a method to avoid weak storey mechanisms in ...

8th International Conference on Behavior of Steel Structures in Seismic Areas Shanghai, China, July 1-3, 2015

A METHOD TO AVOID WEAK STOREY MECHANISMS IN CONCENTRICALLY BRACED FRAMES Daniel B. Merczel*, Jean-Marie Aribert*, Hugues Somja*, Mohammed Hjiaj*, János Lógó** * University INSA de Rennes, Dept. of Civil Engineering, France e-mails: [email protected], [email protected], [email protected], [email protected] ** Budapest University of Technology and Economics, Dept. of Structural Mechanics, Hungary e-mail: [email protected] Keywords: concentrically braced frame, seismic design, weak storey, plastic analysis. Abstract. This article deals with concentrically braced frames that are prone to exhibit weak storey behaviour in the case of a seismic event. The introduction of the key features of seismic design of braced frames according to Eurocode 8 is followed by the description of an alternative new redesign method that aims to prevent the occurrence of weak stories. The efficiency of the new method is proved by numerical examples.

1 INTRODUCTION Concentrically braced frames (hereinafter referred to as CBF-s) are among the most common structural systems for resisting lateral forces. They represent an economical structural form for providing lateral resistance for various types of actions. For wind effects, low return period seismic actions or for seismic actions in buildings of a high importance class the design behaviour is elastic. Due to their geometry the CBF-s counteract the horizontal forces by truss action that entails primarily axial forces in the members. The large resistance and stiffness limits the drifts and vibrations in the braced buildings, therefore provides occupancy comfort and impedes damages in the non-structural parts. In classical design CBF-s are expected to develop a global plastic mechanism when subjected to earthquakes. In the global mechanism all the braces undergo plastic deformations while the columns and beams are non-dissipative elements. Conversely, recent seismic events and numerical simulations pointed out, that some CBF-s develop an unfavourable plastic collapse mechanism when subjected to earthquake excitation even if the design has been made for such effects via the application of the corresponding standards such as Eurocode 8. This effect is often referred to as the weak storey mechanism. In the weak storey phenomenon the plastic deformations and drifts are localized on one or a limited number of storeys. Evidently, the dissipative and drift capacity of the weak storeys is by far inferior to the capacity of the whole CBF. Furthermore, the weak storey behaviour incorporates the significant bending of the columns, which facilitates collapse. The weak storey behaviour is unfavourable as it results in low seismic performance and may even lead to early collapse, therefore it needs to be prevented. In the following, the basic requirements for CBF-s of the Eurocode 8 design standard and the criteria of the new Robust Seismic Brace Design method (RSBD) will be discussed. The performance of CBF-s designed according to EC8 and reinforced by the RSBD method will be presented.

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

2 EUROCODE 8 DESIGN OF CBF-S In the design of building structures Eurocode 8 [1] provides two basic concepts for the seismic analysis. Earthquake resistant buildings can either have a low-dissipative or a dissipative behaviour. In a dissipative structure controlled inelastic deformations are expected to dissipate significant energy and damp the response of the structure. The behaviour factor, q, that accounts for this effect is 4 for CBF-s, which is the lower boundary of high ductility class (DCH) structures. In Eurocode 8 the criteria for the verification of dissipative structures follow the capacity design philosophy. Plastic deformations are strictly required to occur in dissipative members that are to be designed with sufficient ductility, whereas the yield or collapse of non-dissipative zones is to be evaded. To ensure the elastic behaviour of non-dissipative zones their seismic requirements are amplified in the design with a sufficient overstrength. In CBF-s the braces are meant to be the dissipative members. Therefore CBF-s are designed so that the yield of the diagonals takes place before the failure of the connections or the columns or beams. For the sake of homogeneous dissipative behaviour a simultaneous yield of the braces on every storey has to be ensured. Eurocode 8 aims to promote this by a condition given for the overstrength factors, Ωi, realized on the different storeys, making them closely uniform. It needs to be verified that the maximum overstrength does not differ from the minimum by more than 25%.

 max  1.25 

(1)

where

  min i , max  max  i with  i 

N pl , Rd ,i N br , Ed ,i

(2)

For the design of columns and beams Eurocode 8 requires the fulfilment of the following condition:

Nb,Rd  M Ed   N Ed ,G  1.1   ov    N Ed ,E

(3)

where Nb,Rd(MEd) is the design buckling resistance taking into consideration the simultaneous bending, NEd,G and NEd,E are the axial actions from the non-seismic and the seismic actions respectively. The overstrength factor, Ω, accounts for the reserve in the dissipative members. It ensures that the resistance of the non-dissipative members is adequate until the first plastic deformations occur. The γov factor accounts for the random variability of the material properties. Its recommended value is:

 ov  1.25

(4)

The factor 1.1 represents the increase of the yield stress of the dissipative members due to strain hardening. In addition Eurocode 8 defines limitations to the relative slenderness:

  2.0

(5)

1.3  

(6)

and for X-braced configurations also:

The upper bound defined in Eq. (5) is imposed to prevent the rapid degradation of the resistance of the braces. Furthermore this limits the plastic out-of-plane deformation of gusset plates which are prone to low-cycle fatigue fracture. The lower bound Eq. (6) assures the sufficient flexibility of the diagonals in compression. With its requirements, Eurocode 8 assumes and attempts to promote the development of a global plastic mechanism, see Figure 1/a.

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

Figure 1/a. Global plastic mechanism.

Figure 1/b. Weak storey mechanism.

This mechanism allows the largest possible dissipation in the structure as every brace participates in the dissipation of the seismic input energy and therefore the global mechanism provides the largest reduction of the internal forces in the structure. Conversely, in a localised weak storey mechanism, see Figure 1/b the dissipation capacity is reduced, which makes the use of a relatively high behaviour factor (q=4) questionable. In other words larger seismic forces can be expected. As the Eurocode 8 criteria are based on the elastic response of the structure subjected to the reduced-intensity seismic action, they may not be adapted in a random but probable inelastic response, such as the appearance of a weak storey. Firstly, in an elastic analysis, due to the truss action of the bracing, only negligible bending can be expected. Conversely, the columns in a weak storey response are subjected to significant bending, so the requirement imposed on the axial resistance of the columns, Eq. (3), may not be satisfactory to provide adequate column sections. In a weak storey response the axial forces of the braces may also be different from the ones obtained by elastic analysis, therefore the uniformity condition of the storey overstrength factors, Eq. (2), may be violated. The criteria of the Eurocode 8 presented so far refer to and therefore may only be valid in the reduced-intensity seismic action used for the design. In order to take into account the inelastic response in case of a stronger earthquake, the use of other new criteria are necessary.

3 THE ROBUST SEISMIC BRACE DESIGN METHOD In order to establish an appropriate redesign criterion instead of using a perfect and elastic model exhibiting global behaviour, a plastic model penalized with imperfections and their consequential effects should be used. First we assume that the structure exhibits weak storey phenomenon on one storey. Moreover we suppose that the bars are deteriorated by tensile yield and the deformation of the mid-section. This results in having triangular-shape braces, which significantly decreases the lateral stiffness of the considered storey and devolves significant loading on the columns, see [2]. The loss of stiffness significantly changes the modes of vibration of the CBF. If the deterioration and the stiffness loss is elevated the expected response of the multi-storey building is as depicted in Figure 2. The accelerations coming from such a displaced shape can be well approximated by only a constant acceleration above the weak storey. The horizontal loads determined from this acceleration are therefore proportional to the masses attributed to the storeys. If the masses are equal on every storey, the equivalent load pattern is constant above and zero under the weak storey. In an n-storey CBF n different load patterns can be analysed each ranging from the top down to a particular storey. For each load pattern, the load parameter corresponding to the formation of the global, λglob, and the local, λloc, plastic mechanism can be computed i.e. n parameter pairs can be calculated. If the local load multiplier is larger than the global, then according to the kinematic principle of plasticity, the occurrence of a storey collapse is prevented. So, in order to secure that all the storeys have enough reserve so that no localized collapse shall occur, the following criterion has to be met:

loc ,i  1.0 where i=1..n glob,i

(7)

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

Figure 2. Inelastic weak storey response of a CBF and plastic yield mechanisms of first criterion

The introduced criterion is necessary to evade weak storey collapse but not sufficient to ensure that the difference of the maximum interstorey drifts on all the soreys is within an acceptable range. Let us divide the local multiplier into two parts i.e. the participation of the brace(s) and the participation of the columns. Let us denote the former by λbr and the later by λcol. The diagrams of Figure 3 depict two theoretical cases where the load multiplier is plotted against the anticipated lateral interstorey drift response of a storey. The thin lines depict the initial response of the storey, the thick lines are anticipated when the triangular brace deterioration is present. When the resistance provided only by the brace is high, i.e. λbr is close to λglob, large inelastic deformations are not attained, the brace deterioration is relatively small, see on the left. Conversely, in the case of the diagram on the right when significant involvement of the columns is needed to respect the first criterion of Eq. (7), the inelastic displacements and the deterioration are larger. The objective of a second criterion is to prevent such inelastic displacement differences between storeys that otherwise satisfy the first criterion. Regarding the drift corresponding to the yield of the brace, dbr, if the yield stress is the same on every storey, then this drift also has to be closely the same constant on every storey. In order to promote a uniform response on every floor, it needs to be ensured, that the inelastic drift corresponding to the attainment of the global mechanism, dglob, is also closely the same on every floor. Therefore, the following ratio has to be quasi constant for all the storeys: d br ,i d glob,i

 const

(8)

Let us consider the dashed lines defining the secant stiffness of the inelastic response. The ratio of the stiffness values, denoted by β, describes the relative stiffness loss resulted by the deterioration of the brace.

br ,i

glob,i

d br , i

d glob,i

 i

(9)

As it has been mentioned before, significant localised decrease of the stiffness modifies the modal behaviour in a way that promotes the development of the weak storey. Consequently, in order to prevent the occurrence of the weak storey, it is required that the β shall also be the same on every storey. By combining Eq. (8) and (9) the so called Brace Performance Ratio (BPR) can be expressed, which can be well approximated by the ratio of load multipliers, given that the left side of the equation below is considered to be quasi constant on every storey.

BPRi 

br ,i d  i br ,i  const. glob,i d glob,i

(10)

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

loc

loc glob

glob

col

col

br br

d

d

del =dbr dglob

del =dbr

dglob

Figure 3. Theoretical elasto – plastic load multiplier displacement curves

As a second design criterion, additional to Eq. (7), it is required that the maximum and minimum BPR do not differ by more than 0.1. It is obvious that if the BPR is close to 1.0, then the bracing is overdesigned, column resistance is barely needed and the storey is not involved in the plastic dissipation. Therefore, it is recommended that the maximum BPR shall not exceed 0.9. It has to be noted, that the 0.1 and 0.9 constants have been determined empirically, using numerous examples.

BPRmin  0.1  BPRmax

(11)

With the application of the kinematic principle of plastic analysis, the formulae presented below can be derived for the computation of the load multipliers. It is noted, that the formulae were also elaborated to account for second order effects.

glob,i 

N

pl , br ,i

 cos i

k     mpk   l  k i  l 1  n

m

loc ,i 

(12)

i

N pl ,br ,i  cos i   M red ,i , j  j 1

n

 mpk

1 Hi

(13)

k i

br ,i 

N pl ,br ,i  cos i n

 mpk

(14)

k i

In the formulas Npl,br is the plastic axial resistance of the brace, α is the angle of the brace with the horizontal, mp is a vector indicating the mass differences of the storeys, Mred is the moment reserve of the columns with consideration to the simultaneous axial force and H is the storey height.

4 NUMERICAL MODELLING In order to determine the seismic performance of the various CBF buildings an incremental dynamic analysis (IDA) program has been carried out. The nonlinear time history analysis computations have been executed with the use of FinelG finite element analysis software. For the analysis seven artificial accelerograms were selected. The length of each excitation is 20 seconds and the spectra of the ground motions were fitted to the design spectrum. For the sake of the IDA the acceleration records were scaled by multiplying the peak ground acceleration by a scale factor ranging from 0.1 up to 2.0. To account for geometry and material nonlinearities, the members were divided into multiple corrotational beam elements and the Giuffré – Mengotto – Pinto [3] cyclic law has been used. The masses and simultaneous

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

loads were applied in the intersections of the beams and columns. The initial imperfection of the braces has been considered. In order to validate the accuracy of the nonlinear finite element models the results of an experimental test, carried out by Black et al [4], has been adopted. A bar with a 4 in × 4 in × 0.25 in square hollow section and 10 feet length is loaded with an alternating and monotonically growingamplitude imposed displacement sequence. In Figure 4 the measured and the computed behaviour of the brace is presented. As it can be seen, the curves are mostly overlapping one another. The figure shows that the performance of the hysteretic model is good as the stiffness, the ultimate strength and the degradation phenomenon are simulated successfully.

Figure 4. Experimental and numerical brace response

5 WORKED EXAMPLES For numerical validation of the proposed method several multi-storey CBF-s were designed. The buildings are identical in plan. The spacing of two adjacent columns is 6 metres in both directions. There are four bays in both directions giving 24 m × 24 m plan layout. Out of the four bays the inner two are braced concentrically on the four facades, see Figure 5. The elevation of the buildings differs in terms of storey number. The number of storeys is six, eight and ten in the buildings denoted by CBF6, CBF8 and CBF10 respectively. The letter M stands for 20% mass increase in the upper half of the 8storey building. The storey height is 3 metres on every floor in each building. The columns in the braced bays are pinned at the base, but otherwise continuous throughout the whole height and they are positioned in a way that their strong axis participates in the lateral resistance. Consequently the lateral load resisting system is the same in both directions. All the other columns are pinned at both ends. The dead load is a uniform loading of 6.77 kN/m2. The buildings are considered to have an office occupation that gives 3 kN/m2 evenly distributed loading on the slabs. The ψ2 combination factor was taken to be 0.3 therefore 30% of the live loading is concurrent with the seismic action. The earthquake acceleration response spectrum is considered to be the type one in EC8 assuming B type soil conditions. The design ground acceleration is taken to be 0.25g. pinned column

continous column

n×3m

3 2

4×6m = 24m

n

1 4×6m = 24m

Figure 5. View and plan of CBF buildings

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

In the following tables the column and brace cross sections, the storey overstrength factors and the results for the evaluation of Eq. (7) and (11) are presented. The differences between the initial Eurocode 8 designs and the RSBD reinforced designs as well as the violation of the RSBD criteria by the Eurocode designs are highlighted by bold letters. Table 1. Eurocode 8 and RSBD CBF designs, analysis results

CBF10

CBF8 - M

CBF6

building and storey 6 5 4 3 2 1 8 7 6 5 4 3 2 1 10 9 8 7 6 5 4 3 2 1

Eurocode 8 column

brace (SHS)

Ω

HEA 160 90×5 1.00 HEA 180 100×6 0.98 HEB 200 100×8 1.04 HEB 220 100×10 1.04 HEB 240 100×10 1.01 HEB 280 120×10 1.00 HEA 160 100×5 1.00 HEA 200 100×8 1.08 HEB 200 100×10 1.03 HEB 240 140×7.1 1.02 HEB 280 120×10 1.02 HEB 320 140×10 1.09 HEM 240 140×10 1.00 HEM 260 150×10 1.00 HEA 160 100×5 0.98 HEA 180 100×8 1.00 HEB 200 100×10 1.00 HEB 220 120×10 1.11 HEB 260 120×10 1.02 HEB 280 140×10 1.07 HEB 320 140×10 1.01 HEM 240 140×12.5 1.09 HEM 260 140×12.5 1.03 HD320×198 150×12.5 0.98

RSBD

loc ,i glob ,i

BPR

0.71 0.95 1.20 1.35 1.28 1.01 0.65 1.01 1.08 1.22 1.37 1.61 1.58 1.15 0.54 0.80 0.96 1.20 1.19 1.32 1.29 1.47 1.38 0.99

0.70 HEA 200 100×6 1.32 0.81 HEA 200 100×6.3 1.00 0.91 HEB 200 100×8 1.00 100×10 1.02 0.96 HEB 220 0.90 HEB 240 100×10 0.96 0.90 HEB 280 120×10 0.97 0.54 HEA 240 100×8 1.44 0.76 HEA 240 1.20 110×8 0.77 HEB 200 100×10 1.11 0.78 HEB 240 140×7.1 1.02 0.79 HEB 280 120×10 1.02 140×10 1.09 0.88 HEB 320 0.78 HEM240 140×10 1.00 0.77 HEM260 150×10 1.00 0.45 HEA 220 100×10 1.73 0.62 HEA 220 100×10 1.27 0.68 HEB 200 110×10 1.19 0.84 HEB 220 120×10 1.06 0.76 HEB 260 120×10 1.02 0.85 HEB 280 140×10 1.03 0.77 HEB 320 140×10 0.99 150×10 0.98 0.88 HEM 240 0.79 HEM 260 140×12.5 1.03 0.70 HD320×198 150×12.5 1.00

column

brace (SHS)

Ω

loc ,i glob ,i

BPR

1.01 1.14 1.20 1.29 1.23 0.97

0.89 0.84 0.87 0.90 0.86 0.86

1.06 1.27 1.14 1.15 1.30 1.53 1.50 1.09

0.76 0.81 0.79 0.74 0.75 0.83 0.74 0.73

0.99 1.15 1.15 1.10 1.13 1.22 1.20 1.27 1.34 0.99

0.79 0.78 0.80 0.76 0.73 0.77 0.70 0.71 0.75 0.72

In the diagrams of Figure 5 the evolution of the attained maximum interstorey drift ratio (IDR) is plotted against the growing seismic intensity defined by the scale factor. The IDR is the lateral interstorey drift divided by the storey height.

Daniel B. Merczel, Jean-Marie Aribert, Hugues Somja, Mohammed Hjiaj, János Lógó

Figure 5. IDA diagrams of analysed CBF-s.

6 CONCLUSION As it can be observed in the diagrams of Figure 5, the Eurocode 8 designs exhibit broader families of curves i.e. the difference between the largest and smallest IDR is generally larger than of the reinforced buildings. The significant discrepancies occur even below scale factor 1.0, which is the design intensity. The 2% IDR, indicated by a solid line is considered as an acceptance limit and as it can be seen the Eurocode 8 designs exceed this limit below the design seismic intensity. Consequently we may conclude, that the CBF-s considered in the present article are susceptible to exhibit the weak storey behaviour and also to early collapse if designed according to Eurocode 8 provisions. The RSBD method resulted in reinforcement of the storeys that are found to be susceptible to weak storey behaviour by the IDA analysis. Taking a look at Table 1 we may see that the redesigns strongly violate the uniformity condition imposed on the storey overstrength factors by Eq. (1). The performance of the RSBD designs is favourable even in the case of the irregular masses in the 8-storey CBF. The band of the curves is rather narrow and early collapse is not attained. In fact, in the taller 8 and 10 storey buildings the obtained dimensions may be decreased as the acceptance criterion is not or only exceeded beyond 1.5 times the design intensity. Therefore, the RSBD method introduced in this article can be considered as a reliable alternative in design to prevent the occurrence of weak storeys and to strongly enhance the seismic performance of CBF-s.

REFERENCES [1]

EN 1998-1:2004 Eurocode 8 “Design of structures for earthquake resistance – Part 1: General rules, seismic actions and rules for buildings”

[2]

Merczel D. B, Somja H., Aribert J-M, Lógó, J, “On the behavior of concentrically braced frames subjected to seismic loading”, Periodica Politechnica Civil Engineering, 57/2, 113-122, 2013.

[3]

Menegotto M., Pinto P., “Method of analysis for cyclically loaded reinforced concrete plane frames including changes in geometry and nonelastic behaviour of elements under combined normal force and bending” IABSE, Lisbon. 1973. Black G. R., Wenger B. A., Popov E. P., “Inelastic buckling of steel struts under cyclic load reversals” UCB/EERC-80/40, Earthquake Engineering Research Center, Berkeley, CA. 1980.

[4]