Analytical mechanical model for laminated rubber bearings including crack effect

Abstract

The buckling phenomenon and deformation distribution of the laminated rubber bearing, taking into account the crack effect, are predicted by the period structure assumption and transfer matrix method (TMM). A concentrated flexural spring is utilized to simulate the local flexural stiffness deterioration that arises from the crack. The solutions of the individual rubber layer derived from Haringx’s column theory are formulated in the development of the crack model, and the inside steel shim is modeled as a rigid body whose transfer relation is then combined with that of the rubber layer. The resulting total transfer matrix, along with the boundary conditions, is applied to determine the horizontal stiffness and critical buckling load of the bearing with a single, or multiple cracks. Experimental data demonstrates the validity of the proposed analytical model. The results show that for a single crack, the least decrease in critical buckling load occurs when the crack is vertically centered along the height of the bearing; when the crack is closer to the top or bottom of the bearing, the critical buckling load may reduce by about half for a high level of deterioration. Two or multiple cracks make the critical buckling load decreased rapidly, especially for cracks in different rubber layers; nevertheless, not only the nearer these cracks approach the bearing center but also the much closer these cracks are together, the less the critical buckling load decreases.

Keywords

laminated rubber bearing transfer matrix method crack buckling periodic structure

Introduction

Base isolation, which often adopts multilayer elastomeric bearings, provides an effective and economic approach to mitigate the vibration transmitted to the superstructure. Elastomeric laminated rubber isolation bearings are arranged with rubber layers periodically bonded to rigid steel shims, therefore not only offering sufficient axial stiffness to sustain the weight of the system but also reserving necessary flexibility in the horizontal direction (Kelly and Takhirov, 2007; Tsai and Hsueh, 2001; Warn et al., 2007). The buckling analysis of the bearings under compressive force is well known, and has drawn the attention of many researchers. Stanton et al. (1990) as well as Imbimbo and Kelly (1997a, 1997b), for example, investigated the critical buckling load and buckling behavior of laminated rubber isolators according to the theory of Haringx’s equivalent continuous column (Haringx, 1949; Kelly, 1997), by considering the shear and flexural stiffness of the bearing. This theory has also been adopted, together with the stiffness matrix method, by Chang (2002) to predict the critical buckling force of the laminated rubber bearings for various geometric arrangements and boundary conditions. Koh and Kelly (1988, 1989) introduced a simple two-spring model to study the mechanical characteristics and stability behavior for the laminated rubber isolator. Further, Kikuchi et al. (2010) extended the Koh-Kelly model to three dimensions by adding a series of shear springs and axial springs. Mazda and Shiojiri (1993) used the transfer matrix method (TMM) to predict the buckling behavior of laminated rubber bearings. Utilizing TMM, Ellakany et al. (2004), Ellakany and Tablia (2010) investigated the static and free vibration of composite structures with a shear elastic interface. It was promising that TMM exhibited the advantages of simplicity and fewer iterations. The general problem of the combination of the rubber bearing and other elastomers was given by Wang et al. (2024) and Xu et al. (2023) through experimental study and finite element method. The refined numerical modeling for the rubber bearings considering the hysteretic strength degradation was developed and verified from experimental study performed by Zheng et al. (2022).

Cracks often occur in the bearings because of corrosion and aging of the rubber, rust or pitting of the steel shims, cavitation as well as other factors such as an earthquake (Yin et al., 2024). Chou and Huang (2011) predicted the crack initiation and propagation in rubber bearings, indicating that fatigue cracks initiated first at the outermost boundary between rubber and steel plates. Different damages including the cracks were collected and labeled for condition classification to improve the damage identification of rubber bearings (Cui et al., 2021). Zhang et al. (2022) treated the damage isolation bearing as a periodic structure to propose a damage detection scheme. The model assumed the change in shear modulus of the rubber layer (Ding et al., 2019) due to the ageing of the rubber material. Various identification frameworks to detect the deterioration of rubber bearings due to crack damage, rubber rupture and bearing void were proposed (Gao et al., 2023; Zeng et al., 2024; Zhang et al., 2024). Matsuzaki (2022) investigated the effects of capacity hierarchy and the ageing deterioration of rubber bearings on the predominant ultimate failure mode, and presented that the seismic safety could be improved by setting an adequate capacity hierarchy. However, to date, scarce work has been performed on the buckling problem of cracked laminated rubber isolators. Whether the buckling of the bearing displays high sensitivity to the local crack is undetermined, therefore it is quite valuable to take into account the influences of the cracks on the horizontal stiffness and the critical buckling load of the bearing.

The buckling analysis of the cracked laminated rubber isolators may be performed identical to that of a cracked composite column whereas governed by the low shear stiffness. Zapata-Medina et al. (2010) studied the buckling of a cracked uniform Timoshenko column considering the influence of shear deformation based on closed formulas. Arboleda-Monsalve et al. (2007) examined the buckling problem of cracked Timoshenko beam-columns with generalized boundary conditions, and presented the variation of critical buckling loads derived from different models such as the simple Bernoulli-Euler beam and the relatively more sophisticated Rayleigh, Shear as well as Timoshenko beam. Takahashi (1999) investigated the stability of a non-uniform cracked Timoshenko beam including a compressive load and a tangential follower force distributed over the center line. However, these buckling analyses have been confined to the Timoshenko column with only one single crack. The influence of multiple cracks on the buckling and stability of Timoshenko beam-columns are investigated by Fan and Zheng (2003) through modified Fourier series. Caddemi et al. (2013) have applied distribution theory to analyze the stability of the uniform Timoshenko column in appearance of multiple cracks under tensile or compressive forces. Li (2001, 2003) studied the decrease of the critical buckling force, due to an arbitrary number of cracks, of multi-step uniform columns with shear deformation and non-uniform columns subjected to compressive loads, by utilizing the transfer matrix approach. This approach allows for the critical buckling load to be determined conveniently through a determinant of order two.

In the present work, the influences of the deterioration level and number of the crack on the bearing buckling and deformation distribution are examined. The interest lies in determination, in terms of the crack location, of which section is the most unfavorable and what the critical buckling load variation results from a single, two or multiple cracks. The same combination of periodic structure assumption and the transfer matrix approach is applied as in the study of laminated rubber bearings without any damage (Ding et al., 2017). The closed-form solutions of the individual rubber layer behaving as a Haringx’s column proposed by Chang (2002) are formulated in the development of the crack model, and the inside steel shims, described using the rigid body assumption, are modeled to establish the system transfer matrix. The horizontal stiffness and critical buckling load of the laminated rubber bearing are assessed by analyzing the horizontal force and the lateral displacement at the upper surface of the isolator, derived from the system transfer matrix. The accuracy of the specific model is validated through recuperating the cracked model into the perfect model.

Theoretical derivation of cracked laminated rubber bearing

Assumptions

The typical periodic element of the laminated rubber isolator consists of a rubber layer and a steel shim, as depicted in Figure 1. The number of the rubber layers is $N + 1$ , and the number of the steel layers is $N$ , so the total number of the periodic elements

Figure 1.

Typical laminated rubber bearing. (Figure from Chang (2002)).

is $N$ . It is assumed that the cracks with negligible height owing to the compressive loads are only present in the rubber layers and also the number of cracks in the i-th rubber layer is $n_{i}$ . The cracks can be modeled as concentrated reductions of the flexural stiffness, accomplished by additional flexural springs of stiffnesses $k_{i j} (j = 1, 2, . . ., n_{i})$ to describe the discontinuity of rotation, as shown in Figure 2, located at the section points $x_{i 1}$ , $x_{i 2}$ , … , $x_{i n_{i}}$ such that $0 < x_{i 11} < x_{i 21} < . . . < x_{i n_{i} 1} < t_{r}$ , and $t_{r}$ is the height of one rubber layer. The $x$ -axis represents the central axis whose origin is located at the lower end of the i-th undeformed rubber layer $(i = 1, 2, . . ., N + 1)$ . It is observed from Figure 2 that the cracked rubber layer is separated into $n_{i} + 1$ segments through the $n_{i}$ cracks. $x_{i j 1}$ and $x_{i j 0}$ represent the upper and lower boundaries of the j-th rubber segment in the i-th rubber layer, respectively. Other notations below with the subscripts ij1 and ij0 denote the corresponding variables at $x_{i j 1}$ and $x_{i j 0}$ , respectively. The stiffness of each flexural spring varies from zero for completely flexural deteriorated section to infinity for nondeteriorated one.

Figure 2.

Cracked rubber layer: (a) the i-th rubber layer with n_i cracks; (b) a flexural spring to simulate a crack.

Governing equations of one rubber layer with cracks

Inspired by Chang’s methodology (2002), herein the Haringx’ theory is applied to develop the model of the first rubber segment in the i-th rubber layer for $0 < x < x_{i 1}$ . It is assumed that the rubber segment is a continuous column, and in addition the plane section is perpendicular to the undeformed axis and always plane but not

perpendicular to the deformed axis after deformation, such that the corresponding deformed configuration of the first rubber segment is shown in Figure 3(a). The deflection can therefore be defined in terms of lateral displacement of the central line, $v (x)$ , and the rotation of the plane with respect to the undeformed axis, $φ (x)$ , as displayed in Figure 3(b). The deformed cross section produces two internal forces: the shear force $V (x)$ and the bending moment $M (x)$ . Then the force quantities can be related to the deflection variables so as to obtain the constitutive equations through

M (x) = E I \frac{\partial φ (x)}{\partial x}

(1a)

V (x) = G A [\frac{\partial v (x)}{\partial x} - φ (x)]

(1b)

Here,

E I

is the effective flexural rigidity, while

G

and

A

are the shear modulus and the total shear area, respectively.

Figure 3.

Loads and deformation of the first rubber segment in the i-th rubber layer: (a) full configuration; (b) cross section. (Figure adapted from Chang (2002)).

Herein only small rotation $φ (x)$ is studied for a heavy structure with a low center of gravity, and therefore application of equation (1) into the shear and bending moment equations of the deformed configuration displayed in Figure 3(b), yields the following equations

E I \frac{\partial φ (x)}{\partial x} + P [v (x) - v_{i 10}] + M_{i 10} - V_{i 10} x = 0

(2a)

G A [\frac{\partial v (x)}{\partial x} - φ (x)] - P φ (x) + V_{i 10} = 0

(2b)

where

P

V_{i 10}

and

M_{i 10}

are the axial force, the horizontal force and the moment at the lowest boundary

x_{i 10}

of the first rubber segment in the i-th rubber layer, respectively; and

v_{i 10}

denotes the lateral displacement at

x_{i 10}

After some manipulation, $\partial v (x) / \partial x$ can be written from equation (2b) as

\frac{\partial v (x)}{\partial x} = \frac{P + G A}{G A} φ (x) - \frac{V_{i 10}}{G A}

(3)

Substituting $\partial φ (x) / \partial x$ , obtained directly by differentiating the function $φ (x)$ with respect to $x$ through equation (3), into equation (2a) and also taking into account the relationship between equation (3) and the partial derivative of equation (2a) lead to a pair of differential equations

\frac{\partial^{2} v (x)}{\partial x^{2}} + α^{2} v (x) = α^{2} v_{i 10} - \frac{α^{2} M_{i 10}}{P} + \frac{α^{2} V_{i 10} x}{P}

(4a)

\frac{\partial^{2} φ (x)}{\partial x^{2}} + α^{2} φ (x) = \frac{α^{2} V_{i 10}}{P}

(4b)

where

α

is a parameter given by

α^{2} = \frac{P (P + G A)}{E I G A}

(5)

By applying the relationship between $v (x)$ and $φ (x)$ , the general expression of the solutions to the above differential equations can be found in the form

v (x) = A \cos (α x) + B \sin (α x) + v_{i 10} - \frac{M_{i 10}}{P} + \frac{V_{i 10}}{P} x

(6a)

φ (x) = α β B \cos (α x) - α β A \sin (α x) + \frac{V_{i 10}}{P}

(6b)

in which A and B are constants to be derived according to the boundary conditions and

β

is denoted by

β = \frac{G A}{P + G A}

(7)

The bending moment and shear force in the cross section are hence rewritten in terms of equations (1), (2), and (6) as follows:

M (x) = - P A \cos (α x) - P B \sin (α x)

(8a)

V (x) = P α β B \cos (α x) - P α β A \sin (α x)

(8b)

As shown in Figure 3, the deformation of the lower point $x_{i 10}$ is described by a lateral displacement $v_{i 10}$ and rotation $φ_{i 10}$ . Therefore, the boundary conditions are given by

v (x_{i 10}) = v_{i 10}

(9a)

φ (x_{i 10}) = φ_{i 10}

(9b)

Substituting equation (6) into equation (9) gives the solutions $A$ and $B$ as follows

A = \frac{M_{i 10}}{P}

(10a)

B = \frac{P φ_{i 10} - V_{i 10}}{P α β}

(10b)

Accordingly, inserting equation (10) into equations (6) and (8), yields the deflection and the internal force

v (x) = v_{i 10} + \frac{\sin (α x)}{α β} φ_{i 10} + [\frac{x}{P} - \frac{\sin (α x)}{P α β}] V_{i 10} + [\frac{\cos (α x)}{P} - \frac{1}{P}] M_{i 10}

(11a)

φ (x) = \cos (α x) φ_{i 10} + [\frac{1}{P} - \frac{\cos (α x)}{P}] V_{i 10} - \frac{α β \sin (α x)}{P} M_{i 10}

(11b)

V (x) = P \cos (α x) φ_{i 10} - \cos (α x) V_{i 10} - α β \sin (α x) M_{i 10}

(11c)

M (x) = - \frac{P \sin (α x)}{α β} φ_{i 10} + \frac{\sin (α x)}{α β} V_{i 10} - \cos (α x) M_{i 10}

(11d)

Then, equations (11a)-(11d) are combined to obtain the matrix expression:

Y_{i 1} (x) = T_{i 1} (x) Y_{i 10}

(12)

where

Y_{i 1} (x)

and

Y_{i 10}

represent the state vectors of the position

x

and lower boundary of the rubber segment, respectively, which are defined as

Y_{i 1} (x) = \{\begin{array}{l} v (x) \\ φ (x) \\ V (x) \\ M (x) \end{array}\}

(13a)

Y_{i 10} = {\begin{cases} v_{i 10} \\ φ_{i 10} \\ V_{i 10} \\ M_{i 10} \end{cases}}

(13b)

T_{i 1} (x)

T_{i 1} (x) = [\begin{array}{l} 1 & \frac{\sin (α x)}{α β} & \frac{x}{P} - \frac{\sin (α x)}{P α β} & \frac{\cos (α x)}{P} - \frac{1}{P} \\ 0 & \cos (α x) & \frac{1}{P} - \frac{\cos (α x)}{P} & \frac{- α β \sin (α x)}{P} \\ 0 & P \cos (α x) & - \cos (α x) & - α β \sin (α x) \\ 0 & - \frac{P \sin (α x)}{α β} & \frac{\sin (α x)}{α β} & - \cos (α x) \end{array}]

(14)

In particular, while

x = x_{i 11}

, the state vector

Y_{i 1} (x_{i 11})

at the upper boundary of the first rubber segment for the i-th rubber layer, can be expressed in the form

Y_{i 1} (x_{i 11}) = \{\begin{array}{l} v_{i 11} \\ φ_{i 11} \\ V_{i 11} \\ M_{i 11} \end{array}\} = Y_{i 11}

(15)

and

T_{i 1} (x) = T_{i 1} (x_{i 11}) = T_{i 11}

therefore indicates the connections between the deflections and forces of the upper and lower boundaries for the first rubber segment.

As shown in Figure 2, at the interface of the first and second rubber segments, the displacement, shear force and bending moment are needed to be continuous and the rotation has a jump, which provides the following compatibility and equilibrium conditions at the cracked section

{\begin{cases} v (x_{i 20}) \\ φ (x_{i 20}) \\ V (x_{i 20}) \\ M (x_{i 20}) \end{cases}} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & k_{i 1}^{- 1} \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}] {\begin{cases} v (x_{i 11}) \\ φ (x_{i 11}) \\ V (x_{i 11}) \\ M (x_{i 11}) \end{cases}}

(16)

Making use of equations (16) and (12) leads to

{\begin{cases} v (x_{i 20}) \\ φ (x_{i 20}) \\ V (x_{i 20}) \\ M (x_{i 20}) \end{cases}} = T_{i 1 K} {\begin{cases} v (x_{i 10}) \\ φ (x_{i 10}) \\ V (x_{i 10}) \\ M (x_{i 10}) \end{cases}}

(17)

in which

T_{i 1 K} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & k_{i 1}^{- 1} \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}] T_{i 11}

(18)

The equation of the ( $n_{i} + 1$ )-th segment for the i-th rubber layer can be generated by assembling equations (12) and (17) repeatedly to get the following form:

Y_{i, n_{i} + 1, 1} = \{\begin{array}{l} v (x_{i, n_{i} + 1, 1}) \\ φ (x_{i, n_{i} + 1, 1}) \\ V (x_{i, n_{i} + 1, 1}) \\ M (x_{i, n_{i} + 1, 1}) \end{array}\} = T_{i} \{\begin{array}{l} v (x_{i 10}) \\ φ (x_{i 10}) \\ V (x_{i 10}) \\ M (x_{i 10}) \end{array}\} = T_{i} Y_{i 10} (i = 1, 2, . . ., N + 1)

(19)

where

T_{i} = T_{i, n_{i} + 1, 1} T_{i, n_{i}, K} . . . T_{i j K} . . . T_{i 2 K} T_{i 1 K} (i = 1, 2, . . ., N + 1; j = 1, 2, . . ., n_{i})

(20a)

T_{ij K} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & k_{i j}^{- 1} \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}] T_{ij 1}

(20b)

T_{i j 1} = T_{i 1} (x_{i j 1})

(20c)

A flow diagram explaining the details of the TMM for cracks in one rubber layer is also shown in Figure 4. Especially, if a crack is located at the upper end of the i-th rubber layer, $T_{i}$ should be replaced by

T_{i}^{t} = [\begin{array}{l} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & k_{i t}^{- 1} \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & - 1 \end{array}] T_{i}

(21)

Figure 4.

Flow diagram of the TMM for cracks in the i-th rubber layer.

The equation for the crack at the lower end or both ends of the i-th rubber layer can be obtained similarly.

Transfer matrix for a cracked periodic element

As described above, the i-th periodic element shown in Figure 5 is provided with the i-th rubber layer and the i-th steel shim, and it is assumed that the rubber layer contributes completely to the deformations of the isolator, as opposed to the steel shim, that is considered to exhibit only the motion form of a rigid body (Chang, 2002). In this manner, the deflections and forces at the upper surface of the i-th rigid steel shim shown in Figure 5 are given by

{\bar{v}}_{i, n_{i} + 1, 1} = v_{i, n_{i} + 1, 1} + t_{s} φ_{i, n_{i} + 1, 1}

(22a)

{\bar{φ}}_{i, n_{i} + 1, 1} = φ_{i, n_{i} + 1, 1}

(22b)

{\bar{V}}_{i, n_{i} + 1, 1} = V_{i, n_{i} + 1, 1}

(22c)

{\bar{M}}_{i, n_{i} + 1, 1} = - V_{i, n_{i} + 1, 1} t_{s} + M_{i, n_{i} + 1, 1} - P t_{s} φ_{i, n_{i} + 1, 1}

(22d)

Figure 5.

Load and deflection notation: (a) a cracked rubber layer; (b) a cracked rubber layer bonded with a steel shim. (Figure adapted from Chang (2002)).

where $t_{s}$ denotes the height of the steel shim. Then the state vector ${\bar{Y}}_{i, n_{i} + 1, 1}$ at the upper surface of the i-th steel shim would be described in terms of the state vector $Y_{i, n_{i} + 1, 1}$ at the upper surface of the i-th rubber layer as the following matrix form:

{\bar{Y}}_{i, n_{i} + 1, 1} = T_{S} (t_{s}) Y_{i, n_{i} + 1, 1}

(23)

in which

{\bar{Y}}_{i, n_{i} + 1, 1} = {\begin{cases} {\bar{v}}_{i, n_{i} + 1, 1} \\ {\bar{φ}}_{i, n_{i} + 1, 1} \\ {\bar{V}}_{i, n_{i} + 1, 1} \\ {\bar{M}}_{i, n_{i} + 1, 1} \end{cases}}

(24a)

T_{S} (t_{s}) = [\begin{array}{l} 1 & t_{s} & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & - P t_{s} & - t_{s} & 1 \end{array}]

(24b)

Combining equations (19) and (23), the state vector at the upper surface of the steel shim can be connected with that at the lower end of the i-th rubber layer through

{\bar{Y}}_{i, n_{i} + 1, 1} = T^{'} {\bar{Y}}_{i 10}

(25)

where

T^{'} = T_{S} (t_{s}) T_{i}

(26a)

and

{\bar{Y}}_{i 10} = {\begin{cases} {\bar{v}}_{i 10} \\ {\bar{φ}}_{i 10} \\ {\bar{V}}_{i 10} \\ {\bar{M}}_{i 10} \end{cases}}

(26b)

which represents the state vector at the lower end of the i-th rubber layer and provides that

{\bar{Y}}_{i 10} = Y_{i 10}

Taking into account the compatibility as well as the equilibrium conditions at the interface between the i-th and the i+1-th periodic elements lead to:

{\bar{Y}}_{i, n_{i} + 1, 1} = J {\bar{Y}}_{i + 1, 10}

(27)

where

J = [\begin{array}{l} I & 0 \\ 0 & - I \end{array}]

(28)

and

I

defines a two by two identity matrix.

Using equations (27) and (25) in order to eliminate ${\bar{Y}}_{i, n_{i} + 1, 1}$ yields

{\bar{Y}}_{i + 1, 10} = J^{- 1} T_{S} (t_{s}) T_{i} {\bar{Y}}_{i 10} = T_{T i} {\bar{Y}}_{i 10}

(29)

where

T_{T i} = J^{- 1} T_{S} (t_{s}) T_{i}

describes the transfer matrix of the i-th periodic element. It is interesting to notice that the transfer matrix determines the relationship between the state vector at the lower end of the rubber layer in the (i+1)-th element and that in the i-th element.

A typical laminated rubber isolator, displayed in Figure 1, generally includes N periodic elements and an extra rubber layer, as well as one upper and lower rigid end steel plates, such that the system transfer matrix can be conveniently written in the form

T_{T} = T_{S} (t_{s 0}) T_{N + 1} T_{T N} . . . T_{T i} . . . T_{T 2} T_{T 1} T_{S} (t_{s 0})

(30)

where

t_{s 0}

defines the height of each end steel plate.

Application of the system transfer matrix $T_{T}$ yields the following formulation of the considered entire laminated rubber bearing

{\bar{Y}}_{N + 1, n_{N + 1} + 1, 1} = T_{T} {\bar{Y}}_{110}

(31a)

{\begin{cases} {\bar{v}}_{N + 1, n_{N + 1} + 1, 1} \\ {\bar{φ}}_{N + 1, n_{N + 1} + 1, 1} \\ {\bar{V}}_{N + 1, n_{N + 1} + 1, 1} \\ {\bar{M}}_{N + 1, n_{N + 1} + 1, 1} \end{cases}} = [\begin{array}{l} T_{T 11} & T_{T 12} & T_{T 13} & T_{T 14} \\ T_{T 21} & T_{T 22} & T_{T 23} & T_{T 24} \\ T_{T 31} & T_{T 32} & T_{T 33} & T_{T 34} \\ T_{T 41} & T_{T 42} & T_{T 43} & T_{T 44} \end{array}] {\begin{cases} {\bar{v}}_{110} \\ {\bar{φ}}_{110} \\ {\bar{V}}_{110} \\ {\bar{M}}_{110} \end{cases}}

(31b)

where

{\bar{Y}}_{N + 1, n_{N + 1} + 1, 1}

represents the state vector at the upper surface of the cracked isolator.

Solution of cracked laminated rubber bearings

In most situations, the laminated rubber isolator is restrained against any displacement and cross section rotation at the bottom end, while it is permitted to move in the horizontal direction but yet constrained from rotation at the top end. An axial compressive load $P$ and a horizontal force $V$ are applied at the upper surface. Accordingly, imposing the set boundary conditions into equation (31b) can get the unknown quantity of displacements and forces at both ends of the bearing as follows

{\bar{M}}_{110} = - {T_{T 24}}^{- 1} T_{T 23} {\bar{V}}_{110}

(32a)

{\bar{v}}_{N + 1, n_{N + 1} + 1, 1} = T_{T 13} {\bar{V}}_{110} + T_{T 14} {\bar{M}}_{110}

(32b)

{\bar{V}}_{N + 1, n_{N + 1} + 1, 1} = T_{T 33} {\bar{V}}_{110} + T_{T 34} {\bar{M}}_{110}

(32c)

{\bar{M}}_{N + 1, n_{N + 1} + 1, 1} = T_{T 43} {\bar{V}}_{110} + T_{T 44} {\bar{M}}_{110}

(32d)

So far the state vectors at the both ends, which can be used to assess the mechanical behaviour of the whole cracked laminated rubber isolation bearing, have been determined. In other words, the horizontal stiffness can be easily obtained from the ratio of the applied horizontal force to the lateral displacement at the upper surface of the isolation bearing, by using equation (32). For the case of a high center of gravity, large-angle deformations may occur, hence the application of this derivation may be invalid.

Numerical examples

The typical circular laminated rubber isolation bearing is considered in the following numerical analysis. The number of rubber layers is 20 with $G = 0.611$ MPa and $t_{r} = 10$ mm, while the number of steel shims is 19 with $t_{s} = 2$ mm. Each upper and lower end steel plate is characterized by the thickness $t_{s 0} = 21$ mm. The isolation bearing has an overall diameter of 300 mm, which mainly consists of the diameter $D$ , 280 mm of the rigid steel shim, as well as a supplemental protective rubber cover, 10 mm.

For the purpose of highlighting the contribution of the rubber protective cover to the shear rigidity of the analysis composite model, the overall shear area $A$ of the isolation bearing can be therefore modified as (Chang, 2002; Ding et al., 2017)

A = A_{1} + A_{2} \frac{t_{r}}{t_{r} + t_{s}}

(33)

where

A_{1}

denotes the area of the steel shim while

A_{2}

indicates the cross sectional area of the cover rubber.

Due to the single circular rubber layer reinforced with two rigid steel shims, the flexural rigidity is represented according to Gent and Meinecke’s theory (1970) as follows:

E I = E_{Y} I (1 + \frac{2}{3} S^{2})

(34)

where

E_{Y}

denotes the effective modulus. Assuming that the rubber material herein has the feature of incompressibility, it follows that

E_{Y} = 3 G

I = π D^{4} / 64

is the moment of inertia of the cross section, while

S = D / 4 t_{r}

is the shape factor of the circular isolation bearing.

A single crack

The performance of a bearing with only a single crack is first investigated both in terms of horizontal stiffness and critical buckling load and in terms of the deformation. The variations of the horizontal stiffness with various compressive forces for a single crack in the entire bearing are plotted in Figure 6, where different crack locations are considered and each crack is located at the center of each rubber layer, namely $x_{i 1} / t_{r} = 1 / 2$ . Note that $R_{i 1} = k_{i 1} / (E I / h_{r})$ defines the dimensionless flexural stiffness of the crack section. Figure 6(a) is obtained by setting $R_{i 1} = 0.01$ ; hence it represents the limit behaviour of the bearing for fully deteriorated section corresponding to $R_{i 1} \to 0$ . Figure 6(d) is obtained by considering $R_{i 1} = 1000$ ; hence it can be considered representative of the limit behaviour of the bearing for non-deteriorated section corresponding to $R_{i 1} \to \infty$ . The curves in Figure 6(b) and (c) are related to different values of $R_{i 1}$ , which represent different crack deterioration levels. It can be seen from Figure 6 that regardless of the crack location and the deterioration level, the horizontal stiffness is always decreased as the compressive force is increased. When the horizontal stiffness is equal to zero, the bearing is prone to instability and accordingly the corresponding compressive load is referred to as the critical buckling load $p_{c r}$ . Furthermore, it clearly shows that the horizontal stiffness and critical buckling load are more sensitive to the crack location if the flexural stiffness is small enough. In particular, when $R_{i 1} = 0.01$ , the critical load varies from $p_{c r} = 9.44$ for $i = 10$ or $i = 11$ to $p_{c r} = 4.80$ for $i = 1$ or $i = 20$ , which indicates a 49% reduction in the critical load capacity. It is important to take into account the reduction of the critical buckling load due to the influence of the crack locations.

Figure 6.

Variation of horizontal stiffness with compression force for a single crack and different crack locations: (a) $R_{i 1} = 0.01$ ; (b) $R_{i 1} = 1$ ; (c) $R_{i 1} = 5$ ; (d) $R_{i 1} = 1000$ .

Comparing the four graphs in Figure 6, one can find that the horizontal stiffness and critical load display little sensitivity to the section deterioration level for $i = 10$ or $i = 11$ . In other words, the horizontal stiffness and critical load are slightly changed in this case even if the section is fully deteriorated. However, if the crack section is at the top or bottom rubber layer ( $i = 1$ or $i = 20$ ), a drastic decrease of the horizontal stiffness and critical load is observed as $R_{i 1}$ decreases. Physically, the bearing exhibits the clamped-clamped with sidesway uninhibited boundary conditions, leading to an inflection point at the center of the bearing with non-deteriorated section, when tending to buckle. The inflection point acts as one hinge, where the crack effect disappears. Whereas, if the crack approaches the top or bottom section, the bearing converges to the pinned-clamped with sidesway uninhibited boundary conditions, providing the more weakened constraint to become a more unstable mechanism.

A comparison between the present result for the particular case $R_{i 1} = 1000$ and the experimental data measured by Tsai and Hsueh (2001) is displayed in Figure 7. In the experiment, a pair of identical bearings as a series system were adopted with the geometrical and material properties similar to the numerical study described just before Section 3.1. The lower end of the lower bearing was fixed and the upper end of the upper bearing was connected to a vertical actuator, applied a constant compression force. The horizontal actuator, connected with the specimens at the junction of the two bearing, moved sinusoidally by giving displacement amplitude and frequency to a controller. The application of the measured horizontal force and displacement history gives the horizontal stiffness under a specified compression force. The figure demonstrates that the numerical prediction is in good agreement with the experimental result, validating the accuracy of the proposed model, due to that the bearing model with multiple cracks is formulated no matter what the crack position or number, while the numerical result is established based on the cracked model, by setting the crack flexural stiffness $R_{i 1} = 1000$ .

Figure 7.

Comparison with numerical and experimental results.

The influences of the crack location and deterioration level on the critical buckling load are analyzed in an explicit and comprehensive manner in Figure 8, which again highlights that under the same deterioration level, a high value of the critical buckling load is generated if the crack position appears at the center of the bearing and substantially decreases as the crack section shifts toward the upper or lower end of the isolation bearing.

Figure 8.

Influence of crack location and deterioration level on the critical buckling load for a single crack at $x_{i 1} / t_{r} = 1 / 2$ .

When $P = 4 G A$ and $V = 0.2 G A$ , the variations of the internal deformations, that is, the lateral displacement $v (h_{x} / h) = \bar{v} / h$ , and the rotation $φ (h_{x} / h) = \bar{φ} \times 180 / π$ , over the bearing height $h_{x} / h$ are displayed in Figures 9 and 10 for each crack at $x_{i 1} / t_{r} = 1 / 2$ . These figures reveal that the higher the section deteriorates, the more the deformations increase. Even though the horizontal stiffness and critical load are identical, as discussed above, if the crack sections are at the symmetric locations of the bearing, the internal deformations of the bearing are not exactly the same for the same deterioration level. This is particularly evident for $R_{i 1} = 0.01$ , $i = 1$ and $R_{i 1} = 0.01$ , $i = 20$ , with one lateral displacement curve upwards concave and the other upwards convex.

Figure 9.

Distribution of lateral displacement $v (h_{x} / h)$ over bearing height for different crack locations: (a), (b) $R_{i 1} = 0.01$ ; (c), (d) $R_{i 1} = 1$ ; (e) , (f) $R_{i 1} = 5$ ; (g), (h) $R_{i 1} = 1000$ .

Figure 10.

Distribution of rotation $φ (h_{x} / h)$ over bearing height for different crack locations: (a), (b) $R_{i 1} = 0.01$ ; (c), (d) $R_{i 1} = 1$ ; (e), (f) $R_{i 1} = 5$ ; (g), (h) $R_{i 1} = 1000$ .

A single crack in different locations ( $x_{i 1} / t_{r} = 9 / 10$ , $x_{i 1} / t_{r} = 1 / 2$ , i.e. $x_{i 1} / t_{r} = 5 / 10$ and $x_{i 1} / t_{r} = 1 / 10$ ) of a rubber layer is also considered. The horizontal stiffnesses obtained for the different crack locations at the rubber layer $i = 10$ and $i = 1$ are displayed in Figure 11 for $R_{i 1} = 0.01$ and $R_{i 1} = 1$ . In the plots, even though the crack occurs in one rubber layer, the horizontal stiffness and critical buckling load are higher but more insensitive as the crack is nearer to the center of the bearing. To illustrate this behavior, it is shown by comparing Figure 11(a) with (c) that when $i = 10$ and $x_{10, 1} = 1 / 10$ , the critical load varies from $p_{c r} = 9.66$ for $R_{i 1} = 1$ to $p_{c r} = 9.01$ for $R_{i 1} = 0.01$ , leading to a 7.2% reduction in the critical load capacity; while when $i = 10$ and $x_{10, 1} = 9 / 10$ , the critical load varies from $p_{c r} = 9.84$ for $R_{i 1} = 1$ to $p_{c r} = 9.75$ for $R_{i 1} = 0.01$ , leading to a 0.9% reduction in the critical load capacity. In addition, for $i = 10$ and $R_{i 1} = 0.01$ , the critical load decreases by 8.2%, varying from $p_{c r} = 9.75$ for $x_{10, 1} = 9 / 10$ to $p_{c r} = 9.01$ for $x_{10, 1} = 1 / 10$ . Hence, different crack locations at one rubber layer should also be focused on especially for a large deteriorated level.

Figure 11.

Horizontal stiffnesses for varying crack location in a rubber layer for a single crack: (a) $i = 10$ , $R_{i 1} = 0.01$ ; (b) $i = 1$ , $R_{i 1} = 0.01$ ; (c) $i = 10$ , $R_{i 1} = 1$ ; (d) $i = 1$ , $R_{i 1} = 1$ .

Two cracks

The analysis is performed for two cracks in different rubber layers and within a rubber layer. The purpose is to analyze the effects of two cracks on the horizontal stiffness, critical buckling load and deformation and compare the results of bearings with a single and two cracks.

Two cracks in different rubber layers

The horizontal stiffnesses and critical buckling loads are computed for the bearings with two crack sections in different rubber layers but in the middle of each rubber layer (

x_{i, 1} / t_{r} = 1 / 2

), listed in Table 1. In Table 1, each “×” or each “√” denotes a crack and two “×” or two “√” in a column represent a configuration. In order to highlight the unfavorable crack locations, two sets of crack locations are analyzed. One set is two symmetrical cracks centralized in the 10th and 11th rubber layers and decentralized in the 1st and 20th rubber layers; the other set is two general cracks located in the 14th and 17th rubber layers as well as the 13th and 18th rubber layers. Of the second set, the 14th rubber layer is selected by shifting the 13th rubber layer off the bearing center, while the 17th rubber layer is selected by shifting the 18th rubber layer toward the bearing center. Over all, the configuration of cracks in the 11th and 20th rubber layers is compared with the first and second sets and the configuration of cracks in the 8th and 18th rubber layers is compared with the second set. In this paper, the same deterioration level for every crack in a configuration is considered. For example, for the configuration of cracks occurred in the 10th and 11th rubber layer,

R_{10, 1} = R_{11, 1}

Table 1.

Configurations for two cracks in two different rubber layers.

The horizontal stiffnesses obtained for the six configurations with two different deterioration levels are displayed in Figure 12. For the same compressive force, the horizontal stiffness for the bearing with two centralized cracks in the 10th and 11th rubber layers is higher than that for the bearing with one crack in the 11th rubber layer and the other in the 20th rubber layer. Of the first set and the configuration of cracks in the 11th and 20th rubber layers, the horizontal stiffness for the bearing with the two decentralized cracks in the 1st and 20th rubber layers is the lowest. For the second set, the horizontal stiffness for the bearing with cracks in the 14th and 17th rubber layers is higher than in the 13th and 18th rubber layers. It is interesting to find that the horizontal stiffness for the bearing with cracks in the 13th and 18th rubber layers is then higher than in the 11th and 20th rubber layers, which again shows that the decentralized cracks make the horizontal stiffness low. Even though the 8th and 13th rubber layers are symmetrical in the bearing, the horizontal stiffnesses of the configurations of cracks in the 13th and 18th rubber layers and the 8th and 18th are not the same, and the latter is lower than the former. Therefore, it can be concluded from these observations that not only the nearer the two cracks approach the bearing center but also the much closer the two cracks are together, making the equivalent compressive length shorter, the higher the horizontal stiffnesses are.

Figure 12.

Variation of horizontal stiffness for two cracks in different rubber layers: (a) $R_{i 1} = 0.01$ ; (b) $R_{i 1} = 1$ .

The critical buckling loads obtained for the bearing with two crack sections in two different rubber layers for the same deterioration level are displayed in Figure 13. Notice that the top curve is obtained by taking the two cracks respectively centralized in the 10th and 11th rubber layers. As the crack locations move apart, the critical buckling load decreases. The lowest curve corresponds to the configuration by taking the two cracks respectively decentralized in the 1st and 20th rubber layers. Therefore, the two curves can be considered representative of the limit critical buckling load of the bearing with two cracks in different rubber layers. Moreover, the critical buckling load approaches zero (0.19 when $R_{1, 1} = R_{20, 1} = 0.01$ ) for high levels of deterioration if the cracks simultaneously appear at the top and bottom rubber layers (the 1st and 20th rubber layers). It is important to note that even when the two crack sections are located at the middle (the 10th and 11th rubber layers) of the bearing, the critical buckling load is rapidly reduced to reach a low value equal to only about 1/6 times the maximum critical buckling load value as $R_{10, 1}$ and $R_{11, 1}$ decrease from 1 to 0.01. Finally, comparison between Figures 8 and 13 shows that when $R_{i 1} = 0.01$ , the critical load varies from $p_{c r} = 9.44$ for a single crack in the $i = 10$ or $i = 11$ rubber layer to $p_{c r} = 1.58$ for two cracks in the $i = 10$ and $i = 11$ rubber layers, and the critical load varies from $p_{c r} = 4.80$ for a single crack in the $i = 1$ or $i = 20$ rubber layer to $p_{c r} = 0.19$ for two cracks in the $i = 1$ and $i = 20$ rubber layers, indicating a significant reduction in the critical load capacity. The two cracks render the buckling mechanism of the bearing much more complex than for a single crack. Therefore much attention should be paid to the buckling problem if two cracks appear in the bearing, especially for two decentralized cracks.

Figure 13.

Influence of crack location and deterioration level on the critical buckling load for two cracks in different rubber layers.

The internal deformation profiles for two crack sections with $R_{i 1} = 0.1$ over the bearing height are shown in Figure 14, where $P = 0.01 G A$ and $V = 0.2 G A$ . Note that a very small compressive force is chosen here because under these cases the smallest critical buckling load is only $0.19 G A$ , as displayed in Figure 13. Comparing with the deformation of the bearing with non-deteriorated section (Figures 9(g) and (h), and 10(g) and (h)), significant differences in the displacement and rotation are noticed when the crack sections are decentralized at the top and bottom of the rubber layers. It is necessary to consider the crack effect for effectively predicting the maximum deformation of the practical damaged bearings.

Figure 14.

Distribution of deformation over bearing height for different crack locations: (a), (b) lateral displacement $v (h_{x} / h)$ ; (c), (d) rotation $φ (h_{x} / h)$ .

Two cracks in one rubber layer

The horizontal stiffness for the bearing with two crack sections within one rubber layer for $R_{i, 1} = R_{i, 2} = 1$ is shown in Figure 15. In order to avoid confusion, the symmetrical locations of the cracks are not listed. The displayed examples emphasize that regardless of the cracked rubber layer, the horizontal stiffness becomes always high when the two cracks move toward the bearing center simultaneously. Compared with Figures 6(b) and 11(c) and (d), adding a crack again to the cracked rubber layer generally causes the decrease of the horizontal stiffness and critical buckling load.

Figure 15.

Variation of horizontal stiffness with compression force for two cracks in one rubber layer: (a) $i = 10$ ; (b) $i = 8$ ; (c) $i = 3$ ; (d) $i = 1$ .

Take $i = 1$ for example, the critical buckling load is 6.38 and 6.28 respectively for a single crack at $x_{1, 1} = 9 / 10$ and $x_{1, 1} = 5 / 10$ , while it becomes 5.73 for both the cracks at the same location in the 1st rubber layer. The critical buckling loads are demonstrated in Figure 16. For a large deteriorated level, the effect of two crack locations in one rubber layer is more obvious. Comparison with Figure 8 further highlights the decrease of the critical buckling load after adding a crack, especially for a large deteriorated level.

Figure 16.

Influence of crack location and deterioration level on the critical buckling load for two cracks in one rubber layer.

Multiple cracks

The influence of three cracks in different locations on the horizontal stiffness and critical buckling load of the bearing is evaluated. A comparison is carried out among the performance of bearings with only a single crack, two cracks and three cracks.

Three cracks in three different rubber layers

Examples of three cracks in the middle of three different rubber layers are listed in Table 2. As in Table 1, each “×” or each “√” denotes a crack and three “×” or three “√” in a column represent a configuration (similarly hereinafter). The horizontal stiffness and critical buckling load of the bearings for these configurations are plotted in Figures 17 and 18. Both the figures demonstrate that the horizontal stiffness and critical buckling load increases as the three cracks move toward the bearing center simultaneously. Also, the much closer the three cracks approach together, the larger the horizontal stiffness and critical buckling load are. In other words, if two cracks are kept the same, adding another crack that approaches the bearing center nearer and also makes all the cracks much closer, gives rise to a larger horizontal stiffness and critical buckling load. These patterns are very similar to that exhibited in the two-crack configurations.

Table 2.

Configurations for three cracks in three different rubber layers.

Figure 17.

Variation of horizontal stiffness for three cracks in three different rubber layers for the same deterioration level $R_{i 1} = 1$ .

Figure 18.

Influence of crack location and deterioration level on the critical buckling load for three cracks in three different rubber layers.

For comparison, the critical buckling loads of configurations for a single, two and three cracks at the middle of each rubber layer are listed in Table 3. It further highlights that a bearing with a, two or multiple cracks much closer to the bearing center exhibits large critical buckling load for the same deterioration level. Moreover, a combination of centralized cracks gives larger critical buckling load than decentralized cracks. The critical buckling load decreases from 9.68 for two centralized cracks in the 10th and 11th rubber layers to 9.08 for three centralized cracks in the 10th, 11th and 12th rubber layers, even greater than that for two decentralized cracks in the 1st and 10th rubber layer and also for a single crack in the 1st rubber layer.

Table 3.

Comparison of configurations for three cracks in three rubber layers and configurations for a single and two cracks.

Cracked rubber layer	10	10, 11	10, 11, 12	10, 11, 20	1	1, 10	1, 10, 11	1, 10, 20
Critical buckling load for $R_{i 1} = 1$	9.78	9.68	9.08	4.85	6.27	5.37	4.85	3.86

Three cracks in two rubber layers

To illustrate the behavior of bearings with three cracks in two rubber layers listed in Table 4, the horizontal stiffness and critical buckling load are shown in Figures 19 and 20, where the symmetrical locations of the cracks are not marked. The horizontal stiffness and critical buckling load display the similar variation trend to that demonstrated in the configurations for three cracks in three rubber layers as the crack locations move far from the bearing center, as expected.

Table 4.

Configurations for three cracks in two rubber layers.

Figure 19.

Variation of horizontal stiffness for three cracks in two rubber layers for the same deterioration level $R_{i 1} = R_{i 2} = 1$ .

Figure 20.

Influence of crack location and deterioration level on the critical buckling load for three cracks in two rubber layers.

Comparison of configurations for a single, two cracks and three cracks in two rubber layers is listed in Table 5. Adding a crack into a cracked rubber layer definitely leads to the decrease of the critical buckling load, but the decrease rate depends on the added crack location. For a single crack (

i = 11

;

x_{11, 1} / t_{r} = 5 / 10

), the decrease rate of.

Table 5.

Comparison of configurations for three cracks in two rubber layers and configurations for a single and two cracks.

Cracked rubber layer	11	10, 11	11, 11	10, 11, 11	10, 11, 12
Crack location in the rubber layer	$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{5}{10}$
		$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{11, 2}}{t_{r}} = \frac{9}{10}$	$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$
		$\frac{x_{11, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{11, 2}}{t_{r}} = \frac{9}{10}$	$\frac{x_{11, 2}}{t_{r}} = \frac{9}{10}$	$\frac{x_{12, 1}}{t_{r}} = \frac{5}{10}$
Critical buckling load for $R_{i 1} = 1$	9.78	9.68	9.63	9.44	9.08

Adding another crack ( $i = 10$ ; $x_{10, 1} / t_{r} = 5 / 10$ ) in a non-cracked rubber layer is smaller than that of adding another crack ( $i = 11$ ; $x_{11, 2} / t_{r} = 9 / 10$ ) in the cracked rubber layer, whereas for two cracks in the middle of respective rubber layer ( $i = 10$ and $i = 11$ ), the decrease rate of adding another crack ( $i = 12$ ) in the middle of a non-cracked rubber layer is larger than that of adding another crack ( $i = 11$ ; $x_{11, 2} / t_{r} = 9 / 10$ ) in the cracked rubber layer. The results are expected and follow that if one or two cracks remain the same, the addition of another crack near the bearing center, making all the cracks more gathered, results in a larger critical buckling load than that far from the bearing center.

Three cracks in one rubber layer

Similar calculations are repeated for three cracks within only one rubber layer. The results are shown in Figures 21 and 22, where $i = 10$ is considered. Note that not only the three cracks approach the bearing center nearer but also the three crack become much closer to each other, the horizontal stiffness and critical buckling load are much larger. This is consistent with the behavior demonstrated in Sections 3.3.1 and 3.3.2.

Figure 21.

Variation of horizontal stiffness for three cracks within one rubber layer for the same deterioration level $R_{i 1} = R_{i 2} = R_{i 3} = 0.1$ .

Figure 22.

Influence of crack location and deterioration level on the critical buckling load for three cracks within one rubber layer.

The critical buckling loads of the bearings with one, two and three cracks within one rubber layer (

i = 10

) are given in Table 6. Compared with one crack in the middle (

x_{10, 1} / t_{r} = 5 / 10

) of the rubber layer and two cracks at

x_{10, 1} / t_{r} = 1 / 10

and

x_{10, 2} / t_{r} = 5 / 10

, adding another crack again to form three cracks in the same rubber layer, significantly reduces the critical buckling load. A comparison between the two three-crack configurations indicates that the addition of the third crack location

x_{10, 3} / t_{r} = 6 / 10

renders the final three cracks much more gathered, thus leading to the critical buckling load higher than

x_{10, 3} / t_{r} = 9 / 10

, which is in accordance to the results presented in Sections 3.2.1-3.3.2.

Table 6.

Comparison of configurations for three cracks in one rubber layer and configurations for a single and two cracks.

Cracked rubber layer	10	10, 10	10, 10, 10	10, 10, 10
Crack location in the rubber layer	$\frac{x_{10, 1}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{1}{10} \frac{x_{10, 2}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{1}{10} \frac{x_{10, 2}}{t_{r}} = \frac{5}{10}$	$\frac{x_{10, 1}}{t_{r}} = \frac{1}{10} \frac{x_{10, 2}}{t_{r}} = \frac{5}{10}$
Crack location in the rubber layer	$\frac{x_{10, 1}}{t_{r}} = \frac{5}{10}$		$\frac{x_{10, 3}}{t_{r}} = \frac{6}{10}$	$\frac{x_{10, 3}}{t_{r}} = \frac{9}{10}$
Critical buckling load for $R_{i 1} = 1$	9.78	9.63	7.87	6.65

Conclusions

A period structure assumption and transfer matrix formulation, along with the closed-form solutions based on the Haringx column theory for the rubber layer as well as the motion principle of the rigid body for the steel shim, are derived to analyze the effects of a single or multiple cracks, on the buckling and deformation distribution of a laminated rubber isolation bearing. The proposed approach presents an efficient tool for evaluating the horizontal stiffness and critical buckling load of the bearings with an arbitrary number of cracks, by only changing the number of transfer matrices above and below the crack section and the height of each segmented rubber layer with cracks. The results indicate that the horizontal stiffness and critical load depend on the number, location and level of deterioration. For a single crack section, there exists the least decrease in critical buckling load when the crack is at the center of the bearing, and moving from the situation of a bearing with the crack located at the center to the situation at the top or bottom, gives a reduction of about one-half of the critical buckling load for a high level of deterioration. However, if the crack appears at the center of the bearing, the critical buckling load is not sensitive to the level of deterioration. Furthermore, the crack leads to an increase in both the lateral displacement and the internal rotation. Cracks in two or multiple rubber layers drastically reduce the critical buckling load especially for cracks in different rubber layers. Whereas if one or two cracks already exist in the bearing, the addition of another crack around the bearing center, making the final cracks more gathered, results in a less decrease in critical buckling load. A consideration of the crack effect is necessary for effectively predicting the mechanical properties of the damaged bearings. The verification of the methodology and model still stand for based on the limited set of test results, therefore further investigation is needed and recommended.

Footnotes

ORCID iD

Yong-Chang Yang

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship and/or publication of this article: The authors gratefully acknowledge the financial support for this research provided by the National Natural Science Foundation of China (Grant No. 51908521, 52478319 and 52178370).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References

Arboleda-Monsalve

Zapata-Medina

Aristizabal-Ochoa

(2007) Stability and natural frequencies of a weakened timoshenko beam-column with generalized end conditions under constant axial load. Journal of Sound and Vibration 307(1-2): 89–112.

Caddemi

Caliò

Cannizzaro

(2013) The influence of multiple cracks on tensile and compressive buckling of shear deformable beams. International Journal of Solids and Structures 50(20-21): 3166–3183.

Chang

(2002) Modeling of laminated rubber bearings using an analytical stiffness matrix. International Journal of Solids and Structures 39(24): 6055–6078.

Chou

Huang

(2011) Crack initiation and propagation in circular rubber bearings subjected to cyclic compression. Journal of Applied Polymer Science 121: 1747–1756.

Cui

Chen

, et al. (2021) Geometric attention regularization enhancing convolutional neural networks for bridge rubber bearing damage assessment. Journal of Performance of Constructed Facilities 35(5): 04021061.

Ding

Zhu

(2017) Analysis of mechanical properties of laminated rubber bearings based on transfer matrix method. Composite Structures 159: 390–396.

Ding

Zhu

(2019) Seismic response and vibration transmission haracteristics of laminated rubber bearings with a single disorder. Journal of Engineering Mechanics 145(12): 04019093.

Ellakany

Tablia

(2010) A numerical model for static and free vibration analysis of elastic composite beams with end shear restraint. Meccanica 45: 463–474.

Ellakany

Elawadly

Alhamaky

(2004) A combined transfer matrix and analogue beam method for free vibration analysis of composite beams. Journal of Sound and Vibration 277: 765–781.

10.

Fan

Zheng

(2003) Stability of a cracked timoshenko beam column by modified fourier series. Journal of Sound and Vibration 264(2): 475–484.

11.

Gao

Ren

Liu

, et al. (2023) Study on deterioration identification method of rubber bearings for bridges based on YOLOv4 deep learning algorithm. Bridge Structures 19: 175–188.

12.

Gent

Meinecke

(1970) Compression, bending and shear of bonded rubber blocks. Polymer Engineering and Science 10: 48–53.

13.

Haringx

(1949) On highly compressible helical springs and rubber rods, and their application for vibration-free mountings, III. Philips Research Reports 4: 206–220.

14.

Imbimbo

Kelly

(1997a) Stability of isolators at large horizontal displacements. Earthquake Spectra 13(3): 415–430.

15.

Imbimbo

Kelly

(1997b) Stability aspects of elastomeric isolators. Earthquake Spectra 13(3): 431–449.

16.

Kelly

(1997) Earthquake-Resistant Design with Rubber. London: Springer.

17.

Kelly

Takhirov

(2007) Tension buckling in multilayer elastomeric isolation bearings. Journal of Mechanics of Materials and Structures 2(8): 1591–1605.

18.

Kikuchi

Nakamura

Aiken

(2010) Three-dimensional analysis for square seismic isolation bearings under large shear deformations and high axial loads. Earthquake Engineering & Structural Dynamics 39(13): 1513–1531.

19.

Koh

Kelly

(1988) A simple mechanical model for elastomeric bearings used in base isolation. International Journal of Mechanical Sciences 30(12): 933–943.

20.

Koh

Kelly

(1989) Modeling of seismic isolation bearings including shear deformation and stability effects. Applied Mechanics Reviews 42: S113–S120.

21.

(2001) Buckling of multi-step cracked columns with shear deformation. Engineering Structures 23(4): 356–364.

22.

(2003) Classes of exact solutions for buckling of multi-step non-uniform columns with an arbitrary number of cracks subjected to concentrated and distributed axial loads. International Journal of Engineering Science 41(6): 569–586.

23.

Matsuzaki

(2022) Time-dependent seismic reliability of isolated bridges considering ageing deterioration of lead rubber bearings. Structure and Infrastructure Engineering 18(10-11): 1526–1541.

24.

Mazda

Shiojiri

(1993) Simplified analysis of laminated rubber bearings by transfer matrix method. Journal of Structural Engineering B 39: 203–209.

25.

Stanton

Scroggins

Taylor

, et al. (1990) Stability of laminated elastomeric bearings. Journal of Engineering Mechanics 116(6): 1351–1371.

26.

Takahashi

(1999) Vibration and stability of non-uniform cracked timoshenko beam subjected to follower force. Computers & Structures 71(5): 585–591.

27.

Tsai

Hsueh

(2001) Mechanical properties of isolation bearings identified by a viscoelastic model. International Journal of Solids and Structures 38(1): 53–74.

28.

Wang

Yuan

Tan

, et al. (2024) Experimental study and numerical model of a new passive adaptive isolation bearing. Engineering Structures 308: 118044.

29.

Warn

Whittaker

Constantinou

(2007) Vertical stiffness of elastomeric and lead-rubber seismic isolation bearings. Journal of Structural Engineering 133(9): 1227–1236.

30.

Zhao

Guan

, et al. (2023) Research on the response of rubber bearings with rotation through experiments and finite element analysis. Iranian Journal of Science and Technology, Transactions of Civil Engineering 47: 3441–3451.

31.

Yin

, et al. (2024) Gaussian process regression driven rapid life-cycle based seismic fragility and risk assessment of laminated rubber bearings supported highway bridges subjected to multiple uncertainty sources. Engineering Structures 316: 118615.

32.

Zapata-Medina

Arboleda-Monsalve

Aristizabal-Ochoa

(2010) Static stability formulas of a weakened timoshenko column: effects of shear deformations. Journal of Engineering Mechanics 136(12): 1528–1536.

33.

Zeng

Xiong

, et al. (2024) A multitask framework with self-attention mechanism for simultaneous detection of axial pressure, shear deformation, and rupture damage in rubber bearings. Structural Health Monitoring-An International Journal 77: 14759217241283725.

34.

Zhang

Zhu

Weng

(2022) Damage detection of laminated rubber bearings based on the antiresonance frequencies of periodic structure and its sensitivity analysis. Shock and Vibration 2022: 4297011–4297018.

35.

Zhang

Yang

, et al. (2024) Seismic damage assessment of bonded versus unbonded laminated rubber bearings: a deep learning perspective. Engineering Structures 321: 118996.

36.

Zheng

Wang

Tan

, et al. (2022) Numerical modeling and experimental validation of Sliding-LRBs considering hysteretic strength degradation. Engineering Structures 262: 114374.