IS 1893:2002 CRITERIA FOR EARTHQUAKE RESISTANT DESIGN OF STRUCTURES

PART1 GENERAL PROVISIONS AND BUILDINGS

The Code is now split into five parts

Part 1 - General provisions and buildings

Part 2 - Liquid retaining tanks - Elevated and ground supported
Part 3 - Bridges and retaining walls
Part 4 - Industrial structures including stack like structures
Part 5 - Dams and embankments

Part 1 contains provisions that are general in nature and applicable to all
structures. Also, it contains provisions that are specific to buildings only.

The important changes as compared to IS:1893-1984 are as follows:

1. Seismic zone map is revised with only four seismic zones. Zone I is
upgraded to Zone II. Killari area is enhanced to Zone III. Bellary isolated
zone
is removed. East coast is enhanced to Zone III and connected with Zone III of
Godavari Graben area.

2. Seismic zone factor is changed reflecting more realistic value of peak

ground acceleration.
3. Response acceleration spectra are now specified for three types of
founding strata viz. Hard, Medium and Soft.

4. The empirical formula for calculating fundamental natural period T=0.1n

for moment resisting frames without bracing or shear walls is replaced with
Ta=0.075h0.075 for RC framed buildings. This formula applies to bare frames
e.g. in industrial plant buildings. The formula for framed buildings with
in-filled masonry walls is Ta = 0.09h/d0.5 where h and d are the height and
base dimension of the building along the considered direction of earthquake.

5. Revised procedure first calculates the actual force that may be

experienced by the structure during the probable maximum earthquake, if it
were
to remain elastic. Then response reduction due to ductile deformation or
frictional energy dissipation in the cracks is applied via `response reduction
factor' R in place of the earlier performance factor K. The list of building
systems and the corresponding values of R is more exhaustive.

The code procedures for calculating base shear VB are summarized below:

IS:1893-1984
------------

VB = K.C.αh.W

where

K = Performance factor 1.0 for SMRF (IS:4326 detail) and 1.6 for OMRF (IS:456
detail)

C = Fundamental time period dependant coefficient

= 1.0 for T <= 0.35 sec and 0.5/T(2/3) for T > 0.35 sec

αh = β.I.α0

β = Soil-foundation system dependant coefficient

= 1.2 for isolated footings without tie beams in medium soils, piles in soft
soils, combined or isolated footings with tie beams in soft soils
= 1.5 isolated footings without tie beams in soft soils

I = Importance factor
= 1.5 for hospitals, schools, cinema halls, monumental structures, telephone
exchanges, radio, fire, railway power stations and 1.0 for others

α0 = Zone dependant design seismic coefficient

Zone II III IV V
α0 0.02 0.04 0.05 0.08

W = Seismic weight of building

= Dead load + appropriate amount of live load
= Dead load + 25% for LL up to 3 kN/sq.m
50% for LL > 3 kN/sq.m
0% for LL on roof
IS:1893-2002
------------

VB = Ah.W

[Z/2].[Sa/g]
Ah = ------------
[R/I]

Z = Zero period acceleration value for the Maximum Considered Earthquake

Zone II III IV V
Z 0.10 0.16 0.24 0.36

Sa/g = Spectral acceleration coefficient for Hard, Medium or Soft soil, 5%

damping
= 2.5 for T <= 0.40 and 1.00/T for T > 0.40 (Hard: GP,GW,SP,SW,SC with
N>30)
= 2.5 for T <= 0.55 and 1.36/T for T > 0.55 (Medium: All with 10<N<30 SP
with N>15)
= 2.5 for T <= 0.67 and 1.67/T for T > 0.67 (Soft: All except SP with
N<10)

I = Importance factor
= 1.5 for hospitals, schools, cinema halls, monumental structures,
telephone
exchanges, television, radio, fire, railway power stations and 1.0 for
others
R = Response reduction factor

Ordinary RC Moment Resisting Frame (OMRF) 3

Special RC Moment Resisting Frame (SMRF) 5
Ordinary RC Shear Walls 3
Ductile RC Shear Walls 3

Dual Systems with frames carrying >25% of VB

Ordinary RC Shear Walls with OMRF 3

Ordinary RC Shear Walls with OMRF 4
Ductile Shear Walls with OMRF 4.5
Ductile Shear Walls with SMRF 5

W = Seismic weight of building

= Dead load + appropriate amount of live load
= Dead load + 25% for LL up to 3 kN/sq.m
50% for LL > 3 kN/sq.m
0% for LL on roof

Comparative Values of Maximum Base Shear

----------------------------------------

IS:1893-1984 IS:1893-2002 Percent Increase

Zone VB(SMRF) VB(OMRF) VB(SMRF) VB(OMRF) SMRF OMRF

II 0.02W 0.032W 0.025W 0.042W 25.0 31.3
III 0.04W 0.064W 0.040W 0.067W 0.0 4.7
IV 0.05W 0.080W 0.060W 0.100W 20.0 25.0
V 0.08W 0.128W 0.090W 0.150W 12.5 17.2

Elimination of C results in higher force up to 11%, 25% and 43% for hard,
medium and soft soils respectively in the peak region and elimination of beta
results in lower force for soft soils to the extent of 20%.

6. Accidental torsion is introduced:

edi = 1.5 esi + 0.05 bi or

= esi - 0.05 bi whichever governs

edi = Design eccentricity at floor i

esi = Static eccentricity at floor i defined as the distance between center

of mass and center of rigidity

bi = Plan dimension of floor i perpendicular to the direction of force

7. Definition and treatment of irregularities is elaborated:

In Plan
-------
i) Torsional : If floor diaphragms are rigid in their own plane and maximum
storey drift at one end is > 1.2*average storey drift

ii) Re-entrant corners : if projection beyond re-entrant corner is > 15% of

plan dimension in that direction

iii) Diaphragm discontinuity: if open areas > 50% of gross enclosed area or
change in effective diaphragm stiffness from one storey to next > 50%

iv) Out-of-plane offsets: discontinuities in lateral load resisting paths

v) Non-parallel systems

In Elevation
------------
i) Soft-storey: Lateral stiffness < 70% of that in in the storey above or
< 80% of the average lateral stiffness of three storeys above

ii) Mass: seismic weight of any storey except roof < 200% of adjacent storeys

iii) Geometic: horizontal dimension of a lateral force resisting element >

150% of that in adjacent storey

iv) In plane discontinuity: In plane offset of a lateral force resisting

element > length of that element

v) Weak-storey having lateral strength < 80% of that in the storey above

Additional requirements for some of the irregularities are specified:

Soft Storey:
The columns and beams of the soft storey are to be designed for 2.5 times the
storey shears and moments calculated under seismic loads besides the columns
designed and detailed for the calculated storey shears and moments, shear
walls
placed symmetrical in both directions of the building as far away from the
center
of the building as feasbible to be designed exclusively for 1.5 times the
lateral
storey shear force calculated as before.

Non-Parallel Systems:
Earthquake effects about the two orthogonal axes must be combined:

a. Ex ± 0.3Ey
b. Ey ± 0.3Ex

8. More load combinations are required:

Basic Combination Expanded Combinations

1) 1.5(D+L)

2) 1.2(D+L±E)
1.2(D+L+EXP), 1.2(D+L+EXN), 1.2(D+L-EXP), 1.2(D+L-EXN),
1.2(D+L+EYP), 1.2(D+L+EYN), 1.2(D+L-EYP), 1.2(D+L-EYN)

3) 1.5(D±E)
1.5(D+EXP), 1.5(D+EXN), 1.5(D-EXP), 1.5(D-EXN),
1.5(D+EYP), 1.5(D+EYN), 1.5(D-EYP), 1.5(D-EYN)

4) 0.9D±1.5E
0.9D+1.5EXP, 0.9D+1.5EXN, 0.9D-1.5EXP, 0.9D-1.5EXN,
0.9D+1.5EYP, 0.9D+1.5EYN, 0.9D-1.5EYP, 0.9D-1.5EYN

where

D : Dead
L : Live
E : Earthquake
X : along X axis
Y : along Y axis
P : Positive eccentricity
N : Negative eccentricity

This results in 25 actual combinations but can be reduced rationally.

Limits of static analysis

-------------------------
Zone II Zone III Zone IV Zone V
Regular buildings <90 m <90 m <40 m <40 m
Irregular buildings <40 m <40 m <12 m <12 m

Example
The data for this example is as follows:

Storeys 2 Concrete grade 25 Seismic zone 3

Foundation depth mm 1500 Slab cover mm 25 Frame type OMRF
Ground storey height mm 3000 Beam cover mm 50 Soil type
Medium
Typical storey height mm 3000 Column cover mm 50 Importance factor 1.0

----------------------------------------------------------------------
Column Data Beam Data Slab Data
----------------------------------------------------------------------
Mark Cx Cy Mark bw D Load Mark t Loads
mm mm mm mm kN/m mm kN/sq.m
Dead Live
----------------------------------------------------------------------
C1 300 300 B1 300 500 6.25 S1 150 1.25 5.00
B1R 300 500 2.25 S1R 150 3.25 1.00
B1G 300 400 6.25
----------------------------------------------------------------------
Loads (kN):
Roof Floor Ground
----------------------- ---------------------- ----------------------
Slab 0.15*25*25 = 93.75 0.15*25*25 = 93.75 = 0
Finish 3.25*25 = 81.25 1.25*25 = 31.25 = 0
Beams 0.3*0.5*25*20 = 75 0.3*0.5*25*20 = 75 0.3*0.4*25*20 = 60
Walls 2.25*20 = 45 6.25*20 = 125 6.25*20 = 125
Columns 0.3*0.3*3*25*4= 27 0.3*0.3*3*25*4= 27 0.3*0.3*1.5*25*4 =
13.5
(below) --- ---
-----
Dead 322 352
198.5

Live 1*25 = 25 5*25 = 125 = 0

Analysis & Design of Floor Slab A1B2 :

Dead load (0.15*25+1.25) = 5 kN/m2

Live load = 5 kN/m2
Factored Load 1.5*5+1.5*5 = 15 kN/m2

Aspect ratio = 5.0 / 5.0 = 1.0

M+ = 0.056*15*52 = 21.0 kN-m/m

As+ = 504 sq.mm/m
Mu+ = 0.87*415*504*(150-25)*(1-(504*415/1000*125*25)/106 = 21.2 kN-m/m (OK)

Gravity Analysis & Design of Beam A1B1 :

Direct dead UDL on beam = (75+125)/4 = 50.00 kN

Triangular dead load from slab = 125/4 = 31.25 kN
Triangular live load from slab = 125/4 = 31.25 kN
Total = 112.50 kN

Simply supported moment :

Dead = 50*5/8 + 31.25*5/6 = 57.29 kN-m
Live = 31.25*5/6 = 26.04 kN-m
Total = 83.33 kN-m

Fixed end moment :

Dead = 50*5/12 + 31.25*5*5/48 = 37.11 kN-m
Live = 31.25*5*5/48 = 16.28 kN-m
Total = 53.39 kN-m

Factored fixed end moment = 1.5*37.11 + 1.5*16.28 = 80.08 kN-m

Substitute Frame Properties:

Gross properties of sections will be used in the analysis. For beams,

effective
width of flange is taken as lo/12+bw+3*Df where lo is the distance between
points
of contraflexure, assumed as 0.7 times beam span and Df is the flange depth.

Gross Moment of Inertia of Beams:

Overall depth = 500 mm

Web width = 300 mm
Flange depth = 150 mm
Flange width = 0.7*5000/12 + 300 + 3*150 = 1042 mm

Depth of CG from top = 300*500*250 + (1042-300)*150*75

------------------------------- = 176 mm
300*500 + (1042-300)*150

Icg =
0.3*0.53/12 + 0.3*0.5*(0.25-0.176)2 + (1.042-0.3)*0.153/12 +
(1.042-0.3)*0.15*(0.176 -0.075)2 = 0.00529 m4

Gross Moment of Inertia of Columns = 0.3*0.33/12 = 0.000675 m4

Moments in upper and lower columns of a single bay symmetrical frame are

MF * KU
Mc = -------------------------
(KU+KL+0.5*KB/2)

where:

MF = Fixed end beam moment = 80.08 kN-m

KU = I/L of upper column = 0.000675 / 3 = 0.000225 m3
KL = I/L of lower column = 0.000675 / 3 = 0.000225 m3
KB = I/L of beam = 0.005290 / 5 = 0.001058 m3

80.08 * 0.000225
Mc = ------------------------- = 18.4 kN-m
(2*0.000225+0.001058/2)

Factored shear in columns = 18.4*2/3 = 12.27 kN

Factored moment at beam ends = 2*18.4 = 36.80 kN-

m

Factored moment at mid-span = 1.5*83.33 - 36.8 = 88.20 kN-m

Factored shear at beam ends = 1.5*112.5 / 2 = 84.38 kN

Reinforcement in beam:

Longitudinal steel

As(min) = (0.85/415)*300*(500-50) = 277 sq.mm

Mu(min) = 0.87*415*277*450*(1-277*415/300*450*25)/1000000 = 43.47 kN-m

Required top steel at beam supports < 277 sq.mm

As(mid) = 553 sq.mm

Mu(mid) = 0.87*415*553*450*(1-553*415/1042*450*25)/1000000 = 88.1 kN-m (OK)

Stirrups

Minimum Asv/sv = 0.4*300/415 = 0.289

Shear capacity of concrete = 0.36*300*450/1000 = 48.60 kN

Shear capacity of stirrups = 0.87*415*0.289*450/1000 = 46.95 kN
Total shear capacity = 95.55 kN < 84.38 kN
(OK)
Required sv/Asv > 1/0.289 = 3.46

Seismic Analysis

Seismic Weight

Roof Floor Ground

------------ ------------- --------------
Slab+finishes 175.0 125.0 0.0
Beams 75.0 75.0 60.0
Walls above 45.0 125/2 = 62.5 125/2 = 62.5
Walls below 125/2 = 62.5 125/2 = 62.5 0.0
Columns above 0.0 27/2 = 13.5 27/2 = 13.5
Columns below 27/2 = 13.5 27/2 = 13.5 13.5/2 = 6.75
Live 0.0 0.5*125 = 62.5 0.0
---- ----- ------
wi (kN) 371.0 414.5 142.75

Time Period Ta

h = 7.5 m
d = 5.0 m

Ta = 0.09 * h / sqrt(d)
= 0.09 * 7.5 / sqrt(5) = 0.302 second

Design horizontal seismic coefficient Ah

ZISa/g 0.16*1.0*2.5
Ah = ------ = ------------ = 0.0667
2R 2*3.0

where

Z Zone factor = 0.16

I Importance factor = 1.0
R Response reduction factor = 3.0 for OMRF
Sa/g Response acceleration coefficient = 2.5

Base Shear VB

VB = Ah * Σwi = 0.0667*(371.0 + 414.5 + 142.75) = 61.91 kN

Vertical Distribution of Base Shear Qi

Level wi hi wi*hi2 Qi
4 371.00 7.5 20868.8 61.91*20868.8/29583.6 = 43.67 kN
3 414.50 4.5 8393.6 61.91* 8393.6/29583.6 = 17.57 kN
2 142.75 1.5 321.2 61.91* 321.2/29583.6 = 0.67 kN
1 0.00 0.0 0.0 0.00 kN
-------- -----
29583.6 61.91 kN
 Seismic Loads and Bending Moments

Storey Shear Column Moment Beam Shear Column Axial

kN kN-m kN kN

2 43.67 43.67*3/2*4=16.38 16.38*2/5 = 6.55 =

6.55
1 61.24 61.24*3/2*4=22.97 (16.38+22.97)*2/5 = 15.74 (6.55+15.74) =
22.29

Effect of Eccentricity in Plan

Due to biaxial symmetry of loads and layout, center of mass and shear center
are
located at (2.5,2.5) meters with respect to origin at lower left corner. Thus

xcm = ycm = xsc = ysc = 2.5 m

Static eccentricity esi

Frames along x-axis esiy = (xcm-xsc) = (2.5-2.5) = 0.00 m
Frames along y-axis esix = (ycm-ysc) = (2.5-2.5) = 0.00 m

Accidental eccentricity .05bi

Frames along x-axis = 0.05*biy = 0.05 * 5 = 0.25 m
Frames along y-axis = 0.05*bix = 0.05 * 5 = 0.25 m

Design eccentricity edi

Frames along x-axis = ediy = esiy+0.05*biy = 0 + 0.25 = 0.25 m
Frames along y-axis = edix = esix+0.05*bix = 0 + 0.25 = 0.25 m

Radius of Gyration of Strength

rk = (ΣVxi,j*(ysci-yi,j)2/ΣVxi,j + ΣVyi,j*(xsci-xi,j)2/ΣVyi,j)1/2

where
Vxi,j/ΣVxi,j=fraction of storey shear along x direction at level i in column
j=1/4
Vyi,j/ΣVyi,j=fraction of storey shear along y direction at level i in column
j=1/4
xi,j =x ordinate of j-th column at level i = 0 and 5 m
yi,j =y ordinate of j-th column at level i = 0 and 5 m

rk2 = 2*(2.5-0)2/4 + 2*(2.5-5)2/4 + 2*(2.5-0)2/4 + 2*(2.5-5)2/4 = 12.5 m2

Magnification Factor at all levels

Frames along x-axis = δxi = 1 + ediy*yi,k/rk2 = 1 + 2.5*0.25/12.5 = 1.05
Frames along y-axis = δyi = 1 + edix*xi,k/rk2 = 1 + 2.5*0.25/12.5 = 1.05

Analysis and Design of Columns of First Storey for D + L + Ex

Unfactored Axial Load in Columns

Due to dead loads (322+352)/4 = 168.50 kN
Due to live loads ( 25+125)/4 = 37.50 kN
Due to earthquake 22.29 kN

Unfactored Moment in Columns

Due to dead loads 18.4*37.11/80.08 = 8.53 kN-m
Due to live loads 18.4*16.28/80.08 = 3.74 kN-m
Due to earthquake 22.97 kN-m

Unfactored Shear in Columns

Due to dead loads 2*8.53/3 = 5.69 kN
Due to live loads 2*3.74/3 = 2.49 kN
Due to earthquake 61.24/4 = 15.31 kN

Pu = 1.2 * (168.5 + 37.5 + 22.29*1.05) = 275.29 kN

Mux = 1.2 * (8.53 + 3.74) = 14.72 kN-m
Muy = 1.2 * (8.53 + 3.74 + 22.97*1.05) = 36.39 kN-m
Vux = 1.2 * (5.69 + 2.49 + 61.24*1.05/4) = 24.10 kN
Vuy = 1.2 * (5.69 + 2.49) = 9.82 kN

Check for As = 1080 sq.mm

p 1080 * 100
--- = -------------- = 0.048
fck 300 * 300 * 25

Puz = 0.45fc.b.D + 0.75fy.As

= 0.45*25*300*300 + 0.75*415*1080 = 1348650 N

Pu 275290
--- = -------- = 0.204
Puz 1348650

αn = 1 + (.204 - .2) / 0.6 = 1.0067

Refer Chart 45 of SP-16

Pu 275290
------- = ---------- = 0.122
fck.b.D 25*300*300

Mux1
-------- = 0.09
fck.b.D2

Mux1 = Muy1 = 0.09*25*300*3002 = 60750000 N-mm

(Mux/Mux1)αn+(Muy/Muy1)αn < 1

(14.72/60.75)1.0067 + (36.39/60.75)1.0067 = 0.241 + 0.598 = 0.837 < 1 (OK)

Column-beam Intersection at top of first storey:

Unfactored Moments at Beam Ends

Due to dead loads 36.8*121.875/(168.75*1.5) = 17.72 kN-m

Due to live loads 36.8*46.875/(168.75*1.5) = 6.82 kN-m
Due to earthquake 16.38 + 22.97 = 39.35 kN-m

Combinations for moment

DL+E -1.2*17.72 - 1.2*6.82 - 1.2*1.05*39.35 = -79.03 kN-m

DL-E -1.2*17.72 - 1.2*6.82 + 1.2*1.05*39.35 = 20.13 kN-m
D+E -1.5*17.72 - 1.5*1.05*39.35 = -88.56 kN-m
D-E -0.9*17.72 + 1.5*1.05*39.35 = 46.03 kN-m

Unfactored Shears at Beam Ends

Due to dead loads 81.25/2 = 40.63 kN

Due to live loads 31.25/2 = 15.63 kN
Due to earthquake 2*39.35/5 = 15.74 kN

Combinations for shear

DL+E 1.2*40.63 + 1.2*15.63 + 1.2*1.05*15.74 = 87.34 kN

E 1.5*1.05*15.74 = 24.79 kN

Design for hogging moment = -88.56 kN-m

As = 513 sq.mm
Mu = 0.87*415*513*450*(1-513*415/300*450*25)/1000000 = 78.1 kN-m

Design for sagging moment = 46.03 kN-m

As = 277 sq.mm
Mu = 0.87*415*277*450*(1-277*415/1042*450*25)/1000000 = 44.6 kN-m (OK)

Minimum Shear capacity of beam = 95.55 kN (OK)

TORSIONAL PROVISIONS IN IS:1893(2002)

New clauses were introduced in the revised Indian seismic code for torsion of
symmetric as well as asymmetric buildings with rigid diaphragms. The treatment
of torsional provisions is elaborated here along with a solved example.
Clause 7.9.2 of IS:1893(2000) reads as follows:

The design eccentricity, edi to be used at floor i shall be taken as:

edi = 1.5esi + 0.05bi or

= esi - 0.05bi

whichever of these gives the more severe effect in the shear of any frame

where

esi = Static eccentricity at floor i defined as the distance between centre of

mass
and centre of rigidity

bi = Floor plan dimension of floor i, perpendicular to the direction of force

Under seismic loads, structures experience lateral forces acting, in general,

at
a design eccentricity edi with respect to a neutral point, such that
deflections on
the side towards edi are higher than those on the other side of the neutral
point.
The sides towards and away from edi are known as the flexible and stiff sides
respectively.

Multiplier 1.5 on esi in the first equation is the dynamic amplification

factor to
account for possible coupling of torsional and lateral modes of vibration and
depends on the ratio of frequencies in the two modes. When frequencies in the
two
modes are far apart, dynamic amplification factor is 1.0 as in the second
equation.

Accidental eccentricity due to possible variations of live load, stiffness and

ground motion along the width of building is given by 0.05bi. Obviously, this
factor
can take positive or negative value.

Two cases are possible:

1. Lateral-torsional mode coupling occurs and accidental eccentricity is in

the
same direction as the static eccentricity which is reflected by the first
equation.
In general, this is the governing case for members on the flexible side.

2. Lateral-torsional mode coupling does not occur and the static eccentricity
is in
the direction opposite to the static eccentricity which is what the second
part of
the equation implies. In general, this is the governing case for members on
the
stiff side.

Seismic force acting at the code specified design eccentricity results in

torques
at various floor levels. There are two approaches to account for this effect.

1. Floor Torques about Centers of Rigidity:

Static eccentricity is defined as the distance between the center of mass and
the
center of rigidity at a given floor. Centers of rigidity are points on each
floor of
a multistoreyed building such that lateral loads applied through them do not
cause
rotation of any of the floors [1].

In order to locate centers of rigidity, the following procedure is adopted:

1. The structural models constrained to deflect only in the direction of

applied
seismic loads along x and y axes are analyzed.

2. Free body diagram of each floor is taken along with storey shears vi+1,j
and
vi,j above and below that floor respectively where subscript i refers to
storey and
subscript j refers to shear resisting element of that storey.

3. The point of intersection of resultants of net storey shears (vi,j -

vi+1,j)
along the orthogonal axes is the center of rigidity for the storey.

A pair of design eccentricities and the resulting floor torques at each storey
can
now be calculated. These floor torques are applied to a three-dimensional
frame
model taking due care of the fact that 3D frame analysis accounts for static
eccentricity 1.0 esi automatically.

2. Storey Torsion about Shear Center:

Static eccentricity is defined as the distance between the center of

cumulative mass
from roof down to the level under consideration and shear center at that
level.

In order to locate shear center, the following procedure is adopted:

1. The structural models constrained to deflect only in the direction of

applied
seismic loads along x and y axes are analyzed.

2. Free body diagram of the substructure from roof down to the level being
considered
is taken along with shears Vxi,j and Vyi,j at the cut where subscript i refers
to level and subscript j refers to shear resisting element at that level.

3. The point of intersection of resultants of shears Vxi,j and Vyi,j defines

shear
center at that level.
A pair of design eccentricities at the level under consideration can now be
calculated.

Analogous to the quantity (1+6e/d) commonly used to calculate maximum pressure

in
an eccentrically loaded footing, magnification factors on all forces and
moments
obtained in step 1 above are given by:

δxi = 1 + ediy*yi,k/rk2
δyi = 1 + edix*xi,k/rk2

where

δxi = Magnification factor for frames in x direction at level i

δyi = Magnification factor for frames in y direction at level i

Vxi,j = Shear along x direction at level i in column j

Vyi,j = Shear along y direction at level i in column j

edix = Maximum additive design eccentricity at level i along x axis

ediy = Maximum additive design eccentricity at level i along y axis

xsci = x ordinate of shear center at level i = ΣVyi,j*xi,j / ΣVyi,j

ysci = y ordinate of shear center at level i = ΣVxi,j*yi,j / ΣVxi,j

xi,j = x ordinate of j-th column at level i

yi,j = y ordinate of j-th column at level i

rk = Radius of gyration of stiffness

= (ΣVxi,j*(ysci-yi,j)2/ΣVxi,j + ΣVyi,j*(xsci-xi,j)2/ΣVyi,j)1/2

It must be emphasized that, in general, location of shear center is different

from
center of rigidity. Tso [2] has shown equivalence of the two procedures given
above.

The second approach is computationally simpler and will be used to illustrate

the
effect of torsion on the two storeyed building shown below consisting of
ground,
first floor and roof levels.
The data for this example is as follows:

----------------------------------------------------------------------
Column Data Beam Data Slab Data
----------------------------------------------------------------------
Mark Cx Cy Mark bw D Wall Mark t Loads
mm mm mm mm Load mm Dead Live
kN/m kN/sq.m
----------------------------------------------------------------------
C1 300 300 B1 300 500 6.25 S1 150 1.25 5.00
B1R 300 500 2.25 S1R 150 3.25 1.00
B1G 300 400 6.25
----------------------------------------------------------------------

Here, suffixes R and G on beam and slab marks refer to roof and ground levels
respectively. Storey heights are 3.0 m and foundation depth is 1.5 m below
ground.

Seismic Weight wi

Seismic weight at a particular level consists of:

- Dead loads of slab and beams including finishes at the level

- Proportional dead loads of walls and columns above and below
- Appropriate amount of live loads at the level as per code

The following table gives seismic weights at various levels along with x
ordinate
of center of gravity, xcg measured from lower left corner which will be used
later
for calculating center of mass. Due to symmetry in vertical direction, ycg is
located
at 2.5 m for all of the components. Density of reinforced concrete is taken as
25 kN/m3 for this example.

Roof Floor Ground xcg

m
---------------- ----------------- ----------------
-----
Slab+finishes 280.00 200.00 0.00
4.000
Beams 116.25 116.25 93.00
4.161
Walls above 69.75 193.75/2 = 96.88 193.75/2 = 96.88
4.161
Walls below 193.75/2 = 96.88 193.75/2 = 96.88 0.00
4.161
Columns above 0.00 40.50/2 = 20.25 40.50/2 = 20.25
4.333
Columns below 40.50/2 = 20.25 40.50/2 = 20.25 20.25/2 = 10.13
4.333
Live 0.00 0.5*200 = 100.00 0.00
4.000
------ ------
------
wi (kN) 583.13 650.51 220.26

Center of Mass

Centers of mass at various levels with respect to origin at lower left corner
are
calculated as :

xcm = Σwj*xcgj/Σwj
ycm = Σwj*ycgj/Σwj

where subscript j refers to each component of seismic weight from roof

downwards
to the level under consideration.

xcm at Roof Level

=(280*4+(116.25+69.75+96.88)*4.161+20.25*4.333)/583.13
=4.090 m

xcm at First Floor Level

=(4.090*583.13+200*4+(116.25+2*96.88)*4.161+2*20.25*4.333+100*4)/
(583.13+650.51)
=4.094 m

xcm at Ground Level

=((583.13+650.51)*4.094+(93+96.88)*4.161+(20.25+10.13)*4.333)/
(583.13+650.51+220.26)
=4.108 m

ycm = 2.5 m at all levels

Seismic Analyses on Constrained Models

In lieu of analyses, it will be assumed here that all columns share storey
shears
equally at all levels.

Shear Center

Following the assumption of storey shears being shared equally by all columns,
values of Vxi,j/ΣVxi,j and Vyi,j/ΣVyi,j are all equal to 1/6. Thus at all
levels,

xsc = ΣVyi,j*xi,j/ΣVyi,j = 2*(0+5+8)/6 = 4.333 m

ysc = ΣVxi,j*yi,j/ΣVxi,j = 3*(0+5)/6 = 2.500 m

Static Eccentricity esi

The distance between center of mass and shear center gives static eccentricity
along each of the axes at various levels.

Along x-axis: esix = xcm - xsc

Roof : 4.090 - 4.333 = -0.243 m
Floor : 4.094 - 4.333 = -0.239 m
Ground : 4.108 - 4.333 = -0.225 m

Along y-axis: esiy = ycm - ysc = 2.5 - 2.5 = 0 m (at all levels)

Accidental Eccentricity bi

Along x-axis = 0.05*bix = 0.05 * 8 = 0.40 m

Along y-axis = 0.05*biy = 0.05 * 5 = 0.25 m

Maximum Design Eccentricity edimax

Algebraic addition of static and accidental eccentricities gives maximum value

of
design eccentricity.

Along x-axis = 1.5*esix + 0.05*bix

Roof : -1.5*0.243 - 0.40 = -0.765 m

Floor : -1.5*0.239 - 0.40 = -0.759 m
Ground : -1.5*0.225 - 0.40 = -0.738 m

Along y-axis = 1.5*esiy + 0.05*biy = 0 + 0.25 = 0.25 m

Minimum Design Eccentricity edimin

Algebraic subtraction of accidental eccentricity from static eccentricity

gives
minimum value of design eccentricity.

Along x-axis = ediminx = esix - 0.05*bix

Roof : -0.243 + 0.40 = 0.157 m

Floor : -0.239 + 0.40 = 0.161 m
Ground : -0.225 + 0.40 = 0.175 m
Along y-axis = ediminy = esiy - 0.05*biy = 0 - 0.25 = -0.25 m

Radius of Gyration of Strength

rk = (ΣVxi,j*(ysci-yi,j)2/ΣVxi,j + ΣVyi,j*(xsci-xi,j)2/ΣVyi,j)1/2

The quantities Vxi,j/ΣVxi,j and Vyi,j/ΣVyi,j being equal to 1/6 at all levels,
rk2 = {3*(2.5-0)2+3*(2.5-5)2+2*(4.333-0)2+2*(4.333-5)2+2*(4.333-8)2}/6 = 17.14 m2

Magnification Factors

δxi = 1 + ediy*yi,k/rk2

Frame along grid 1

Roof : 1+(0-4.333)*(-0.765)/17.14 = 1.193 or 1+(0-4.333)*0.157/17.14=0.960
Floor : 1+(0-4.333)*(-0.759)/17.14 = 1.192 or 1+(0-4.333)*0.161/17.14=0.959
Ground : 1+(0-4.333)*(-0.738)/17.14 = 1.186 or 1+(0-4.333)*0.175/17.14=0.956
Maximum: 1.193

Frame along grid 2

Roof : 1+(5-4.333)*(-0.765)/17.14 = 0.970 or 1+(5-4.333)*0.157/17.14=1.006
Floor : 1+(5-4.333)*(-0.759)/17.14 = 0.971 or 1+(5-4.333)*0.161/17.14=1.006
Ground : 1+(5-4.333)*(-0.738)/17.14 = 0.971 or 1+(5-4.333)*0.175/17.14=1.007
Maximum: 1.007

Frame along grid 3

Roof : 1+(8-4.333)*(-0.765)/17.14 = 0.836 or 1+(8-4.333)*0.157/17.14=1.034
Floor : 1+(8-4.333)*(-0.759)/17.14 = 0.838 or 1+(8-4.333)*0.161/17.14=1.034
Ground : 1+(8-4.333)*(-0.738)/17.14 = 0.842 or 1+(8-4.333)*0.175/17.14=1.038
Maximum: 1.038

As expected, frame along grid 1 which is farthest from shear center and on the
flexible side experiences maximum magnification factor.

δyi = 1 + edix*xi,k/rk2
= 1 + 0.25*2.5/17.14 = 1.037 for frames along A and B at all levels.

Storey-wise magnification factors or, conservatively, the maximum value of

magnification factor for each frame in x and y direction is applied to all
actions
found from seismic analysis of constrained model.

REFERENCES

1. Jain SK and Murty SVR, "Proposed Changes in Indian Seismic Code,IS:1893

(Part 1) 2002". IITK-GSDMA Project on Building Codes.

2. Tso WK, "Static Eccentricity Concept for Torsional Moment Estimations".

Journal of Structural Engineering, ASCE, Vol. 116, No. 5 pp 1199-1212.