Mathematical modelling of yielding shear panel device

Thin-Walled Structures 59 (2012) 153–161

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Mathematical modelling of yielding shear panel device Md Raquibul Hossain a,1, Mahmud Ashraf b,n a b

School of Civil Engineering, The University of Queensland, QLD 4072, Australia School of Engineering and Information Technology, The University of New South Wales, Canberra, ACT 2600, Australia

a r t i c l e i n f o

abstract

Article history: Received 22 November 2011 Received in revised form 27 April 2012 Accepted 27 April 2012 Available online 5 July 2012

Passive control devices are used in structures to dissipate seismic energy with a view to minimise damages to the major structural components. A relatively new device, yielding shear panel device (YSPD), to exploit the shear deformation of thin metal plates to absorb energy has recently been developed. YSPD is formed by welding a thin steel plate within a steel square hollow section (SHS) and its behaviour is determined by complex interactions among the thin diaphragm plate, the surrounding SHS and the boundary conditions. Available experimental results show that YSPD produces nonlinear pinching hysteretic force–displacement response when subjected to cyclic loading. This paper proposes a mathematical model to predict the hysteretic response using easily available parameters, i.e. the geometry of the YSPD and the properties of the material. Results obtained from the verified finite element (FE) models are used to calibrate the proposed models, which should facilitate the full scale modelling of building frames retrofitted with YSPD. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Bouc-Wen-Baber-Noori (BWBN) model Cyclic loading Earthquake Seismic energy dissipation Yielding shear panel device

1. Introduction Earthquake is one of the major catastrophic events for destroying cities and its inhabitants. During the last few decades researchers from around the globe put extensive efforts to achieve sustainable solution to diminish the dire effects caused by earthquakes, which have led to the innovation of various control devices. Passive energy dissipation technique is one of the widely adopted concepts to absorb earthquake energy without causing significant damages to the main structural elements. Passive energy dissipation devices are generally simple, easy-torehabilitate, economical and do not rely on external power, which could be a major concern during an earthquake. Metal yielding devices are widely used due to their simplicity in both design and implementation and the devices dissipate energy by taking advantage of the material’s stable hysteretic behaviour. Added damping and stiffness (ADAS) device [1], triangular added damping and stiffness (TADAS) device [2] and steel plate shear wall (SPSW) [3] are the most common metal devices used for passive energy dissipation. Recent catastrophic earthquakes clearly demonstrate our vulnerability towards this natural event and indicate the need for developing more cost-effective, easy-to-use yet reliable devices to minimise the loss of lives in the event of an earthquake. Yielding n

Corresponding author. Tel.: þ61 7 3365 3655. E-mail addresses: [email protected] (M.R. Hossain), [email protected], [email protected] (M. Ashraf). 1 Tel.: þ61 7 3365 8395. 0263-8231/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.tws.2012.04.018

shear panel device (YSPD) [4,5] is a newly proposed energy dissipating device that exploits the in-plane shear deformation of a steel plate to dissipate energy. YSPD is composed of a steel diaphragm plate welded inside a square hollow steel section (SHS). The SHS provides the necessary supporting boundary for the diaphragm plate as well as allows the device to be appropriately connected to both the V-brace and the beam using bolted connection. Fig. 1 shows geometric details of an YSPD and the proposed bolted connection to facilitate easy rehabilitation. Hossain et al. [6] proposed a finite element modelling technique for YSPD using general purpose finite element package ANSYS [7]. Results obtained from FE simulations have been used to propose a bilinear force–displacement model, which relies on the appropriate estimation of the initial stiffness and the yield strength of a given YSPD. The bilinear model has recently been modified into a tri-linear model [8]. An appropriate hysteretic mathematical model for YSPD is required to evaluate the performance of structures equipped with YSPDs. Bouc-Wen model [9] is widely used as a macroscopic model to simulate the response of energy dissipating devices [10,11], in which, a single nonlinear differential equation simulates the hysteresis response of the target energy dissipating device by connecting the restoring force to the corresponding deformation. Appropriately defined parameters allow the BoucWen model to generate smooth hysteretic force–displacement curves without considering any effects of pinching and degradation of strength and stiffness. Baber and Noori [12] modified the Bouc-Wen model to include strength degradation, stiffness degradation and pinching characteristics of a nonlinear system, which

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Thickness = T Thickness = t S

d

D

Elevation

T o p V ie w

Fig. 1. (a) Yielding shear panel device (YSPD) with the bolted connection and (b) schematic diagram showing the geometric parameters of YSPD [5].

is known as ‘Bouc-Wen-Baber-Noori (BWBN) model’. The BWBN model could be used to appropriately simulate the hysteretic response of YSPD to include the pinching characteristics observed both in laboratory testing and in FE simulation [5,6,13]. This paper concentrates on the appropriate identification of the BWBN model parameters for YSPD using the results obtained from a total 130 FE models. The FE modelling technique has been verified and all necessary details are reported elsewhere [6]. Suitable relationships among the model parameters and the physical parameters of YSPD are proposed herein to obtain hysteretic load–deformation response of YSPD using the knowledge of easily available geometric and material properties.

2. Bouc-Wen-Baber-Noori (BWBN) model in the context of YSPD

Fig. 2(b). kt is the tangential stiffness of YSPD after the tension field is formed, whilst a ¼kt/F1. The hysteretic model may be visualised as a combination of two springs connected parallel to each other. The elastic spring has a stiffness equal to the tangential stiffness kt, whilst the response of the non-degrading pinching hysteretic spring depends on the hysteretic parameter z, which is defined by the following first order nonlinear differential equation [12,15]: n z_ ¼ d_ hðzÞA½g þ bsngðd_ zÞ9z9

where A, b, g and n are hysteretic model parameters that control the shape of the curve and h(z) is the pinching inducing function. Value of sngðd_ zÞ depends on the sign of ðd_ zÞ and the value becomes þ1 if ðd_ zÞ is positive or becomes  1 if ðd_ zÞ is negative. Foliente [15,16] proposed a pinching inducing function, as shown in Eqs. (3) and (4), to incorporate the effect of residual force. _

2

hðzÞ ¼ 1:0z1 eððz sgnðd Þqzu Þ The load applied to the loading end flange is supported by the bolded connection at the support end as shown in Fig. 2(a). Following the concept of BWBN model [9,12], the restoring force F, as shown in Fig. 2(b), produced in the YSPD may be expressed as F ¼ F e þ F h ¼ aF 1 d þ ð1aÞF 1 z

ð1Þ

where Fe and Fh are the elastic and the hysteretic component of the restoring force respectively. Eq. (1) provides a nonlinear force– displacement (F–d) relationship based on parameters F1, a and z. F1 is defined as the force representing the intersecting point of the bilinear envelope at a unit displacement of d1 as shown in

ð2Þ

zu ¼

 1 1=n bþg

=z22 Þ

ð3Þ



ð4Þ

where 0 r z1 r 1 controls the severity of pinching, z2 causes the pinching region to spread, zu is the ultimate value of z and q is a model parameter. z1 and z2 may be expressed by Eqs. (5) and (6) as functions of hysteretic energy e,

z1 ¼ z1o ½1:0eðpeÞ 

ð5Þ

z2 ¼ ðc0 þ dc eÞðl þ z1 Þ

ð6Þ

M.R. Hossain, M. Ashraf / Thin-Walled Structures 59 (2012) 153–161

155

F

k

Support end flange

Loading end flange

Force (F)

,δ

F Displacement (δ) Fig. 2. (a) Deformed shape of YSPD [14] and (b) nonlinear force–displacement (F–d) relationship of YSPD.

where p controls the rate of initial drop in the slope, z1o is the total slip, c0 contributes to the amount of pinching, dc controls the rate of pinching spread and l controls the rate of change of z2 as z1 changes. The rate of change of hysteretic energy can be expressed as

20

e_ ¼ ð1aÞF 1 zd_

10

15

3. Parametric study based on the developed FE models of YSPDs The notion of this research is based on the assumption that the BWBN model should be able to represent the force–displacement relationship of a YSPD if the model parameters are appropriately correlated with the relevant physical parameters. An extensive and thorough parametric study has been carried out as part of the current research to establish appropriate relationships between the BWBN model and the geometric and material properties associated with YSPD. Nonlinear finite element models of YSPD were developed using ANSYS [7] and their performance was verified against available test results; all necessary details of the FE modelling technique are available in Hossain et al. [6]. The FE models used in the current study have the same orientation as those would occur in a practical V-brace assembly. Material properties conform to those for structural grade hot-rolled steel sheet as given in AS/NZS 1594:2002 [17]. A bilinear kinematic hardening model (E¼ 200 GPa and Et ¼2 GPa) is used to simulate the material behaviour in the developed FE models [6]. A total of 130 FE models have been analysed for both monotonic and cyclic loading. A total displacement of 20 mm is applied for the monotonic loading whilst a four cycle loading history of amplitudes 1, 5, 10 and 20 mm is applied for the cyclic loading as shown in Fig. 3. Table 1 shows the considered ranges for physical parameters of YSPDs used in the parametric study. Finite element results are used for estimating the model parameters. Yao et al. [18] used Matlab [19] for estimating the parameters of the Bouc-Wen model to represent a magnetorheological energy dissipating damper. Amongst the available various built-in optimisation tools in Matlab [19], the least square technique is used in the current research to estimate the model parameters for representing YSPDs. A new function is developed to represent the BWBN model and the nonlinear least-squares optimisation method is used to identify an appropriate value for each parameter. A set of parametric values for each FE model is obtained following this optimisation process. 4. Parameter estimation for YSPDs Parameters required for the non-degrading pinching hysteretic BWBN model can be divided into the following two

Displacement (mm)

ð7Þ

5 0 -5 -10 -15 -20 Time Fig. 3. Displacement history applied for cyclic loading.

Table 1 Range of physical parameters used in FE simulation. Parameter

Range

Thickness of SHS plate (t), mm Thickness of diaphragm plate (T), mm Size of SHS (D), mm Size of diaphragm plate (d), mm Yield strength of SHS plate (fy, SHS), MPa Yield strength of diaphragm plate (fy, dia), MPa

4, 5, 6 2, 3, 4 100, 110, 120 (D  2  T) 250, 300, 350 250, 300, 350

Note: Designation rule adopted for YSPD is ‘‘YSPD D  T  t  fy, example: YSPD 100  4  2 250/350.

dia/fy, SHS’’.

For

categories—hysteretic parameters (kt, F1, A, b, g, n) and pinching parameters (q, z1o, p, c0, dc, l). Hysteretic parameters controls the shape of the curve, whilst the pinching parameters define the pinching inducing function h(z). Ma et al. [20] identified a functional redundancy among the parameters used in BWBN model. A convenient way to remove this redundancy is to assign a constant value of 1 to the parameter A. Constantinou and Adnane [21] proposed further simplification by assuming A/(b þ g)¼1, which eventually yields (b þ g) ¼1. Ni et al. [22] showed any combination of b and g to satisfy (b þ g) ¼1 produces a similar hysteretic response for softening hysteresis loops. To satisfy the aforementioned constraint, as proposed by Constantinou and Adnane [21], A/(b þ g) ¼1, both b and g are assumed to have a magnitude of 0.5 in the current study. Thus the unknown hysteretic parameters required to define the shape of

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the force–displacement curve reduce to kt, F1 and n; these three parameters can be calibrated using the monotonic force– displacement response of YSPDs. The proposed simplified model with A¼1 and b ¼ g ¼0.5 will start with an initial slope of F1, which will gradually decrease depending on the value of n to match the force–displacement response of a YSPD. 4.1. Calibration of hysteretic parameters 4.1.1. Tangential stiffness, kt Appropriate formulation of the force–displacement response of a YSPD, especially when the diaphragm plate undergoes into the post-buckling regime, is highly significant to identify its energy dissipation characteristics. A tension field is developed after the shear buckling of diaphragm plate. Fig. 4 shows a typical state of stress acting on the diaphragm plate with a tension field inclined at an assumed angle a. Change in shear force dF as a result of this post-buckling tension field stresses can be expressed by Eq. (8). 1 dF ¼ dsty ðsin2aÞtd 2

The diaphragm plate is assumed to be simply supported with undeformed flanges for calculating the tangential stiffness kt, whilst this boundary of the diaphragm plate is allowed to move due to deformation of the SHS flanges. Test results and FE models showed that the tangential stiffness of a YSPD is somewhat smaller than the value obtained using Eq. (11); this reduction depends on the thickness of both the SHS and the diaphragm plate as well as the size of the YSPD. A stiffness reduction factor is proposed herein to account for the observed discrepancy between the theoretical and the experimental stiffness. The magnitude of the proposed reduction factor increases with increasing SHS thickness (T), whilst decreases with an increase in both the size of the diaphragm (d) and the diaphragm plate thickness (t). A dimensionless stiffness reduction c factor f c T a =t b d is assumed to account for an appropriate boundary condition where T, t and d are measured in mm. Proposed constants fc, a, b and c can be calibrated by regression analysis of the results obtained from the parametric study. Eq. (12) may be used to obtain the tangential stiffness of a YSPD, kt ¼

1 f Ta Et t bc c 4 t d

ð12Þ

ð8Þ

Kharrazi et al. [23] computed the post bucking shear deformation by equating the work done by the post-buckling component of the shear force to the strain energy produced by the tension field. Using the same technique, the incremental post-buckling shear deformation can be expressed by Eq. (9), whilst Eq. (10) may be derived using Eqs. (8) and (9) to compute the tangential stiffness of the YSPD at the post-buckling state. dd ¼ 2dsty d=Et sin2a

ð9Þ

1 2 kt ¼ Et t sin 2a 4

ð10Þ

Eq. (10) shows that the shear deformation in the post-buckling regime depends on the inclination of the tension field a. A number of previous research reports intended to quantify an appropriate magnitude for the angle of inclination in tension field action. CAN/CSA-S16-01 [24] suggests a limit varying between 381 and 451 for the inclination angle of pin ended strips in steel plate walls. Shishkin et al. [25] reviewed all available relevant literature and reported that the inclination angle magnitudes in the range 38– 501 produced similar results for the ultimate capacity of steel plate shear walls. An inclination angle of 451 is hence considered reasonable for the tension strips developed within the square diaphragm plate of a YSPD. With a ¼ 45o the post-buckling stiffness may be expressed by the simple expression given as follows: 1 kt ¼ Et t 4

ð11Þ

y

Regression analysis produced the following magnitudes for the considered constants fc ¼57510, a¼1.33, b¼0.5 and c¼3.0. Fig. 5 shows good agreement between the tangential stiffness calculated using Eq. (12) and those obtained from the finite element analysis.

4.1.2. Calibration of parameters F1 and n BWBN model parameters F1 and n determine the shape of the force–displacement response. For a simply supported diaphragm plate the force–displacement response is bilinear and in such a case, the parameter n becomes infinity, whilst F1 may be calculated using the following equation: 1 F 1 ¼ pffiffiffitdf y,dia 3

The actual value of F1 is observed to be significantly affected by other relevant parameters and hence a multiplying factor is proposed to be used with Eq. (13) to obtain an accurate prediction for F1. The proposed multiplication factor is assumed to have the c same form as f c T a =t b d for simplicity and this allows including the effects of the boundary condition and the material strength. Another multiplying factor (fy, SHS/fy, dia)h is introduced to incorporate the effect of difference in strengths of the SHS plate and that of the diaphragm plate. Hence the parameter F1 can be predicted using the following equation 1 f Ta F 1 ¼ pffiffiffi tdf y,dia  bc c  t d 3

σxx= σty.sin2 α

f y,SHS f y,dia

!h ð14Þ

1.00

σxy= ½.σty.sin2 α

0.75

σyx= ½.σty.sin2 α σyy= σty.cos α 2

kt , kN/mm (FE)

σty

ð13Þ

R2 = 0.9631 0.50

0.25

α

x

0.00 0.00

0.25

0.50

0.75

1.00

kt , kN/mm (Equation 12)

Fig. 4. Tension field stresses in diaphragm plate after buckling.

Fig. 5. Correlation of the tangential stiffness (kt) calculated using Eq. (12) and those obtained from finite element analysis.

M.R. Hossain, M. Ashraf / Thin-Walled Structures 59 (2012) 153–161

The parameter estimation technique is used in the current study to obtain a specific combination of magnitudes for F1 and n for each finite element model. The proposed constants for F1 are predicted from the regression analysis as fc ¼2.065, a ¼0.25, b¼0.0, c¼0.25 and h¼ 0.8. Fig. 6(a) shows the relationship between for the magnitudes of F1 estimated from the finite element models and those calculated according to Eq. (14). It is obvious for Fig. 6(a) that the proposed constants can predict F1 with a very high degree of accuracy. The shape parameter n depends on the physical parameters, the boundary condition and the strength characteristics of both the SHS and the diaphragm plate. To maintain uniformity in the proposed calibration technique, the value of n may be predicted using the following equation: !h f y,SHS f Ta n ¼ bc c  r 2:0 ð15Þ f y,dia t d The constants are obtained from the regression analysis as fc ¼ 6.78  109, a¼3.0, b¼2.5, c¼5.5 and h¼  0.8. It is worth mentioning that the results obtained from the regression analysis showed that a magnitude with n42 generate unusually sharp force–displacement curve and hence an upper limit of 2 has been proposed for n. Fig. 6(b) compares the magnitudes of n calculated following Eq. (15) with those obtained from the FE models. There is a bit of scatter observed in Fig. 6(b) but the R2 value demonstrates the reliable accuracy of the proposed equation in predicting n.

4.1.3. Performance of the proposed equations for monotonic response The BWBN parameters kt, F1 and n define the hysteretic behaviour of a force–displacement curve without considering the pinching effect. This section compares the monotonic force–displacement response obtained from the FE models to those predicted using the proposed parameters. The pinching parameters will be incorporated to predict the cyclic response of YSPDs in Section 4.2. Table 2 summarises the proposed magnitudes for the constants to determine the hysteretic parameters using Eqs. (12), (14) and (15). Fig. 7 shows comparisons between the force–displacement response obtained using finite element simulations and those obtained using the BWBN model predictions. The proposed Eqs. (12), (14) and (15) have been used to determine the BWBN model parameters kt, F1 and n. The comparisons clearly demonstrate good agreement between the analytical results and the results obtained using FE simulation. Table 3 compares the amount of energy required to achieve specified displacements during the course of monotonic loading in FE simulations to those obtained from the proposed model predictions. The ratio of the amount of energy required to attain different displacements

indicates that the developed model may be used to predict energy dissipation with reasonable accuracy.

4.2. Calibration of the pinching parameters 4.2.1. Pinching parameters for hysteretic response Pinching parameters q, z1o, p, c0, dc, l, as described in Section 2, determine the magnitude and the spread of pinching due to the buckling of the diaphragm plate. Ma et al. [20] conducted both global and local sensitivity analysis of the BWBN parameters and it was reported that z1o is the most sensitive pinching parameter, whilst dc and l are the least sensitive. In the current research appropriate pinching parameters are estimated using the results obtained from FE simulations exploiting the parameter estimation technique available in Matlab [19]. All pinching parameters are considered as functions of the easily available physical parameters of YSPD, i.e. diaphragm plate thickness T, SHS plate thickness t and size of the diaphragm d. To maintain uniformity in formulations, the c general form is taken as f c T a =t b d where T, t and d are measured in mm and fc is a constant. Appropriate magnitudes for all these constants are obtained by regression analysis; Table 4 lists the values for the pinching parameters. Fig. 8 compares the pinching parameters q, z1o, p, c0 estimated from FE analysis to those c calculated using the proposed closed form expression of f c T a =t b d . q and c0 showed good agreement between the estimated and the proposed magnitudes, whilst z1o showed some scatter as its magnitude varies within a very small range and a notable scatter was observed for p due to the same reason. The remaining two parameters dc and l, which do not play a vital role in pinching modelling, are also estimated in a similar fashion and the proposed constants for these parameters are reported in Table 4.

4.2.2. Comparison based on cyclic response Fig. 9 shows comparisons between the FE simulations and the BWBN model predictions for the cyclic response of YSPDs. Overall, the comparison shows good agreement between the FE results and the predicted results using the proposed analytic technique. Table 5 compares the average energy dissipated in FE simulation Table 2 Proposed constants the hysteretic parameters. Pinching parameter

fc

A

b

c

h

kt F1 n

57,510 2.065 6.78  109

1.33 0.25 3.00

0.50 0.00 2.50

3.00 0.25 5.50

– 0.8  0.8

125

3 n (Estimated from FEA)

F1 , kN (Estimated from FEA)

157

100 R2 = 0.9854

75 50 25 0

2

R2 = 0.9025

1

0 0

25

50

75

F1 , kN (Equation 14)

100

125

0

1 2 n (Equation 15)

3

Fig. 6. (a) Correlation between F1 calculated according to Eq. (14) and those obtained from finite element analysis and (b) correlation between n calculated using Eq. (15) and those obtained from finite element analysis.

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M.R. Hossain, M. Ashraf / Thin-Walled Structures 59 (2012) 153–161

80

100 YSPD 100×4×t-250/250

40

t = 2 mm

t = 3 mm

Force (kN)

t = 3 mm

60 t = 2 mm

40

20

20

FE Simulation

FE Simulation

Model Prediction

Model Prediction

0

0 0

5

10 15 Displacement (mm)

20

0

100

20

YSPD 110×6×t-250/350

60

t = 4 mm

100

t = 4 mm

t = 3 mm

80

t = 3 mm

60

t = 2 mm

t = 2 mm

40 20

Force (kN)

80 Force (kN)

5 10 15 Displacement (mm)

120 YSPD 110×4×t-350/300

40

FE Simulation

FE Simulation

20

Model Prediction

Model Prediction

0

0 0

5 10 15 Displacement (mm)

20

0

100

5 10 15 Displacement (mm)

20

120 YSPD 120×5×t-300/300

YSPD 120×6×t-350/350

t = 4 mm

t = 4 mm

100

80 60 t = 2 mm

40 20

Force (kN)

t = 3 mm

Force (kN)

t = 4 mm

80

60 Force (kN)

YSPD 100×5×t-250/350

t = 4 mm

80

t = 3 mm

60

t = 2 mm

40

FE Simulation

FE Simulation

20

Model Prediction

Model Prediction

0

0 0

5 10 15 Displacement (mm)

20

0

5 10 15 Displacement (mm)

20

Fig. 7. Comparison of the force–displacement response obtained from FE models to those obtained using the proposed formulations.

Table 3 Comparison of the amount of energy required to attain specified displacements during the course of monotonic loading. YSPD designation (D  t)

Ratio of average energy for different displacements (BWBN model/FE)

Table 4 Proposed constants fc, a, b and c for the pinching parameters. Pinching parameter

fc

a

b

c

q

1.07  106 1.0756 0.0435 8.67  10  7 1.00  10  5 3.64  1013

1.6138 0.0167  1.0140 0.0000 0.0000 1.4422

0.9250 0.0180 0.0210 1.6480 0.0000 6.8280

3.5690 0.0280  0.1242  3.1420 0.0000 7.0760

5 mm

10 mm

15 mm

20 mm

z1o

0.97 1.03 1.01 0.98 0.97 0.92 1.03 0.93 0.84

0.97 1.03 1.03 0.99 1.00 0.97 1.02 0.96 0.90

0.97 1.01 1.01 0.99 0.99 0.98 1.01 0.96 0.92

0.98 1.00 1.00 0.99 0.99 0.98 1.01 0.97 0.93

c0 dc l

p 100  2 100  3 100  4 110  2 110  3 110  4 120  2 120  3 120  4

to the amount of energy absorbed by hysteretic response predicted using the BWBN model.

5. Validation of the proposed model against test results The BWBN model is capable of considering nonlinearity and pinching effects observed in the test and FE results of YSPDs. The magnitudes for the parameters used in this pinching hysteretic

model are estimated based on a large number of validated FE models by exploiting the parameter estimation technique available in MATLAB [19]. These estimated values formed the basis for proposing a simple and unified formulation for all model parameters. Force–displacement response of YSPDs may be predicted mathematically using these formulations by identifying appropriate values for the parameters of the BWBN model. The proposed formulations are validated herein by comparing the available test results with those obtained using the BWBN model. Fig. 10 shows the comparison for the monotonic response of YSPD, which demonstrates overall good agreement between the test and the analytically predicted force–displacement response. Fig. 11 compares the test results for cyclic response of YSPDs to those obtained using BWBN model. The overall agreement may be considered acceptable and the proposed formulations may be used

M.R. Hossain, M. Ashraf / Thin-Walled Structures 59 (2012) 153–161

(Estimated from FEA)

1.05

1.00 R2 = 0.8486 0.50

0.00 0.00

0.50 1.00 q (Calculated as closed form)

R2 = 0.5676 0.95 0.9

0.85 0.85

1.50

ζ

ψ (Estimated from FEA)

0.03 2

R = 0.4223 0.02 0.01

0 0.000

0.90 0.95 1.00 (Calculated as closed form)

1.05

1

0.04 p (Estimated from FEA)

1

ζ

q (Estimated from FEA)

1.50

159

0.010 0.020 0.030 p (Calculated as closed form)

0.75 0.5 0.25

0 0.00

0.040

R2 = 0.8924

0.25 0.50 0.75 ψ (Calculated as closed form)

1.00

Fig. 8. Correlation of pinching parameters calculated using the proposed closed form expression to those obtained from FE analysis.

60

60 YSPD 110×5×2-250/250

YSPD 100×4×2-300/300

30 Force (kN)

Force (kN)

30 0 -30

0 -30

FE Simulation

FE Simulation Model Prediction

Model Prediction

-60 -25

-12.5 0 12.5 Displacement (mm)

-60 -25

25

120

YSPD 110×4×3-350/300

40 Force (kN)

Force (kN)

60 0 -60

0 -40 FE Simulation

FE Simulation Model Prediction

-12.5 0 12.5 Displacement (mm)

Model Prediction

-80 -25

25

120

25

YSPD 120×5×4-350/350

60 Force (kN)

60 Force (kN)

-12.5 0 12.5 Displacement (mm)

120 YSPD 100×6×4-350/300

0 -60

0 -60 FE Simulation

FE Simulation

-120 -25

25

80 YSPD 120×6×3-250/350

-120 -25

-12.5 0 12.5 Displacement (mm)

Model Prediction

-12.5 0 12.5 Displacement (mm)

25

Model Prediction

-120 -25

-12.5 0 12.5 Displacement (mm)

25

Fig. 9. Comparisons of the force–displacement response between finite element simulations and model predictions of YSPDs.

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in FE modelling technique to evaluate the performance of building frames retrofitted with YSPDs. Appropriate analytic model will enable designers to avoid going into all minor details of modelling a YSPD and hence will save significant computational time.

Table 5 Comparison of energy required for cyclic loading. YSPD size (D)

Ratio of average energy for different diaphragm thicknesses (t) (BWBN model/FE) 2 mm

3 mm

4 mm

1.11 1.09 0.97

1.15 1.10 0.96

1.18 1.13 0.97

6. Conclusion Yielding shear panel device is a novel passive control device developed to mitigate the seismic vulnerability of structures.

80

100 YSPD 100×4×3

YSPD 120×5×3

60 Force (kN)

Force (kN)

80 60 40 Test Result

20

40 Test Result

20

FE Simulation BWBN Model

FE Simulation BWBN Model

0

0 0

5 10 Displacement (mm)

0

15

100

15

YSPD 120×5×4

80 Force (kN)

80 Force (kN)

5 10 Displacement (mm)

100 YSPD 100×4×4

60 40 Test Result FE Simulation BWBN Model

20

60 40 Test Result FE Simulation BWBN Model

20

0

0 0

5 10 15 Displacement (mm)

20

0

5 10 Displacement (mm)

15

Fig. 10. Force–displacement response of YSPDs under monotonic loading.

60

80 YSPD 100×4×3

YSPD 120×5×3

40 Force (kN)

Force (kN)

30 0 -30 -60 -10

-5 0 5 Displacement (mm)

0 -40

Test result FE simulation Model Prediction

-80 -20

10

80

20

Force (kN)

60

0

-80 -10

-10 0 10 Displacement (mm)

YSPD 120×5×4

40

-40

Test result FE simulation Model Prediction

120 YSPD 100×4×4

Force (kN)

100 110 120

-60

Test result FE simulation Model Prediction

-5 0 5 Displacement (mm)

0

10

-120 -20

Test result FE simulation Model Prediction

-10 0 10 Displacement (mm)

Fig. 11. Force–displacement response of YSPDs under cyclic loading.

20

M.R. Hossain, M. Ashraf / Thin-Walled Structures 59 (2012) 153–161

A finite element modelling technique for YSPD has recently been proposed [6] and the current study exploits the results obtained from the FE models to devise a mathematical model. This proposed mathematical formulation will allow evaluating YSPDs performance in energy absorption through simulation as well as to develop design methods to identify appropriate size, location and numbers of YSPDs that are required for seismic retrofitting. The BWBN model has been adopted as a pinching hysteretic model for YSPD. A thorough parametric study using ANSYS has been conducted to estimate appropriate magnitudes for the relevant key parameters. A unified, simple formulation technique has been adopted for all the BWBN model parameters so that the easily obtainable physical properties such as plate thicknesses, size of YSPD and material strength could be used in mathematical modelling. The proposed model has been validated against both the available test results and those obtained from finite element simulations. The proposed mathematical model should enable designers to incorporate the effects of YSPDs without going through a detailed modelling and hence should provide considerable saving in the design process. References [1] Whittaker AS, Bertero VV, Thompson CL, Alonso LJ. Seismic testing of steel plate energy dissipation devices. Earthquake Spectra 1991;7:563–604. [2] Tsai KC, Hong CP. Steel triangular plate energy absorber for earthquakeresistant buildings. In: Constructional steel design: world developments. Elsevier Applied Science; 1992. p. 529–40. [3] Timler P, Kulak GL. Experimental study of steel plate shear walls. Structural engineering report no. 114. Department of Civil Engineering, University of Alberta; 1983. [4] Williams M, Albermani F. Monotonic and cyclic tests on shear diaphragm dissipators for steel frames. Advanced Steel Construction 2006;2:1–21. [5] Chan RWK, Albermani F, Williams MS. Evaluation of yielding shear panel device for passive energy dissipation. Journal of Constructional Steel Research 2009;65:260–8. [6] Hossain MR, Ashraf M, Albermani F. Numerical modelling of yielding shear panel device for passive energy dissipation. Thin-Walled Structures 2011; 49:1032–44.

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