長方形彈性層黏著於剛性板的傾斜分析

行政院國家科學委員會專題研究計畫 成果報告

長方形彈性層黏著於剛性板的傾斜分析

計畫類別： 個別型計畫

計畫編號： NSC93-2211-E-011-015-

執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 國立臺灣科技大學營建工程系

計畫主持人： 蔡相全

計畫參與人員： 黃俊智,林育民

報告類型： 精簡報告

處理方式： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢

中 華 民 國 94 年 9 月 15 日

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摘要

橡膠層墊是由多層橡膠片黏著於薄鋼片之間所構成，由於在水平方向具有低勁度，

橡膠層墊用於橋樑工程以容許因溫度變化所產生的伸縮，也被用於基底隔震以減少建築 物因地震造成的振動。橡膠層墊須具有垂直向的剛性以承受上部結構的載重，而垂向的 剛性則來自於橡膠片的側向變位受制於所黏著的鋼片，如果橡膠片的黏著應力超過其強 度，橡膠片和鋼片的黏界面將開裂，導致橡膠承墊不穩定，而無法支承上部結構，因此 在橡膠層墊的設計中，極重視受束制橡膠片承受軸壓與彎矩時的勁度和黏著應力。本研 究計劃的目的乃是去探討長方形橡膠層墊受到彎矩的傾斜勁度與所產生的黏著應力。研 究所用的分析模型是一長方形彈性層，其上下面分別黏著於一剛性板，根據兩個變形的 基本假設，利用彈性力學理論，求解出受剛性板束制之長方形彈性層的傾斜勁度公式與 黏著面的應力公式。所用的第一個變形假設為﹕平行於剛性板的平面斷面，在變形後仍 然保持一平面。第二個變形假設為﹕垂直於剛性板的直線，在變形後成為一拋物線。所 推導出理論公式適用範圍將不受柏松比與形狀係數的限制。本計劃亦利用有限元素分析 法來証明所推導出理論公式的精確性。

關鍵詞﹕橡膠層墊，傾斜勁度，黏著應力。

Abstract

An elastic layer bonded between two rigid plates has higher tilting stiffness than the elastic layer without bonding. While the finite element method can be applied to calculate the stiffness, the tilting stiffness of bonded rectangular layers derived through a theoretical approach in this paper provides a convenient way for parametric study. Based on two kinematics assumptions, the governing equation for the mean pressure is derived from the equilibrium equations. Using the approximate shear boundary condition, the mean pressure is solved and the tilting stiffness of the bonded rectangular layer is then established in an explicit single-series form. Through the solved pressure, the horizontal displacements are derived from the corresponding equilibrium equations, from which the shear stress on the bonding surface can be found. The error of using the approximate shear boundary condition is negligible for the tilting stiffness, but becomes significant for the horizontal displacements and bonding shear stresses near the edges of the rectangular layers.

Keywords: Bonded elastic layer; Elastomeric bearing; Seismic isolation.

1. Introduction

When an elastic layer is bonded between two rigid plates, the rigid plates can restrict the lateral expansion of the elastic layer and result in the bonded elastic layer having higher compression stiffness and tilting stiffness than the elastic layer without bonding. The effect

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becomes more dramatic when Poisson’s ratio of the elastic layer is near 0.5. This characteristic has been adopted in the design of laminated elastomeric bearings that consist of many elastomeric layers bonded to interleaving steel plates. Laminated elastomeric bearings, employed in many fields such as seismic isolation, can provide high vertical rigidity to sustain gravity loading, while still providing the same horizontal flexibility as elastome r.

To analyze the stiffness of bonded layers, two kinematics assumptions are usually adopted: (i) planes parallel to the rigid bonding plates before deformation remain planar after loading; (ii) lines normal to the rigid bonding plates before deformation become parabolic after loading. Gent and Lindley (1959) derived the compression stiffness of an incompressible elastic layer for infinite-strip shape and circular shape. Subsequently, Gent and Meinecke (1970) extended this method to analyze the compression stiffness and tilting stiffness of incompressible elastic layers for square and other shapes.

Although rubber can be treated as incompressible in some analyses, the assumption of incompressibility tends to overestimate the compression stiffness and tilting stiffness of the bonded rubber layer when the layer's shape factor (defined as the ratio of the one bonded area to the force-free area) is high. Kelly (1997) developed a ‘pressure solution’ approach to derive the compression stiffness and the tilting stiffness considering the effect of bulk compressibility. The solutions are available for the layers of infinite-strip shape (Chalhoub and Kelly, 1991), circular shape (Chalhoub and Kelly, 1990) and square shape (Kelly, 1997).

Lindley (1979a) applied an energy method to derive the compression stiffness of the infinite-strip and circular shapes as well as the tilting stiffness of the infinite-strip shape (Lindley, 1979b) for the material of any Poisson's ratio. Koh and Kelly (1989) utilized a

‘variable transform’ approach to derive the compression stiffness of the square shape for compressible material. Koh and Lim (2001) extended this approach to solve the compression stiffness of the rectangular shape.

The stiffness of bonded layers is related to the vertical stress, which can be derived from the mean pressure. Tsai and Lee (1998 and 1999) developed a pressure approach to derive the compression stiffness and tilting stiffness of bonded elastic layers in infinite-strip, circular and square shapes. Recently, Tsai (2003) presented an approach to analyze the tilting stiffness of the circular shape by solving displacement directly. These solutions are accurate for the material of any Poisson's ratio.

To reduce the weight and the cost of laminated elastomeric bearings, the steel plates can be replaced by fiber reinforcement. In contrast to the steel reinforcement that is assumed to be rigid, the fiber reinforcement is flexible in extension. The compression stiffness and tilting stiffness of fiber-reinforced bearings in infinite-strip, circular and rectangular shapes are derived by assuming the elastomeric layer is incompressible and the reinforcement is flexible (Kelly, 1999; Tsai and Kelly, 2001, 2002a, 2002b). Recently, bulk compressibility is also included in the stiffness analysis of fiber-reinforced bearings of the infinite-strip shape (Kelly,

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2002; Kelly and Takhirov, 2002).

Recently, Tsai (2005) applied the approximate shear boundary condition to solve the pressure distribution of the bonded rectangular layer under compression, from which the compression stiffness of the bonded rectangular layer is derived in an explicit single-series form. In this paper, the pressure approach by Tsai (2005) is extended to solve the tilting stiffness of the rectangular layers bonded between rigid plates. The displacements of the elastic layer and the bonding shear stresses on the interface of the elastic layer and the rigid plate are also derived. To verify the exactness of the theoretical solutions, finite element analyses are carried out, where the eight-node solid elements with incompatible bending modes is applied to model the elastic layer.

2. Solution of pressure

A rectangular layer of linearly elastic, homogeneous and isotropic material bonded between two rigid plates is shown in Fig. 1 where a Cartesian coordinate system (x, y, z) is located at the center of the layer. The elastic layer has a thickness of t, a width of 2a along the x-axis, and a length of 2b along the y-axis. Under a pure bending moment, the top and bottom rigid plates rotate about the y axis to form an angle φ. The displacements of the elastic layer, u, v and w along the x, y and z directions respectively, can be assumed to have the form

4 ) 1 )(

, ( ) , ,

( ₂²

t y z x u z y x

u = − (1)

4 ) 1 )(

, ( ) , ,

( ₂²

t y z x v z y x

v = − (2)

xz z

y x

w ρ

) 1 , ,

( = (3)

where ρ is the radius of curvature of the rotation, defined as

ρ =φt (4)

Eqs. (1) and (2) represent the assumption of quadratic deformation on vertical lines; Eq. (3) represents the assumption that planes parallel to the rigid plates remain planar.

For isotropic elastic material, the mean pressure p has the following relation with the displacements

) (

) , ,

(x y z u_,x v_,y w_,z

p =−κ + + (5)

in which κ is the bulk modulus and the commas imply differentiation with respect to the indicated coordinate. The effective pressure p is the average pressure through the thickness, defined as

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∫−

= ^/2

2

/ ( , , )

) 1 ,

( ^t

t p x y z dz y t

x

p (6)

which becomes, when using the displacement assumptions in Eqs. (1) to (3), ρ

κ

v x p u

y

x+ −

−

= ( )

3 2

,

, (7)

Integrating the equilibrium equations in the x and y directions through the thickness leads to

κ ν

x yy

xx

u p u t

u_{, , 2} ^,

) 2 1 ( 2

3 12

= −

−

+ (8)

κ ν

y yy

xx

v p v t

v_{, , 2} ^,

) 2 1 ( 2

3 12

= −

−

+ (9)

in which ν is Poisson’s ratio. Differentiate Eqs. (8) and (9) with respect to x and y, respectively, and add them up, which yields

κ ρ α

α x

p p

p_,xx + _,yy − ² =− ² (10)

with

ν α ν

−

= −

1 ) 2 1 ( 6 1

t (11)

Differentiating of Eq. (10) twice with respect to x and y, respectively, and adding them up gives

0 ) (

2 _{, ,} ² _{, ,}

,xxxx + pxxyy +pyyyy− pxx+ pyy =

p α (12)

At the edges of the elastic layer, the normal stress is zero, i.e. σ_x(a,y,z)=0 and 0

) , , (x b z =

σy . Taking integration through the thickness of the layer, these boundary conditions indicate

) , 3 (

) 2 1 ( ) 2 ,

(a y u_, a y

p _x

ν κ − ν

= (13)

) , 3 (

) 2 1 ( ) 2 ,

(x b v_, x b

p _y

ν κ − ν

= (14)

The horizontal shear stress τ_xy vanishes at the edges. In order to be able to solve the pressure explicitly, the τ_xy terms in the x-direction equilibrium equation at y=b and the y-direction equilibrium equation at x=a are neglected. In other words, τ_xy,x(a,y,z)≈0 and τ_xy,y(x,b,z)≈0, which means

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0 ) , ( ) ,

( _,

, a y +v a y =

u_{yx xx} (15)

0 ) , ( ) ,

( _,

, xb +v xb =

u_{yy xy} (16)

From Eqs. (7), (9) and (15), the equilibrium equation in the y direction at x=a becomes )]

, 6 ( ) , ( 3 [

) 2 1 ( ) 2 ,

( _{, 2}

, v a y

y t a v y

a

p_y = − _yy −

ν

κ ν (17)

From Eqs. (7), (8) and (16), the equilibrium equation in the x direction at y=b becomes )]

, 6 ( ) , ( 3 [

) 2 1 ( ) 2 ,

( _{, 2}

, u x b

b t x u b

x

p_x = − _xx −

ν

κ ν (18)

By using Eq. (7), Eqs. (13) and (14) become ρ

κ ν

ν p a y a y

a v_y

2 3 ) , ( ) 2 1 ( 2

) 1 ( ) 3 ,

, ( −

−

− −

= (19)

ρ κ

ν

ν p xb x b

x u_x

2 3 ) , ( ) 2 1 ( 2

) 1 ( ) 3 ,

, ( −

−

− −

= (20)

The approximate boundary conditions for the pressure can be established by combining Eq.

(17) with Eq.(19), and Eq. (18) with Eq. (20):

ν ρ κ

ν a

y t a t p

y a

p_,yy( , )−6₂(1− ) ( , )= 6₂(1−2 ) (21)

ν ρ κ

ν x

b t x t p

b x

p_,xx( , )− 6₂ (1− ) ( , )= 6₂(1−2 ) (22) The pressure at the corners can be found from Eqs. (7), (13) and (14),

ν ρ

κ a

b a

p(± , )= m (1−2 ) (23)

By virtue of symmetric characteristics about the x and y axes, the solution of p( yx, ) can be assumed to have the following series form

] sin ) ( cos

) ( [ )

2 1 ( ) , (

1

x y

f y x

x f y

x

p _n

n

n n

n γ γ

ρ κ ν

κ ∑^∞

= +

+

−

−

= (24)

where f_n(x) is an odd function, f_n(y) is an even function, and

n b

n π

γ )

2 ( −1

= (25)

na

n π

γ = (26)

Substituting Eq. (24) into Eq. (12) gives

0 ) ( ) (

) ( ) 2

( )

( ^{2 2} _, ^{4 2 2}

, x − + f x + + f x =

f_nxxxx γ_n α _nxx γ_n α γ_{n n} (27)

0 ) ( ) (

) ( ) 2

( )

( ^{2 2} _, ^{4 2 2}

, y − + f y + + f y =

f_{n yyyy} γ_n α _nyy γ_n α γ_{n n} (28)

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By using Eq. (24), the boundary conditions in Eqs. (21) and (22) become )

1 ( 6

) 2 1 ( 12 ) 1 ) (

( _{2 2}

ν γ

ν ν γ

ρ + −

−

= −

t b a a

f

n n

n

n (29)

) 1 ( 6

) 2 1 ( 12 ) 1 ) (

( _{2 2}

ν γ

ν ν γ

ρ + −

−

= −

t a b a

f

n n

n

n (30)

Assigning x=a to Eq. (10) and using Eqs. (24) and (29) leads to )]

1 ( 6

) 2 1 ( ) 12 (

4 ) [ 1 ) (

( ^{2 2 2} _{2 2}

, γ ν

ν α ν

γ γ να

ρ + −

+ − +

− −

= b t

a a f

n n

n n xx

n (31)

Assigning y=b to Eq. (10) and using Eqs. (24) and (30) leads to )]

1 ( 6

) 2 1 ( ) 12 (

4 ) [ 1 ) (

( ^{2 2 2} _{2 2}

, γ ν

ν α ν

γ γ να

ρ + −

+ − +

− −

= a t

b a f

n n

n n yy

n (32)

From Eqs. (27), (29) and (31), f can be solved as _n sinh } sinh sinh

]sinh ) 1 ( 6

) 2 1 ( 1 3 ) {[

1 4 (

)

( _{2 2}

a x a

x t

b x a

f

n n n

n n

n n

n γ

γ β

β ν

γ

ν γ

ν ρ +

− + + −

− −

= (33)

with

2

2 α

γ

βn = n + (34)

From Eqs. (28), (30) and (32), f_n can be solved as cosh } cosh cosh

]cosh ) 1 ( 6

) 2 1 ( 1 3 ) {[

1 4 (

)

( _{2 2}

b y b

y t

a y a

f

n n n

n n

n n

n γ

γ β

β ν

γ

ν γ

ν ρ +

− + + −

− −

= (35)

with

2

2 α

γ

βn = n + (36)

Accordingly, the pressure solution is

sin } cosh ]

cosh cosh

)cosh ) 1 ( 6

) 2 1 ( 1 3 [(

]cos sinh

sinh sinh

)sinh ) 1 ( 6

) 2 1 ( 1 3 [(

{ ) 1 ( 4

) 2 1 ) (

, (

2 2

2 2

1

a x b

y b

y t

b y a

x a

x t

a x y

x p

n n n

n n

n n

n n n

n n

n n

n n

γ γ γ

γ β

β ν

γ

ν

γ γ γ

γ β

β ν

γ

νρ ρ ν κ ν

− + + + −

−

+

− + + + −

−

− +

−

−

= ∑^∞

=

(37)

3. Effective bending modulus

According to the elementary beam theory, the effective tilting stiffness of the elastic layer is defined as

∫ ∫− −

= ^b

b a

a zz

eff xdxdy

EI) ρ σ

( (38)

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in which σ is the average of the vertical normal stress _zz σ through the thickness, _zz ) ]

2 1 [ ( 1 1 ^/2

2

∫− / +

− −

= +

= ^t

t zz

zz

x p dz E

t ν κ ρ

ν σ ν

σ (39)

where E is the elastic modulus. For clarification, define the effective bending modulus as

y eff

b I

E (EI)

= (40)

where I_y =(4/3)ba³ is the moment of inertia of the rectangular area about the y axis. By using the pressure solution in Eq. (37), the normalized effective bending modulus can be found as

tanh ]}

) ( ) tanh ) 1 ( 6

) 2 1 ( 1 3 tanh (

) ( [ tanh ] 2) [( 1

1

tanh ] )) 1 ( 6

) 2 1 ( 1 3 tanh (

) [ ( { 1 ) 2 1 )(

1 ( 1 12

2 2

2 2 2

1 2 2 2

2

a a

a a

t a

a

a a

n

b b t

b b n

E E

n n

n n

n n

n

n n

n n

n n

n n b

β β

β β

ν γ

ν γ

γ

γ γ

π

β β ν

γ

ν γ

γ π

ν ν

ν

−

− +

− −

− − + −

− +

− −

− − + +

= ∑^∞

= (41)

This equation reveals that the normalized effective bending modulus is a function of Poisson’s ratioν , the aspect ratio a/b and the shape factor S that is defined as

) (a b t S ab

= + (42)

for the bonded rectangular layers.

When the aspect ratioa/b→0, Eq. (41) becomes tanh )]

1 )

( 1 3 (1 2 1 1 3 1 [

1

2 2

2 a a a

E E_b

α α

α ν

ν

ν + −

+ −

= − (43)

which is the same as the effective bending modulus of the infinite-strip layer derived by Tsai and Lee (1999). When Poisson’s ratio ν →0.5, Eq. (41) becomes

) )]}

( 1 1 tanh

( 1 2

) ) (

1 tanh

)( 1 2 3

) [(

5 . 0 (

)]

tanh 1 ( tanh 2

3) 2

[(

6 { 1

2 2

2 2

2 2 2 4

2 1

2 2

2 2 2 4

2 2 4

a a

a a a

t t n

b

b b b t

t n

a t E

E

n n

n n n

n n n

n n

n n

n b

γ γ

γ γ γ

γ γ

γ γ γ γ

γ π

−

− +

+ −

− + +

− + −

+ +

= ∑^∞

=

(44)

The effective bending modulus calculated from Eq. (41) by using the first 50 terms of the series is plotted as a function of Poisson’s ratio ν in Fig. 2 for a/b=0.5,1,2 and

20 ,

=2

S to compare with the finite element solution. The present solution is close to the finite element solution, which indicates that applying the approximate shear boundary conditions in Eqs. (15) and (16) to derive the tilting stiffness is acceptable. To study the effect of aspect ratio, the ratio of the effective bending modulus of the rectangular layer in Eq. (41) to the effective bending modulus of the infinite-strip layer (a/b =0) in Eq. (43) is plotted

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in Fig. 3 as a function of aspect ratio a / for b S =2 and S =20, which shows that the effect of aspect ratio becomes more significant when the elastic layer has larger Poisson’s ratio. For the higher shape factor (S =20), the effective bending modulus in Eq. (43) can be treated as an approximate form of Eq. (41), if the Poisson’s ratio is smaller than 0.49.

To study the convergence of the series solution in Eq. (41), let E_b^(k) denote the value of

E using the first k terms of the series in Eq. (41). Regarding b E_b⁽⁵⁰⁾ as the converged solution, the ratios of E_b⁽¹⁾ and E_b⁽²⁾ to E_b⁽⁵⁰⁾ are plotted in Fig. 4 as a function of ν or S for the rectangular layer of a/b=0.5. For the range of 1≤S≤100, the maximum error of

) 1 (

Eb is about 12%, and the maximum error of E_b⁽²⁾ is about 4%.

4. Solution of displacements

When using Eq. (8) to solve the displacement u( yx, ), the pressure solution in Eq. (37) and the symmetric property of deformation imply that

] cos ) cosh cosh

cosh (

cos ) cosh cosh

cosh [(

) , (

1 0

x y

C y B

y A

y x

C x B

x A

U y

x u

n n

n n n

n n

n

n n

n n n

n n

γ γ

β δ

γ γ

β δ

+ +

+

+ +

+

= ∑^∞_{= (45)}

where U₀, A_n, B_n,C_n, A_n, B_n and C_n are the constants; the terms of A and_n A_n are the homogeneous solutions of Eq. (8), so that

2

2 12

n t

n = γ +

δ (46)

2

2 12

n t

n = γ +

δ (47)

Substituting Eqs. (37) and (45) into Eq. (8), the following constants can be solved as ρ

8

2 0

U = t (48)

n n n

n n

n b a t

B at

γ β ν γ

ν β

ν ν ν

ρ ]

) 1 ( 6

) 2 1 ( 1 3

sinh [ ) 1 ( ) 2 1 (

) 1 (

2 2 2

− +

− −

−

−

= − (49)

a b

C at

n n

n ν γ

ν

ρ sinh

) 1 ( ) 2 1 ( 2

2 −

− −

= (50)

)] 1 ( 6

) 2 1 ( 1 3 cosh [

) 1 ( ) 2 1 (

) 1 (

2 2 2

ν γ

ν β

ν ν ν

ρ + −

− −

−

−

= −

t b

B t

n n

n

n (51)

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