the REF model except for thehigher, more practical value of K e0.5 5 N/m per radian=θ . Thedeformation results solved by this general force are plotted inFig. 6. Again the deformation is amplified by 10 times for clarity. Inaddition to the deformation of both linear and unilateral founda-tion cases, the identified collapsed regions can be seen in this

Fig. 5. Distribution density of a general distributed force.Fig. 6. Deformation of ring centroid due to the distributed force given in Fig. 5.

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436 435

figure, corresponding roughly to the circumferential regions of theapplied distributed force (more on the left than on the right).

Finally, we remark that with a cut-off harmonic number N¼50,even for the complex applied force cases the iterative computa-tions are completed within seconds on a modern PC. This makesthe proposed methods suitable for parametric design studieswhere the different beam and foundation material properties,geometric dimensions, etc. can be quickly optimized.

5. Conclusion and discussion

In this paper, the analysis of the well-studied REF model hasbeen extended to the unilateral foundation case. This paperfocused on the static deformation problem of a Timoshenko ringresting on a two-parameter foundation with unilateral radialstiffness and subject to arbitrary in-plane force. For the case of alinear foundation, the governing equations have been derived andsolved analytically for both the radial and tangential displace-ments as well as for the section rotation. It is then observed thatfor a unilateral foundation, the ring on the linear foundation canbe treated as supported by a distributed excessive force to thelinear foundation model. Then, the nonlinear unilateral foundationproblem can be solved via the compensation of this excessiveforce, thereby setting up a simple iterative scheme. Comparisonwith FEA results for a concentrated radial force illustrated thevalidity and accuracy of the proposed approach. Then a morecomplex distributed force with both radial and tangential com-ponents is considered and the deformation as well as the deter-mined collapsed region are shown.

This approach achieves the solution to the nonlinear unilateralfoundation problem in a physically-motivated, fast and elegantway. The computation involves simple iterations based on linearanalytical solutions. Direct numerical solution of the governingnonlinear differential equations is avoided. Compared with FEA,the proposed method achieves the nonlinear solution withouttime-consuming modeling and meshing work. Compared with theexisting semi-analytical method in [18], the proposed method iscapable of solving for the deformation response to an arbitrarycomplex distributed force in a unified way. Furthermore, in ourcontinuing work, we shall show how the method can be easilyextended to solve the static contact problemwith arbitrary surfaceprofiles. In a companion paper [19], the authors present theapplication of this iterative approach in the analysis of the

dynamic forced response problem of a ring on a unilateral elasticfoundation. Therein lies the major benefit of the iterativeapproach. It avoids the need for solving the complex, coupled andnonlinear differential equations for the unilateral elastic founda-tion. It can therefore be used to accelerate parametric designsensitivity studies and optimizations to select the proper ring andfoundation materials and geometric properties for the envisagedapplication of the REF.

Appendix A. Coefficients for displacement solutions

Coefficients for solutions of radial direction components

Cur nZrur n Zrur n Zrur

ZD n ZD n ZD n ZDQ

Cu nZrut n Zrut n

ZD n ZD n ZD n ZDQ

C nZr n Zr n

ZD n ZD n ZD n ZDQ

Cur nZrur n Zrur n Zrur

ZD n ZD n ZD n ZDQ

Cu nZrut n Zrut n

ZD n ZD n ZD n ZDQ

C nZr n Zr n

ZD n ZD n ZD n ZDQ

4 2 0

6 4 2 03 1

6 4 2 03 1

6 4 2 04 2 0

6 4 2 03 1

6 4 2 03 1

6 4 2 0 A.1

r c rn c

r c rn c

r c rn c

r s rn s

r s rn s

r s rn s

,

4 2

6 4 2 ,

,

3

6 4 2 ,

,

3

6 4 2 ,

,

4 2

6 4 2 ,

,

3

6 4 2 ,

,

3

6 4 2 ,

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

θ

ϕϕ ϕ

θ

ϕϕ ϕ

=+ +

+ + +

=+

+ + +

=+

+ + +

=+ +

+ + +

= −+

+ + +

= −+

+ + + ( )

Coefficients for solutions of tangential direction components

Cur nZ ur n Z ur n

ZD n ZD n ZD n ZDQ

Cu nZ ut n Z ut n Z ut

ZD n ZD n ZD n ZDQ

C nZ n Z n Z

ZD n ZD n ZD n ZDQ

Cur nZ ur n Z ur n

ZD n ZD n ZD n ZDQ

Cu nZ ut n Z ut n Z ut

ZD n ZD n ZD n ZDQ

C nZ ut n Z ut n Z ut

ZD n ZD n ZD n ZDQ

3 1

6 4 2 04 2 0

6 4 2 04 2 0

6 4 2 03 1

6 4 2 04 2 0

6 4 2 04 2 0

6 4 2 0 A.2

c n c

c n c

c n c

s n s

s n s

s n s

,

3

6 4 2 ,

,

4 2

6 4 2 ,

,

4 2

6 4 2 ,

,

3

6 4 2 ,

,

4 2

6 4 2 ,

,

4 2

6 4 2 ,

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

( )

θ θ

θθ θ θ

ϕθϕ θϕ θϕ

θ θ

θθ θ θ

ϕθ θ θ

=+

+ + +

=+ +

+ + +

=+ +

+ + +

= −+

+ + +

=+ +

+ + +

=+ +

+ + + ( )

θ θ

θ θ

θ θ

θ θ

θ θ

θ θ

C. Wang et al. / International Journal of Mechanical Sciences 101-102 (2015) 429–436436

where

ZD GA EA EI

ZD GA Rh EA K GA REI K K REA EI GA EA EI

ZD K R h EA K GA K R EA GA R hEA K

GA Rh EA K K R EI K GA K REI

REA EI K GA EA EI

ZD K R K R h K R h R EA R hEA Rh EA K

Zrur REA EI

Zrur R Rh EA K GA R EA REI K GA EI

Zrur R GA K R Rh h

Zrut GA EA REI

Zrut GA R hK GA Rh K Rh EA K GA R EA R

Zr GA R EA

Zr R GA RK GA hK hEA K GA EA

Z ur R GA RhEA GA EI EA EI

Z ur GA R hK GA R h K R h EA K GA R EA

GA R hEA

Z ut GA EI R

Z ut GA Rh K GA RhEA K REI EA EI R

Z ut K R h K GA K R GA K R h R h EA K

GA R EA GA R hEA

Z GA RhEA

Z GA RhK K RhEA GA REA GA hEA R

Z K R hK GA K R GA K R h R hEA K

GA R EA GA RhEA

6 4

4 4 4 8

2 4 4

2 4 4

4 4

0 4 4 4 4

4 4

2 4 4 4

0 4 4

3 4 4

1 2 4

3 4

1 2 2 2

3 2 2 2

1 4 2 2 4

2

4 4

2 2 2 2

0 2 4 2 2

4 2

4 2

2 2 2 2 2

0 4 4 2 4

4 2 A.3

r

r r

r r

r r r

r

r r r

r

r r r

2

2 2 3 2

2 2

4 3 2 2 3 2 2

2 2

2 2 2

2 2 2 2

2

2

3 2 2 2 2 3

2

2

3 2 4 3 2 2

3 2

3 3 2 2

2

( )

( )( )

( )

( )

( )

( )( )

( )

ϕ

ϕ

θ

θ

θ

θ

θ

θϕ

θϕ

θϕ

=

= + + −

= + +

− + +

+ +

= − + + − +

=

= + + +

= − +

= − −

= − − −

=

= − − −

= − − −

= − + + +

+

=

= − + +

= + + +

+ +

=

= + − −

= + + +

+ + ( )

θ θ

θ θ θ θ θ θ θ θ

θ θ θ θ θ

θ θ θ θ θ

θ θ θ θ θ

θ θ θ

θ θ

θ θ θ θ θ θ

θ

θ θ

θ θ θ θ θ

θ

θ θ θ θ θ

θ θ θ θ

θ θ θ θ θ

θ

θ

θ θ θ θ θ

θ θ θ

θ θ

θ

θ θ θ θ

θ θ θ

θ θ

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