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Simplified stiffness formulae for elastic layers bonded between rigid plates
Engineering Structures 25 (2003) 1443–1454 www.elsevier.com/locate/engstruct
Simplified stiffness formulae for elastic layers bonded between rigid plates Hsiang-Chuan Tsai ∗, Wei-Jen Pai Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC Received 13 May 2002; received in revised form 10 January 2003; accepted 29 January 2003
Abstract An elastic layer bonded between two rigid plates has higher compressive stiffness and tilting stiffness than those of an unbonded layer. This effect has been adopted in the design of reinforced elastomeric bearings. Although compressive stiffness and tilting stiffness of bonded layers can be theoretically derived, some of the theoretical solutions are cumbersome in practical application. In this paper, simplified formulae for compressive stiffness and tilting stiffness of bonded elastic layers of circular, square and infinite-strip shapes are developed from theoretical solutions. Every set of simplified formulae consists of three approximated equations in which each equation has its applicable range of shape factor and bulk modulus. The divisions between the applicable ranges of different approximated equations are also established so that errors of the simplified formulae can be minimized. 2003 Elsevier Ltd. All rights reserved. Keywords: Elastomeric bearing; Base isolation; Bonded layer
1. Introduction Reinforced elastomeric bearings consist mainly of thin sheets of elastomer layers bonded to interleaving steel shims. Because of their low stiffness in the horizontal direction, reinforced elastomeric bearings are utilized in bridge engineering to allow temperature deformation and in base isolation to reduce building vibration induced by earthquakes. Reinforced elastomeric bearings must possess sufficient rigidity to sustain the gravity loading of superstructures, which makes the compressive stiffness and tilting stiffness of reinforced elastomeric bearings important factors in structural analysis and design. Gent and Lindley [3] derived the compressive stiffness of an incompressible elastic layer bonded between rigid plates for infinite-strip and circular shapes. Subsequently, Gent and Meinecke [4] extended this method to analyze the compressive stiffness and tilting stiffness of incompressible elastic layers for square and other shapes. Although elastomer can be treated as incompressible in some analyses, the assumption of incom-
∗
Corresponding author. E-mail address: [email protected] (H.-C. Tsai).
0141-0296/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0141-0296(03)00044-0
pressibility tends to overestimate the compressive stiffness and tilting stiffness of the bonded elastomer layer when the shape factor of the elastomer layer is high. To include the influence of compressibility on compressive stiffness, Gent and Lindley [3] recommended an ad hoc modification where the inverse of the effective compression modulus for the compressible layer is equal to the sum of the inverse of the effective compression modulus assuming incompressibility and the inverse of the bulk modulus of the layer. Kelly [5] developed a theoretical approach for deriving the compressive stiffness and tilting stiffness taking into consideration the effect of bulk compressibility. The solutions are available for the layers of infinite-strip shape [2], circular shape [1] and square shape [6]. These solutions are very accurate for layers of high shape factor and material of Poisson’s ratio between 0.49 and 0.5, e.g. rubber. Chalhoub and Kelly [1,2] also found that the ad hoc modification recommended by Gent and Lindley [3] was not accurate and proposed another form of modification. All the solutions of the above-mentioned research works are based on two kinematic assumptions and one stress assumption. Without utilizing the stress assumption, Koh and Kelly [7] derived the compression stiffness
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of the bonded square layer by a variable transform approach. Recently, Koh and Lim [8] extended this approach to solve the compression stiffness of the bonded rectangular layer. The compressive stiffness and tilting stiffness derived by Tsai and Lee [9,10] for the bonded layers of infinitestrip shape, circular shape and square shape are shown to be accurate for any elastic layer without limitation on the values of Poisson’s ratio and shape factor. However, not all of these stiffness formulae are convenient for numerical calculation in practical applications. The purpose of this paper is to find out the simplified forms of these stiffness formulae, so that they can be easily utilized in practical applications but without losing accuracy, and to compare these simplified formulae with those proposed by previous studies.
2. Deformation of bonded elastic layers An elastic layer bonded between two rigid plates and subjected to a compression force P in the z direction is shown in Fig. 1, which has a thickness t and a plane are A. The thickness reduction of the layer in the z direction can be expressed as d⫽
Pt AEc
(1)
where Ec is the effective compression modulus. Fig. 2 shows the deformation of the bonded layer when the top and bottom rigid plates are subjected to a moment M in the y direction. The angle q formed by the rigid plates can be expressed as q⫽
Mt IEb
(2)
Fig. 2. Deformed shape of a bonded elastic layer subjected to bending moment.
where I is the moment of inertial of the plane area about the y axis and Eb is the effective flexure modulus. The magnitudes of the effective compression modulus Ec and the effective flexure modulus Eb vary with the geometry and material properties of the elastic layer. The geometry of the bonded layer is represented by the shape factor S, which is defined as the ratio of the plane area A to the perimeter area not bonded to the rigid plates. The material properties for the elastic material such as rubber are usually defined in terms of the shear modulus G and the bulk modulus . Denote a nondimensional parameter c as c⫽
2(1 ⫹ n) ⫽ G 3(1⫺2n)
where n is Poisson’s ratio. When n is near 0.5, the value of c increases rapidly. For example, c = 50 for n = 0.49 and c = 500 for n = 0.499. c becomes infinite when n = 0.5. Theoretical derivation [9,10] shows that E c / G and E b / G are functions of S and c. The ad hoc effective compression modulus adopted by Gent and Lindley [3] that accounts for the effect of bulk compressibility is 1 1 1 ⫽ ⬁⫽ Ec Ec
Fig. 1. Deformed shape of a bonded elastic layer subjected to compressive force.
(3)
(4)
where E⬁c is the effective compression modulus assuming bulk incompressibility. Chalhoub and Kelly [1,2] found that the above equation is not quite correct; the last term of Eq. (4) must be multiplied by 6 / 5 for the infinitestrip layer and 3 / 4 for the circular layer. The ad hoc form of Eq. (4) comes from the concept of serial springs. The stiffness Ec is smaller than the stiffness E⬁c because Ec includes bulk compressibility. Hence, Ec can be treated as a spring which combines in series a spring of E⬁c and an additional spring representing bulk compressibility. Denoting the stiffness of the additional spring as Kc, the simplified formula for the effective compression modulus may have the form
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1 1 1 ⫽ ⫹ Ec E⬁c Kc
(5) ⬁ c
Because E c / G and E / G are function of S and c, K c / G is function of S and c, too. Similarly, the simplified formula for the effective flexure modulus may have the form 1 1 1 ⫽ ⬁⫹ Eb Eb Kb
(6)
where E⬁b is the effective flexure modulus assuming bulk incompressibility and K b / G is a function of S and c. To study the accuracy of the simplified formulae, we use the error of the approximation computed from the simplified formula as an index to show its deviation from the exact value calculated from the theoretical formula. The error is given by 1 subtracted from the ratio of the approximation to the exact value, so that a negative error means that the approximation is smaller than the exact value.
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from which selecting the first three terms gives yI0(y) 1 4 1 ⬇1 ⫹ y2⫺ y 2I1(y) 8 192
(12)
Substituting the above equation into Eq. (8) and neglecting the high-order terms of 1 / c leads to 1 Ec ⬇3(1 ⫹ 2S2)⫺ (1 ⫹ 12S2 ⫹ 48S4) G c
(13)
The effective compression modulus for the circular layer of incompressible material, E⬁c , can be derived by setting c = ⬁ in the above equation, E⬁c ⫽ 3(1 ⫹ 2S2)G
(14)
The form of Kc defined in Eq. (5) for the circular layer can be derived by substituting Eqs. (13) and (14) into Eq. (5) and presenting only the major term of c, Kc 3 ⫽ cfc(S) G 4
(15)
where 3. Effective compression modulus of circular layers The shape factor of a bonded circular layer which has a radius of b and a thickness of t is defined as S⫽
b 2t
(7)
The theoretical solution for the effective compression modulus of bonded circular layers has the closed form [9] 4 Ec ⫽c⫹ ⫺ G 3
2 (c⫺ )2 3 4 (c ⫹ )[yI0(y) / (2I1(y))]⫺1 3
(8)
fc(S) ⫽
[1 ⫹ 1 / (2S2)]2 1 ⫹ 1 / (4S2) ⫹ 1 / (48S4)
(16)
As shown in Fig. 3, f c(S) asymptotically approaches 1. 3 Since f c(10) = 1.0075, we can use Kc = for Sⱖ10, 4 which is the modification recommended by Chalhoub 4 and Kelly [1]. When S = 1.51, fc(S) = and Kc = , 3 which is the ad hoc modification suggested by Gent and Lindley [3]. Since f c(S) increases rapidly with decreasing S when S ⬍ 4, f c(S) cannot be approximated as a constant value for small S. Substituting E⬁c defined in Eq. (14) and Kc defined in Eq. (15) into Eq. (5), we have
where I0 and I1 denote the modified Bessel functions of the first kind of order 0 and order 1, respectively, and y is a nondimensional factor defined as y⫽
12S
冑3c ⫹ 4
(9)
For the case of large bulk modulus, 1 / c is a small amount. According to Eq. (9), the order of y is O(y2) = O(1 / c). When the argument of the Bessel functions is small, the functions may be approximated by I0(y) ⫽ 1 ⫹
y2 y4 y6 ⫹ ⫹ ⫹% 4 64 2304
(10)
and I1(y) ⫽
y y3 y5 y7 ⫹ ⫹ ⫹ ⫹% 2 16 384 18432
(11)
Fig. 3. Functions of fc(S) and fb(S) for effective compression modulus of circular layers.
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冋
Ec 1 4 ⬇ ⫹ 2 G 3(1 ⫹ 2S ) 3cfc(S)
册
⫺1
(17)
The curves of E c / G versus c calculated from Eqs. (13) and (17) are plotted in Fig. 4 for S = 10 and compared with the exact curve calculated from Eq. (8). It can be seen that Eq. (13) only asymptotically approaches the exact curve when c ⬎ 104; however, Eq. (17) not only has the same shape as but also is very close to the exact curve. Also plotted in Fig. 4 is the curve calculated from Eq. (4) using E⬁c defined in Eq. (14), which has a shape similar to the exact curve but with some deviation. For the case of small bulk modulus, c is a small amount and the order of 1 / y becomes O(1 / y2) = O(c). For large values of argument, the Besel functions can be approximated by the asymptotic expansions I0(y) ⫽
ey
冉
ey
冉
冑2πy
and I1(y) ⫽
冑2πy
1⫹
1⫺
冊
1 9 ⫹% ⫹ 8y 128y2
冊
3 15 ⫺% ⫺ 8y 128y2
(18)
(19)
from which selecting the first three terms gives I0(y) 1 3 ⬇1 ⫹ ⫹ I1(y) 2y 8y2
(20)
Substituting the above equation into Eq. (8) and utilizing Eq. (9) yields the following approximated expression for the effective compression modulus
冉 冊
4 Ec ⬇ c⫹ G 3
⫺
(21) 2 4(c⫺ )2 3
8 4 (c⫺ ) ⫹ S冑3c ⫹ 4[8 ⫹ (c ⫹ ) / (16S2)] 3 3
Fig. 4. Effective compression modulus for circular layers of S = 10 calculated by exact and approximated equations.
The curves of E c / G versus c calculated from Eq. (21) for S = 10 are also plotted in Fig. 4, which shows that Eq. (21) is closer to the exact curve than Eq. (17) when c ⬍ 102.5. The errors of Eqs. (17) and (21) versus c for S = 10 are plotted in Fig. 5, which reveals that the values of E c / G calculated from Eq. (17) are very accurate for c ⬎ 103.5 but the error increases abruptly when c ⬍ 103.5. On the other hand, Eq. (21) is very accurate for c ⬍ 102 but the error increases abruptly when c ⬎ 102. This indicates that an accurate simplified formula must be formed by a set of two approximated equations: Eq. (17) which is accurate for larger values of c and Eq. (21) which is accurate for smaller values of c. The extent of c values to which each approximated equation is applicable can be decided by the c value at the intersecting point of the two error curves shown in Fig. 5, which is c = 527. In other words, for the circular layers of S = 10, Eq. (17) is applicable to cⱖ527 and Eq. (21) is applicable to cⱕ527. The maximum error created by these two approximated equations is 2.5%, which occurs at the intersecting point. The applicable range of the approximated equations also changes with shape factors, which means that the c value at the intersecting point varies with S. The same procedure described in Fig. 5 for S = 10 is employed to find the intersecting points of different S values. The c values at the intersecting points of 16 different S values between 1 and 100 are plotted in Fig. 6. It can be seen that the c values at the intersecting points increase with the S values. Their regression curve, which is also plotted in Fig. 6, has the form S ⫽ ⫺0.30 ⫹ 0.44冑c
(22)
This equation clearly defines the division between the applicable ranges of the two approximated equations. Eq. (17) is applicable to the S values smaller than that in Eqs. (22) and (21) is applicable to the S values higher than that in Eq. (22).
Fig. 5. Error curves of two approximated equations of effective compression modulus for circular layers of S = 10.
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S ⫽ 冑168.5⫺0.125c
(25)
which is derived from Eq. (24) by neglecting the highorder terms of 1 / S. Fig. 7 indicates that Eq. (25) is a good approximation of Eq. (24) for ⑀ = 0.00443. Therefore, when Eq. (17) is used to calculate Ec, f c(S)⬇1 can be assumed for the S values which are higher than that in Eq. (25) and the error will not be enhanced much. The simplified formulae for the effective compression modulus of bonded circular layers can then be summarized as (I) for Sⱖ⫺0.30 + 0.44√c Fig. 6. Division equation derived from intersections of error curves for effective compression modulus of circular layers.
When Eq. (17) is used to calculate E c, f c(S) defined in Eq. (16) can be approximated as f c(S)⬇1 for the case of Sⱖ10. The range where f c(S)⬇1 is available can be further extended to smaller S values for larger c values. When cⰇS, the second term of Eq. (17) becomes much less than the first terms; the error caused by setting f c(S)⬇1 becomes negligible on the Ec values. Denote E1 as the value of Ec calculated from Eq. (17) and E2 as the value of Ec calculated from the same equation but using f c(S)⬇1. The difference ratio between E1 and E2 can derived as e⫽
E1⫺E2 (E⬁c / G)[1⫺(1 / fc(S))] ⫽ E1 Ccc ⫹ (E⬁c / G)
(23)
where C c = 3 / 4 is the constant term in Eq. (15). From Eq. (23), we have
冉 冊冋 冉
冊 册
1 1 E⬁c 1 1⫺ ⫺1 c⫽ Cc G e fc(S)
(24)
When S = 10, we can obtain c = 548 from Eq. (22), which gives ⑀ = 0.00443 from Eq. (23). The variation of c with S calculated from Eq. (24) for ⑀ = 0.00443 is plotted in Fig. 7. Also plotted in this figure is the curve
Fig. 7. Divisions among three approximated equations of effective compression modulus for circular layers.
Ec 4 ⫽c⫹ G 3
⫺
2 4(c⫺ )2 3
冋
8 4 (c⫺ ) ⫹ S冑3c ⫹ 4 8 ⫹ (c ⫹ ) / (16S2) 3 3
册
(II) for S ⬍ ⫺0.30 + 0.44√c and Sⱖ√168.5⫺0.125c
冋
册
1 4 Ec ⫽ ⫹ G 3(1 ⫹ 2S2) 3c (III) for S ⬍ ⫺0.30 √168.5⫺0.125c
冋 冉 冊冉
1 Ec ⫽ G 3(1 ⫹ 2S2)
⫺1
+
0.44√c
and
S
⬍
冊册
4 1 ⫹ 1 / (4S2) ⫹ 1 / (48S4) ⫺1 3c [1 ⫹ 1 / (2S2)]2 Fig. 7 depicts the applicable areas of the above three approximated equations. The errors of the simplified formulae versus c are plotted in Fig. 8 for several S values between 1 and 100. For the error curve of a specified S value, the peak error occurs at the intersection point in ⫹
Fig. 8. Errors of simplified formula for effective compression modulus of circular layers.
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Eq. (22). The maximum of the peak errors is 7.3% at S = 1. With increasing S value, the peak error reduces to 2.4% at S = 100. For the cases of S = 1, 2 and 5, the error curves have jumps near c = 103 because of the transition from the third equation to the second equation.
4. Effective flexure modulus of circular layers
冉
⫺
2 3
冊
2
(26)
8[I0(y) / (2yI1(y))⫺(1 / y2)] 4 c ⫹ ⫺4[I0(y) / (2yI1(y))⫺(1 / y2)] 3
where y has been defined in Eq. (9). For small values of y, substituting the first four terms of Eqs. (10) and (11) into Eq. (26) and neglecting the high-order terms of 1 / χ, we have
冉
冊 冉
9 8 8 Eb 7 1 ⫹ S2 ⫹ S4 ⬇ ⫹ 2S2 ⫺ G 2 4c 3 3
冊
(27)
By setting c = ⬁ in the above equation, the effective flexure modulus for incompressible material is derived as
冉
冊
7 ⫹ 2S2 G E ⫽ 2 ⬁ b
(28)
To find Kb in Eq. (6) for the circular layer, substitute Eqs. (27) and (28) into Eq. (6) and keep the major term of χ, Kb 2 ⫽ cfb(S) G 3
Eb 4 ⬇c ⫹ G 3
⫺
The theoretical solution for the effective flexure modulus of bonded circular layers has the closed form [10] 4 Eb ⫽c⫹ ⫺ c G 3
Eqs. (18) and (19) into Eq. (26) and neglecting the highorder terms of c gives the approximated equation of effective flexure modulus for small values of c as (32) 2 8(c⫺ )2 3
冋
3c ⫹ S冑3c ⫹ 4 8 ⫹
5 4 (c ⫹ ) / S2 16 3
册
The E b / G curves versus c calculated from Eqs. (31) and (32) for S = 10 are plotted in Fig. 9 and compared with the exact curve calculated from Eq. (26). The figure indicates that Eq. (32) is a good approximation of Eq. (26) at smaller values of c and Eq. (31) is a good approximation of Eq. (26) at larger values of c. The division between the applicable ranges of these two approximated equations is S ⫽ ⫺0.5 ⫹ 0.61冑c
(33)
which is solved by a procedure similar to that described in the previous section for the effective compression modulus. When Eq. (31) is utilized to calculate Eb, the division between using f b(S)⬇1 and using f b(S) in Eq. (30) can be derived by the same method described in the previous section for the effective compression modulus, S ⫽ 冑440⫺0.333c
(34)
This equation is the approximation of ⑀ = 0.0056, where ⑀ is the difference ratio with definition similar to that in Eq. (23). The curves of Eqs. (33) and (34) are plotted in Fig. 10 and they intersect at S = 15 and c = 646. These two curves form the divisions of the three approximated equations which represent the simplified
(29)
where fb(S) ⫽
[1 ⫹ 7 / (4S2)]2 1 ⫹ 1 / S2 ⫹ 3 / (8S4)
(30)
The variation of fb with S is plotted in Fig. 3. Similar to the function f c(S), f b(S) asymptotically approaches 1 but the converge rate is slower than that of f c(S). Since 2 f b(15) = 1.0111, we can use Kb = for Sⱖ15. Substitut3 ing E⬁b defined in Eq. (28) and Kb defined in Eq. (29) into Eq. (6), we have
冋
册
3 Eb 1 ⫹ ⬇ G 3.5 ⫹ 2S2 2cfb(S)
⫺1
(31)
For large values of y, substituting the first three terms of the series expressions of Bessel functions defined in
Fig. 9. Effective flexure modulus for circular layers of S = 10 calculated by exact and approximated equations.
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between 5.2% and 3% for different S values between 1 and 100.
5. Effective compression modulus of square layers The shape factor of a bonded square layer which has a side length of 2b and a thickness of t is defined as S⫽
Fig. 10. Divisions among three approximated equations of effective flexure modulus for circular layers.
formulae for the effective flexure modulus of bonded circular layers, summarized as (I) for Sⱖ⫺0.50 + 0.61√c Eb 4 ⫽c⫹ ⫺ G 3
2 8(c⫺ )2 3
冋
5 4 3c ⫹ S冑3c ⫹ 4 8 ⫹ (c ⫹ ) / S2 16 3
册
(II) for S ⬍ ⫺0.50 + 0.61√c and Sⱖ√440⫺0.333c
冋
册
3 1 Eb ⫹ ⫽ 2 G 3.5 ⫹ 2S 2c
⫺1
(III) for S ⬍ ⫺0.50 + 0.61√c and S ⬍ √440⫺0.333c
冋
冉冊
1 3 1 ⫹ 1 / S2 ⫹ 3 / (8S4) Eb ⫹ ⫽ G 3.5 ⫹ 2S2 2c [1 ⫹ 7 / (4S2)]2
册
⫺1
The error curves of the simplified formulae versus c are plotted in Fig. 11, which shows that the peak errors are
b 2t
(35)
The theoretical solution for the compressive stiffness of bonded square layers derived by Tsai and Lee [9] is a form of infinite series, of which the coefficients must be solved by numerical computing. To eliminate complex expression and complicated computing from the theoretical solution, a simplified formula is necessary. In the theoretical solution, the first term of the infinite series is the dominant term. For large bulk modulus, the dominant term of the theoretical solution can be reduced to 1 Ec ⬇(3 ⫹ 6.692S2)⫺ (1 ⫹ 10.768S2 G c
(36)
⫹ 54.665S4) The first term of the above equation represents the effective compression modulus of incompressible material, E⬁c . However, this term is derived from the approximated equation of using the first term of the theoretical solution and may be replaced by the exact solution [9] E⬁c ⫽ 3 ⫹ 6.748S2 G
(37)
Substituting the new approximated expression of Ec and the above equation of E⬁c into Eq. (5), we can solve Kc by neglecting the secondary terms, Kc ⫽ 0.833cfc(S) G
(38)
in which fc(S) ⫽
(1 ⫹ (0.445 / S2))2 1 ⫹ (0.197 / S2) ⫹ (0.018 / S4)
(39)
The function f c(S) decays to 1 very fast with increasing S. Since f s(10) = 1.0069, it is reasonable to set K c = 1 , so that Eq. 0.833 when Sⱖ10. Also, fc(1.87) = 0.833 (4) is only available for S = 1.87 where K c = . Substituting E⬁c defined in Eq. (37) and Kc defined in Eq. (38) into Eq. (5), we have Fig. 11. Errors of simplified formula for effective flexure modulus of circular layers.
冋
1 1 Ec ⬇ 2 ⫹ G 3 ⫹ 6.748S 0.833cfc(S)
册
⫺1
(40)
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For small bulk modulus, the dominant term of the theoretical solution can be reduced to 2 (c⫺ )2 Ec 3 4 ⬇c ⫹ ⫺ 1 2 G 3 (c ⫹ )y 3
⫺
(41)
4 (c ⫹ )(1.22y⫺1 ⫹ (4 / y)) 3
2
(III) for S ⬍ ⫺0.33 √153.4⫺0.125c
⫹
in which y is defined as
冪3c ⫹ 4
冋
Ec 1 1 ⫹ ⫽ 2 G 3 ⫹ 6.748S 0.833c +
册
⫺1
0.50√c
and
冋 冉 冊
Ec 1 ⫽ G 3 ⫹ 6.748S2
2 (c⫺ )2 3
y ⫽ 6S
(II) for S ⬍ ⫺0.33 + 0.50√c and Sⱖ√153.4⫺0.125c
(42)
The variations of E c / G with c calculated from Eqs. (40) and (41) for S = 10 are plotted in Fig. 12 and compared with the exact curve calculated from the theoretical solution. The figure reveals that the curve of Eq. (40) is very close to the exact curve, especially when c is large, and the curve of Eq. (41) is accurate for c ⬍ 102.7. Also plotted in Fig. 12 is the curve of Eq. (4), which shows some deviation from the exact curve. The simplified formula for the effective compression modulus of bonded square layers can be summarized as (I) for Sⱖ⫺0.33 + 0.50√c 2 4 (c ⫹ )(c⫺ )2 3 3 4 Ec ⫽c⫹ ⫺ 1 G 3 24S2(c ⫹ ) 3 2 (c⫺ )2 3 ⫺ 4 4 S冑6c ⫹ 8[2.44 ⫹ (c ⫹ ) / (3S2)]⫺(c ⫹ ) 3 3
Fig. 12. Effective compression modulus for square layers of S = 10 calculated by exact and approximated equations.
1 1 ⫹ (0.197 / S2) ⫹ (0.018 / S4) 0.833c (1 ⫹ (0.445 / S2))2
S
⬍
册
⫺1
The divisions among the above three approximated equations are solved by a procedure similar to that illustrated in the circular layer. The errors of the simplified formulae versus c for several shape factors between S = 1 and S = 100 are plotted in Fig. 13, which shows that the maximum error of 6.8% occurs at S = 1. The error is smaller than 2% if Sⱖ5. Koh and Lim [8] present a series solution for the effective compression modulus of bonded rectangular layers, which uses only simple arithmetic operations and can be applied to calculate the effective compression modulus of bonded square layers. The series solution computed using the first four terms is highly accurate, but if using only the first term of the series, the error may be up to 20% [8]. The simplified formulae presented here require different formulae for different ranges of shape factor and Poisson’s ratio, which seems somewhat cumbersome but will ensure that the computed solution is accurate through the full ranges of shape factor and Poisson’s ratio. 6. Effective flexure modulus of square layers Similar to the compressive stiffness formula, the theoretical solution for the tilting stiffness of bonded
Fig. 13. Errors of simplified formula for effective compression modulus of square layers.
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square layers derived by Tsai and Lee [10] is an infiniteseries form and is more complicated than that for the compressive stiffness. The coefficient in each term of the series must be solved by numerical computing. The first term of the infinite series is the dominant term of the theoretical solution. For large bulk modulus, the dominant term of the theoretical solution can be reduced to 1 Eb ⬇(3.4 ⫹ 2.25S2)⫺ (2.31 ⫹ 5.44S2 G c
(43)
⫹ 16.46S4)
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mum value, of which the error at the intersection is 7% and the maximum positive error is less than 3% for S = 10. The variations of E b / G with c calculated from Eq. (47) with C b = 0.375 and Eq. (48) for S = 10 are plotted in Fig. 14 and compared with the exact curve calculated from the theoretical solution. According to these two approximated equations, the first set of simplified formulae for the effective flexure modulus of bonded square layers can be established, (I) for Sⱖ⫺0.04 + 0.45√c
The first term of the above equation represents the effective flexure modulus of incompressible material, E⬁b , but this term is derived from the approximated formula of using the dominant term of the theoretical solution. It may be replaced by the exact solution. E⬁b ⫽ 3.471 ⫹ 2.228S2 G
(44)
Substituting the new expressions of Eb and the above equation of E⬁b into Eq. (6) and keeping the major term of c, we can solve Kb as Kb ⫽ Cbcfb(S) G
(45)
in which C b = 0.3 and (1 ⫹ (1.56 / S2))2 fb(S) ⫽ 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4)
(46)
⬁ b
Substituting E in Eq. (44) and Kb in Eq. (45) into Eq. (6) gives
冋
册
1 1 Eb ⬇ ⫹ G 3.471 ⫹ 2.228S2 Cbcfb(S)
⫺1
(47)
Eb 4 ⫽c⫹ ⫺ G 3
⫺
2 4 (c⫺ )2(c ⫹ ) 3 3 1 2 8S (c ⫹ )[1 ⫹ 6c ⫹ 8 / (12S)] 3
冑
2 3(c⫺ )2 3
冑
4 4 4 S 6c ⫹ 8[2 ⫹ 5(c ⫹ ) / (12S2)] ⫹ (c ⫹ )[0.5 ⫹ 5(c ⫹ ) / (8S2)] 3 3 3
(II) for S ⬍ ⫺0.04 + 0.45√c and Sⱖ√411.2⫺0.1667c
冋
册
8 1 Eb ⫹ ⫽ G 3.471 ⫹ 2.228S2 3c (III) for S ⬍ ⫺0.04 √411.2⫺0.1667c
冋 冉冊
+
⫺1
0.45√c
1 Eb ⫽ G 3.471 ⫹ 2.228S2 ⫹
8 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4) 3c (1 ⫹ (1.56 / S2))2
and
册
S
⬍
⫺1
The errors of the first set of simplified formulae are plotted in Fig. 15, which shows that the peak errors are between 6% and 11% for shape factors between S = 1
For small bulk modulus, the dominant term of the theoretical solution can be reduced to 4 Eb ⬇c ⫹ ⫺ G 3
2 3(c⫺ )2 3 1 1 1 ⫹ (1 / y) (c ⫹ )y2 3
冉
冊
2 3(c⫺ )2 3 1 ⫺ 4 1 ⫹ (0.5 / y) ⫹ (5 / y2) ⫹ (15 / y3) (c ⫹ )y 3
冉
(48)
冊
where y is defined in Eq. (42). If C b = 0.3 is used in Eq. (47), the error at the intersection of the error curves of Eqs. (47) and (48) for S = 10 is 15%, which seems too high. It is found that increasing Cb can reduce the error at the intersection but may create positive error for larger c in Eq. (47). C b = 0.375 is found to be an opti-
Fig. 14. Effective flexure modulus for square layers of S = 10 calculated by exact and approximated equations.
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simplified formulae for the effective flexure modulus of bonded square layers can be established, (I) for Sⱖ⫺0.20 + 0.52√c 4 2 (c⫺ )2(c ⫹ ) 6c ⫹ 8 Eb 3 3 4 1⫺ ⫽c⫹ ⫺ G 3 12S 1 2 8S (c ⫹ ) 3
冉
⫺
冑
冊
2 3(c⫺ )2 3
冑
4 4 4 S 6c ⫹ 8[2 ⫹ (c ⫹ ) / (3S2)] ⫹ (c ⫹ )[1 ⫹ 5(c ⫹ ) / (12S2)] 3 3 3
(II) for S ⬍ ⫺0.20 + 0.52√c and Sⱖ√340⫺0.135c Fig. 15. Errors of the first set of simplified formula for effective flexure modulus of square layers.
and S = 100. These errors seem to be a little higher than the errors of the other simplified formulae. To have a more accurate simplified formula, Cb is no longer a constant and the form of Kb in Eq. (45) has to be changed. Regression analysis gives the new form of Kb as Kb ⫽ 0.3cfb(S) ⫹ 0.677S2 ⫹ 4.965 G
(49)
Substituting E⬁b in Eq. (44) and Kb in Eq. (49) into Eq. (6) gives
冋
1 Eb ⬇ G 3.471 ⫹ 2.228S2
册
⫺1
Fig. 14 shows that the effective flexure modulus calculated from the above equation is very close to the exact value at larger c but becomes higher than the exact value at smaller c. Because the error becomes positive, the error curve of Eq. (50) will not intersect with the error curve of Eq. (48). To have two intersecting curves, Eq. (48) is then modified as 2 3(c⫺ )2 1 3 1⫺ 1 y (c ⫹ )y2 3
冉 冊
2 3(c⫺ )2 3 1 ⫺ 4 1 ⫹ (1 / y) ⫹ (4 / y2) ⫹ (10 / y3) (c ⫹ )y 3
冉
⫹
册
1 0.3c ⫹ 0.677S2 ⫹ 4.965
⫺1
(III) for S ⬍ ⫺0.20 + 0.52√c and S ⬍ √340⫺0.135c
冋 冉
1 Eb ⫽ G 3.471 ⫹ 2.228S2 ⫹ 0.3c ⫹ 4.965
(1 ⫹ (1.56 / S2))2 ⫹ 0.677S2 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4)
冊册
⫺1 ⫺1
(50)
1 ⫹ 0.3cfb(S) ⫹ 0.677S2 ⫹ 4.965
4 Eb ⬇c ⫹ ⫺ G 3
冋
1 Eb ⫽ G 3.471 ⫹ 2.228S2
The errors of the second set of simplified formulae are plotted in Fig. 16. The errors for shape factors between S = 1 and S = 100 are smaller than 4%. It can be seen that the second set is more accurate than the first set of simplified formulae.
(51)
冊
which, as shown in Fig. 14, is higher than the exact curve for larger c but is still accurate for smaller c. According to Eqs. (50) and (51), the second set of
Fig. 16. Errors of the second set of simplified formula for effective flexure modulus of square layers.
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
1453
7. Simplified formulae for infinite-strip layers The shape factor of a bonded infinite-strip layer which has a width of 2b and a thickness of t is defined as b t
S⫽
(52)
The infinite-strip shape is an idealized condition. Its theoretical stiffness formulae are not complicated. But for completeness, the approximated formulae derived by a simplified procedure similar to that mentioned above are described as follows. The effective compression modulus for the bonded layer of infinite-strip shape has the closed form [9] Fig. 17. Errors of simplified formula for effective compression modulus of infinite-strip layers.
2 (c⫺ )2 3 tanhy 4 Ec ⫽c⫹ ⫺ 4 G 3 y c⫹ 3
冉 冊
(53) where y has been defined in Eq. (54). The simplified formula for the effective flexure modulus of bonded infinite-strip layers is summarized as
where y is a nondimensional factor defined as y⫽
6S
(54)
冑3c ⫹ 4
The simplified formula for the effective compression modulus of bonded infinite-strip layers is summarized as (1) for Sⱖ⫺0.60 + 0.67√c
册
⫺1
冋 冉冊
1 Ec ⫽ G 4(1 ⫹ S2) 6 1 ⫹ 5 / (6S2) ⫹ 5 / (24S4) 5c (1 ⫹ (1 / S2))2
冋
册
10 1 Eb ⫹ ⫽ 2 G 4 ⫹ 0.8S 7c
⫺1
册
⫺1
2 (c⫺ )2 3 3 y 4 Eb ⫽c⫹ ⫺ ⫺1 2 4 y tanhy G 3 c⫹ 3
冊
冋 冉冊
1 Eb ⫽ G 4 ⫹ 0.8S2 ⫹
10 1 ⫹ 7 / (2S2) ⫹ 35 / (8S4) 7c (1 ⫹ (5 / S2))2
册
⫺1
The errors of the above simplified formula are plotted in Fig. 18. The maximum error of 5% occurs at S = 2.
The errors of the above simplified formula are plotted in Fig. 17. The maximum error of 8% occurs at S = 1. When Sⱖ5, the error of the simplified formula is less than 3%. The effective flexure modulus for the bonded layer of infinite-strip shape has the closed form [10]
冉 冊冉
册
(III) for S ⬍ ⫺1.05 + 1.07√c and S ⬍ √738⫺0.875c
(III) for S ⬍ ⫺0.60 + 0.67√c and S ⬍ √152⫺0.208c
⫹
2 3(c⫺ )2 3 ⫺ 4 4 S冑3c ⫹ 4 2 ⫹ (c ⫹ ) / (6S2) ⫹ (c ⫹ ) 3 3 (II) for S ⬍ ⫺1.05 + 1.07√c and S ⬍ √738⫺0.875c
(II) for S ⬍ ⫺0.60 + 0.67√c and Sⱖ√152⫺0.208c
冋
4 Eb ⫽c⫹ G 3
冋
2 (c⫺ )2 Ec 3 4 ⫽c⫹ ⫺ G 3 2S冑3c ⫹ 4
1 6 Ec ⫽ ⫹ 2 G 4(1 ⫹ S ) 5c
(I) for Sⱖ⫺1.05 + 1.07√c
(55)
8. Conclusion The simplified formulae for the effective compression modulus and effective flexure modulus of the circular, square and infinite-strip layers are derived. Every set of simplified formulae consists of three approximated equations. Each approximated equation has its applicable range of shape factor and bulk modulus. The divisions between the applicable ranges of different approximated
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layer without limitation on values of Poisson’s ratio and shape factor, the derived simplified formulae, which have low errors with respect to the theoretical solutions, can be applied to the layers of any elastic material and shape factor.
Acknowledgements The support from the National Science Council, Republic of China, under Grant No. NSC 89-2211-E011013 is greatly appreciated.
Fig. 18. Errors of simplified formula for effective flexure modulus of infinite-strip layers.
equations are defined so that the error of the simplified formulae can be minimized. The second and third approximated equations have a form similar to the ad hoc equation using the concept of serial spring, but the third one will be utilized only when the layer has low bulk modulus. For an elastic material such as rubber, the ratio of bulk modulus to shear modulus, c, has an order of 103. If we say c = 1600, only the effective flexure modulus of the square layer for the shape factor S ⬍ 12 will use the third approximated equation. Compared with the second approximated equation, the first approximated equation is applicable to higher shape factor or lower bulk modulus. For c = 1600, the effective compression modulus will use the first approximated equation when S ⬎ 17 for the circular layer, S ⬎ 20 for the square layer and S ⬎ 26 for the infinite-strip layer, while the effective flexure modulus will use the first approximated equation when S ⬎ 24 for the cicular layer, S ⬎ 18 for the square layer and S ⬎ 42 for the infinite-strip layer. Because the theoretical solutions, from which the simplified formulae are derived, are accurate for any elastic
References [1] Chalhaub MS, Kelly JM. Effect of bulk compressibility on the stiffness of cylindrical base isolation bearings. International Journal of Solids and Structures 1990;26:734–60. [2] Chalhoub MS, Kelly JM. Analysis of infinite-strip-shaped base isolator with elastomer bulk compression. Journal of Engineering Mechanics, ASCE 1991;117:1791–805. [3] Gent AN, Lindley PB. The compression of bonded rubber blocks. Proceeding of the Institution of Mechanical Engineers 1959;173:111–7. [4] Gent AN, Meinecke EA. Compression, bending and shear of bonded rubber blocks. Polymer Engineering and Sciences 1970;10:48–53. [5] Kelly JM. Earthquake-resistant design with rubber. London: Springer-Verlag, 1993. [6] Koh CG, Kelly JM. Effects of axial load on elastomeric isolation bearings. Report no. UCB/EERC-86/12, Earthquake Engineering Research Center, University of California, Berkeley, 1987. [7] Koh CG, Kelly JM. Compression stiffness of bonded square layers of nearly incompressible material. Engineering Structures 1989;11:9–15. [8] Koh CG, Lim HL. Analytical solution for compression stiffness of bonded rectangular layers. International Journal of Solids and Structures 2001;38:445–55. [9] Tsai H-C, Lee C-C. Compressive stiffness of elastic layers bonded between rigid plates. International Journal of Solids and Structures 1998;35:3053–69. [10] Tsai H-C, Lee C-C. Tilting stiffness of elastic layers bonded between rigid plates. International Journal of Solids and Structures 1999;36:2485–505.
Simplified stiffness formulae for elastic layers bonded between rigid plates Hsiang-Chuan Tsai ∗, Wei-Jen Pai Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan, ROC Received 13 May 2002; received in revised form 10 January 2003; accepted 29 January 2003
Abstract An elastic layer bonded between two rigid plates has higher compressive stiffness and tilting stiffness than those of an unbonded layer. This effect has been adopted in the design of reinforced elastomeric bearings. Although compressive stiffness and tilting stiffness of bonded layers can be theoretically derived, some of the theoretical solutions are cumbersome in practical application. In this paper, simplified formulae for compressive stiffness and tilting stiffness of bonded elastic layers of circular, square and infinite-strip shapes are developed from theoretical solutions. Every set of simplified formulae consists of three approximated equations in which each equation has its applicable range of shape factor and bulk modulus. The divisions between the applicable ranges of different approximated equations are also established so that errors of the simplified formulae can be minimized. 2003 Elsevier Ltd. All rights reserved. Keywords: Elastomeric bearing; Base isolation; Bonded layer
1. Introduction Reinforced elastomeric bearings consist mainly of thin sheets of elastomer layers bonded to interleaving steel shims. Because of their low stiffness in the horizontal direction, reinforced elastomeric bearings are utilized in bridge engineering to allow temperature deformation and in base isolation to reduce building vibration induced by earthquakes. Reinforced elastomeric bearings must possess sufficient rigidity to sustain the gravity loading of superstructures, which makes the compressive stiffness and tilting stiffness of reinforced elastomeric bearings important factors in structural analysis and design. Gent and Lindley [3] derived the compressive stiffness of an incompressible elastic layer bonded between rigid plates for infinite-strip and circular shapes. Subsequently, Gent and Meinecke [4] extended this method to analyze the compressive stiffness and tilting stiffness of incompressible elastic layers for square and other shapes. Although elastomer can be treated as incompressible in some analyses, the assumption of incom-
∗
Corresponding author. E-mail address: [email protected] (H.-C. Tsai).
0141-0296/03/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0141-0296(03)00044-0
pressibility tends to overestimate the compressive stiffness and tilting stiffness of the bonded elastomer layer when the shape factor of the elastomer layer is high. To include the influence of compressibility on compressive stiffness, Gent and Lindley [3] recommended an ad hoc modification where the inverse of the effective compression modulus for the compressible layer is equal to the sum of the inverse of the effective compression modulus assuming incompressibility and the inverse of the bulk modulus of the layer. Kelly [5] developed a theoretical approach for deriving the compressive stiffness and tilting stiffness taking into consideration the effect of bulk compressibility. The solutions are available for the layers of infinite-strip shape [2], circular shape [1] and square shape [6]. These solutions are very accurate for layers of high shape factor and material of Poisson’s ratio between 0.49 and 0.5, e.g. rubber. Chalhoub and Kelly [1,2] also found that the ad hoc modification recommended by Gent and Lindley [3] was not accurate and proposed another form of modification. All the solutions of the above-mentioned research works are based on two kinematic assumptions and one stress assumption. Without utilizing the stress assumption, Koh and Kelly [7] derived the compression stiffness
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of the bonded square layer by a variable transform approach. Recently, Koh and Lim [8] extended this approach to solve the compression stiffness of the bonded rectangular layer. The compressive stiffness and tilting stiffness derived by Tsai and Lee [9,10] for the bonded layers of infinitestrip shape, circular shape and square shape are shown to be accurate for any elastic layer without limitation on the values of Poisson’s ratio and shape factor. However, not all of these stiffness formulae are convenient for numerical calculation in practical applications. The purpose of this paper is to find out the simplified forms of these stiffness formulae, so that they can be easily utilized in practical applications but without losing accuracy, and to compare these simplified formulae with those proposed by previous studies.
2. Deformation of bonded elastic layers An elastic layer bonded between two rigid plates and subjected to a compression force P in the z direction is shown in Fig. 1, which has a thickness t and a plane are A. The thickness reduction of the layer in the z direction can be expressed as d⫽
Pt AEc
(1)
where Ec is the effective compression modulus. Fig. 2 shows the deformation of the bonded layer when the top and bottom rigid plates are subjected to a moment M in the y direction. The angle q formed by the rigid plates can be expressed as q⫽
Mt IEb
(2)
Fig. 2. Deformed shape of a bonded elastic layer subjected to bending moment.
where I is the moment of inertial of the plane area about the y axis and Eb is the effective flexure modulus. The magnitudes of the effective compression modulus Ec and the effective flexure modulus Eb vary with the geometry and material properties of the elastic layer. The geometry of the bonded layer is represented by the shape factor S, which is defined as the ratio of the plane area A to the perimeter area not bonded to the rigid plates. The material properties for the elastic material such as rubber are usually defined in terms of the shear modulus G and the bulk modulus . Denote a nondimensional parameter c as c⫽
2(1 ⫹ n) ⫽ G 3(1⫺2n)
where n is Poisson’s ratio. When n is near 0.5, the value of c increases rapidly. For example, c = 50 for n = 0.49 and c = 500 for n = 0.499. c becomes infinite when n = 0.5. Theoretical derivation [9,10] shows that E c / G and E b / G are functions of S and c. The ad hoc effective compression modulus adopted by Gent and Lindley [3] that accounts for the effect of bulk compressibility is 1 1 1 ⫽ ⬁⫽ Ec Ec
Fig. 1. Deformed shape of a bonded elastic layer subjected to compressive force.
(3)
(4)
where E⬁c is the effective compression modulus assuming bulk incompressibility. Chalhoub and Kelly [1,2] found that the above equation is not quite correct; the last term of Eq. (4) must be multiplied by 6 / 5 for the infinitestrip layer and 3 / 4 for the circular layer. The ad hoc form of Eq. (4) comes from the concept of serial springs. The stiffness Ec is smaller than the stiffness E⬁c because Ec includes bulk compressibility. Hence, Ec can be treated as a spring which combines in series a spring of E⬁c and an additional spring representing bulk compressibility. Denoting the stiffness of the additional spring as Kc, the simplified formula for the effective compression modulus may have the form
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
1 1 1 ⫽ ⫹ Ec E⬁c Kc
(5) ⬁ c
Because E c / G and E / G are function of S and c, K c / G is function of S and c, too. Similarly, the simplified formula for the effective flexure modulus may have the form 1 1 1 ⫽ ⬁⫹ Eb Eb Kb
(6)
where E⬁b is the effective flexure modulus assuming bulk incompressibility and K b / G is a function of S and c. To study the accuracy of the simplified formulae, we use the error of the approximation computed from the simplified formula as an index to show its deviation from the exact value calculated from the theoretical formula. The error is given by 1 subtracted from the ratio of the approximation to the exact value, so that a negative error means that the approximation is smaller than the exact value.
1445
from which selecting the first three terms gives yI0(y) 1 4 1 ⬇1 ⫹ y2⫺ y 2I1(y) 8 192
(12)
Substituting the above equation into Eq. (8) and neglecting the high-order terms of 1 / c leads to 1 Ec ⬇3(1 ⫹ 2S2)⫺ (1 ⫹ 12S2 ⫹ 48S4) G c
(13)
The effective compression modulus for the circular layer of incompressible material, E⬁c , can be derived by setting c = ⬁ in the above equation, E⬁c ⫽ 3(1 ⫹ 2S2)G
(14)
The form of Kc defined in Eq. (5) for the circular layer can be derived by substituting Eqs. (13) and (14) into Eq. (5) and presenting only the major term of c, Kc 3 ⫽ cfc(S) G 4
(15)
where 3. Effective compression modulus of circular layers The shape factor of a bonded circular layer which has a radius of b and a thickness of t is defined as S⫽
b 2t
(7)
The theoretical solution for the effective compression modulus of bonded circular layers has the closed form [9] 4 Ec ⫽c⫹ ⫺ G 3
2 (c⫺ )2 3 4 (c ⫹ )[yI0(y) / (2I1(y))]⫺1 3
(8)
fc(S) ⫽
[1 ⫹ 1 / (2S2)]2 1 ⫹ 1 / (4S2) ⫹ 1 / (48S4)
(16)
As shown in Fig. 3, f c(S) asymptotically approaches 1. 3 Since f c(10) = 1.0075, we can use Kc = for Sⱖ10, 4 which is the modification recommended by Chalhoub 4 and Kelly [1]. When S = 1.51, fc(S) = and Kc = , 3 which is the ad hoc modification suggested by Gent and Lindley [3]. Since f c(S) increases rapidly with decreasing S when S ⬍ 4, f c(S) cannot be approximated as a constant value for small S. Substituting E⬁c defined in Eq. (14) and Kc defined in Eq. (15) into Eq. (5), we have
where I0 and I1 denote the modified Bessel functions of the first kind of order 0 and order 1, respectively, and y is a nondimensional factor defined as y⫽
12S
冑3c ⫹ 4
(9)
For the case of large bulk modulus, 1 / c is a small amount. According to Eq. (9), the order of y is O(y2) = O(1 / c). When the argument of the Bessel functions is small, the functions may be approximated by I0(y) ⫽ 1 ⫹
y2 y4 y6 ⫹ ⫹ ⫹% 4 64 2304
(10)
and I1(y) ⫽
y y3 y5 y7 ⫹ ⫹ ⫹ ⫹% 2 16 384 18432
(11)
Fig. 3. Functions of fc(S) and fb(S) for effective compression modulus of circular layers.
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H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
冋
Ec 1 4 ⬇ ⫹ 2 G 3(1 ⫹ 2S ) 3cfc(S)
册
⫺1
(17)
The curves of E c / G versus c calculated from Eqs. (13) and (17) are plotted in Fig. 4 for S = 10 and compared with the exact curve calculated from Eq. (8). It can be seen that Eq. (13) only asymptotically approaches the exact curve when c ⬎ 104; however, Eq. (17) not only has the same shape as but also is very close to the exact curve. Also plotted in Fig. 4 is the curve calculated from Eq. (4) using E⬁c defined in Eq. (14), which has a shape similar to the exact curve but with some deviation. For the case of small bulk modulus, c is a small amount and the order of 1 / y becomes O(1 / y2) = O(c). For large values of argument, the Besel functions can be approximated by the asymptotic expansions I0(y) ⫽
ey
冉
ey
冉
冑2πy
and I1(y) ⫽
冑2πy
1⫹
1⫺
冊
1 9 ⫹% ⫹ 8y 128y2
冊
3 15 ⫺% ⫺ 8y 128y2
(18)
(19)
from which selecting the first three terms gives I0(y) 1 3 ⬇1 ⫹ ⫹ I1(y) 2y 8y2
(20)
Substituting the above equation into Eq. (8) and utilizing Eq. (9) yields the following approximated expression for the effective compression modulus
冉 冊
4 Ec ⬇ c⫹ G 3
⫺
(21) 2 4(c⫺ )2 3
8 4 (c⫺ ) ⫹ S冑3c ⫹ 4[8 ⫹ (c ⫹ ) / (16S2)] 3 3
Fig. 4. Effective compression modulus for circular layers of S = 10 calculated by exact and approximated equations.
The curves of E c / G versus c calculated from Eq. (21) for S = 10 are also plotted in Fig. 4, which shows that Eq. (21) is closer to the exact curve than Eq. (17) when c ⬍ 102.5. The errors of Eqs. (17) and (21) versus c for S = 10 are plotted in Fig. 5, which reveals that the values of E c / G calculated from Eq. (17) are very accurate for c ⬎ 103.5 but the error increases abruptly when c ⬍ 103.5. On the other hand, Eq. (21) is very accurate for c ⬍ 102 but the error increases abruptly when c ⬎ 102. This indicates that an accurate simplified formula must be formed by a set of two approximated equations: Eq. (17) which is accurate for larger values of c and Eq. (21) which is accurate for smaller values of c. The extent of c values to which each approximated equation is applicable can be decided by the c value at the intersecting point of the two error curves shown in Fig. 5, which is c = 527. In other words, for the circular layers of S = 10, Eq. (17) is applicable to cⱖ527 and Eq. (21) is applicable to cⱕ527. The maximum error created by these two approximated equations is 2.5%, which occurs at the intersecting point. The applicable range of the approximated equations also changes with shape factors, which means that the c value at the intersecting point varies with S. The same procedure described in Fig. 5 for S = 10 is employed to find the intersecting points of different S values. The c values at the intersecting points of 16 different S values between 1 and 100 are plotted in Fig. 6. It can be seen that the c values at the intersecting points increase with the S values. Their regression curve, which is also plotted in Fig. 6, has the form S ⫽ ⫺0.30 ⫹ 0.44冑c
(22)
This equation clearly defines the division between the applicable ranges of the two approximated equations. Eq. (17) is applicable to the S values smaller than that in Eqs. (22) and (21) is applicable to the S values higher than that in Eq. (22).
Fig. 5. Error curves of two approximated equations of effective compression modulus for circular layers of S = 10.
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
1447
S ⫽ 冑168.5⫺0.125c
(25)
which is derived from Eq. (24) by neglecting the highorder terms of 1 / S. Fig. 7 indicates that Eq. (25) is a good approximation of Eq. (24) for ⑀ = 0.00443. Therefore, when Eq. (17) is used to calculate Ec, f c(S)⬇1 can be assumed for the S values which are higher than that in Eq. (25) and the error will not be enhanced much. The simplified formulae for the effective compression modulus of bonded circular layers can then be summarized as (I) for Sⱖ⫺0.30 + 0.44√c Fig. 6. Division equation derived from intersections of error curves for effective compression modulus of circular layers.
When Eq. (17) is used to calculate E c, f c(S) defined in Eq. (16) can be approximated as f c(S)⬇1 for the case of Sⱖ10. The range where f c(S)⬇1 is available can be further extended to smaller S values for larger c values. When cⰇS, the second term of Eq. (17) becomes much less than the first terms; the error caused by setting f c(S)⬇1 becomes negligible on the Ec values. Denote E1 as the value of Ec calculated from Eq. (17) and E2 as the value of Ec calculated from the same equation but using f c(S)⬇1. The difference ratio between E1 and E2 can derived as e⫽
E1⫺E2 (E⬁c / G)[1⫺(1 / fc(S))] ⫽ E1 Ccc ⫹ (E⬁c / G)
(23)
where C c = 3 / 4 is the constant term in Eq. (15). From Eq. (23), we have
冉 冊冋 冉
冊 册
1 1 E⬁c 1 1⫺ ⫺1 c⫽ Cc G e fc(S)
(24)
When S = 10, we can obtain c = 548 from Eq. (22), which gives ⑀ = 0.00443 from Eq. (23). The variation of c with S calculated from Eq. (24) for ⑀ = 0.00443 is plotted in Fig. 7. Also plotted in this figure is the curve
Fig. 7. Divisions among three approximated equations of effective compression modulus for circular layers.
Ec 4 ⫽c⫹ G 3
⫺
2 4(c⫺ )2 3
冋
8 4 (c⫺ ) ⫹ S冑3c ⫹ 4 8 ⫹ (c ⫹ ) / (16S2) 3 3
册
(II) for S ⬍ ⫺0.30 + 0.44√c and Sⱖ√168.5⫺0.125c
冋
册
1 4 Ec ⫽ ⫹ G 3(1 ⫹ 2S2) 3c (III) for S ⬍ ⫺0.30 √168.5⫺0.125c
冋 冉 冊冉
1 Ec ⫽ G 3(1 ⫹ 2S2)
⫺1
+
0.44√c
and
S
⬍
冊册
4 1 ⫹ 1 / (4S2) ⫹ 1 / (48S4) ⫺1 3c [1 ⫹ 1 / (2S2)]2 Fig. 7 depicts the applicable areas of the above three approximated equations. The errors of the simplified formulae versus c are plotted in Fig. 8 for several S values between 1 and 100. For the error curve of a specified S value, the peak error occurs at the intersection point in ⫹
Fig. 8. Errors of simplified formula for effective compression modulus of circular layers.
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Eq. (22). The maximum of the peak errors is 7.3% at S = 1. With increasing S value, the peak error reduces to 2.4% at S = 100. For the cases of S = 1, 2 and 5, the error curves have jumps near c = 103 because of the transition from the third equation to the second equation.
4. Effective flexure modulus of circular layers
冉
⫺
2 3
冊
2
(26)
8[I0(y) / (2yI1(y))⫺(1 / y2)] 4 c ⫹ ⫺4[I0(y) / (2yI1(y))⫺(1 / y2)] 3
where y has been defined in Eq. (9). For small values of y, substituting the first four terms of Eqs. (10) and (11) into Eq. (26) and neglecting the high-order terms of 1 / χ, we have
冉
冊 冉
9 8 8 Eb 7 1 ⫹ S2 ⫹ S4 ⬇ ⫹ 2S2 ⫺ G 2 4c 3 3
冊
(27)
By setting c = ⬁ in the above equation, the effective flexure modulus for incompressible material is derived as
冉
冊
7 ⫹ 2S2 G E ⫽ 2 ⬁ b
(28)
To find Kb in Eq. (6) for the circular layer, substitute Eqs. (27) and (28) into Eq. (6) and keep the major term of χ, Kb 2 ⫽ cfb(S) G 3
Eb 4 ⬇c ⫹ G 3
⫺
The theoretical solution for the effective flexure modulus of bonded circular layers has the closed form [10] 4 Eb ⫽c⫹ ⫺ c G 3
Eqs. (18) and (19) into Eq. (26) and neglecting the highorder terms of c gives the approximated equation of effective flexure modulus for small values of c as (32) 2 8(c⫺ )2 3
冋
3c ⫹ S冑3c ⫹ 4 8 ⫹
5 4 (c ⫹ ) / S2 16 3
册
The E b / G curves versus c calculated from Eqs. (31) and (32) for S = 10 are plotted in Fig. 9 and compared with the exact curve calculated from Eq. (26). The figure indicates that Eq. (32) is a good approximation of Eq. (26) at smaller values of c and Eq. (31) is a good approximation of Eq. (26) at larger values of c. The division between the applicable ranges of these two approximated equations is S ⫽ ⫺0.5 ⫹ 0.61冑c
(33)
which is solved by a procedure similar to that described in the previous section for the effective compression modulus. When Eq. (31) is utilized to calculate Eb, the division between using f b(S)⬇1 and using f b(S) in Eq. (30) can be derived by the same method described in the previous section for the effective compression modulus, S ⫽ 冑440⫺0.333c
(34)
This equation is the approximation of ⑀ = 0.0056, where ⑀ is the difference ratio with definition similar to that in Eq. (23). The curves of Eqs. (33) and (34) are plotted in Fig. 10 and they intersect at S = 15 and c = 646. These two curves form the divisions of the three approximated equations which represent the simplified
(29)
where fb(S) ⫽
[1 ⫹ 7 / (4S2)]2 1 ⫹ 1 / S2 ⫹ 3 / (8S4)
(30)
The variation of fb with S is plotted in Fig. 3. Similar to the function f c(S), f b(S) asymptotically approaches 1 but the converge rate is slower than that of f c(S). Since 2 f b(15) = 1.0111, we can use Kb = for Sⱖ15. Substitut3 ing E⬁b defined in Eq. (28) and Kb defined in Eq. (29) into Eq. (6), we have
冋
册
3 Eb 1 ⫹ ⬇ G 3.5 ⫹ 2S2 2cfb(S)
⫺1
(31)
For large values of y, substituting the first three terms of the series expressions of Bessel functions defined in
Fig. 9. Effective flexure modulus for circular layers of S = 10 calculated by exact and approximated equations.
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
1449
between 5.2% and 3% for different S values between 1 and 100.
5. Effective compression modulus of square layers The shape factor of a bonded square layer which has a side length of 2b and a thickness of t is defined as S⫽
Fig. 10. Divisions among three approximated equations of effective flexure modulus for circular layers.
formulae for the effective flexure modulus of bonded circular layers, summarized as (I) for Sⱖ⫺0.50 + 0.61√c Eb 4 ⫽c⫹ ⫺ G 3
2 8(c⫺ )2 3
冋
5 4 3c ⫹ S冑3c ⫹ 4 8 ⫹ (c ⫹ ) / S2 16 3
册
(II) for S ⬍ ⫺0.50 + 0.61√c and Sⱖ√440⫺0.333c
冋
册
3 1 Eb ⫹ ⫽ 2 G 3.5 ⫹ 2S 2c
⫺1
(III) for S ⬍ ⫺0.50 + 0.61√c and S ⬍ √440⫺0.333c
冋
冉冊
1 3 1 ⫹ 1 / S2 ⫹ 3 / (8S4) Eb ⫹ ⫽ G 3.5 ⫹ 2S2 2c [1 ⫹ 7 / (4S2)]2
册
⫺1
The error curves of the simplified formulae versus c are plotted in Fig. 11, which shows that the peak errors are
b 2t
(35)
The theoretical solution for the compressive stiffness of bonded square layers derived by Tsai and Lee [9] is a form of infinite series, of which the coefficients must be solved by numerical computing. To eliminate complex expression and complicated computing from the theoretical solution, a simplified formula is necessary. In the theoretical solution, the first term of the infinite series is the dominant term. For large bulk modulus, the dominant term of the theoretical solution can be reduced to 1 Ec ⬇(3 ⫹ 6.692S2)⫺ (1 ⫹ 10.768S2 G c
(36)
⫹ 54.665S4) The first term of the above equation represents the effective compression modulus of incompressible material, E⬁c . However, this term is derived from the approximated equation of using the first term of the theoretical solution and may be replaced by the exact solution [9] E⬁c ⫽ 3 ⫹ 6.748S2 G
(37)
Substituting the new approximated expression of Ec and the above equation of E⬁c into Eq. (5), we can solve Kc by neglecting the secondary terms, Kc ⫽ 0.833cfc(S) G
(38)
in which fc(S) ⫽
(1 ⫹ (0.445 / S2))2 1 ⫹ (0.197 / S2) ⫹ (0.018 / S4)
(39)
The function f c(S) decays to 1 very fast with increasing S. Since f s(10) = 1.0069, it is reasonable to set K c = 1 , so that Eq. 0.833 when Sⱖ10. Also, fc(1.87) = 0.833 (4) is only available for S = 1.87 where K c = . Substituting E⬁c defined in Eq. (37) and Kc defined in Eq. (38) into Eq. (5), we have Fig. 11. Errors of simplified formula for effective flexure modulus of circular layers.
冋
1 1 Ec ⬇ 2 ⫹ G 3 ⫹ 6.748S 0.833cfc(S)
册
⫺1
(40)
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For small bulk modulus, the dominant term of the theoretical solution can be reduced to 2 (c⫺ )2 Ec 3 4 ⬇c ⫹ ⫺ 1 2 G 3 (c ⫹ )y 3
⫺
(41)
4 (c ⫹ )(1.22y⫺1 ⫹ (4 / y)) 3
2
(III) for S ⬍ ⫺0.33 √153.4⫺0.125c
⫹
in which y is defined as
冪3c ⫹ 4
冋
Ec 1 1 ⫹ ⫽ 2 G 3 ⫹ 6.748S 0.833c +
册
⫺1
0.50√c
and
冋 冉 冊
Ec 1 ⫽ G 3 ⫹ 6.748S2
2 (c⫺ )2 3
y ⫽ 6S
(II) for S ⬍ ⫺0.33 + 0.50√c and Sⱖ√153.4⫺0.125c
(42)
The variations of E c / G with c calculated from Eqs. (40) and (41) for S = 10 are plotted in Fig. 12 and compared with the exact curve calculated from the theoretical solution. The figure reveals that the curve of Eq. (40) is very close to the exact curve, especially when c is large, and the curve of Eq. (41) is accurate for c ⬍ 102.7. Also plotted in Fig. 12 is the curve of Eq. (4), which shows some deviation from the exact curve. The simplified formula for the effective compression modulus of bonded square layers can be summarized as (I) for Sⱖ⫺0.33 + 0.50√c 2 4 (c ⫹ )(c⫺ )2 3 3 4 Ec ⫽c⫹ ⫺ 1 G 3 24S2(c ⫹ ) 3 2 (c⫺ )2 3 ⫺ 4 4 S冑6c ⫹ 8[2.44 ⫹ (c ⫹ ) / (3S2)]⫺(c ⫹ ) 3 3
Fig. 12. Effective compression modulus for square layers of S = 10 calculated by exact and approximated equations.
1 1 ⫹ (0.197 / S2) ⫹ (0.018 / S4) 0.833c (1 ⫹ (0.445 / S2))2
S
⬍
册
⫺1
The divisions among the above three approximated equations are solved by a procedure similar to that illustrated in the circular layer. The errors of the simplified formulae versus c for several shape factors between S = 1 and S = 100 are plotted in Fig. 13, which shows that the maximum error of 6.8% occurs at S = 1. The error is smaller than 2% if Sⱖ5. Koh and Lim [8] present a series solution for the effective compression modulus of bonded rectangular layers, which uses only simple arithmetic operations and can be applied to calculate the effective compression modulus of bonded square layers. The series solution computed using the first four terms is highly accurate, but if using only the first term of the series, the error may be up to 20% [8]. The simplified formulae presented here require different formulae for different ranges of shape factor and Poisson’s ratio, which seems somewhat cumbersome but will ensure that the computed solution is accurate through the full ranges of shape factor and Poisson’s ratio. 6. Effective flexure modulus of square layers Similar to the compressive stiffness formula, the theoretical solution for the tilting stiffness of bonded
Fig. 13. Errors of simplified formula for effective compression modulus of square layers.
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
square layers derived by Tsai and Lee [10] is an infiniteseries form and is more complicated than that for the compressive stiffness. The coefficient in each term of the series must be solved by numerical computing. The first term of the infinite series is the dominant term of the theoretical solution. For large bulk modulus, the dominant term of the theoretical solution can be reduced to 1 Eb ⬇(3.4 ⫹ 2.25S2)⫺ (2.31 ⫹ 5.44S2 G c
(43)
⫹ 16.46S4)
1451
mum value, of which the error at the intersection is 7% and the maximum positive error is less than 3% for S = 10. The variations of E b / G with c calculated from Eq. (47) with C b = 0.375 and Eq. (48) for S = 10 are plotted in Fig. 14 and compared with the exact curve calculated from the theoretical solution. According to these two approximated equations, the first set of simplified formulae for the effective flexure modulus of bonded square layers can be established, (I) for Sⱖ⫺0.04 + 0.45√c
The first term of the above equation represents the effective flexure modulus of incompressible material, E⬁b , but this term is derived from the approximated formula of using the dominant term of the theoretical solution. It may be replaced by the exact solution. E⬁b ⫽ 3.471 ⫹ 2.228S2 G
(44)
Substituting the new expressions of Eb and the above equation of E⬁b into Eq. (6) and keeping the major term of c, we can solve Kb as Kb ⫽ Cbcfb(S) G
(45)
in which C b = 0.3 and (1 ⫹ (1.56 / S2))2 fb(S) ⫽ 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4)
(46)
⬁ b
Substituting E in Eq. (44) and Kb in Eq. (45) into Eq. (6) gives
冋
册
1 1 Eb ⬇ ⫹ G 3.471 ⫹ 2.228S2 Cbcfb(S)
⫺1
(47)
Eb 4 ⫽c⫹ ⫺ G 3
⫺
2 4 (c⫺ )2(c ⫹ ) 3 3 1 2 8S (c ⫹ )[1 ⫹ 6c ⫹ 8 / (12S)] 3
冑
2 3(c⫺ )2 3
冑
4 4 4 S 6c ⫹ 8[2 ⫹ 5(c ⫹ ) / (12S2)] ⫹ (c ⫹ )[0.5 ⫹ 5(c ⫹ ) / (8S2)] 3 3 3
(II) for S ⬍ ⫺0.04 + 0.45√c and Sⱖ√411.2⫺0.1667c
冋
册
8 1 Eb ⫹ ⫽ G 3.471 ⫹ 2.228S2 3c (III) for S ⬍ ⫺0.04 √411.2⫺0.1667c
冋 冉冊
+
⫺1
0.45√c
1 Eb ⫽ G 3.471 ⫹ 2.228S2 ⫹
8 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4) 3c (1 ⫹ (1.56 / S2))2
and
册
S
⬍
⫺1
The errors of the first set of simplified formulae are plotted in Fig. 15, which shows that the peak errors are between 6% and 11% for shape factors between S = 1
For small bulk modulus, the dominant term of the theoretical solution can be reduced to 4 Eb ⬇c ⫹ ⫺ G 3
2 3(c⫺ )2 3 1 1 1 ⫹ (1 / y) (c ⫹ )y2 3
冉
冊
2 3(c⫺ )2 3 1 ⫺ 4 1 ⫹ (0.5 / y) ⫹ (5 / y2) ⫹ (15 / y3) (c ⫹ )y 3
冉
(48)
冊
where y is defined in Eq. (42). If C b = 0.3 is used in Eq. (47), the error at the intersection of the error curves of Eqs. (47) and (48) for S = 10 is 15%, which seems too high. It is found that increasing Cb can reduce the error at the intersection but may create positive error for larger c in Eq. (47). C b = 0.375 is found to be an opti-
Fig. 14. Effective flexure modulus for square layers of S = 10 calculated by exact and approximated equations.
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simplified formulae for the effective flexure modulus of bonded square layers can be established, (I) for Sⱖ⫺0.20 + 0.52√c 4 2 (c⫺ )2(c ⫹ ) 6c ⫹ 8 Eb 3 3 4 1⫺ ⫽c⫹ ⫺ G 3 12S 1 2 8S (c ⫹ ) 3
冉
⫺
冑
冊
2 3(c⫺ )2 3
冑
4 4 4 S 6c ⫹ 8[2 ⫹ (c ⫹ ) / (3S2)] ⫹ (c ⫹ )[1 ⫹ 5(c ⫹ ) / (12S2)] 3 3 3
(II) for S ⬍ ⫺0.20 + 0.52√c and Sⱖ√340⫺0.135c Fig. 15. Errors of the first set of simplified formula for effective flexure modulus of square layers.
and S = 100. These errors seem to be a little higher than the errors of the other simplified formulae. To have a more accurate simplified formula, Cb is no longer a constant and the form of Kb in Eq. (45) has to be changed. Regression analysis gives the new form of Kb as Kb ⫽ 0.3cfb(S) ⫹ 0.677S2 ⫹ 4.965 G
(49)
Substituting E⬁b in Eq. (44) and Kb in Eq. (49) into Eq. (6) gives
冋
1 Eb ⬇ G 3.471 ⫹ 2.228S2
册
⫺1
Fig. 14 shows that the effective flexure modulus calculated from the above equation is very close to the exact value at larger c but becomes higher than the exact value at smaller c. Because the error becomes positive, the error curve of Eq. (50) will not intersect with the error curve of Eq. (48). To have two intersecting curves, Eq. (48) is then modified as 2 3(c⫺ )2 1 3 1⫺ 1 y (c ⫹ )y2 3
冉 冊
2 3(c⫺ )2 3 1 ⫺ 4 1 ⫹ (1 / y) ⫹ (4 / y2) ⫹ (10 / y3) (c ⫹ )y 3
冉
⫹
册
1 0.3c ⫹ 0.677S2 ⫹ 4.965
⫺1
(III) for S ⬍ ⫺0.20 + 0.52√c and S ⬍ √340⫺0.135c
冋 冉
1 Eb ⫽ G 3.471 ⫹ 2.228S2 ⫹ 0.3c ⫹ 4.965
(1 ⫹ (1.56 / S2))2 ⫹ 0.677S2 1 ⫹ (0.33 / S2) ⫹ (0.14 / S4)
冊册
⫺1 ⫺1
(50)
1 ⫹ 0.3cfb(S) ⫹ 0.677S2 ⫹ 4.965
4 Eb ⬇c ⫹ ⫺ G 3
冋
1 Eb ⫽ G 3.471 ⫹ 2.228S2
The errors of the second set of simplified formulae are plotted in Fig. 16. The errors for shape factors between S = 1 and S = 100 are smaller than 4%. It can be seen that the second set is more accurate than the first set of simplified formulae.
(51)
冊
which, as shown in Fig. 14, is higher than the exact curve for larger c but is still accurate for smaller c. According to Eqs. (50) and (51), the second set of
Fig. 16. Errors of the second set of simplified formula for effective flexure modulus of square layers.
H.-C. Tsai, W.-J. Pai / Engineering Structures 25 (2003) 1443–1454
1453
7. Simplified formulae for infinite-strip layers The shape factor of a bonded infinite-strip layer which has a width of 2b and a thickness of t is defined as b t
S⫽
(52)
The infinite-strip shape is an idealized condition. Its theoretical stiffness formulae are not complicated. But for completeness, the approximated formulae derived by a simplified procedure similar to that mentioned above are described as follows. The effective compression modulus for the bonded layer of infinite-strip shape has the closed form [9] Fig. 17. Errors of simplified formula for effective compression modulus of infinite-strip layers.
2 (c⫺ )2 3 tanhy 4 Ec ⫽c⫹ ⫺ 4 G 3 y c⫹ 3
冉 冊
(53) where y has been defined in Eq. (54). The simplified formula for the effective flexure modulus of bonded infinite-strip layers is summarized as
where y is a nondimensional factor defined as y⫽
6S
(54)
冑3c ⫹ 4
The simplified formula for the effective compression modulus of bonded infinite-strip layers is summarized as (1) for Sⱖ⫺0.60 + 0.67√c
册
⫺1
冋 冉冊
1 Ec ⫽ G 4(1 ⫹ S2) 6 1 ⫹ 5 / (6S2) ⫹ 5 / (24S4) 5c (1 ⫹ (1 / S2))2
冋
册
10 1 Eb ⫹ ⫽ 2 G 4 ⫹ 0.8S 7c
⫺1
册
⫺1
2 (c⫺ )2 3 3 y 4 Eb ⫽c⫹ ⫺ ⫺1 2 4 y tanhy G 3 c⫹ 3
冊
冋 冉冊
1 Eb ⫽ G 4 ⫹ 0.8S2 ⫹
10 1 ⫹ 7 / (2S2) ⫹ 35 / (8S4) 7c (1 ⫹ (5 / S2))2
册
⫺1
The errors of the above simplified formula are plotted in Fig. 18. The maximum error of 5% occurs at S = 2.
The errors of the above simplified formula are plotted in Fig. 17. The maximum error of 8% occurs at S = 1. When Sⱖ5, the error of the simplified formula is less than 3%. The effective flexure modulus for the bonded layer of infinite-strip shape has the closed form [10]
冉 冊冉
册
(III) for S ⬍ ⫺1.05 + 1.07√c and S ⬍ √738⫺0.875c
(III) for S ⬍ ⫺0.60 + 0.67√c and S ⬍ √152⫺0.208c
⫹
2 3(c⫺ )2 3 ⫺ 4 4 S冑3c ⫹ 4 2 ⫹ (c ⫹ ) / (6S2) ⫹ (c ⫹ ) 3 3 (II) for S ⬍ ⫺1.05 + 1.07√c and S ⬍ √738⫺0.875c
(II) for S ⬍ ⫺0.60 + 0.67√c and Sⱖ√152⫺0.208c
冋
4 Eb ⫽c⫹ G 3
冋
2 (c⫺ )2 Ec 3 4 ⫽c⫹ ⫺ G 3 2S冑3c ⫹ 4
1 6 Ec ⫽ ⫹ 2 G 4(1 ⫹ S ) 5c
(I) for Sⱖ⫺1.05 + 1.07√c
(55)
8. Conclusion The simplified formulae for the effective compression modulus and effective flexure modulus of the circular, square and infinite-strip layers are derived. Every set of simplified formulae consists of three approximated equations. Each approximated equation has its applicable range of shape factor and bulk modulus. The divisions between the applicable ranges of different approximated
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layer without limitation on values of Poisson’s ratio and shape factor, the derived simplified formulae, which have low errors with respect to the theoretical solutions, can be applied to the layers of any elastic material and shape factor.
Acknowledgements The support from the National Science Council, Republic of China, under Grant No. NSC 89-2211-E011013 is greatly appreciated.
Fig. 18. Errors of simplified formula for effective flexure modulus of infinite-strip layers.
equations are defined so that the error of the simplified formulae can be minimized. The second and third approximated equations have a form similar to the ad hoc equation using the concept of serial spring, but the third one will be utilized only when the layer has low bulk modulus. For an elastic material such as rubber, the ratio of bulk modulus to shear modulus, c, has an order of 103. If we say c = 1600, only the effective flexure modulus of the square layer for the shape factor S ⬍ 12 will use the third approximated equation. Compared with the second approximated equation, the first approximated equation is applicable to higher shape factor or lower bulk modulus. For c = 1600, the effective compression modulus will use the first approximated equation when S ⬎ 17 for the circular layer, S ⬎ 20 for the square layer and S ⬎ 26 for the infinite-strip layer, while the effective flexure modulus will use the first approximated equation when S ⬎ 24 for the cicular layer, S ⬎ 18 for the square layer and S ⬎ 42 for the infinite-strip layer. Because the theoretical solutions, from which the simplified formulae are derived, are accurate for any elastic
References [1] Chalhaub MS, Kelly JM. Effect of bulk compressibility on the stiffness of cylindrical base isolation bearings. International Journal of Solids and Structures 1990;26:734–60. [2] Chalhoub MS, Kelly JM. Analysis of infinite-strip-shaped base isolator with elastomer bulk compression. Journal of Engineering Mechanics, ASCE 1991;117:1791–805. [3] Gent AN, Lindley PB. The compression of bonded rubber blocks. Proceeding of the Institution of Mechanical Engineers 1959;173:111–7. [4] Gent AN, Meinecke EA. Compression, bending and shear of bonded rubber blocks. Polymer Engineering and Sciences 1970;10:48–53. [5] Kelly JM. Earthquake-resistant design with rubber. London: Springer-Verlag, 1993. [6] Koh CG, Kelly JM. Effects of axial load on elastomeric isolation bearings. Report no. UCB/EERC-86/12, Earthquake Engineering Research Center, University of California, Berkeley, 1987. [7] Koh CG, Kelly JM. Compression stiffness of bonded square layers of nearly incompressible material. Engineering Structures 1989;11:9–15. [8] Koh CG, Lim HL. Analytical solution for compression stiffness of bonded rectangular layers. International Journal of Solids and Structures 2001;38:445–55. [9] Tsai H-C, Lee C-C. Compressive stiffness of elastic layers bonded between rigid plates. International Journal of Solids and Structures 1998;35:3053–69. [10] Tsai H-C, Lee C-C. Tilting stiffness of elastic layers bonded between rigid plates. International Journal of Solids and Structures 1999;36:2485–505.