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Numerical model for the in-plane seismic capacity evaluation of tall plasterboard internal partitions
Thin-Walled Structures 122 (2018) 572–584
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Full length article
Numerical model for the in-plane seismic capacity evaluation of tall plasterboard internal partitions
MARK
⁎
Crescenzo Petronea,1, Orsola Coppolaa, Gennaro Magliuloa, , Pauline Lopezb, Gaetano Manfredia a b
University of Naples Federico II, Department of Structures for Engineering and Architecture, via Claudio 21, 80125 Naples, Italy Siniat International, 500 Rue Marcel Demonque - Zone Agroparc, CS 70088, 84915 Avignon Cedex 9, France
A R T I C L E I N F O
A B S T R A C T
Keywords: Nonstructural components Plasterboard partitions Numerical model FEM analysis Seismic performance
A finite element model capable to capture the interstory drift which causes the failure of a 5 m high plasterboard partition with steel studs is defined by employing the Direct Strength Method (DSM) to assess the occurrence of different buckling failure modes. The model is validated comparing the numerical behavior with the experimental evidence of a quasi-static test campaign on a plasterboard partition. It is concluded that the model well catches the interstory drifts which cause either the local or the global buckling in the partition. This conclusion is also confirmed by the strain trend in the steel studs.
1. Introduction Nowadays the seismic performance of nonstructural components is recognized to be a key issue in the framework of the Performance-Based Earthquake Engineering (PBEE). Four main issues motivate several studies related both to the evaluation of the seismic capacity and the demand of nonstructural components. Firstly, nonstructural components can cause injuries or deaths after an earthquake; for instance the 64% of the fatalities caused by 1995 Great Hanshin Earthquake was due to the compression (suffocation) of the human body [1]. Such a phenomenon could be caused by the damage to nonstructural components, that may also obstruct the way out from the damaged building. Moreover, nonstructural components generally exhibit damage for low seismic demand levels. The seismic performance of nonstructural components is especially important in frequent, and less intense, earthquakes, in which their damage can cause the inoperability of several buildings. Furthermore, the cost connected to nonstructural components represents the largest portion of a building construction. Taghavi and Miranda [2] evidenced that structural cost typically represents a small portion of the total cost of a building construction, corresponding to 18% for offices, 13% for hotels and 8% for hospitals. Finally, nonstructural components may participate in the lateral resisting system of the primary structure, i.e. varying its lateral strength and stiffness. Plasterboard internal partitions with steel studs are very common nonstructural components, since they are typically employed in several
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building typologies all over the world. Different experimental studies aiming at the evaluation of the seismic capacity of such components are available in literature [3,4]. The seismic evaluation is typically expressed in terms of the Engineering Demand Parameter (EDP) that is required to reach a certain Damage State (DS). The interstory drift is typically selected as the EDP; the aim of these studies is therefore to evaluate the interstory drift that induces different damage states, i.e. from minor to major damage states, in the partition systems. In Kanvinde and Deierlein [5] a macro-model is presented in order to evaluate the seismic behavior of plasterboard partitions. The authors propose an analytical model to determine the lateral shear strength and initial elastic stiffness of wood and gypsum wall panels. In such a case, a uniaxial spring model is defined, by a series of parameters defining the backbone curve, which represents the nonlinear monotonic response that envelopes the cyclic response, and the cyclic nonlinear response including strength and stiffness degradation and pinching phenomenon. The parameters validation is performed by using experimental tests on full-scale wall panels. In Davies et al. [4,6] a description of the experimental results of full-scale tests performed on several cold-formed steel-framed gypsum partitions is reported. The experimental data, including different partition wall configurations, in terms of wall dimensions, material type, testing protocol and boundary conditions are used in order to create seismic fragility curves for such nonstructural partition walls. Moreover, on the base of experimental results, a trilinear hysteretic macro-model is proposed to reproduce the in-plane mechanical behavior of the partitions: it allowed to include partitions in
Corresponding author. E-mail addresses: [email protected] (C. Petrone), [email protected] (O. Coppola), [email protected] (G. Magliulo), [email protected] (P. Lopez), [email protected] (G. Manfredi). 1 Present address: Willis Group Limited, 51 Lime Street, 20th floor, London EC3M 7DQ, UK. http://dx.doi.org/10.1016/j.tws.2017.10.047 Received 7 June 2016; Received in revised form 24 October 2017; Accepted 31 October 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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plane to the main structure. A simple modelling technique is proposed and validated. The validation is performed comparing the numerical behavior of a specific specimen with the experimental results achieved in a quasi-static test campaign conducted at the Laboratory of the Department of Structures for Engineering and Architecture at the University of Naples Federico II. The quasi-static test is performed on a 5 m tall plasterboard internal partition [16], which is representative of the typical partitions used in industrial and commercial buildings in the European countries.
the model of an existing steel building in order to demonstrate the effect on the seismic behavior of the whole structure. A numerical macromodel for plasterboard partition is proposed by Wood and Hutchinson [7]; a pinching material model, available in OpenSees [8], is used in order to reproduce the in-plane behavior of the partitions. The 24 parameters of the model are calibrated by a large number of experimental data obtained from about fifty tests performed on plasterboard partition walls [4]. Telue and Mahendran [9] conducted experimental studies on cold formed steel walls lined with plasterboard. These studies point out that the stud compression strength increases when the steel internal frame is covered by plasterboard on one or both sides. A finite element model was developed and validated using experimental results. However, the behavior of the walls is investigated under compressive vertical loads. Fiorino et al. [10] proposed a seismic design procedure for sheathed cold-formed steel shear walls: through linear dynamic and nonlinear static analysis three nomographs are obtained in order to determine the features of a seismic resistant shear wall. An iterative procedure for the prediction of the pushover response curve of sheathed cold-formed steel shear walls is also presented. A computational model was proposed by Buonopane, Tun and Schafer [11] for the prediction of the seismic behavior of sheathed cold formed steel shear walls, based on experimental tests performed both on shear walls [12] and on screwed panel-to-stud connections [13]. Buonopane et al. [14] also developed a numerical model for the in-plane lateral behavior of cold-formed steel shear walls with wood panels, which captures the nonlinear behavior at the interface between cover panels and fasteners. Indeed, full scale lateral load tests on thirteen shear walls exhibited collapse due to the failure of the fasteners. The modeled walls, having a height of 2.74 m and widths of either 1.22 or 2.44 m, were able to predict the cyclic response of the tested specimens, up to the peak point of lateral strength. However, these research studies focused on steel stud shear walls, whose characteristics, e.g. stud typology, restraint at the base, failure mode, are different than in internal partition walls. The evaluation of the seismic capacity of a plasterboard partition system is typically pursued through the experimental method. The development of a FEM numerical model of such a component would allow to investigate the seismic behavior of a generic partition, taking into account all its features, e.g. studs geometrical and mechanical properties, layer of plasterboards, etc. A detailed numerical model of coldformed steel-framed gypsum partition walls is proposed by Rahmanishamsi, Soroushian and Maragakis [15]. The model, built in OpenSees, provides nonlinear behavior of studs and tracks, and nonlinear behavior of gypsum to stud and stud to track connections. The gypsum boards are simulated by linear shell elements. The contact between boards/tracks and concrete slabs as well as the contact between adjacent boards are also considered in the model. The model, validated by the comparison with experimental results on gypsum partitions walls, is able to predict the trend of the response and the observed damage mechanism. The failure of the tested partitions typically consists in the failure of gypsum-to-tracks/studs screws connections and/or crushing of gypsum boards. The developed numerical model does not incorporate failure due to buckling of the steel stud, which is the typical failure mode in tall plasterboard partition walls, as shown by Petrone, Magliulo, Lopez and Manfredi [16]. Moreover, the experimental tests used for numerical validation are performed on partition walls representative of US partition systems, whose features might be different than European ones. The aim of this research study is the definition of a finite element model able to evaluate the interstory drift that induces the failure of tall plasterboard partitions representative of European partition systems commonly mounted in commercial and industrial buildings. Even if partition walls can be subjected to both the in-plane interstory drift and the out-of-plane acceleration, this study is focused only on the in-plane seismic performance assessment of partitions in existing structures. It is therefore assumed that the investigated partitions would not fail in the out-of-plane direction and that they are rigidly connected in their own
2. Methodology 2.1. Experimental test on the plasterboard partition specimen In this Section the experimental test performed on the considered plasterboard partition specimen is illustrated. The test campaign, exhaustively reported in a previous paper [16], is here briefly described. Particular attention is given to the description of the tested partition and the mounting procedure in order to justify the finite element modelling of the specimen, included in Section 2.2. The test system consists of a steel frame setup, the specimen, i.e. a plasterboard partition, a hydraulic actuator and a reaction wall (Fig. 1). The steel test frame is conceived as a statically indeterminate scheme, since it is not designed to simulate the performance of the structure itself but just in order to transfer the load provided by the hydraulic jack to the partition without absorbing lateral forces. Moreover, since the reaction wall cannot reach the height of the system, the actuator is placed at the middle height of the test setup (Fig. 1). In this way, a given displacement produced by the actuator is doubled at the top of the setup, assuming a rigid behavior of the vertical column. The different elements are connected by pin connections, according to the assumed mechanism. The specimen is 5.0 m high and 5.13 m wide and it is constituted, according to the mounting sequence, by: - two horizontal U guides made of 0.6 mm thick galvanized steel, screwed, both at bottom and at top, in the wooden beams by 200 mm spaced screws; - two vertical U guides made of 0.6 mm thick galvanized steel, screwed in the wooden beams by 500 mm spaced screws; - five C-shaped studs made of 0.6 mm thick galvanized steel, 900 mm spaced; they are placed in the horizontal guides without any mechanical connection (see Fig. 2a); - two steel plates (Fig. 2b), with a rectangular cross-section 100 mm × 0.6 mm, connected to the studs at two different heights of the partition, i.e. 1200 mm and 3800 mm from the base; the steel plates
Fig. 1. Global view of test setup.
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Fig. 2. Specimen mounting: (a) studs arranged in the horizontal guide, (b) steel plate connected to the stud, (c) panel screwed to the stud and steel plate, (d) paper and joint compound.
are connected to the studs with a single screw; - one 18 mm gypsum plasterboard layer for each side of the partition; the plasterboards are connected to the studs and to the steel plates by 250 mm spaced screws (see Fig. 2c); they are assembled in three rows so as to define two horizontal joints at 1200 mm and 3800 mm from the base (see Fig. 3); the joints are sealed with paper and joint compound (Fig. 2d).
partitions are built according to the installation conditions that are generally used in real buildings. A cyclic test is performed in displacement control, according to the testing protocol provided by FEMA 461 [17]. Different measuring instruments are used in order to monitor the specimen behavior during the cyclic tests. The monitoring system provides the following instrumentation typologies:
The partition features have been assessed according to an accurate survey of typical partition systems in European countries and the
- two displacement laser sensors, placed at half the height of the column and at the top of the same column, respectively, in order to monitor top in-plane displacement and verify the rigid movement of the vertical column; the monitored column is the one on the opposite side with respect to the actuator (left side in Fig. 3); - two wire potentiometers, placed in parallel with respect to the laser sensors; - two displacement transducers placed at the two edges of the top horizontal beam, which measure out-of-plane displacements, in order to validate the planarity of the motion; - eleven strain gauges, divided between the steel studs (SG1–SG5 in Fig. 4) and the plasterboards. The relationship among the top force and the top displacement, resulting from the quasi static test conducted on the partition, is shown in Fig. 5. It can be seen that the specimen exhibits a slightly nonsymmetric behavior: in the positive quarter, i.e. the pushing direction, the force reaches its maximum value corresponding to a 20.2 mm displacement (0.40% drift), while in the negative quarter the force reaches the maximum value at 25.2 mm displacement (0.50% drift). The specimen starts undergoing inelastic deformation and losing linearity at a 11.0 mm displacement (0.22% drift): some sounds denote the screws
Fig. 3. View of panel arrangement.
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Fig. 4. Instrumentation and actuator.
specimen is exhibited (Fig. 6b), i.e. the partition collapses due to the buckling of the studs. At this displacement value a significant strength degradation is visible on the hysteretic curve.
150 100
2.2. Definition of a finite element model for the partition
Force [kN]
50
A model of the tested partition is defined in order to investigate the in-plane behavior through the numerical method. SAP2000 [18] program is adopted to perform finite element analyses. The numerical model of the specimen is defined as a 2-D plane model, in order to reduce the computational effort. However, this assumption does not jeopardize the results thanks to the symmetry of the system with respect to the plane in which the partition is modeled. The complete model of the partition in SAP2000 is shown in Fig. 7a. The whole system is 5.00 m high and 5.13 m wide. It is characterized by 90 cm spaced steel studs and a single layer of plasterboards with a double thickness, i.e. 36 mm. The studs are modeled with frame elements with C-shaped cross-section; the boards, modeled as thin linear shell elements, are arranged in three distinguished horizontal rows, defining two horizontal joints; the panels of the first horizontal row are 1.2 m high, whereas the central panels are 2.6 m high and the boards at the top are 1.2 m high. In order to reproduce the actual installation conditions of the boards, horizontal and vertical gaps are included between the plasterboards and the adjacent elements both in the horizontal and in the vertical directions (Fig. 7b). The plasterboards are properly meshed with 25 cm × 25 cm shell elements: the mesh allows introducing the panel-to-stud screw connections, according to their actual spacing. Indeed, the screws, which connect the plasterboard
0 -50 -100 -150 -150
-100
-5 50 0 50 displacem nt [ m] Top p displacemen n [mm]
100
150
Fig. 5. Hysteretic curve exhibited by the partition under the selected test protocol.
bearing the connected plasterboards, the paper installed between the adjacent panels starts cracking (Fig. 6a) and a minor drop of gypsum is observed. Corresponding to a 20.2 mm top displacement, the paper between the different panels completely cracks; at this displacement, the maximum force is recorded. Corresponding to a 68.4 mm displacement (1.37% drift), a global out-of-plane curvature of the
Fig. 6. Partition damage: (a) visible opening on the paper of the lateral panel (0.22% drift) and (b) global out-of-plane curvature of the specimen (1.37% drift).
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Nonlinear lin nk
Vertical ggap p
Horizo ontal gap
(a)
(b)
Fig. 7. (a) Finite element model of the tested partition in SAP2000 and (b) detail of stud-to-panel connection evidencing the horizontal and the vertical gaps.
Table 1 (a) Steel mechanical properties and (b) gypsum wallboards mechanical properties, based on experimental tests [19].
Non-linear link in z-direction
Steel properties
Gypsum properties
j-node Tensile yielding strength,
301.0
fs [N / mm2]
Compression strength,
8.18
fg [N / mm2] Tensile strength,
1.42
fg, t [N / mm2] Young modulus, E [N / mm2]
Z
Non-linear link in x-direction X
(a)
i-node
210,000
Young’s modulus,
3601
E [N / mm2] (b)
Table 1b. Such properties are evaluated through experimental tests on both the steel studs and the gypsum boards used for the tested specimen, here omitted for the sake of brevity. Both the gypsum and the steel materials are modeled with a linear elastic material. The low stress level in the boards justifies such an assumption, as evidenced in Section 3.1; moreover, a linear behavior of the stud can be assumed up to failure, since the collapse mechanism is governed by the elastic buckling failure. The occurrence of the steel stud buckling is checked after the analysis, comparing the demand with the buckling capacity of the stud (see Section 3.2). The presence of the paper and the compound between adjacent plasterboards is neglected in the model. This choice is performed in order to both reduce the computational effort and define a simple model. It should not influence the evaluation of the interstory drift that causes the collapse of the partition. Indeed, the paper and the compound typically crack at low interstory drift demand level, much earlier than the collapse of the partition is exhibited. Finally, it should be noted that, despite the large number of elements, the model of the partition is quite simple. The nonlinearity is lumped in the panel-to-stud screwed connections; this is widely supported by the experimental evidence that showed severely damaged screwed connections before the partition failure; the failure occurred due to the buckling of the partition, which is not included in the model. The occurrence of such a failure mechanism is a-posteriori checked; it is based on the internal forces acting on the stud for a given level of displacement demand (see Section 3.2).
Fig. 8. Stud-to-panel screw connection scheme.
(node j in Fig. 8) either to the stud (node i in Fig. 8) or to the surrounding frame, are 25 cm spaced. They are modeled as nonlinear springs, i.e. NLLINK objects in SAP2000, whose backbone curve is defined in Section 2.3. These links act along the two translational directions in the plane of the partition as evidenced in Fig. 8. A single link is representative of the behavior of two screws, which connect the two plasterboard layers either to the stud or the surrounding frame. Two steel plates are placed at the two horizontal joints between the plasterboard panels; they are modeled by 100 mm × 1.2 mm rectangular cross section horizontal frames between two consecutive studs; internal hinges are placed at the intersections of the steel plate with the studs in order to reproduce the actual constraint given by a single screw connecting the steel plate to the stud. Each plasterboard is connected to the steel plate through three screws. The whole system is surrounded by a 4-hinged steel frame, representative of the steel test setup. Internal hinges are provided at the end of the beam elements in order to simulate the statically indeterminate scheme. The base horizontal steel beam is externally restrained with several hinges, which fix the base of the specimen. The steel studs are only connected to the plasterboards through nonlinear links (Fig. 8). They are not connected to the steel setup, neither at the base nor at the top. The steel material adopted for studs, horizontal plates and test frame is characterized by the mechanical properties shown in Table 1a, whereas the gypsum wallboards mechanical properties are listed in 576
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Fig. 9. (a) Schematic view of the tested specimen dimensions (in mm) and (b) assembled specimen before testing.
experimental and in the numerical force-displacement curve. The yielding displacement dy can be clearly evaluated as follows:
2.3. Calibration of the screwed connection backbone curve In order to define the mechanical behavior of the connections, some experimental tests are performed on the screwed connection adopted in the considered partition. The tested connection refers to the specific configuration of the specimen; the test setup is shown in Fig. 9: two back-to-back studs are connected to two 18 mm thick single layer plasterboards through two screws for each side. The self-drilling screws are characterized by a 3.5 mm diameter and a 35 mm length. The screws strength is evaluated in terms of Rockwell hardness, whose value is around 44. In order to include the real behavior of the screws in the model, a tri-linear spring (NLLINK in SAP2000) is introduced in the numerical model of the partition. The tri-linear envelope is evidenced in Fig. 10, where:
dy =
Fy k
(2)
The third branch of the envelope is simply obtained assuming a linear envelope from the capping point, i.e. the point characterized by the maximum force Fmax, to the ultimate point. The forces of the modeled connection are obtained scaling down the forces by a factor of two, since the tri-linear curve in Fig. 10 is representative of the behavior of four screws, whereas the nonlinear spring included in the model corresponds to two screws. 3. Analysis results and discussion
- Fmax is the maximum force reached during the experimental test and dmax is the corresponding displacement; - Fu and du are the ultimate force and displacement reached at the specimen failure, respectively; - Fy value is obtained by imposing two conditions: (a) the initial stiffness k, i.e. the slope of the first branch of the tri-linear curve, is evaluated according to Schafer [20] as:
k=
0.4Fmax d 0.4
The numerical model of the specimen described in the previous Section is subjected to a large-displacement nonlinear static analysis in displacement control through the SAP2000 program [18]. A monotonic load is applied to the model, since the nonlinear behavior of the partition is lumped in the panel-to-stud screwed connections and only the monotonic experimental curve is available for the considered connection system. The top displacement is applied in 20 consecutive steps reaching a 111 mm maximum displacement, i.e. 2.2% interstory drift. The results of the performed analysis in terms of internal forces, strain values and pushover curve are listed in the following Sections, in order to compare the numerical behavior of the modeled partition to the experimental evidence of the tested specimen.
(1)
in which d0.4 is the displacement value that corresponds to 0.4 Fmax force; (b) the dissipated energy up to Fmax is the same both in the
5
(F
,d
3.1. Global behavior of the partition
)
(F ,d )
m max max
4
u u
(F ,d )
In case the partition is pushed towards the right hand side in Fig. 7, it can be noted that the force is transferred to the base through the plasterboard panels (Fig. 11). The stress trends highlight that the compression stresses in the plasterboards (Fig. 11a) are concentrated in a diagonal strut, i.e. from the top left to the bottom right of each panel. The maximum stress values are close to 1.0 N/mm2 at 111 mm top displacement. In Fig. 11b a tensile diagonal strut is visible in each plasterboard panel from the bottom left to the top right; the maximum tension stress value is about 0.9 N/mm2. The low level of stresses justifies the modelling of the gypsum material with a linear elastic behavior (Section 2.2). In turn the panels transfer the load to the studs through the screws; the studs are therefore subjected to both bending moment and compression axial force. In the stud the compression force reaches a maximum value equal to 1.0 kN. It is confirmed by the local plastic
Force [kN]
y y
3 2 1
Ex xperimental currve N Numerical curve
0 0
2
4
6
8
10
Relative displacement [mm] Fig. 10. Experimental and tri-linear backbone curves of the screwed connection.
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0.00
1.00
-0.08
0.92
-0.15
0.85
-0.23
0.77
-0.31
0.69
-0.38
0.62
-0.46
0.54
-0.54
0.46
-0.62
0.38
-0.69
0.31
-0.77
0.23
-0.85
0.15
-0.92
0.08
-1.00
0.00
(a)
(b) 2
Fig. 11. (a) Compression and (b) tension stresses (in N/mm ) diagram on plasterboards at last step of the analysis.
of cold-formed steel stud wall whose behavior is influenced by the buckling of the studs, when subjected to compression and/or bending moment. The DSM provides three different buckling failure modes [21] of the cold-formed steel elements: - local buckling that involves a distortion of a portion of the crosssection; the half-wavelength of the local buckling mode is less than or equal to the largest characteristic dimension of the compressed member of the cross section (Fig. 15a); - distortional buckling (Fig. 15b) that produces a significant distortion of the cross-section: usually the flanges buckle presenting relative rotation with respect to the undeformed condition of the web, i.e. the section tends to "open" or to "close". The half-wavelength of distortional buckling is usually included between the local and the global buckling half-wavelength; - global buckling, or “Euler” buckling, that involves the translation and/or the rotation of the entire cross-section (Fig. 15c). The halfwavelength is determined by both the stud length and its restraint condition.
Fig. 12. Stud damage due to compression at the partition bottom.
deformation in the studs of the tested partition due to the contact at their base with the base horizontal guide (Fig. 12). The bending moment diagram on studs reveals a concentration of stress values crossing the two horizontal joints (red circle in Fig. 13a). In these zones the high stress values can justify the concentration of damage, which is experimentally pointed out exactly over and under the two horizontal joints (Fig. 13b). Furthermore, the numerical deformed shape (Fig. 14a vs Fig. 7b) points out a relative displacement between plasterboards, also evidenced in the experimental test on the partition (Fig. 14b). The results of the performed analysis are remarkable, since the behavior of the numerical model seems to reproduce quite accurately the experimental evidence.
The occurrence of the stud buckling is influenced: (a) by the mechanical and geometrical characteristics of the studs, (b) by both the sheathing system and (c) the board-to-stud connections, that provide a bracing restraint to the stud. The influence of the panels and the panelto-stud connections are modeled through elastic springs that restraint the steel stud. Three different springs, i.e. two translational ones and a rotational one, are introduced at the connection location (Fig. 16). The “kx” spring represents the contribution of the boards to the inplane lateral stiffness, taking into account the diaphragm effect of the boards and the shear stiffness of the screwed connections [22]. The shear stiffness of the connection is evaluated upon experimental tests, detailed in Section 2.3. The “ky” and “kϕ” springs are respectively representative of the out-of-plane stiffness and the rotational restraint given by the presence of the panels. The stiffness values (kx, ky and kϕ) can be evaluated through closedform formulas provided by Vieira and Schafer [20,23], even if an experimental evaluation is preferred. In the paper, experimental values for kx and ky are considered. The evaluation of the kx in-plane stiffness is explained in Section 2.3, since the backbone curve is introduced in the numerical model of the
3.2. The Direct Strength Method applied to the partitions In this Section the occurrence of the partition buckling is assessed according to the Direct Strength Method (DSM) proposed by Schafer [20]. This method is generally used to design cold-formed steel stud walls braced by sheathing connected to the stud; the partition system investigated in this paper can be included in such a structural system typology. The DSM aims at developing a reliable method for the design 578
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Fig. 13. (a) Bending moment diagram at a 68 mm top displacement and crossing horizontal joints (in red circles) and (b) observed damage on stud below the horizontal joint. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
partition. ky value is defined according to out-of-plane tests performed on the partition [24], omitted for the sake of brevity. The application of the DSM method consists in determining the axial forces and bending moments that produce the instability of the stud. The nominal axial (Pn) and flexural (Mn) strength of the stud can be assessed by using the expressions provided by AISI-S100-07 [25], based on the modeled stud cross section through the CUFSM software [26]. In particular, three resisting axial force and bending moment values are defined, i.e. one for each instability mode of failure. Since the aim of this work is to define a suitable numerical model of sheathed steel stud partitions in order to verify the seismic behavior of such components, the DSM is used as a-posteriori checking method. A domain for each buckling mode is defined (Fig. 17) to check its occurrence for a given level of displacement demand. The internal forces acting on the studs, in terms of axial force and bending moment, are compared to the limit curves. In Fig. 18 and Fig. 19 the buckling occurrence is verified for the different studs of the tested partition at different interstory drift levels. The local and the distortional domain are overlapped for the tested partition; a single limit curve is then plotted representing both local and distortional buckling (indicated for brevity as “Local instability”), as shown in Fig. 18 and Fig. 19. In Fig. 18 the occurrence of the different buckling modes is checked
Fig. 15. Three different buckling failure modes: (a) local buckling, (b) distortional buckling and (c) global buckling.
in stud no. 3 (see Fig. 7a), i.e. the stud that first exhibits buckling. Up to the 0.40% interstory drift (Fig. 18d) the internal stresses in the stud no. 3 are in the safe zone; at a 0.52% interstory drift the partition local instability occurs (Fig. 18e). When the partition reaches an interstory drift equal to 1.37% (Fig. 18n), the stud no. 3 globally buckles. The axial force-bending moment diagrams are plotted also for the
Fig. 14. (a) Deformed shape of the numerical model and (b) detail of the board overlap in the joint.
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underestimated. This phenomenon is due to the non-inclusion of the paper and the compound in the model (see Section 2.2). However, this approximation is limited up to the failure of the paper and the compound that occurs at 20 mm top displacement, i.e. well before the failure of the specimen. Finally, the experimental evidence demonstrates that at 68 mm displacement (1.37% drift) the specimen starts showing a global out-of-plane curvature (Fig. 6b). It can be deduced that the model well catches the interstory displacement required to induce global buckling failure mode of the specimen. At that point the force acting on the partition is also well predicted by the model. The positions of the cross sections of the studs where the internal forces exceed the global instability domain demonstrate that global buckling occurs exactly below the lowest horizontal joint and over the other horizontal joint, confirming the experimental evidence (Fig. 6b). In short, the proposed modelling approach is able to capture the collapse interstory drift, while its use is not recommended to assess the strength of the partition. It should be highlighted that several failure modes can occur in case a partition is subjected to in-plane loads and nonlinearity can be concentrated in different components and/or connections between components according to the material used, geometric features of the specimen, among many other influencing factors. In this specific model, the nonlinearity is concentrated in the panel-to-stud screw connections based on the experimental evidence of the tested tall partition walls [16]. Therefore the good prediction of the collapse interstory drift ratios might not be valid for partitions characterized by different geometrical and mechanical features, e.g. load-bearing stud walls.
Fig. 16. Model of the stud in the DSM: (a) stud section with springs at two sides and (b) details of the springs.
Bending Moment, M
M
ng
Local instability domain Distortional instability domain Global instability domain
M
M
nd
Di sto rti o
nl
Lo cal
buc
kli n
Safe zone
na l
Gl ob al bu ck lin bu ck li
go ccu rs
ng oc cu rs
P
nl
go cc
ur s
3.4. Experimental-numerical strain comparison
P
Axial Force, P
nd
A comparison in terms of strains between the numerical and the experimental models is illustrated in this Section. The strains recorded by SG1, SG2, SG4 and SG5 strain gauges (Fig. 4) at different top displacement levels are compared to the strains resulting from the model. The numerical strains are evaluated from the analysis of the stud cross section subjected to the bending moment and the axial force, assuming linear elastic behavior. The comparison is performed considering both the positive and the negative peak strains achieved at each cycle of the experimental test (Fig. 21). From the comparison of the strains in stud no. 1 (Fig. 21), the main difference is evidenced for small relative displacements, i.e. up to the paper cracking (blue marker), due to the absence of the paper and the compound between the plasterboards. Indeed, for low displacement demand level, the plasterboards absorb the total lateral load and the stud is lightly loaded. This phenomenon is not caught by the model, since the paper and the compound between adjacent plasterboards are not included in the model. Beyond the blue marker in Fig. 21, the steel stud is stressed and the experimental curve exhibits a slope increase. It should be noted that the slope of the strain-displacement curve is very close to the numerical one. For a relative displacement close to 30 mm, represented by the green marker in the plot, the experimental curve denotes an abrupt variation of the slope, which can be associated to the local buckling failure of the stud, confirming the numerical results for studs nos. 1 and 5. Obviously, this slope variation is not caught by the model, since the buckling verification is performed at the end of the analysis. Finally, the global buckling of the stud is clearly visible on the experimental curve for a displacement close to 68 mm. The numerical model predicts that at the same displacement three out of five studs exhibit global buckling, which induce a global failure of the whole partition. The portion of the numerical curve after 68 mm should be neglected, since global buckling failure is checked at the end of the analysis. Similar observations can be performed on the strain trends in stud no. 5 in Fig. 22. Finally, the strains are generally overestimated, due to the neglected modelling of the paper and the compound between adjacent plasterboards. However, the interstory displacement required to induce both local and global
P
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Fig. 17. Local, distortional and global buckling domain.
other studs at 1.37% interstory drift in Fig. 19. Stud no. 2 and stud no. 4 globally buckle at this step, whereas in stud no. 1 and stud no. 5 only local buckling occurs. Nevertheless, the whole partition can be considered to be failed at this step. The DSM provides that the capacity of the stud is conceived as the minimum between local, distortional and global buckling load capacity. It is therefore implicit that the steel studs are designed in order not to exhibit any of the three considered buckling failure modes. However, in this case the three distinct buckling failure modes are considered since a numerical-experimental comparison is performed. 3.3. Experimental-numerical damage comparison A comparison between the experimental damage of the tested partition and the prediction of the numerical model is performed in this Section. The comparison between the numerical pushover curve and the experimental force-displacement backbone curve is shown in Fig. 20. In the numerical model, nonlinear behavior of the screws occurs for a top displacement equal to 10 mm (green marker in Fig. 20); the failure of the partition (red marker in Fig. 20) occurs at 1.37% interstory drift, considering the occurrence of global instability in a single stud as the partition failure. Beyond this point the curve is plotted as a dotted line since it is not representative of the partition behavior. The experimental curve exhibits an initial stiffness similar to the stiffness recorded in the numerical model. The screw bearing mechanism occurs at a 11 mm displacement, which is similar to the displacement required to yield some screws in the numerical model. Beyond a 10 mm top displacement, the numerical curve does not match the experimental one: both the strength and the stiffness are 580
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Fig. 19. DSM verification on stud 1 (a), 2 (b), 4 (c) and 5 (d) at 1.37% interstory drift.
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widely supported by the experimental evidence, which showed severely damaged screwed connections before the partition failure. A tri-linear force-displacement backbone curve is assigned to the screwed connections, matching the experimental results of monotonic tests on such a connection. The failure of the partition due to elastic global buckling is a-posteriori checked, based on the internal forces acting in the steel studs. The numerical results of a monotonic test on the defined model evidence tension and compression struts in the plasterboards due to the applied top displacement. The low stress values both in tension and in compression justify the adoption of a linear elastic material for the boards. The bending moment diagram on studs reveals large demand crossing the two horizontal joints between the plasterboards. Such an evidence can justify the damage experimentally pointed out in the steel stud over and under the two horizontal joints.
instabilities are well predicted. 4. Conclusions The research study deals with the definition of a finite element model capable to capture the interstory drift which causes the failure of a 5 m high plasterboard partition, representative of European partition systems. The assessment is performed comparing the numerical behavior of a specific specimen with the experimental evidence of a quasistatic test campaign conducted at the Laboratory of the Department of Structures for Engineering and Architecture at the University of Naples Federico II. A simple model of the tested partition is defined. Both the steel studs and the plasterboards are modeled with linear elastic elements. The nonlinearity is lumped in the panel-to-stud screwed connections; this is 582
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model for the prediction of the instability failure drift through an a posteriori check. The strains are generally overestimated in the model, due to the neglected modelling of the paper and the compound between adjacent plasterboards. The numerical model, herein shown, represents the initial stage for the modelling of internal plasterboard partitions, since it only allows the determination of the instability failure in-plane drift, neglecting the out-of-plane behavior of the partition. Furthermore, its nonlinear behavior is only lumped at panel-to-stud screw connections. Nevertheless, the presented model allows determining the buckling collapse drift of tall internal plasterboard partitions, which is a key step in performance based earthquake engineering.
The Direct Strength Method (DSM) is applied to assess the occurrence of different buckling failure modes, i.e. local, distortional and global failure modes, in the studs. This method allows considering the restraining effect given by both the presence of the plasterboards and the screwed connections, through the presence of linear springs on the steel stud cross section. The method evidences that both local and distortional instability failure modes occur at a 0.52% interstory drift, whereas the global buckling is exhibited at an interstory drift equal to 1.37%. The experimental test evidences that for a 1.37% drift the specimen starts showing a global out-of-plane curvature: it can be therefore deduced that the model well catches the global buckling failure mode of the specimen. The experimental curve is well reproduced by the numerical one in terms of initial stiffness, whereas the model underestimates the strength of the partition. However, this underestimation, due to the non-inclusion of the paper and the compound in the model, is limited up to the failure of the paper and the compound, which occurs much earlier than the failure of the specimen. The trends of the experimental strain in the steel studs at different interstory drifts confirm the suitability of the
Acknowledgements This research study has been funded both by the Italian Department of Civil Protection in the frame of the national project DPC - ReLUIS 2015 - RS8 and by Siniat International (AGM10004LP). The support provided by Eng. Marco Russo during the execution of the analyses is 583
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[12] P. Liu, K.D. Peterman, B.W. Schafer, Test Report on Cold-Formed Steel Shear Walls. Research Report, CFS-NEES, RR03, Access at 〈www.ce.jhu.edu/cfsnees〉, 2012. [13] K.D. Peterman, B.W. Schafer, Hysteretic Shear Response of Fasteners Connecting Sheathing to Cold-formed Steel Studs. Research Report, CFS-NEES, RR04, Access at 〈www.ce.jhu.edu/cfsnees〉, 2013. [14] S.G. Buonopane, G. Bian, T.H. Tun, B.W. Schafer, Computationally efficient fastener-based models of cold-formed steel shear walls with wood sheating, J. Constr. Steel Res. 110 (2015) 137–148. [15] E. Rahmanishamsi, S. Soroushian, E.M. Maragakis, Analytical model for the inplane seismic performance of cold-formed steel-framed gypsum partition walls, Earthq. Eng. Struct. Dyn. 45 (2016) 619–634. [16] C. Petrone, G. Magliulo, P. Lopez, G. Manfredi, Seismic fragility of plasterboard partitions via in-plane quasi-static tests, Earthq. Eng. Struct. Dyn. 44 (2015) 2589–2606. [17] Federal Emergency Management Agency (FEMA), Interim Protocols for Determining Seismic Performance Characteristics of Structural and Nonstructural Components through Laboratory Testing. Report No. FEMA 461, Washington DC, USA., 2007. [18] CSI Computer & Structures Inc., SAP2000, Linear and Nonlinear Static and Dynamic Analysis of Three-Dimensional Structures, Computer & Structures, Inc, Berkeley, California, 2004. [19] C. Petrone, G. Magliulo, G. Manfredi, Mechanical properties of plasterboards: experimental tests and statistical analysis, J. Mater. Civ. Eng. 28 (11) (2016) 04016129. [20] B.W. Schafer, Sheathing Braced Design of Wall Studs – Final Report, Johns Hopkins University, Baltimore, MD, US, 2013. [21] Direct Strength Method (DSM) – Design Guide., American Iron and Steel Institute, 2006. [22] L.C.M. Vieira, B.W. Schafer, Lateral stiffness and strength of sheathing braced coldformed steel stud walls, Eng. Struct. 37 (2012) 205–213. [23] L.C.M. Vieira, B.W. Schafer, Behavior and design of sheathed cold-formed steel stud walls under compression, J. Struct. Eng.-ASCE 139 (5) (2013) 772–786. [24] C. Petrone, G. Magliulo, G. Manfredi, Out-of-plane seismic performance of plasterboard partition walls via quasi-static tests, Bull. N. Z. Soc. Earthq. Eng. 49 (1) (2016) 125–137. [25] AISI-S100, North American Specification for the Design of Cold-Formed Steel Structural Member, American Iron and Steel Institute, Canada, 2007. [26] CUFSM, Elastic Buckling Analysis of Thin-walled Members – 〈www.ce.jhu.edu/ bschafer/cufsm〉.
gratefully acknowledged. References [1] E. Ikuta, M. Miyano, Study of damage to the human body caused by earthquakes: development of a mannequin for thoracic compression experiments and cyber mannequin using the finite element method, in: R. Spence, E. So, C. Scawthorn (Eds.), Human Casualties in Earthquakes, Springer, Netherlands, 2011, pp. 275–289. [2] S. Taghavi, E. Miranda, Response Assessment of Nonstructural Building Elements, PEER Report 2003/05, College of Engineering, University of California Berkeley, USA, 2003. [3] G. Magliulo, C. Petrone, V. Capozzi, G. Maddaloni, P. Lopez, G. Manfredi, Seismic performance evaluation of plasterboard partitions via shake table tests, Bull. Earthq. Eng. 12 (4) (2014) 1657–1677. [4] R. Retamales, R. Davies, G. Mosqueda, A. Filiatrault, Experimental seismic fragility of cold-formed steel framed gypsum partition walls, J. Struct. Eng. 139 (8) (2013) 1285–1293. [5] A.M. Kanvinde, G.G. Deierlein, Analytical models for the seismic performance of gypsum drywall partitions, Earthq. Spectra 22 (2) (2006) 391–411. [6] R. Davies, R. Retamales, G. Mosqueda, A. Filiatrault, Experimental Seismic Evaluation, Model Parameterization, and Effects of Cold-Formed Steel-Framed Gypsum Partition Walls on the Seismic Performance of an Essential Facility. Technical Report MCEER-11-0005, University at Buffalo, State University of New York, 2011. [7] R.L. Wood, T. Hutchinson, A Numerical Model for Capturing the In-plane Seismic Response of Interior Metal Stud Partition Walls. Technical Report MCEER-12-0007, University of California, San Diego, 2012. [8] F. McKenna, G.L. Fenves, OpenSees Manual 〈http://opensees.berkeley.edu〉, Pacific Earthquake Engineering Research Center, Berkeley, California, 2013. [9] Y. Telue, M. Mahendran, Behaviour and design of cold-formed steel wall frames lined with plasterboard on both sides, Eng. Struct. 26 (5) (2004) 567–579. [10] L. Fiorino, O. Iuorio, R. Landolfo, Sheathed cold-formed steel housing: a seismic design procedure, Thin-Walled Struct. 47 (8–9) (2009) 919–930. [11] S.G. Buonopane, T.H. Tun, B.W. Schafer, Fastener-based computational models for prediction of seismic behavior of CFS shear walls, in: Proceedings of the 10th National Conference in Earthquake Engineering, Earthquake Engineering Research Institute, Anchorage, AK, July 2014, 2014.
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Full length article
Numerical model for the in-plane seismic capacity evaluation of tall plasterboard internal partitions
MARK
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Crescenzo Petronea,1, Orsola Coppolaa, Gennaro Magliuloa, , Pauline Lopezb, Gaetano Manfredia a b
University of Naples Federico II, Department of Structures for Engineering and Architecture, via Claudio 21, 80125 Naples, Italy Siniat International, 500 Rue Marcel Demonque - Zone Agroparc, CS 70088, 84915 Avignon Cedex 9, France
A R T I C L E I N F O
A B S T R A C T
Keywords: Nonstructural components Plasterboard partitions Numerical model FEM analysis Seismic performance
A finite element model capable to capture the interstory drift which causes the failure of a 5 m high plasterboard partition with steel studs is defined by employing the Direct Strength Method (DSM) to assess the occurrence of different buckling failure modes. The model is validated comparing the numerical behavior with the experimental evidence of a quasi-static test campaign on a plasterboard partition. It is concluded that the model well catches the interstory drifts which cause either the local or the global buckling in the partition. This conclusion is also confirmed by the strain trend in the steel studs.
1. Introduction Nowadays the seismic performance of nonstructural components is recognized to be a key issue in the framework of the Performance-Based Earthquake Engineering (PBEE). Four main issues motivate several studies related both to the evaluation of the seismic capacity and the demand of nonstructural components. Firstly, nonstructural components can cause injuries or deaths after an earthquake; for instance the 64% of the fatalities caused by 1995 Great Hanshin Earthquake was due to the compression (suffocation) of the human body [1]. Such a phenomenon could be caused by the damage to nonstructural components, that may also obstruct the way out from the damaged building. Moreover, nonstructural components generally exhibit damage for low seismic demand levels. The seismic performance of nonstructural components is especially important in frequent, and less intense, earthquakes, in which their damage can cause the inoperability of several buildings. Furthermore, the cost connected to nonstructural components represents the largest portion of a building construction. Taghavi and Miranda [2] evidenced that structural cost typically represents a small portion of the total cost of a building construction, corresponding to 18% for offices, 13% for hotels and 8% for hospitals. Finally, nonstructural components may participate in the lateral resisting system of the primary structure, i.e. varying its lateral strength and stiffness. Plasterboard internal partitions with steel studs are very common nonstructural components, since they are typically employed in several
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building typologies all over the world. Different experimental studies aiming at the evaluation of the seismic capacity of such components are available in literature [3,4]. The seismic evaluation is typically expressed in terms of the Engineering Demand Parameter (EDP) that is required to reach a certain Damage State (DS). The interstory drift is typically selected as the EDP; the aim of these studies is therefore to evaluate the interstory drift that induces different damage states, i.e. from minor to major damage states, in the partition systems. In Kanvinde and Deierlein [5] a macro-model is presented in order to evaluate the seismic behavior of plasterboard partitions. The authors propose an analytical model to determine the lateral shear strength and initial elastic stiffness of wood and gypsum wall panels. In such a case, a uniaxial spring model is defined, by a series of parameters defining the backbone curve, which represents the nonlinear monotonic response that envelopes the cyclic response, and the cyclic nonlinear response including strength and stiffness degradation and pinching phenomenon. The parameters validation is performed by using experimental tests on full-scale wall panels. In Davies et al. [4,6] a description of the experimental results of full-scale tests performed on several cold-formed steel-framed gypsum partitions is reported. The experimental data, including different partition wall configurations, in terms of wall dimensions, material type, testing protocol and boundary conditions are used in order to create seismic fragility curves for such nonstructural partition walls. Moreover, on the base of experimental results, a trilinear hysteretic macro-model is proposed to reproduce the in-plane mechanical behavior of the partitions: it allowed to include partitions in
Corresponding author. E-mail addresses: [email protected] (C. Petrone), [email protected] (O. Coppola), [email protected] (G. Magliulo), [email protected] (P. Lopez), [email protected] (G. Manfredi). 1 Present address: Willis Group Limited, 51 Lime Street, 20th floor, London EC3M 7DQ, UK. http://dx.doi.org/10.1016/j.tws.2017.10.047 Received 7 June 2016; Received in revised form 24 October 2017; Accepted 31 October 2017 0263-8231/ © 2017 Elsevier Ltd. All rights reserved.
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plane to the main structure. A simple modelling technique is proposed and validated. The validation is performed comparing the numerical behavior of a specific specimen with the experimental results achieved in a quasi-static test campaign conducted at the Laboratory of the Department of Structures for Engineering and Architecture at the University of Naples Federico II. The quasi-static test is performed on a 5 m tall plasterboard internal partition [16], which is representative of the typical partitions used in industrial and commercial buildings in the European countries.
the model of an existing steel building in order to demonstrate the effect on the seismic behavior of the whole structure. A numerical macromodel for plasterboard partition is proposed by Wood and Hutchinson [7]; a pinching material model, available in OpenSees [8], is used in order to reproduce the in-plane behavior of the partitions. The 24 parameters of the model are calibrated by a large number of experimental data obtained from about fifty tests performed on plasterboard partition walls [4]. Telue and Mahendran [9] conducted experimental studies on cold formed steel walls lined with plasterboard. These studies point out that the stud compression strength increases when the steel internal frame is covered by plasterboard on one or both sides. A finite element model was developed and validated using experimental results. However, the behavior of the walls is investigated under compressive vertical loads. Fiorino et al. [10] proposed a seismic design procedure for sheathed cold-formed steel shear walls: through linear dynamic and nonlinear static analysis three nomographs are obtained in order to determine the features of a seismic resistant shear wall. An iterative procedure for the prediction of the pushover response curve of sheathed cold-formed steel shear walls is also presented. A computational model was proposed by Buonopane, Tun and Schafer [11] for the prediction of the seismic behavior of sheathed cold formed steel shear walls, based on experimental tests performed both on shear walls [12] and on screwed panel-to-stud connections [13]. Buonopane et al. [14] also developed a numerical model for the in-plane lateral behavior of cold-formed steel shear walls with wood panels, which captures the nonlinear behavior at the interface between cover panels and fasteners. Indeed, full scale lateral load tests on thirteen shear walls exhibited collapse due to the failure of the fasteners. The modeled walls, having a height of 2.74 m and widths of either 1.22 or 2.44 m, were able to predict the cyclic response of the tested specimens, up to the peak point of lateral strength. However, these research studies focused on steel stud shear walls, whose characteristics, e.g. stud typology, restraint at the base, failure mode, are different than in internal partition walls. The evaluation of the seismic capacity of a plasterboard partition system is typically pursued through the experimental method. The development of a FEM numerical model of such a component would allow to investigate the seismic behavior of a generic partition, taking into account all its features, e.g. studs geometrical and mechanical properties, layer of plasterboards, etc. A detailed numerical model of coldformed steel-framed gypsum partition walls is proposed by Rahmanishamsi, Soroushian and Maragakis [15]. The model, built in OpenSees, provides nonlinear behavior of studs and tracks, and nonlinear behavior of gypsum to stud and stud to track connections. The gypsum boards are simulated by linear shell elements. The contact between boards/tracks and concrete slabs as well as the contact between adjacent boards are also considered in the model. The model, validated by the comparison with experimental results on gypsum partitions walls, is able to predict the trend of the response and the observed damage mechanism. The failure of the tested partitions typically consists in the failure of gypsum-to-tracks/studs screws connections and/or crushing of gypsum boards. The developed numerical model does not incorporate failure due to buckling of the steel stud, which is the typical failure mode in tall plasterboard partition walls, as shown by Petrone, Magliulo, Lopez and Manfredi [16]. Moreover, the experimental tests used for numerical validation are performed on partition walls representative of US partition systems, whose features might be different than European ones. The aim of this research study is the definition of a finite element model able to evaluate the interstory drift that induces the failure of tall plasterboard partitions representative of European partition systems commonly mounted in commercial and industrial buildings. Even if partition walls can be subjected to both the in-plane interstory drift and the out-of-plane acceleration, this study is focused only on the in-plane seismic performance assessment of partitions in existing structures. It is therefore assumed that the investigated partitions would not fail in the out-of-plane direction and that they are rigidly connected in their own
2. Methodology 2.1. Experimental test on the plasterboard partition specimen In this Section the experimental test performed on the considered plasterboard partition specimen is illustrated. The test campaign, exhaustively reported in a previous paper [16], is here briefly described. Particular attention is given to the description of the tested partition and the mounting procedure in order to justify the finite element modelling of the specimen, included in Section 2.2. The test system consists of a steel frame setup, the specimen, i.e. a plasterboard partition, a hydraulic actuator and a reaction wall (Fig. 1). The steel test frame is conceived as a statically indeterminate scheme, since it is not designed to simulate the performance of the structure itself but just in order to transfer the load provided by the hydraulic jack to the partition without absorbing lateral forces. Moreover, since the reaction wall cannot reach the height of the system, the actuator is placed at the middle height of the test setup (Fig. 1). In this way, a given displacement produced by the actuator is doubled at the top of the setup, assuming a rigid behavior of the vertical column. The different elements are connected by pin connections, according to the assumed mechanism. The specimen is 5.0 m high and 5.13 m wide and it is constituted, according to the mounting sequence, by: - two horizontal U guides made of 0.6 mm thick galvanized steel, screwed, both at bottom and at top, in the wooden beams by 200 mm spaced screws; - two vertical U guides made of 0.6 mm thick galvanized steel, screwed in the wooden beams by 500 mm spaced screws; - five C-shaped studs made of 0.6 mm thick galvanized steel, 900 mm spaced; they are placed in the horizontal guides without any mechanical connection (see Fig. 2a); - two steel plates (Fig. 2b), with a rectangular cross-section 100 mm × 0.6 mm, connected to the studs at two different heights of the partition, i.e. 1200 mm and 3800 mm from the base; the steel plates
Fig. 1. Global view of test setup.
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Fig. 2. Specimen mounting: (a) studs arranged in the horizontal guide, (b) steel plate connected to the stud, (c) panel screwed to the stud and steel plate, (d) paper and joint compound.
are connected to the studs with a single screw; - one 18 mm gypsum plasterboard layer for each side of the partition; the plasterboards are connected to the studs and to the steel plates by 250 mm spaced screws (see Fig. 2c); they are assembled in three rows so as to define two horizontal joints at 1200 mm and 3800 mm from the base (see Fig. 3); the joints are sealed with paper and joint compound (Fig. 2d).
partitions are built according to the installation conditions that are generally used in real buildings. A cyclic test is performed in displacement control, according to the testing protocol provided by FEMA 461 [17]. Different measuring instruments are used in order to monitor the specimen behavior during the cyclic tests. The monitoring system provides the following instrumentation typologies:
The partition features have been assessed according to an accurate survey of typical partition systems in European countries and the
- two displacement laser sensors, placed at half the height of the column and at the top of the same column, respectively, in order to monitor top in-plane displacement and verify the rigid movement of the vertical column; the monitored column is the one on the opposite side with respect to the actuator (left side in Fig. 3); - two wire potentiometers, placed in parallel with respect to the laser sensors; - two displacement transducers placed at the two edges of the top horizontal beam, which measure out-of-plane displacements, in order to validate the planarity of the motion; - eleven strain gauges, divided between the steel studs (SG1–SG5 in Fig. 4) and the plasterboards. The relationship among the top force and the top displacement, resulting from the quasi static test conducted on the partition, is shown in Fig. 5. It can be seen that the specimen exhibits a slightly nonsymmetric behavior: in the positive quarter, i.e. the pushing direction, the force reaches its maximum value corresponding to a 20.2 mm displacement (0.40% drift), while in the negative quarter the force reaches the maximum value at 25.2 mm displacement (0.50% drift). The specimen starts undergoing inelastic deformation and losing linearity at a 11.0 mm displacement (0.22% drift): some sounds denote the screws
Fig. 3. View of panel arrangement.
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Fig. 4. Instrumentation and actuator.
specimen is exhibited (Fig. 6b), i.e. the partition collapses due to the buckling of the studs. At this displacement value a significant strength degradation is visible on the hysteretic curve.
150 100
2.2. Definition of a finite element model for the partition
Force [kN]
50
A model of the tested partition is defined in order to investigate the in-plane behavior through the numerical method. SAP2000 [18] program is adopted to perform finite element analyses. The numerical model of the specimen is defined as a 2-D plane model, in order to reduce the computational effort. However, this assumption does not jeopardize the results thanks to the symmetry of the system with respect to the plane in which the partition is modeled. The complete model of the partition in SAP2000 is shown in Fig. 7a. The whole system is 5.00 m high and 5.13 m wide. It is characterized by 90 cm spaced steel studs and a single layer of plasterboards with a double thickness, i.e. 36 mm. The studs are modeled with frame elements with C-shaped cross-section; the boards, modeled as thin linear shell elements, are arranged in three distinguished horizontal rows, defining two horizontal joints; the panels of the first horizontal row are 1.2 m high, whereas the central panels are 2.6 m high and the boards at the top are 1.2 m high. In order to reproduce the actual installation conditions of the boards, horizontal and vertical gaps are included between the plasterboards and the adjacent elements both in the horizontal and in the vertical directions (Fig. 7b). The plasterboards are properly meshed with 25 cm × 25 cm shell elements: the mesh allows introducing the panel-to-stud screw connections, according to their actual spacing. Indeed, the screws, which connect the plasterboard
0 -50 -100 -150 -150
-100
-5 50 0 50 displacem nt [ m] Top p displacemen n [mm]
100
150
Fig. 5. Hysteretic curve exhibited by the partition under the selected test protocol.
bearing the connected plasterboards, the paper installed between the adjacent panels starts cracking (Fig. 6a) and a minor drop of gypsum is observed. Corresponding to a 20.2 mm top displacement, the paper between the different panels completely cracks; at this displacement, the maximum force is recorded. Corresponding to a 68.4 mm displacement (1.37% drift), a global out-of-plane curvature of the
Fig. 6. Partition damage: (a) visible opening on the paper of the lateral panel (0.22% drift) and (b) global out-of-plane curvature of the specimen (1.37% drift).
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Nonlinear lin nk
Vertical ggap p
Horizo ontal gap
(a)
(b)
Fig. 7. (a) Finite element model of the tested partition in SAP2000 and (b) detail of stud-to-panel connection evidencing the horizontal and the vertical gaps.
Table 1 (a) Steel mechanical properties and (b) gypsum wallboards mechanical properties, based on experimental tests [19].
Non-linear link in z-direction
Steel properties
Gypsum properties
j-node Tensile yielding strength,
301.0
fs [N / mm2]
Compression strength,
8.18
fg [N / mm2] Tensile strength,
1.42
fg, t [N / mm2] Young modulus, E [N / mm2]
Z
Non-linear link in x-direction X
(a)
i-node
210,000
Young’s modulus,
3601
E [N / mm2] (b)
Table 1b. Such properties are evaluated through experimental tests on both the steel studs and the gypsum boards used for the tested specimen, here omitted for the sake of brevity. Both the gypsum and the steel materials are modeled with a linear elastic material. The low stress level in the boards justifies such an assumption, as evidenced in Section 3.1; moreover, a linear behavior of the stud can be assumed up to failure, since the collapse mechanism is governed by the elastic buckling failure. The occurrence of the steel stud buckling is checked after the analysis, comparing the demand with the buckling capacity of the stud (see Section 3.2). The presence of the paper and the compound between adjacent plasterboards is neglected in the model. This choice is performed in order to both reduce the computational effort and define a simple model. It should not influence the evaluation of the interstory drift that causes the collapse of the partition. Indeed, the paper and the compound typically crack at low interstory drift demand level, much earlier than the collapse of the partition is exhibited. Finally, it should be noted that, despite the large number of elements, the model of the partition is quite simple. The nonlinearity is lumped in the panel-to-stud screwed connections; this is widely supported by the experimental evidence that showed severely damaged screwed connections before the partition failure; the failure occurred due to the buckling of the partition, which is not included in the model. The occurrence of such a failure mechanism is a-posteriori checked; it is based on the internal forces acting on the stud for a given level of displacement demand (see Section 3.2).
Fig. 8. Stud-to-panel screw connection scheme.
(node j in Fig. 8) either to the stud (node i in Fig. 8) or to the surrounding frame, are 25 cm spaced. They are modeled as nonlinear springs, i.e. NLLINK objects in SAP2000, whose backbone curve is defined in Section 2.3. These links act along the two translational directions in the plane of the partition as evidenced in Fig. 8. A single link is representative of the behavior of two screws, which connect the two plasterboard layers either to the stud or the surrounding frame. Two steel plates are placed at the two horizontal joints between the plasterboard panels; they are modeled by 100 mm × 1.2 mm rectangular cross section horizontal frames between two consecutive studs; internal hinges are placed at the intersections of the steel plate with the studs in order to reproduce the actual constraint given by a single screw connecting the steel plate to the stud. Each plasterboard is connected to the steel plate through three screws. The whole system is surrounded by a 4-hinged steel frame, representative of the steel test setup. Internal hinges are provided at the end of the beam elements in order to simulate the statically indeterminate scheme. The base horizontal steel beam is externally restrained with several hinges, which fix the base of the specimen. The steel studs are only connected to the plasterboards through nonlinear links (Fig. 8). They are not connected to the steel setup, neither at the base nor at the top. The steel material adopted for studs, horizontal plates and test frame is characterized by the mechanical properties shown in Table 1a, whereas the gypsum wallboards mechanical properties are listed in 576
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Fig. 9. (a) Schematic view of the tested specimen dimensions (in mm) and (b) assembled specimen before testing.
experimental and in the numerical force-displacement curve. The yielding displacement dy can be clearly evaluated as follows:
2.3. Calibration of the screwed connection backbone curve In order to define the mechanical behavior of the connections, some experimental tests are performed on the screwed connection adopted in the considered partition. The tested connection refers to the specific configuration of the specimen; the test setup is shown in Fig. 9: two back-to-back studs are connected to two 18 mm thick single layer plasterboards through two screws for each side. The self-drilling screws are characterized by a 3.5 mm diameter and a 35 mm length. The screws strength is evaluated in terms of Rockwell hardness, whose value is around 44. In order to include the real behavior of the screws in the model, a tri-linear spring (NLLINK in SAP2000) is introduced in the numerical model of the partition. The tri-linear envelope is evidenced in Fig. 10, where:
dy =
Fy k
(2)
The third branch of the envelope is simply obtained assuming a linear envelope from the capping point, i.e. the point characterized by the maximum force Fmax, to the ultimate point. The forces of the modeled connection are obtained scaling down the forces by a factor of two, since the tri-linear curve in Fig. 10 is representative of the behavior of four screws, whereas the nonlinear spring included in the model corresponds to two screws. 3. Analysis results and discussion
- Fmax is the maximum force reached during the experimental test and dmax is the corresponding displacement; - Fu and du are the ultimate force and displacement reached at the specimen failure, respectively; - Fy value is obtained by imposing two conditions: (a) the initial stiffness k, i.e. the slope of the first branch of the tri-linear curve, is evaluated according to Schafer [20] as:
k=
0.4Fmax d 0.4
The numerical model of the specimen described in the previous Section is subjected to a large-displacement nonlinear static analysis in displacement control through the SAP2000 program [18]. A monotonic load is applied to the model, since the nonlinear behavior of the partition is lumped in the panel-to-stud screwed connections and only the monotonic experimental curve is available for the considered connection system. The top displacement is applied in 20 consecutive steps reaching a 111 mm maximum displacement, i.e. 2.2% interstory drift. The results of the performed analysis in terms of internal forces, strain values and pushover curve are listed in the following Sections, in order to compare the numerical behavior of the modeled partition to the experimental evidence of the tested specimen.
(1)
in which d0.4 is the displacement value that corresponds to 0.4 Fmax force; (b) the dissipated energy up to Fmax is the same both in the
5
(F
,d
3.1. Global behavior of the partition
)
(F ,d )
m max max
4
u u
(F ,d )
In case the partition is pushed towards the right hand side in Fig. 7, it can be noted that the force is transferred to the base through the plasterboard panels (Fig. 11). The stress trends highlight that the compression stresses in the plasterboards (Fig. 11a) are concentrated in a diagonal strut, i.e. from the top left to the bottom right of each panel. The maximum stress values are close to 1.0 N/mm2 at 111 mm top displacement. In Fig. 11b a tensile diagonal strut is visible in each plasterboard panel from the bottom left to the top right; the maximum tension stress value is about 0.9 N/mm2. The low level of stresses justifies the modelling of the gypsum material with a linear elastic behavior (Section 2.2). In turn the panels transfer the load to the studs through the screws; the studs are therefore subjected to both bending moment and compression axial force. In the stud the compression force reaches a maximum value equal to 1.0 kN. It is confirmed by the local plastic
Force [kN]
y y
3 2 1
Ex xperimental currve N Numerical curve
0 0
2
4
6
8
10
Relative displacement [mm] Fig. 10. Experimental and tri-linear backbone curves of the screwed connection.
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0.00
1.00
-0.08
0.92
-0.15
0.85
-0.23
0.77
-0.31
0.69
-0.38
0.62
-0.46
0.54
-0.54
0.46
-0.62
0.38
-0.69
0.31
-0.77
0.23
-0.85
0.15
-0.92
0.08
-1.00
0.00
(a)
(b) 2
Fig. 11. (a) Compression and (b) tension stresses (in N/mm ) diagram on plasterboards at last step of the analysis.
of cold-formed steel stud wall whose behavior is influenced by the buckling of the studs, when subjected to compression and/or bending moment. The DSM provides three different buckling failure modes [21] of the cold-formed steel elements: - local buckling that involves a distortion of a portion of the crosssection; the half-wavelength of the local buckling mode is less than or equal to the largest characteristic dimension of the compressed member of the cross section (Fig. 15a); - distortional buckling (Fig. 15b) that produces a significant distortion of the cross-section: usually the flanges buckle presenting relative rotation with respect to the undeformed condition of the web, i.e. the section tends to "open" or to "close". The half-wavelength of distortional buckling is usually included between the local and the global buckling half-wavelength; - global buckling, or “Euler” buckling, that involves the translation and/or the rotation of the entire cross-section (Fig. 15c). The halfwavelength is determined by both the stud length and its restraint condition.
Fig. 12. Stud damage due to compression at the partition bottom.
deformation in the studs of the tested partition due to the contact at their base with the base horizontal guide (Fig. 12). The bending moment diagram on studs reveals a concentration of stress values crossing the two horizontal joints (red circle in Fig. 13a). In these zones the high stress values can justify the concentration of damage, which is experimentally pointed out exactly over and under the two horizontal joints (Fig. 13b). Furthermore, the numerical deformed shape (Fig. 14a vs Fig. 7b) points out a relative displacement between plasterboards, also evidenced in the experimental test on the partition (Fig. 14b). The results of the performed analysis are remarkable, since the behavior of the numerical model seems to reproduce quite accurately the experimental evidence.
The occurrence of the stud buckling is influenced: (a) by the mechanical and geometrical characteristics of the studs, (b) by both the sheathing system and (c) the board-to-stud connections, that provide a bracing restraint to the stud. The influence of the panels and the panelto-stud connections are modeled through elastic springs that restraint the steel stud. Three different springs, i.e. two translational ones and a rotational one, are introduced at the connection location (Fig. 16). The “kx” spring represents the contribution of the boards to the inplane lateral stiffness, taking into account the diaphragm effect of the boards and the shear stiffness of the screwed connections [22]. The shear stiffness of the connection is evaluated upon experimental tests, detailed in Section 2.3. The “ky” and “kϕ” springs are respectively representative of the out-of-plane stiffness and the rotational restraint given by the presence of the panels. The stiffness values (kx, ky and kϕ) can be evaluated through closedform formulas provided by Vieira and Schafer [20,23], even if an experimental evaluation is preferred. In the paper, experimental values for kx and ky are considered. The evaluation of the kx in-plane stiffness is explained in Section 2.3, since the backbone curve is introduced in the numerical model of the
3.2. The Direct Strength Method applied to the partitions In this Section the occurrence of the partition buckling is assessed according to the Direct Strength Method (DSM) proposed by Schafer [20]. This method is generally used to design cold-formed steel stud walls braced by sheathing connected to the stud; the partition system investigated in this paper can be included in such a structural system typology. The DSM aims at developing a reliable method for the design 578
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Fig. 13. (a) Bending moment diagram at a 68 mm top displacement and crossing horizontal joints (in red circles) and (b) observed damage on stud below the horizontal joint. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).
partition. ky value is defined according to out-of-plane tests performed on the partition [24], omitted for the sake of brevity. The application of the DSM method consists in determining the axial forces and bending moments that produce the instability of the stud. The nominal axial (Pn) and flexural (Mn) strength of the stud can be assessed by using the expressions provided by AISI-S100-07 [25], based on the modeled stud cross section through the CUFSM software [26]. In particular, three resisting axial force and bending moment values are defined, i.e. one for each instability mode of failure. Since the aim of this work is to define a suitable numerical model of sheathed steel stud partitions in order to verify the seismic behavior of such components, the DSM is used as a-posteriori checking method. A domain for each buckling mode is defined (Fig. 17) to check its occurrence for a given level of displacement demand. The internal forces acting on the studs, in terms of axial force and bending moment, are compared to the limit curves. In Fig. 18 and Fig. 19 the buckling occurrence is verified for the different studs of the tested partition at different interstory drift levels. The local and the distortional domain are overlapped for the tested partition; a single limit curve is then plotted representing both local and distortional buckling (indicated for brevity as “Local instability”), as shown in Fig. 18 and Fig. 19. In Fig. 18 the occurrence of the different buckling modes is checked
Fig. 15. Three different buckling failure modes: (a) local buckling, (b) distortional buckling and (c) global buckling.
in stud no. 3 (see Fig. 7a), i.e. the stud that first exhibits buckling. Up to the 0.40% interstory drift (Fig. 18d) the internal stresses in the stud no. 3 are in the safe zone; at a 0.52% interstory drift the partition local instability occurs (Fig. 18e). When the partition reaches an interstory drift equal to 1.37% (Fig. 18n), the stud no. 3 globally buckles. The axial force-bending moment diagrams are plotted also for the
Fig. 14. (a) Deformed shape of the numerical model and (b) detail of the board overlap in the joint.
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underestimated. This phenomenon is due to the non-inclusion of the paper and the compound in the model (see Section 2.2). However, this approximation is limited up to the failure of the paper and the compound that occurs at 20 mm top displacement, i.e. well before the failure of the specimen. Finally, the experimental evidence demonstrates that at 68 mm displacement (1.37% drift) the specimen starts showing a global out-of-plane curvature (Fig. 6b). It can be deduced that the model well catches the interstory displacement required to induce global buckling failure mode of the specimen. At that point the force acting on the partition is also well predicted by the model. The positions of the cross sections of the studs where the internal forces exceed the global instability domain demonstrate that global buckling occurs exactly below the lowest horizontal joint and over the other horizontal joint, confirming the experimental evidence (Fig. 6b). In short, the proposed modelling approach is able to capture the collapse interstory drift, while its use is not recommended to assess the strength of the partition. It should be highlighted that several failure modes can occur in case a partition is subjected to in-plane loads and nonlinearity can be concentrated in different components and/or connections between components according to the material used, geometric features of the specimen, among many other influencing factors. In this specific model, the nonlinearity is concentrated in the panel-to-stud screw connections based on the experimental evidence of the tested tall partition walls [16]. Therefore the good prediction of the collapse interstory drift ratios might not be valid for partitions characterized by different geometrical and mechanical features, e.g. load-bearing stud walls.
Fig. 16. Model of the stud in the DSM: (a) stud section with springs at two sides and (b) details of the springs.
Bending Moment, M
M
ng
Local instability domain Distortional instability domain Global instability domain
M
M
nd
Di sto rti o
nl
Lo cal
buc
kli n
Safe zone
na l
Gl ob al bu ck lin bu ck li
go ccu rs
ng oc cu rs
P
nl
go cc
ur s
3.4. Experimental-numerical strain comparison
P
Axial Force, P
nd
A comparison in terms of strains between the numerical and the experimental models is illustrated in this Section. The strains recorded by SG1, SG2, SG4 and SG5 strain gauges (Fig. 4) at different top displacement levels are compared to the strains resulting from the model. The numerical strains are evaluated from the analysis of the stud cross section subjected to the bending moment and the axial force, assuming linear elastic behavior. The comparison is performed considering both the positive and the negative peak strains achieved at each cycle of the experimental test (Fig. 21). From the comparison of the strains in stud no. 1 (Fig. 21), the main difference is evidenced for small relative displacements, i.e. up to the paper cracking (blue marker), due to the absence of the paper and the compound between the plasterboards. Indeed, for low displacement demand level, the plasterboards absorb the total lateral load and the stud is lightly loaded. This phenomenon is not caught by the model, since the paper and the compound between adjacent plasterboards are not included in the model. Beyond the blue marker in Fig. 21, the steel stud is stressed and the experimental curve exhibits a slope increase. It should be noted that the slope of the strain-displacement curve is very close to the numerical one. For a relative displacement close to 30 mm, represented by the green marker in the plot, the experimental curve denotes an abrupt variation of the slope, which can be associated to the local buckling failure of the stud, confirming the numerical results for studs nos. 1 and 5. Obviously, this slope variation is not caught by the model, since the buckling verification is performed at the end of the analysis. Finally, the global buckling of the stud is clearly visible on the experimental curve for a displacement close to 68 mm. The numerical model predicts that at the same displacement three out of five studs exhibit global buckling, which induce a global failure of the whole partition. The portion of the numerical curve after 68 mm should be neglected, since global buckling failure is checked at the end of the analysis. Similar observations can be performed on the strain trends in stud no. 5 in Fig. 22. Finally, the strains are generally overestimated, due to the neglected modelling of the paper and the compound between adjacent plasterboards. However, the interstory displacement required to induce both local and global
P
ng
Fig. 17. Local, distortional and global buckling domain.
other studs at 1.37% interstory drift in Fig. 19. Stud no. 2 and stud no. 4 globally buckle at this step, whereas in stud no. 1 and stud no. 5 only local buckling occurs. Nevertheless, the whole partition can be considered to be failed at this step. The DSM provides that the capacity of the stud is conceived as the minimum between local, distortional and global buckling load capacity. It is therefore implicit that the steel studs are designed in order not to exhibit any of the three considered buckling failure modes. However, in this case the three distinct buckling failure modes are considered since a numerical-experimental comparison is performed. 3.3. Experimental-numerical damage comparison A comparison between the experimental damage of the tested partition and the prediction of the numerical model is performed in this Section. The comparison between the numerical pushover curve and the experimental force-displacement backbone curve is shown in Fig. 20. In the numerical model, nonlinear behavior of the screws occurs for a top displacement equal to 10 mm (green marker in Fig. 20); the failure of the partition (red marker in Fig. 20) occurs at 1.37% interstory drift, considering the occurrence of global instability in a single stud as the partition failure. Beyond this point the curve is plotted as a dotted line since it is not representative of the partition behavior. The experimental curve exhibits an initial stiffness similar to the stiffness recorded in the numerical model. The screw bearing mechanism occurs at a 11 mm displacement, which is similar to the displacement required to yield some screws in the numerical model. Beyond a 10 mm top displacement, the numerical curve does not match the experimental one: both the strength and the stiffness are 580
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0.3
0.2
stud3 δ=0.12% 0.1
0.3 Loc. inst . Glob. inst. (P ,M)
0.2
stud3 δ=0.135% 0.1
0 10
20 P [kN]
30
40
10
30
40
0
stud3 δ=0.405%
0
0.2
stud3 δ=0.525%
30
40
10
(d)
20 P [kN]
30
stud3 δ=0.765% 0.1
10
0 30
0.2
stud3 δ=0.885%
40
Loc. inst . Glob. inst. (P ,M)
0.2
stud3 δ=1%
0 10
(g)
20 P [kN]
30
40
0
10
(h)
stud3 δ=1.13% 0.1
0.2
stud3 δ=1.24% 0.1
0 30
40
40
Loc. inst . Glob. inst. (P ,M)
0.2
stud3 δ=1.37% 0.1
0 20 P [kN]
30
0.3 Loc. inst . Glob. inst. (P ,M) M[kNm]
M[kNm]
0.2
20 P [kN]
(i)
0.3 Loc. inst . Glob. inst. (P ,M)
40
0.1
0
0.3
30
0.3 Loc. inst . Glob. inst. (P ,M)
0 20 P [kN]
20 P [kN]
(f)
0.1
(l)
stud3 δ=0.645%
0
M[kNm]
M[kNm]
0.2
10
0.2
40
0.3
0
Loc. inst . Glob. inst. (P ,M)
(e) Loc. inst . Glob. inst. (P ,M)
40
0 0
0.3
30
0.1
0 20 P [kN]
20 P [kN]
0.3 Loc. inst . Glob. inst. (P ,M)
0.1
10
10
(c)
M[kNm]
M[kNm]
M[kNm]
0.2
0.1
M[kNm]
20 P [kN]
0.3 Loc. inst . Glob. inst. (P ,M)
0
stud3 δ=0.285%
(b)
0.3
10
0.2
0 0
(a)
0
Loc. inst . Glob. inst. (P ,M)
0.1
0 0
M[kNm]
M[kNm]
Loc. inst . Glob. inst. (P ,M) M[kNm]
M[kNm]
0.3
0 0
10
20 P [kN]
30
40
0
(m) Fig. 18. Verification on stud no. 3 according to DSM up to 1.37% interstory drift.
581
10
20 P [kN]
(n)
30
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Fig. 19. DSM verification on stud 1 (a), 2 (b), 4 (c) and 5 (d) at 1.37% interstory drift.
0.3 Loc. instt . Glob. insst. (P ,M)
0.2
M[kNm]
M[kNm]
0.3
stud1 δ=1.37% % 0.1
Loc. inst . Glob. inst. (P ,M)
0.2
studd2 δ=1.37% 0.1
0
0 0
10
20 P [kN]
30
40
0
10
(a)
30
400
(b)
0.3
0.3 Loc. instt . Glob. insst. (P ,M)
0.2
M[kNm]
M[kNm]
20 P [kN]
stud4 δ=1.37% % 0.1
Loc. inst . Glob. inst. (P ,M)
0.2
studd5 δ=1.37% 0.1
0
0 0
10
20 P [kN]
30
40
0
(c)
10
20 P [kN]
30
400
(d) Fig. 20. Numerical pushover curve and experimental force-displacement curve. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).
150 100
Force [kN]
50 Numeer. backbone curve c Experr.backbone currve Screw w yielding Globaal instability
0 -50 -100 -150
-150
-10 00
-50 0 5 50 Top displacement [mm]
100
150
widely supported by the experimental evidence, which showed severely damaged screwed connections before the partition failure. A tri-linear force-displacement backbone curve is assigned to the screwed connections, matching the experimental results of monotonic tests on such a connection. The failure of the partition due to elastic global buckling is a-posteriori checked, based on the internal forces acting in the steel studs. The numerical results of a monotonic test on the defined model evidence tension and compression struts in the plasterboards due to the applied top displacement. The low stress values both in tension and in compression justify the adoption of a linear elastic material for the boards. The bending moment diagram on studs reveals large demand crossing the two horizontal joints between the plasterboards. Such an evidence can justify the damage experimentally pointed out in the steel stud over and under the two horizontal joints.
instabilities are well predicted. 4. Conclusions The research study deals with the definition of a finite element model capable to capture the interstory drift which causes the failure of a 5 m high plasterboard partition, representative of European partition systems. The assessment is performed comparing the numerical behavior of a specific specimen with the experimental evidence of a quasistatic test campaign conducted at the Laboratory of the Department of Structures for Engineering and Architecture at the University of Naples Federico II. A simple model of the tested partition is defined. Both the steel studs and the plasterboards are modeled with linear elastic elements. The nonlinearity is lumped in the panel-to-stud screwed connections; this is 582
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-3
x 10
x 10
2
2 Num. model SG1-P os SG1-Neg P aper crack Local inst . Global inst.
1.8 1.6
1.6 1.4
1.2
Strain[-]
Strain[-]
1.4
Num. model SG5-P os SG5-Neg P aper crack Local inst . Global inst.
1.8
1 0.8
1.2 1 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
20
40
60
80
100
120
0
20
Relative displacement [mm] [ ]
40
60
80
100
120
Relative displacement [mm]
Fig. 21. Strain recording (SG1 and SG5) trends vs relative displacement demand on stud no. 1. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article). -3
-3
x 10
x 10
2
2 Num. model SG2-P os SG2-Neg P aper crack Local inst . Global inst.
1.8 1.6
1.6 1.4
1.2
Strain[-]
Strain[-]
1.4
Num. model SG4-P os SG4-Neg Paper crack Local inst . Global inst.
1.8
1 0.8
1.2 1 0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0 0
20
40
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Fig. 22. Strain recording (SG2 and SG4) trends vs relative displacement demand on stud no. 5.
model for the prediction of the instability failure drift through an a posteriori check. The strains are generally overestimated in the model, due to the neglected modelling of the paper and the compound between adjacent plasterboards. The numerical model, herein shown, represents the initial stage for the modelling of internal plasterboard partitions, since it only allows the determination of the instability failure in-plane drift, neglecting the out-of-plane behavior of the partition. Furthermore, its nonlinear behavior is only lumped at panel-to-stud screw connections. Nevertheless, the presented model allows determining the buckling collapse drift of tall internal plasterboard partitions, which is a key step in performance based earthquake engineering.
The Direct Strength Method (DSM) is applied to assess the occurrence of different buckling failure modes, i.e. local, distortional and global failure modes, in the studs. This method allows considering the restraining effect given by both the presence of the plasterboards and the screwed connections, through the presence of linear springs on the steel stud cross section. The method evidences that both local and distortional instability failure modes occur at a 0.52% interstory drift, whereas the global buckling is exhibited at an interstory drift equal to 1.37%. The experimental test evidences that for a 1.37% drift the specimen starts showing a global out-of-plane curvature: it can be therefore deduced that the model well catches the global buckling failure mode of the specimen. The experimental curve is well reproduced by the numerical one in terms of initial stiffness, whereas the model underestimates the strength of the partition. However, this underestimation, due to the non-inclusion of the paper and the compound in the model, is limited up to the failure of the paper and the compound, which occurs much earlier than the failure of the specimen. The trends of the experimental strain in the steel studs at different interstory drifts confirm the suitability of the
Acknowledgements This research study has been funded both by the Italian Department of Civil Protection in the frame of the national project DPC - ReLUIS 2015 - RS8 and by Siniat International (AGM10004LP). The support provided by Eng. Marco Russo during the execution of the analyses is 583
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