Abstract
In a liquefied nature gas (LNG) plant, a large concrete framed structure supporting multiunits of heavy compressors was designed in accordance with ACI 31808 code requirements and ASCE 705seismic loads. When applied to new structures, provisions of ACI 318 are intended to provide Life Safety (LS) performance for the Design Basis Earthquake (DBE). Due to the important function of the compressors, this study will perform a seismic assessment of the Table Top Pedestal to assure adequate capacity of preventing collapse from Maximum Considered Earthquake (MCE) and only having limited structural damage under a moderate (MOD) 20%@50 years earthquake event. In this study, the Linear Dynamic Procedure (LDP) of ASCE 4113 is used for the seismic performance evaluation of the table top structure. The structural modeling parameters and acceptance criteria of structural performances are based on Chapter 10 of ASCE 4113. Soilstructural interaction and PDelta effects are considered in the analysis process. The response spectra of the three levels of seismic hazards of DBE, MCE and MOD earthquake were developed for response spectra analysis. The procedures presented in this study can be used as a general guideline for PerformanceBased Design of most reinforced concrete structures located in industrial plants.
1. Introduction
In a liquefied nature gas (LNG) plant, it requires multiple compressor units of methane, ethylene and propane compressors in order to convert natural gas to LNG by cooling the liquid temperature to –270 °F. A typical concrete framed structure (also referred to as Table Top Pedestal) supporting two trains of methane compressors as shown in Fig. 1 has been designed by Jan [1] in accordance with ACI 31808[2] and ASCE 705 [3] code requirements. The maximum considered earthquake (MCE) in the plant site has a 2 % probability of exceedance in 50 years (a return period of 2475 years). The design base earthquake (DBE) is twothirds of the of MCE. It is equivalent to an earthquake having a 10 % probability in 50 years (a return period of 475 years). For ordinary structures, life safety under a DBE hazard event are ensured by designing the structure for the effects of codeprescribed earthquake forces and by conforming to material design and detailing requirement set forth in the code. Due to the important function of the compressors, this study will perform a seismic assessment of the Table Top Pedestal to assure adequate capacity of preventing collapse from MCE and only having limited structural damage under a moderate (MOD) earthquake event of 20%@50 years (a return period of 225 years). The ground response spectra of the three levels of earthquake MCE, DBE and MOD are shown in Fig. 2.
Structural analysis of the Table Top Pedestal may be performed by using either linear dynamic procedure (LDP) or nonlinear dynamic analysis procedure (NDP) as provided in ASCE 4113 [4] for performancebased seismic evaluations of structures. NDP is normally required for a tall building whose height exceeding the limit of building code (ASCE 7). Extensive international research and developments have been carried out on the tall building design in USA [5], Canada [6], Japan [7], China [8], Turkey [9] and Greece [10]. The structural response obtained from the nonlinear dynamic analysis results represent true structural performances under the earthquake event. However, NDP requires considerable efforts in mathematic modelling and tedious analysis process and interpretation of analysis results. An industrial structure such as the table top pedestal has little irregularity in the structure layout in general, thus the linear dynamic procedure is appropriate to be used for seismic performance evaluation of the structure. In general, LDP yields a more conservative design result than NDP design result as presented in Reference [11]. Mathematical modelling and analysis process for the linear dynamic procedure can be accomplished efficiently by using commercial available general finite element programs. Therefore, the linear dynamic procedure is adopted in this paper for carrying out the seismic performance assessment of the pedestal. The LDP is a practical and efficient approach for seismic performance based retrofitting of existing structures and is recommended to be used for analysis of new structures in the industrial plants.
Fig. 1Finite element model – table top pedestal
Fig. 2Earthquake ground response spectra
2. The existing compressor table top structure
The threedimensional finite element model of the concrete table top pedestal is shown in Fig. 1. The mathematic model consists of 1230 plate elements for deck slab, 2400 plate elements for mat and 80 line elements for 40 columns. The soil stiffness is represented by Winkler spring elements. In order to consider the effects of concrete cracking, shrinkage, and reinforcement slip, the effective component stiffness is reduced per Table 105 of ASCE 41. In this study, the modulus of elasticity of concrete ${E}_{c}$ for all columns and deck slab are reduced to 0.3${E}_{c}$. The structural design parameters in the original structural design [1] are summarized as follows:
Concrete structure:
– Mat: 93 ft×164 ft×4 ft, with #9 @ 12” top/bottom, total weight = 9200 kips,
– Deck size: 87 ft×128 ft×3.3 ft with #9 @8” top/bottom, total weight = 5500 kips,
– Columns: 403.3 ft squares, total weight = 1200 kips,
– Concrete maximum compressive strength $f\mathrm{\text{'}}c=$ 4ksi,
– Yield strength of reinforcing steel = 60 ksi,
– Critical damping ratio of concrete structure ${\beta}_{c}=$ 4 %.
Equipment:
– The table top supports two trains of methane compressors. Each train has one compressor and one gas turbine.
– Compressor: weight = 522 kips/unit, operating speed =104.5 cps,
– Gas turbine: weight = 370 kip/unit, operating speed = 64.7 cps.
Soil properties:
– Allowable soil bearing pressure = 2.5 ksf,
– Ultimate soil bearing pressure = 5 ksf,
– Critical damping ratio of soil ${\beta}_{c}=$ 10 %.
Seismic design parameters used in the original structural design:
– Soil site class “D”,
– Occupancy category III,
– Seismic design category “D”,
– Occupancy important factor of structure $I=$ 1.25,
– Response modification factor $R=$ 8 for special reinforced concrete moment frame per ASCE 7.
– Mapped Acceleration parameters:
• At short period (0.2 s) ${S}_{s}=$ 1.12,
• At one second period ${S}_{1}=$ 0.48.
– Design spectral acceleration parameters:
• At short period (0.2 s) ${S}_{DS}=$ 0.79,
• At 1second period ${S}_{D1}=$ 0.49.
3. Linear dynamic analysis procedure (LDP)
In this study, the modal response spectrum method was used to evaluate structural responses. The linearly elastic response spectra shown in Fig. 2 are not modified to account for anticipated nonlinear response. It is expected that the LDP will produce displacements that approximate maximum displacements expected during the selected Seismic Hazard Level but will produce internal forces that exceed those that would be obtained in a yielding building. Calculated internal forces typically exceed those that the building can sustain because of anticipated inelastic response of components. These forces are evaluated through the acceptance criteria of ASCE 4113 Section 7.5.2, which include modification factors to account for anticipated inelastic response demands and capacities. In the criteria, a component is classified as either deformation controlled (ductile) or force controlled (nonductile) elements. Moreover, structural elements are categorized as primary or secondary components. A primary component resists earthquake forces while a structural component not designed to resist earthquake forces is categorized as secondary component. In LDP, deformation and force controlled actions are evaluated using following criteria:
Deformationcontrolled actions: $mk{Q}_{CE}\ge {Q}_{UD}$,
Forcecontrolled actions: $k{Q}_{CL}\ge {Q}_{UF}$,
where: $m$ is the component demand modification factor to account for the expected ductility related to this action at the selected structural performance level, $k$ is the knowledge factor to account for the uncertainty of collection of asbuilt data per 6.2.4 of ASCE 4113, ${Q}_{CE}$ is the expected strength of a component at the deformation level under consideration for deformationcontrolled actions, ${Q}_{UD}$ is the deformationcontrolled design action due to gravity loads (${Q}_{G}$) and earthquake loads (${Q}_{E}$), ${Q}_{CL}$ is the lowerbound strength of a component at the deformation level under consideration for forcecontrolled actions, ${Q}_{UF}$ is force controlled design action determined by:
where: ${C}_{1}$ is the modification factor that relates the expected maximum inelastic displacements to displacements calculated using linear elastic response, ${C}_{2}$ is a modification factor that considers the effect of pinched hysteresis shapes, cyclic stiffness degradation, and strength deterioration on maximum displacement response, $J$ is the force delivery reduction factor, calculated as the smallest demand capacity ratio (DCR = ${Q}_{UD}$/${Q}_{CE}$) of all components in the load path delivering forces to the component being examined.
4. Seismic performance objectives
As previous discussed, there are three seismic hazard levels considered in this paper. The corresponding structural performance levels per ASCE 4313 guidelines are as followings:
Severe overall structural damage occurs under 2%@50 years (MCE) seismic hazard event. It is corresponding to the Collapse Prevention (CP) Structural Performance Level.
Moderate overall structural damage occurs under 10%@50 years (DBE) seismic hazard event. It is corresponding to the Life Safety (LS) Structural Performance Level.
Light overall structural damage occurs under 20%@50 years (MOD) seismic hazard event. It is corresponding to the Immediate Occupancy (IO) Structural Performance Level.
The 5 % damped acceleration spectra for the three seismic hazards are shown in Fig. 2. The component demand modification factors corresponding to the three structural performance levels are evaluated per ASCE 4113.
5. Dynamic analysis and analysis results
The linear dynamic analysis procedure described in Section 3 is used in this study. The elastic response spectrum analyses for horizontal excitations in the orthogonal $X$ and $Y$ axes are performed for the three levels of earthquake events MCE, DBE and MOD. The seismic responses in the $X$ and $Y$ direction, ${Q}_{Ex}$ and ${Q}_{Ey}$, are then combined with gravity load ${Q}_{G}$ as:
Deformationcontrolled actions:
Forcecontrolled actions:
where: ${C}_{1}{C}_{2}=$ 1.1 from Table 73 of ASCE 41 for period T between 0.3 and 1.0 seconds, $J=$ 2.0 for MCE, 1.5 for DBE and 1.0 for MOD earthquake event based on section 7.5.2.1.2 of ASCE 4113.
The following two gravity load conditions are considered in all above load combinations:
– ${Q}_{G}=$ 1.1(${Q}_{D}+{Q}_{L}$) (Notes: ${Q}_{D}=$ Dead loads, ${Q}_{L}=$ Live loads),
– ${Q}_{G}=$ 0.9${Q}_{D}$.
Based on the analysis results, the capacity demands of columns and deck slab under gravity loads are less than 80 % of nominal capacity of the components. Therefore, the vertical seismic effects are not considered in this study.
The fundamental natural frequencies of the soil supported table top structure are 2.57 cps and 2.80 cps in the two horizontal directions and 4.43 cps in the vertical direction. From the response spectra analysis, the maximum accelerations at the top deck slab are shown in Table 1. All acceleration responses are within 150 % of peak spectra accelerations. They appear to be within the acceptable floor acceleration limits for nonstructural components.
Table 1Acceleration responses at top deck slab
Peak Spectra Accel. (g)  Earthquake $X$Excitation  Earthquake $Y$Excitation  
$X$Horiz. Accel. (g)  $Y$Horiz. Accel. (g)  $Z$Vertical Accel. (g)  $X$Horiz. Accel. (g)  $Y$Horiz. Accel. (g)  $Z$Vertical Accel. (g)  
DBE  0.79  0.94  0.41  0.38  0.38  1.16  0.50 
MCE  1.18  1.41  0.61  0.54  0.69  1.75  0.73 
MOD  0.39  0.47  0.20  0.19  0.23  0.58  0.28 
The maximum displacement at the top deck slab are shown in Table 2. The story height between the top of deck slab to the top of mat is 23 ft. The allowable drifts are set to be 3 %, 2 % and 1 % of story height for MCE, DBE and MOD earthquake levels, respectively. All displacements shown in Table 2 are within the allowable limits.
The dynamic analysis results of internal forces of columns and deck slab are summarized in Tables 3 and 4.
Table 2Displacement responses at top deck slab
Allowable drift (in)  Earthquake $X$excitation  Earthquake $Y$excitation  
$X$horiz. disp (in)  $Y$horiz. disp (in)  $Z$vertical disp (in)  $X$horiz. disp (in)  $Y$horiz. disp (in)  $Z$vertical disp (in)  
DBE  5.52  1.15  0.46  0.34  0.61  1.68  0.38 
MCE  8.28  1.75  0.68  0.51  0.92  2.52  0.58 
MOD  2.76  0.58  0.23  0.17  0.31  0.84  0.19 
Table 3Column internal forces
Loads  $P$ (kips)  ${M}_{x}$ (kipsft)  ${M}_{y}$ (kipsft)  ${V}_{x}$ (kips)  ${V}_{y}$ (kips)  Remarks  
DBE  1.1 ${Q}_{D+L}$  401  13  5  9  18  ${P}_{max}$ 
1.1 ${Q}_{D+L}+E$  260  3275  1776  140  262  Flexural  
682  2181  1966  161  171  ${P}_{max}$  
0.9 ${Q}_{D}+E$  18  2801  1863  144  214  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  461  921  2201  203  86  Flexural  
571  1407  1144  94  110  ${P}_{max}$  
0.9 $D+E$/${C}_{1}{C}_{2}J$  94  1412  1331  108  110  Flexural  
MCE  1.1 ${Q}_{D+L}+E$  291  4963  2441  192  398  Flexural 
822  3164  3011  245  247  ${P}_{max}$  
0.9 ${Q}_{D}+E$  –69  4266  2618  202  327  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  478  1009  2456  227  98  Flexural  
592  1557  1303  107  122  ${P}_{max}$  
0.9 $D+E$/${C}_{1}{C}_{2}J$  71  1612  1475  120  126  Flexural  
MOD  1.1 ${Q}_{D+L}+E$  231  1584  1109  88  127  Flexural 
541  1198  923  76  94  ${P}_{max}$  
0.9 ${Q}_{D}+E$  104  1334  1106  87  102  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  401  832  1072  86  65  Flexural  
540  1407  1198  76  94  ${P}_{max}$  
0.9 $D+E$/${C}_{1}{C}_{2}J$  126  1134  1131  92  88  Flexural 
Table 4Top deck slab element stress resultants
Loads  ${M}_{x}$ (kipsft/ft)  ${M}_{y}$ (kipsft/ft)  ${M}_{xy}$ (kipsft/ft)  ${V}_{x}$ (kips/ft)  ${V}_{y}$ (kips/ft)  Remarks  
DBE  1.1 ${Q}_{D+L}+E$  100  140  60  90  90  Flexural 
0.9 ${Q}_{D}+E$  90  90  50  75  120  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  60  90  30  50  50  Flexural  
0.9 $D+E$/${C}_{1}{C}_{2}J$  60  60  25  40  40  Flexural  
MCE  1.1 ${Q}_{D+L}+E$  120  180  70  100  100  Flexural 
0.9 ${Q}_{D}+E$  100  150  60  120  120  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  80  100  40  60  60  Flexural  
0.9 $D+E$/${C}_{1}{C}_{2}J$  70  80  30  45  45  Flexural  
MOD  1.1 ${Q}_{D+L}+E$  75  70  25  55  50  Flexural 
0.9 ${Q}_{D}+E$  60  60  20  50  45  Flexural  
$D+E$/${C}_{1}{C}_{2}J$  50  60  24  35  35  Flexural  
0.9 $D+E$/${C}_{1}{C}_{2}J$  40  50  20  30  30  Flexural 
6. Seismic performance assessments
The steps for seismic performance evaluation of the concrete structural components in the linear dynamic analysis procedure are as following:
Step 1: Classify components as a Primary or Secondary components.
All columns and the deck slab of the table top structure are considered as primary components. They are required to resist seismic forces and accommodate deformations for the structure to achieve the selected Performed Level.
Step 2: Classify components as deformationcontrolled or forcecontrolled actions.
Different actions for a same element can be classified in different category. For instance, shears in beams and columns are considered as forcecontrolled actions. However, bending moments in columns and beams can be considered as force or deformationcontrolled actions depending on the amplitude of the axial force.
Step 3: Determine component properties.
The LowerBound material properties of the table top structure are concrete compressive strength ${f}_{c}^{\text{'}}=$ 4 ksi and reinforcing steel yield strength ${f}_{y}=$ 60 ksi. The expected strength properties are ${f}_{ce}^{\text{'}}=$ 1.5·4 = 6 ksi and ${f}_{ye}=$ 1.25·60 = 75 ksi per Table 101 of ASCE 4113. Where evaluating the behavior of deformationcontrolled actions, the expected material strengths are used. Where evaluating the forcecontrolled actions, the lowerbound material strengths are used. Calculations of component design strength are in accordance with ACI 31808 with an exception that the strength reduction factor $\phi $ is taken as unity.
All column crosssections are 40 in squares with 28#9 longitudinal reinforcement and 6#4 shear tie legs with 135° seismic hooks. The spacing of transvers reinforcement ${s}_{o}$ is 5 in. The effective depth of column ${d}_{c}=$ 36 in. The following are calculated column properties and strength:
– Gross area ${A}_{g}=$ 1600 in^{2}, Total longitudinal reinforcing area = 28 in^{2},
– Nominal axial load strength at zero eccentricity ${P}_{o}=$ 1600·4 = 6400 kips,
– Flexural reinforcement ratio ${\rho}_{s}=$ 0.0175,
– Arear of shear reinforcement ${A}_{v}=$ 6·0.2 = 1.2 in^{2},
– Shear reinforcement ratio ${\rho}_{v}=$ 1.2/(40·5) = 0.006.
– Shear strength of column: ${V}_{s}={A}_{v}\xb7{f}_{y}\xb7{d}_{c}/{s}_{o}=$ 1.2·60·36/5 = 605 kips, ${V}_{c}=$ 2·$\sqrt{f\text{'}c}$·40·36 = 182 kips,
– Total shear strength ${V}_{o}={V}_{c}+{V}_{s}=$ 787 kips,
The deck slab has a uniforms thickness $t=$ 40 in. It was provided with #9 @ 8 in longitudinal reinforcements at top and bottom faces. The following are calculated slab properties and strength:
– Longitudinal reinforcing area = 1.33 in^{2}/ft top and bottom,
– Flexural reinforcement ratio ${\rho}_{s}=$ 0.0031.
Step 4: Determine component demand modification factor $m$ for deformationcontrolled actions.
Determination of $m$factors for columns are based on the criteria provided in Tables 109 of ASCE 4113. The failure mode of columns is considered as condition $i$ (flexure failure) since the column transvers reinforcement ratio ${\rho}_{v}=$ 0.0065 > 0.002 and the ratio of shear ties spacing to column depth ${s}_{o}$/${d}_{c}=$ 5/37 = 0.135 < 0.5. The evaluated $m$factors are shown in Table 5. In the table, $P$ is the axial force action.
Table 5Column mmodification factors
$P$ (kips)  $P/{A}_{g}f\text{'}c$  ${p}_{v}$  $m$factors  
DBE  682  0.107  0.006  2.49 
MCE  822  0.128  0.006  2.94 
MOD  540  0.084  0.006  2.02 
Determination of $m$factors for deck slab are based on the criteria provided in Table 1016 of ASCE 41. The evaluated $m$factors are shown in Table 6. In the table, ${V}_{g}$ is the gravity shear acting on the critical shear sections and ${V}_{o}$ is the direct punching shear strength defined by ACI 318.
Step 5: Check component capacities for deformationcontrolled actions.
The acceptance criteria for deformationcontrolled actions is:
For the table top structure, the knowledge factor is 1.0 since the structural design meets the benchmark requirement of Tables 4 and 5 ASCE 4113. Calculations of component strength are based on the expected strength material properties. The allowable $m$modification factors for columns and slab are calculated in Tables 5 and 6. The deformationcontrolled actions caused by the combination of gravity load ${Q}_{G}$ and earthquake forces ${Q}_{E}$ are presented in Tables 3 and 4. The expected strength ${Q}_{E}$ of component deformationcontrolled action of an element are determined considering all coexisting actions on the component from load combinations as discussed in Section 3. The component capacity checks are summarized in the Tables 7 and 8. In Table 7, $P$ is column axial force and ${M}_{u}$ is algebra sum of the biaxial bending moments for the square columns. In Table 8, ${M}_{u}$ is bending moments with consideration of twisting moment in the slab. All components demand capacity ratios (DCR) are less than the allowable $m$modification factors and therefore, achieve the seismic performance objectives.
Table 6Top deck slab mmodification factors
Column locations  Seismic levels  $Vg$ (kips)  $Vo$ (kips)  DCR $Vg$/$Vo$  Continuity reinforcement  $m$factors 
Corner  DBE  312  906  0.34  yes  2.15 
MCE  312  906  0.34  yes  2.48  
MOD  312  906  0.34  yes  1.15  
Side  DBE  380  1750  0.22  yes  2.45 
MCE  380  1750  0.22  yes  2.93  
MOD  380  1750  0.22  yes  1.45  
Interior  DBE  401  2331  0.17  yes  2.54 
MCE  401  2331  0.17  yes  3.08  
LOW  401  2331  0.17  yes  1.58 
Table 7Summary of column capacity check for deformationcontrolled actions
°  Load combinations  $P$ (kips)  ${M}_{u}$ (${M}_{x}+{M}_{y}$) (kipsft)  ${M}_{o}$ (kipsft)  DCR (${M}_{u}$/${M}_{o}$)  Allowable $m$factor  Remarks 
DBE (LS)  1.1$\left(D\right)+E$  260  5051  3300  1.53  2.49  Acceptable 
0.9$\left(D\right)+E$  18  4767  2900  1.64  2.49  Acceptable  
MCE (CP)  1.1$\left(D\right)+E$  294  7404  3500  2.12  2.94  Acceptable 
0.9$\left(D\right)+E$  –194  6884  2800  2.46  2.94  Acceptable  
MOD (IO)  1.1$\left(D\right)+E$  303  2693  3500  0.77  2.02  Acceptable 
0.9$\left(D\right)+E$  187  2440  3100  0.79  2.02  Acceptable 
Table 8Summary of top deck slab capacity check for deformationcontrolled actions units
Load combinations  ${M}_{u}$ (${M}_{y}+{M}_{xy}$) (kipsft/ft)  ${M}_{n}$ (kipsft/ft)  DCR ${M}_{u}$/${M}_{n}$  $m$factor  Remarks  
DBE (LS)  1.1$\left(D\right)+E$  200  432  0.46  2.15  Acceptable 
0.9$\left(D\right)+E$  140  432  0.32  2.15  Acceptable  
MCE (CP)  1.1$\left(D\right)+E$  250  432  0.58  2.48  Acceptable 
0.9$\left(D\right)+E$  210  432  0.49  2.48  Acceptable  
MOD (IO)  1.1$\left(D\right)+E$  95  432  0.22  1.15  Acceptable 
0.9$\left(D\right)+E$  80  432  0.19  1.15  Acceptable 
Step 6: Check component capacities for forcecontrolled actions.
The acceptance criteria for forcecontrolled actions is:
Axial forces and shears in columns and slabs are forcecontrolled actions. Calculations of component strength are based on the lower bound material properties. Tables 9 and 10 present summaries of capacity checks for columns and top deck slab, respectively. The demand capacity ratios (DCR) are less than 1.0. Therefore, the capacity of the table top pedestal meets the seismic performance objectives.
Table 9Summary of column capacity check for forcecontrolled actions
Loads  $P$ (kips)  $Po$ (kips)  $DCR=P/Po$  $V$ (kips)  $Vo$ (kips)  $DCR=V/Vo$  
$D+\frac{E}{{C}_{1}{C}_{2}J}$  DBE  571  6000  0.095  145  189  0.77 
MCE  564  6000  0.094  162  189  0.86  
MOD  540  6000  0.090  121  189  0.64 
Table 10Summary of top deck slab capacity check for forcecontrolled actions
Loads  Beam shear (kips/ft)  Punching shear (kips)  
$V$ (kips)  $Vo$ (kips)  $DCR=V/Vo$  $Vu$  $Vo$  $DCR=Vu/Vo$  
D+E/C1C2J  DBE  71  162  0.44  571  906  0.63 
MCE  141  162  0.87  592  906  0.65  
MOD  64  162  0.39  540  906  0.60 
Per ASCE 4113, the capacity of soil bearing pressure shall be evaluated as forcecontrolled actions. The calculated soil bearing pressure are in allowable limits as shown in Table 11.
Table 11Foundation bearing capacity
Loads  Soil bearing pressure (ksf)  Allowable bearing pressure (ksf)  
Gravity  Gravity  1.75  2.5 
D+E/C1C2J  DBE  2.75  5.0 
MCE  2.90  5.0  
MOD  2.56  5.0 
7. Conclusions
Based on the assessment of seismic performance presented in this study, it is concluded that the concrete compressor table top structure can provide life safety structure performances under a design basis earthquake event, has adequate capacity of preventing collapse from a maximum considered earthquake event, and has only limited structural damage under a moderate earthquake event.
The Linear Dynamic Procedure provided in ASCE 4113 is a practical procedure for seismic performance assessment of industrial plant structures. This procedure can achieve performancebased seismic design without requiring highly sophisticated nonlinear dynamic analysis program which is still under research and development. The analysis steps presented in this study for using linear dynamic analysis procedure can be applied for seismic performancebased retrofitting of reinforced concrete structures.
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