a finite strip loaded by a bonded-rivet of a different material

A ®nite strip loaded by a bonded-rivet of a di�erentmaterial

K.C. Ho, K.T. Chau *

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received 24 January 1997; received in revised form 6 May 1998

Abstract

This paper investigates the maximum stress concentrations in a ®nite strip loaded by a bonded elastic rivet by

using the complex variable method in conjunction with the least-square boundary collocation method (BCM). Therivet-load is modeled by a uniform distributed body force; and the resultant rivet-force is acting along the transversedirection. The accuracy of the BCM is checked by comparing the results to those of the ®nite element method for aspeci®c ®nite geometry of a strip and by the exact solution for the case of an in®nite plane. Numerical results show

that the maximum shear and hoop stresses at the interface decrease with increasing b/R, where b is half of the widthof the strip and R is the radius of the rivet. The maximum shear stress at the interface increases with z= m2/m1(where m1 and m2 are the shear moduli of the strip and rivet respectively) while the maximum hoop stress decreases

with z. For ze1, the maximum normal bond stress at the interface decreases initially to a local minimum beforerising to a steady value as b/R further increases. As b/R increases, the angular location of maximum stressoccurrence ymax, which is measured from the direction of resultant rivet-force, increases from about 368 0428 to 908(the in®nite plane limit) for the shear bond stress, and jumps suddenly from a roughly constant value (508 0558) to08 (the in®nite plane limit) for the normal bond stress. Similar sudden shifts in the angular location of maximumstress are also observed in the hoop stress at the interface. # 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Finite strip; BCM; Stress

1. Introduction

Stress concentration in rivet-loaded strips is one of

the classical problems in the mathematical theory oflinear elasticity (e.g. Howland [1, 2], Howland and

Stevenson [3], and Theocaris [4]). Many engineering

structures, such as aircraft and steel buildings, are

composed of rivet-strip systems. Recently, a new appli-cation of rivet-loaded strip emerges in the curtain wall

design of buildings. In particular, rock panels such as

granite and marble panels have been used extensivelyin forming the external cladding walls in modern build-

ings. In order to provide aesthetic and prestigious

appearance, the rock panels are normally connected at

their edges by rivets from the inside of the buildings.

The rivets are commonly bonded to the pre-drilled

holes inside the rock panels by epoxy, and the rock

panels are then sealed at their four edges. Although

these panels are not load-bearing, they should with-

stand wind pressure and self weight. Brittle failure of

these rock panels, however, occurs frequently at the

interface between the panel and the rivet because of

the high stress concentration there. As the rock panels

are very expensive, an optimum design of the thickness

of them is essential. Therefore, the maximum stress

concentration in the rock panels expressed as a func-

tion of the ratio of rivet size to strip width should pro-

vide important information in the design of rivet-panel

systems used in curtain walls.

Computers and Structures 70 (1999) 203±218

0045-7949/99/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved.

PII: S0045-7949(98 )00147-3

PERGAMON

* Corresponding author. Tel.: +852 2766 6015; Fax: +852

2334 6389; E-mail: [email protected].

In this study, the rivet-rock panel system is modeled

as a ®nite elastic strip loaded by a bonded rivet of adi�erent material. The analytical solution of this pro-posed problem is extremely di�cult, if not impossible,

to obtain. Therefore, we propose to investigate,numerically, the stress concentration in a ®nite striploaded by a bonded rivet, by using the boundary collo-

cation method (BCM).The BCM is, in essence, a special form of the weighted

residual method, in which the general solution of thegoverning equation is known and a number of points orso-called collocation points are chosen on the boundary

such that the boundary conditions are satis®ed eitherexactly at these points or approximately among these

points. One of the main advantages of the BCM is thatit requires less computational e�ort in comparison withother numerical methods, such as the ®nite element

method (FEM). For example, by solving nine speci®ctwo-dimensional harmonic boundary value problems,Kolodziej and Kleiber [5] concluded that the BCM gives

more accurate results than those by the FEM when thesame computational e�ort is used. Kolodziej [6] have

reviewed brie¯y the application of the BCM in themechanics of continuous media and provided manyvaluable references in the ®eld up to 1985. As summar-

ized by Kolodziej [6], the earliest use of the BCM isprobably in 1934 by Slater in solving electromagnetic

problems and it has been applied to elastostatics pro-blems, ¯uid ¯ow problems, and vibration problems inplates and membranes since then. The BCM has also

been applied to stress concentration problems in in®niteperforated plates with circular holes by Goldberg andJabbour [7] and Slot and Yalch [8], and with circular in-

clusions by Lee and Conway [9]. Newman [10] also cal-culated the stress intensity factor in an in®nite plate

containing both cracks and holes. The stress concen-trations in perforated square plates are evaluated bySchlack and Little [11] and Redekop [12] when the plate

is subject to edge loads and by Wang and Lin [13] whenit is subject to internal pressure. For ®nite rectangularperforated plates containing both cracks and holes, the

BCM has been applied by Woo et al. [14], Wang etal. [15] and Woo and Chan [16]. For pin-loaded rivet

problems, the BCM has been applied to determine thecontact stresses in orthotropic plates of in®nite extent byHyer and Klang [17] and of ®nite extent by Madenci et

al. [18]. However, to the best of our knowledge, thestress concentration problems of a ®nite strip subject to

bonded-rivet load have not been considered before bythe BCM or by any other methods.Therefore, we investigate in this study the stress dis-

tributions in a ®nite strip loaded by a bonded rivetusing the BCM. The rivet-load is modeled here by auniform distributed body force; such a body-force-

approach was ®rst proposed by Hyer and Klang [17].The general solutions to the problems are ®rst formu-

lated in terms of the two Muskhelishvili's [19] analytic

functions for the strip where the body force is zeroand, in terms of the three Stevenson's [20] analytic

functions, for the rivet where the body force exists.The general solution for these analytic functions are

further expressed in terms of a power series of thecomplex variable z � x� iy with unknown coe�cients

(where x and y are the Cartesian coordinates shown inFig. 1 and i � ��ÿ1p �. Then, an approximate solution is

sought by using the BCM such that the solutionssatisfy both the continuity condition on the bonded

surface between the rivet and the strip and the bound-ary conditions on the strip surfaces, either pointwise or

in an average sense. In particular, when the number ofboundary points chosen equals exactly the total num-

ber of unknown coe�cients assumed in the series sol-ution, the boundary and continuity conditions are

satis®ed exactly on these chosen points. This variant ofthe BCM is normally referred to as the point matching

method (PMM). However, when the number of thechosen points exceeds the number of coe�cients in the

series solution, the continuity and boundary conditionscan be satis®ed approximately using the least squares

method (LSM), which is used to minimize the errorsamong the collocation points along the boundaries.

The ®rst publication on the LSM can be traced backto that by Legendre [21], although the original idea

was ®rst proposed by Gauss [22] in 1795 when he triedto calculate the planetary orbits on the basis of his

Fig. 1. A sketch for a ®nite strip (domain 1) loaded by a

bonded elastic rivet (domain 2) in a state of uniform body

force f0 with resultant force P= f0pR2 and simply-supported

at two ends.

K. Ho, K. Chau / Computers and Structures 70 (1999) 203±218204

measurements of planetary movements (see the histori-cal summary by Stigler [23]). The studies by Lee and

Conway [9] and France [24] suggest that the result byLSM is, in general, slightly better than those by thePMM. Hence, the LSM is employed in the present

study.In this study, we will focus on the e�ect of the shear

modulus ratio z= m2/m1 of the rivet to the strip

(where m2 and m1 are the shear moduli of the rivet andthe strip, respectively) and the geometric ratio b/R(where b is half of the width of the strip and R is the

radius of the rivet) on the magnitude of the stress con-centration at the rivet-strip interface.

2. Complex variable method in plane elastostatic

problems

The theory of the complex variable method appliedin plane elasticity was well established by Kolosov [25]and Muskhelishvili [19], although Goursat [26] and

Filon [27] were probably the ®rst to introduce thisidea. The formalism by Kolosov [25] andMuskhelishvili [19] is, however, restricted to plane elas-tostatic problems with no body force. The complex

variable technique to plane elastic problems with non-zero body force was probably ®rst derived byStevenson [20], and similar derivations can also be

found in textbooks of elasticity (e.g. Refs [28, 29]). Inparticular, for two-dimensional elasticity problems, itis advantageous to introduce the following complex

body force F :

F � Fx � iFy �1�where i � ��ÿ1p

and Fx and Fy are the components of

the body force along the x- and y-axes in a Cartesiancoordinate. If the body force F is conservative (i.e.@Fx/@y= @Fy/@x), it can be expressed by a complex

potential V(z, �z ) as:

F � ÿrV �2a�where z= x+ iy and �z =xÿ iy is the complex conju-

gate of z and

r � @

@x� i

@

@y�2b�

is the complex di�erential operator in Cartesian coor-

dinate. It can be shown that the stresses and displace-ments in the polar coordinate (r and y) of a solidsubjected to a complex body force F can be expressed

by three analytic functions F(z), C(z) and W(z, �z ) as(Refs [20, 28, 29]):

srr � syy � 2�F 0�z� � F 0�z�� 4

1� k@W

@z; �3a�

syy ÿ srr � 2isry � 2

��zF 00�z� �C 0�z� � 1ÿ k

1� k@ �W

@z

�e2iy;

�3b�

2m�ur � iny� ��kF�z�ÿ zF 0�z�ÿC�z�ÿ 1ÿ k

1� kW

�eÿiy

�3c�and

@W

@z� V �3d�

where the complex potential W(z, �z ) can be obtained ifthe body force in the solid is known (see the discussionby Milne-Thomson [29]) and the superimposed ``bar''denotes the conjugate of the complex quantity; s rr, s ry

and s yy are the radial, shear and tangential stresses re-spectively; ur and n y are radial and tangential displace-ments respectively; m and n are the shear modulus and

Poisson's ratio of the material; and, ®nally, k is equalto 3±4n for plane strain condition and (3ÿ n)/(1+ n)for plane stress condition. In the Cartesian coordinate,

the stresses and displacements of the material can berelated to those polar components given in 3(a)±(c) by:

sxx � syy � srr � syy; �4a�

syy ÿ sxx � 2isxy � �syy ÿ srr � 2isry�eÿ2iy; �4b�

ux � iny � �ur � iny�eiy �4c�where s xx, s xy and s yy are the stress components andux and n y are the displacement components in theCartesian coordinate. Note that the stresses and displa-cements given in Eqs. (3) and (4) satisfy both the equi-

librium and compatibility equations automatically.It is important to ensure the stresses and displace-

ments given in Eq. (3) are single-value in applying the

complex variable method in plane elastic problems.The single-valueness of these three analytic functionsF(z) and C(z) and W(z, �z ) can, in general, be assured

if appropriate forms are assumed, which will be furtherdiscussed in a later section.

3. Statement of the strip problem

As discussed in the Introduction, the rivet±rockpanel problem is modeled as a ®nite elastic striploaded by a bonded elastic rivet in this study. The

strip is assumed to be deformed under plane straincondition since the strip is actually obtained as a typi-cal cross section of the rock panels near the rivet-con-

nections. As shown in Fig. 1, we consider a ®nite stripABCDEFA of length 2a and width 2b containing a cir-cular bonded rivet GHIJG of radius R. In addition,

K. Ho, K. Chau / Computers and Structures 70 (1999) 203±218 205

the strip region is referred to as domain 1 while therivet region as domain 2 for easy reference, as shown

in Fig. 1.When a loading of resultant P (force per unit thick-

ness) acts on the rivet, it is assumed to be distributed

as a uniform body force f0 (force per unit volume) inthe x-direction such that P= f0(pR

2). This body-forceapproach for rivet action was ®rst proposed by Hyer

and Klang [17]. Although the two ends of the strip(EF and BC in Fig. 1) are, in actual site condition,connected to other rock panels by rivets, it should be

accurate enough to assume that the strip is simply sup-ported at the ends BC and EF. Thus, we set s yy tozero and apply a parabolic shear stress distribution ony=2a (as shown in Fig. 1).

Since both displacements and stresses are requiredto be continuous at the bonded interface between therivet and the strip on r= R, we have a mixed bound-

ary value problem. As this work is motivated by itspossible application to a steel rivet±rock panel system,di�erent elastic constants have to be assigned to the

strip and the rivet. That is, the elastic constants for thestrip are m1 and k1 and those for the rivet are m2 andk2. Expressed in mathematical terms, the continuity

conditions between the rivet and the strip are:

s�1�rr ÿ s�2�rr ÿ i�s�1�ry ÿ s�2�ry � � 0; �5a�u�1�r ÿ u�2�r � i�n�1�y ÿ n�2�y � � 0 �5b�on r= R. The boundary conditions on the tractionfree edges BAF and CDE are

s�1�xx ÿ is�1�xy � 0 �5c�on x=2b, while the simply supported conditionsalong BC and EF are

s�1�yy � i

�s�1�xy2

3f0pR2

8b3�b2 ÿ x2�

�� 0 �5d�

on y=2a, where the numbers 1 and 2 in the parenth-eses denote, as remarked earlier, the domains of thestrip and the rivet, respectively. Since the strip is

assumed to be simply supported at y=2a, thus onlya parabolic distribution of the shear stresses s xy isimposed, as shown in the second term on the left-hand

side of Eq. (5d).

4. General solution of the strip problem

In view of the single-value requirement, di�erent sol-ution forms have to be used for F(z) and C(z) forsimply- and multi-connected domains. In particular,

for the strip (domain 1 in Fig. 1), we have a multi-con-nected domain with a single pole at z=0; and the gen-eral solution forms for F1(z) and C1(z) are (e.g.

Refs [19, 30]):

F1�z� � ÿ f0R2

2�1�k1� ln z� f0

�XM1

n�0A2nz

2n�XN1

n�1

Aÿ2nz2n

�; �6a�

C1�z� � f0R2k1

2�1� k1� ln z� f0

�XM1

n�1B2nz

2n �XN1

n�1

Bÿ2nz2n

��6b�

where A 2n, A ÿ2n, B 2n and B ÿ2n are unknown coe�-

cients to be determined (note that all these coe�cients

have been scaled by f0) and the subscript ``1'' for F,C and k denotes the strip. As discussed by

Muskhelishvili [19], the constant term A0 accounts for

the rigid body displacement. M1 and N1 are the num-

ber of series terms retained in our approximate sol-

utions; but, more generally, both M1 and N 1 should

approach in®nity if an exact solution is wanted. In ad-

dition, to simplify our later calculation using the LSM

we have truncated the series solutions for F1(z) and

C1(z) to the same order. Since our strip problem is

symmetric about the x-axis and anti-symmetric about

the y-axis, only coe�cients of 2n needed to be retained

and A 2n, . . . , B ÿ2n are all real coe�cients. For the

rivet (domain 2 shown in Fig. 1), we have a simply-

connected circular domain, the general solution forms

for F2(z) and C2(z) can be simpli®ed to (Refs [19, 30]):

F2�z� � f0XM2

n�1C2nz

2n; �7a�

C2�z� � f0XM2

n�1D2nz

2n �7b�

where C 2n and D 2n are unknown coe�cients to be

determined (again, these have been scaled by f0) and

the subscript ``2'' for F and C denotes the rivet.

Again, due to symmetry, only the 2n terms needed to

be considered and C 2n and D 2n are real coe�cients.

As discussed before, the series solutions have been

truncated to ®nite terms.

As discussed in Section 3, the rivet load can be mod-

eled by the body force Fx= f0 and Fy=0. Substitution

of these into Eqs. (2) and (3d) leads to two di�erential

relations between f0, V2 and W2. Integration of these

expressions gives:

V2 � ÿ f0�z� �z�2

; �8a�

W2 � ÿ f02

�z2

2� z �z

�: �8b�

Consequently, the stresses and displacements in the

rivet can be approximated in terms of C 2n and D 2n by

substitution of Eqs. (7) and (8) into Eq. (3). Similarly,

the stresses and displacements in the strip can be writ-

ten approximately in terms of the unknown coe�cients


A 2n, A ÿ2n, B 2n and B ÿ2n by substitution of Eq. (6)into Eq. (3) and setting W to zero.

5. Boundary collocation method

In this section, the unknown coe�cients A0, A 2n,A ÿ2n, B 2n, B ÿ2n, C 2n and D 2n will be determined bythe BCM in a way that the continuity and boundary

conditions (Eq. (5)) will be satis®ed approximately.The number of the unknown coe�cients M1, M2 andN1 to be retained in the series solutions depends

mainly on the accuracy of approximation required andthe convergence of the result expected. Our main taskis to solve the unknown coe�cients so that the bound-

ary conditions in Eq. (5) are satis®ed as closely aspossible. Since our problem is symmetric about the x-axis, only half of the strip is considered (i.e. ABCD in

Fig. 1). In applying the BCM, we choose some closely-spaced collocation points along the boundaries of thehalf-strip ABCD and the half-rivet GHI. The totalnumber of the collocation points Nc chosen must be

larger than or equal to the total number of unknowncoe�cients Nu=2(M1+M2+N1)+1. As discussedin the Introduction, we have the PMM if Nc=Nu; and

we have the LSM if Nc>Nu. And only the applicationof LSM will be considered here. Since our truncatedseries solutions are approximations only, when we

express the stresses and displacements at these colloca-tion points, the boundary conditions need not be satis-®ed exactly if the LSM is used. That is, errors in the

boundary conditions are allowed at these collocationpoints. In other words, the right-hand side of Eq. (5) isno longer zero, but instead is replaced by some errorfunctions. These error expressions have four di�erent

forms, depending on which parts of the boundary thatour collocation points are on:

1. for the collocation points za={b+ iy on ABÿb+ iy on CD for

ae ye0,

E1 � fs�1�xx�F1;C1� ÿ is�1�xy �F1;C1�g=f0

�ÿ R2

2�1� k1��1� k1�za � za� j za j2 �z3a

j za j4( )

�XM1

n�1f2n�z2nÿ1a � z2nÿ1a ÿ �2nÿ 1� j za j2 z2nÿ3a �A2n

ÿ 2nz2nÿ1a B2ng ÿXN1

n�1

� 2n

� j za j2 �z2n�1a � z2n�1a � � �2n� 1�z2n�3a

j za j4�n�1��(

�Aÿ2n ÿ 2nz2n�1a

j za j2�2n�1� Bÿ2n)

& �9a�

where vzv is the magnitude of the complex number z;

2. for the collocation points zb=x+ ia on BC for

ÿbE xE b,

E2 � s�1�yy �F1;C1� � i

�s�1�xy �F1;C1�

�� 3f0pR2

8b3�b2 ÿ x2�

��.f0

� R2

2�1� k1�� k1 ÿ 1� j zb j2 zbÿ j zb j2 zb � z3b

j zb j4�

� i3pR2

8b3

�b2 ÿ z2b � z2b � 2 j zb j2

4

��XM1

n�1f2n�z2nÿ1b � z2nÿ1b � �2nÿ 1� j zb j2 z2nÿ3b �A2n

� 2nz2nÿ1b B2ng �XN1

n�1

� 2n

�ÿ j zb j2 �z2n�1b � z2n�1b � � �2n� 1�z2n�3b

j zb j4�n�1��(

�Aÿ2n ÿ 2nz2n�1b

j zb j2�2n�1� Bÿ2n)

; �9b�

3. for the collocation points zc=R(cos y+ i sin y) onGHI for 0E yE p,

E3 � fs�1�rr �F1;C1� ÿ s�2�rr �F2;C2;W2� ÿ i�s�1�ry �F1;C1�

ÿ s�2�ry �F2;C2;W2��g=f0

�ÿ R2

2�1� k1�� 1� k1�zc � 2zc

j zc j2�

� 1

2

�zc � 2zc

1� k2

��XM1

n�12n�2�1ÿ n�z2nÿ1c

�� z2nÿ1c �A2n ÿ 2nz2n�1c

j zc j2 B2ng

ÿXN1

n�12n

�z2n�1c � 2�n� 1�z2n�1c

j zc j2�2n�1��Aÿ2n

(

ÿ 2nz2nÿ1c

j zc j4n Bÿ2n

)ÿXM2

n�12n�2�1ÿ n�z2nÿ1c

�� z2nÿ1c �C2n ÿ 2nz2n�1c

j zc j2 D2ng; �9c�


E4 � msfu�1�r �F1;C1� ÿ u�2�r �F2;C2;W2� � i�n�1�y �F1;C1�

ÿ n�2�y �F2;C2;W2��g=f0

� mszc2m1 j zc j

ÿ R2

2�1� k1��k1�ln zc � ln zc� ÿ z2c

j zc j2��

� k1A0 �XM1

n�1f�k1z2nc ÿ 2n j zc j2 z2�nÿ1�c �A2n

ÿ z2nc B2ng �XN1

n�1

�k1 j zc j2 z2nc � 2nz2�n�1�c

j zc j2�2n�1��(

�Aÿ2n ÿ z2ncj zc j4n Bÿ2n

��

� mszc2m2 j zc j

XM2

n�1�ÿ�k2z2nc ÿ 2n j zc j2 z2�nÿ1�c �C2n

(

� z2nc D2n� ÿ �1ÿ k2�zc2�1� k2�

�zc2� zc

��9d�

where ms equals m1 if m2em1 and equals m2 if

m1>m2, and it is introduced into the displacement

error function such that the displacement error term

has a comparable order with the stress errors given

in Eq. (9)(a)±(d). In addition, all error functions in

Eq. (9) are normalized with respect to f0.

Since Nc>Nu, the boundary condition can only be

satis®ed in an average sense using the LSM. In particu-

lar, we de®ne the following total error as the sum of

the squares of error given in Eq. (9a±d) for all the col-

location points:

SE2 �Xn1j�1f�Re�E1�zaj��2 � �Im�E1�zaj��2g

�Xn2j�1f�Re�E2�zbj��2 � �Im�E2�zbj��2g

�Xn3j�1f�Re�E3�zcj��2 � �Im�E3�zcj��2

� �Re�E4�zcj��2 � �Im�E4�zcj��2g �10�where z aj, z bj and z cj denote the jth collocation point

on AB/CD, BC and GHI respectively, and n1, n2 and

n3 are the number of the collocation points on these

corresponding boundaries. Re( ) and Im( ) denote the

real and imaginary parts of ( ), respectively. Once the

collocation points are ®xed, SE 2 is a real function in

terms of only the unknown real coe�cients A0, A 2n,

. . . , D 2n.

We now apply the LSM to minimize SE 2 in Eq. (10)

in order that boundary conditions are approximately

satis®ed. Di�erentiation of SE 2 with respect to A0

then setting the result to zero, we obtain the following

equation:

@�SE2�@A0

� 0: �11�

Similarly, di�erentiating SE 2 with respect to the otherunknown coe�cients A 2n, . . . , D 2n and setting the cor-

responding results to zero, we can obtain a systemtotal of 2(M1+M2+N1)+1 linear algebraicequations for all the unknown coe�cients. In matrix

form, the system of equations is:

�S�fXg � fYg �12�where [S] is a [2(M1+M2+N1)+1]� [2(M1+M2

+N1)+1] matrix with known entities, {Y} is a[2(M1+M2+N1)+1]�1 vector of known elementsand {X}={A0, A 2n, . . . , D 2n}

T is the vector for theunknown coe�cients.

The system of equations given in Eq. (12) can besolved by using the singular value decomposition(SVD) technique which has been highly recommended

for the inversion of a nearly singular matrix by Presset al. [31]. In particular, the square matrix [S] isdecomposed to the form [U][W][V]T by the SVD where

[U] and [V] are orthogonal square matrices and [W] isa square matrix with diagonal elements wi only. Thus,the inverse of [S] is [S]ÿ1= [V][W]ÿ1[U]T where [W]ÿ1

is a square matrix with diagonal elements 1/wi only.Once the system in Eq. (12) is solved, the stresses anddisplacements in both the strip and the rivet can bedetermined by substituting the coe�cients A0, A 2n,

. . . , D 2n into Eqs. (3) and (4). Before carrying out adetailed parametric study, we ®rst verify, in the nextsection, our results by the BCM with the results by the

FEM for a particular case of ®nite strip and by theexact solution of an in®nite plane loaded by a bondedrivet when a= b>>R.

6. Numerical veri®cations of the BCM

Exact solutions for our rivet-loaded strip problem,of course, do not exist; however, when b/R and a/R

approach in®nity, the exact solution can easily be de-rived (see Ref. [32]). For the sake of completeness, thederivation of this exact solution is summarized brie¯yin the Appendix. An in®nite plate under plane stress

condition (i.e. k=(3ÿ n)/(1+ n)) is modeled here bya ®nite square plate containing a very small rivet andonly half of the square plate needed to be considered

in view of the symmetry property. The number of theequally-spaced collocation points used along theboundaries GHI, AB and BC are 901, 251 and 101, re-

spectively. In particular, the following geometric andmaterial parameters are used: z= m2/m1=10, b/R=10, a/b=1 and n1= n2=1/3. Fig. 2 plots the nor-


mal bond stress s rr/(P/R), shear bond stress s ry/(P/R)

and hoop stress s yy/(P/R) on the circular boundary(r= R) of the square plate vs the polar angle y shownin Fig. 1. The solid and dotted lines are the results of

the BCM and the exact solution given in theAppendix, respectively. Although only a large square

plate with a small rivet is considered using the BCM,our results agree very well with the exact solution forthe in®nite plate.

To further illustrate the validity of our solutionscheme for ®nite strips, we compare here the results of

two speci®c strip problems to those obtained by theFEM. In particular, the geometric and material par-

ameters used in the two examples are: z=100, b/R=5/3 and z=0.1, b/R=8, and with the length-to-width ratio a/b=5 and n1= n2=1/3 for both pro-

blems. The strip is assumed to be simply-supportedunder plane stress condition. The stress variations

along the interface are given by the solid lines inFigs. 3 and 4. The ®nite element analysis was done by

SAP90 [33] using 72 three-node triangular and 2310four-node quadrangular isoparametric elements. Tobetter model the stress concentration, we use a ®ner

mesh around the circular interface between the stripand the rivet. The stresses at the circular boundary of

the strip can be interpolated from the outputs of the

element forces and stresses by SAP90. The predictionsof s rr/(P/R), s ry/(P/R) and s yy/(P/R) in the strip bythe FEM on the circular interface are plotted as dottedlines in Figs. 3 and 4. As shown in these ®gures, the

results of the FEM agree very well with those of theBCM. Thus, the BCM is accurate even when thewidth-to-diameter ratio is small (i.e. b/R=5/3).

7. Numerical results

As discussed in the Introduction, the strip isassumed to be simply-supported under plane straincondition in this section. In addition, the geometricratio a/b of the strip is ®xed at 5 and the number of

collocation points along the boundaries of the strip isthe same as those in Section 6 for all of our numericalresults to be discussed. Figs. 5 and 6 show the e�ect of

Poisson's ratio on the angular variations of the normalstress s rr/(P/R), shear stress s ry/(P/R) and hoop stresss yy/(P/R) for z equal to 50 and 1, respectively, with

four di�erent combinations of Poisson's ratio for therivet and strip. Note that compressive stresses areplotted as negative values in these plots. In view of the

Fig. 2. The normalized normal stress s rr/(P/R), shear stress s ry/(P/R) and hoop stress s yy/(P/R) vs y at the circular boundary of a

large square plate under plane stress condition. Together with the exact solution of a rivet-loaded in®nite plane (dotted lines), the

solutions by the BCM (solid lines) are given for the shear modulus ratio z= m2/m1=10, width-to-diameter ratio b/R=10, length-

to-width ratio a/b=1 and Poisson's ratio of both materials n1= n2=1/3.


symmetry of the strip problem about the x-axis, only

half of the stress distributions are plotted (i.e.

08E yE1808). The solid, dotted and dashed lines are

for the normalized normal, shear and hoop stresses, re-

spectively. We have set the Poisson's ratio to either 0.1

or 0.4 for both the strip and rivet; and the four combi-

nations of Poisson's ratio considered in Figs. 5 and 6

should include most extreme situations for steel rivets

and rock panels. For the case of a sti� rivet shown in

Fig. 5 (z=50), a higher value of n1 gives a lower bond

shear stress concentration for y between 408 and 1408,and the decrease is most profound at y=908; the nor-

mal bond stress also decreases with n1 for 08E yE508and 1308E yE1808, and the e�ect is most profound at

y equal to 08 and 1808. However, the peak stresses for

both cases are quite insensitive to the Poisson's ratio.

On the contrary, the hoop stress distribution as well as

its maximum value increases with the Poisson's ratio

of the strip n1. All s rr/(P/R), s ry/(P/R) and s yy/(P/R)

are, however, insensitive to the change of the Poisson's

ratio of the rivet n2. When the rivet and strip material

are the same (z=1), the e�ect of Poisson's ratio

shown in Fig. 6 are relatively insigni®cant compared to

the results shown in Fig. 5.

We now consider the stress concentration in our

strip problem by varying the values of z and b/R with

n1= n2=1/3. All other conditions are the same as

those given in Figs. 5 and 6. The e�ect of b/R on the

angular variations of the normal stress s rr/(P/R), shear

stress s ry/(P/R) and hoop stress s yy/(P/R) on the cir-

cular boundary are illustrated in Figs. 7 and 8 for zequal to 10/3 and 1/2, respectively. The graphs for

z=10/3 should closely resemble the shear modulus

ratio in the case of a steel rivet in a marble panel. As

expected, both Figs. 7 and 8 show that the maximum

stress concentrations, in general, decrease with b/R.

When the rivet is sti�er than the strip (e.g. z=10/3),

the shear stress is the largest among the stress concen-

trations (see Fig. 7); however, when the rivet is softer

than the strip, the hoop stress becomes the largest

among the stress concentrations (see Fig. 8). It is inter-

esting to note that for z=10/3 both the hoop and nor-

mal bond stresses change sign, i.e. from tension to

compression or vice versa, as b/R changes from 8 to

1.5, as shown in Fig. 7. The e�ect of z on s rr/(P/R),

s ry/(P/R) and s yy/(P/R) is shown in Fig. 9. In general,

the maximum shear bond stress increases with z while

the maximum hoop stress decreases with z.


strip under plane stress condition. Solutions by the BCM (solid lines) and by the FEM (dotted lines) are given for the shear mod-

ulus ratio z= m2/m1=100, width-to-diameter ratio b/R=5/3, length-to-width ratio a/b=5 and Poisson's ratio of both materials

n1= n2=1/3.



strip. Solutions by the BCM (solid lines) and by the FEM (dotted lines) are given for the shear modulus ratio z= m2/m1=0.1 and

width-to-diameter ratio b/R=8. All other conditions are the same as those given in Fig. 3.

Fig. 5. The normalized normal stress s rr/(P/R) (solid lines), shear stress s ry/(P/R) (dotted lines) and hoop stress s yy/(P/R) (dashed

lines) at the circular boundary of the strip vs y with 4 di�erent combinations of the Poisson's ratios for the strip and the rivet. The

shear modulus ratio used is z= m2/m1=50. The strip is simply-supported under plane strain conditions with a width-to-diameter

ratio b/R=5/3 and length-to-width ratio a/b=5.


Fig. 6. Same as Fig. 5 except that z= m2/m1=1.


lines) at the circular boundary of the strip vs y with various width-to-diameter ratios b/R for the shear modulus ratio z= m2/m1=10/3. The strip is simply-supported under plane strain conditions with a length-to-width ratio a/b=5 and Poisson's ratios of

both materials n1= n2=1/3.


Fig. 8. Same as Fig. 7 except that z= m2/m1=1/2.


lines) at the circular boundary of the strip vs y with various shear modulus ratio z for width-to-diameter ratio b/R=2. All other

conditions are the same as those given in Fig. 7.


For design purposes, it is essential to estimate the

largest stress concentration in the strip for a given set

of z and b/R. Therefore, the maximum stress concen-

trations [s ry/(P/R)]max, [s rr/(P/R)]max and [s yy/(P/

R)]max and the angular locations of their occurrence on

the circular boundary ymax are plotted vs b/R for var-

ious values of z in Figs. 10±12, respectively. Although

the maximum stresses can either be positive or be

negative, only their absolute values are shown in

Figs. 10±12. As observed earlier in Figs. 7 and 8, the

stress concentrations [s ry/(P/R)]max and [s yy/(P/R)]max

decrease with b/R (shown in Figs. 10 and 12, respect-

ively). On the contrary, when ze1, the maximum nor-

mal stress shown in Fig. 11 decreases initially to a

Fig. 10. The maximum normalized shear stress s ry/(P/R) and their angular locations of occurrence ymax in the ®rst quadrant

08E yE908 on the circular boundary of the strip vs width-to-diameter ratio b/R with various shear modulus ratios z. Only absolute

values of the maximum stresses are shown. All other conditions are the same as those given in Fig. 7.


local minimum, then rises slightly to a steady value as

b/R further increases. All the maximum stress concen-

trations shown in Figs. 10±12 approach the in®nite

plane limit as b/R increases to about 5. In general, the

maximum shear stress increases with z while the maxi-

mum hoop stress decreases with z; however, the maxi-

mum normal stress may either increase or decrease

with z, depending on the value of b/R, as shown in

Fig. 11.

The angle in the ®rst quadrant (i.e. 08E yE908), atwhich the stress maximum occurs, is denoted by ymax

and is also given in Figs. 10±12. As b/R increases, the

Fig. 11. The maximum normalized normal stress s rr/(P/R) and their angular locations of occurrence ymax in the ®rst quadrant

08E yE908 on the circular boundary of the strip vs width-to-diameter ratio b/R with various shear modulus ratios z. All other con-

ditions are the same as those given in Fig. 7.


angular location of maximum stress occurrence ymax

increases from about 368 0428, depending on the

value z, to 908 (the in®nite plane limit) for the shear

bond stress, and jumps suddenly from a roughly con-

stant value (508 0558, depending on the value of z) to08 (the in®nite plane limit) for the normal bond stress.

A similar sudden shift in the location of maximum

stress is also observed in hoop stress at the interface of

the strip and rivet.

For a speci®ed rivet size, the graphs given in

Figs. 10±12 should provide useful information for the

design of the optimum thickness of rock panels to be

used such that the allowable stresses of the rock panels

will not be exceeded under the design load.

Fig. 12. The maximum normalized hoop stress s yy/(P/R) and their angular locations of occurrence ymax in the ®rst quadrant

08E yE908 on the circular boundary of the strip vs width-to-diameter ratio b/R with various shear modulus ratios z. All other con-

ditions are the same as those given in Fig. 7.


8. Conclusion

The stress concentrations in a ®nite strip, which isloaded by a bonded elastic rivet of a di�erent material,are obtained by using the boundary collocation

method (BCM). The ®nite strip is assumed to besimply-supported under plane strain conditions. Therivet load is modeled by a uniform body force distribu-

ted on the rivet. By the complex variable technique,the stresses and displacements of the strip and the rivetare ®rst expressed in terms of an analytic complex

series with unknown coe�cients. The BCM is used to®nd the coe�cients in this series solution such that theboundary conditions are satis®ed approximately usinga least squares approach. Comparisons of the results

with the exact solution in the case of an in®nite planeand with those by the ®nite element method for thecase of a ®nite strip lend credence to the validity of

our proposed method (see Figs. 2±4).We found that the e�ect of Poisson's ratio on the

maximum stress concentration is not very signi®cant,

except for the case of hoop stress when the strip isloaded by a very sti� rivet. For such a case, [s yy/(P/R)]max increases with the Poisson's ratio of the strip. In

addition, numerical results show, as expected, that smal-ler values of b/R usually induce larger stress concen-tration at the rivet±strip interface. For design purposes,the maximum stress concentrations [s ry/(P/R)]max, [s rr/

(P/R)]max and [s yy/(P/R)]max and the angular locationsof their occurrence on the circular boundary ymax areplotted vs b/R for various values of z in Figs. 10±12, re-

spectively. In general, the maximum shear stressincreases with z while the maximum hoop stressdecreases with z; however, the maximum normal stress

may either increase or decrease with z, depending on thevalue of b/R. As b/R increases, the angle ymax, at whichthe maximum stress concentration occurs, increasesfrom about 368 0428, depending on the value z, to 908(the in®nite plane limit) for the shear bond stress, andjumps suddenly from a roughly constant value(508 0558, depending on the value of z) to 08 (the in®-

nite plane limit) for the normal bond stress. Similar sud-den shifts in the location of maximum stress are alsoobserved in hoop stress at the interface. For a ®xed rivet

size, the graphs given in Figs. 10±12 should provide use-ful information for the design of the optimum thicknessof rock panels if the allowable stresses of the rock panels

and the design load are given.

Acknowledgements

This study was supported by RGC research grant

through The Hong Kong Polytechnic University Grantno. 0350-334-A3-310. The problem was inspired by dis-cussions with Prof S. L. Chan. This work constitutes

part of the MPhil thesis of K.C. Ho under the supervi-sion of K.T. Chau.

Appendix A

Consider an elastic rivet of radius R bonded to anin®nite plane, and the rivet load is modeled by a uni-form body force f0 (force per unit volume) distributedalong the x-direction. The continuity condition at the

interface r= R between the plane and the rivet are thesame as those given in Eq. (5a, b), and the decay con-dition requires the stresses to vanish at in®nity.

Employing the complex variable method, the generalsolutions F1(z) and C1(z) for the in®nite plane or``domain 1'' are the same as those given in Eq. (6a, b)

except that A 2n and B 2n are all equal to zero for ne1due to the decay condition and that N 1 should tend toin®nity. The general solutions for F2(z) and C2(z) are

the same as in Eq. (7a, b) but with M2 tending towardin®nity, and W2 is given by Eq. (8b). For both therivet and in®nite plane, the stresses and displacementscan be obtained by substituting the corresponding ana-

lytic functions into Eq. (3). Substitution of these result-ing stresses and displacements into the boundarycondition (5a, b) on z= Re iy and setting the coe�-

cient of each e i(2n+1)y to zero (for ÿ1< n<1)leads to the following results:

A0 � R2

�lnR

1� k1� 1ÿ k2 ÿ z2zk1�k2 � z�

�;

Bÿ2 � R4

2

�1

1� k1ÿ 1

2�k2 � z��;

C2 � 1

2

�1

1� k2ÿ 1

2�k2 � z��;

Aÿ2n � Bÿ2nÿ2 � C2n�2 � D2n � 0 for n � 1 �A1�

where all symbols have the same de®nitions as thosegiven in Section 2 and z= m2/m1 is the shear modulus

ratio of the rivet to the plane. Finally, substitution ofeqn (A1) into Eq. (3), gives the following stress com-ponents of the in®nite plane in polar coordinate (r and

y):

s�1�rr � ÿf0R

2

�k1 � 3

k1 � 1

�R

r

�

��

1

k2 � zÿ 2

k1 � 1

��R

r

�3�cos y; �A2�


s�1�ry �f0R

2

�k1 ÿ 1

k1 � 1

�R

r

�

ÿ�

1

k2 � zÿ 2

k1 � 1

��R

r

�3�sin y; �A3�

s�1�yy �f0R

2

�k1 ÿ 1

k1 � 1

�R

r

�

��

1

k2 � zÿ 2

k1 � 1

��R

r

�3�cos y: �A4�

This exact solution is the same as those derived by Hoand Chau [32], although the Airy's stress function wasused in that paper, instead of the complex variablemethod used here.

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