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Theoretical and experimental study on the cyclic behaviour of X braced steel frames
Engineering Structures 46 (2013) 763–773
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Theoretical and experimental study on the cyclic behaviour of X braced steel frames Giovanni Metelli ⇑ University of Brescia, Department of Civil Engineering, Architecture, Land and Environment, Via Branze 43, 25123 Brescia, Italy
a r t i c l e
i n f o
Article history: Received 17 February 2012 Revised 13 July 2012 Accepted 17 August 2012
Keywords: Concentrically braced frames Critical load Slenderness Effective length factor Seismic behaviour
a b s t r a c t The paper presents a theoretical and experimental study on the slenderness of the diagonal members of concentrically X braced steel frames. According to Eurocode 8 and Italian Standard, the non-dimensional slenderness of the diagonal members should be lower than an upper limit, in order to provide sufficient hysteretic energy dissipation, and greater than a lower limit, to avoid overloading the columns in the prebuckling stage. This seismic design provision requires the realistic evaluation of the diagonal member slenderness. A reduced scale test bench was designed to study X braced steel frames with different geometrical characteristics of diagonal members and with restraints as similar as possible to the theoretical ones. The results confirm the theoretical prediction of the effective length factor of the diagonal members which exhibited good hysteretic behaviour with large inelastic deformation capacity under cyclic loading. The low values of the effective length factor suggest the use of plates to realise the diagonal member instead of the classical angles as shown in the design example. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In steel multi-storey buildings the lateral actions due to wind or earthquake are generally withstood by braced frames, among which the most common are the concentrically X braced frames designed with only the tension diagonal active. Such a structural system is widely used because of its cost effectiveness, reduced construction time and the simplicity of its design. The behaviour induced by seismic actions depends on the ductility and the dissipative capabilities of the structure, whose inelastic deformations are provided by braces buckling and yielding when concentrically braced frames are used [1,2]. Furthermore, if Capacity Design (CD) based codes are adopted, the brace is designed by assuming that the dissipative zones are mainly positioned along the diagonals, whose yielding must take place before the failure of connections failure and the formation of any inelastic mechanism in the beams, columns, and foundations [3,4]. In X braced frames with the diagonals connected at the intersection node, the phase during which the compression brace is also active precedes the yield phase of the tension brace. So, if the buckling resistance is too high, an initial shear action peak can occur and consequently, with the view of applying the Capacity Design criterion, an increase in the resistance of the beams, columns, foundations and connections may be requested. ⇑ Tel.: +39 0303711234; fax: +39 0303711312. E-mail address: [email protected] 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.08.021
In the initial load phase (Fig. 1a), as the axial stiffness of the braces is usually very small compared to that of the columns and beams, both diagonals are subject to the axial action N ¼ V=ð2 cos #Þ. As a consequence, when the first critical load occurs in the compression brace the storey shear force V is equal to V max;1 ¼ 2N cr cos #. Once the compression brace has buckled, the storey shear Vmax,2 is carried mainly by the yielded tension brace V max;2 ¼ N y cos # (Fig. 1b). Thus, in order to avoid the peak load in the first cycle which can overload the columns and brace joints, the condition Vmax,1 < Vmax,2, (i.e. Ncr < 0.5Ny), must be respected. For this reason, both Eurocode 8 [3] and the Italian Standard [4] require the non-dimensional slenderness of the braces to be greater than 1.3. In such a case in fact, as the reduction factor v for the relevant buckling mode calculated according the Eurocode 3 [5] reaches the maximum of 0.47, the buckling resistance (Nb,Rd = vAfyk/cM1) is less than half of the yield strength (NRd = Afyk/cM0). Eurocode 8 and the Italian Standards also set at 2 the upper limit for the non-dimensional slenderness of the braces, to guarantee sufficient hysteretic dissipation of the frame. To respect both the lower and the upper slenderness limits, a realistic evaluation of the brace’s effective lengths is needed, contrary to usual design practice which considers safe an evaluation by excess of slenderness when member shall be verified against buckling. In the scientific literature the problem of the peak shear strength of X braced frames during the initial cycle of a seismic event does not seem analysed in sufficient depth. The reason is maybe to be found in the fact that the American Standard [6] does
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Fig. 1. Maximum shear action in the first cycle (a) and in the subsequent cycles (b).
not prescribe a lower limit to the slenderness. Several test were performed on braces with different cross section showing an increased energy dissipation when the brace slenderness is decreased while the ductility level at fracture can be markedly reduced in RHS profiles due to the local buckling and fracture at plastic hinge [1,2,7]. An equation, in good agreement with test results, was also proposed to evaluate the post buckling resistance of the brace for increasing ductility level, thus allowing to estimate the shear strength of braced frames. An experimental research confirmed the limited energy dissipation of tension only concentrically braced frame [8] with a loss of stiffness and strength to lateral displacement which can lead to the formation of soft-story mechanism in multi-story buildings due to its limited capacity to distribute the inelastic deformation demand over the structure height [2]. Analytical [9,10] and experimental [11] studies on X braced frames showed that the effective length of the compressive diagonal is lower the half diagonal length because of the restraint provided by the complementary brace, whose stiffness is increased by the presence of a tensile action. The present research focused on the experimental evaluation of the brace’s effective length in order to validate the theoretical formulations proposed in literature. The test results herein presented shed light on the role of the lower-bond value imposed by standards on the brace slenderness to limit the peak shear strength of the X braced frames during the early cycles of a seismic event. A design example of a multi-storey building with X-braced frames in a low-seismic zone is also illustrated to show the necessity to adopt for the diagonals cross sections with very small gyratory radiuses for the respect of slenderness limits, so as to recommend the use of simple plates instead of the classical angles.
2. Buckling of X braced frames The correct evaluation of the effective length must be made both for the in-plane and the out-of-plane buckling. The latter should preferably be avoided as it can cause serious damage to the adjacent non-structural elements, such as infills and claddings. With regards to the in-plane buckling it’s important to highlight that the braces’ boundary conditions change during the seismic event. At the beginning both the braces are subject to the same axial load, so the compression brace’s buckling is hindered by the second order effect induced by the tension brace. Once the load is reversed after the yielding of the tension brace and the buckling of the compression brace during the first cycle, the rotational constraint at the braces intersection point is more flexible, compared to the first cycle, because the complementary brace is still plastically deformed due to the buckling. For this reason it cannot carry any axial load until it is straightened out thus
providing no beneficial second order effect to the buckling of the other diagonal. The effective lengths’ values can be calculated once the critical load value is evaluated through the buckling analysis of the brace [12,13]. 2.1. In-plane effective length In-plane brace buckling occurs without any displacement of the diagonals’ intersection node and with its u1 rotation hindered by the flexural stiffness of the tension brace. First the calculation for X braces with fixed-end restraints (Fig. 2a) will be presented. The critical load Pcr is obtained setting equal to zero the rotational stiffness M11 of the semi-diagonal members converging to the intersection node (Appendix A) [12,13]:
M11 ¼ 2
4EJ 3I1 ðaLÞ 3I1 ðam LÞ þ L 4I21 ðaLÞ I22 ðaLÞ 4I21 ðam LÞ I22 ðam LÞ
! ¼0
ð1Þ
E being the modulus of elasticity of the steel, J the brace moment of inertia, I1 and I2 the stability functions defined in Appendix A, L the length of the semi-diagonal, a = (P/EJ)0.5, P the axial load in the compression brace and mP the axial load in the other brace. The maximum value of the critical load is obtained when the complementary brace is loaded by a tensile force of the same value (m = 1). In this case the minimum value of aL that satisfies Eq. (1) is 5.61. This value allows to calculate the critical load Pcr and the slenderness factor b, defined as the ratio between the in-plane effective length L0,ip and the length L of the semi-diagonal:
Pcr ¼ EJ
b¼
aL L
2 ¼
p2 EJ ðbLÞ2
con b ¼
p aL
L0;ip p ¼ ¼ 0:56 L aL
ð2Þ
ð3Þ
If the complementary brace is unloaded, a lower value of the critical load is obtained. As a result, Eq. (1) is satisfied by aL = 5.325, corresponding to a slenderness factor b = 0.59. Also in the case of pinned-end restraints (Fig. 2b), the critical load value is obtained by setting equal to zero the second order rotational stiffness M11 of braces’ connecting node (Appendix A):
M11 ¼ 2
3EJ 1 1 ¼0 þ L I1 ðaLÞ I1 ðam LÞ
ð4Þ
In the case of tensile action in the complementary brace (m = 1) Eq. (4) is satisfied when aL = 3.927, corresponding to a slenderness factor b = 0.80, while in the case of no tensile action
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Fig. 2. In plane brace deformation due to the rotation u1 of the internal node, for fixed-end (a) and pinned-end diagonals (b).
(m = 0) Eq. (4) is satisfied if aL = 3.127, providing a slenderness factor b = 0.843. These theoretical values of the slenderness factor are consistent with the results found in literature [9]. 2.2. Out-of-plane effective length During the first load cycle, the first buckling mode of the compression brace (Fig. 3a) cannot occur because the presence, even modest, of tensile action in the other brace prevents the intersection point from moving out-of-plane [10,11]. The buckled shaped is thus characterised by the out-of-plane rotation of the braces’ intersection node, rotation hindered only by the torsional stiffness of the opposite brace (Fig. 3b). Therefore the effective length L0,op is always less than the length L of the semi-diagonal (L0,op = 0.7L for fixed ends and L0,op = L for pinned ends). In the load cycles following the first, since one brace has buckled and shows an inelastic deformation, it hinders the compression brace’s first form of out-of-plane buckling just with its flexional stiffness. The critical load value can be obtained by setting equal to zero the second order translational stiffness H11 of the connecting node (Appendix A). In the case of fixed-end braces (Fig. 4a), the equation to determine the critical load is:
H11 ¼ 2
12EJ L3
3 4I21 ðaLÞ I22 ðaLÞ
! þ
24EJ L3
¼0
Fig. 4. Out of plane deformation of the brace for fixed-end (a) and pinned-end (b).
Table 1 Effective length factors b = L0/L of the braces. Restraint
In-plane
Lower limit Upper limit
Out-of-plane
Pinned end
Fixed end
Pinned end
Fixed end
0.800 0.843
0.560 0.590
<1.0 1.425
<0.7 0.7125
ð5Þ
fulfilled if aL = 4.409, corresponding to an effective length factor b = 0.7125.
In the case of braces with pinned ends (Fig. 4b) the stability equation, expressed by Eq. (6), is satisfied by aL = 2.205, corresponding to an effective length factor b = 1.425:
H11 ¼ 2
3EJ L3
1 6EJ þ 3 ¼0 I1 ðaLÞ I3 ðaLÞ L
ð6Þ
In Table 1 the values of the effective length factor for in-plane and out-of-plane buckling are reported for both end restraint conditions of the diagonal members. 3. Test bench
Fig. 3. Out of plane buckled shapes.
In order to investigate the validity of the theoretical formulations on the effective length, a test bench was built to carry out on a reduced scale a series of tests on X braced frames with boundary conditions as close as possible to the ideal theoretical ones. The test bench consisted of a square frame 621 mm wide, made of four pairs of external rods (1 in Fig. 5a) within which the two diagonal braces were positioned (2 in Fig. 5a). The external rods were hinged to each other (3 in Fig. 5a), while the braces were fixed at both ends by joints that prevent them from rotating (4 in Fig. 5a and Fig. 5b). The two vertical joints were connected directly to the Instron test machine, while the joints’ rotation at
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Fig. 5. (a) Test bench and (b) pin’s detail.
the ends of the horizontal brace was impeded by a coupling sleeve with an extension rod (5 in Fig. 5a). The two diagonal braces were manufactured separately and subsequently welded together at the mid-length to guarantee the correct positioning within the bench. The applied vertical force F is measured by the test machine’s load cell, while the axial forces in the external rods are measured by strain-gauges. The axial force in the diagonal braces is calculated as a function of the applied force F, of the force in the external rods Fc and of the variable angle # between the external rod and the vertical brace (Fig. 6). The effect, albeit modest, of the friction in the pins linking the joints to the external rods was also taken into account, evaluated through a preliminary test.
Vertical brace force :
F dv ¼ F 2F c cos #
Horizontal brace force :
F do ¼ 2F c sin #
ð7Þ ð8Þ
Fig. 7. Test set up: detail of the LVDTs position for the measurement of the brace axial deformation (a) and of the brace buckling deflection (b).
Angle between the vertical brace and the external rod : L do =2 # ¼ arctan L dv =2
ð9Þ
The axial deformation of the vertical brace was measured with an LVDT inductive transducer, fixed to the two joints’ pins by means of two alloy extension rods (a in Fig. 7a), while a second LVDT, linking the two sliding parts of the coupling sleeve, measures the axial deformation of the horizontal brace (b in Fig. 7a). Strain gauges were applied to the diagonal braces to correctly measure the value of the critical load, while two LVDTs were positioned at the middle of the semi-diagonal to measure its in-plane buckling deflection (c and d in Fig. 7b). Further details of the test bench are to be found in [14].
4. Test programme
Fig. 6. Forces in the frame elements and frame deformation.
Three tests were carried out on braces with a rectangular crosssection whose geometrical characteristics are shown in Table 2. The specimens were designed to buckle in the frame plane, with a non-dimensional slenderness kip ¼ 1:3 calculated assuming the effective length value b as gradually reaching the theoretical one. Thus, as the non-dimensional slenderness is defined as kip ¼ kip =k1 , being k1 = p(Es/fyk)0.5 the slenderness corresponding to the yield strength, which is equal to 86.8 for S275 steel chosen for the test, the specimens were designed with an in plane slenderness kp = 113 which corresponds to a in plane thickness ip ffiffiffiffiffiffi tip ¼ bL 12=kip . Specimens 1 and 2 were designed in accordance with the current professional practice, with an evaluation by excess of the effective lengths: in the plane a fixed-end constraint for the diagonals was assumed, along with a pinned constraint at the intersection point (L0,ip = 0.7L), thus ignoring the stiffening contribution of
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G. Metelli / Engineering Structures 46 (2013) 763–773 Table 2 Design geometrical characteristics of the specimens. Specimen
kip
kip
L0,ip = bL (mm)
tip (mm)
kop
kop
L0,op = bL (mm)
top (mm)
1 2 3
1.3 1.3 1.3
113 113 113
0.70L 0.70L 0.56L
7.5 7.5 5.8
1.3 1.3 1.3
113 113 113
1.25L 1.00L 0.71L
13.4 10.7 7.5
L = 350 mm: length of the semi-diagonal; tip: in-plane brace thickness; top: out-of-plane brace thickness; Lp 0,ip ffiffiffiffiffiffiand L0,op: in-plane and out-of-plane effective pffiffiffiffiffiffilength; k ¼ k=k1 : non-dimensional slenderness being k1 = p(Es/fy.nom)0.5 = 86.8 for S275 steel; in plane slenderness kip ¼ L0;ip 12=tip ; out-of plane slenderness kop ¼ L0;op 12=top .
Table 3 Mean mechanical proprieties of the steel used for the braces. Yield strength Ultimate strength Elongation after fracture Elongation at maximum tensile force Modulus of elasticity
fym (MPa) ftm (MPa) A5 (%) Agt (%) Esm (MPa)
318.6 455.9 38.7 19.7 213,100
Test carried out in according to EN ISO 6892-1:2009 [15].
specimen 1 (L0,op = 1.25L). The design of specimen 3 was based on the theoretical values of the effective lengths, assuming L0,ip = 0.56L and L0,op = 0.71L for in-plane and out-of-plane buckling respectively. S275 steel was adopted for the braces, with an average yield strength fym of 318 MPa and an average ultimate strength ftm of 456 MPa (Table 3). The experimental tests were carried out applying to the braces an axial deformation of increasing amplitude. The first cycle was characterised by a deformation of 4 mm, 4.35 times the nominal yield elongation dy = 0.92 mm (dy = 2Lfyk/Es, L = 350 mm being the length of the semi-diagonal; Fig. 8). The brace deformation was then applied stepwise in an increase of 2 mm between the subsequent cycles up to the brace fracture. 5. Experimental results
Fig. 8. Geometrical characteristics of the specimen.
the tension brace. For the out-of-plane buckling a fixed-end restraint was assumed and the stiffening effect of the other brace was ignored (L0,op = L), adopting a greater safety coefficient for
The main aim of the tests was to confirm the theoretical values of the effective lengths. For this reason special care was taken during the first two cycle loadings. The first cycle started applying a tensile force in the vertical brace, thus compressing the horizontal one. All three tests confirmed the in-plane effective length theoretical value L0,ip = 0.56L, the semi-length of the brace being L = 350 mm. The out-of-plane buckling did not occur even in the subsequent load cycles up to the brace fracture. As an example, the results of the second test are illustrated in detail. In Figs. 9 and 10 the axial force F versus the axial deformation d curves of both braces are shown for the first two load cycles. The horizontal brace buckled for an axial force Ncr = 20.75 kN (Fig. 9), when the vertical brace was loaded by a tensile force
Fig. 9. Specimen 2: axial force Fdo–deformation do curve of the horizontal brace during the first and the second cycles.
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Fig. 10. Specimen 2: axial force Fdv–deformation dv curve of the vertical brace during the first and the second cycles.
Fig. 11. Specimen 2: buckled shape of the horizontal brace during the first cycle (a) and failure of the vertical brace (b).
Fdv = 22 kN (Fig. 10). The effective length, the in plane moment of inertia being J = 338 mm4, was L0,ip = 185 mm calculated by means of the expression of the Euler load (Eq. (2)) and the slenderness factor was equal to b = L0,ip/L = 185/350 = 0.53, value that practically coincides with the theoretical one (b = 0.56), considering measurement approximations as inevitable in these experimental conditions. Fig. 11a shows the critical deformation of the horizontal brace in the first load cycle, while Fig. 12 presents the pictures of the deformations corresponding to the points indicated by Fi in the diagrams of Figs. 9 and 10. In the second load cycle the vertical brace buckled, after having yielded in tension, when the horizontal member still showed a plastic – buckling deformation with a compression force of 3.7 kN (Fig. 9). The critical load was Ncr = 14.4 kN (Fig. 10), value equal to 70% of that of the first cycle. The fracture took place in the plastic hinge during the 18th tensile cycle of the vertical brace (Fig. 11b), with a total deformation of 37 mm (e = 37/700 = 0.053), equal to roughly 40 times the yield elongation (Fig. 13). The main results of the three tests are presented in Table 4. In Figs. 13 and 14, the typical hysteretic behaviour of an axially loaded member is shown for both braces of specimen 2. In compression, Baushinger’s effect and the inelastic deformations accumulated during the previous cycles decreased markedly the buckling resis-
tance, while in tension the brace developed its yield strength during each subsequent cycle only if a larger elongation was applied. In tension the braces developed also some strain hardening. With regards to the post-buckling strength of the braces, the compression resistance C 0u was measured for a ductility level l = d/dy equal to 3, after buckling occurred during the first cycle, as proposed in [1]. For a brace elongation equal to 2.75 mm (= 3dy), the tested specimens exhibited a post-buckling strength C 0u varying between 31% and 43% of the nominal yield strength for the horizontal brace of specimens 3 and 1 respectively (Table 4). Earlier experimental and theoretical studies [1] showed a lower postbuckling strength ranging between 15% and 30% of the brace yield strength. The experimental results allow also to evaluate the overall hysteretic behaviour of X braced frames, as shown in the sketch of Fig. 15. The test frame is a self-balanced system and the storey shear load V can be defined as the component of the applied force F on the direction of the frame’s external rods (Fig. 15). The interstorey displacement dx is obtained by measuring the axial deformation dv and do of the braces.
V ¼ F cos #
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ d2v þ d2o
ð11Þ
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Fig. 12. Specimen 2: brace deformation during the first and the second cycle at the point Fì indicated in Figs. 9 and 10.
Fig. 13. Specimen 2: axial force Fdv–deformation dv curve of the vertical brace up to the failure.
Fig. 14. Specimen 2: axial force Fdo–deformation do curve of the horizontal brace up to the failure.
Table 4 Test results. Specimen
1 2 3a
1° cycle
12° cycle
Ncr (kN)
Horizontal brace L0,ip (mm)
b
Nmax (kN)
do,max
C 0u (kN)
Ncr (kN)
Vertical brace L0,ip (mm)
b
Nmax (kN)
dv,max
C 0u (kN)
V/Vy
V/Vy
27.4 20.7 6.3
187 185 204
0.54 0.53 0.59
33.62 26.70 16.71
33.9 37.3 41.3
11.6 9.0 3.7
23.4 14.4 4.2
205 223 249
0.59 0.64 0.72
35.04 26.94 17.83
35.0 37.0 41.9
10.5 8.2 4.2
2.11 2.13 1.68
1.69 1.73 1.71
Ncr: Experimental critical load; L0,ip: in-plane effective length; b: in-plane effective length factor; Nmax: maximum axial force; dmax: maximum elongation. a The test on specimen 3 was interrupted before the brace fracture.
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In Fig. 16a and Fig. 16b the dimensionless shear action V/Vy is plotted as a function of the inter-storey drift, which is the ratio between the inter-storey displacement dx and the inter-storey height h = 621.5 mm. Vy is the nominal plastic shear strength of the brace, defined as the shear action which yields the diagonal assuming a tension-only braced frame (Vy = A fyk cos #). It is worthwhile to point out that specimen 3 did not show the typical shear peak strength of specimens 1 and 2, because it was designed with an in-plane slenderness ratio kip equal to 1.3 assuming the theoretical effective length factor b equal to 0.56L. Even though the tested X braced frames exhibited a typical pinched hysteretic behaviour with a lack of stiffness under load reversal due to the inelastic deformation accumulated in both diagonals, the failure did not occur up to a 7% drift, value well beyond the specified code limits generally adopted for the ultimate limit state (ULS).
Fig. 15. X braced frame: storey shear and drift.
6. Design example The application of the effective length values exposed above leads to the design of unusual X braced frames. As an example, here presented are the main outcomes in the design of a typical multi-storey building, realised in accordance to the Eurocode 8 [3] and Italian Standard [4]. A seven-storey office-building with a regular grid of columns and beams is considered. The plan layout of the structure is shown in Fig. 17. The seismic-resistant action is entrusted to eight tension-only X braced frames, positioned symmetrically at the corners of the building. The X braced frames’ design was carried out for a typical lowseismic zone in Northern Italy (ground type C, ag = 0.11g). Nevertheless, it is the author’s belief that the considerations exposed for the choice of the type of braces’ cross-section have general validity. The gravity loads considered in buildings were: floor dead load of 4.00 kN/m2, floor live load of 3.00 kN/m2, roof snow load of 1.50 kN/m2 and exterior walls of 8.00 kN/m. As the building conforms to the criteria for regularity in plan, the modal analysis may be performed using two separate planar models for the earthquake directed along the two perpendicular directions. Furthermore, as the seismic behaviour in both directions is the same, only the case of an earthquake in the X direction was considered, amplifying the seismic action by 30% to keep into account accidental torsional effects. A behaviour factor q equal to 4 was assumed for the reduction of the elastic response spectrum at ultimate limit state. In Fig. 18 are reported the values of the seismic masses applied to the X brace frame’s nodes and the values of the static vertical loads used for the modal analysis, carried out considering only the tension braces (Eurocode 8 §6.3 and Italian Standard §7.5.5) and modelling each member with hinged connections. This assumption is considered as conservative for the evaluation of the seismic response of the building because the columns are usually continuous and the beam–column joints may be rigid. The dimensioning of the X braced frame was performed including also the second-order effects (or P–D effects) in accordance with the Italian Standard and Eurocode 8 (par. 4.4.2.2), by multiplying the relevant seismic action effects by the factor 1/(1 h), being h the inter-storey drift sensitivity coefficient:
h¼
Fig. 16. Hysteretic brace response; dimensionless storey shear V/Vy–drift curve for specimen 2 (a) and specimen 3 (b).
P dr V h
ð12Þ
where dr is the inter-story displacement, calculated by multiplying the inter-storey displacement determined with the design response spectrum at the ultimate limit state by the behaviour factor q, thus equal to the displacement calculated with the elastic spectrum, V is the total storey shear computed by the design response spectrum and P is the total gravity load at and above the storey considered. In Table 5 the NEd forces in the diagonals, the dr drifts for each storey, the over-strength coefficients Xi and the factor h of the 2nd order effects are reported. It is worthwhile to point out that the factor h resulted always less than 0.2, as it must be to have the possibility of keeping into account second order effects simply by increasing the horizontal seismic action effects by the factor 1/(1 h). Controlling the parameter h is particularly arduous when designing buildings in low seismic zones where the effects of the seismic actions are modest so that the dimensioning of the diagonals according to the resistance leads to X braced frames which are too flexible to impede dynamic instability. It is thus necessary to oversize the diagonals, as is evident from the modest values of the r stresses, reported in Table 5. To reach the same results more quickly, normally it would be more practical to carry-out the design adopting the value of q = 2 for the behaviour factor [14].
G. Metelli / Engineering Structures 46 (2013) 763–773
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Fig. 17. Structure lay-out: floor plan view and front view of the X braced frames (units in m).
Fig. 18. X braced frame of the design example: masses and the loads applied for the modal response analysis.
The effective length L0 = bL of the diagonal was calculated with reference to the semi-length L, considering the 2.7 m length as being with a constant cross section (Fig. 19). Fixed end restraints of the braces were assumed with an in-plane effective length L0,ip of 1.51 m (L0,ip = 0.56 2.7 = 1.51 m). To respect the code limits to the non-dimensional slenderness ð1:3 < k < 2:0Þ, the in-plane slenderness kip for S235 steel must be between 122 and 188 so that the gyratory radius must remain in the range between 8 and 12 mm. With such small values of the gyratory radius it is recommendable to use for the diagonals simple plates, which allow to easily calibrate the resistance according to the storey shear, thus minimizing the variance of the overstrength coefficients Xi (Eurocode 8 §6.7.3), as shown in Table 5. In the example, the thickness tip of the in-plane diagonal must be between 28 mm and 43 mm.
To avoid out-of-plane buckling, the relative slenderness must be lower than the in-plane one. Considering fixed-end constraints, assuming thus L0,op = 0.7125 2.7 = 1.93 m, an out-of-plane thickness top > tip L0,op/L0,ip = 1.3tip must be chosen. This condition is normally satisfied by thicknesses designed on the basis of the resistance requirement. To notice that for the concentric braces S235 steel was chosen to have a greater rapport between stiffness and resistance. In fact, while to respect both the inter-storey drift at the damage limitation state (DLS) and the h factor limit to control the dynamic instability an adequate brace stiffness is required, at the same time the brace resistance can be reduced because it is normally overabundant with regards to the force demand in low-seismicity areas. Adopting a CD procedure, an oversizing of the diagonals
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Table 5 ULS: effects of seismic actions in the designed X braced frame. Storey
P (kN)
dr =h (%)
dr/h (%)
NEd (kN)
V (kN)
h
N 0Ed (%)
Ad (mm2)
r (MPa)
Xi
7 6 5 4 3 2 1
600 1535 2470 3405 4340 5275 6210
0.17 0.17 0.17 0.16 0.15 0.14 0.12
0.68 0.69 0.67 0.64 0.59 0.55 0.49
70 111 132 149 171 198 231
57.3 90.9 107.8 122.1 140.4 162.4 180.5
0.071 0.117 0.154 0.178 0.183 0.177 0.168
75 126 156 181 210 241 278
690 1320 1740 2070 2400 2700 2790
109 95 89 88 87 89 100
2.05 2.35 2.50 2.56 2.56 2.51 2.25
0
0
0
P = Total gravity load at and above the storey considered; dr =h = drift calculated with the design response spectrum at ULS; dr =h ¼ ðdr =hÞ q; NEd = first order axial force in the braces; h: inter-storey drift sensitivity coefficient; N 0Ed ¼ N Ed =ð1 hÞ second order axial force; V = total seismic storey shear; Ad = area of the brace cross-section; r = tension brace; Xi = NEd/Npl.R = overstrength coefficient; r = tensile stress in the diagonal member.
Fig. 19. Brace details (units in mm).
Fig. 20. Detail of the connection of the diagonal to the gusset plate: welded end connection (a); bolted end-connection (b); internal-node connection (c).
implies a further increase of the resistance demand for nondissipative elements. The building design is governed by ULS resistance verification, as for the DLS the maximum story displacement dr is equal to 0.2% h (fourth floor), a value considerably lower than the 0.5% h limit required in the case of infills rigidly connected to the structure. Finally, in Fig. 19 details of brace connection is shown for fixedend restraints. Both welded and bolted types of connection have been designed according to the CD criterion in order to test a full scale X braced frames with plates as diagonal members (Fig. 20). The recent test results on full scale [16] have confirmed the validity of the theoretical formulation also achieved with reduced scale specimens. Furthermore the hysteretic behaviour of X braced frame was characterised by large inelastic deformation (beyond 4% drift) without the brace failure even after several cyclic loadings.
7. Concluding remarks The respect of both the upper-bound and the lower-bound values of non-dimensional slenderness required by the Italian Standard and Eurocode 8 for the X braced frames’ diagonals implies the necessity of a realistic evaluation of the effective lengths, contrary to current design practice that considers as safe an estimate by excess of slenderness. For this reason, the international standards should clearly specify that in the design process of the bracing members the assumed effective length b depends on the boundary condition offered by end connections as well as on the tensile force acting in the complementary member. The experimental results demonstrated the accuracy of the theoretical effective lengths whose values are by far smaller than those used for the calculations in the current design practice. The specimen designed with a brace slenderness ratio equal to 1.3
G. Metelli / Engineering Structures 46 (2013) 763–773
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Fig. A1. Stiffness of the semi-diagonal member.
did not show the peak shear strength during the first cycle thus avoiding the overloading of the non-dissipative elements. Furthermore, the reduced scale X braced frames exhibited good inelastic deformation capacity under cyclic loading with and ultimate elongation of the diagonals greater the 40 times the yield one. The design example of a multi-storey building with X-braced frames shows the necessity to adopt for the diagonals cross sections with very small gyratory radiuses, so as to recommend the use of simple plates. The use of plates, laid with the greater side perpendicular to the frame, allows to easily avoid out-of-plane buckling, that can cause serious damage to wall partitions and cladding. Furthermore bracing plates solve the problem of calibrating the diagonals’ section according to the resistance demand avoiding diagonals oversizing, inevitable when adopting the usual solutions, like angles. Acknowledgments The author sincerely thanks Prof. Piero Gelfi for his highly esteemed help in the research work. Financial support for this study was provided by the University of Brescia. The cooperation in carrying out the tests of Mr. Luca Martinelli, technician of the P. Pisa Laboratory of the University of Brescia, and of Eng. Alex Cappa is gratefully acknowledged. Appendix A. Second order stiffness [12,13] In Fig. A1 the rotational M11 stiffness and the translational H11 stiffness of the brace are shown for pinned-end or fixed-end. A.1. Compressive axial force Stability function
3 1 1 6 1 1 I2 ðaLÞ ¼ aL aL tanðaLÞ aL sinðaLÞ aL tanðaLÞ I3 ðaLÞ ¼ aL
I1 ðaLÞ ¼
sffiffiffiffiffi P : being aL ¼ EJ
A.2. Tensile axial force By substituting P with P, a2 with a2 and a with ia in the stability function of the case with a compressive axial force, the stability functions become:
3 1 1 6 1 1 I2 ðaLÞ ¼ aL tanhðaLÞ aL aL aL sinhðaLÞ tanhðaLÞ I3 ðaLÞ ¼ aL
I1 ðaLÞ ¼
being sinðiaLÞ ¼ i sinhðaLÞ; cosðiaLÞ ¼ coshðaLÞ: References [1] Tremblay R. Inelastic seismic response of steel bracing members. J Constr Steel Res 2002;58:665–701. [2] Tremblay R. Seismic behaviour and design of concentrically braced steel frames. Eng J, AISC, 2001;40. [3] Eurocode 8: design of structures for earthquake resistance – part 1: general rules, seismic actions and rules for buildings, EN 1998-1, CEN; December, 2004. [4] D.M. 14.1.2008. Norme tecniche per le costruzioni. [5] Eurocode 3: design of steel structures – part 1–1: general rules and rules for buildings, EN 1993-1-1, CEN; 2005. [6] AISC, seismic provisions for structural steel buildings ANSI/AISC 341-05, 341s1-05, Chicago, Illinois; March 9, 2005. [7] Tremblay R, Archambault MH, Filiatrault A. Response of concentrically braced steel frames made with rectangular hollow bracing members. J Struct Eng 2003;129(12):1626–36. [8] Filiatrault A, Tremblay R. Design of tension-only concentrically braced steel frames for seismic induced impact loading. Eng Struct 1998;20(12):1087–96. [9] Sabelli R, Hohbach D. Design of cross-braced frames for predictable buckling behavior. J Struct Eng 1999;125(2):163–8. [10] Picard A. e Beaulieu D. Design of diagonal cross-bracings. Part 1: theoretical study. Eng J, AISC, third quarter; 1987. [11] Picard A, e Beaulieu D. Design of diagonal cross-bracings. Part 2: experimental study. Eng J, AISC, third quarter; 1987. [12] Bazant ZP, Cedolin L. Stability of structures: elastic, inelastic, fracture, and damage theories. New York, Oxford: Oxford Univ. Press; 1991. [13] Timoshenko, Gere. Theory of elastic stability. McGraw Hill Kogakusha Ldt.; 1961. [14] Gelfi P, Metelli G, Cappa A. Studio Teorico e Sperimentale sul comportamento ciclico di controventi concentrici a X. Technical report n.1, Università degli Studi di Brescia, DICATA; 2008 [in Italian]. [15] EN ISO 6892-1:2009. Metallic materials – tensile testing – part 1: method of test at room temperature. [16] Metelli G, Marchina E, Gelfi P. Studio sperimentale sul comportamento ciclico di controventi concentrici a X. In: Proceedings of the XXIII Congresso CTA. 9– 12 Ottobre 2011, Lacco Ameno, Ischia, NA; 2011 [in Italian].
Contents lists available at SciVerse ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Theoretical and experimental study on the cyclic behaviour of X braced steel frames Giovanni Metelli ⇑ University of Brescia, Department of Civil Engineering, Architecture, Land and Environment, Via Branze 43, 25123 Brescia, Italy
a r t i c l e
i n f o
Article history: Received 17 February 2012 Revised 13 July 2012 Accepted 17 August 2012
Keywords: Concentrically braced frames Critical load Slenderness Effective length factor Seismic behaviour
a b s t r a c t The paper presents a theoretical and experimental study on the slenderness of the diagonal members of concentrically X braced steel frames. According to Eurocode 8 and Italian Standard, the non-dimensional slenderness of the diagonal members should be lower than an upper limit, in order to provide sufficient hysteretic energy dissipation, and greater than a lower limit, to avoid overloading the columns in the prebuckling stage. This seismic design provision requires the realistic evaluation of the diagonal member slenderness. A reduced scale test bench was designed to study X braced steel frames with different geometrical characteristics of diagonal members and with restraints as similar as possible to the theoretical ones. The results confirm the theoretical prediction of the effective length factor of the diagonal members which exhibited good hysteretic behaviour with large inelastic deformation capacity under cyclic loading. The low values of the effective length factor suggest the use of plates to realise the diagonal member instead of the classical angles as shown in the design example. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In steel multi-storey buildings the lateral actions due to wind or earthquake are generally withstood by braced frames, among which the most common are the concentrically X braced frames designed with only the tension diagonal active. Such a structural system is widely used because of its cost effectiveness, reduced construction time and the simplicity of its design. The behaviour induced by seismic actions depends on the ductility and the dissipative capabilities of the structure, whose inelastic deformations are provided by braces buckling and yielding when concentrically braced frames are used [1,2]. Furthermore, if Capacity Design (CD) based codes are adopted, the brace is designed by assuming that the dissipative zones are mainly positioned along the diagonals, whose yielding must take place before the failure of connections failure and the formation of any inelastic mechanism in the beams, columns, and foundations [3,4]. In X braced frames with the diagonals connected at the intersection node, the phase during which the compression brace is also active precedes the yield phase of the tension brace. So, if the buckling resistance is too high, an initial shear action peak can occur and consequently, with the view of applying the Capacity Design criterion, an increase in the resistance of the beams, columns, foundations and connections may be requested. ⇑ Tel.: +39 0303711234; fax: +39 0303711312. E-mail address: [email protected] 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.08.021
In the initial load phase (Fig. 1a), as the axial stiffness of the braces is usually very small compared to that of the columns and beams, both diagonals are subject to the axial action N ¼ V=ð2 cos #Þ. As a consequence, when the first critical load occurs in the compression brace the storey shear force V is equal to V max;1 ¼ 2N cr cos #. Once the compression brace has buckled, the storey shear Vmax,2 is carried mainly by the yielded tension brace V max;2 ¼ N y cos # (Fig. 1b). Thus, in order to avoid the peak load in the first cycle which can overload the columns and brace joints, the condition Vmax,1 < Vmax,2, (i.e. Ncr < 0.5Ny), must be respected. For this reason, both Eurocode 8 [3] and the Italian Standard [4] require the non-dimensional slenderness of the braces to be greater than 1.3. In such a case in fact, as the reduction factor v for the relevant buckling mode calculated according the Eurocode 3 [5] reaches the maximum of 0.47, the buckling resistance (Nb,Rd = vAfyk/cM1) is less than half of the yield strength (NRd = Afyk/cM0). Eurocode 8 and the Italian Standards also set at 2 the upper limit for the non-dimensional slenderness of the braces, to guarantee sufficient hysteretic dissipation of the frame. To respect both the lower and the upper slenderness limits, a realistic evaluation of the brace’s effective lengths is needed, contrary to usual design practice which considers safe an evaluation by excess of slenderness when member shall be verified against buckling. In the scientific literature the problem of the peak shear strength of X braced frames during the initial cycle of a seismic event does not seem analysed in sufficient depth. The reason is maybe to be found in the fact that the American Standard [6] does
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Fig. 1. Maximum shear action in the first cycle (a) and in the subsequent cycles (b).
not prescribe a lower limit to the slenderness. Several test were performed on braces with different cross section showing an increased energy dissipation when the brace slenderness is decreased while the ductility level at fracture can be markedly reduced in RHS profiles due to the local buckling and fracture at plastic hinge [1,2,7]. An equation, in good agreement with test results, was also proposed to evaluate the post buckling resistance of the brace for increasing ductility level, thus allowing to estimate the shear strength of braced frames. An experimental research confirmed the limited energy dissipation of tension only concentrically braced frame [8] with a loss of stiffness and strength to lateral displacement which can lead to the formation of soft-story mechanism in multi-story buildings due to its limited capacity to distribute the inelastic deformation demand over the structure height [2]. Analytical [9,10] and experimental [11] studies on X braced frames showed that the effective length of the compressive diagonal is lower the half diagonal length because of the restraint provided by the complementary brace, whose stiffness is increased by the presence of a tensile action. The present research focused on the experimental evaluation of the brace’s effective length in order to validate the theoretical formulations proposed in literature. The test results herein presented shed light on the role of the lower-bond value imposed by standards on the brace slenderness to limit the peak shear strength of the X braced frames during the early cycles of a seismic event. A design example of a multi-storey building with X-braced frames in a low-seismic zone is also illustrated to show the necessity to adopt for the diagonals cross sections with very small gyratory radiuses for the respect of slenderness limits, so as to recommend the use of simple plates instead of the classical angles.
2. Buckling of X braced frames The correct evaluation of the effective length must be made both for the in-plane and the out-of-plane buckling. The latter should preferably be avoided as it can cause serious damage to the adjacent non-structural elements, such as infills and claddings. With regards to the in-plane buckling it’s important to highlight that the braces’ boundary conditions change during the seismic event. At the beginning both the braces are subject to the same axial load, so the compression brace’s buckling is hindered by the second order effect induced by the tension brace. Once the load is reversed after the yielding of the tension brace and the buckling of the compression brace during the first cycle, the rotational constraint at the braces intersection point is more flexible, compared to the first cycle, because the complementary brace is still plastically deformed due to the buckling. For this reason it cannot carry any axial load until it is straightened out thus
providing no beneficial second order effect to the buckling of the other diagonal. The effective lengths’ values can be calculated once the critical load value is evaluated through the buckling analysis of the brace [12,13]. 2.1. In-plane effective length In-plane brace buckling occurs without any displacement of the diagonals’ intersection node and with its u1 rotation hindered by the flexural stiffness of the tension brace. First the calculation for X braces with fixed-end restraints (Fig. 2a) will be presented. The critical load Pcr is obtained setting equal to zero the rotational stiffness M11 of the semi-diagonal members converging to the intersection node (Appendix A) [12,13]:
M11 ¼ 2
4EJ 3I1 ðaLÞ 3I1 ðam LÞ þ L 4I21 ðaLÞ I22 ðaLÞ 4I21 ðam LÞ I22 ðam LÞ
! ¼0
ð1Þ
E being the modulus of elasticity of the steel, J the brace moment of inertia, I1 and I2 the stability functions defined in Appendix A, L the length of the semi-diagonal, a = (P/EJ)0.5, P the axial load in the compression brace and mP the axial load in the other brace. The maximum value of the critical load is obtained when the complementary brace is loaded by a tensile force of the same value (m = 1). In this case the minimum value of aL that satisfies Eq. (1) is 5.61. This value allows to calculate the critical load Pcr and the slenderness factor b, defined as the ratio between the in-plane effective length L0,ip and the length L of the semi-diagonal:
Pcr ¼ EJ
b¼
aL L
2 ¼
p2 EJ ðbLÞ2
con b ¼
p aL
L0;ip p ¼ ¼ 0:56 L aL
ð2Þ
ð3Þ
If the complementary brace is unloaded, a lower value of the critical load is obtained. As a result, Eq. (1) is satisfied by aL = 5.325, corresponding to a slenderness factor b = 0.59. Also in the case of pinned-end restraints (Fig. 2b), the critical load value is obtained by setting equal to zero the second order rotational stiffness M11 of braces’ connecting node (Appendix A):
M11 ¼ 2
3EJ 1 1 ¼0 þ L I1 ðaLÞ I1 ðam LÞ
ð4Þ
In the case of tensile action in the complementary brace (m = 1) Eq. (4) is satisfied when aL = 3.927, corresponding to a slenderness factor b = 0.80, while in the case of no tensile action
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Fig. 2. In plane brace deformation due to the rotation u1 of the internal node, for fixed-end (a) and pinned-end diagonals (b).
(m = 0) Eq. (4) is satisfied if aL = 3.127, providing a slenderness factor b = 0.843. These theoretical values of the slenderness factor are consistent with the results found in literature [9]. 2.2. Out-of-plane effective length During the first load cycle, the first buckling mode of the compression brace (Fig. 3a) cannot occur because the presence, even modest, of tensile action in the other brace prevents the intersection point from moving out-of-plane [10,11]. The buckled shaped is thus characterised by the out-of-plane rotation of the braces’ intersection node, rotation hindered only by the torsional stiffness of the opposite brace (Fig. 3b). Therefore the effective length L0,op is always less than the length L of the semi-diagonal (L0,op = 0.7L for fixed ends and L0,op = L for pinned ends). In the load cycles following the first, since one brace has buckled and shows an inelastic deformation, it hinders the compression brace’s first form of out-of-plane buckling just with its flexional stiffness. The critical load value can be obtained by setting equal to zero the second order translational stiffness H11 of the connecting node (Appendix A). In the case of fixed-end braces (Fig. 4a), the equation to determine the critical load is:
H11 ¼ 2
12EJ L3
3 4I21 ðaLÞ I22 ðaLÞ
! þ
24EJ L3
¼0
Fig. 4. Out of plane deformation of the brace for fixed-end (a) and pinned-end (b).
Table 1 Effective length factors b = L0/L of the braces. Restraint
In-plane
Lower limit Upper limit
Out-of-plane
Pinned end
Fixed end
Pinned end
Fixed end
0.800 0.843
0.560 0.590
<1.0 1.425
<0.7 0.7125
ð5Þ
fulfilled if aL = 4.409, corresponding to an effective length factor b = 0.7125.
In the case of braces with pinned ends (Fig. 4b) the stability equation, expressed by Eq. (6), is satisfied by aL = 2.205, corresponding to an effective length factor b = 1.425:
H11 ¼ 2
3EJ L3
1 6EJ þ 3 ¼0 I1 ðaLÞ I3 ðaLÞ L
ð6Þ
In Table 1 the values of the effective length factor for in-plane and out-of-plane buckling are reported for both end restraint conditions of the diagonal members. 3. Test bench
Fig. 3. Out of plane buckled shapes.
In order to investigate the validity of the theoretical formulations on the effective length, a test bench was built to carry out on a reduced scale a series of tests on X braced frames with boundary conditions as close as possible to the ideal theoretical ones. The test bench consisted of a square frame 621 mm wide, made of four pairs of external rods (1 in Fig. 5a) within which the two diagonal braces were positioned (2 in Fig. 5a). The external rods were hinged to each other (3 in Fig. 5a), while the braces were fixed at both ends by joints that prevent them from rotating (4 in Fig. 5a and Fig. 5b). The two vertical joints were connected directly to the Instron test machine, while the joints’ rotation at
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Fig. 5. (a) Test bench and (b) pin’s detail.
the ends of the horizontal brace was impeded by a coupling sleeve with an extension rod (5 in Fig. 5a). The two diagonal braces were manufactured separately and subsequently welded together at the mid-length to guarantee the correct positioning within the bench. The applied vertical force F is measured by the test machine’s load cell, while the axial forces in the external rods are measured by strain-gauges. The axial force in the diagonal braces is calculated as a function of the applied force F, of the force in the external rods Fc and of the variable angle # between the external rod and the vertical brace (Fig. 6). The effect, albeit modest, of the friction in the pins linking the joints to the external rods was also taken into account, evaluated through a preliminary test.
Vertical brace force :
F dv ¼ F 2F c cos #
Horizontal brace force :
F do ¼ 2F c sin #
ð7Þ ð8Þ
Fig. 7. Test set up: detail of the LVDTs position for the measurement of the brace axial deformation (a) and of the brace buckling deflection (b).
Angle between the vertical brace and the external rod : L do =2 # ¼ arctan L dv =2
ð9Þ
The axial deformation of the vertical brace was measured with an LVDT inductive transducer, fixed to the two joints’ pins by means of two alloy extension rods (a in Fig. 7a), while a second LVDT, linking the two sliding parts of the coupling sleeve, measures the axial deformation of the horizontal brace (b in Fig. 7a). Strain gauges were applied to the diagonal braces to correctly measure the value of the critical load, while two LVDTs were positioned at the middle of the semi-diagonal to measure its in-plane buckling deflection (c and d in Fig. 7b). Further details of the test bench are to be found in [14].
4. Test programme
Fig. 6. Forces in the frame elements and frame deformation.
Three tests were carried out on braces with a rectangular crosssection whose geometrical characteristics are shown in Table 2. The specimens were designed to buckle in the frame plane, with a non-dimensional slenderness kip ¼ 1:3 calculated assuming the effective length value b as gradually reaching the theoretical one. Thus, as the non-dimensional slenderness is defined as kip ¼ kip =k1 , being k1 = p(Es/fyk)0.5 the slenderness corresponding to the yield strength, which is equal to 86.8 for S275 steel chosen for the test, the specimens were designed with an in plane slenderness kp = 113 which corresponds to a in plane thickness ip ffiffiffiffiffiffi tip ¼ bL 12=kip . Specimens 1 and 2 were designed in accordance with the current professional practice, with an evaluation by excess of the effective lengths: in the plane a fixed-end constraint for the diagonals was assumed, along with a pinned constraint at the intersection point (L0,ip = 0.7L), thus ignoring the stiffening contribution of
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G. Metelli / Engineering Structures 46 (2013) 763–773 Table 2 Design geometrical characteristics of the specimens. Specimen
kip
kip
L0,ip = bL (mm)
tip (mm)
kop
kop
L0,op = bL (mm)
top (mm)
1 2 3
1.3 1.3 1.3
113 113 113
0.70L 0.70L 0.56L
7.5 7.5 5.8
1.3 1.3 1.3
113 113 113
1.25L 1.00L 0.71L
13.4 10.7 7.5
L = 350 mm: length of the semi-diagonal; tip: in-plane brace thickness; top: out-of-plane brace thickness; Lp 0,ip ffiffiffiffiffiffiand L0,op: in-plane and out-of-plane effective pffiffiffiffiffiffilength; k ¼ k=k1 : non-dimensional slenderness being k1 = p(Es/fy.nom)0.5 = 86.8 for S275 steel; in plane slenderness kip ¼ L0;ip 12=tip ; out-of plane slenderness kop ¼ L0;op 12=top .
Table 3 Mean mechanical proprieties of the steel used for the braces. Yield strength Ultimate strength Elongation after fracture Elongation at maximum tensile force Modulus of elasticity
fym (MPa) ftm (MPa) A5 (%) Agt (%) Esm (MPa)
318.6 455.9 38.7 19.7 213,100
Test carried out in according to EN ISO 6892-1:2009 [15].
specimen 1 (L0,op = 1.25L). The design of specimen 3 was based on the theoretical values of the effective lengths, assuming L0,ip = 0.56L and L0,op = 0.71L for in-plane and out-of-plane buckling respectively. S275 steel was adopted for the braces, with an average yield strength fym of 318 MPa and an average ultimate strength ftm of 456 MPa (Table 3). The experimental tests were carried out applying to the braces an axial deformation of increasing amplitude. The first cycle was characterised by a deformation of 4 mm, 4.35 times the nominal yield elongation dy = 0.92 mm (dy = 2Lfyk/Es, L = 350 mm being the length of the semi-diagonal; Fig. 8). The brace deformation was then applied stepwise in an increase of 2 mm between the subsequent cycles up to the brace fracture. 5. Experimental results
Fig. 8. Geometrical characteristics of the specimen.
the tension brace. For the out-of-plane buckling a fixed-end restraint was assumed and the stiffening effect of the other brace was ignored (L0,op = L), adopting a greater safety coefficient for
The main aim of the tests was to confirm the theoretical values of the effective lengths. For this reason special care was taken during the first two cycle loadings. The first cycle started applying a tensile force in the vertical brace, thus compressing the horizontal one. All three tests confirmed the in-plane effective length theoretical value L0,ip = 0.56L, the semi-length of the brace being L = 350 mm. The out-of-plane buckling did not occur even in the subsequent load cycles up to the brace fracture. As an example, the results of the second test are illustrated in detail. In Figs. 9 and 10 the axial force F versus the axial deformation d curves of both braces are shown for the first two load cycles. The horizontal brace buckled for an axial force Ncr = 20.75 kN (Fig. 9), when the vertical brace was loaded by a tensile force
Fig. 9. Specimen 2: axial force Fdo–deformation do curve of the horizontal brace during the first and the second cycles.
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Fig. 10. Specimen 2: axial force Fdv–deformation dv curve of the vertical brace during the first and the second cycles.
Fig. 11. Specimen 2: buckled shape of the horizontal brace during the first cycle (a) and failure of the vertical brace (b).
Fdv = 22 kN (Fig. 10). The effective length, the in plane moment of inertia being J = 338 mm4, was L0,ip = 185 mm calculated by means of the expression of the Euler load (Eq. (2)) and the slenderness factor was equal to b = L0,ip/L = 185/350 = 0.53, value that practically coincides with the theoretical one (b = 0.56), considering measurement approximations as inevitable in these experimental conditions. Fig. 11a shows the critical deformation of the horizontal brace in the first load cycle, while Fig. 12 presents the pictures of the deformations corresponding to the points indicated by Fi in the diagrams of Figs. 9 and 10. In the second load cycle the vertical brace buckled, after having yielded in tension, when the horizontal member still showed a plastic – buckling deformation with a compression force of 3.7 kN (Fig. 9). The critical load was Ncr = 14.4 kN (Fig. 10), value equal to 70% of that of the first cycle. The fracture took place in the plastic hinge during the 18th tensile cycle of the vertical brace (Fig. 11b), with a total deformation of 37 mm (e = 37/700 = 0.053), equal to roughly 40 times the yield elongation (Fig. 13). The main results of the three tests are presented in Table 4. In Figs. 13 and 14, the typical hysteretic behaviour of an axially loaded member is shown for both braces of specimen 2. In compression, Baushinger’s effect and the inelastic deformations accumulated during the previous cycles decreased markedly the buckling resis-
tance, while in tension the brace developed its yield strength during each subsequent cycle only if a larger elongation was applied. In tension the braces developed also some strain hardening. With regards to the post-buckling strength of the braces, the compression resistance C 0u was measured for a ductility level l = d/dy equal to 3, after buckling occurred during the first cycle, as proposed in [1]. For a brace elongation equal to 2.75 mm (= 3dy), the tested specimens exhibited a post-buckling strength C 0u varying between 31% and 43% of the nominal yield strength for the horizontal brace of specimens 3 and 1 respectively (Table 4). Earlier experimental and theoretical studies [1] showed a lower postbuckling strength ranging between 15% and 30% of the brace yield strength. The experimental results allow also to evaluate the overall hysteretic behaviour of X braced frames, as shown in the sketch of Fig. 15. The test frame is a self-balanced system and the storey shear load V can be defined as the component of the applied force F on the direction of the frame’s external rods (Fig. 15). The interstorey displacement dx is obtained by measuring the axial deformation dv and do of the braces.
V ¼ F cos #
ð10Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dx ¼ d2v þ d2o
ð11Þ
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Fig. 12. Specimen 2: brace deformation during the first and the second cycle at the point Fì indicated in Figs. 9 and 10.
Fig. 13. Specimen 2: axial force Fdv–deformation dv curve of the vertical brace up to the failure.
Fig. 14. Specimen 2: axial force Fdo–deformation do curve of the horizontal brace up to the failure.
Table 4 Test results. Specimen
1 2 3a
1° cycle
12° cycle
Ncr (kN)
Horizontal brace L0,ip (mm)
b
Nmax (kN)
do,max
C 0u (kN)
Ncr (kN)
Vertical brace L0,ip (mm)
b
Nmax (kN)
dv,max
C 0u (kN)
V/Vy
V/Vy
27.4 20.7 6.3
187 185 204
0.54 0.53 0.59
33.62 26.70 16.71
33.9 37.3 41.3
11.6 9.0 3.7
23.4 14.4 4.2
205 223 249
0.59 0.64 0.72
35.04 26.94 17.83
35.0 37.0 41.9
10.5 8.2 4.2
2.11 2.13 1.68
1.69 1.73 1.71
Ncr: Experimental critical load; L0,ip: in-plane effective length; b: in-plane effective length factor; Nmax: maximum axial force; dmax: maximum elongation. a The test on specimen 3 was interrupted before the brace fracture.
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In Fig. 16a and Fig. 16b the dimensionless shear action V/Vy is plotted as a function of the inter-storey drift, which is the ratio between the inter-storey displacement dx and the inter-storey height h = 621.5 mm. Vy is the nominal plastic shear strength of the brace, defined as the shear action which yields the diagonal assuming a tension-only braced frame (Vy = A fyk cos #). It is worthwhile to point out that specimen 3 did not show the typical shear peak strength of specimens 1 and 2, because it was designed with an in-plane slenderness ratio kip equal to 1.3 assuming the theoretical effective length factor b equal to 0.56L. Even though the tested X braced frames exhibited a typical pinched hysteretic behaviour with a lack of stiffness under load reversal due to the inelastic deformation accumulated in both diagonals, the failure did not occur up to a 7% drift, value well beyond the specified code limits generally adopted for the ultimate limit state (ULS).
Fig. 15. X braced frame: storey shear and drift.
6. Design example The application of the effective length values exposed above leads to the design of unusual X braced frames. As an example, here presented are the main outcomes in the design of a typical multi-storey building, realised in accordance to the Eurocode 8 [3] and Italian Standard [4]. A seven-storey office-building with a regular grid of columns and beams is considered. The plan layout of the structure is shown in Fig. 17. The seismic-resistant action is entrusted to eight tension-only X braced frames, positioned symmetrically at the corners of the building. The X braced frames’ design was carried out for a typical lowseismic zone in Northern Italy (ground type C, ag = 0.11g). Nevertheless, it is the author’s belief that the considerations exposed for the choice of the type of braces’ cross-section have general validity. The gravity loads considered in buildings were: floor dead load of 4.00 kN/m2, floor live load of 3.00 kN/m2, roof snow load of 1.50 kN/m2 and exterior walls of 8.00 kN/m. As the building conforms to the criteria for regularity in plan, the modal analysis may be performed using two separate planar models for the earthquake directed along the two perpendicular directions. Furthermore, as the seismic behaviour in both directions is the same, only the case of an earthquake in the X direction was considered, amplifying the seismic action by 30% to keep into account accidental torsional effects. A behaviour factor q equal to 4 was assumed for the reduction of the elastic response spectrum at ultimate limit state. In Fig. 18 are reported the values of the seismic masses applied to the X brace frame’s nodes and the values of the static vertical loads used for the modal analysis, carried out considering only the tension braces (Eurocode 8 §6.3 and Italian Standard §7.5.5) and modelling each member with hinged connections. This assumption is considered as conservative for the evaluation of the seismic response of the building because the columns are usually continuous and the beam–column joints may be rigid. The dimensioning of the X braced frame was performed including also the second-order effects (or P–D effects) in accordance with the Italian Standard and Eurocode 8 (par. 4.4.2.2), by multiplying the relevant seismic action effects by the factor 1/(1 h), being h the inter-storey drift sensitivity coefficient:
h¼
Fig. 16. Hysteretic brace response; dimensionless storey shear V/Vy–drift curve for specimen 2 (a) and specimen 3 (b).
P dr V h
ð12Þ
where dr is the inter-story displacement, calculated by multiplying the inter-storey displacement determined with the design response spectrum at the ultimate limit state by the behaviour factor q, thus equal to the displacement calculated with the elastic spectrum, V is the total storey shear computed by the design response spectrum and P is the total gravity load at and above the storey considered. In Table 5 the NEd forces in the diagonals, the dr drifts for each storey, the over-strength coefficients Xi and the factor h of the 2nd order effects are reported. It is worthwhile to point out that the factor h resulted always less than 0.2, as it must be to have the possibility of keeping into account second order effects simply by increasing the horizontal seismic action effects by the factor 1/(1 h). Controlling the parameter h is particularly arduous when designing buildings in low seismic zones where the effects of the seismic actions are modest so that the dimensioning of the diagonals according to the resistance leads to X braced frames which are too flexible to impede dynamic instability. It is thus necessary to oversize the diagonals, as is evident from the modest values of the r stresses, reported in Table 5. To reach the same results more quickly, normally it would be more practical to carry-out the design adopting the value of q = 2 for the behaviour factor [14].
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Fig. 17. Structure lay-out: floor plan view and front view of the X braced frames (units in m).
Fig. 18. X braced frame of the design example: masses and the loads applied for the modal response analysis.
The effective length L0 = bL of the diagonal was calculated with reference to the semi-length L, considering the 2.7 m length as being with a constant cross section (Fig. 19). Fixed end restraints of the braces were assumed with an in-plane effective length L0,ip of 1.51 m (L0,ip = 0.56 2.7 = 1.51 m). To respect the code limits to the non-dimensional slenderness ð1:3 < k < 2:0Þ, the in-plane slenderness kip for S235 steel must be between 122 and 188 so that the gyratory radius must remain in the range between 8 and 12 mm. With such small values of the gyratory radius it is recommendable to use for the diagonals simple plates, which allow to easily calibrate the resistance according to the storey shear, thus minimizing the variance of the overstrength coefficients Xi (Eurocode 8 §6.7.3), as shown in Table 5. In the example, the thickness tip of the in-plane diagonal must be between 28 mm and 43 mm.
To avoid out-of-plane buckling, the relative slenderness must be lower than the in-plane one. Considering fixed-end constraints, assuming thus L0,op = 0.7125 2.7 = 1.93 m, an out-of-plane thickness top > tip L0,op/L0,ip = 1.3tip must be chosen. This condition is normally satisfied by thicknesses designed on the basis of the resistance requirement. To notice that for the concentric braces S235 steel was chosen to have a greater rapport between stiffness and resistance. In fact, while to respect both the inter-storey drift at the damage limitation state (DLS) and the h factor limit to control the dynamic instability an adequate brace stiffness is required, at the same time the brace resistance can be reduced because it is normally overabundant with regards to the force demand in low-seismicity areas. Adopting a CD procedure, an oversizing of the diagonals
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Table 5 ULS: effects of seismic actions in the designed X braced frame. Storey
P (kN)
dr =h (%)
dr/h (%)
NEd (kN)
V (kN)
h
N 0Ed (%)
Ad (mm2)
r (MPa)
Xi
7 6 5 4 3 2 1
600 1535 2470 3405 4340 5275 6210
0.17 0.17 0.17 0.16 0.15 0.14 0.12
0.68 0.69 0.67 0.64 0.59 0.55 0.49
70 111 132 149 171 198 231
57.3 90.9 107.8 122.1 140.4 162.4 180.5
0.071 0.117 0.154 0.178 0.183 0.177 0.168
75 126 156 181 210 241 278
690 1320 1740 2070 2400 2700 2790
109 95 89 88 87 89 100
2.05 2.35 2.50 2.56 2.56 2.51 2.25
0
0
0
P = Total gravity load at and above the storey considered; dr =h = drift calculated with the design response spectrum at ULS; dr =h ¼ ðdr =hÞ q; NEd = first order axial force in the braces; h: inter-storey drift sensitivity coefficient; N 0Ed ¼ N Ed =ð1 hÞ second order axial force; V = total seismic storey shear; Ad = area of the brace cross-section; r = tension brace; Xi = NEd/Npl.R = overstrength coefficient; r = tensile stress in the diagonal member.
Fig. 19. Brace details (units in mm).
Fig. 20. Detail of the connection of the diagonal to the gusset plate: welded end connection (a); bolted end-connection (b); internal-node connection (c).
implies a further increase of the resistance demand for nondissipative elements. The building design is governed by ULS resistance verification, as for the DLS the maximum story displacement dr is equal to 0.2% h (fourth floor), a value considerably lower than the 0.5% h limit required in the case of infills rigidly connected to the structure. Finally, in Fig. 19 details of brace connection is shown for fixedend restraints. Both welded and bolted types of connection have been designed according to the CD criterion in order to test a full scale X braced frames with plates as diagonal members (Fig. 20). The recent test results on full scale [16] have confirmed the validity of the theoretical formulation also achieved with reduced scale specimens. Furthermore the hysteretic behaviour of X braced frame was characterised by large inelastic deformation (beyond 4% drift) without the brace failure even after several cyclic loadings.
7. Concluding remarks The respect of both the upper-bound and the lower-bound values of non-dimensional slenderness required by the Italian Standard and Eurocode 8 for the X braced frames’ diagonals implies the necessity of a realistic evaluation of the effective lengths, contrary to current design practice that considers as safe an estimate by excess of slenderness. For this reason, the international standards should clearly specify that in the design process of the bracing members the assumed effective length b depends on the boundary condition offered by end connections as well as on the tensile force acting in the complementary member. The experimental results demonstrated the accuracy of the theoretical effective lengths whose values are by far smaller than those used for the calculations in the current design practice. The specimen designed with a brace slenderness ratio equal to 1.3
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Fig. A1. Stiffness of the semi-diagonal member.
did not show the peak shear strength during the first cycle thus avoiding the overloading of the non-dissipative elements. Furthermore, the reduced scale X braced frames exhibited good inelastic deformation capacity under cyclic loading with and ultimate elongation of the diagonals greater the 40 times the yield one. The design example of a multi-storey building with X-braced frames shows the necessity to adopt for the diagonals cross sections with very small gyratory radiuses, so as to recommend the use of simple plates. The use of plates, laid with the greater side perpendicular to the frame, allows to easily avoid out-of-plane buckling, that can cause serious damage to wall partitions and cladding. Furthermore bracing plates solve the problem of calibrating the diagonals’ section according to the resistance demand avoiding diagonals oversizing, inevitable when adopting the usual solutions, like angles. Acknowledgments The author sincerely thanks Prof. Piero Gelfi for his highly esteemed help in the research work. Financial support for this study was provided by the University of Brescia. The cooperation in carrying out the tests of Mr. Luca Martinelli, technician of the P. Pisa Laboratory of the University of Brescia, and of Eng. Alex Cappa is gratefully acknowledged. Appendix A. Second order stiffness [12,13] In Fig. A1 the rotational M11 stiffness and the translational H11 stiffness of the brace are shown for pinned-end or fixed-end. A.1. Compressive axial force Stability function
3 1 1 6 1 1 I2 ðaLÞ ¼ aL aL tanðaLÞ aL sinðaLÞ aL tanðaLÞ I3 ðaLÞ ¼ aL
I1 ðaLÞ ¼
sffiffiffiffiffi P : being aL ¼ EJ
A.2. Tensile axial force By substituting P with P, a2 with a2 and a with ia in the stability function of the case with a compressive axial force, the stability functions become:
3 1 1 6 1 1 I2 ðaLÞ ¼ aL tanhðaLÞ aL aL aL sinhðaLÞ tanhðaLÞ I3 ðaLÞ ¼ aL
I1 ðaLÞ ¼
being sinðiaLÞ ¼ i sinhðaLÞ; cosðiaLÞ ¼ coshðaLÞ: References [1] Tremblay R. Inelastic seismic response of steel bracing members. J Constr Steel Res 2002;58:665–701. [2] Tremblay R. Seismic behaviour and design of concentrically braced steel frames. Eng J, AISC, 2001;40. [3] Eurocode 8: design of structures for earthquake resistance – part 1: general rules, seismic actions and rules for buildings, EN 1998-1, CEN; December, 2004. [4] D.M. 14.1.2008. Norme tecniche per le costruzioni. [5] Eurocode 3: design of steel structures – part 1–1: general rules and rules for buildings, EN 1993-1-1, CEN; 2005. [6] AISC, seismic provisions for structural steel buildings ANSI/AISC 341-05, 341s1-05, Chicago, Illinois; March 9, 2005. [7] Tremblay R, Archambault MH, Filiatrault A. Response of concentrically braced steel frames made with rectangular hollow bracing members. J Struct Eng 2003;129(12):1626–36. [8] Filiatrault A, Tremblay R. Design of tension-only concentrically braced steel frames for seismic induced impact loading. Eng Struct 1998;20(12):1087–96. [9] Sabelli R, Hohbach D. Design of cross-braced frames for predictable buckling behavior. J Struct Eng 1999;125(2):163–8. [10] Picard A. e Beaulieu D. Design of diagonal cross-bracings. Part 1: theoretical study. Eng J, AISC, third quarter; 1987. [11] Picard A, e Beaulieu D. Design of diagonal cross-bracings. Part 2: experimental study. Eng J, AISC, third quarter; 1987. [12] Bazant ZP, Cedolin L. Stability of structures: elastic, inelastic, fracture, and damage theories. New York, Oxford: Oxford Univ. Press; 1991. [13] Timoshenko, Gere. Theory of elastic stability. McGraw Hill Kogakusha Ldt.; 1961. [14] Gelfi P, Metelli G, Cappa A. Studio Teorico e Sperimentale sul comportamento ciclico di controventi concentrici a X. Technical report n.1, Università degli Studi di Brescia, DICATA; 2008 [in Italian]. [15] EN ISO 6892-1:2009. Metallic materials – tensile testing – part 1: method of test at room temperature. [16] Metelli G, Marchina E, Gelfi P. Studio sperimentale sul comportamento ciclico di controventi concentrici a X. In: Proceedings of the XXIII Congresso CTA. 9– 12 Ottobre 2011, Lacco Ameno, Ischia, NA; 2011 [in Italian].