ARTICLE IN PRESS

International Journal of Mechanical Sciences 50 (2008) 1035– 1041

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International Journal of Mechanical Sciences

0020-74

doi:10.1

� Tel.:

E-m

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

Tensionless contact of a finite beam resting on Reissner foundation

Yin Zhang �

State Key Laboratory of Nonlinear Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, China

a r t i c l e i n f o

Article history:

Received 8 January 2007

Received in revised form

2 February 2008

Accepted 16 February 2008Available online 23 February 2008

Keywords:

Tensionless contact

Beam

Reissner foundation

Lift-off

03/$ - see front matter & 2008 Elsevier Ltd. A

016/j.ijmecsci.2008.02.006

+86 10 62648720.

ail address: zhangyin@lnm.imech.ac.cn

a b s t r a c t

A more generalized model of a beam resting on a tensionless Reissner foundation is presented.

Compared with the Winkler foundation model, the Reissner foundation model is a much improved one.

In the Winkler foundation model, there is no shear stress inside the foundation layer and the foundation

is assumed to consist of closely spaced, independent springs. The presence of shear stress inside

Reissner foundation makes the springs no longer independent and the foundation to deform as a whole.

Mathematically, the governing equation of a beam on Reissner foundation is sixth order differential

equation compared with fourth order of Winkler one. Because of this order change of the governing

equation, new boundary conditions are needed and related discussion is presented. The presence of the

shear stress inside the tensionless Reissner foundation together with the unknown feature of contact

area/length makes the problem much more difficult than that of Winkler foundation. In the model

presented here, the effects of beam dimension, gap distance, loading asymmetry and foundation shear

stress on the contact length are all incorporated and studied. As the beam length increases, the results of

a finite beam with zero gap distance converge asymptotically to those obtained by the previous model

for an infinitely long beam.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

As noted by Timoshenko and Woinowsky-Krieger [1], thecorners of a laterally loaded, simply supported rectangular plate ingeneral have the tendency of separating from its supports and theseparating of the structure from the support is often called lifting-off [2–10]. The support of Timoshenko and Woinowsky-Krieger isassumed to react to both tension and compression [1]. While, suchassumption is motivated more by the desire for mathematicalsimplicity than by physical reality [4]. The support material isoften a rather complex medium which may react only tocompression [11] and is thus often referred to as tensionlessfoundation [3,9,12] or unilateral support [13,14]. The contactbetween the structure and the tensionless support with lifting-offis often named as unbonded contact [2,5,15], which emphasizesthe property of lifting-off; or as tensionless contact [3,6,7,9,10],which emphasizes the property of the unilateral response of thesupport. There are two ways of formulating the tensionless/unbonded contact problem: integral formulation [2,6,7,9,16] anddifferential formulation [3,10,12–15]. If the support is modeled asan elastic half-space, in essence it is to solve a Boussinesq problemand its formulation can only be integral [2,16]. If the support ismodeled as an elastic foundation, its formulation can be either

ll rights reserved.

integral [6,7,9] or differential [3,10,12–15]. Compared with thedifferential formulation, the integral formulation and the iterativealgorithm needed to solve the problem are rather lengthyandcomplex. On the modeling aspect, the elastic half-space model isalso a mathematically much more difficult elasticity problem thanthe elastic foundation one [11].

The elastic foundation model simplifies the elastic continuumproblem by assuming certain relationship between the supportreaction/pressure to its surface displacement. For example, theWinkler foundation model is to treat the support as if it consists ofclosely spaced, independent springs and its pressure is directlyrelated with the spring elongation/compression. However, suchassumption may sometimes cause the erroneous results, espe-cially on the stresses inside the elastic continuum. A vividexample given by Johnson [17] demonstrates that the contactpressure profile of a sphere derived from the elastic half-spacemodel and the Winkler foundation model is ellipsoidal andparaboloidal, respectively. Li and Dempsey [5], Akbarov andKocaturk [8] did the comparative study on the tensionless contactof a plate with a support which is modeled as the Winklerfoundation and the elastic half-space, respectively and the resultsobtained from these two modelings are significantly different.Different elastic foundation models such as Filonenko-Borodich,Hetenyi, Pasternak, Vlasov, Reissner foundation models aredeveloped as an effort to better capture the characteristics of anelastic continuum. For the characteristics and development ofdifferent elastic foundation models, reader should refer to Kerr’s

ARTICLE IN PRESS

Undeformed beam and elastic foundation position

P

ZX

X1 X2

L1 L2

Wo

L

H

P

H

Lift-off (no contact)

Contact

No contact

Contact

Fig. 1. (a). The schematic diagram and related coordinate system of the continuous

contact of a beam under a point load P. (b) The discontinuous contact scenario.

Y. Zhang / International Journal of Mechanical Sciences 50 (2008) 1035–10411036

papers [11,18] for details. Among those models, Vlasov [19]and Reissner [20] models are the ones of approaching theproblem from a continuum point of view. In spite of its conceptualelegance, Vlasov model faces a difficulty in determining anunknown parameter controlling the decay of stress insidethe foundation, which results in complex iterative computa-tion [21]. Reissner starts his model by examining the governingequations and boundary conditions of an elastic half-spaceand obtains relatively simple (as compared with elastic half-space model) pressure-surface displacement relation by assumingthe in-plane (the plane perpendicular to the foundationdepth direction) stresses are zero [20]. Reissner’s assumptionleads to a conclusion that the shear stresses are constantthroughout the depth of the foundation [11], which is physi-cally unrealistic, especially for the thick foundationlayer. However, ‘‘in view of the fact that foundation modelsare introduced to study the response of the foundationsurface to loads and not the stresses caused within the founda-tion, this particular deficiency may in general be of no seriousconsequence.’’ [11].

The above studies of tensionless contact [2,4–10,12–16] treatthe support either as the Winkler foundation or as an elastic half-space. Weitsman [3] conducted a comparative study and derivedthe analytical solutions for an infinitely long beam resting on atensionless Winkler foundation and Reissner foundation, respec-tively. However, it has been realized that the boundary conditionsused by Weitsman [3] for an infinitely long beam can beproblematic for the finite beam/cylinder case [10,15]. The non-symmetric solution terms are thrown away by Weitsman [3] withthe assumption that the tensionless contact of an infinitely longbeam is symmetric, which as shown later in this paper, is a veryreasonable assumption. However, when the beam is finite andloading is asymmetric, this assumption is invalid. Also the gapdistance between the structure and support in above studies[2–9,12–16] is zero. In the microelectromechanical systems(MEMS) area, the microstructures contact the substrate due tocapillary force, external pressure or electrostatic force and there isa gap distance between the microstructure and the substrate[22–24]. For MEMS devices contact study, it is thus important toincorporate the gap distance as a parameter into the modelcapable of describing the finite-sized structures. This paperpresents a more generalized model on the tensionless contactof a beam on the Reissner foundation, which incorporates thebeam dimension, gap distance and loading asymmetry asimportant parameters in the model. A parameter l whichindicates the contribution of shear stress inside the Reissnerfoundation is introduced and varied to show its influence on thecontact. The problem formulation is a differential one which inessence retains the mathematical simplicity of the elasticfoundation model.

2. Equations of equilibrium and its solutions

As shown in Fig. 1(a), a hinged–hinged beam is separatedfrom the Reissner foundation with the gap distance of Wo. Ef andGf are Young’s modulus and shear modulus of the Reissnerfoundation, respectively. P is the concentrated load and thecoordinate system starts at the loading point. L1 and L2 are thedistances from the loading point to the beam left and right ends,respectively. L ¼ L1 þ L2 is the beam length and EI is the beamflexural rigidity. H is the layer thickness of the Reissnerfoundation. X1 is leftside contact length and X2 is the rightsidecontact length (X1 and X2 are positive numbers). The beam willseparate from the foundation once the beam deflection W is lessthan the gap distance Wo. The beam deflection W is divided into

the following three regions and the governing equation for eachregion holds as follows [3,10]:

EId4W1

dX4¼ 0; W1oWo; �L1oXo� X1

EIGf H3

12E2f

d6V

dX6�

DH

Ef

d4V

dX4þ

Gf H2

3Ef

d2V

dX2� V

¼ �H

EfPdðXÞ þ

Gf H3

12E2f

Pd2dðXÞ

dX2; W2XWo; L1pXpX2

EId4W3

dX4¼ 0; W3oWo; X2oXoL2

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

(1)

V ¼W2 �Wo and d is the Dirac delta function. The abovegoverning equation does not account for the beam self weight,which can be modeled as a uniform load [3]. The above equationalso implicitly assumes the so-called continuous contact scenario[3]. For the beam with self weight or uniformly distributeddownward load, the beam can have the contact scenario as shownin Fig. 1(b). As the contact regions in Fig. 1(b) are separated fromone another, the contact scenario is called discontinuous contact[2]. However, the concentrated load required for the discontin-uous contact to occur is so large and the beam will collapse at amuch smaller concentrated load. Therefore, it is of no practical useto analyze such contact scenario [3]. To nondimensionalize Eq. (1),the following quantities are introduced

k ¼Ef

H; b4

¼k

4EI; x ¼ bX,

w ¼ bW ; v ¼ bV ; l2¼

H2b2Gf

48Ef(2)

ARTICLE IN PRESS

Y. Zhang / International Journal of Mechanical Sciences 50 (2008) 1035–1041 1037

With the substitution of v ¼ w2 �wo, the dimensionless form ofEq. (1) now is as follows:

d4w1

dx4¼ 0; w1owo; �l1oxp� x1

l2 d6w2

dx6�

1

4

d4w2

dx4þ 16l2 d2w2

dx2�w2 þwo

¼ �FdðxÞ þ 4l2Fd00ðxÞ; w2Xwo; �x1pxo� x2

d4w3

dx4¼ 0; w3owo; x2pxol2

8>>>>>>>>>>>><>>>>>>>>>>>>:

(3)

It is interesting to notice that in the above equation, thedimensionless gap distance wo plays exactly the same role asthe beam weight in Weitsman’s governing equation [3]. x1, x2, l1,l2, l, wo, wi (i ¼ 1;2;3) are defined as

x1 ¼ bX1; x2 ¼ bX2; l1 ¼ bL1; l2 ¼ bL2,

l ¼ bL; wo ¼ bWo; wi ¼ bWi ði ¼ 1;2;3Þ (4)

The characteristic equation of w2 in Eq. (3) is as follows:

f6�

1

4l2f4þ 16f2

�1

l2¼ 0 (5)

By letting D ¼ f2, Eq. (5) can be written as a cubic equation

D3 �1

4l2D2 þ 16D�

1

l2¼ 0 (6)

The three roots of Eq. (6) can be obtained from Cardan solution[25]. In this case, the three roots are always that one is real andthe other two are complex conjugates. Therefore, the six roots ofEq. (5) which are the square roots of those three roots of Eq. (6)are obtained as follows:

f1;2 ¼ �a; f3;4;5;6 ¼ �Z� gi (7)

a, Z, g are three real number. So there are two real roots and fourcomplex conjugate roots for Eq. (5). These six roots give thesolution form of homogeneous part of w2. The solutions of Eq. (3)are now given as the following three equations:

w1 ¼ B1x3þ B2x

2þ B3xþ B4 (8)

w2 ¼ A1 coshðaxÞ þ A2 sinhðaxÞ þ A3 sinhðZxÞ sinðgxÞ

þ A4 sinhðZxÞ cosðgxÞ þ A5 coshðZxÞ sinðgxÞ

þ A6 coshðZxÞ cosðgxÞ þ A sinh jaxj

þ B coshðZxÞ sin jgxj þ D sinhðZxÞ cos jgxj þwo (9)

w3 ¼ C1x3þ C2x

2þ C3xþ C4 (10)

Here Bi, Ci (i ¼ 1–4) and Aj (j ¼ 1–6) are the unknown constants tobe determined. Because the separation points (x1 and x2) are alsounknown, there are total 16 unknowns to be determined. The firstsix terms in Eq. (9) are the homogeneous solution and the lastthree terms are the particular solution. A, B and D are the threeconstants given as follows:

A ¼4Fðg4 � Z4Þ

aD; B ¼

Fða4 � Z4 þ 10Z2g2 � 5g4Þ

gD,

D ¼Fð�a4 þ g4 � 10Z2g2 þ 5Z4Þ

ZD(11)

With the following definition of D as

D ¼ ðZ2 þ g2Þ½a4 � 2a2ðZ2 � g2Þ þ ðZ2 þ g2Þ2� (12)

Here it would be instructive to have a comparison with Weits-man’s formulation/solution on this problem. Geometrically,Weitsman’s beam length is infinite and the gap distance is zero(wo ¼ 0) [3]. Weitsman’s governing equation is only for thecontact part and his solution is the following:

w2 ¼ A01 coshðaxÞ þ A03 sinhðZxÞ sinðgxÞ þ A06 coshðZxÞ

� cosðgxÞ þ A sinh jaxj þ B coshðZxÞ sin jgxj

þ D sinhðZxÞ cos jgxj (13)

A01, A03 and A06 are three unknown constants to be determined. Thereason for the above solution is due to the assumption that thecontact area is symmetric to the concentrated load, which is alsoselected as the coordinate origin. The functions associated withA01, A03, A06, A, B and D are all even functions. Those terms of oddfunctions are thrown away because of this symmetry assumption.For a beam with finite length and the concentrated load off thecenter, Weitsman’s solution above cannot be valid any more andthose terms associated with odd functions survive. However, asdiscussed later, when the beam length is large, those odd functionterms do not contribute much to the solution, in other words,Weitsman’s solution is a very good approximation.

3. Boundary/matching conditions

As mentioned above, there are total 16 unknowns to bedetermined (Bi, Ci (i ¼ 1–4), Aj (j ¼ 1–6) and x1, x2). Therefore, 16boundary/matching conditions are needed for the problem solving.

At x ¼ �x1, x2, the beam separates from the elastic foundationand different governing equations apply as reflected from Eq. (1)or Eq. (3). Certain conditions must be satisfied for w1, w2 and w3

to match each other. The following two equations give the eightmatching conditions:

w1ð�x1Þ ¼ w2ð�x1Þ;dw1ð�x1Þ

dx¼

dw2ð�x1Þ

dx,

w2ðx2Þ ¼ w3ðx2Þ;dw2ðx2Þ

dx¼

dw3ðx2Þ

dx(14)

d2w1ð�x1Þ

dx2¼

d2w2ð�x1Þ

dx2;

d3w1ð�x1Þ

dx3¼

d3w2ð�x1Þ

dx3,

d2w2ðx2Þ

dx2¼

d2w3ðx2Þ

dx2;

d3w2ðx2Þ

dx3¼

d3w3ðx2Þ

dx3(15)

Eq. (14) is the geometric conditions which require the continuity ofthe beam deflection and its slope at the separation points.Physically, Eq. (14) plays the role of compatibility condition toguarantee that there is no beam fracture/breaking at the separationpoints. Eq. (15) is the natural boundary conditions which requirethe continuity of bending moment and shear force. Also at theseparation points, the following two matching conditions hold:

d4w2ð�x1Þ

dx4¼ 0;

d4w2ðx2Þ

dx4¼ 0 (16)

At the separation points there is no interaction between the beamand foundation, Eq. (16) physically states that the pressure stress atthe separation points is zero [3]. There are two displacementconstraint conditions at x ¼ �x1, x2

w2ð�x1Þ ¼ wo ðor w1ð�x1Þ ¼ woÞ,

w2ðx2Þ ¼ wo ðor w3ðx2Þ ¼ woÞ (17)

Eq. (17) prescribes the separation rule that the beam lifts-off fromthe tensionless foundation once its deflection is less than the gap

ARTICLE IN PRESS

l=200 l=40 l=20

0 0.5 1 1.5 2 2.5 31.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

ξ 2

Symmetric loading and wo=0

λ

Fig. 2. l versus contact lengths of a hinged–hinged beam with the three different

lengths (l) under a symmetric loading (l1 ¼ 0:5l). The gap distance is wo ¼ 0.

Y. Zhang / International Journal of Mechanical Sciences 50 (2008) 1035–10411038

distance. For a hinged–hinged beam, the boundary conditions atthe ends are as follows:

w1ð�l1Þ ¼ 0;d2w1ð�l1Þ

dx2¼ 0,

w3ðl2Þ ¼ 0;d2w3ðl2Þ

dx2¼ 0 (18)

Eqs. (14)–(18) offer 16 boundary/matching conditions in total. Bysubstituting the solution forms of w1, w2 and w3 into those 16boundary/matching conditions, those 16 unknowns can be found.Because of the unknown feature of x1 and x2, solving these 16unknowns is a nonlinear problem and Newton–Raphson method isapplied here.

Again, here let us have a discussion on Weitsman’s boundary/matching conditions. For Weitsman’s case, there are only fourunknowns, A01, A03, A06 and x1 (x1 ¼ x2 due to the symmetryassumption). Weitsman’s four boundary conditions are as follows[3]:

d2w2ðx1Þ

dx2¼ 0;

d3w2ðx1Þ

dx3¼ 0;

d4w2ðx1Þ

dx4¼ 0,

Gf

24

d5w2ðx1Þ

dx5þ

16dw2ðx1Þ

dx

" #¼ 0 (19)

The first three boundary conditions indicate no bending moment,shear force and pressure stress at the separation point. Comparedwith Eq. (19), Eq. (15) does not indicate the vanishing of bendingmoment and shear force at the separation point. Kerr derives thematching conditions of a plate on a Pasternak foundation by avariational approach and his matching conditions are very similarto those in Eqs. (14) and (15), which explicitly negate thevanishing of bending moment and shear force as the matchingconditions [15]. The last boundary condition in Eq. (19) indicatesthat at separation point there is no shear stress. Here it is worthdiscussing this last boundary conditions. This no shear stresscondition at the separation point is originally given by Reissner asa ‘‘simple and reasonable condition’’ without a proof [20]. Kerr’sanalysis shows that this no shear stress condition ‘‘is valid only inthe very special case when the foundation material under thefinite plate is separated along the cylindrical boundary from thesurrounding foundation materials or walls’’ [11]. The moregeneralized boundary condition as a substitution for the no shearstress condition given by Kerr is as follows [11]:

Dr4V þ4Ef

HV � q ¼ 0 (20)

V ¼W �Wo as defined before. r2 ¼ q2=qX2þ q2=qY2 is the

Laplacian operator and r4 ¼ ðr2Þ2. For one-dimensional case,

r4 ¼ d4=dX4. D is the plate/beam flexural rigidity and q is auniformly distributed external load. Here the beam is weightless,so q ¼ 0. Also at X2 and �X1, V ¼ 0. Therefore, the dimensionlessboundary condition derived from Eq. (20) is as follows:

d4v

dx4¼

d4w2

dx4¼ 0 (21)

This boundary condition has already been used in Eq. (19). It isnoticed that Weitsman’s boundary conditions of Eq. (19) startwith the second derivative and there is no information for thedeflection and slope at the separation point. For the study of thetensionless Winkler foundation, Weitsman prescribes the vanish-ing of the deflection and slope as the boundary conditions [3]. It isalso noticed that when Gf ¼ 0, which is the limit case of Winklerfoundation, the fourth boundary condition of no shear stress in Eq.(19) is automatically satisfied. So there is some inconsistencyhere. Weitsman’s formulation focuses only on the contact part andthe formulation presented here considers both contact and lifting-

off parts. However, in the next section Weitsman’s formulation isdemonstrated to be an approximation with high accuracy whenthe beam length is very large.

4. Results and discussion

Fig. 2 shows l versus the contact lengths of hinged–hingedbeams with different lengths. The gap distance wo is zero. Theconcentrated load F is symmetric, i.e., at the center of the beam, sothe left and right contact lengths are the same (x1 ¼ x2) due to thesymmetry. l is the parameter defined in Eq. (2), which physicallyindicates the shear stress contribution to the total foundationpressure as compared with the normal one. As mentioned before,the foundation becomes Winkler foundation when l or Gf is zero.Fig. 2 shows that the beam with longer length has smaller contactlength because longer beam means larger flexurality and morebeam parts lift-off from the foundation. The contact lengthdecreases with the increase of l and later converges when l islarge enough. For an infinitely long beam with zero gap distance,Weitsman [3] obtains x1 ¼ p=2 � 1:5708 when l ¼ 0 and x1 ¼

p=2ffiffiffi2p� 1:1107 as l!1. For the beam of l ¼ 200 computed

here, x1 ¼ 1:58 when l ¼ 0 and x1 ¼ 1:12 as l!1. By furtherincreasing the beam length l, the contact length asymptoticallyconverges to Weitsman’s results. When the beam with finitelength and under an asymmetric loading, the left and rightcontact lengths are in general different from each other, i.e.,x1ax2. Fig. 3 shows the beam left and right contact lengths underan asymmetric loading at l1 ¼ 0:6l and the gap distance is alsozero. Clearly from Fig. 3, the short beam (l ¼ 20) is more sensitiveto the asymmetric loading and its left and right contact lengthsare different from each other. When the beam length increases,the difference between left and right contact lengths shrinks.When l ¼ 200 and l ¼ 2, x1 ¼ 1:124 and x2 ¼ 1:127; the differencebetween the left and right contact lengths now is very small. Thecontact length of the beam under a symmetric loading is x1 ¼

x2 ¼ 1:125 when l ¼ 200 and l ¼ 2. This in essence demonstratesthat the Weitsman’s symmetric assumption for an infinitely longbeam as discussed above is an approximation with high accuracy.It is noticed that in both Figs. 2 and 3 of wo ¼ 0, the contactlengths are independent of the magnitude of F. This interestingand somewhat surprising result is due to the lifting-off mechan-ism, which also occurs in the Winkler foundation [3,6,9,10] and

ARTICLE IN PRESS

Left Contact Length Right Contact Length

0 0.5 1 1.5 2 2.5 31

1.5

Dim

ensi

onle

ss C

onta

ct L

engt

h

l=200

Left Contact Length Right Contact Length

0 0.5 1 1.5 2 2.5 31

1.5

2

l=40

Left Contact Length Right Contact Length

0 0.5 1 1.5 2 2.5 31

1.5

2l=20

λ

Fig. 3. l versus contact length of a hinged–hinged beam with the three different lengths (l) under an asymmetric loading (l1 ¼ 0:6l). The gap distance is wo ¼ 0.

-10 -8 -6 -4 -2 0 2 4 6 8 10 4

3

2

1

0

-1

-2

-3

-4

-5

-6 x 10-3 Symmetric loading: l=20 and λ=2

Undeformed elastic foundationposition

F=0.01F=0.02Separation point

w

ξ

Fig. 4. The deflection of the beam with l ¼ 20 and l ¼ 2 under the symmetric

loadings of F ¼ 0:01 and 0.02, respectively.

-12 -10 -8 -6 -4 -2 0 2 4 6 8 4

2

0

-2

-4

-6

-8 x 10-3

w

Asymmetric loading: l=20 and λ=2

F=0.01F=0.02Separation point

Undeformed elastic foundationposition

ξ

Fig. 5. The beam deflection with l ¼ 20 and l ¼ 2 under the asymmetric loadings

of F ¼ 0:01 and 0.02, respectively. The asymmetric loadings are both located at

l1 ¼ 0:6l.

Y. Zhang / International Journal of Mechanical Sciences 50 (2008) 1035–1041 1039

elastic half-space cases [2]. To explain this, let us examine thebeam deflection under different loads. Fig. 4 plots the beamdeflections under two symmetric loadings of F ¼ 0:01 and 0:02,respectively. Although the loads are different, clearly form Fig. 4the beam lifts-off/separates from the elastic foundation at theexactly same points. The larger load F just pushes the beamdeeper into the foundation without the change of contact length.Fig. 5 plots the beam deflections under two asymmetric loadingsof F ¼ 0:01 and 0:02 (both located at l1 ¼ 0:6l), respectively. Thesame scenario happens again that the larger load just pushes thebeam deeper into the foundation without changing either the leftor right contact length.

Fig. 6 plots the dimensionless contact length of the beam withl ¼ 20 and wo ¼ 0:1 under the different symmetric loads of F as a

function of l. Unlike those in Figs. 2 and 3, the contact lengths inFig. 6 are no longer independent on the magnitude of F. For acomparison reason, the contact length of the same size beam withwo ¼ 0 is also plotted in Fig. 6. Fig. 6 shows that larger load F haslarger contact length for the beam with wo ¼ 0:1. It is also noticedthat as F increases, the curves of wo ¼ 0:1 approaches the curve ofwo ¼ 0. The curve of F ¼ 0:4 and wo ¼ 0:1 is very close to that ofwo ¼ 0. Again, here it will be instructive for us to examine thebeam deflection under different loads when the gap distance isnonzero. Fig. 7 plots the deflections of the beam with l ¼ 20, wo ¼

0:1 under the load F ¼ 0:01 for l ¼ 0:1 and 2, respectively. Thecontact length is x1 ¼ x2 ¼ 1:3293 when l ¼ 0:1 and x1 ¼ x2 ¼

0:916 when l ¼ 2. As a comparison, Fig. 8 plots the deflection of

ARTICLE IN PRESS

0 0.5 1 1.5 2 2.5 3

F=0.01, wo=0.1

F=0.4, wo=0.1

wo=0

F=0.1, wo=0.1

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

Dim

ensi

onle

ss C

onta

ct L

engt

h

l=20:Comparison of Zero Gap and Nonzero Gap Cases

λ

Fig. 6. l versus contact length of a hinged–hinged beam of l ¼ 20 and wo ¼ 0:1

under different symmetric loadings and their comparison with l ¼ 20 and wo ¼ 0

case.

λ=0.1λ=2.0Separation Point

0.12

0.1

0.08

0.06

0.04

0.02

0

-0.02

ξ

w

F=0.01, wo=0.1, l=20:Beam Deflection At Different λ

ξ2=0.916

ξ2=1.3293

10-10 8642-8 -6 -4 -2 0

Fig. 7. The deflections of the beam with l ¼ 20, wo ¼ 0:1, l ¼ 0:1 and 2:0 under a

smaller symmetric load of F ¼ 0:01.

λ=0.1λ=2.0Separation Point

-10 -8 -6 -4 -2 0 2 4 6 8 100.3

0.25

0.2

0.15

0.1

0.05

0

-0.05

-0.1

-0.15

ξ

w

F=0.4, wo=0.1, l=20:Beam Deflection At Different λ

ξ2=1.737

ξ2=1.19

Fig. 8. The deflections of the beam with l ¼ 20, wo ¼ 0:1, l ¼ 0:1 and 2:0 under a

larger symmetric load of F ¼ 0:4.

Y. Zhang / International Journal of Mechanical Sciences 50 (2008) 1035–10411040

the beam with l ¼ 20, wo ¼ 0:1 under a much larger load F ¼ 0:4for l ¼ 0:1 and 2. The contact length now is x1 ¼ x2 ¼ 1:737 whenl ¼ 0:1 and x1 ¼ x2 ¼ 1:19 when l ¼ 2. The deflections in Figs. 7and 8 are dramatically different from one another no matter whatl value is. Figs. 7 and 8 demonstrate that for the finite beam withnonzero gap distance the contact length is dependent on themagnitude of F. Here it will be more reasonable to use F=wo ratherthan F as a parameter to evaluate the contact. When wo ¼ 0, nomatter how small F is, F=wo !1 and this explains why thecontact length of beam with zero gap distance is independent ofthe loading magnitude. For the beam with a fixed nonzero wo,F=wo becomes larger and larger with the increase of F. This can beused to explain why in Fig. 6 those curves of wo ¼ 0:1 approachthat of wo ¼ 0 with F increasing.

A hinged–hinged beam is used here as the study case. Othertypes of beam end boundary conditions such as free–free,clamped–clamped cantilever ones, etc., can also be studied withthe change of Eq. (18).

5. Conclusion

The contact lengths and deflections of a finite beam with zero/nonzero gap distance, different lengths and symmetric/asym-metric loading on a Reissner foundation are discussed. The beamlength and gap distance are vital on determining the contactlength. For a relatively short beam, the loading location is alsovery important and the asymmetric loading results in thedifference of left and right contact lengths. When the gap distanceis zero and the beam length becomes larger and larger, thedifference between the symmetric and asymmetric loadingsbecomes smaller and smaller; the contact length approaches theone obtained by Weitsman for an infinitely long beam. When thegap distance is zero, the contact length of a finite beam is againdemonstrated to be independent of the load magnitude as that ofan infinitely long beam. But that of a finite beam with nonzero gapdistance is demonstrated to be dependent on the load magnitude.Therefore, F=wo is proposed as a parameter to evaluate the contactlength. For the fixed nonzero gap distance case, the beam contactlength approaches that of the zero gap distance case when F

approaches infinity (so is F=wo).

Acknowledgments

This work is supported by both the National Natural ScienceFoundation of China (NSFC, Grant nos. 10502050 and 10721202)and the Scientific Research Foundation for the Returned OverseasChinese Scholars, State Education Ministry of China.

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