SHEAR WALL

References :

1. Hand book of Concrete Engineering by Lintel

2. Advanced Reinforced Concrete Design by P C Varghese
 Introduction
 Shear walls are concrete walls specially designed in
bldg. to resist lateral forces that are produced in
plane of wall due to Wind, Earthquake etc.

 Shear walls are generally provided in tall bldg. to

avoid total collapse of bldg. under lateral forces.
They are usually provided between columns, in stair-
wells, lift well, toilet ,utility shafts,etc.
Important properties of shear wall :

1. Good ductility under reversible / repeated

over loads.

2. Less bending tensile stresses due to lateral

loads.

3. Located symmetrical to avoid torsional

stresses.

4. Stiffness of shear wall is high in its own plane.

Typical arrangement of shear wall in building
• There is no limitation concern to the
geometrical shape of shear wall systems.

• The triangle, rectangle, angle, channel

and wide flanges are the common types
of geometrical forms.
 Types of Shear Walls
• Simple Rectangular Shear Wall

• Flanged Shear Wall

• Framed Shear Wall (with / without infilled wall)

• Coupled Shear Wall

• Column Supported Shear Wall

• Core Type Shear wall

 Simple shear wall

Shear wall with Flanges

Bar Bell type

Coupled shear wall

Column supported
Types of shear wall

Core type
Behavior of Shear Wall under Lateral Loads

• It is assumed that floors are infinitely stiff in

their plane and do not deformed.

• Deformation of frame is in Shear mode.

• Deformation of shear wall is in bending mode

rather than shear mode.
Deformation shape of shear wall under lateral loading
Deformation shape of Frame under lateral loading
Frame & Shear Wall Interaction
• Shear Wall
 
: Vertical systems cantilever from the ground.
 
• Braced frames are like trusses, walls act like deep beams.
 
 

   
• Buildings that carry gravity loads using bearing walls,
typically also use the walls as shear walls.
 
The walls must be design to serve both duties.
    

Reactions:    
Lateral Loads Only Gravity Only Lateral + Gravity

• The lateral loads induce two types of motion:

tipping and sliding.
• Tipping is rotation and sliding is translation.
• Vertical reactions
    counteract tipping.
• The reactions form a moment, resisting the rotation.

• Note the downward tension reaction.

• The gravity  loads also counteract tipping.
• The wall can be viewed as a pre-stressed beam.
• The pre-stressing effect of gravity is generally beneficial.
(since it is usually costly to make foundations that can resist tension uplift)

• The combined compression of gravity plus overturning can be

very high.
• The lateral forces create an overturning moment, while the
gravity loads create a resisting moment.
• To avoid uplift forces on the foundation, the resisting moment
must be larger than the overturning moment.
• The resisting moment typically accounts only for dead loads.
• Horizontal reactions counteract sliding.
• The sum of the reactions for the wall are called the wall's
base shear
Vertical Normal Stress:
Lateral Loads Only Gravity Only Lateral + Gravity

• Gravity increases compression stresses and reduce

tension stresses.
• For gravity and lateral acting together, the distribution
of stresses is asymmetric, with most of the wall acting in
compression.
• The compressive effect of gravity increases the
compression and decreases the tension.
Shear Stress: Lateral Loads Only

The stresses increases moving down the wall

1. Simple Rectangle and Bar Bell type free standing Shear-Walls

Simple rectangular wall Bar bell type wall

* Subjected to bending and shear under action of vertical and horizontal

shear along its length.

Bar Bell Type walls : min. steel is put over inner 0.7 to 0.8 length L and
remaining steel is placed at end for a length 0.15 to 0.12 L on either side.
These are stronger and more ductile than the simple rectangular type.

Disadvantage

* During earthquake attract & dissipates a lot of energy by cracking, which

is difficult to repair.
 2. Coupled Shear Wall

coupling beams

Two structural walls are joined together by relatively short spandrel

beams to increase the stiffness of wall & structure dissipates energy
by yielding the coupling beams.
The walls should satisfy the following requirement
(a) The system should develop hinges only in coupling beam before
shear failure.
(b) The coupling beam should be designed to have good energy
dissipation characteristics.
Action of coupling beam is as shown, Refer Fig.1
The beam will bend in double curvature due to displacement & shear will
reduce the axial force.
If MP = Magnitude of plastic moment . Then,
N = Total reaction is given by
2M P
N   no.hinges. formed
Length.of .beam

Fig.1 : Action of coupled shear walls as energy dissipation device

(a) external forces & reactions of str. (b) action of coupling beams.
The diagonal steel (as shown in fig.2) is provided because even a large
amount of transverse steel for ductility is not effective.

Refer Cl.9.5 of IS:13920-1993 for design criterion.

 3. Rigid Frames with Shear Walls
Fig.3 Shows interaction of simple shear walls and rigid frames of a tall bldg.
In shear mode, frame & wall will deflect in bending mode. The interaction
reduces max.moment but max. shear increases, which will increase tendency
of shear failure.

Fig.3

a) action between frame & wall b) shears in wall c) moments in wall

4. Framed Shear Walls and Infilled Shear Walls

Framed walls are casted monolithically
Infilled walls are constructed by casting frames first and
infilling it with masonry or conc. block later.
 5. Column supported shear walls
This type of walls are constructed when ,
Shear walls are to be discontinued at floor level.

6. Core Type of Shear Walls

In some bldg. to withstand lateral loads, vertical core is provided
this can be sometimes elevators and other service areas. This type have
good resistance against torsion.
Classification according to behavior :

(a)Shear- shear walls –In which deflection and strength are

controlled by shear. Thus are usually low rise walls.

(b)Ordinary Moment walls- In which deflection and strength

are controlled by flexure. Thus are usually high rise shear
walls.

(c)Ductile Moment shear walls- These are special walls

meant for seismic region.
LOADS ON SHEAR WALLS
Centre of rigidity and Centre of Mass

1. Lateral Stiffness ( K) : It is defined as force required (applied at top of shear

wall) to produce unit lateral displacement.

2. Centre of Rigidity : It is defined as point on the horizontal plane through which

the lateral load should pass in order that there will be no rigid body rotation.
It’s coordinates are given by eqn.
K i x i K i y i
xr  and yr 
K i K i

3. Centre of Mass (i.e. C.G.)

Further, as lateral forces (due to earthquake) is proportional to mass,
 mi x i mi y i
x and y 
m i m i

Two cases arises from combination of centre of stiffness & centre of mass :
1. If both coincide, no torsion.
2. If do not coincide, twisting moment produced.
Principle of Shear Wall Analysis
Assumptions :
1. All horizontal loads are taken by various shear walls and not by frames.
2. Where there is no torsion, load is taken by each shear wall in proportion to its
stiffness as below,
EL = F1 + F2 + F3 - - - - - -
Where, EL – Earthquake Load &
F1,F2,F3 ---- Forces on various shear walls
F1 = K 1 
Where,  ---- displacement at top &
K1 ---- Lateral stiffness of shear wall
Hence, K 1  + K2  + K3  + --- = EL
EL

K1  K 2  K 3    
K1
F 1 = K1   EL
K 1
 STIFFNESS OF WALL
There are three types of deflections to be considered :
W
Stiffness = Force required at top for unit deflection 

WH 3
1] 1 bending  (as cantilever )
3EI
WH
2] 2 shear 
CAG

Where , C = Shape factor ( 0.8 for rectangle )

W = Load applied
E
G  (assume  = 0.22 )
2(1   )
3] 3 is due to foundation rocking (rotation) ( Ref. Fig. next slide )

L1 / 2 L1 / 2
Moment due to
 Bxxdxc  B  dx = (B L3 γ θ / 12)
2
rotation θ
M  x
 L1 / 2  L1 / 2
γ = Modulus of sub-grade reaction
 Let, R = ( B L3) / 12 is the moment produce due to unit rotation of foundation.
Rotation due to moment, WH
WH 
R
Hence, deflection produced = (Rotation )x (H)
WH 2
 3 rocking 
R
Hence, Total  = 1+ 2+ 3
Therefore, Lateral stiffness K =
W
includes bending, shear & rotation

DESIGN OF RECTANGULAR & FLANGED SHEAR WALL
The design-detailing shall be done as per IS 13920 (1993)
General dimensions
1. Thickness of wall ( t ) NOT < 150 mm.
2. For flanged wall ,effective extension of flange width beyond face of web
should be least of the following. (refer following fig.)
a. ½ dist. to a adjacent shear wall
b. 1/10th of total wall height
c. Actual width “L”

Boundary element

Plain shear walls with boundary element.

3.The portion along wall edges specially enlarged & strengthened by longitudinal and
transverse r/f (like column) is called Boundary element.
This should be provided, when comp. stress in extreme fibre exceeds
0.2 fck .& when comp. stress is less than 0.15 fck the boundary element is
discontinued.

(Note : If special confining steel is provided then boundary elements are not required)

Special confining
reinforcement
shall be provided
over the full
height of a
column

Special Confining Reinforcement for Columns under discontinue

Following rules are to be observed for detailing of steel

1. Walls are to be provided with r/f in 2 orthogonal directions.

The min. steel ratios for each of the vertical and horizontal directions should be
> 0.0025
As
   0.0025
Ac ( gross )

2. If factored shear stress (v) exceeds 0.25fck or if the thickness of wall exceeds
200 mm, then r/f should be provided on both faces of wall.

3. Dia. of bar should not exceed 1/10th of thickness of wall.

4. The max. spacing should not exceeds L/5,3t or 450 mm, where L is length of
wall.
REINFORCEMENT FOR SHEAR
Nominal shear stress is calculated as,
Vu
v 
td
Where, d = Effective width ( = 0.8L for rectangular section)
Vu = Factored shear Force
Nominal shear stress v > c max. [ IS : 456 (2000) Table 20]
or > c max = 0.63 fck
• Shear taken by concrete is same as beam shear. ( Table 9 of IS 456 assuming
0.25% steel ) & if necessary increase it’s value by following multiplying factor ‘’
3Pu
  1 (but not more than 1.5)
Ac fck
Where Pu --- Total axial load
 --- Multiplying factor.
Shear capacity of concrete and steel is given by
Vc =  c t d
Vs = Vu – Vc
The steel necessary to resist the shear is determined from following formula -

0.87 fyAs.d Where, Vs = Vu - c x t x d = S.F. resisted by horiz. shear r/f

Vs 
Sv As = area of horiz. shear r/f
Sv = spacing of shear r/f

NOTE : Vertical steel provided in wall for shear should not be less than horiz. steel.

Adequacy of Boundary Element

The max. axial load on Boundary element due to vertical load and moment, is
Mu  Muv
P = Sum of factored Gravity Loads +
c
where, Mu = Factored moment on the whole wall

Muv = Moment of resistance provided by the rectangular wall

( i.e. excluding the boundary element ) [Ref. IS:13920-1993, Appendix A]

c = C / C distance between boundary element

 NOTE :

1. Load factor for gravity loads = 0.8 if gravity loads tend to add to strength of wall.

2. The boundary element is designed as column with vertical steel not less than
0.8% & not greater than 4%.

3.The Bar Bells of shear wall should be provided as shown in fig.

SHEAR WALL
Required Development Splice and Anchorage

The splicing of vertical flexure steel should be avoided as far as

possible in region of flexural-yielding.

Splicing can be extended to a distance of :

1. Length of wall “L” above the base of wall or

2. 1 / 6 height of wall

# If splicing is needed, not more than 1/3 of steel should be spliced at

such a section.
# Splicing of adjacent bars should be staggered a min. of 600 mm.
 Modeling of Shear Walls

• Wide Column Modeling

• Modeling Using Shell Element

Wide column Element for Modeling
of Shear Wall
 Stiffness matrix for shear wall
considering shear deformation
 AE AE 
 L 0 0  0 0 
L
 12 EI 6 EI 12 EI 6 EI 
 0 0  3 
 L3 L2 L L 2

6 EI
1 
 0  4    EI 0
6 EI
 2  4    EI 
K L2 L L L
 1     AE 0 0
AE
0 0


 L L 
 0 12 EI 6 EI 12 EI 6 EI
  0  2 
 L3 L2 L3 L 
 6 EI EI 6 EI EI 
 0 2  0 4   
 L2 L L2 L
Where
12 EI
 2
L GA
K = Global stiffness matrix


G= Shear modulus of elasticity

A= Effective shear area

E= Modulus of elasticity
L= Length of member
I= Moment of Inertia
Modeling of finite size joints
 Stiffness of member with FSJC
T
K F KF
   

Where
1 dl 0 0 
0 1 0 0 
F  
 0 0 1  bl 
 
0 0 0 1 

bl = Length of rigid offset at the left end.

dl =Length of rigid offset at the right end.

Shear Wall Analysis :
•Effect of slab rigidity in Frame analysis.
.

•Effect of mass system on Frame.

.

•Effect of FSJC in Frame analysis.

.

•Effect of slab eccentricity in Frame analysis.

.

•Effect of soil embedment.

.

•Seismic/ Wind analysis of Shear Wall - building.

 Effect of Slab Rigidity
• Analysis of bare frame
• Analysis of frame with slab

Data:
1. Size of frame – 4m x 4m.
2. Height of frame – 3m.
3. Grade of concrete – M20.
4. Thickness of slab – 100mm, 125mm and 150mm.
5. Beam size – 300 x 400 mm.
6.Ten various sizes of column.
 Comparison of deflection in bare
frame and frame with slab
Deflection of frame (mm)
Stiffness
Column
of
Sr. No size Without 100mm 125mm 150mm
Column
(mm) slab Thick slab Thick slab Thick slab
(N/mm)

1 230x300 5140 6.31 6.19 6.13 6.05

2 230x350 8166 4.42 4.31 4.25 4.18
3 230x400 12200 3.31 3.22 3.17 3.11
4 230x450 17300 2.61 2.53 2.48 2.42
5 230x500 23800 2.11 2.05 2.01 1.96
6 230x550 31700 1.75 1.70 1.66 1.63
7 230x600 41100 1.48 1.43 1.40 1.37
8 230x650 52300 1.26 1.22 1.20 1.17
9 230x700 65300 1.08 1.05 1.03 1.01
10 230x750 80300 0.93 0.91 0.98 0.87
 Comparison of deflection in X-dirction for frame with or without slab

9.00E+04

8.00E+04

7.00E+04

6.00E+04
Stiffness(N/mm)

With out slab

5.00E+04
100mm THK. slab
125mm Thk slab
4.00E+04
150mm Thk slab
3.00E+04

2.00E+04

1.00E+04

0.00E+00
0 1 2 3 4 5 6 7
Deflection(mm)
 Comparison of Time period in frame
Time period of frames in (sec) for diff.
Stiffness slab Thickness
Sr. Column size of
no (mm) Column
(N/mm) 100 mm 125mm 150mm

1 230 x 300 5140 0.100 0.110 0.119

2 230 x 350 8166 0.083 0.091 0.101
3 230 x 400 12200 0.072 0.079 0.085
4 230 x 450 17300 0.063 0.069 0.075
5 230 x 500 23800 0.057 0.063 0.067
6 230 x 550 31700 0.052 0.057 0.061
7 230 x 600 41100 0.048 0.052 0.056
8 230 x 650 52300 0.044 0.048 0.052
9 230 x 700 65300 0.041 0.045 0.048
 Time period of frame in x-Direction

90000

80000

70000

60000
Stiffness(N/mm)

50000 100mm Thk slab

125mm Thk slab
40000 150mm Thk slab

30000

20000

10000

0
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13
Time period(sec)
 Conclusions
• Horizontal deflection reduces by increase
in slab thickness. But reduction is very
small.

• Time period increase by increase in slab

thickness .

• Increase in column stiffness slab rigidity

effect reduce.
 Effect of Mass System
• Distributes mass system.
• Lump mass system.
 Comparison of Time periods in
distributed mass system and lump mass
system
Sr. Time period of frame (sec)
No Frame With slab Frame With slab
Column
Frame (lump mass) ( distributed mass)
size
(mm) Without
slab 100mm 125mm 150mm 100mm 125mm 150mm
Thick Thick Thick Thick Thick Thick

1 230x300 0.109 0.147 0.154 0.160 0.147 0.154 0.160

2 230x350 0.093 0.124 0.129 0.133 0.124 0.129 0.133
3 230x400 0.082 0.108 0.112 0.115 0.108 0.112 0.115
4 230x450 0.074 0.096 0.100 0.102 0.096 0.100 0.102
5 230x500 0.068 0.088 0.090 0.092 0.088 0.091 0.093
6 230x550 0.063 0.080 0.083 0.085 0.081 0.083 0.085
7 230x600 0.059 0.075 0.077 0.078 0.075 0.077 0.079
8 230x650 0.055 0.069 0.072 0.073 0.070 0.072 0.073
9 230x700 0.051 0.065 0.067 0.068 0.065 0.067 0.069
10 230x750 0.048 0.061 0.063 0.064 0.061 0.063 0.064
Effect of FSJC
 Comparison of deflection in model
with or without FSJC
Deflection in mm
100mm 125mm 150mm
Sr Column Without slab
Thick. slab Thick slab Thick slab
. size
no (mm)

C.C FSJC C.C FSJC C.C FSJC C.C FSJC

1 230 x 300 6.31 5.61 6.19 5.50 6.13 5.45 6.05 5.37
2 230 x 350 4.42 3.93 4.31 3.91 4.25 3.79 4.18 3.73
3 230 x 400 3.31 2.95 3.22 2.87 3.17 2.82 3.11 2.77
4 230 x 450 2.61 2.32 2.53 2.25 2.48 2.20 2.42 2.15
5 230 x 500 2.11 1.87 2.05 1.81 2.01 1.78 1.96 1.74
6 230 x 550 1.75 1.55 1.70 1.49 1.66 1.47 1.63 1.43
7 230 x 600 1.48 1.29 1.43 1.26 1.40 1.24 1.37 1.21
8 230 x 650 1.26 1.10 1.22 1.07 1.20 1.05 1.17 1.03
9 230 x 700 1.08 0.95 1.05 0.92 1.03 0.90 1.01 0.89
 Comparison of deflection for frame without slab

90000

80000

70000

60000
Stiffness(N/mm)

50000
C.C model
FSJC model
40000

30000

20000

10000

0
0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00
Deflection(mm)
Comparison of time period in model
with or without FSJC
Time period (Sec)
100mm Thick 125mm Thick 150mm Thick
Stiffnes
Column slab slab slab
Sr. s of
size
no Column
(mm) C.C FSJC C.C FSJC C.C FSJC
(N/mm)
model model model model model model

1 230 x 300 5140 0.100 0.091 0.110 0.101 0.119 0.110

2 230 x 350 8170 0.083 0.076 0.091 0.084 0.101 0.091
3 230 x 400 12200 0.072 0.066 0.079 0.073 0.085 0.078
4 230 x 450 17300 0.063 0.058 0.070 0.064 0.075 0.069
5 230 x 500 23800 0.057 0.052 0.063 0.058 0.067 0.062
6 230 x 550 31700 0.052 0.048 0.057 0.052 0.061 0.056
7 230 x 600 41100 0.048 0.044 0.052 0.048 0.056 0.052
8 230 x 650 52300 0.044 0.040 0.048 0.044 0.052 0.048
9 230 x 700 65300 0.041 0.037 0.045 0.041 0.048 0.045
 Comparison of Time period in Frame with
100mm thick slab
90000

80000

70000

60000
C.C model
Stiffness(N/mm)

50000 FSJC model

40000

30000

20000

10000

0
0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11
Time period(sec)
 Observations
• The B.M in the column can be reduced
by 10 % with accounting the FSJC.

• The deflection of frame can be reduced

by 10 to 15% due to increase in stiffness
of frame.

• From the results obtained from analysis,

time period of the structure reduces due
to increase in the stiffness of frame.
 Effect of Eccentricity

• Generally in modeling members are

connected center to center .

• The model in software does behave like

the actual structure.

• There are some problems related to the

eccentricity of the members.
 At the time of modeling following point
should take in account:
• Eccentricity of slab.

• Eccentricity of column.

• Eccentricity of beam.

• Finite size joint correction

Various types of eccentricities
• These eccentricity affects the
1) Natural frequency.
2) Deflection.
3) B.M and S.F.
4) Torsion.

• Normally designers do not account these

effects.
 C / C model:
• In C / C modeling the center of slab and center
of beam are same.
• Beam behave as rectangular section.
 Model taking slab eccentricity:
• Beam behave as flange section.

• Rigid links are used in the modeling

• Eccentricity change with depth of slab and
depth of beam .

Eccentricity (e) = (Db-Ds)/2

Where,
Ds- Depth of slab.
Db- Depth of beam

• Length of rigid link is equal to eccentricity.

Mathematical model of eccentric slab
 Comparison of Deflection in Models with
and without eccentricity
Deflection of Model in mm
100mm Thick 125mm Thick 150mm Thick
Stiffness
Column slab slab slab
Sr. of
size
no Column
(mm) C.C Ecce. C.C Ecce. C.C Ecce.
(N/mm)
model model model model model model

1 230 x 300 5140 6.19 6.26 6.13 6.23 6.05 6.19

2 230 x 350 8166 4.31 4.38 4.25 4.35 4.18 4.31
3 230 x 400 12200 3.22 3.28 3.17 3.25 3.11 3.22
4 230 x 450 17300 2.53 2.58 2.48 2.55 2.42 2.52
5 230 x 500 23800 2.05 2.09 2.01 2.07 1.96 2.04
6 230 x 550 31700 1.70 1.73 1.66 1.72 1.63 1.70
7 230 x 600 41100 1.43 1.46 1.40 1.45 1.37 1.43
8 230 x 650 52300 1.22 1.24 1.20 1.23 1.17 1.22
9 230 x 700 65300 1.05 1.07 1.03 1.06 1.01 1.05
 Comparison of deflection in model of 100
mm thick slab with and without slab
eccentricity
90000

80000

70000

60000
Stiffness(N/mm)

50000
C/C model
Slab eccentricite model
40000

30000

20000

10000

0
0 1 2 3 4 5 6 7
Deflection(mm)
 Comparison of Time Period in model with
and without eccentricity
100mm Thick slab 125mm Thick slab 150mm Thick slab
Stiffness
Column size of
(mm) Column
Sr. (N/mm) C.C Ecce. C.C Ecce. C.C Ecce.
no model model model model model model
1 230 x 300 5140 0.0995 0.1004 0.1100 0.1118 0.1194 0.1223
2 230 x 350 8170 0.0830 0.0840 0.0914 0.0934 0.0989 0.1021
3 230 x 400 12200 0.0717 0.0727 0.0787 0.0808 0.0848 0.0882
4 230 x 450 17300 0.0634 0.0644 0.0695 0.0715 0.0745 0.0781
5 230 x 500 23800 0.0571 0.0580 0.0625 0.0644 0.0671 0.0703
6 230 x 550 31700 0.0520 0.0528 0.0569 0.0587 0.0610 0.0640
7 230 x 600 41100 0.0477 0.0485 0.0523 0.0538 0.0560 0.0588
8 230 x 650 52300 0.0441 0.0447 0.0483 0.0497 0.0518 0.0542
9 230 x 700 65300 0.0409 0.0414 0.0448 0.0461 0.0481 0.0503
10 230 x 750 80300 0.0380 0.0385 0.0418 0.0428 0.0449 0.0468
 Comparison of Time period in model of 100mm
thick slab with and without slab eccentricity
90000

80000

70000

60000
Stiffness(N/mm)

50000 C/C model

40000 Model considering slab

eccentricity
30000

20000

10000

0
0 0.02 0.04 0.06 0.08 0.1 0.12
Tim e period(sec)
 Observations
• Deflection of the frame for the lateral load
increases by considering eccentricity.

• Deflection in beam for vertical load is reduces,

because Moment of inertia of beam increases
due to the flange section.

• Time period of model also increases, because

the mass is lumped at some higher level than
normal model.

• There are no considerable changes in the B.M

for horizontal load.
 Spring Modeling

Idealized spring model for rigid Footing

• Spring model is more realistic.

• It depends upon soil condition at the site.

• It also changes with respect to water

table, size of footing, depth of footing.

• There are special methods of calculating

the spring stiffness .
 Different Approaches of spring
modeling:
• FEMA-273

• FEMA-356

• ATC-40
 FEMA- 273 approach:
Parameters requires for calculate spring stiffness.

• Type of soil.

• Size of footing.

• Depth of footing.
 Soil classifications:

Designation SBC of soil Shear modulus (G)

Soil type
kN/m2 kN/m2
S1 3240 5330000
S2 1640 3700000 Hard soil
S3 880 533000
S4 440 55600 Medium
S5 245 37000 soil
S6 150 4630
S7 100 2710 Soft soil
S8 50 742
 Spring constants for footing in
FEMA 273

k   k o
Where
k 0 = Stiffness coefficient for the equivalent
circular footing

 = Foundation shape correction factor.

 = Embedment factor.
Stiffness coefficient for the equivalent
circular footing :
Displacement degree of freedom k0

Vertical translation 4GR

1 
Horizontal translation 8GR
2 
Torsion rotation 16GR 3
3

Rocking rotation 8GR 3

3(1   )
Rectangular footing:
 Equivalent radius of footing :

Degree of freedom

Translation Rocking Torsion

About About About z-

x-axis y-axis axis
1 1 1
1
 BL  2  BL

3


4 B L

3

4

 BL B  L
2 2
  4

Equivalent    
    3   3   6 
radius, R
Graph shows Shape factor for
footing
Graph shows Embedment factor for
footing
 FEMA-356 Approach

K emb   .K sur
Where

Kemb= Spring constant at specific depth

Ksur=Spring constant at Surface.

β = Embedment factor.
Spring constant at surface:

GB  L
0.65

K X , sur  3.4   1.2
2     B  

GB  L
0.65
L 
K Y , sur  3.4   0.4  0.8
2    B  B 

GB  L
0.75

K Z , sur  1.55   0.8
1    B 
 GB 3  L 
K XX , sur  0.4   0.1
1    B  

GB 3   
L
2.4

K YY , sur  0.47   0.034
1   B 

  
L
2.45

K ZZ , sur  GB 0.53   0.51
3

 B 

Where
G = shear modulus of soil.
B = Width of footing.
L =Length of footing.
ν = Poisson ratio
Correction factor for embedment:

 x  1  0.21
D  
 1  1.6
hd  B  L  
0.4

 2  
 B    BL  

Y   x
 2


 z  1 
1 D

B  

 d  B  L   
3
 2 2.6  1 0.32 
 21 B  L    BL  
 

d  2d  d 
0.2
B
 xx  1  2.5 1    
B  B D L 
 d 
0. 6
     
d
1.9
d
0.6

 yy  1  1.4  1.5  3.7    

L   L   D  

0.9
 B  d 
 zz  1  2.61   
 L  B 

Where
D = Depth of foundation up to bottom of footing.

d = Thickness of footing.

h = Depth of foundation up to the center of footing.

 ATC-40 Approach

K emb  e.K sur

Where

Kemb= Spring constant at specific depth

Ksur=Spring constant at Surface.

e = Embedment factor.
Spring constant at surface:
GL  B 
0.85
GL   B 
K X , sur   2  2.5    0.11  L 
2    L   0.75     

GL  B 
0.85

K Y , sur  2  2.5  
2     L  

GL  B 
0.75

K Z , sur  0.73  1.54  

1     L  
 0.25
G 0.75  L   B
K xx ,sur  Ix    2.4  0.5 
1  B  L

G 0.75
 L 
0.15

K yy , sur  I Y 3  
1    B  

Where
G = shear modulus of soil.
B = Width of footing.
L =Length of footing.
 = Poisson ratio.
 Correction factor for embedment:
  d  
0.4

 0. 5 
   D  16 L  B  d  
 2D    2 
e x  1  0.15   1  0.52  2
 
  L     LB  
   
   

  d  
0.4

 0.5 
   D  16 L  B  d  
 2D    2 
e y  1  0.15   1  0.52 2
 
  B     BL  
   
   

   2 L  2 B   
0.67
 D B  
e z  1  0.095 1  1.3  1  0.2  d 
 B L    LB  
 d  2d  d 
0.20
B 
0.50

e xx  1  2.52 1      
B  B D  L  

 2d 
0.60
 1.9
 2d   d 
0.60

e yy  1  0.92  1.5      
 L    L  D 
Winkler Spring model
 Comparison of spring model
approaches
Comprising Parameters:

• Size of The Footing.

• Depth of The Foundation.

• Thickness of Footing.
 Size of The Footing
Data:

• Depth of the foundation = 2 m.

• Thickness of the footing = 0.5m.

• Shear modulus of soil = 4630 kN/m2

• Poisson’s ratio = 0.35.

Variation of Kx with respect to area of
footing.
Area FEMA-356 FEMA-273
ATC-40 (kN/m)
(m2) (kN/m) (kN/m)
1.00 48783.72 50247.51 21425.45
2.25 57196.46 58866.51 32138.18
4.00 65280.10 67144.26 42850.91
6.25 73138.56 75188.95 53563.63
9.00 80832.49 83063.53 64228.85
12.25 88400.21 90807.73 74933.66
16.00 95867.54 98448.26 85638.47
20.25 103252.79 106004.07 96343.28
25.00 110569.42 113489.10 107048.09
30.25 117827.71 120913.94 117752.90
36.00 125035.64 128286.85 128552.72
42.25 132199.60 135614.42 139265.45
49.00 139324.75 142902.00 149978.17
 Graph show variation of Kx with respect
to the area of the footing.
Variation of stiffness in spring (Kx)

160000.00

140000.00

120000.00
STIFFNESS(kN/m)

100000.00 ATC-40
FEMA-356
80000.00 FEMA-273

60000.00

40000.00

20000.00

0.00
0.00 10.00 20.00 30.00 40.00 50.00 60.00
AREA(sq.m)
Variation of Kxx with respect to area of
footing. Area ATC-40 FEMA-356 FEMA-273
(m2) (kN/m) (kN/m) (kN/m)
1.00 12567.58 13887.81 6862.13
2.25 27886.38 30848.54 23159.70
4.00 52432.43 58048.70 54897.07
6.25 88608.65 98159.43 107220.85
9.00 138817.98 153851.89 185277.63
12.25 205463.35 227797.24 294214.01
16.00 290947.68 322666.62 439176.60
20.25 397673.91 441131.20 625311.99
25.00 528044.96 585862.12 857766.79
30.25 684463.76 759530.55 1141687.60
36.00 869333.24 964807.62 1482221.02
42.25 1085056.33 1204364.51 1884513.65
49.00 1334035.96 1480872.36 2353712.08
 Graph show variation of Kxx with
respect to the area of the footing.
ATC-40
Variation of stiffness in spring (Kxx) FEMA-356
FEMA-273
2500000.00

2000000.00
STIFFNESS(kN/m)

1500000.00

1000000.00

500000.00

0.00
0.00 10.00 20.00 30.00 40.00 50.00 60.00
AREA(sq.m)
 Depth of the foundation:

Data:

• Size of the footing = 2m x 2m.

• Thickness of the footing = 0.5m.

• Shear modulus of soil = 55600 kN/m2

• Poisson’s ratio = 0.35.

 Variation of spring stiffness with respect to depth in FEMA-273

1200000

1000000

800000 1m
Stiffness (kN/m)

1.5m
2m
600000 2.5m
3m
3.5m
400000 4m

200000

0
T-x T-y T-z R-x R-y Rz
 Variation of spring stiffness with respect to depth in FEMA-356

1400000

1200000

1000000
1m
1.5m
Stiffness (kN/m)

800000
2m
2.5m
600000 3m
3.5m
4m
400000

200000

0
T-x T-y T-z R-x R-y Rz
 Variation of spring stiffness with respect to depth in ATC-40

1200000

1000000

800000 1m
Stiffness (kN/m)

1.5m
2m
600000 2.5m
3m
3.5m
400000 4m

200000

0
T-x T-y T-z R-x R-y
 Variation of Kx with depth

1200000

1000000

800000
Stiffness (kN/m)

ATC-40
600000 FEMA-356
FEMA-273

400000

200000

0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Depth (m)
 Effect of soil embedment

Options to model the support condition

at the base:
• Model taking fixity at top.

• Model taking fixity at bottom.

• Hinge supported model.

• Hinge and roller model.

• Spring model.

• FEM model of Footing.

 Data:
• Size of frame – 4m x 4m.

• Height of frame – 4.5m

• Grade of concrete – M20.

• Thickness of slab – 125mm.

• Ten various sizes of column.

• Size of beam -230mm x 400mm

• Size of footing – 1500 x1000mm

Model taking fixity at the bottom of the footing
Model taking fixity at the top of the Footing

Model Taking Hinge at Base

Hinge and Roller Model

Spring Model
Mathematical model of Footing
spring constant for footing in different
types of soil

Spring stiffness for the various

Degree of freedom types of soil (knew/m)

Hard Medium Soft

Translation along x-axis 54435876 567849 47287

Translation along y-axis 56371286 588038 48968
Translation along z-axis 38668592 403372 33590
Rocking about x-axis 20212944 210852 17558
Rocking about y-axis 28588871 298225 24834
Torsion about z-axis 34373160 358564 29859
 Deflection of frame without plinth beam
for different support condition
Deflection of frame in X-direction for various support condition (mm)
Column
Sr. size Spring Spring Spring FEM FEM FEM
no (mm) FB FT Hinge H&R (H) (M) (S) (H) (M) (S)
1 230 x 300 13.77 13.76 55.01 7.94 13.77 14.23 19.16 15.63 17.08 30.66
2 230 x 350 9.40 9.39 37.87 5.51 9.40 9.88 14.85 10.71 12.21 25.21
3 230 x 400 6.89 6.88 28.25 4.11 6.89 7.40 12.41 7.90 9.45 21.90

4 230 x 450 5.33 5.32 22.42 3.23 5.33 5.86 10.92 6.16 7.77 19.72

5 230 x 500 4.28 4.27 18.69 2.63 4.28 4.84 9.97 5.00 6.68 18.20
6 230 x 550 3.53 3.52 16.19 2.19 3.53 4.13 9.32 4.18 5.94 17.11
7 230 x 600 2.97 2.97 14.45 1.86 2.97 3.61 8.87 3.58 5.41 16.30
8 230 x 650 2.54 2.53 13.20 1.61 2.54 3.22 8.54 3.11 5.03 15.69

9 230 x 700 2.19 2.18 12.28 1.40 2.19 2.91 8.30 2.74 4.74 15.22
10 230 x 750 1.91 1.90 11.59 1.23 1.91 2.66 8.11 2.44 4.52 14.85
Deflection of frame without plinth beam
for different support condition

90000 FT

80000 FB
H&R
70000
SPRING (H)
60000 FEM(H)
50000 SPRING (M)
SPRING (S)
40000
FEM(M)
30000
FEM(S)
20000 Hinge
10000

0
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Deflection (mm)
 Deflection of frame with plinth beam for
different support condition
Deflection of frame in X-direction for various support condition (mm)
Sr
Column
. Spring Spring Spring FEM FEM FEM
size FB FT Hinge H&R
no (H) (M) (S) (H) (M) (S)

1 230 x 300 8.42 8.42 10.88 7.00 8.42 8.68 11.55 8.85 9.49 15.62
2 230 x 350 6.30 6.29 8.74 5.00 6.29 6.57 9.52 6.69 7.39 13.65
3 230 x 400 4.99 4.98 7.51 3.81 4.98 5.28 8.34 5.37 6.14 12.53
4 230 x 450 4.11 4.10 6.75 3.03 4.10 4.42 7.61 4.47 5.33 11.84
5 230 x 500 3.46 3.46 6.25 2.50 3.46 3.82 7.31 3.83 4.78 11.39
6 230 x 550 2.97 2.96 5.89 2.10 2.97 3.36 6.80 3.34 4.38 11.08
7 230 x 600 2.58 2.57 5.64 1.80 2.58 3.10 6.57 2.95 4.09 10.86
8 230 x 650 2.26 2.25 5.45 1.56 2.25 2.73 6.40 2.64 3.87 10.70
9 230 x 700 1.99 1.98 5.31 1.36 1.98 2.51 6.27 2.37 3.69 10.57
10 230 x 750 1.76 1.74 5.20 1.20 1.75 2.32 6.17 2.15 3.55 10.48
 Deflection of frame with plinth beam for different
support condition

90000 FB
80000 FT
Hinge
70000
H&R
60000
Stiffness(N/mm)

Spring H
50000 Spring M
40000 Spring S
30000 FEM H
20000 FEM M
FEM S
10000
0
0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 18.00
Defflection(mm)
 Comparison between Frame with plinth beam
and without plinth beam in various support
condition
90000

80000

70000

60000 FEM H
Stiffness(N/mm)

50000 FEM H Plinth

40000 Hinge

Hinge Plinth
30000

20000

10000

0
0.00 10.00 20.00 30.00 40.00 50.00 60.00
Deflection (m m )
 Time period of frame without plinth beam
for different support condition
Time period of frame in X-direction for various Support conditions
Sr. Column Spring Spring Spring FEM FEM FEM
no size FB FT Hinge H&R (H) (M) (S) (H) (M) (S)

1 230 x 300 0.166 0.166 0.332 0.126 0.166 0.169 0.196 0.177 0.185 0.248

2 230 x 350 0.137 0.137 0.276 0.105 0.137 0.141 0.173 0.147 0.157 0.225

3 230 x 400 0.118 0.118 0.238 0.091 0.118 0.122 0.158 0.126 0.138 0.210

4 230 x 450 0.103 0.103 0.212 0.080 0.103 0.108 0.149 0.111 0.125 0.200

5 230 x 500 0.093 0.093 0.194 0.073 0.093 0.099 0.142 0.100 0.116 0.192

6 230 x 550 0.082 0.082 0.180 0.066 0.084 0.091 0.137 0.092 0.109 0.186

7 230 x 600 0.077 0.077 0.170 0.061 0.077 0.085 0.134 0.085 0.104 0.182

8 230 x 650 0.071 0.071 0.163 0.057 0.071 0.080 0.132 0.079 0.101 0.179

9 230 x 700 0.066 0.066 0.157 0.053 0.066 0.076 0.130 0.074 0.098 0.176

10 230 x 750 0.062 0.062 0.153 0.050 0.062 0.073 0.128 0.070 0.095 0.174
Time period of frame without plinth beam
for different support condition
90000

80000
FT
70000 FB
H&R
60000
Stiffness(N/mm)

SPRING (H)
50000 FEM(H)

40000 SPRING (M)

SPRING (S)
30000 FEM(M)
20000 FEM(S)
Hinge
10000

0
0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500
Tim e period(sec)
Time period of frame with plinth beam for
different support condition
Time period of frame in X-direction for various Support conditions
Sr.
Column size Spring Spring Spring FEM FEM FEM
no FB FT Hinge H&R
(H) (M) (S) (H) (M) (S)

1 230 x 300 0.130 0.130 0.150 0.119 0.130 0.132 0.153 0.133 0.138 0.179

2 230 x 350 0.112 0.112 0.132 0.100 0.112 0.115 0.139 0.116 0.122 0.168

3 230 x 400 0.100 0.100 0.123 0.087 0.100 0.103 0.131 0.104 0.111 0.161

4 230 x 450 0.091 0.091 0.116 0.078 0.091 0.094 0.125 0.095 0.104 0.157

5 230 x 500 0.083 0.083 0.112 0.071 0.083 0.088 0.121 0.088 0.098 0.154

6 230 x 550 0.077 0.077 0.109 0.065 0.077 0.082 0.118 0.082 0.094 0.152

7 230 x 600 0.072 0.072 0.106 0.060 0.072 0.078 0.116 0.077 0.091 0.151

8 230 x 650 0.067 0.067 0.105 0.056 0.067 0.074 0.115 0.073 0.088 0.149

9 230 x 700 0.063 0.063 0.103 0.052 0.063 0.071 0.114 0.069 0.086 0.149

10 230 x 750 0.059 0.059 0.102 0.049 0.059 0.068 0.113 0.066 0.085 0.148
 Time period of frame with plinth beam for
different support condition

90000

80000
FB
70000
FT
Hinge
60000
H&R
Stiffness(N/mm)

50000 Spring H
Spring M
40000 Spring S
FEM H
30000
FEM M
FEM S
20000

10000

0
0.0300 0.0500 0.0700 0.0900 0.1100 0.1300 0.1500 0.1700 0.1900
Tim e period(sec)
 Comparison between Frame with plinth beam
and without plinth beam in various support
condition
90000

80000

70000

60000 FEM H
Stiffness(N/mm)

FEM H Plinth
50000
Hinge
40000
Hinge Plinth
30000

20000

10000

0
0.0000 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500
Tim e period (sec)
 B.M in column without plinth beam for
different support condition
Max. B.M in column for various Support conditions
Sr. Column Spring Spring Spring FEM FEM FEM
no size FB FT Hinge H&R (H) (M) (S) (H) (M) (S)

1 230 x 300 51.02 53.01 100.00 38.61 53.01 52.61 51.35 49.10 48.06 57.61

2 230 x 350 52.06 54.60 100.00 37.94 54.60 53.96 51.97 50.14 48.58 59.95

3 230 x 400 53.55 56.56 100.00 39.71 56.55 55.60 52.64 51.60 49.37 62.28

4 230 x 450 55.46 58.83 100.00 41.74 58.82 57.48 53.31 53.46 50.42 64.41

5 230 x 500 57.71 61.36 100.00 43.97 61.34 59.53 53.96 55.64 51.64 66.27

6 230 x 550 60.21 64.06 100.00 46.31 64.04 61.68 54.56 58.05 52.96 67.84

7 230 x 600 62.86 66.86 100.00 48.68 66.83 63.86 55.09 60.61 54.31 69.14

8 230 x 650 65.56 69.67 100.00 51.03 69.63 66.00 55.56 63.22 55.84 70.20

9 230 x 700 68.25 72.42 100.00 53.29 72.37 68.05 55.96 65.81 56.91 71.06

10 230 x 750 70.85 75.06 100.00 55.44 75.01 69.98 56.30 68.33 58.09 71.75
 B.M in column without plinth beam for
different support condition
90000

80000

70000
FT
60000
FB
Stiffness(N/mm)

H&R
50000
SPRING (H)
FEM(H)
40000
SPRING (M)
30000 SPRING (S)
Hinge
20000 FEM(M)
FEM(S)
10000

0
35.00 45.00 55.00 65.00 75.00 85.00 95.00 105.00
B.M(kN.m)
 B.M in column with plinth beam for
different support condition
Max. B.M in column for various Support conditions
Sr. Column Spring Spring Spring FEM FEM FEM
no size FB FT Hinge H&R (H) (M) (S) (H) (M) (S)

1 230 x 300 34.61 38.06 40.00 38.58 38.06 38.09 38.26 34.33 34.19 34.39

2 230 x 350 34.53 37.78 41.15 39.62 37.78 37.85 38.36 34.10 33.79 34.09

3 230 x 400 34.98 38.09 42.41 41.06 38.08 37.90 38.37 34.34 33.72 34.09

4 230 x 450 35.96 39.24 43.71 42.83 39.23 38.86 38.31 35.09 33.98 34.31

5 230 x 500 37.41 40.78 44.98 44.85 40.78 40.14 38.22 36.28 34.52 34.69

6 230 x 550 39.24 42.66 46.16 47.03 42.64 41.64 38.09 37.84 35.26 35.15

7 230 x 600 41.34 44.76 47.23 49.28 44.74 43.29 37.96 39.84 36.13 35.63

8 230 x 650 43.60 46.99 48.18 51.52 46.97 45.00 37.84 41.67 37.07 36.10

9 230 x 700 45.93 49.27 49.00 53.70 49.24 46.70 37.72 43.74 38.02 36.53

10 230 x 750 48.25 51.54 49.70 55.78 51.50 48.35 37.69 45.82 38.96 36.93
 B.M in column with plinth beam for
different support condition
90000

80000

70000

60000 FB
Stiffness(N/mm)

FT
50000 Hinge
H&R
40000
Spring H
Spring M
30000
Spring S
20000 FEM H
FEM M
10000 FEM S

0
30.00 35.00 40.00 45.00 50.00 55.00 60.00
B.M(kN.m )
 Comparison between Frame with plinth beam
and without plinth beam in various support
condition
B.M in Column

90000
80000
70000 FEM H
FEM H Plinth
Stiffness(N/mm)

60000
Hinge
50000
Hinge Plinth
40000
30000
20000
10000
0
20.00 40.00 60.00 80.00 100.00 120.00
B.M (kN.m )
 Conclusions
• The deflection and Time period of frame
with hinged support are near about same
as that of frame with soft soil model.

• Deflection and Time period of Hinged and

Roller model are less than any other
model.

• There is large variation between the spring

model and FEM model of same soil,
actually FEM model are used only for the
large size of the footing.
 Seismic Analysis of Building
• Symmetrical building

• Semi-symmetrical building

• Unsymmetrical building
symmetrical model
 Design data for the building
Type of structure Reinforced concrete
structure (G + 9)
Zone III
Response reduction factor 5
Importance factor 1
Soil condition Hard, Medium, soft
Floor to floor height 3m
Depth of foundation 2m
Depth of slab 140mm
External wall 230mm
Internal wall 150mm
Shear wall 200mm
Grade of concrete M20, M25,M30
Wide column element modeling Shell element modeling
of shear wall of shear wall
comparison of axial force in member for
different element
4000.00

3500.00

3000.00 shell element

Axial force (kN)

2500.00 wide column element

2000.00

1500.00

1000.00

500.00

0.00
Axial Axial

1482 1472
comparisons of Shear forces and B.M in
member for different element
s hell elem ent
w ide colum n e le m ent
45

40

35
S .F ,A x ia l ( k N )

30

25

20

15

10

0
s hear B.M s hear B.M

1482 1472
 Effect of support condition in static
analysis in shell element model
2000.00

1800.00

1600.00

1400.00
Axial force (kN)

1200.00

1000.00
Axial
800.00

600.00

400.00

200.00

0.00
FT FB Hinge H&R ATC- FEMA- FEMA- ATC- FEMA- FEMA- ATC- FEMA- FEMA-
40 SH 356 273 40 SM 356 273 40 SS 356 273
SH SH SM SM SS SS
Support condition
 Comparison of B.M and S.F in member
1472 for different support condition

60.00 FT
FB
Hinge
50.00 H&R
ATC-40 SH
B.M and S.F (kN.M , kN)

40.00 FEMA-356 SH
FEMA-273 SH
ATC-40 SM
30.00 FEMA-356 SM
FEMA-273 SM

20.00 ATC-40 SS
FEMA-356 SS
FEMA-273 SS
10.00

0.00
shear B.M
 Response Spectrum Analysis of
Building
• Response spectrum analysis done for three different
types of soil condition.

• Total seismic weight of building =13552 kN

• Fundamental time period of building = 0.896 sec

• Shear wall is modeled using both element wide column

element and shell element

• Analysis is for different support conditions.

 Comparison of time period in shell
element model and wide column element
model
1.2

0.8
Time period (Sec)

0.6
Wide colum n elem ent
0.4 Shell elem ent

0.2

0
1 2 3 4 5 6 7 8 9 10 11 12
Mode no
 Comparison of story deflection in shell
element model and wide column element
model
7.00

6.00

5.00
Deflection (mm)

4.00

Shell elem ent

3.00
w ide colum n elem ent

2.00

1.00

0.00
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
 Time period for different support
condition in Symmetrical model
Symmetrical model (time period in Sec):
mode
no FT FB Hinge H&R spring H spring M spring S
1 1.0693 1.0547 1.0581 1.0000 1.0564 1.2159 1.7283
2 1.0692 1.0547 1.0581 0.9998 1.0564 1.2159 1.7283
3 0.7544 0.7450 0.7469 0.7087 0.7459 0.8212 0.8732
4 0.3114 0.3072 0.3087 0.2917 0.3077 0.3355 0.3637
5 0.3114 0.3072 0.3087 0.2916 0.3077 0.3354 0.3633
6 0.2229 0.2200 0.2207 0.2089 0.2203 0.2404 0.2999
7 0.1522 0.1503 0.1511 0.1433 0.1505 0.1615 0.2555
8 0.1522 0.1503 0.1511 0.1432 0.1505 0.1614 0.1770
9 0.1392 0.1379 0.1379 0.1337 0.1382 0.1609 0.1770
10 0.1107 0.1093 0.1097 0.1040 0.1094 0.1180 0.1687
11 0.0917 0.0907 0.0912 0.0874 0.0908 0.1114 0.1617
12 0.0917 0.0907 0.0912 0.0874 0.0908 0.1056 0.1616
 Comparison of time period in
symmetrical model for different support
condition
2.00
Fix at bottom
1.80 Fix at top
hinge
1.60 H&R
FEMA-356 SH
1.40 FEMA-273 SH
ATC-40 SH
1.20 FEMA-356 SM
Time period (sec)

FEMA-273 SM
1.00
ATC-40 SM
0.80 FEMA-356 SS
FEMA-273 SS
0.60 ATC-40 SS

0.40

0.20

0.00
1 2 3 4 5
Mode No
 Story deflections in symmetrical building
for hard strata
7.00

6.00

FT
5.00 FB
hINGE
Deflection(mm)

4.00 H &R
ATC-40 SH
3.00 FEMA-273 SH
FEMA-356 SH

2.00

1.00

0.00
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
Floor level
Story deflections in symmetrical building
for medium strata
10.00

9.00

8.00
FT
7.00 FB
Deflection(mm)

6.00 hINGE
H &R
5.00 ATC-40 SM
FEMA-273 SM
4.00
FEMA-356 SM
3.00

2.00

1.00

0.00
f

d
h

d
t
oo

1s
3r

2n
9t

8t

7t

6t

5t

4t

un
R

ro
G
Floor level
Story deflections in symmetrical building
for soft strata
18.00
FT

FB
16.00
HINGE
14.00 H &R

ATC-40 SS
12.00
FEMA-273 SS
Deflection(mm)

10.00 FEMA-356 SS

8.00

6.00

4.00

2.00

0.00
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
Floor level
 Forces in members for symmetrical
building in hard strata
FT
18 FB
Hinge
16 H&R
ATC-40 SH
14
FEMA-356 SH
B.M/ S.F/Tosion ( kN,m)

FEMA-273 SH
12

10

0
B.M S.F Torsion B.M S.F Torsion

1472 (C) 1482( C )

 Forces in members for symmetrical
building in medium strata
FT
Sym m etrical m odel
FB

25 Hinge
H&R

20 ATC-40 SM
B.M/S.F/Torsion (kN , m)

FEMA-356 SM
FEMA-273 SM
15

10

0
B.M S.F Torsion B.M S.F Torsion

1472 (C) 1482( C )

 Forces in members for symmetrical
building in soft strata
30 FT
FB
Hinge
25
H&R
ATC-40 SS
B.M/S.F/Torsion (kN,m)

20 FEMA-356 SS
FEMA-273 SS

15

10

0
B.M S.F Torsion B.M S.F Torsion

1472 (C) 1482 ( C )

Semi-symmetrical model
 Design data for the building
Type of structure Reinforced concrete
structure (G + 9)
Zone III
Response reduction factor 5
Importance factor 1
Soil condition Hard, Medium, soft
Floor to floor height 3m
Depth of foundation 2m
Depth of slab 160mm
External wall 230mm
Internal wall 150mm
Shear wall 200mm
Grade of concrete M20, M25,M30
Mathematical model of Semi-symmetrical building
 Effect of support condition in static
analysis in shell element model
3500.00
Axial
3000.00

2500.00
Axial force (kN)

2000.00

1500.00

1000.00

500.00

0.00
FT FB Hinge H&R ATC- FE FE ATC- FE FE ATC- FE FE
40 SH MA- MA- 40 SM MA- MA- 40 SS MA- MA-
356 273 356 273 356 273
SH SH SM SM SS SS
Support condition
 Comparison of B.M and S.F in member 63
for different support condition
160.00

140.00

120.00 FT
FB
B.M and S.F (kN.M , kN)

100.00 Hinge
H&R
ATC-40 SH
80.00
FEMA-356 SH
FEMA-273 SH
60.00
ATC-40 SM
FEMA-356 SM
40.00
FEMA-273 SM
ATC-40 SS
20.00 FEMA-356 SS
FEMA-273 SS
0.00
shear B.M
 Response Spectrum Analysis of
Building
• Response spectrum analysis done for three different
types of soil condition.

• Total seismic weight of building = 78505 kN

• Fundamental time period of building in X-dir. = 0.732

• Fundamental time period of building in Y-dir = 0.545

• Analysis is for different support conditions.

 Time period for different support
condition in Semi-symmetrical model
Semi-symmetrical (Time period in sec):
Mode no FB FT H &R H Spring H Spring M Spring S
1 1.1800 1.1700 1.2011 1.2100 1.1793 1.2453 1.7481
2 1.0969 1.0802 1.0477 1.0913 1.0892 1.2317 1.4569
3 0.9523 0.9391 0.9208 0.9519 0.9456 1.0398 1.1616
4 0.3946 0.3911 0.4043 0.4062 0.3914 0.3962 0.4187
5 0.3337 0.3289 0.3230 0.3340 0.3314 0.3522 0.4160
6 0.2979 0.2940 0.2937 0.2996 0.2954 0.3103 0.3893
7 0.2271 0.2252 0.2328 0.2334 0.2253 0.2274 0.3453
8 0.1726 0.1702 0.1689 0.1731 0.1712 0.2100 0.2859
9 0.1659 0.1639 0.1628 0.1640 0.1645 0.1826 0.2413
10 0.1609 0.1591 0.1593 0.1631 0.1595 0.1795 0.2356
11 0.1587 0.1574 0.1580 0.1612 0.1575 0.1638 0.2178
12 0.1549 0.1533 0.1487 0.1536 0.1537 0.1630 0.2087
 Comparison of time period in semi-symmetrical
model for different support condition
2.0000

1.8000 FB
FT
1.6000 H &R
H
ATC-40 SH
1.4000 FEMA-273 SH
FEMA-356 SH
ATC-40 SM
Time period (sec)

1.2000
FEMA-273 SM
FEMA-356 SM
1.0000 ATC-40 SS
FEMA-273 SS
0.8000 FEMA-356 SS

0.6000

0.4000

0.2000

0.0000
1 2 3 4 5
Mode no
 Story deflections in semi-symmetrical
building for hard strata
7
FT
6 FB
hINGE
5
H&R
Deflection(mm)

4 ATC-40 SH
FEMA-273 SH
3
FEMA-356 SH
2

0
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
Floor level
 Story deflections in semi-symmetrical
building for medium strata
10.00

9.00

8.00
FT
7.00 FB
hINGE
Deflection(mm)

6.00 H&R
ATC-40 SM
5.00 FEMA-273 SM
FEMA-356 SM
4.00

3.00

2.00

1.00

0.00
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
Floor level
 Story deflections in semi-symmetrical
building for soft strata
18.00

16.00

14.00

12.00
FT
Deflection(mm)

10.00 FB
hINGE
8.00
H&R
ATC-40 SS
6.00
FEMA-273 SS
FEMA-356 SS
4.00

2.00

0.00
Roof 9th 8th 7th 6th 5th 4th 3rd 2nd 1st Ground
Floor level
 Forces in members for semi-symmetrical
building in hard strata
35

30

25
B.M/S.F/Torsion (kN,m)

FT
20 FB
Hinge
15 H&R
ATC-40 SH
10 FEMA-356 SH
FEMA-273 SH
5

0
B.M S.F Torsion B.M S.F Torsion

63 (C) 439 ( C )
Forces in members for semi-symmetrical
building in medium strata
50 FT
FB
45
Hinge
40
H&R

35 ATC-40 SM
B.M/S.F/torsion (kN ,m)

FEMA-356 SM
30
FEMA-273 SM
25

20

15

10

0
B.M S.F Torsion B.M S.F Torsion

63 (C) 439 (C)

Forces in members for semi-symmetrical
building in soft strata
70

60

50
FT
B.M/S.F/Torsion (kN,m)

FB
40 Hinge
H&R
30 ATC-40 SS
EFMA-356 SS
FEMA-273 SS
20

10

0
B.M S.F Torsion B.M S.F Torsion

63 (C) 439( C )
unsymmetrical model
 Design data for the building
Type of structure Reinforced concrete
structure (G + 9)
Zone III
Response reduction factor 5
Importance factor 1
Soil condition Hard, Medium, soft
Floor to floor height 3m
Depth of foundation 2m
Depth of slab 160mm
External wall 230mm
Internal wall 150mm
Shear wall 200mm
Grade of concrete M20, M25,M30
Mathematical model of unsymmetrical building
 Effect of support condition in static
analysis in shell element model

2500.00
Axial

2000.00
Axial force (kN)

1500.00

1000.00

500.00

0.00
FT FB Hinge H&R ATC- FE FE ATC- FE FE ATC- FE FE
40 SH MA- MA- 40 SM MA- MA- 40 SS MA- MA-
356 273 356 273 356 273
SH SH SM SM SS SS
Support condition
 Comparison of B.M and S.F in member 33
for different support condition
120.00

100.00

FT
FB
B.M and S.F (kN.M , kN)

80.00
Hinge
H&R
ATC-40 SH
60.00
FEMA-356 SH
FEMA-273 SH
ATC-40 SM
40.00 FEMA-356 SM
FEMA-273 SM
ATC-40 SS
20.00 FEMA-356 SS
FEMA-273 SS

0.00
shear B.M
 Response Spectrum Analysis of
Building
• Response spectrum analysis done for three different
types of soil condition.

• Total seismic weight of building = 44167 kN

• Fundamental time period of building in X-dir. = 0.668

• Fundamental time period of building in Y-dir = 0.732

• Analysis is for different support conditions.

 Time period for different support
condition in unsymmetrical model
Unsymmetrical (Time period in sec)
Mode no FB FT H &R H Spring H Spring M Spring S
1 1.5973 1.5838 1.6952 1.7050 1.5855 1.6339 1.8748
2 1.2452 1.2309 1.2179 1.2425 1.2461 1.4700 1.8566
3 0.9055 0.8938 0.8740 0.8962 0.9090 1.1705 1.3228
4 0.5139 0.5098 0.5473 0.5485 0.5102 0.5155 0.5340
5 0.3529 0.3488 0.3475 0.3540 0.3525 0.3838 0.4158
6 0.2858 0.2836 0.3021 0.3025 0.2838 0.2863 0.3646
7 0.2331 0.2303 0.2281 0.2317 0.2334 0.2646 0.2985
8 0.1883 0.1869 0.1974 0.1975 0.1870 0.1917 0.2925
9 0.1711 0.1693 0.1701 0.1723 0.1706 0.1884 0.2852
10 0.1594 0.1575 0.1517 0.1575 0.1580 0.1797 0.2561
11 0.1417 0.1402 0.1398 0.1403 0.1406 0.1663 0.2246
12 0.1348 0.1338 0.1356 0.1397 0.1339 0.1564 0.2127
 Comparison of time period in Unsymmetrical model
for different support condition

Fix at bottom
Fix at top
2.0000 H& R
Hinge
1.8000
ATC-40 SH
1.6000 FEMA-273 SH
FEMA-356 SH
1.4000
ATC-40 SM
Time period (sec)

1.2000 FEMA-273 SM
1.0000 FEMA-356 SM
ATC-40 SS
0.8000 FEMA-273 SS
0.6000 FEMA-356 SS

0.4000
0.2000
0.0000
1 2 3 4 5
Mode no
 Story deflections in Unsymmetrical
building for hard strata

7.00

6.00
FT

5.00 FB
HINGE
Deflection(mm)

4.00 H& R
ATC-40 SH
3.00 FEMA-273 SH
FEMA-356
2.00

1.00

0.00
f

d
h

d
t
oo

1s
3r

2n
9t

8t

7t

6t

5t

4t

un
R

ro
G
Floor level
 Story deflections in Unsymmetrical
building for medium strata

12.00
FT
10.00 FB
HINGE
H&R
8.00
Deflection(mm)

ATC-40 SM
FEMA-273 SM
6.00
FEMA-356 SM

4.00

2.00

0.00
f

d
h

d
oo

1s
3r

2n
9t

8t

7t

6t

5t

4t

un
R

ro
G
Floor level
 Story deflections in unsymmetrical
building for soft strata
18.00

16.00

14.00 FT
FB
12.00 HINGE
H&R
Deflection(mm)

ATC-40 SS
10.00
FEMA-273 SS
FEMA-356 SS
8.00

6.00

4.00

2.00

0.00

Floor level
 Forces in members for Unsymmetrical
building in hard strata

25

FT

FB
20
Hinge
B .M /S.F /T o rsio n (kN ,m )

H&R

15 ATC-40 SH

FEMA-356 SH
FEMA-273 SH
10

0
B.M S.F Torsion B.M S.F Tors ion

33(C) 223 ( C )
 Forces in members for unsymmetrical
building in medium strata

35

30

25
B.M/S.F/Torsion (kN,m)

FT
20 FB

Hinge
15
H&R
10 ATC-40 SM

FEMA-356
5 SM
FEMA-273
SM
0
B.M S.F Torsion B.M S.F Torsion

33(C) 223 ( C)
 Forces in members for unsymmetrical
building in soft strata

45

40

35
FT
B.M/S.F/Torsion (kN,m)

30 FB
Hinge
25
H&R
20
ATC-40 SS
15 FEMA-356 SS
FEMA-273 SS
10

0
B.M S.F Torsion B.M S.F Torsion

33(C) 223 ( C )
 Appling forces in principal direction

• In this method seismic forces will be

applied in principal direction.

• This system is available in SAP 2000 by

specifying excitation angle in response
spectrum case.

• The more realistic results will get.

Formulation for calculating the principal
direction of structure

2
Ix  Iy  Ix  Iy 
I max      I xy
min 2  2 

 2 I xy
tan 2 
Ix  Iy
Unsymmetrical building with shear wall.
 Member Forces
Member No:213

Combination
s  Axial V2 V3 Torsion M2 M3

Max Principal 74.86 13.72 19.89 1.96 24.16 22.52

SRSS 74.86 13.70 19.87 1.95 24.15 22.46
100+30+30 73.69 13.00 17.49 1.69 21.24 21.30
100-30+30 45.59 7.57 15.38 0.49 10.26 12.38
100-30-30 28.85 7.57 15.38 0.79 10.26 12.38
100+30-30 56.95 13.00 17.49 0.79 21.24 21.30
-100+30+30 -28.85 -7.57 -15.38 1.69 -10.26 -12.38
-100-30+30 -56.95 -13.00 -17.49 -0.79 -21.24 -21.30
-100-30-30 -73.69 -13.00 -17.49 -1.69 -21.24 -21.30
-100+30-30 -45.59 -7.57 -15.38 -0.79 -10.26 -12.38
Shell No:213

Combinations
  F11 F22 F12 M11 M22 M12
Max Principal 21.36 106.79 51.74 3.21 16.04 1.59
SRSS 21.36 106.79 51.74 3.20 16.04 1.59
100+30+30 19.22 96.10 50.51 2.96 14.78 1.47
100-30+30 9.50 47.52 30.63 1.60 8.02 0.81
100-30-30 8.40 41.96 28.71 1.60 8.02 0.81
100+30-30 18.12 90.54 48.59 2.96 14.78 1.47
-100+30+30 -8.40 -41.96 -28.71 -1.60 -8.02 -0.81
-100-30+30 -18.12 -90.54 -48.59 -2.96 -14.78 -1.47
-100-30-30 -19.22 -96.10 -50.81 -2.96 -14.78 -1.47
-100+30-30 -9.50 -47.52 -30.63 -1.60 -8.02 -0.81
L-Shape building.
 Member Forces
Member No:02

Combinations  Axial V2 V3 Torsion M2 M3

Max Principal 33.00 6.82 6.82 0.00 12.07 12.07
SRSS 33.00 6.82 6.82 0.00 12.07 12.07
100+30+30 30.98 6.27 6.27 0.00 11.09 11.09
100-30+30 17.00 3.37 3.37 0.00 5.97 5.97
100-30-30 15.58 3.37 3.37 0.00 5.97 5.97
100+30-30 29.61 6.27 6.27 0.00 11.09 11.09

-100+30+30 -15.58 -3.37 -3.37 0.00 -5.97 -5.97

-100-30+30 -29.55 -6.27 -6.27 0.00 -11.09 -11.09

-100-30-30 -30.98 -6.27 -6.27 0.00 -11.09 -11.09

-100+30-30 -17.01 -3.37 -3.37 0.00 -5.97 -5.97

Member No:74

Combinations Axial V2 V3 Torsion M2 M3

Max Principal 18.82 10.20 10.20 0.00 16.65 16.65
SRSS 18.82 10.20 10.20 0.00 16.65 16.65
100+30+30 6.98 9.37 9.37 0.00 15.31 15.31
100-30+30 6.67 5.05 5.05 0.00 8.25 8.25
100-30-30 -4.59 5.05 5.05 0.00 8.25 8.25
100+30-30 -3.97 9.37 9.37 0.00 15.31 15.31
-100+30+30 4.90 -5.05 -5.05 0.00 -8.25 -8.25
-100-30+30 3.97 -9.37 -9.37 0.00 -15.31 -15.31
-100-30-30 -6.98 -9.37 -9.37 0.00 -15.31 -15.31
-100+30-30 -6.36 -5.05 -5.05 0.00 -8.24 -8.24
 Conclusions
• Increase in slab thickness increases the
rigidity of floor at that level.

• In the building model it is better to use

lump mass instead of distributed mass.

• Considering effect of finite joint

correction the bending moment is
reduced around 10%.
• Considering effect of finite joint correction
reduction in deflection is around 10 to 15 %.

• In the building model by taking slab

eccentricity in consideration the horizontal
deflection increases.

• Vertical deflection reduces and time period of

the building increases by considering slab
eccentricity.
 It is better to use FEMA-356 approach instead
of using FEMA-273 and ATC-40, it is newly
revised.

 The spring constant variation depends on

depth of foundation, thickness of footing in
FEMA-356. These parameters are not
considered in FEMA-273 and ATC-40.

 In case of single frame there is much variation

in time period, deflection and B.M in the model
by introducing various support conditions.
 By introducing the spring model in building we
can reduce the B.M and S.F in the members.

 The time period of the structure increases

when soil-structure interaction is considered.

 Horizontal deflection increase as the soil effect

is considered in the analysis.

 Applying forces in principal axis is the better

option. Due to this the number of load
combinations are reduces.
 References:
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