International Journal of Civil, Structural, Environmental and Infrastructure Engineering Research and Development (IJCSEIERD) ISSN 2249-6866 Vol.2, Issue 3, Sep 2012 76-97 © TJPRC Pvt. Ltd.,
COMPARATIVE EVALUATION OF ELASTIC DESIGN AND PERFORMANCE BASED PLASTIC DESIGN METHOD FOR A STEEL MOMENT RESISTING FRAME 1 1 2,2
SEJAL P. DALAL, 2ANDEEP A. VASANWALA & 2ATUL K. DESAI
Assistant Professor, Civil Engineering Department SVIT, Vasad, Gujarat, India
Associate Professor Applied Mechanics Department, SVNIT, Surat, Gujarat, India
ABSTRACT Presented in this paper is the comparative evaluation of Performance Based Plastic Design (PBPD) method Elastic Design (ED) method in terms of lateral force distribution, design, strength and economy. For this, a steel moment resisting frame is first designed using the Elastic Design method and then using the Performance Based Plastic Design Method. The Lateral forces in the Elastic Design method are calculated using the Elastic Design Spectra and all the structural members are designed as elastic beam-columns based on Limit State Design Philosophy. The Lateral forces in the Performance Based Plastic Design Method are calculated using the inelastic spectral acceleration which is obtained by applying proper reduction factors. Results prove the superiority of the PBPD method over the Elastic Design method in terms of safety and overall economy.
KEYWORDS: Elastic Design Method, Performance Based Plastic Design Method, Target and Yield Mechanism.
INTRODUCTION When struck by severe ground motions, the structures designed by elastic design procedures have been found to undergo inelastic deformations in a somewhat ‘uncontrolled’ manner. The inelastic activity, which may include severe yielding and buckling of structural members and connections, can be unevenly and widely distributed in the structure. This may result in a rather undesirable and unpredictable response, sometimes total collapse, or difficult and costly repair work at best. While the elastic design practice has served the profession rather well in the past, societal demands are pushing the practice to achieving higher levels of performance, safety and economy, including life-cycle costs. For the practice to move in that direction, design factors, such as determination of appropriate design lateral forces and member strength hierarchy, selection of desirable yield mechanism, structure strength and drift, etc., for specified hazard levels should become part of the design process right from the start. One such method known as the PBPD method for Earthquake Resistant Design of Structures, which accounts for inelastic structural behavior directly, has been developed by Goel et al, 2001. The PBPD method is a displacement based method in which a predetermined failure pattern is used at certain
77
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
points of a structure based on “strong column –weak beam concept”. Results of extensive inelastic static and dynamic analyses have proven the validity of the method. The method has been successfully applied to steel Moment Frame (Goel et al , 2008,2010 ), Eccentrically Braced Frame (Chao and Goel, 2006a) , Composite buckling restrained braced frame (Dasgupta,2004),Concentrically Braced Frame (Chao and Goel, 2006b) and Special Truss Moment Frame (Goel and Chao, 2008). In all cases, the frames developed the desired strong column–weak beam yield mechanisms as intended, and the storey drifts/ductility demands were well within the selected design values, thus meeting the selected performance objectives. An excellent literature review of the method has been done by Dalal et al (2010). The PBPD method is a direct design method that uses pre-selected target drift and yield mechanisms as key performance objectives that determine the degree and distribution of expected structural damage. It is based on the formulations derived from the capacity-spectrum method using Newmark–Hall (1982) reduction factors for the inelastic demand spectrum. The design base shear for a specified hazard level is calculated by equating the work needed to push the structure monotonically up to the target drift to the energy required by an equivalent Elasto Plastic Single Degree of Freedom system to achieve the same state. Plastic design is performed to detail the frame members and connections in order to achieve the intended yield mechanism and behavior. The current Indian Standard code (IS800:2007) uses the limit state procedure (which is a force based design) for design of steel structures to ensure a good earthquake resistant design which at times may fail in case of a severe earthquake as it is based on elastic analysis. The dead and imposed loads are calculated using IS875, (parts I to V) and the seismic loads are calculated using IS1893:2002 based on Elastic Design Spectrum. A steel moment resisting frame has been designed using the PBPD method in accordance with the IS 800: 2007 code by Dalal et al (2011) in which the cross sections of beams have been reduced at certain pre defined locations so that it forms the “weak link” during an earthquake and plastic hinges are formed. The steel moment resisting frame as shown in figure 1 has the parameters given in the Table 1and is subjected to dead load and imposed load as given in Table 2.The gravity and the seismic load calculations are shown in Section 1.1 and distributed as shown in figure 2.
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Figure 1: Plan and Elevation of the Steel moment Resisting Frame
78
79
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
Table 1:Design Parameters Type of structure
4-Storey Steel Moment Frame.
Number of stories
4
Bays in X direction
6 bays at 9 m
Bays in Y direction
4 bays at 9 m
Floor height
4 m for all floors and 4.25 m for first floor
Building Height Materials
16.25 m Structural steel with fy = 250 N/mm2
Floor Seismic Weight for Roof
9720 KN
Floor Seismic Weight for Level 4
9331 KN
Floor Seismic Weight for Level 3
9429 KN
Floor Seismic Weight for Level 3
9525 KN
Seismic zone factor, Z
Zone 4
Soil Profile Type
Type 2 Medium
Importance factor, I
1
T
0.939 sec Table 2:Design Loads
Dead Load (KN/m2) Live Load (KN/m2) Mass Seismic Load(KN/m2) Dead Load due to exterior Curtain Wall (KN/m)
Typical Roof 4
Level 4 4.5
Level 3 4.5
Level 2 4.5
1
2.5
2.5
2.5
5
4.8
4.85
4.9
3.75 ( 3 m Wall height)
4.5 ( 4 m Wall height)
4.5 ( 4 m Wall height)
5 ( 4.25 m Wall height)
Calculation Of Gravity Loads And Seismic Loads 1.
Floor Seismic Dead Weight (full structure) Roof = (5 x 54 x 36)
= 9720 KN
Level 4 =
(4.8 x 54 x 36)
= 9331 KN
Level 3 =
(4.85 x 54 x 36) = 9429 KN
Level 2 =
(4.9 x 54 x 36)
= 9525 KN
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Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Full structure 2.
= 38005 KN
Beam Load Calculation (for exterior moment frame)
The exterior beams take the dead load from 4.8 meter of slab (which accounts for 0.3 meter of overhang. The live load is calculated based on a 4.5 meter tributary width only. Uniformly distributed loads on the exterior beams: 1.
Roof From slab (dead load) = 4 x 4.8
= 19.2 KN/m
From exterior curtain wall (dead load)
= 3.75 KN/m
From slab (live load) = 1 x 4.5
= 4.5 KN/m
Total Load Combination w3 (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL = 1.4 x (19.2 +3.75) + 0.35 x 4.5 ≈ 34 KN/m 2.
Level 4 and 3 From slab (dead load) = 4.5 x 4.8
= 21.6 KN/m
From exterior curtain wall (dead load)
= 4.5 KN/m
From slab (live load) = 2.5 x 4.5
= 11.25 KN/m
Total Load Combination w2 (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL = 1.4 x (21.6 +4.5) + 0.35 x 11.25 ≈ 40 KN/m 3.
Level 2 From slab (dead load) = 4.5 x 4.8
= 21.6 KN/m
From exterior curtain wall (dead load)
= 5 KN/m
From slab (live load) = 2.5 x 4.5
= 11.25 KN/m
Total Load Combination w1 (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL = 1.4 x (21.6 +5) + 0.35 x 11.25 ≈ 41 KN/m 3.
Concentrated Load at Column Lines (values for one exterior moment frame)
4.
Roof Dead load (exterior column lines) = 4.8 x 1.56 x 4 + 2 x 4.8 x 3.75 x 1.2
= 73 KN
Live load (exterior column lines) = 4.5 x 1.2 x 1
= 5.4 KN
Dead load (interior column lines) = 0 + 9 x 3.75 x 1.2
= 40.5 KN
Live load (interior column lines) = 0
= 0 KN
Total Load Combination (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL
5.
L5 (roof) = 1.4 (73) + 0.35(5.4)
= 104 KN
L6 (roof) = 1.4 (40.5) + 0.35(0)
= 56.7 KN
Level 4 and 3 Dead load (exterior column lines) = 4.8 x 1.56 x 4.25 + 2 x 4.8 x 4 x 1.2
= 78 KN
Live load (exterior column lines) = 4.5 x 1.2 x 2.5
= 13.5 KN
Dead load (interior column lines) = 0 x 9 x 4 x 1.2
= 43.2 KN
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Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
Live load (interior column lines) = 0
=0 KN
Total Load Combination (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL
6.
L3 (level 3, 4) = 1.4(78) + 0.35(13.5)
= 114 KN
L4 (level 3, 4) = 1.4(43.2)+ 0.35(0)
=60.48 KN
Level 2 Dead load (exterior column lines) =4.8 x 1.56 x 4.25 + 2 x 4.8 x4.25 x 1.2
= 80 KN
Live load (exterior column lines) =4.5 x 1.2 x 2.5
= 13.5 KN
Dead load (interior column lines) =0 x 9 x 4.25 x 1.2
= 46 KN
Live load (interior column lines)=0
=0 KN
Total Load Combination (Clause 3.5.1 IS 800: 2007) = 1.4 DL + 0.35 LL L1 (level 2) = 1.4 (80) + 0.35(13.5)
= 114 KN
L2 (level 2) = 1.4 (46) + 0.35(0)
= 64.4 KN
DESIGN OF THE MOMENT RESISTING FRAME USING ELASTIC DESIGN METHOD The linear static procedure has been a traditional structural analysis method for earthquake resistant design but it does not represent the nonlinear behavior of the dynamic response of a structure caused by an earthquake ground motion. In the current Indian Standard Seismic design practice (IS18932000), we first obtain design base shear from code-specified spectral acceleration assuming the structures to behave elastically (which depends on the soil factors and the time period of the structure), and reducing it by force reduction factor, R, depending upon available ductility of the structural system. The design forces are also adjusted for the importance of specific structures by using an occupancy importance factor, I. The step by step procedure of the elastic design of this frame is shown in the following section.
82
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Figure 2 : Distribution of gravity forces. Step 1 : Calculation of Design Parameters Based on the IS1893-2002, this frame is found having seismic design parameters as shown in Table 3 Table 3 : The seismic parameters of the frame as per IS1893-2002 Natural time Period "T" Soil Type
0.939 seconds II
Elastic Spectral Acceleration " Sa / g" Importance Factor "I" Zone Factor " Z"
1.448 1 0.36
Response Reduction Factor "R" 5 Total Seismic Weight of the Building "W" 38005 Step 2 : Calculation of design seismic base shear and lateral forces on each floor. The total design lateral force or design seismic base shear ( V b ) along any principal direction shall be determined by the following expression
V
A W
Equation 1
83
V
0.052 38005
1981 KN
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
where A
0.052
Equation 2
This Base shear V b is distributed along the height of the Building as follows: Q
V
∑!"#
Equation 3
Where V b= Base shear Ah= Design Horizontal Seismic Coefficient W = Total seismic weight of the structure Z = Zone factor I = Importance factor R = Response reduction factor Sa/g = Average response acceleration coefficient Qi = Design lateral force at floor i Wi = Seismic weight of floor i hi = Height of floor i measured from base n = Number of storeys in the building is the number of levels at which the masses are located.
Table 4 : Lateral force Distribution in floors
Roof
Level 4
Level 3
Level 2
ÎŁ Sum
Seismic Weight of floor i "Wi"
9720
9331
9429
9525
38005
Design Lateral Force at Level i "Qi"
1063.63
580.25
265.94
71.3
1981.1
Step 3: Static Analysis of the frame The moment resisting frame was applied the above lateral forces along with gravity loads as shown in figure 2 and static analysis was carried out. Based on the obtained values of axial force, shear force and bending moment, all the structural members of this frame were designed using the elastic design approach. As all the beams and columns have all the three components viz. shear, axial and moment, they all need to be designed as beam-columns. The step by step procedure of design of the structural members is shown in succeeding sections.
84
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Step 4: Design Compressive Stress for axially loaded compression members (Member buckling resistance in compression) The buckling resistance in compression (or the Compressive Load Capacity) of all the members should be less than the maximum value of axial force experienced by the member. Common hot rolled members used for carrying axial compression usually fail by flexural buckling. The buckling strength of these members is affected by residual stresses, initial bow and accidental eccentricities of load. To account for all these factors, the strength of members subjected to axial compression is dependent on Imperfection Factor "α" defined by buckling class a, b, c or d as given in table 7 of IS800-2007 code (Clause 7.1.2.1). To calculate buckling resistance in compression of members, following procedure should be implemented.
λ
$%& ' * + ()
π ,
Equation 4
0.5-1 . α/λ 0 0.21 . λ 2
χ f=> N>
3
4 56
χ%&
9.:
7λ8
;
γ?@
Equation 5
Equation 6
Equation 7
f=> A
Where λ = Non Dimensional Effective Slenderness ratio KL/r = Effective slenderness ratio χ = Stress Reduction factor for different buckling class, slenderness ratio and yield stress f cd = Design Compressive Strength A = Effective Sectional Area N d = Design Compressive Load Capacity Step 5 Safety against Shear The Shear strength of all the members should be less than the maximum value of shear force. This design shear resistance " Vn " of members shall be calculated using Clause 6.4 VA
>BCDE @! FGCH%& √J
Equation 8
85
V>
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame K!
Equation 9
γ?@
Where Vn = shear resistance dsection = depth of the section tweb = thickness of web Vd = shear strength Step 6 Safety against Flexure To ensure safety of members against flexure, it is important that the flexural strength of members is less than the maximum value of bending moment. The flexural strength of members can be calculated using clause 8.2.There are two cases to be kept in mind while calculating the flexural strength (i)
When the factored shear force is less than 0.6 times Shear Resistance “Vd”
(ii)
When the factored shear force is more than 0.6 times Shear Resistance “Vd”
(i)If, the factored shear force is less than 0.6 times the shear capacity then, Design Flexural Strength shall be calculated as M>
C %&
Equation 10
γ?@
Where Md= Design Flexural Strength Ze = Elastic Section Modulii of the section (ii)If, the factored shear force is more than 0.6 times the shear capacity then, Clause 9.2.2 should be satisfied. The Design Flexural Strength is dependent on whether the section is compact or semi compact. When the ratio of width to thickness of flange is less than 15.7, the section is compact otherwise it is semi compact. (a)Section is semi compact M> %MN O
PQN
C %&
Equation 11
γ?@
Where Md for shear = Design moment capacity of section considering high shear force effect (b) Section is Plastic or compact M> %MN O
PQN
M> PRQOF = 0 βO
PQN
6M> PRQOF = 0 M% SRQOF = 8 T 1.2
C %&
γ?@
Equation 12
86
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
M> PRQOF =
C %&
Equation 13
UV BE D %&
Equation 14
γ?@
M% SRQOF =
γ?@
Where Md
for shear
Md
elastic
Md
plastic
=Design moment capacity of the whole section considering high shear force effect
=Design moment capacity of the whole section disregarding high shear force effect =Plastic Design strength of the area of the cross section excluding the shear area considering
partial safety factor Where βO
W
PQN
2V 0 1X V>
Step 7 Check for Combined effects The members subjected to combined effect of axial force, shear force and bending moment need to satisfy the following equation Y
YZ
[&
.[
!Z&
T 1.0
Equation 15
Where N = Factored applied axial force Nd = Design Strength in Tension obtained as Agfy/γmo Ag = Gross Cross Sectional Area My = Factored applied moments
Mdy =Design reduced flexural strength obtained as β ZSf] /γ_M Step 8 Resistance to lateral torsional buckling As the frame undergoes seismic forces, it is important to check its resistance to lateral torsional buckling as per clause 8.2.2.1. The bending Moment capacity of the section against torsional effects "Mt" should be less than the maximum bending moment coming on the section. To calculate " Mt" ,following steps should be followed f=N λ`h `h
3.3π ,
6`)a ⁄N& 8
i%
%&
D* H
c1 .
3
d
'
`)a⁄N& e ⁄Fe
+ f
d.g
0.5j1 . α`h /λ`h 0 0.21 . λ`h k
Equation 16
Equation 17 Equation 18
87
χ`h f
>
MF
3
l )a 5j )a 7λ)a k
9.:
χ`h f] ⁄γ_M
β β ZS f
m
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
T 1.0
Equation 19 Equation 20
>
Equation 21
Where L LT = 0.9 L= Effective length for lateral torsional buckling hf = centre to centre distance between flanges tf = thickness of flanges f cr,b = Extreme fibre bending compressive stress λ LT =Non dimensional slenderness ratio α LT=the imperfection parameter =0.49 for welded section and 0.21 for rolled section χ LT = Bending stress reduction factor for lateral torsion buckling fbd = Design bending compressive strength Mt = Bending Moment Capacity for torsional effects Step 9 Check with Interaction Formula (should be less than 1) Ratio of moments (negative /positive) "ψy" Member resistance to combined bending and axial compression K]
C_] Y
YDZ &
1 . 6λ] 0 0.28 Y
Y
DZ &
0.6 . 0.4ψ] q 0.4
. K]
r?& [ [Z
T 1 . 0.8 Y
Y
DZ &
T 1.0
Equation 22 Equation 23 Equation 24
C my = Equivalent uniform moment factor ( as per clause 9.3.2.2 and table 18 of IS 800-2007) The details of member sections thus obtained by the above mentioned steps of Elastic Design method as done in the current design practice is briefed in Table 5.
88
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Table 5 : The Design Sections obtained when the frame is designed by Elastic Design method Sr. No.
Level
Section
Section Modulus (cm3)
Weight ( kg/m)
Beams 1
Roof
ISMB600 with 320 x 32 mm cover plates
8930
313
2
Level 4
ISMB600 with 320 x 32 mm cover plates
8930
313
3
Level 3
ISMB600 with 320 x 32 mm cover plates
8930
313
4
Level 2
ISMB600 with 320 x 32 mm cover plates
8930
313
Exterior Columns 1
Roof
ISWB600
3539
133.7
2
Level 4
ISWB600A with 400 x 16 mm cover plates
7502
245.6
3
Level 3
ISWB600A with 400 x 16 mm cover plates
7502
245.6
4
Level 2
ISWB600A with 400 x 16 mm cover plates
7502
245.6
Interior Columns 1
Roof
ISMB600 with 320 x 32 mm cover plates
8930
313
2
Level 4
ISMB600 with 320 x 32 mm cover plates
8930
313
3
Level 3
ISWB600A with 400 x 16 mm cover plates
7502
245.6
4
Level 2
ISWB600A with 400 x 16 mm cover plates
7502
245.6
DESIGN OF THE MOMENT RESISTING FRAME USING THE PBPD METHOD In this section, the steel moment frame building is designed using the Performance Based Plastic Design Methodology in accordance with the IS 800:2007 code. The PBPD method is a direct design method where drift and yield mechanism, e.g. strong column–weak beam condition, are built in the design process from the very start. The design base shear for a specified hazard is calculated based on the reduction factors “Rµ”proposed by Newmark (1982). Also, a new distribution of lateral design forces is used that is based on relative distribution of maximum storey shears consistent with inelastic dynamic response results (Chao et al., 2007). Plastic design is then performed to detail the frame members and connections in order to achieve the intended yield mechanism and behavior. Thus, determination of design base shear,
89
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
lateral force distribution and plastic design are three main components of the PBPD method, which will be discussed in the following sections. The step by step procedure of the design of the steel moment resisting frame using the PBPD method is as follows Step 1 : Calculate gravity loading and Seismic loading for the structure “W”. Step 2 :Select an appropriate vertical distribution of forces based on the mode shapes obtained from Modal Analysis.(usually mode shape 1 is chosen
Table 6 :The seismic Parameters of the frame for PBPD method Seismic zone factor, Z
0.36
Soil Profile Type
Type 2 Medium
Importance factor, I SQ
1
APRQOF =
0.64 g
T
0.939 sec
Yield drift ratio θ]
1%
Target drift ratio θt
Inelastic drift ratio θS Ductility factor uv
2%
θw
θt 0 θ]
xy
1% 2.0
Reduction Factor due to Ductility Rµ
2.0
Energy Modification Factor γ
0.75
α
0.942
Design Base shear Vb
8321 KN
Actual base shear for each floor
1040 KN
90
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
Step 3 : Select a desired Target Yield Mechanism for design earthquake hazard.
∆p
∆p
∆p
∆p
L'
L'
L'
γp
γp
θp θp
θp
Mpc
Mpc
θp
θp
θp
Mpc
Mpc L
L
γp
γp
θp
θp
L'
L
L
Figure 3 : The Target Yield Mechanism for the structure. Step 4 : Calculate the shear distribution factor “βi” of each floor. β
W
∑!"# z
z! !
X
d.{gh|9.
Equation 25
where γp = The rotation at the plastic hinge [ γS
' ′ + θS ] `
`
L'= The distance between the two plastic hinges of the beam = 0.8375L βi = Shear distribution factor at level i wj= seismic weight at level j hj= height of level j from base wn= seismic weight at the top level hn= height of roof level from base T = fundamental time period Step 5 : Calculate α. }
/∑••‚3/~• 0 ~•53 1€• 1 W∑„
ƒ„ …„
†"# ƒ† …†
d.{g‡ |9.
X
'
ˆ‰ Š‹ ‡ Œ
+
Equation 26
91
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
Step 6 : Calculate Story shear “V” 7•5i• 5 •‘’“ ”„•–“—˜”™
•
Ž
š›
Equation 27
Step 7 : Calculate the Lateral force “Fn” of Roof Floor. œ•
•
∑/ž” 7ž”Ÿ# 1
Equation 28
Step 8 : Calculate the Lateral force “Fi” of each level. œ• /~• 0 ~•53 1
ϥ
Equation 29
Where Fn =Lateral Force at roof level ( nth level ) Fi =Lateral Force at ith level Step 9 : Calculate the required beam moment capacity “Mu” at each level ¡
where ¢¨
~•
¢£
~•
∑„ ”"# ¤” …”7 ¥‰™ ¦ ¦
∑„ ”"#'ž” , +
Equation 30
1.1•′€3 4
Mpc= required plastic moment of columns in the first story of the 1-bay model (V' =V/No. of bays) Mpb = required moment strengths at the top floor level βiMpb = required moment strengths at level i. Step 10: Calculate the design beam moment “Mdesign” by applying proper factors Mdesign= 1.1 Mu/0.75 Step 11 :
Equation 31
Calculate the section modulus “Zp” required.
Zp = Mdesign/ fy Step 12 : Design of Beams using the Reduced Beam Section (RBS). Select RBS dimensions a, b, and c (figure 4) subject to the following limits: 0.5©ª T « T 0.75©ª
0.65-£ T © T 0.85-£ 0.1©ª T ® T 0.25©ª
Where bf = width of beam flange
Equation 32
92
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
db= depth of beam section ZRBS = plastic section modulus at center of the reduced beam section, Zp = plastic section modulus for full beam cross-section, tf = thickness of beam flange. a = distance from face of column to start of RBS cut b = length of RBS cut c = depth of cut at the center RBS section at the center of reduced beam section. Step 13 : Check the compactness of the RBS ¯¢ 0 2®²ª 6-£ 0 ²ª 8
¯°±’
Equation 33
The ratio of ZRBS/Zp should be near to the assumption 0.75 The Moment Capacity MRBS of this RBS beam is calculated as ³´µ¶ ª· ž¸
°±’
‘¹º
Equation 34
Step 14 : Checking the Moments and Shears of the beams Checking Shear Compute the probable maximum moment at the center of the RBS, “MprRBS”: MprRBS = Cpr Ry fy ZRBS
Equation 35
Ry = ratio of the expected yield stress to the specified minimum yield stress (=1.1) Cpr = factor to account for the peak connection strength, including strain hardening, local restraint, additional reinforcement, and other connection condition (=1.15) •°±’
¥‰»´µ¶
2
•′°±’
¼′
.
¢¿°±’
À′
/¡½¾1¼′
0
Equation 36
/Á-Â1À′ 2
V RBS = Probable Positive Shear Force at center of RBS V' RBS = Probable Negative Shear Force at center of RBS This shear VRBS or V' RBS should be less than Vn as per the Clause 8.4.1 IS 800:2007 VA
ÃÄ %& √J
FG %& √J
>H FG %& √J
Where Vn = nominal plastic shear resistance of section
Equation 37
93
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
Figure 4: The details of the Reduced Beam Section
CHECKING MOMENTS Compute the probable maximum moment, “MfcRBS� at the face of the column. If it is larger than the expected moment capacity of the beam section at that location, the beam section will need to be further reduced. The expected moment at the face of the column is computed as follows:
94
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
M%=
M′%=
MSN
Å
MSN
Å
Å
Å
.V
Å
. V′
SÆ and
Å
Equation 38
SÆ
The moment carrying capacity of the section is given by Clause 8.2.1.2 IS 800 : 2007 MS
ZS f ] β γ_M
Where βb=1.0 for plastic and compact sections γmo = Partial safety factor = 1.0 Sh =The distance from a column face to the center of RBS cut = a + b/2 This “MfcRBS” should not exceed “Mp”. Step 15:
Calculation of Design Axial Force and Moments for the Exterior and Interior Columns
Exterior Column Tree F`
Z 4∑!"#6[U*ÈÉÊ 8 ;54∑!"#Ë6KÈÉÊ 8' Ì 5 D+ Í;5j[UD k
∑!"# ψ
Equation 39
Where ψ
6β 0 β 53 8
∑A‚36β
0 β 53 8
FL = sum of lateral forces in columns Interior Column Tree F`
Z 4 ∑!"#6[U*ÈÉÊ8 ;54∑!"#Ë6KÈÉÊ 5K′ÈÉÊ 8' Ì 5 D + Í;5j [UD k
∑!"# ψ
Equation 40
Step 16: Design of Exterior and Interior Columns Y
YZ
[&
.[
!Z&
T 1.0
Where γpi = The rotation at the plastic hinge L'= The distance between the two plastic hinges of the beam = 0.8375L fy = tensile strength of steel FL = sum of lateral forces in columns
Equation 41
95
Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
Sh =The distance from a column face to the center of RBS cut = a + b/2 N = Factored applied axial force = (V rbs + V' rbs + Point load + load from upper floor) Nd = Design Strength in Tension obtained as Agfy/γmo Ag = Gross Cross Sectional Area My = Factored applied moments = [(V rbs + V' rbs )( S h + dc / 2 ) +2 M Mndy =Design reduced flexural strength obtained as β ZS f] /γ_M
prbs
]
The details of member sections thus obtained by the above mentioned steps of PBPD method are briefed in Table 6. Table 6 : The Design Sections obtained when the frame is designed by PBPD method
Sr. No.
Level
Section Modulus (cm3)
Weight ( kg/m)
ISWB600 B and RBS at 400 mm from column face with a = 150 mm b = 500 mm c = 40 mm
Z p = 4341 Z RBS = 3252
145
Section Beams
1
Roof
2
Level 4
ISWB600 A with 400 x 10 mm cover plates and RBS at 400 mm from column face with a = 150 mm b = 500 mm c = 40 mm
Z p = 6426 Z RBS = 4863
197
3
Level 3
ISMB600 with 320 x 20 mm cover plates and RBS at 400 mm from column face with a = 150 mm b = 500 mm c = 40 mm
Z p = 7478 Z RBS = 4963
223
4
Level 2
ISMB600 with 320 x 25 mm cover plates and RBS at 400 mm from column face with a = 150 mm b = 500 mm c = 40 mm
Z p = 8510 Z RBS = 5526
248
Exterior Columns 1
Roof
ISWB600 A with 400 x 10 mm cover plates
6130
207
2
Level 4
ISWB600 A with 400 x 20 mm cover plates
8420
270
3
Level 3
ISWB600 A with 400 x 20 mm cover plates
8420
270
4
Level 2
ISWB600 A with 400 x 20 mm cover plates
8420
270
Interior Columns 1
Roof
ISWB600 A with 400 x 25 mm cover plates
10591
302
2
Level 4
ISWB600 A with 400 x 40 mm cover plates
14581
397
3
Level 3
ISWB600 A with 400 x 40 mm cover plates
14581
397
4
Level 2
ISWB600 A with 400 x 40 mm cover plates
14581
397
Sejal P. Dalal, Andeep A. Vasanwala & Atul K. Desai
96
CONCLUSIONS 1.
In the Elastic Design method as all the structural members (i.e. both beams and columns) have all the three components viz. shear, axial and moment, they are all designed as beam-columns. The design is simple and hassle free.Whereas in the PBPD method, the beams are provided with hinges (weak links) at predetermined points. The beams are to be designed for flexure and shear. In addition these, weak links are also to be designed separately and checked. This makes the design lengthy.
2.
The failure pattern and failure points in the PBPD method are predetermined and fixed prior to the design whereas in the elastic design method, the failure can occur anywhere in the structure.
3.
For the same values of dead and live load, the values of shear force, axial force and bending moment in the PBPD and elastic design frame is totally different. This is probably due to the variation in lateral force. Also, there is 35 % decrease in weight of beams, 17 % increase in weight of exterior columns and 33 % increase in weight of interior columns of PBPD method as compared to Elastic Design method. Hence, it could be seen that the “strong column weak beam” concept is satisfied in PBPD method.
4.
The overall cost of the structure remains same because in the PBPD method, even if the reduction in weights due to RBS is not considered, still the overall weight of the structure is just 15% higher than Elastic Design Method. This fact can make the PBPD method more acceptable and admirable to designers.
REFERENCES 1.
Chao S H, Goel S C, Lee S S (2007) : “A seismic design lateral force distribution based on inelastic state of structures”, Earthquake Spectra 23: 3, 547–569.
2.
Chao S H, Goel S C. (2006a) : “Performance-based design of eccentrically braced frames using target drift and yield mechanism”, AISC Engineering Journal Third quarter: 173–200.
3.
Chao S H, Goel S C. (2006b): “A seismic design method for steel concentric braced frames (CBF) for enhanced performance”. In Proceedings of Fourth International Conference on Earthquake Engineering, Taipei, Taiwan, 12–13 October, Paper No. 227
4.
Chao S H, Goel S C. (2008): “Performance-based plastic design of seismic resistant special truss moment frames”, AISC Engineering Journal Second quarter: 127–150.
5.
Dalal S P , Vasanwala S A , Desai A K : (2011) , “Performance based Plastic Design of Structure :
A Review “ in International Journal of Civil and Structural Engineering , Volume
1 No. 4 -2011 , pp 795-803. 6.
Dalal S P , Vasanwala S A , Desai A K : (2012) , “Applying Performance based Plastic Design method to steel moment resisting frame in accordance with the Indian Standard Code” in International Journal of Engineering and Technology , Volume 2 No.3 –March 2012 , pp 409-418, IJET publications, United Kingdom.
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Comparative Evaluation of Elastic Design and Performance Based Plastic Design Method for A Steel Moment Resisting Frame
7.
Dasgupta P, Goel SC, Parra-Montesinos G. (2004): “Performance-based seismic design and behavior of a composite buckling restrained braced frame (BRBF)”. In Proceedings of Thirteenth World Conference on Earthquake Engineering, Vancouver, Canada, 1–6 August 2004, Paper No. 497
8. IS-1893:2000: “Criteria for Earthquake Resistant Design of Structures”. 9. IS-800:2007: “General Construction in Steel Code of Practice”. 10. IS-875 Parts I to V: “ Indian Standard Code of Practice for design loads other than earthquake) for buildings and structures”. 11. Lee S S, Goel S C. (2001): “Performance-Based design of steel moment frames using target drift and yield mechanism.” Research Report no. UMCEE 01-17, Dept. of Civil and Environmental Engineering, University of Michigan, Ann Arbor, MI. 12. Newmark N M, Hall WJ. (1982): “Earthquake Spectra and Design”, Engineering Monographs on Earthquake Criteria, Structural Design, and Strong Motion Records, Vol 3, Earthquake Engineering Research Institute, University of California, Berkeley, CA.