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Cyclic behaviour, deformability and rigidity of stiffened steel shear panels
Journal of Constructional Steel Research 63 (2007) 554–563 www.elsevier.com/locate/jcsr
Cyclic behaviour, deformability and rigidity of stiffened steel shear panels M.M. Alinia ∗ , M. Dastfan Faculty of Civil Engineering, Amirkabir University of Technology, Tehran, Iran Received 1 December 2005; accepted 20 June 2006
Abstract Shear panels play an important role in improving the seismic behaviour of structures. They generally occur as thin steel plate shear walls (TSPSW) or shear panels created within the web of link beams in eccentrically braced frame (EBF) structures. The post-buckling capacity, deformability and energy dissipation of shear panels are now widely accepted by structural engineers and has resulted in more economical designs. Comparing the behaviour of unstiffened panels with that of heavily stiffened panels shows that unstiffened panels provide a more ductile response while heavily stiffened panels have a wider yield area, which in turn results in higher energy dissipation. Considering these two extreme cases, simultaneous maximum ductility and energy dissipation response cannot be expected. In this numerical research the effect of stiffening upon the ultimate strength of shear panels is investigated. Then, the cyclic behaviour of stiffened and unstiffened shear panels is studied. Finally, with regard to the smaller areas contained within the hysteretic loops of unstiffened panels (due to their pinching records), the optimal stiffening needed to provide both the desirable energy dissipation and ductility is investigated. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Cyclic behaviour; Deformability; Rigidity; Shear panels
1. Introduction Shear panels are widely used as the webs of link beams in eccentrically braced frames (EBFs) and as earthquake resisting thin steel plate shear walls within building frames. EBFs address the desire for a laterally stiff framing system with significant energy dissipation capability to accommodate large seismic forces. A typical EBF consists of link beams, eccentric braces and columns (see Becker and Ishler [1]). The eccentric connection of braces to link beam introduces high shear forces in the web of the link beams. The short segment of the frame where these forces are concentrated is called the shear panel (otherwise known as shear link). The excellent ductility of shear yielding is the main reason for using shear panels. Since the panel portion of the beam element is the “fuse” that determines the strength of other elements, such as the braces and columns, its capacity should be determined conservatively, and based on the actual yield strength of the material. The ∗ Corresponding address: Department of Civil and Environmental Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875 4413, Iran. Tel.: +98 21 6641 8008; fax: +98 21 6454 3232. E-mail address: [email protected] (M.M. Alinia).
c 2006 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2006.06.005
shear panel portion of the beam is the most critical element of an EBF. Beam stiffeners are used to prevent buckling and to ensure ductile shear yielding of the web. Stiffeners are required at each end of the link and at regular intervals within the link. UBC 2211.10.7 requires full depth web stiffeners on both sides of the beam web at the brace end of the link beam. A recent study by Alinia [2] shows that for an optimum design, the transverse stiffeners should divide the length of panels into portions equal to or less than their widths. The main function of steel plate shear walls is to resist horizontal storey shear and overturning moments caused by lateral loads (see Astaneh-Asl [3]). In general steel plate shear wall systems consist of a steel plate wall, two boundary columns and horizontal floor beams. The steel plate wall and columns act compositely as a vertical plate girder. The columns act as flanges of the vertical plate girder and the steel plate wall acts as its web. The horizontal floor beams act, more-or-less, as transverse stiffeners in a plate girder. A properly designed and detailed shear wall is very ductile and has a relatively large energy dissipation capacity. Consequently, steel plate shear walls are very efficient and economical lateral load resisting systems. Also, steel shear wall systems have relatively high initial stiffness, and are thus very effective in limiting the lateral drift of structures.
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Nomenclature a b D E G Is J t α β φ ν
Width of panel Height of panel Flexural rigidity of isotropic plate (= Et 3 /12(1− ν 2 )) Young’s modulus Shear modulus of supporting members (= E/2(1 + ν)) Second moment of area of stiffeners Torsional constant of supporting members Thickness of panel Relative stiffness factor of supporting members Stiffener rigidity ratio Aspect ratio of panel (= a/b) Poisson’s ratio
In the early applications of steel plate shear walls in the United States and Japan, the walls had numerous vertical and horizontal stiffeners. The main purpose of the stiffeners was to prevent elastic and overall buckling; and to increase the shear buckling strength of the wall. In today’s steel fabrication shops welding stiffeners to steel walls can be costly and time-consuming. In recent years the research and testing of realistic specimens have indicated that the steel plate alone, due to its post-buckling loading capacity, performs in a very ductile, desirable and efficient manner. As a result, in most recent applications of steel plate shear walls in the United States and Canada, unstiffened steel plates have been used. A number of researchers in United States, Japan, Canada and United Kingdom have studied the behaviour of steel shear walls. Timler and Kulak [4], Kulak [5], and Driver et al. [6], conducted monotonic and cyclic tests on unstiffened steel plate shear walls. Researchers at the University of British Columbia completed a series of cyclic and shaking table tests on steel plate shear walls (see Rezai et al. [7,8]). In these studies cyclic shear loads were applied to two single-storey specimens. The boundary frames in the specimens were moment frames resulting in a “dual” structural system. These specimens exhibited good ductility and over-strength factors. The tests also demonstrated that the infill steel plates significantly reduced demand on the moment-resisting frame by producing redundant diagonal storey braces, which alleviated the rotation demand on the beam-to-column connections. More recently Lubell et al. [9] presented results from a study on the performance of unstiffened thin steel plate shear walls for medium- and high-rise buildings. The post-buckling strength of the panels was relied upon for most of the frame shear resistance, similar to the slender web of a plate girder. Experimental testing was conducted on two single- and one four-storey steel shear wall specimens, under cyclic quasistatic loading. Each specimen consisted of a single bay with column-to-column and floor-to-floor dimensions of 900 mm, representing a 1/4-scale model of a typical office building
Fig. 1. Assumed stress–strain relationship of mild steel.
core. Identification of load–deformation characteristics and the stresses induced in the structural components were the primary objectives of the testing program. Good energy dissipation and displacement ductility capacities were achieved. Primary inelastic damage modes included yielding of the infill plates combined with column yielding in the single-panel tests and yielding of the columns in the multi-storey frame. In Japan Takanashi et al. [10] and Mimura and Akiyama [11] conducted some of the earliest tests of steel shear walls. Takanashi conducted cyclic tests on fourteen one- and twostorey specimens with different thicknesses. Compared to typical building dimensions the specimens could be considered to be 1/4-scale of prototype walls. With the exception of one specimen all specimens had either vertical or both vertical and horizontal stiffeners welded on one or both sides of the steel plate. The boundary frames were pin-connected frames. The specimens were loaded along their diagonals to create almost pure shear in the panels. The behaviour of the specimens was very ductile and drift angles in some cases exceeded 0.10 rad. The shear strengths of the specimens were predicted well by the Von-Mises yield criterion for pure shear. The two twostorey specimens tested by Takanashi [10] were designed to represent shear walls for high-rise buildings. The specimens were full scale. One specimen represented walls with openings and one without. Once again the shear yield strength predicted by Von-Mises yield criterion was in close agreement with the test results. The researchers concluded that conventional beam theory could be used to calculate the stiffness and strength of stiffened shear walls. Sugii and Yamada [12] reported the results of cyclic and monotonic tests on fourteen steel plate shear walls. The specimens were 1/10-scale and two stories in height. The boundary frame was a rigid composite frame with steel Isections encased in rectangular reinforced concrete sections. All specimens showed pinching of the hysteresis loops due to buckling of the plate. In the United Kingdom Sabouri and Roberts [13] and Roberts [14] reported on the results of 16 tests of steel shear panels loaded diagonally. The small-scale specimens in these tests consisted of steel plates placed within a pin-connected
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Fig. 2. Boundary conditions for a typical panel.
frame and connected to it using bolts. Some panels had perforations. The cyclic load was applied along the diagonal axis which resulted in pure shear. The tests indicated that all panels possessed adequate ductility and sustained four large inelastic cycles. Typical hysteresis loops showed specimens reaching a ductility of more than seven without any decrease in strength. One of the interesting aspects of this test program was the investigation on the effects of perforations in the wall on strength and stiffness. The researchers concluded that the strength and stiffness decreases linearly with the increase in (1 − D/d), where D is the opening diameter and d is the panel depth. In the United States Elgaaly and Caccese [15,16] conducted a number of studies on steel plate shear walls. The experimental part of their research included cyclic testing of six three-storey one-bay specimens subjected to cyclic horizontal load at roof level. The specimens were about 1/4-scale and the steel plate shear walls did not have stiffeners. The studies also included valuable analytical research and resulted in the development of analytical models of hysteresis behaviour of steel plate shear
Fig. 3. Percentage increase in shear buckling strength versus rigidity of the boundary members; Ref. [20].
walls. Based on the behaviour of these six specimens they concluded that when an un-stiffened thin plate is used as a shear wall, inelastic behaviour commences by yielding of the wall, and the strength of the system is governed by plastic hinge formation in the columns. They also concluded that when
Fig. 4. Load–displacement curves for stiffened shear panels.
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Fig. 5. Mode shapes and stress patterns for some typical models.
relatively thick plates are used the failure mode is governed by column instability, and that only a negligible increase occurs in the strength of the system due to increased thickness of the wall. In general the researchers recommended the use of thinner, un-stiffened plates so that yielding of the plate occurs before column buckling. In recent years low-yield point (LYP) steel plates have been developed in Japan and used successfully as steel plate shear walls. Low-yield steel has a yield point about half that of mild steel, with much greater ductility and ultimate elongation of more than twice that of mild steel. It has been demonstrated in Japan that such steel can be used very effectively in energy dissipating structural elements. Rai [17] conducted cyclic load tests on low-yield aluminium alloy shear panels, performed to determine the onset and effect of inelastic web buckling on the load–deformation behaviour. According to the report, yielding of the aluminium shear panels could be used as a means to dissipate energy. Although low-yield steel seems very effective in achieving optimum performance of shear walls, it is not widely produced in most countries. The aim of the present research was therefore to achieve optimum behaviour of shear walls using regular mild steel and stiffeners.
2. Method of study 2.1. Modelling and material properties The cyclic behaviour of stiffened panels under shear loading was studied here by means of nonlinear Finite Element analysis. Large-deflection effects were taken into account. The nonlinear computations were performed using ANSYS code. SHELL43 elements were used to model panels, stiffeners, and surrounding members. The elements were connected through the mean surface of each plate by using multipoint constraints. Nonlinearity of material and geometry were included in the models. The plate material was considered to be elastoplastic; the assumed stress–strain relationship of the material is shown in Fig. 1. Young’s modulus and other material properties were as follows: (E 1 = 210 GPa, σ y = 240 MPa
E 2 = 2.1 GPa, and ν = 0.3).
The Von-Mises yield criteria, known to be the most suitable for mild steel, was used in this research (see Takanashi et al. [10] and Mimura and Akiyama [11]).
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Table 1 Geometrical specifications of panels, stiffeners and buckling loads—all dimensions are in (mm) and (N) Model no.
Panel dimensions Length Depth
Is Thickness
Number of stiffeners Longitudinal Transverse
β
Shear buckling load Unstiffened Stiffened
1 2 3 4 5 6
300 300 300 300 300 300
300 300 300 300 300 300
1 1 1 1 1 1
0 6 28 60 210 280
0 3 3 3 3 3
0 3 3 3 3 3
0 1 4 9 30 40
8,664 8,664 8,664 8,664 8,664 8,664
8,664 11,670 22,220 32,080 69,550 167,670
7 8 9
300 300 300
300 300 300
1 1 1
25 50 100
5 5 5
5 5 5
6 12 24
8,664 8,664 8,664
31,150 49,950 84,080
10 11 12
450 450 450
450 450 450
1 1 1
155 260 310
3 3 3
3 3 3
15 25 30
5,749 5,749 5,749
39,130 54,150 61,200
13 14 15
3,000 3,000 3,000
3,000 3,000 3,000
10 10 10
700,000 1,400,000 2,100,000
3 3 3
3 3 3
10 20 30
86,000 86,000 86,000
411,000 652,000 1,091,000
16
2,000
3,000
10
1,370,000
3
5
30
97,000
1,074,000
at the top of the upper boundary member. This node was coupled to other adjacent nodes to prevent stress concentration and local failure. The finite element analysis was displacement controlled; restraint details are shown in Fig. 2. 3. Discussion of results 3.1. Monotonic loading
Fig. 6. Percentage increase in the ultimate strength due to stiffening.
In order to investigate the effect of stiffening upon the ultimate strength of shear panels, a series of simulations were carried out under monotonic shear loading. First a square unstiffened panel (300 × 300 × 0.5 mm) was analysed and its ultimate strength computed. Rectangular bar type boundary members were connected to the edges of the plate, having a typical shear wall rigidity value of α = 1000; where (see Fig. 3) α=
Fig. 7. Time history of drift applied to models.
2.2. Loading and restraints A convergence study was performed and a mesh with 576 elements was chosen for the unstiffened panel (see Alinia and Dastfan [18] and Zienkiewicz [19]). The shear load was transferred into the panel by applying displacement to a node
GJ . Et 3 /12(1 − ν 2 )
(1)
The panel was then stiffened either by only longitudinal or by combinations of longitudinal and transverse stiffeners. The post-buckling nonlinear load–displacement curves of these panels are shown in Fig. 4. The stiffeners were rectangular steel bars (10×10 mm) and provided sufficient rigidity to form nodal lines and prevent global buckling of the panels [2]. The number and position of stiffeners are illustrated by small sketches beside each curve. Fig. 4 shows that using stiffeners increases the shear buckling strength of the plates more significantly than their ultimate strengths. This is in agreement with the results of previous investigations carried out by the authors [20,21]. Fig. 5 presents the buckling modes and stress distributions for some typical models defined in Fig. 4. The formation of local buckling modes are clearly visible; the red areas in the stress pattern diagrams represent the yield zones which lie within the diagonal tension fields, while the blue points represent the least stressed areas.
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Fig. 8. Shear force–drift curves for model (a): Weak boundary members.
Fig. 9. Shear force–drift curves for model (b): Medium boundary members.
Fig. 10. Shear force–drift curves for model (c): Strong boundary members.
The percentage increase in the ultimate strength of panels, for various stiffening patterns, is shown in Fig. 6. Figs. 4 and 6 show that stiffening shear panels increase the ultimate strength, and that this increase is almost linear with the number of stiffeners. It is also shown that uni-directional stiffeners produce higher strength than bi-directional stiffeners. For example, strengthening a panel with two longitudinal stiffeners (pattern 2H0V) increases the strength by about 100%, whereas
one transverse and one longitudinal stiffener (1H1V) increase the strength by only 80%. 3.2. Cyclic loading 3.2.1. Unstiffened panels The analysis of unstiffened panels under cyclic shear loading was carried out for square panels (300 × 300 × 1 mm) having
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Fig. 11. Time history of drift applied to models 1–16.
Fig. 12. First buckling mode of two typical models.
three different boundary members selected from Fig. 3: a— weak (α = 34), b—medium (α = 268) and c—strong (α = 2150). The boundary conditions and modelling of a typical panel are shown in Fig. 2. It should be noted that the rigidity of the boundary members has little effect on the post-buckling capacity of shear panels [20,21]. The panels were subjected to cyclic loading by applying a very small horizontal displacement to the top nodes. Fig. 7 shows the time history of drift applied to the models. The cyclic displacements were applied according to the loading history, which began with very small values of overall drift which were increased gradually until failure of the panels. The loading history was established according to the specifications for qualifying cyclic tests of beam-to-column and link-to-column connections in seismic provisions for structural steel buildings, AISC 97 [22]. During the first few loading cycles the thin panels buckled in the first mode (one large half-wave in the middle portion). In subsequent cycles, due to the formation of some plastic areas, the buckling mode entered the second and later the third mode. In order to assess the deformability and the energy dissipation capacity of panels with different boundary stiffness, their hysteresis shear force–drift curves are plotted in Figs. 8–10. According to these figures the hysteresis curve lifts upward as the boundary members get stronger; this confirms that the
Fig. 13. Shear force–displacement curves for three typical models.
post-buckling capacity of panels remains almost unaffected. The area covered by the envelopes of the hysteresis curves represents the amount of energy absorbed by the panels. It can be seen that the increase in the energy absorption is due primarily to the increase in the elastic buckling capacities. The main point to be noticed in Figs. 8–10, is the “pinched shape” of the hysteresis curves. This pinched shape is because of the unstiffened, slender nature and ductile behaviour of the panels, and their buckling prior to yielding of the material. Even in the third model, which had very strong boundary members, this effect is visible. It can be concluded that the stiffness of boundary members has little effect, neither on the rigidity nor on the deformability of panels, and that the only ways to avoid pinching and increase energy absorption capacity are either to thicken the plate, which is uneconomical, or to use stiffeners. Alternatively low-yield point steel (LYPS), which causes the formation of yield zones before buckling, can be used. The provision of stiffeners can delay overall buckling, and/or force local buckling of the sub-panels. However, using too many stiffeners reduces the deformability of panels; this is
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Fig. 14. Shear force–displacement envelope curves for models 1–6.
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Fig. 15. Shear force–displacement envelope curves for models 7–9.
a disadvantage for steel shear walls used for strengthening structures against seismic loadings. The optimum amount of stiffening can be defined as that which causes local buckling and material yielding to occur simultaneously. Such stiffening recovers the pinching and produces sufficient concurrent deformability and energy dissipation. 3.2.2. Stiffened panels The effect of stiffeners upon the cyclic behaviour of shear panels is discussed in this section. Sixteen models detailed in Table 1 were considered. Models 1–6 were selected to study the cyclic loading behaviour of stiffened panels with varying stiffener rigidities, by analyzing their hysteresis load–displacement curves and comparing their envelope curves. Models 7–9 were selected to study the effects of stiffener spacing. Models 10–15 were chosen to investigate the influence of panel dimensions. The first fifteen models were all square in dimension; model 16 was used to confirm the results for a different panel aspect ratio. In Table 1 Is is the second moment of area of stiffeners, and the parameter β is defined as the relative rigidity of the stiffener to that of the panel: β = E Is /bD.
Fig. 16. Shear force–displacement envelope curves for models 10–12.
(2)
All boundary members incorporated in the selected models were stiff enough (α > 500), to act as more-or-less clamped edges. The time history of drift applied to these models is shown in Fig. 11. Fig. 12 shows the first buckling mode of two typical models, one with no stiffener (model 1) and one with 3 + 3 stiffeners (model 5). The hysteresis shear load–displacement curves of three typical models are shown in Fig. 13. Fig. 14 provides a comparison of the envelopes of hysteresis curves for models 1–6. Results presented for models 1–6 (see Figs. 13 and 14 and Table 1) show that increasing the rigidity of the stiffeners increases the elastic shear buckling load, which exceeds the yield load for model 5 i.e. the pinching effect disappears. Thus, by incorporating sufficient stiffeners, the area covered by the envelope curves is increased and the panel dissipates more energy. Model 5 had a stiffener rigidity ratio of 30. In this case the envelope of the hysteresis load–displacement curves forms an oval shape, and there is no sign of pinching. Further increase
Fig. 17. Shear force–displacement envelope curves for models 13–15.
in the rigidity of the stiffeners, as for model 6, does not significantly improve the cyclic behaviour, and unfavourably reduces deformability. In order to investigate the required amount of stiffeners to prevent pinching, eight more panels with different geometries and stiffeners were modelled and analyzed. Models 7–9 consisted of thin square plates (300 × 300 × 1 mm) with 5 longitudinal and 5 transverse stiffeners. The stiffener rigidity ratios for these models were 6, 12, and 24 respectively (see Table 1). Envelopes of the hysteresis load–displacement curves for these three models are presented in Fig. 15. The difference between models 7–9 and 1–6 is the number of stiffeners attached to the panels. Models 10–12 were made of 450 × 450 mm plates with the thickness kept at 1 mm; they were therefore more slender than
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While incorporating stiffeners in mild steel panels can prevent pinching of hysteresis curves and increase energy dissipation, care must be taken not to stiffen too much, and lose deformability. To investigate the effects of panel aspect ratio, model 16 was selected. This rectangular model had an aspect ratio of 1.5, and had three longitudinal plus five transverse stiffeners (see Table 1). Fig. 18 shows the first buckling mode shape of this panel while Fig. 19 presents its hysteresis shear load–displacement response curve. Figs. 18 and 19 show that although there was overall buckling of the panel, there was no pinching effect in the hysteresis curve. Thus, once more, with a stiffener rigidity ratio of 30, concurrent suitable energy dissipation and deformability was accomplished. 4. Conclusions
Fig. 18. First mode of buckling—Model 16.
Fig. 19. Shear force–displacement curve for model 16.
the previous models. Stiffener rigidity ratios for these models were 15, 25 and 30, respectively. The corresponding curves for these models are plotted in Fig. 16. Models 13–15 were similar to models 2–6, except that their dimensions were multiplied by 10; the number of stiffeners and their rigidity ratios were kept constant. These models were selected to study the effects of dimensions and sizing. The envelopes of the hysteresis curves for models 13–15 are shown in Fig. 17. The results presented in Table 1 and Figs. 13–17 show that when the stiffener rigidity ratio is set to about 30, no pinching is observed in the hysteresis load–displacement curves. Overall buckling is therefore delayed so that yielding occurs prior to buckling. The area covered by the envelope curves increases and more energy is dissipated. Also, due to the considerable post-buckling out-of-plane displacements, the panels still possess sufficient deformability. Further increase in the rigidity of the stiffeners would force local buckling of panels. In this case even more energy would be dissipated, but the displacement would be significantly reduced; hence the panel would lose its deformability and the response modification factor would be decreased.
Thin unstiffened steel shear panels have a very ductile behaviour, and buckle during the early stages of lateral loading; consequently they do not have great energy dissipation capacity. This buckling phenomenon was observed in all unstiffened models, regardless of the rigidity of their boundary members. The use of stiffeners in mild steel panels is the most common method to prevent pinching of the hysteresis curves, and to increase energy dissipation capacity. On the other hand, too much stiffening leads to a loss of structural deformability. It is concluded that an optimum amount of stiffeners should be used to achieve both sufficient rigidity and deformability. A stiffener rigidity ratio of about 30 seems to provide this ideal behaviour. The use of low-yield steel, or other materials with a lower yield point than that of mild steel, may reduce the need for stiffeners. However, more research should be carried out in this area. References [1] Becker R, Ishler M. Steel tips, seismic design practice for eccentrically braced frames. Canada: Structural Steel Education Council; 1996. [2] Alinia MM. A study into optimization of stiffeners in plates subjected to shear loading. Int J Thin Walled Structures 2005;43(5):845–60. [3] Astaneh-Asl A. Steel tips, seismic behaviour and design of steel shear walls. Canada: Structural Steel Education Council; 2001. [4] Timler PA, Kulak GL. Experimental study of steel plate shear walls. Structural engineering report no. 114. Canada: University of Alberta; 1983. [5] Kulak GL. Unstiffened steel plate shear walls. In: Narayanan R, Roberts TM, editors. Structures subjected to repeated loading-stability and strength. London: Elsevier Applied Science Publications; 1991. p. 237–76 [chapter 9]. [6] Driver RG, Kulak GL, Kennedy DJL, Elwi AE. Seismic performance of steel plate shear walls based on a large-scale multi-storey test. In: Proceedings, 11th world conference on earthquake engineering. 1996. p. 8. [7] Rezai M, Ventura CE, Prion HGL, Lubbell AS. Unstiffened steel plate shear walls: Shake table testing. In: Proceedings, sixth US national conf. on earthquake engrg. 1998. [8] Rezai M, Ventura CE, Prion HGL. Numerical investigation of thin unstiffened steel plate shear walls. In: Proceedings, 12th world conf. on earthquake engineering. 2000. [9] Lubell AS, Prion HGL, Ventura CE, Rezai M. Unstiffened steel plate shear wall performance under cyclic loading. J Struct Eng 2000;126(4): 453–60. [10] Takanashi Y, Takemoto T, Tagaki M. Experimental study on thin
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[11]
[12]
[13]
[14] [15]
steel shear walls and particular bracing under alternative horizontal load Preliminary Report. In: IABSE. symp. on resistance and ultimate deformability of structures acted on by well-defined repeated loads. 1973. Mimura H, Akiyama H. Load-deflection relationship of earthquake–resistant steel shear walls with a developed diagonal tension field. Trans AIJ 1977;(October):260. Sugii K, Yamada M. Steel panel shear walls with and without concrete covering. In: Proceedings, 11th world conference on earthquake engrg. 1996. Sabouri-Ghomi S, Roberts TM. Nonlinear dynamic analysis of steel plate shear walls including shear and bending deformations. Eng Struct 1992; 14(5):309–17. Roberts TM. Seismic resistance of steel plate shear walls. Eng Struct 1995;17(5):344–51. Elgaaly M, Caccese V. Post-buckling behaviour of steel-plate shear walls under cyclic loads. J Struct Eng, ASCE 1993;119(2):588–605.
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[16] Caccese V, Elgaaly M. Experimental study of thin steel-plate shear walls under cyclic load. J Struct Eng, ASCE 1993;119(2):573–87. [17] Rai DC. Inelastic cyclic buckling of aluminium shear panels. J Eng Mech 2002;128(11):1233–7. [18] Alinia MM, Dastfan M. Behaviour of thin steel plate shear walls regarding frame members. Int J Const Steel Research 2006;62(7):730–8. doi:10.1016/j.jcsr. 2005.11.007. [19] Zienkiewicz OC. The finite element method. 3rd ed. London: McGrawHill Ltd.; 1977. p. 260, 292–95. [20] Alinia MM, Dastfan M. Effect of surrounding members on shear buckling of panels. Int J Thin Walled Structures 2005 [submitted for publication]. [21] Alinia MM, Dastfan M. The effects of surrounding members on postbuckling behaviour of thin steel plate shear walls. In: Shen ZY et al., editors. Advances in steel structures, vol. 2. 2005. p. 1427–32. [22] AISC 97. Seismic provisions for structural steel buildings. Chicago: American Institute of Steel Construction; 1997.
Cyclic behaviour, deformability and rigidity of stiffened steel shear panels M.M. Alinia ∗ , M. Dastfan Faculty of Civil Engineering, Amirkabir University of Technology, Tehran, Iran Received 1 December 2005; accepted 20 June 2006
Abstract Shear panels play an important role in improving the seismic behaviour of structures. They generally occur as thin steel plate shear walls (TSPSW) or shear panels created within the web of link beams in eccentrically braced frame (EBF) structures. The post-buckling capacity, deformability and energy dissipation of shear panels are now widely accepted by structural engineers and has resulted in more economical designs. Comparing the behaviour of unstiffened panels with that of heavily stiffened panels shows that unstiffened panels provide a more ductile response while heavily stiffened panels have a wider yield area, which in turn results in higher energy dissipation. Considering these two extreme cases, simultaneous maximum ductility and energy dissipation response cannot be expected. In this numerical research the effect of stiffening upon the ultimate strength of shear panels is investigated. Then, the cyclic behaviour of stiffened and unstiffened shear panels is studied. Finally, with regard to the smaller areas contained within the hysteretic loops of unstiffened panels (due to their pinching records), the optimal stiffening needed to provide both the desirable energy dissipation and ductility is investigated. c 2006 Elsevier Ltd. All rights reserved.
Keywords: Cyclic behaviour; Deformability; Rigidity; Shear panels
1. Introduction Shear panels are widely used as the webs of link beams in eccentrically braced frames (EBFs) and as earthquake resisting thin steel plate shear walls within building frames. EBFs address the desire for a laterally stiff framing system with significant energy dissipation capability to accommodate large seismic forces. A typical EBF consists of link beams, eccentric braces and columns (see Becker and Ishler [1]). The eccentric connection of braces to link beam introduces high shear forces in the web of the link beams. The short segment of the frame where these forces are concentrated is called the shear panel (otherwise known as shear link). The excellent ductility of shear yielding is the main reason for using shear panels. Since the panel portion of the beam element is the “fuse” that determines the strength of other elements, such as the braces and columns, its capacity should be determined conservatively, and based on the actual yield strength of the material. The ∗ Corresponding address: Department of Civil and Environmental Engineering, Amirkabir University of Technology, 424 Hafez Ave., Tehran 15875 4413, Iran. Tel.: +98 21 6641 8008; fax: +98 21 6454 3232. E-mail address: [email protected] (M.M. Alinia).
c 2006 Elsevier Ltd. All rights reserved. 0143-974X/$ - see front matter doi:10.1016/j.jcsr.2006.06.005
shear panel portion of the beam is the most critical element of an EBF. Beam stiffeners are used to prevent buckling and to ensure ductile shear yielding of the web. Stiffeners are required at each end of the link and at regular intervals within the link. UBC 2211.10.7 requires full depth web stiffeners on both sides of the beam web at the brace end of the link beam. A recent study by Alinia [2] shows that for an optimum design, the transverse stiffeners should divide the length of panels into portions equal to or less than their widths. The main function of steel plate shear walls is to resist horizontal storey shear and overturning moments caused by lateral loads (see Astaneh-Asl [3]). In general steel plate shear wall systems consist of a steel plate wall, two boundary columns and horizontal floor beams. The steel plate wall and columns act compositely as a vertical plate girder. The columns act as flanges of the vertical plate girder and the steel plate wall acts as its web. The horizontal floor beams act, more-or-less, as transverse stiffeners in a plate girder. A properly designed and detailed shear wall is very ductile and has a relatively large energy dissipation capacity. Consequently, steel plate shear walls are very efficient and economical lateral load resisting systems. Also, steel shear wall systems have relatively high initial stiffness, and are thus very effective in limiting the lateral drift of structures.
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Nomenclature a b D E G Is J t α β φ ν
Width of panel Height of panel Flexural rigidity of isotropic plate (= Et 3 /12(1− ν 2 )) Young’s modulus Shear modulus of supporting members (= E/2(1 + ν)) Second moment of area of stiffeners Torsional constant of supporting members Thickness of panel Relative stiffness factor of supporting members Stiffener rigidity ratio Aspect ratio of panel (= a/b) Poisson’s ratio
In the early applications of steel plate shear walls in the United States and Japan, the walls had numerous vertical and horizontal stiffeners. The main purpose of the stiffeners was to prevent elastic and overall buckling; and to increase the shear buckling strength of the wall. In today’s steel fabrication shops welding stiffeners to steel walls can be costly and time-consuming. In recent years the research and testing of realistic specimens have indicated that the steel plate alone, due to its post-buckling loading capacity, performs in a very ductile, desirable and efficient manner. As a result, in most recent applications of steel plate shear walls in the United States and Canada, unstiffened steel plates have been used. A number of researchers in United States, Japan, Canada and United Kingdom have studied the behaviour of steel shear walls. Timler and Kulak [4], Kulak [5], and Driver et al. [6], conducted monotonic and cyclic tests on unstiffened steel plate shear walls. Researchers at the University of British Columbia completed a series of cyclic and shaking table tests on steel plate shear walls (see Rezai et al. [7,8]). In these studies cyclic shear loads were applied to two single-storey specimens. The boundary frames in the specimens were moment frames resulting in a “dual” structural system. These specimens exhibited good ductility and over-strength factors. The tests also demonstrated that the infill steel plates significantly reduced demand on the moment-resisting frame by producing redundant diagonal storey braces, which alleviated the rotation demand on the beam-to-column connections. More recently Lubell et al. [9] presented results from a study on the performance of unstiffened thin steel plate shear walls for medium- and high-rise buildings. The post-buckling strength of the panels was relied upon for most of the frame shear resistance, similar to the slender web of a plate girder. Experimental testing was conducted on two single- and one four-storey steel shear wall specimens, under cyclic quasistatic loading. Each specimen consisted of a single bay with column-to-column and floor-to-floor dimensions of 900 mm, representing a 1/4-scale model of a typical office building
Fig. 1. Assumed stress–strain relationship of mild steel.
core. Identification of load–deformation characteristics and the stresses induced in the structural components were the primary objectives of the testing program. Good energy dissipation and displacement ductility capacities were achieved. Primary inelastic damage modes included yielding of the infill plates combined with column yielding in the single-panel tests and yielding of the columns in the multi-storey frame. In Japan Takanashi et al. [10] and Mimura and Akiyama [11] conducted some of the earliest tests of steel shear walls. Takanashi conducted cyclic tests on fourteen one- and twostorey specimens with different thicknesses. Compared to typical building dimensions the specimens could be considered to be 1/4-scale of prototype walls. With the exception of one specimen all specimens had either vertical or both vertical and horizontal stiffeners welded on one or both sides of the steel plate. The boundary frames were pin-connected frames. The specimens were loaded along their diagonals to create almost pure shear in the panels. The behaviour of the specimens was very ductile and drift angles in some cases exceeded 0.10 rad. The shear strengths of the specimens were predicted well by the Von-Mises yield criterion for pure shear. The two twostorey specimens tested by Takanashi [10] were designed to represent shear walls for high-rise buildings. The specimens were full scale. One specimen represented walls with openings and one without. Once again the shear yield strength predicted by Von-Mises yield criterion was in close agreement with the test results. The researchers concluded that conventional beam theory could be used to calculate the stiffness and strength of stiffened shear walls. Sugii and Yamada [12] reported the results of cyclic and monotonic tests on fourteen steel plate shear walls. The specimens were 1/10-scale and two stories in height. The boundary frame was a rigid composite frame with steel Isections encased in rectangular reinforced concrete sections. All specimens showed pinching of the hysteresis loops due to buckling of the plate. In the United Kingdom Sabouri and Roberts [13] and Roberts [14] reported on the results of 16 tests of steel shear panels loaded diagonally. The small-scale specimens in these tests consisted of steel plates placed within a pin-connected
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Fig. 2. Boundary conditions for a typical panel.
frame and connected to it using bolts. Some panels had perforations. The cyclic load was applied along the diagonal axis which resulted in pure shear. The tests indicated that all panels possessed adequate ductility and sustained four large inelastic cycles. Typical hysteresis loops showed specimens reaching a ductility of more than seven without any decrease in strength. One of the interesting aspects of this test program was the investigation on the effects of perforations in the wall on strength and stiffness. The researchers concluded that the strength and stiffness decreases linearly with the increase in (1 − D/d), where D is the opening diameter and d is the panel depth. In the United States Elgaaly and Caccese [15,16] conducted a number of studies on steel plate shear walls. The experimental part of their research included cyclic testing of six three-storey one-bay specimens subjected to cyclic horizontal load at roof level. The specimens were about 1/4-scale and the steel plate shear walls did not have stiffeners. The studies also included valuable analytical research and resulted in the development of analytical models of hysteresis behaviour of steel plate shear
Fig. 3. Percentage increase in shear buckling strength versus rigidity of the boundary members; Ref. [20].
walls. Based on the behaviour of these six specimens they concluded that when an un-stiffened thin plate is used as a shear wall, inelastic behaviour commences by yielding of the wall, and the strength of the system is governed by plastic hinge formation in the columns. They also concluded that when
Fig. 4. Load–displacement curves for stiffened shear panels.
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Fig. 5. Mode shapes and stress patterns for some typical models.
relatively thick plates are used the failure mode is governed by column instability, and that only a negligible increase occurs in the strength of the system due to increased thickness of the wall. In general the researchers recommended the use of thinner, un-stiffened plates so that yielding of the plate occurs before column buckling. In recent years low-yield point (LYP) steel plates have been developed in Japan and used successfully as steel plate shear walls. Low-yield steel has a yield point about half that of mild steel, with much greater ductility and ultimate elongation of more than twice that of mild steel. It has been demonstrated in Japan that such steel can be used very effectively in energy dissipating structural elements. Rai [17] conducted cyclic load tests on low-yield aluminium alloy shear panels, performed to determine the onset and effect of inelastic web buckling on the load–deformation behaviour. According to the report, yielding of the aluminium shear panels could be used as a means to dissipate energy. Although low-yield steel seems very effective in achieving optimum performance of shear walls, it is not widely produced in most countries. The aim of the present research was therefore to achieve optimum behaviour of shear walls using regular mild steel and stiffeners.
2. Method of study 2.1. Modelling and material properties The cyclic behaviour of stiffened panels under shear loading was studied here by means of nonlinear Finite Element analysis. Large-deflection effects were taken into account. The nonlinear computations were performed using ANSYS code. SHELL43 elements were used to model panels, stiffeners, and surrounding members. The elements were connected through the mean surface of each plate by using multipoint constraints. Nonlinearity of material and geometry were included in the models. The plate material was considered to be elastoplastic; the assumed stress–strain relationship of the material is shown in Fig. 1. Young’s modulus and other material properties were as follows: (E 1 = 210 GPa, σ y = 240 MPa
E 2 = 2.1 GPa, and ν = 0.3).
The Von-Mises yield criteria, known to be the most suitable for mild steel, was used in this research (see Takanashi et al. [10] and Mimura and Akiyama [11]).
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Table 1 Geometrical specifications of panels, stiffeners and buckling loads—all dimensions are in (mm) and (N) Model no.
Panel dimensions Length Depth
Is Thickness
Number of stiffeners Longitudinal Transverse
β
Shear buckling load Unstiffened Stiffened
1 2 3 4 5 6
300 300 300 300 300 300
300 300 300 300 300 300
1 1 1 1 1 1
0 6 28 60 210 280
0 3 3 3 3 3
0 3 3 3 3 3
0 1 4 9 30 40
8,664 8,664 8,664 8,664 8,664 8,664
8,664 11,670 22,220 32,080 69,550 167,670
7 8 9
300 300 300
300 300 300
1 1 1
25 50 100
5 5 5
5 5 5
6 12 24
8,664 8,664 8,664
31,150 49,950 84,080
10 11 12
450 450 450
450 450 450
1 1 1
155 260 310
3 3 3
3 3 3
15 25 30
5,749 5,749 5,749
39,130 54,150 61,200
13 14 15
3,000 3,000 3,000
3,000 3,000 3,000
10 10 10
700,000 1,400,000 2,100,000
3 3 3
3 3 3
10 20 30
86,000 86,000 86,000
411,000 652,000 1,091,000
16
2,000
3,000
10
1,370,000
3
5
30
97,000
1,074,000
at the top of the upper boundary member. This node was coupled to other adjacent nodes to prevent stress concentration and local failure. The finite element analysis was displacement controlled; restraint details are shown in Fig. 2. 3. Discussion of results 3.1. Monotonic loading
Fig. 6. Percentage increase in the ultimate strength due to stiffening.
In order to investigate the effect of stiffening upon the ultimate strength of shear panels, a series of simulations were carried out under monotonic shear loading. First a square unstiffened panel (300 × 300 × 0.5 mm) was analysed and its ultimate strength computed. Rectangular bar type boundary members were connected to the edges of the plate, having a typical shear wall rigidity value of α = 1000; where (see Fig. 3) α=
Fig. 7. Time history of drift applied to models.
2.2. Loading and restraints A convergence study was performed and a mesh with 576 elements was chosen for the unstiffened panel (see Alinia and Dastfan [18] and Zienkiewicz [19]). The shear load was transferred into the panel by applying displacement to a node
GJ . Et 3 /12(1 − ν 2 )
(1)
The panel was then stiffened either by only longitudinal or by combinations of longitudinal and transverse stiffeners. The post-buckling nonlinear load–displacement curves of these panels are shown in Fig. 4. The stiffeners were rectangular steel bars (10×10 mm) and provided sufficient rigidity to form nodal lines and prevent global buckling of the panels [2]. The number and position of stiffeners are illustrated by small sketches beside each curve. Fig. 4 shows that using stiffeners increases the shear buckling strength of the plates more significantly than their ultimate strengths. This is in agreement with the results of previous investigations carried out by the authors [20,21]. Fig. 5 presents the buckling modes and stress distributions for some typical models defined in Fig. 4. The formation of local buckling modes are clearly visible; the red areas in the stress pattern diagrams represent the yield zones which lie within the diagonal tension fields, while the blue points represent the least stressed areas.
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Fig. 8. Shear force–drift curves for model (a): Weak boundary members.
Fig. 9. Shear force–drift curves for model (b): Medium boundary members.
Fig. 10. Shear force–drift curves for model (c): Strong boundary members.
The percentage increase in the ultimate strength of panels, for various stiffening patterns, is shown in Fig. 6. Figs. 4 and 6 show that stiffening shear panels increase the ultimate strength, and that this increase is almost linear with the number of stiffeners. It is also shown that uni-directional stiffeners produce higher strength than bi-directional stiffeners. For example, strengthening a panel with two longitudinal stiffeners (pattern 2H0V) increases the strength by about 100%, whereas
one transverse and one longitudinal stiffener (1H1V) increase the strength by only 80%. 3.2. Cyclic loading 3.2.1. Unstiffened panels The analysis of unstiffened panels under cyclic shear loading was carried out for square panels (300 × 300 × 1 mm) having
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Fig. 11. Time history of drift applied to models 1–16.
Fig. 12. First buckling mode of two typical models.
three different boundary members selected from Fig. 3: a— weak (α = 34), b—medium (α = 268) and c—strong (α = 2150). The boundary conditions and modelling of a typical panel are shown in Fig. 2. It should be noted that the rigidity of the boundary members has little effect on the post-buckling capacity of shear panels [20,21]. The panels were subjected to cyclic loading by applying a very small horizontal displacement to the top nodes. Fig. 7 shows the time history of drift applied to the models. The cyclic displacements were applied according to the loading history, which began with very small values of overall drift which were increased gradually until failure of the panels. The loading history was established according to the specifications for qualifying cyclic tests of beam-to-column and link-to-column connections in seismic provisions for structural steel buildings, AISC 97 [22]. During the first few loading cycles the thin panels buckled in the first mode (one large half-wave in the middle portion). In subsequent cycles, due to the formation of some plastic areas, the buckling mode entered the second and later the third mode. In order to assess the deformability and the energy dissipation capacity of panels with different boundary stiffness, their hysteresis shear force–drift curves are plotted in Figs. 8–10. According to these figures the hysteresis curve lifts upward as the boundary members get stronger; this confirms that the
Fig. 13. Shear force–displacement curves for three typical models.
post-buckling capacity of panels remains almost unaffected. The area covered by the envelopes of the hysteresis curves represents the amount of energy absorbed by the panels. It can be seen that the increase in the energy absorption is due primarily to the increase in the elastic buckling capacities. The main point to be noticed in Figs. 8–10, is the “pinched shape” of the hysteresis curves. This pinched shape is because of the unstiffened, slender nature and ductile behaviour of the panels, and their buckling prior to yielding of the material. Even in the third model, which had very strong boundary members, this effect is visible. It can be concluded that the stiffness of boundary members has little effect, neither on the rigidity nor on the deformability of panels, and that the only ways to avoid pinching and increase energy absorption capacity are either to thicken the plate, which is uneconomical, or to use stiffeners. Alternatively low-yield point steel (LYPS), which causes the formation of yield zones before buckling, can be used. The provision of stiffeners can delay overall buckling, and/or force local buckling of the sub-panels. However, using too many stiffeners reduces the deformability of panels; this is
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Fig. 14. Shear force–displacement envelope curves for models 1–6.
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Fig. 15. Shear force–displacement envelope curves for models 7–9.
a disadvantage for steel shear walls used for strengthening structures against seismic loadings. The optimum amount of stiffening can be defined as that which causes local buckling and material yielding to occur simultaneously. Such stiffening recovers the pinching and produces sufficient concurrent deformability and energy dissipation. 3.2.2. Stiffened panels The effect of stiffeners upon the cyclic behaviour of shear panels is discussed in this section. Sixteen models detailed in Table 1 were considered. Models 1–6 were selected to study the cyclic loading behaviour of stiffened panels with varying stiffener rigidities, by analyzing their hysteresis load–displacement curves and comparing their envelope curves. Models 7–9 were selected to study the effects of stiffener spacing. Models 10–15 were chosen to investigate the influence of panel dimensions. The first fifteen models were all square in dimension; model 16 was used to confirm the results for a different panel aspect ratio. In Table 1 Is is the second moment of area of stiffeners, and the parameter β is defined as the relative rigidity of the stiffener to that of the panel: β = E Is /bD.
Fig. 16. Shear force–displacement envelope curves for models 10–12.
(2)
All boundary members incorporated in the selected models were stiff enough (α > 500), to act as more-or-less clamped edges. The time history of drift applied to these models is shown in Fig. 11. Fig. 12 shows the first buckling mode of two typical models, one with no stiffener (model 1) and one with 3 + 3 stiffeners (model 5). The hysteresis shear load–displacement curves of three typical models are shown in Fig. 13. Fig. 14 provides a comparison of the envelopes of hysteresis curves for models 1–6. Results presented for models 1–6 (see Figs. 13 and 14 and Table 1) show that increasing the rigidity of the stiffeners increases the elastic shear buckling load, which exceeds the yield load for model 5 i.e. the pinching effect disappears. Thus, by incorporating sufficient stiffeners, the area covered by the envelope curves is increased and the panel dissipates more energy. Model 5 had a stiffener rigidity ratio of 30. In this case the envelope of the hysteresis load–displacement curves forms an oval shape, and there is no sign of pinching. Further increase
Fig. 17. Shear force–displacement envelope curves for models 13–15.
in the rigidity of the stiffeners, as for model 6, does not significantly improve the cyclic behaviour, and unfavourably reduces deformability. In order to investigate the required amount of stiffeners to prevent pinching, eight more panels with different geometries and stiffeners were modelled and analyzed. Models 7–9 consisted of thin square plates (300 × 300 × 1 mm) with 5 longitudinal and 5 transverse stiffeners. The stiffener rigidity ratios for these models were 6, 12, and 24 respectively (see Table 1). Envelopes of the hysteresis load–displacement curves for these three models are presented in Fig. 15. The difference between models 7–9 and 1–6 is the number of stiffeners attached to the panels. Models 10–12 were made of 450 × 450 mm plates with the thickness kept at 1 mm; they were therefore more slender than
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While incorporating stiffeners in mild steel panels can prevent pinching of hysteresis curves and increase energy dissipation, care must be taken not to stiffen too much, and lose deformability. To investigate the effects of panel aspect ratio, model 16 was selected. This rectangular model had an aspect ratio of 1.5, and had three longitudinal plus five transverse stiffeners (see Table 1). Fig. 18 shows the first buckling mode shape of this panel while Fig. 19 presents its hysteresis shear load–displacement response curve. Figs. 18 and 19 show that although there was overall buckling of the panel, there was no pinching effect in the hysteresis curve. Thus, once more, with a stiffener rigidity ratio of 30, concurrent suitable energy dissipation and deformability was accomplished. 4. Conclusions
Fig. 18. First mode of buckling—Model 16.
Fig. 19. Shear force–displacement curve for model 16.
the previous models. Stiffener rigidity ratios for these models were 15, 25 and 30, respectively. The corresponding curves for these models are plotted in Fig. 16. Models 13–15 were similar to models 2–6, except that their dimensions were multiplied by 10; the number of stiffeners and their rigidity ratios were kept constant. These models were selected to study the effects of dimensions and sizing. The envelopes of the hysteresis curves for models 13–15 are shown in Fig. 17. The results presented in Table 1 and Figs. 13–17 show that when the stiffener rigidity ratio is set to about 30, no pinching is observed in the hysteresis load–displacement curves. Overall buckling is therefore delayed so that yielding occurs prior to buckling. The area covered by the envelope curves increases and more energy is dissipated. Also, due to the considerable post-buckling out-of-plane displacements, the panels still possess sufficient deformability. Further increase in the rigidity of the stiffeners would force local buckling of panels. In this case even more energy would be dissipated, but the displacement would be significantly reduced; hence the panel would lose its deformability and the response modification factor would be decreased.
Thin unstiffened steel shear panels have a very ductile behaviour, and buckle during the early stages of lateral loading; consequently they do not have great energy dissipation capacity. This buckling phenomenon was observed in all unstiffened models, regardless of the rigidity of their boundary members. The use of stiffeners in mild steel panels is the most common method to prevent pinching of the hysteresis curves, and to increase energy dissipation capacity. On the other hand, too much stiffening leads to a loss of structural deformability. It is concluded that an optimum amount of stiffeners should be used to achieve both sufficient rigidity and deformability. A stiffener rigidity ratio of about 30 seems to provide this ideal behaviour. The use of low-yield steel, or other materials with a lower yield point than that of mild steel, may reduce the need for stiffeners. However, more research should be carried out in this area. References [1] Becker R, Ishler M. Steel tips, seismic design practice for eccentrically braced frames. Canada: Structural Steel Education Council; 1996. [2] Alinia MM. A study into optimization of stiffeners in plates subjected to shear loading. Int J Thin Walled Structures 2005;43(5):845–60. [3] Astaneh-Asl A. Steel tips, seismic behaviour and design of steel shear walls. Canada: Structural Steel Education Council; 2001. [4] Timler PA, Kulak GL. Experimental study of steel plate shear walls. Structural engineering report no. 114. Canada: University of Alberta; 1983. [5] Kulak GL. Unstiffened steel plate shear walls. In: Narayanan R, Roberts TM, editors. Structures subjected to repeated loading-stability and strength. London: Elsevier Applied Science Publications; 1991. p. 237–76 [chapter 9]. [6] Driver RG, Kulak GL, Kennedy DJL, Elwi AE. Seismic performance of steel plate shear walls based on a large-scale multi-storey test. In: Proceedings, 11th world conference on earthquake engineering. 1996. p. 8. [7] Rezai M, Ventura CE, Prion HGL, Lubbell AS. Unstiffened steel plate shear walls: Shake table testing. In: Proceedings, sixth US national conf. on earthquake engrg. 1998. [8] Rezai M, Ventura CE, Prion HGL. Numerical investigation of thin unstiffened steel plate shear walls. In: Proceedings, 12th world conf. on earthquake engineering. 2000. [9] Lubell AS, Prion HGL, Ventura CE, Rezai M. Unstiffened steel plate shear wall performance under cyclic loading. J Struct Eng 2000;126(4): 453–60. [10] Takanashi Y, Takemoto T, Tagaki M. Experimental study on thin
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[16] Caccese V, Elgaaly M. Experimental study of thin steel-plate shear walls under cyclic load. J Struct Eng, ASCE 1993;119(2):573–87. [17] Rai DC. Inelastic cyclic buckling of aluminium shear panels. J Eng Mech 2002;128(11):1233–7. [18] Alinia MM, Dastfan M. Behaviour of thin steel plate shear walls regarding frame members. Int J Const Steel Research 2006;62(7):730–8. doi:10.1016/j.jcsr. 2005.11.007. [19] Zienkiewicz OC. The finite element method. 3rd ed. London: McGrawHill Ltd.; 1977. p. 260, 292–95. [20] Alinia MM, Dastfan M. Effect of surrounding members on shear buckling of panels. Int J Thin Walled Structures 2005 [submitted for publication]. [21] Alinia MM, Dastfan M. The effects of surrounding members on postbuckling behaviour of thin steel plate shear walls. In: Shen ZY et al., editors. Advances in steel structures, vol. 2. 2005. p. 1427–32. [22] AISC 97. Seismic provisions for structural steel buildings. Chicago: American Institute of Steel Construction; 1997.