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Performance-based assessment of response reduction factor of RC-elevated water tank considering soil flexibility: a case study

  • Kashyap N. Patel
  • Jignesh A. Amin
Open Access
Original Research
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Abstract

The seismic design codes/standards of most countries include the nonlinear response of a structure implicitly through a response reduction/modification factor (R). It is the factor by which the actual base shear should be reduced to find the design base shear during design basic earthquake considering nonlinear behavior and deformation limits of structures. In the present study, attempts are made to determine the ‘R’ factors of four existing RC staging elevated water tanks, which are designed as per draft Indian standards for seismic design of liquid and RC designs and having a ductile detailing considering the effects of soil flexibility. The elevated RC water tanks are analyzed using displacement controlled non-linear static pushover analysis to evaluate the base shear capacity and ductility of tank considering soil flexibility. The ‘R’ factor is obtained for four realistic designs of elevated RC water tanks having different capacities at two performance levels. The evaluated values of ‘R’ factor are compared with the values suggested in the design code. The results of the study show that the flexibility of supporting soil has considerable effect on response reduction factor, period and overall performance of water tank, indicating that idealization of fixity at base may be seriously mistaken for soft soils. All the studied water tanks were designed with higher safety margin than that of specified in Indian Standards.

Keywords

Response reduction factor Soil flexibility Ductility factor Strength factor Pushover analysis 

Introduction

Elevated RC staging water tanks have an essential role in water distribution network. Therefore, they should remain operational after earthquakes for essential needs. Various deficiencies for the poor seismic performance of elevated water tanks during past earthquakes such as improper structural design of elements and joints, low reserve strength and ductility, and the effect of soil flexibility have been reported in the previous studies (Rai 2003; Dutta et al. 2000). Current force-based design procedure adopted by most seismic design codes allows the seismic design of structures to be based on static or dynamic analyses of elastic models of the structure using elastic design spectra. The codes anticipate that structures will undergo inelastic deformations under strong seismic events; therefore, such inelastic behavior is usually incorporated into the design by dividing the elastic spectra by a factor R, which reduces the spectrum from its original elastic demand level to a design level. The values of the R factor basically depend on the structural ductility factor and over-strength. For a structure supported on flexible foundation, the consideration of the soil flexibility in analysis increases the elastic period and damping of the structure–foundation system and subsequently it affect the structural ductility. This implies that the response reduction factors, which are currently not affected by the soil flexibility, could be altered. The values of response reduction factor of RC elevated water tank are given in IITK-GSDM guidelines (IITK-GSDM guidelines for Seismic design of liquid storage tanks 2007) and IS 1893 (Part-II) for Seismic design of liquid storage tanks (Draft IS 1893 2006; BIS IS 1893 2014), which is derived empirically based on engineering judgment. The fourth revision of seismic design code was published in 1984. After Bhuj earthquake in 2001 and considering the fact that significant advancements have taken place in the field of earthquake engineering, fifth revision of IS 1893 (Part-1), general provision for earthquake resistant buildings and structures was published in 2002 (IS 1893 2002). However, the draft code for seismic design of liquid storage tanks IS 1893 (Part-2) was prepared in 2006 and final version was published in 2014. IS1893:1984 recommended the single degree of freedom idealization of tank; whereas draft as well as the final version of IS 1893 (Part-II) 2014 recommended two-degree of freedom idealization for analysis of elevated tanks and the flexibility of bracing beam is included in the evaluation of lateral stiffness of tank staging. The values of response reduction factor of elevated water tank adopted by different codes/standards are summarized in Table 1. The recent revision of IS 1893 (Part-2): 2014 had increased the value “R” factor of special moment resisting RC staging water tank to 4 from the 2.5 suggested in 2006 version of the draft code. The existing literature in this area does not provide any specific basis on which a value of 4 is assigned for ductile moment resisting RC staging elevated water tanks in the IS 1893 (Part-2): 2014.
Table 1

Values of ‘R’ from different codes

Codes

R value

IBC 2000/FEMA 368 (IBC 2000; FEMA 368 2000)

1.5–3.0

AWWA D110 (1995)

2–2.75

ACI 350.3 (2001)

2.0–4.75

RCC frame support IITK-GSDM guidelines (SMRF)/Draft code IS:1893 (Part-II)

2.5

RCC frame support IS:1893 (Part-2):2014 (SMRF)

4

Different aspects of the response reduction factor of various structural systems have been investigated and essential weaknesses have been pointed out by many researcher. Mondal et al. (2013) estimated the real values of response reduction factor of actual RC moment frame structure designed and detailed using the Indian codes for earthquake and RC designs and for ductile detailing. Authors concluded that codes recommend higher than real value of ‘R’ for RC frame. Tamboli and Amin (2015) evaluated the ‘R’ factor of RC frame strengthened using the different types of bracing systems and concluded that type and arrangement of bracing systems have significant impact on the ‘R’ factor. Masoudi et al. (2012) discuss the seismic behavior and failure mechanism of RC frame and shaft-supported tanks under severe earthquakes considering the P–∆ effects by performing linear and nonlinear response history analyses. Ghateh et al. (2015) presented a methodical approach to determine the response modification factors for total 48 elevated tanks of different capacities and RC frame dimensions commonly used in industry. They have suggested not to use the same response reduction factor for tanks having a different staging height and capacity. Patel et al. (2014) evaluated component-wise response reduction factor of elevated water tanks having equal staging height and different capacities. They concluded that the value of response reduction factor for RC staging tank is significantly influenced by time period, capacity of tank, and seismic zone. The possible serious effects of neglecting soil flexibility and their effects on structural safety have been evaluated by many researchers (Dutta et al. 2004; Halabian and Erfani 2013; Livaoglu and Dogangun 2007). Dutta et al. (2004) studied the effect of soil flexibility on the dynamic characteristics of RC frame staging tanks having different configurations and concluded that analysis without considering soil flexibility may lead to lower or higher estimation of seismic base shear of RC frame staging tanks considering fluid–structure interaction. Halabian and Erfani (2013) evaluated the effect of foundation flexibility and structural strength on response reduction factor of RC frame buildings.

The aim of the present study is component-wise evaluation of response reduction factor of the four existing realistic RC water tanks considering the effects of soil flexibility and comparing these values with the values given in the seismic design code. All tanks are located in different parts of Gujarat, India and were designed and detailed as per the Indian codal provisions IS: 456-2000 (BIS IS 456 2000), IITK-GSDM guidelines/Draft IS 1893 (Part-II) and IS: 13920-1993 (BIS IS 13920 1993) by different design engineers.

Concept of ‘R’ factor

The ‘R’ factors are essential seismic design tools, which indicate the level of inelasticity expected in structural systems during an earthquake event. This factor intended to account for both damping and ductility inherent in structural systems at the displacements great enough to approach the maximum displacement of the systems. The ‘R’ factor reflects the capability of structure to dissipate energy through inelastic behavior. The response reduction factor ‘R’ depends on various components such as overstrength factor (Rs), ductility factor (Rµ), redundancy factor (RR) and damping factor (Rζ). According to ATC-19 (1995), it is described as
$$R = R_{s} \times R_{\mu } \times R_{R} \times R_{\zeta } .$$
(1)

Strength factor (R s)

Strength factor (Rs) accounts for the yielding of a structure at load higher than the design load due to various partial safety factors, strain hardening, oversized members and confinement of concrete. Non-structural elements also contribute to the overstrength. The overstrength factor generally varies with seismic zones, height of structure and design gravity loads/capacity of water tank. The strength factor (Rs) is defined as a ratio of maximum base shear (V0) to the design base shear (Vd),
$$R_{\text{s}} = {\raise0.7ex\hbox{${V_{ 0} }$} \!\mathord{\left/ {\vphantom {{V_{ 0} } {V_{\text{d}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${V_{\text{d}} }$}}.$$
(2)

Ductility factor (R µ)

The ductility factor (Rµ) is a measure of the global nonlinear response of a structural system in terms of its plastic deformation capacity. Miranda and Bertero (1994) summarized and reworked the RµµT relationships developed by a number of researchers including Riddell and Newmark (1979), Newmark and Hall (1982) and Krawinkler and Nasser (1992) for soft, alluvium and rock soil sites. In the present study, the Rµµ–T relationships established by Miranda and Bertero (Riddell and Newmark 1979) are used. The Miranda and Bertero’s equations for ductility factor shown below were developed using 124 ground motions recorded on a different soil conditions, and assumed 5% of critical damping:
$$R_{\mu } = \frac{\mu - 1}{\varPhi } + 1,$$
(3)
where µ is the displacement ductility ratio generally defined as the ratio of maximum displacement to the displacement at yield. For rock sites,
$$\varPhi = 1 + \frac{1}{10T - \mu T} - \frac{1}{2T}e^{{ - 1.5\left( {\ln \left( T \right) - 0.6} \right)^{2} }} .$$
(4)
For alluvium sites,
$$\varPhi = 1 + \frac{1}{12T - \mu T} - \frac{2}{5T}e^{{ - 2\left( {\ln \left( T \right) - 0.2} \right)^{2} }} .$$
(5)
For soft soil sites,
$$\varPhi = 1 + \frac{{T_{\text{g}} }}{3T} - \frac{{3T_{\text{g}} }}{4T}e^{{ - 3\left( {\ln \left( {{\raise0.7ex\hbox{$T$} \!\mathord{\left/ {\vphantom {T {T_{\text{g}} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${T_{\text{g}} }$}}} \right) - 0.25} \right)^{2} }} ,$$
(6)
where Tg is the predominant time period of the ground motion.

Redundancy factor (R R)

Yielding at one location in the structure does not indicate yielding of the whole structure. Hence, the load distribution, due to redundancy of the structure, provides additional safety margin. RC structure with multiple lateral load-resisting frames is normally considered as redundant structure, because each of the seismic frames is designed and detailed to transfer the seismic forces to the soil. Following the conservative assumption, RR = 1.0 is used in this study.

Damping factor (R ζ)

The damping factor (Rζ) takes into accounts the effect of ‘added’ viscous damping for a structures having supplementary energy dissipating devices. Without any such devices, the damping factor is generally taken as one.

Description of the existing water tank considered in this study

In the present study, ‘R’ factors of four existing RC-elevated water tanks having capacities of 140, 480, 1000 and 2200 m3 are evaluated with and without considering the flexibility of soil. All investigated water tanks are located in different parts of Gujarat, India and were designed and detailed using the code provisions IS: 456-2000, Draft IS1893 (Part-II) 2006 and IS: 13920-1993 by different design engineers. Live load on roof slab/top dome is assumed as 0.5 kN/m2. Damping in the structures is considered to be 5% of the critical. The typical configurations of staging system of study water tanks are shown in Fig. 1. All the water tanks had a raft foundation and supported on medium soil. The brief structural details and description of considered elevated water tanks are shown in Tables 2 and 3.
Fig. 1

Plan configuration of study water tanks

Table 2

Detailed descriptions of RC tanks and members

Capacity (m3)

140

480

Earthquake zone

Zone-III

Zone-III

Type of soil

Medium soil

Medium soil

Diameter of staging (m)

6

9

Height of staging (m)

18

18

Tie beam levels

Plinth + 4.5 m c/c

Plinth + 4.5 m c/c

No. of columns

6

8

Top dome (mm)

100

100

Top ring beam (mm)

170 × 300

350 × 300

Cylindrical wall (mm)

170

170

Bottom ring beam (mm)

300 × 300

400 × 300

Lower circular ring beam (mm)

400 × 800

500 × 1000

Bottom dome (mm)

170

175

Conical dome (mm)

170

200

Braces (mm)

300 × 530

300 × 600

Columns (mm)

500

600

Reinforcement in tie beam and plinth beam

4–16 #

6–16 #

Reinforcement in columns

12–16 #

12–16 #

Reinforcement in lower circular beam

6–20 + 2–16#

3–25 + 2–20 #

Table 3

Detailed descriptions of RC tanks and members

Capacity(m3)

1000

2200

Zone

Zone-V

Zone-III

Soil type

Hard soil

Medium soil

Height of staging (m)

20

23

Diameter of staging (m)

16.25

22.75

Height of container (m)

5

7

Wall thickness (mm)

230

250

Top slab thickness (mm)

170

175

Bottom slab thickness (mm)

230

250

Height of staging (m)

15

18.4

Tie beam levels (m)

Plinth + 5 m c/c

Plinth + 4.6 m

Column size (mm)

650

600

Reinforcement in column (No-dia)

16–20 mm

8–20 mm

Plinth beam (mm × mm)

400 × 600

300 × 500

Tie of beam (mm × mm)

400 × 600

300 × 500

Bottom slab beam (mm × mm)

350 × 1200

350 × 950

No. of column

12

24

Length of column (m)

5

4.6

Reinforcement in columns (No-dia)

16–20#

8–20#

Ground beam 1,2

2–16# + 4–20#(top)

6–16#(bottom)

2–16# + 3–20#(top)

4–16#(bottom)

Ground beam 3,4

4–16# + 4–20#(top),4–16# + 2–20#(bottom)

2–16# + 3–20#(top),4–16#(bottom)

Ground beam 5,6

2–16# + 5–20#(top),4–20#(bottom)

Tie beam 1,2

4–16# + 3–20#(top)4–16# + 2–20#(bottom)

2–16# + 4–20#(top)4-–16#(bottom)

Tie beam 3,4

4–16# + 4–20#(top),4–16# + 3–20#(bottom)

2–16# + 5–20#(top),4–16#(bottom)

Tie beam 5,6

5–20#(top),4-–20#(bottom)

Lower ring beam B1,B2

4–20# + 6–25#(top),5–25# + 3–12#(bottom)

2–20# + 2–16#(top),5–16# + 3–10#(bottom)

Lower ring beam B3,B4,B5

4–20# + 8–25# + (top),10-25#

4–20# + 5–25# (top),6–20#

Lower ring beam B6,B7,B8

2–16# + 6+20# + (top),6–20#

3–20# + (top),7–20#

Lower ring beam B9,B10,B11

4–16# + 5–20#(top),8–20#(bottom)

2–20#(top),4–20#(bottom)

Lower ring beam B12,B13,B14

4–20# + 3–25#(top),5–20#(bottom)

Modeling of soil flexibility

The behavior of the structures depends mainly on its mass, stiffness, supporting soil properties and foundation type. When the lateral forces, such as seismic forces, act on these systems, the displacement of structure and soil is not independent of each other. The process by which the response of the soil affects the motion of the structure and the motion of the structure affects the response of the soil is termed as soil–structure interaction. Conventional structural designs neglect the effects of soil flexibility. Neglecting soil flexibility is acceptable for light structures such as low rise buildings and simple rigid-retaining walls on relatively stiff soil. The behavior of soil can be conveniently simulated using a set of elastic springs. The underneath soil flexibility can be modeled with equivalent translation, rocking and torsional elastic stiffness based on soil properties and using equations given by Whitman and Richart (Ghazetas 1983) (Table 4). In this study, three different properties of soil stratum, i.e., hard, medium and soft soil conditions were considered to study the effect of soil flexibility. The linear elastic material behavior was assumed for soil. The elastic properties and calculated equivalent spring constants of considered soils in different direction are shown in Table 5.
Table 4

Equivalent spring stiffness for raft foundation

Mode

Vertical

Horizontal

Rocking

Torsion

Stiffness

\(\frac{{4{\text{G}}R}}{1 - \mu }\)

\(\frac{{8{\text{G}}R}}{2 - \mu }\)

\(\frac{{8{\text{G}}R^{3} }}{3(1 - \mu )}\)

\(\frac{{16{\text{G}}R^{3} }}{3}\)

Mass ratio

\(\frac{m(1 - \mu ) }{{4\rho R^{3} }}\)

\(\frac{m(2 - \mu ) }{{8\rho R^{3} }}\)

\(\frac{{3I_{x} (2 - \mu ) }}{{8\rho R ^{3} }}\)

\(\frac{{I_{z} }}{{\rho R ^{5} }}\)

Damping ratio

\(\frac{0.425}{{\bar{m}^{0.5} }}\)

\(\frac{0.29}{{\bar{m}^{0.5} }}\)

\(\frac{0.15}{{(1 + \bar{m})\bar{m}^{0.5} }}\)

\(\frac{0.50}{{(1 + 2\bar{m})}}\)

Ix, Iz = mass moments of inertia around horizontal and vertical axes, respectively; damping ratio = C/Ccr, where Ccr = 2(Km) 1/2 or Ccr =2(KI)1/2 for translational or rotational modes of vibration. With I = Ix or Iz for rocking or torsion, respectively

Table 5

Elastic soil properties and spring stiffness considered in water tank

Type of soil

Degrees of freedom

Spring constant 140 m3

Spring constant 480 m3

Spring constant 1000 m3

Spring constant 2200 m3

(kN/m/m2)

(kN/m/m2)

(kN/m/m2)

(kN/m/m2)

Hard

Horizontal

42096.2

37678.88

15852.45

12533.36

Vertical

63144.3

56518.32

23778.69

18800.04

Rocking

604674.18

677375.95

1613908

2043851

Torsion

604674.18

677375.95

1613908

2043851

Medium

Horizontal

24177.77

21640.7

9104.78

7198.47

Vertical

36266.66

32461.05

13657.18

10797.72

Rocking

347291.9

389047.91

926941.1

1173877

Torsion

347291.9

389047.91

926941.1

1173877

Soft

Horizontal

5766.29

5161.21

2171.45

1716.807

Vertical

8649.68

7741.82

3257.18

2575.211

Rocking

83827.6

92786.29

221071.6

279964.7

Torsion

83827.6

92786.29

221071.6

279964.7

Pushover analysis of finite element model of study water tanks

SAP 2000 V15 (Structural Analyis software 2000) software is used to perform nonlinear static pushover analysis of study water tanks. The RC beams and columns are modeled as three-dimensional frame elements with centerline dimension. Different parameters such as weight of staging, weight of container, convective mass, impulsive mass, height of impulsive mass, height of convective mass and C.G of container are computed as per guidelines given by IS 1893 (Part-II) guidelines for seismic design of water tanks. The convective mass and impulsive mass are assigned at the appropriate height of the structure as per the codal provisions, whereas self-weight of the container is assigned as uniformly distributed load on the top circular beam. Walls and domes/slabs are assumed to behave as rigid diaphragms. The diaphragm action of a slab is considered by assigning a rigid link at the floor level of container. Damping ratio of 5% is assumed. Flexural (M3), axial biaxial moment (PM2M3) plastic hinges are assigned to each ends of the beams and columns, respectively, where the resultant moments under gravity and lateral loads are maximum. Shear hinges were assigned at the ends of a beam. The idealized force–deformation curve and acceptance criteria for various performance levels in beams and columns have been considered as per FEMA 356 guidelines. The non-linear load–deformation relation (FEMA 356 (2000)) adopted for the pushover analysis is shown in Fig. 2. It is described by linear response from A (unloaded condition) to an effective yield B, then a linear response at reduced stiffness from point B to C, then sudden reduction in lateral load resistance to point D, then response at reduced resistance to E, and final loss of resistance thereafter. The slope from point B to C, ignoring effects of gravity loads acting through lateral displacements, shall be taken as 0–10% of the initial slope unless an alternate slope is justified by experiment or analysis. Point C shall have an ordinate equal to the strength of the component and an abscissa equal to the deformation at which significant strength degradation begins. Line DE represents the residual strength of the member. Three performance levels, Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP) are considered in this study as specified in guidelines such as FEMA-356 and ATC-40. Description of the expected damage condition of the RC frame staging as par FEMA 356 seismic performance levels has been summarized in Table 6.
Fig. 2

Performance level for pushover analysis

Table 6

Seismic performance levels (based on Table C1-3, FEMA 356)

 

Immediate occupancy (IO)

Life safety (LS)

Collapse prevention (CP)

Overall

Light

Moderate

Severe

Damage

Extensive cracking and hinge formation in ductile elements. Limited cracking and/or splice failure in some non-ductile columns. Severe damage in short columns.

Extensive damage to beams. Spalling of cover and shear cracking (< 1/8″width) for ductile columns. Minor spalling in nonductile columns. Joint cracks < 1/8″ wide.

Minor hairline cracking.

Limited yielding possible at a few locations. No crushing (strains below 0.003).

Lateral drift limit

1% transient; negligible

permanent

1% transient;

2% permanent

1% transient;

4% permanent

Figure 3 shows the modeling of a typical tank for the pushover analysis. In pushover analysis, first a ‘gravity push’ was applied with full dead load, convective mass, impulsive mass and 25% of live load. The pushover analysis is load controlled up to the first yielding and thereafter it is displacement controlled. The node associated with the center of mass of container is used as target node to apply the push loading. From the pushover analysis, the base shear (V) versus roof displacement (Δroof) curve of the structure, usually called static pushover curve, is obtained.
Fig. 3

Finite element model of study tanks for pushover analysis

The definition of response reduction factor ‘R’ is integrated to the selected performance limit state of the structure. The Indian standard IS: 1893 does not describe the limit state corresponding to which values of ‘R’ factor are suggested in this code. In this study, two performance limits are considered for the computation of ‘R’ of study water tanks. From past two decades, the concept of performance-based seismic design has come into practice. Performance-based provisions describe different performance limits for various types of members (i.e., ATC-40 (1996), FEMA 356). The performance limits can be classified as global/structural limits and local/element limits. The global limits are generally based on vertical load capacity, lateral load capacity and lateral drift of structure. In this study, the Performance Limit 1 or PL1 corresponds to the life safety limit state of RC frame member defined in FEMA-356. The PL1 is defined at the member level in terms of the allowable plastic hinge rotation at member ends. Tables 7 and 8 provide the local deformation limits specified by FEMA-356 in terms of allowable plastic hinge rotations of beam and column element, respectively, in a RC special moment-resisting frame (SMRF) element at different performance levels for flexural failures of an element. The Performance Limit 2 or PL2 is defined as the point corresponding to the maximum base shear on the force–displacement relationship curve of structures or in terms of the maximum drift ratio of structure, whichever attained earlier. The limit state PL2 is defined at the structural level in terms of the ultimate capacity of the structural system. The various performance levels in terms of the maximum drift ratio specified in ATC-40 are shown in Table 9. The global performance limit PL2 is defined by a maximum drift ratio of 0.33Vi/Pi at structural stability. For all the water tanks considered in this study, having a maximum base shear (Vi) is nearly 6% of the total seismic weight of tank. Therefore, the allowable drift ratio corresponding to structural stability is roughly equal to 0.02, which is similar or very close to limit corresponding to life safety.
Table 7

Plastic rotation limits for RC beams as per FEMA 356

\(\frac{p - p'}{{p_{bal} }}\)

Trans reinf

\(\frac{v}{{b_{w} d\surd f_{c} }}\)

Acceptance criteria

IO

LS

CP

≤ 0.0

C

≤ 3

0.010

0.02

0.025

≤ 0.0

C

≤ 6

0.005

0.01

0.02

Table 8

Plastic rotation limits for RC columns as per FEMA 356

\(\frac{p - p'}{{p_{\text{bal}} }}\)

Trans reinf

\(\frac{v}{{b_{w} d\surd f_{c} }}\)

Acceptance criteria

IO

LS

CP

≤ 0.1

C

≤ 3

0.005

0.015

0.020

≤ 0.1

C

≥ 6

0.005

0.012

0.016

≥ 0.4

C

≤ 3

0.003

0.012

0.015

≥ 0.4

C

≥ 6

0.003

0.010

0.012

Table 9

Deformation limits for different performance level

Performance level

 

Immediate occupancy

Damage control

Life safety

Structural stability

Maximum drift ratio

0.01

0.01–0.02

0.02

0.33Vi/Pi

The performance limit PL1 is defined at the member/local level in terms of the maximum plastic rotation at the ends of RC frame member. This limit state is monitored continuously at each step of the pushover analysis, and the performance point PL1 is noted when the plastic rotation of any beam or column member reaches to limit states defined as given in Tables 7 and 8, respectively. The second performance limit state PL2 is defined at the global/structural level as the point on the pushover curve corresponding to the maximum base shear or point corresponding to the maximum drift ratio of 0.02, whichever attained earlier. For all considered water tanks, the maximum base shear capacity of structures on pushover curve reached earlier than the maximum drift ratio of 0.02 during NSPA. Therefore, the performance limit state PL2 is considered corresponding to the maximum base shear on pushover/capacity curves in this study. The results of pushover analysis show that the beams of RC frame staging enter into inelastic range before the columns. Generally, design of staging using IS 13920, i.e., capacity design concept leads to strong column weak beam design. The behavior of frame staging elevated water tank supporting a large convective and impulsive mass at the top is completely different than SMRF typically used in building, because in building the mass of the structure is distributed at different story levels.

Computation of ‘R’ factor for the study water tanks

Figures 4, 5, 6 and 7 show the pushover/capacity curves considering the P–∆ effects and their bilinear representations (dotted lines) for study elevated water tanks having a capacity of 140, 480, 1000 and 2200 m3 supported on different soil conditions, respectively. To obtain useful information from the capacity curve, it is approximated by an idealized bilinear relationship using equal energy method. This method assumes that the area enclosed by the curve above the bilinear approximation is equal to the area enclosed by the curve below the bilinear approximation. The yielding of structures is evaluated according to the bilinear representations of the pushover curve. As mentioned earlier, two performance limits PL1 and PL2 are considered for the computation of ‘R’ factor.
Fig. 4

a Pushover curve for 140 m3 tank with fixed base. b Pushover curve for 140 m3 tank on hard soil. c Pushover curve for 140 m3 tank on medium soil. d Pushover curve for 140 m3 tank on soft soil

Fig. 5

a Pushover curve for 480 m3 tank with fixed base. b Pushover curve for 480 m3 tank on hard soil. c Pushover curve for 480 m3 tank on medium soil. d Pushover curve for 480 m3 tank on soft soil

Fig. 6

a Pushover curve for 1000 m3 tank with fixed based. b Pushover curve for 1000 m3 tank on hard soil. c Pushover curve for 1000 m3 tank on medium soil. d Pushover curve for 1000 m3 tank on soft soil

Fig. 7

a Pushover curve for 2200 m3 tank with fixed base. b Pushover curve for 2200 m3 tank on hard soil. c Pushover curve for 2200 m3 on medium soil. d Pushover curve for 2200 m3 tank on soft soil

The parameters such as the maximum base shear (Vo), ultimate displacement (∆u), yield base shear (Vd) and yield displacement (∆y) required for the evaluation of response reduction factor are obtained from the pushover curves or from its bilinear approximation. The components of ‘R’ such as ductility ratio (µ), ductility factor (Rµ), and overstrength factor (Rs) for different water tanks are derived from these parameters corresponding to PL1 and are shown in Tables 10, 11, 12 and 13, respectively. An exact estimation of the impulsive and convective time period is necessary for the assessment of design base shear and realistic ‘R’ factor of a water tank.
Table 10

Component-wise evaluation of ‘R’ based on PL1 for 140 m3 tank (Zone-III)

Type of soil

Ti (sec) (impulsive)

V 0

(kN)

V d

(kN)

Rs = V0/Vd

Δm

(mm)

Δy

(mm)

µ = Δmy

R μ

R R

R

Fixed

1.22

340

70

4.85

211

92

2.29

2.67

1

12.94

Hard

1.31

336

66

5.09

211

103

2.05

2.34

1

11.93

Medium

1.36

336

84

4.01

209

107

1.97

2.23

1

8.92

Soft

1.64

335

85

3.94

211

110

1.91

2.29

1

9.02

Table 11

Component-wise evaluation of ‘R’ based on PL1 for 480 m3 tank (Zone-III)

Type of soil

Ti (sec) (impulsive)

V 0

(kN)

V d

(kN)

Rs = V0/Vd

Δm

(mm)

Δy

(mm)

µ = Δmy

R R

R

Fixed

1.22

663

175

3.78

189

83

2.27

2.61

1

9.86

Hard

1.24

659

173

3.80

190

86

2.20

2.53

1

9.61

Medium

1.25

657

230

2.85

186

92

2.02

2.34

1

6.66

Soft

1.28

653

273

2.39

186

94

1.97

2.35

1

5.61

Table 12

Component-wise evaluation of ‘R’ based on PL1 for 1000 m3 tank (Zone-V)

Type of soil

Ti (sec) (impulsive)

V 0

(kN)

V d

(kN)

Rs = V0/Vd

Δm

(mm)

Δy

(mm)

µ = Δmy

R μ

R R

R

Fixed

1.21

2000

987

2.02

176

83

2.12

2.42

1

4.88

Hard

1.25

1996

954

2.09

179

87

2.05

2.35

1

4.91

Medium

1.29

1996

1232

1.62

180

90

2.00

2.29

1

3.70

Soft

1.35

1988

1446

1.37

194

102

1.90

2.27

1

3.10

Table 13

Component-wise evaluation of ‘R’ based on PL1 for 2200 m3 tank (Zone-III)

Type of soil

Ti (sec) (impulsive)

V 0

(kN)

V d

(kN)

Rs = V0/Vd

Δm

(mm)

Δy

(mm)

µ = Δmy

R μ

R R

R

Fixed

1.55

2505

659

2.79

218

110

1.81

2.05

1

7.79

Hard

1.57

2505

650

3.85

224

120

1.70

1.91

1

7.35

Medium

1.61

2500

835

2.99

235

124

1.70

1.82

1

5.44

Soft

1.70

2485

966

2.57

258

148

1.50

1.75

1

4.49

The ‘R’ values corresponding to PL1 for the tanks resting on medium soil range from 3.17 to 8.92 for the four tanks considered, and are all higher than the IITK-GSDM guidelines/Draft IS 1893 (Part-2) specified value of R (= 2.5) for ductile/special moment resistance RC frame staging water tanks. The ‘R’ values corresponding to PL1 for 140, 480 and 2200 m3 water tanks supported on different soil types range from 4.49 to 11.93, and are all higher than the IS 1893 (Part-2) 2014 specified value of R (= 4) for ductile/special moment resistance RC frame staging water tanks. It is worth to mention that all the studied water tanks are constructed before the year 2014, and therefore, in this study it is assumed that it was designed using IITK-GSDMA guidelines/Draft IS 1893 (Part-2). This indicates that all the study water tanks were designed with higher safety margin than that of specified in Indian standards. It is worth to mention that the strength factor in a water tanks depends on various factors, such as the safety margins specified in the code, partial safety factors for loads and material strength. Furthermore, in same design code, strength factor becomes subjective to the individual designer’s selection of cross-sectional dimension depending on the demand, because the section and the percentage of reinforcement provided for a member are never exactly as per the demand requirements. For example, the same section will be provided for all the columns, although the design requirement usually varies for these. Additionally, the reinforcements provided are typically slightly more than the required due to the availability of discrete rebar sizes at site. This conservative decision imparted through a designer’s choice adds to Rs.

The pushover curves reflect that for all tanks, PL2 is reached after PL1 (i.e., at a larger displacement). According to pushover curves and their bilinearisation, maximum shear is almost the same as that of PL1. Since design shear do not change, reserve strength factor is also the same as in PL1. There is no change in yield displacement values for PL2 as compared to PL1. Ultimate displacement values for PL2 are larger as compared to PL1 values, and therefore, the ductility values are higher for PL2 as compared to PL1. Among the various components of ‘R’ presented in Tables 14, 15, 16 and 17, reserve strength factor Rs remains the same as in PL1, while ductility factor Rµ comes out to be higher, which finally results in higher ‘R’ factors overall. For PL2, ‘R’ ranges from 3.30 to 13.53 for all studied water tanks. Based on PL2, the IS 1893(Part-2): 2002 recommendation is on the conservative side. It should, however, be noted that this limit does not consider any member level behavior such as maximum plastic rotation at the ends. However, The actual value of ‘R’ of study water tanks needed to be even lower than what is evaluated in this study, because of many reasons, such as irregularity in dimensions leading to torsional effects, poor quality control and construction practice, not following ductile detailing requirements exactly as per the standards, and deterioration in concrete with time. If soil flexibility is not taken into account in estimating ‘R’ factor of elevated water tank properly, the accuracy in evaluating seismic base shear and assessing the structural safety for a structure subjected to earthquakes could not be reliable.

The effect of soil flexibility considerably increases the impulsive time period of the elevated water tank, indicating that modification in soil stiffness could have considerable effect on the fundamental period of vibration. It is observed that flexibility of supporting soil has considerable effect on base shear, ductility factor and response reduction factor of water tank. Considering soil flexibility, particularly when the tank supported on relatively soft soil is crucial. The increase in the flexibility of the soil reduces the overstrength factor as well as ductility factor of study water tanks. The consideration of soil flexibility in analysis reduces the values of ‘R’ factor of study water tanks as compared to fixed base condition. The overstrength factor as well as ductility factor reduce with increase in the size of the study water tank. Consideration of flexibility of soft and medium soil in analysis reduces the values of ‘R’ factor as much as 25 and 40% for the considered tanks, respectively, as compared to fixed base condition. The effect of the soil flexibility is the least in case of hard soil.

The response reduction factors of 140, 480 and 2200 m3 water tanks are also evaluated for the higher seismically intensive zone-IV to check its suitability for that zone. Tables 14, 15, and 16 show the response reduction factor and its components for 140, 480 and 2200 m3 water tanks, respectively, for seismic zone-III and -IV. Table 17 shows the response reduction factor and its components for 1000 m3 water tanks for seismic zone-V for PL2. The ‘R’ factors of these three tanks for different supporting soil conditions range from 3 to 8.7 and 3.57 to 9.09 for PL1 and PL2, respectively, for seismic zone-IV. All the evaluated values of ‘R’ factor are higher than the IIITK-GSDMA and Draft IS 1893 (Part-II) guidelines suggested value of R (= 2.5) for ductile/special moment resistance RC frame staging water tanks, indicating that these three water tanks were designed with very high reserve strength than that of specified in Indian standards and found to be safe for even higher seismic zone-IV.
Table 14

Component of ‘R’ for 140 m3 based on PL1 and PL2 for seismic zone-III and IV

140 m3 capacity (zone-III)

140 m3 capacity (zone-IV)

Type of soil

R s

R R

PL1

PL2

R s

R r

PL1

PL2

R µ

R

R µ

R

R µ

R

R µ

R

Fixed

4.85

1

2.67

12.94

2.79

13.53

3.26

1

2.67

8.70

2.79

9.09

Hard

5.1

1

2.34

11.93

2.58

13.15

3.42

1

2.34

8.0

2.58

8.82

Medium

4.01

1

2.23

8.92

2.55

10.19

2.68

1

2.23

5.97

2.55

6.83

Soft

3.94

1

2.29

9.02

2.53

9.96

2.63

1

2.29

6.02

2.53

6.65

Table 15

Component of ‘R’ for 480 m3 based on PL1 and PL2 for seismic zone-III and IV

480 m3 capacity (zone-III)

480 m3 capacity (zone-IV)

Type of soil

R s

iR

PL1

PL2

R s

R R

PL1

PL2

R µ

R

R µ

R

R µ

R

R µ

R

Fixed

3.78

1

2.61

9.86

2.96

11.18

2.52

1

2.61

6.57

2.96

7.45

Hard

3.8

1

2.53

9.61

2.76

10.48

2.53

1

2.53

6.41

2.76

6.98

Medium

2.85

1

2.34

6.66

2.50

7.12

1.90

1

2.34

4.46

2.50

4.75

Soft

2.39

1

2.35

5.61

2.46

5.87

1.59

1

2.35

3.75

2.46

3.91

Table 16

Component of ‘R’ for 2200 m3 based on PL1 and PL2 for seismic zone-III and IV

2200 m3 capacity (zone-III)

2200 m3 capacity (zone- IV)

Type of soil

R s

R R

PL1

PL2

Rs

RR

PL1

PL2

R µ

R

R µ

R

R µ

R

R µ

R

Fixed

3.79

1

2.05

7.79

2.27

8.60

2.54

1

2.05

5.20

2.27

5.76

Hard

3.85

1

1.91

7.35

2.11

8.12

2.57

1

1.91

4.92

2.11

5.42

Medium

2.99

1

1.82

5.44

2.04

6.11

1.99

1

1.82

3.64

2.04

4.05

Soft

2.57

1

1.75

4.49

2.09

5.37

1.71

1

1.75

3.00

2.09

3.57

Table 17

Component of ‘R’ for 1000 m3 based on PL1 and PL2 for seismic zone-V

1000 m3 capacity (zone-V)

Type of soil

R s

R R

PL1

PL2

R µ

R

R µ

R

Fixed

2.02

1

2.42

4.88

2.62

5.30

Hard

2.09

1

2.35

4.91

2.50

5.230

Medium

1.62

1

2.29

3.70

2.50

4.07

Soft

1.37

1

2.27

3.1

2.40

3.30

Figure 8 shows the effect of soil condition on the ductility factor of study tanks. The ductility ratio as well as the ductility factor decreases with increase of the soil flexibility from hard to soft soil due to increase in the flexible behavior of water tank. In case of 2200 m3 water tank, consideration of flexibility of medium and soft soil in analysis reduces ductility factor as much as 11 and 14% as compared to fixed base condition. For all the studied water tanks, consideration of soil flexibility in computation of ‘R’ factor reduces the ductility factor as much as 14% as compared to the ideal fixed base condition.
Fig. 8

Ductility factor of study water tank for different soil conditions

Conclusions

In this study, the ‘R’ factors of four existing RC-elevated water tanks designed using IS 456, Draft IS 1893 *Part-II) and detailed as per IS 13920 are evaluated considering soil flexibility. The focus has been given in methodical assessment of the R factor, consideration of soil flexibility; realistic performance-based limit states at both structure and member levels. The schematic procedure for evaluating the realistic values of response reduction factor of RC staging elevated water tank incorporating the effect of soil flexibility, is also presented. The use of realistic values of response reduction factor may prove useful for safe and economic seismic design of elevated water tank. The significant observations of the present study are summarized as follows:

  • The R values corresponding to PL1 for the tanks resting on medium soil range from 3.17 to 8.92 for the four tanks considered, and all are higher compared to the IITK-GSDM guidelines/Draft IS 1893 (Part-II) suggested value of R (= 2.5) for ductile/special moment resistance RC frame staging water tanks. This indicates that all the studied water tanks were designed with higher safety margin.

  • Based on PL2, R factor ranges from 3.30 to 13.53, indicating the IITK-GSDM guidelines recommendation is on the conservative side.

  • As the soil spring stiffness decrease from hard to soft soil, the impulsive time period of elevated water tank model increases and response reduction factor decreases. This means that the impulsive time period and response reduction factor is also a function of soil flexibility.

  • The overstrength factor as well as ductility factor reduce with the increase of the size of the studied water tanks.

  • The flexibility of supporting soil has considerable effect on displacement ductility and response reduction factor of water tanks. Consideration of flexibility of medium and soft soil in analysis reduces the values of ‘R’ factor as much as 25 and 40% for the considered tanks, respectively, as compared to fixed base condition. The effect of the soil flexibility is the least in case of hard soil. The analysis of water tank with fixed base assumption may lead to underestimation or overestimation of seismic base shear of elevated tanks with any staging configurations.

  • Consideration of flexibility of medium and soft soil during analysis increases yield and ultimate displacement response demands compared to that of fixed base model Consideration of flexibility of medium and soft soil during analysis reduces the displacement ductility ratio (µ) of water tank models as much as 12% and 16% for the considered tanks, respectively, as compared to fixed base condition.

  • The actual value of ‘R’ of study water tanks needed to be even smaller than what is evaluated in this study, because of various reasons, such as poor quality control and workmanship during the construction, irregularity in dimensions leading to torsional effects, and not following the ductile detailing requirements exactly as per the standards which would lead to deterioration in concrete with time.

Notes

Acknowledgements

The authors acknowledge the financial support from a project sanctioned by Gujarat Council on Science and Technology, Government of Gujarat, Gujarat, India (Order No. GUJCOST/MRP/15-16/1905) for successful completion of this work.

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Authors and Affiliations

  1. 1.Sardar Vallabhbhai Patel Institute of TechnologyGujaratIndia

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