comparative study of the seismic vulnerability of steel buildings

COMPARATIVE STUDY OF THE SEISMIC VULNERABILITY OF STEEL BUILDINGS C.A. Bermudez & J.E. Hurtado National University of Colombia

A.H. Barbat & L.G. Pujades Universitat Politècnica de Catalunya

ABSTRACT: This research focuses on seismic vulnerability of steel buildings. Using stochastic methods the seismic vulnerability of both, moment resistance frames and braced frames are studied and their results are compared with each other. Input random variables employed were: peak ground acceleration (PGA), cross sectional area (A), moment of inertia (I), specified minimum yield stress of the type of steel being used (Fy), Modulus of elasticity of steel (E), dead load factor (D), and live load factor (L). The Force-Deformation Relations for Steel Elements or Components of FEMA 356 were used. Output variables studied were: Tension rupture of the anchor rods, Yielding in the gross section of tension members, Flexural buckling, Flexural yielding, Shear yielding, Buckling in members subject to bending and axial compression, and Interstory drift index. Keywords: Seismic vulnerability, steel buildings, stochastic methods, moment frames and braced frames

1. INTRODUCTION The physical seismic vulnerability can be defined as an internal risk factor of a structure exposed to an earthquake and corresponds to its intrinsic predisposition to be damaged by the earthquake. There are many procedures to evaluate the seismic vulnerability and damage. These methods include those that use macroseismic intensities for defining the earthquakes, and some practical applications to urban areas are well described (Barbat et al. 2009). Vulnerability and fragility of existing buildings can be also evaluated from capacity curves, which are force-displacement diagrams, generally corresponding to the maximum response of structures in the fundamental mode of vibration. They allow describing the structural seismic performance, especially the expected damage, and they are obtained by means of nonlinear structural analyses. In seismic urban areas, there are many buildings with different levels of seismic vulnerability and some of them show an inadequate behaviour during earthquakes. For this reason, many recent earthquake engineering studies are oriented towards the development, validation and application of techniques, which increase the seismic capacity of buildings and allow taking better decisions on the seismic vulnerability and risk. Some of these techniques are based on procedures which accurately estimate the seismic capacity and the seismic risk of the buildings. The methodologies of HAZUS 99 and RISK-UE fall into that category, as well as seismic risk studies using computational models for nonlinear structural analysis providing capacity and fragility curves. Faber reviews important issues related to risk assessment in engineering and discusses about the use of stochastic methods in civil engineering analyses (Faber 2008). It has to be pointed out that, when the seismic behaviour of a building is evaluated, uncertainties are high in both the seismic hazard of the site and the vulnerability of the building. The purpose of this study is to perform a probabilistic analysis of buildings designed by using the Load and Resistance Factor Design (LRFD) specification for Structural Steel Buildings (AISC 1994). To do that, two alternative design proposals, namely moment-resisting frames and braced frame structures are considered. Monte Carlo simulation is applied to analyze the seismic safety of these structures. The random nature of the seismic action and that of the parameters describing the strength of the steel structures are taken into

consideration. The seismic action was modelled as a non-stationary signal in time and frequency. In our simulations, the probability distribution for the peak ground acceleration (PGA) has been defined in such a way that corresponds to the design earthquake which has a 10 % of probability of being exceeded. The statistical distributions for the parameters describing the strength of the steel structures are determined from the tolerances that are permitted by the American Institute of Steel Construction (AISC). RUAUMOKO computer code (Carr 2002) was used to perform non-linear dynamic analyses. In these computations the strength degradation functions provided by the Prestandard and Commentary for the Seismic Rehabilitation of Buildings FEMA 356 (FEMA 2000) were used. In order to define the statistical distribution for different loads and resistance parameters, the works of Hurtado (1999) and Marek et al. (1996) have been used. The main limit states reached by the buildings are compared with the code provisions. In this way, the structural behaviour under synthetic seismic events generated according to both the seismic features of the construction site and the seismic hazard level for which the buildings were designed, is evaluated. Thus, this study determines the probability of exceedance of the structural members’ capacity and, consequently, the safety levels of these types of buildings.

2. LIMIT STATES CONSIDERED The primary objective of the LRFD specification is to provide reliability conditions for steel structures under various loading conditions and it uses separate factors for each load and strength conditions. The required strength and the design strength in the LRFD specification are related by the Eqn. 2.1: ∑γ Q

φR

(2.1)

where Qi are the different types of loads, γi are their respective load factors, Rn is the nominal strength and φ is a resistance factor. 2.1. Limit state of tension rupture of the anchor rods The anchor rods are designed according to the limit state of tension rupture at the effective section. The following equation must be verified the 2.2: M

d·T

(2.2)

where MBASE is the maximum moment at the base of the column, d is the distance between the centres of the anchor rods (see Fig. 2.1) and T is the tensile force in anchor rods due to loads. When a base plate has 4 anchor rods, as shown in Fig. 1, the total tensile force is the addition of the tensile forces acting in 2 anchor rods. The tension strength of each anchor rod is given by the Eqn. 2.3: T

0.75 · 0.75F A

(2.3)

where Tb is the tensile strength of the anchor rod, Fub is the specified minimum tensile strength of the anchor rod and Agb is the nominal body area of the anchor rod. For an ASTM A449 quality steel and for diameters lower than 25 mm, Fub = 827.8 MPa. For diameters up to 38 mm, Fub = 724.3 MPa.

d

Col W 12

Base plate

Anchor rod Steel A 449

Figure 2.1. Detail of column anchorage to foundation.

2.2. Limit state of tensile yielding in the gross section The design strength of members under tension involves different structural elements and connection plates. The limit state of tensile yielding in the gross section is intended to prevent excessive elongation of the member. Usually, the portion of the total member length occupied by fastener holes is small. The effect of early yielding at the reduced cross sections on the total member elongation is negligible. The design strength for tensile yielding in the gross section must satisfy the Eqn. 2.4. T

0.90F A

(2.4)

where T is the design strength of members under tension, Fy is the specified minimum yield stress and Ag is the gross area of the member. For steel ASTM A-36, Fy is 248 MPa. 2.3. Limit state of buckling Braces are structural members subjected to compression and they behave as pinned-connected members. When the cross section of these structural members has two symmetry axes, the limit state governing their behaviour is the buckling. In these cases, the following equation must be verified Eqn. 2.5: C

0.85F A F

(2.5)

where C is the compressive force due to the corresponding loading combinations, Ag is the gross section area of the member and FCR is the critical stress calculated by Eqn. 2.6 or Eqn. 2.7. F F λ

#

0.658!" $F

if λ)

1.5

(2.6)

,.-.. /F !#"

if λ) 0 1.5

(2.7)

+

12 56 4 π3

(2.8)

where Fy is the specified yield stress, K is the effective length factor, L is the length of the member, r is the radius of gyration about buckling axis and E is the modulus of elasticity of steel which is 2,0 E+5

MPa for steel ASTM A-36. 2.4. Limit states of flexural yielding and shear yielding in beams The limit state of flexural yielding has to be considered for beams with compact cross section subjected to bending and having the lateral buckling restricted. Their flexural yielding strength limit is given by the equation M

0.9M7

0.9ZF

(2.9)

where MP is the plastic moment, Z is the plastic section modulus about the axis of bending; Fy is the specified minimum yield stress of the steel. The strength for the limit state of shear yielding is given by the Eqn. 2.10: V

0.9⋅0.6F A:

0.9V

(2.10)

where Vn is the nominal shear strength; Aw = d tw is the area subjected to shear; d is the overall height of the beam cross section; tw is the beam web thickness; Fy is the specified minimum yield stress of the steel. 2.5. Limit state of buckling in members subjected to flexure and axial compression The interaction of flexure and compression in beam-columns is governed by the Eqn. 2.11a or Eqn. 2.11b: 7; φ< 7=

-

A;B C A=B

> ? @φ

7; Gφ< 7=

A;B C A=B

>φ

>φ

A;6

C A=6

>φ

A;6

C A=6

1.0

D

1.0

if

7; φ< 7=

E 0.2

(2.11a)

if

7; φ< 7=

H 0.2

(2.11b)

where Pu is the required axial compressive strength; Mux is the required flexural strength about axis x; Muy is the required flexural strength about axis y; Pn is the nominal compressive strength. Mnx is the nominal flexural strength about axis x; Mny is the nominal flexural strength about axis y; φc = 0.85 is the resistance factor for compression and φb =0.9 is the resistance factor for flexure. 2.6. Limit state of lateral deflection The maximum interstory drift of a building, that is the maximum lateral deflection of a floor relative to the lateral deflection of the floor immediately below, divided by the distance between floors, should not exceed the limit established in the corresponding building construction code. The code used in this work is the Colombian code, NSR-98 (AIS 1998). In order to control the damage of steel structures and of their non-structural elements, this code establishes that the maximum interstory drift for frames is 0.01.

3. STUDIED BUILDINGS This study has been focused on the 6 most abundant steel structural subclasses of the 13 defined in HAZUS-99 (FEMA, NIBS, 1999): low-rise, mid-rise and high-rise moment resisting frame and braced frame. Some representative models of each subclass have been designed under certain conditions for its analysis. These include a permanent load of 4,2 kN/m2, a live load of 2.0 kN/m2, a high hazard seismic zone with PGA=0,25g, a soil type E with soil factor S=1,5, and an importance factor I=1,1. It has been confirmed, by the static code method, that all models comply with: 1) the maximum allowed interstory drift index of 1% and, 2) the columns axial compression forces do not exceed 50% of the design

strength. This last, is that one of FEMA 356 (FEMA 2000) imposed in order to consider the column as a deformation-controlled, not a force-controlled one, and it has, consequently, a ductile behavior. Fig. 3.1 to Fig. 3.6 show the shapes that have been chosen for each of the members and the building dimensions. The spacing considered between frames is 9,50 m. Main features of the analyzed models are summarized in Table 4.1. Table 4.1. Weights and fundamental periods of the buildings Number Weight per area Building type of stories [N/m2] Low-rise moment-resisting frame 3 581 Mid-rise moment-resisting frame 7 736 High-rise moment-resisting frame 13 936 Low-rise braced frame 3 545 Mid-rise braced frame 7 527 High-rise braced frame 13 629

5 4

Weight per volume [N/m3] 152 232 297 143 166 200

m=15000 Kg

Slope = 5.55 %

Period [s] 0.64 0.97 1.55 0.27 0.62 1.30

Beams 10 x 17

2.58

m=8000 Kg Beams 14 x 38

3.12

3

m=69000 Kg Beams 14 x 38

3.12

2

m=69000 Kg Beams 14 x 38

3.12

1 COL W 12 x 87

1.46

Figure 3.1. Sketch of the low-rise braced frame

4.8

4.8

4.8

1.46

Figure 3.2. Sketch of the low-rise moment resisting frame

8x4x5/16

Figure 3.3. Sketch of the mid-rise braced frame

Figure 3.4. Sketch of the mid-rise moment resisting frame

Figure 3.5. Sketch of the high-rise braced frame

Figure 3.6. Sketch of the high-rise moment resisting frame

4. PROBABILITY OF EXCEEDANCE OF THE LIMIT STATES The following paragraphs contain information from each of the types of the studied buildings with respect to the probability of exceedance of different variables that include: maximum ground acceleration, interstory drift, limit states with greater probability of exceedance and threshold of various damage states. Note in all of these Figures that the probability of exceedance of the maximum acceleration (first bar) is in all the cases of 10%, which is considered acceptable. Fig. 4.1 shows that the behavior of low-rise braced frames is very reliable. It is only expected the slight damage state. Some limit states have probability of exceedance greater than 10% when it is calculated with respect to the nominal value of resistance, which highlights the need of designing joints considering the overstrength resulting from the large different between actual and nominal value of the yield point Fy. Fig. 4.2 instead shows that the behavior of low-rise moment resisting frames has no the structural reliability that the Load and Resistance Factors Design Method of the AISC aims to attain. The probability that the interstory drift exceeds the acceptable limit is 40%. Probabilities of fracturing the anchor rods and buckling the columns under combined stresses of flexure and axial compression are also very high. The probability of reaching the moderate damage state is 24%.

'acelmax' 'mom_max_8-1' 'mom_min_8-1' 'mom_max_15-1' 'mom_min_15-1' 'mom_min_30-1' 'mom_min_30-2' 'mom_min_32-1' 'mom_min_32-2' 'mom_min_45-1' 'mom_min_45-2' 'mom_min_47-1' 'mom_min_47-2' 'axial_min_64' 'axial_max_65' 'axial_min_66' 'axial_max_67' 'axial_min_68' 'axial_max_69' 'axial_min_70' 'axial_max_71' 'axial_min_72' 'axial_max_73' 'axial_min_74' 'axial_max_75' 'axial_min_76' 'axial_max_77' 'axial_min_78' 'axial_max_79' 'axial_min_80' 'axial_min_84' 'axial_max_85' 'axial_min_86' 'axial_max_87' 'axial_min_88' H1-1a_8-1 H1-1a_8-2 H1-1a_15-1 H1-1a_15-2 H1-1a_11-1 H1-1a_11-2 H1-1a_18-1 H1-1a_18-2 despl abs máx Slight damage Moderate Extensive damage Collapse

Calculate with regard to nominal resistance values


Calculate with regard to nominal resistance values 'ac

_4 2'

in _4 5'

in

'

_4 1'

sm

od .

ve

áx

Figure 4.5. Exceedance probability lity of high-rise high moment resisting ing frames lap

so

'm om 'ace 'm _m lma o m a x x' _ 'm _m 18o m ax 1 ' _ 'm _m 18o m ax 2 ' _1 _ 'm m 9o m ax 1 ' _ 'm _m 19o m ax 2 ' _ _2 'm ma 0-1 om x_ ' 2 'm _m 0-2 om ax ' 'm _m _5om a 1 ' x 'm _m _9om in_ 1' 'm _m 18om in_ 1' 'm _m 18om in_ 2' 'm _m 19om in_ 1' 'm _m 19om in_ 2' _ 2 'm min 0-1 om _2 ' 'm _m 0-2 om in_ ' _m 5-1 in ' F.2 _9-4 1' F.2 6_1 -4 -1 F.2 6_1 -4 -2 F.2 6_5 -4 -1 F.2 6_5 -4 -2 F. 6_9 2- -1 F.2 46_ -4 9-2 F.2 6_1 de -4 6 3 -1 sp _1 l a 3bs 2 m a D. x le D. ve m od D. . s Co e v. lap so

Co

D. se v.

D. m

D. le

ab

F.2 -4 6_ 91 F.2 -4 6_ 92

in _4 6' F.2 -4 6_ 51

ial_ m

ial_ m

ia l_m

in

elm ax

ial_ m

de sp l

'ax

'ax

'ax

'ax

0.1500

Calculate with regard to random resistance values

Figure 4.1. Exceedance probability of low-rise low braced frames

Figure 4.3. Exceedance probability of mid-rise mid braced frames

0.7

Exceedance probability of high-rise moment resisting frames

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

'acelmax' 'mom_max_14-1' 'mom_min_14-1' 'mom_max_27-1' 'mom_min_27-1' 'mom_min_101-1' 'mom_min_101-2' 'axial_min_1' 'axial_min_2' 'axial_min_3' 'axial_min_4' 'axial_min_5' 'axial_min_6' 'axial_min_7' 'axial_min_8' 'axial_min_9' 'axial_min_10' 'axial_min_11' 'axial_min_12' 'axial_min_13' 'axial_min_14' 'axial_min_15' 'axial_min_16' 'axial_min_17' 'axial_min_18' 'axial_min_19' 'axial_min_23' 'axial_max_1' 'axial_max_2' 'axial_max_3' 'axial_max_4' 'axial_max_5' 'axial_max_6' 'mom_min_69-1' 'mom_min_69-2' 'mom_min_71-1' 'mom_min_71-2' 'mom_min_84-1' 'mom_min_84-2' 'mom_min_86-1' 'mom_min_86-2' 'mom_min_99-1' 'mom_min_99-2' 'axial_max_7' 'axial_max_10' H1-1a_14-1 H1-1a_14-2 H1-1a_27-1 H1-1a_27-2 H1-1a_17-1 H1-1a_17-2 H1-1a_30-1 H1-1a_30-2 despl abs máx Slight damage Moderate damage Extensive damage Collapse

'a om celm _ a 'm ma x' om x _ 'm _m 1-1 om in ' _ _ 'm ma 1-1' om x _ 1 'm _m 4-1 om i n_ ' 1 _ 'm ma 4-1' om x _ _ 27 'm mi -1' om n_ _ 27 'm mi -1' om n_ _ 40 'm mi -1' om n_ 9 'm _m 9-1 om i n ' _9 'm _mi 9-2 om n_ ' _m 100 'm om i n_ 1 ' _ 10 'm mi 0-2 om n_ ' _m 101 -1 i de n_1 ' sp 01 l a -2 bs ' m á D. x le v D. e m od D. . se Co v. lap so

'm

0.0000

0.4000

Exceedance probability of low-rise braced frames 0.8000

Exceedance probability of low-rise moment resisting frames

0.3500

0.7000

0.3000

0.6000

0.2500

0.5000

0.2000

0.4000

0.1000 0.1029

0.3000

0.2000

0.0500

0.1000

0.0000

0.0000




Figure 4.2. Exceedance probability of low-rise low moment resisting frames

According to Figure 4.3, while mid-rise mid braced frames show high stiffness, the probability of yielding of the braces mainly under compression stresses is very high. Also the probabilities of yielding at central columns bases and buckling in columns under combined com stresses of flexure and axial compression are very high.

0.9000

Exceedance probability of mid-rise braced frames

0.8000

0.7000

0.6000

0.5000

0.4000

0.3000

0.2000

0.1000


Figure 4.4. Exceedance probability of mid-rise mid moment resisting frames

Fig. 4.4 and 4.5 show that mid-rise rise and high-rise high rise moment resisting frame have a probability of up to 60% of exceeding limit state of tension fracture of anchor rods. It has been found that to maintain a uniform degree of reliability ann additional safety factor of 2.75 2.75 must be used for designing anchorages. Otherwise, it is possible to say these frames exhibit the structural reliability that LRFD aims to attain.

0.8

Exceedance probability of high-rise braced frames



Figure 4.6. Exceedance probability of high-rise high braced frames

Figure 4.6 shows that the high-rise braced frames have small lateral displacements but probabilities close to 40% of exceeding limit state of tension fracture of anchor rods. Many braces under compression stresses have high probabilities of exceedance, mainly when they are calculated with respect to the nominal values of resistance.

5. CONCLUSIONS Low-rise braced buildings, with connections designed considering the possible overstrength of structural members, have a high structural reliability, manifested in minimum lateral deformations and probabilities of exceedance very low. On the other hand, the low-rise moment resisting buildings have a probability of up to 60% of exceeding limit states. For mid-rise and high-rise moment resisting frame it is found that, except for the limit state of tension fracture of anchor rods, the probability of exceedance of strength and serviceability limit states are very low which results in a high structural reliability. For its part, the high-rise braced buildings have deflections very low but high probabilities of exceedance of strength limit states, mainly axial compression of the braces. According with these results, for the steel low- rise buildings, the braced frames are more reliable than the steel moment resisting frames. Instead, for the mid-rise and high-rise buildings, the moment resisting frames are more reliable than the braced frames.

ACKNOWLEDGEMENT This research has been partially funded by the National University of Colombia and by the Science and Innovation Ministry of Spain, by the European Commission and with FEDER grants through the research projects with reference numbers CGL2008-00869/BTE and INTERREG POCTEFA 2007-2013/ 73/08.

REFERENCES AIS. (1998). Normas Colombianas de Diseño y Construcción Sismo Resistente NSR-98. Asociación Colombiana de Ingeniería Sísmica. Bogotá, D. C., Colombia. AISC (1994). Load and Resistance Factor Design (LRFD) Manual of Steel Construction, American Institute of Steel Construction, Chicago, IL, USA. Barbat, A., Carreño, M., Pujades, L., Lantada, N., Cardona, O., Marulanda, M. (2009). Seismic vulnerability and risk evaluation methods for urban areas. A review with application to a pilot area. Structure and Infrastructure Engineering, DOI: 10.1080/15732470802663763. Carr, J. (2002). Ruaumoko3d- Inelastic Dynamic Analysis Program, Dept. of Civil Engineering, University of Canterbury, Christchurch, New Zealand. Faber, M., (2008). Risk Assessment in Engineering Principles, System Representation & Risk Criteria. JCSS Joint Committee of Structural Safety. ISBN 978-3-909386-78-9. FEMA (2000). Prestandard and Commentary for the Seismic Rehabilitation of Buildings FEMA 356, Federal Emergency Management Agency. Washington, D.C., the USA. FEMA, NIBS. (1999). Earthquake Loss Estimation Methodolo-gy (HAZUS 99). Federal Emergency Management Agency y National Institute of Building Sciences. Washington, D.C., USA. Hurtado, J. (1999). Modelación estocástica de la acción sísmica, Centro Internacional de Métodos Numéricos en Ingeniería (CIMNE), Barcelona, España. Marek, P., Gustar, M. and Anagnos, T. (1996). Simulation-Based Reliability Assessment for Structural Engineers. Boca Raton, Florida, USA.