Elastic and Dynamic analysis of a multistorey frame

ELASTIC AND DYNAMIC
ANALYSIS OF A MULTISTOREY
FRAME
By
Nayan Kumar Dutta

What is an Earthquake ?
Sudden movement of the earth’s crust
90% of all earthquakes result from movements on geological
faults – tectonic earthquakes
Earthquakes are generally natural disasters of unpredictable
nature.
But due to human activity such as construction of large dams
and reservoirs, mine blasts, nuclear tests, etc earthquake may be
generated.

Causes of Earthquakes
Plate Tectonics
 Elastic Rebound Theory

Theory of Plate Tectonics
The lithosphere if fragmented
into seven major tectonic
plates and many smaller
ones.
Due to convection current in
viscous mantle these plates
move in different directions and
at different speeds from those of
the neighbouring ones.

Elastic Rebound Theory
Most boundaries of the plates have irregularities and this
leads to a form of stick-slip behaviour.
Once the boundary has locked, continued relative motion
between the plates leads to increasing stress and stored strain
energy in the volume around the fault surface.
This continues until the stress has risen sufficiently to
overcome the strength of the rock, suddenly allowing sliding
over the locked portion of the fault, releasing the stored
energy that spreads out through seismic waves that causes
earthquakes.

Focus and Epicentre
 The point within the earth from which
the earthquake originates is the Focus
or Hypocenter.
 The point on the earth’s surface lying
vertically above the focus is the
Epicenter.
 Distance from epicenter to any point of
interest is called epicentral distance.
 The depth of focus from the epicenter,
is the Focal Depth, shallow focus
earthquakes with focal depths less than
about 70km are the most damaging.

How are Earthquakes Located?
P wave travels faster than S wave.
The difference in arrival time between the two
types of seismic waves can be used to
calculate the distance of the earthquake's
epicentre from the measuring station.
DE = DeltaT x (VP VS) / (VP - VS)
where
DE = Distance to epicentre (km)
DeltaT = Difference between P and S-wave
arrival time (s)
VP = P-wave velocity (km/s)
VS = S-wave velocity (km/s)

A circle with a radius equal to
the distance to the epicentre is
plotted around the seismograph
station. This is then repeated for
the other two stations and the
point where the three circles
intersect is the location of the
earthquake’s epicentre.

Measerement of earthquake(Magnitude &Intensity)
 Magnitude is a quantitative measure of the size of an earthquake. There is only
one magnitude per earthquake.
 Richter defined the magnitude of an earthquake as the logarithm to base 10 of the
maximum seismic wave amplitude (in microns) recorded on a standard
seismograph at a distance of 100 km from the epicentre
 An increase in magnitude (M) by 1.0 means that the amplitude of the earthquake
waves increases 10 times.
 Earthquake magnitude is related to the amount of energy released in the
earthquake.
 log10 E = 11.8 + 1.5M, where, energy, E, is in ergs
 Other magnitude scales are Surface wave magnitude, Body wave magnitude,
Duration magnitude and Moment Magnitude (MW).

Intensity of an Earthquake
 Intensity is a qualtitative measure of the severity of shaking at a location during
an earthquake. The severity of shaking is much higher near the epicenter than
farther away.
 Intensity scales are -
 Rossi-Forel intensity scale (developed in the late 19th century) – ten stages
 Mercalli intensity scale (1902) – twelve stages
 Modified Mercalli Intensity (MMI) scale (1931)
 Medvedev-Spoonheuer-Karnik (MSK) intensity scale (1964)
 Both scales are quite similar and range from I (least perceptive) to XII (most
severe).

Details of MSK Intensity Scale

 Recently in Nepal Earthquake (of 7.8 magnitude) more than 18,000 people died.

So the social consequences of earthquakes, in terms of
human casualties and injuries and direct and indirect
economic losses justify the need to be prepared for
earthquakes.
Earthquakes are still difficult to predict
So,we should prepare for earthquake by making our
buildings with proper structural design procedure.

Indian Codes relevant to Earthquake Engineering
IS 1893 - 2002: Criteria for earthquake resistant design of structures.
IS 13920 - 1993: Code of practice for ductility detailing of reinforced
concrete structures subjected to Seismic forces.
IS 4326 - 1993: Earthquake resistant design and construction of
buildings code of practice.
IS 13828 - 1993: Improving Earthquake resistance of low strength
masonry buildings - Guidelines.
IS 13827 - 1993: Guidelines for improving earthquake resistance -
Earthen Buildings.
Among these codes in our project we have used IS 1893-
2002,IS13920-1993 and IS 875.

Method of seismic Analysis
1. Linear static analysis (or equivalent static analysis) :–
Applicable for regular structures and low rise buildings.
2. Linear dynamic analysis :-–
can be performed either by
response spectrum method (mode superposition method like
SRSS,CQC) or by elastic time history method.
3. Non-linear static analysis:-
4. Non-linear dynamic analysis:-
Describe the actual behaviour of the structure during an
earthquake.

• Slabs forces the beam
to bend with it when
horizontal forces act.

 The philosophy of earthquake
design for structures is:
 In frequent, minor ground shaking -
Main structural members should
not be damaged, other building parts may
have repairable damage
 In occasional, moderate ground
shaking -
Main structural members may
sustain reparable damage, other building
parts may have to be replaced
 In rare, major ground shaking –
Structural members may be
irreparably damaged but the structure
should not collapse
General Goals in Seismic-Resistant Design and Construction
Courtesy: IITK-BMTPC EQ. Tips

 After minor shaking, the building will be fully operational within a short time and the
repair costs will be small.
 After moderate shaking, the building will be operational once the repair and
strengthening of the damaged main members are completed.
 After a strong earthquake, the building may become unsuitable for further use, but will
stand so that inhabitants can be evacuated.
 Important buildings, like hospitals and fire stations, play a critical role in post-
earthquake activities and must remain functional immediately after the earthquake.
 Collapse of dams during earthquakes can cause flooding.
 Damage to sensitive facilities like nuclear power plant, chemical plants, etc. can cause
a further disaster.
 These structures must sustain very little damage and should be designed for a higher
level of earthquake protection.

Seismic Zoning Map of India
 In 1935, GSI prepared a seismic hazard map of three zones depicting
likely damage scenario (severe, moderate, light).
 By evaluating peak horizontal ground acceleration based on
earthquake data from 1904-1950, Jai Krishna developed a 4-zone
seismic map in 1958. The Indian peninsular regions were not given
any seismic consideration as it was considered to be a stable plateau.
 BIS provided a seismic zone map in IS: 1893-1962 with seven zones
based on isoseismals of major earthquakes and average intensity
attenuation relationship. Past smaller earthquakes, trend of principal
tectonic features and local ground conditions were also considered.
 For Zone 0 (intensity less than V) seismic loading on the
structure need not be considered.

 Koyna earthquake of 1967 (magn. 6.5) in Peninsular India caused the
revision of the seismic zone map in IS: 1893-1970. Number of zones
was reduced to five (based on five seismo-tectonic units of the
country).
 Due to the Latur earthquake (magn. 6.2) in 1993, the seismic status of
the Indian peninsular shield was again reviewed.
 The IS 1893-2002 map has only four zones. Zone I has been
enhanced to Zone II.
 Enhanced extent of Zone III (Chennai is now in Zone III, previously
in Zone II).
 This 2002 seismic zone map is not the final word on the seismic
hazard of the country, and hence there can be no sense of
complacency in this regard!
 The decision of the BIS is to have a new revised zoning map using
probabilistic framework.
 To account for new available information, the shapes of some of the
isoseismals were changed and the extent of Zone 0 in the southern part of the
Indian Peninsula was reduced in the seismic zone map of IS: 1893-1966

EARTHQUKE LOAD DESIGN (STATIC EQUIVALENT METHOD)
Z = zone factor = 0.16 (zone-3)
I = importance factor = 1.5
The structure is a special RC moment resisting frame (SMRF)
R = Response reduction factor = 5
Factored dead load on each floor = 3.725 kN/m2
 Live load on each floor = 4 kN/m2
EQ in X direction,
 Ta = 0.508. For type – 3 Soil, Sa/g = 2.5.
Dead Load from beam on each floor = 434.49 kN
Dead Load from column on each floor = 205.8 kN
Dead load from column on ground level = 176.4 kN
Dead load from wall (2nd from 6th floor) = 1556.526 kN
Dead load from wall (ground floor) = 778.31 kN
Dead load from wall (roof level) = 862.31 kN

DEAD LOAD ON ROOF
Dead Load from Slab = 3.445 kN/m2
Dead Load from column = 102.9 kN
Total Load,
On Ground Level = 3040.40 kN
On Floor level (2nd to 6th) = 3846.20 kN
On Roof level (7th) = 2393.90 kN
Horizontal Acceleration (Ah) = (1.5/5) x (0.16/2) x (2.5) = 0.06
Total weight of ll the floor = 28510.9 kN
DESIGN BASE SHEAR
V b = Ah x W = (0.06 x 28510.9) kN = 1710.65 kN
EQ in Z direction,
d = 14.5, Ta = 0.638,
Horizontal Acceleration (Ah) = (1.5/5) x (0.16/2) x (2.13) = 0.05112
DESIGN BASE SHEAR
V b = Ah x W = (0.05112 x 28510.9) KN = 1457.47 kN

40
A plot of the peak value of a response quantity to a ground motion time
history, as a function of the natural vibration period, Tn, or natural frequency,
wn, of the system is called the response spectrum for that quantity.
Each such plot is for SDOF systems having a fixed damping ratio, z, and
several such plots for different values of z are included together in one graph
to cover the range of z values for real structures.
What is Response Spectrum ?
   
   
   zz
zz
zz
,,max,
,,max,
,,max,
nno
nno
nno
TtuTu
TtuTu
TtuTuD






41
Deformation Response Spectrum
El Centro ground motion

42
Response Spectra (z = 2%) for
El Centro ground motion
Deformation Response Spectrum
Pseudo-velocity Response Spectrum
Pseudo-acceleration Response
Spectrum

43
Why do we need three spectra when each of them contains the same
information ?
 Each spectrum provides a physically meaningful quantity.
 Deformation spectrum – provides peak deformation
 Pseudo-velocity spectrum – directly related to peak strain energy of the system
 Pseudo-acceleration spectrum – directly related to the peak equivalent static force and
base shear

44
Tripartite Response Spectrum for El Centro ground motion

45
Mean and mean +1s spectra.
Dashed lines show an idealized design spectrum.

46
As per IS 1893(Part 1) : 2002

 Masses are connected to each other and to a
supporting point by linear springs and viscous dashpots.

48
Each characteristic deflected shape Mode Shape
So for a N-DOF system there exist ‘N’ mode shapes and ‘N’ natural frequencies.
Note: Any ith mode shape has (i-1) nodes (points of zero displacement).
NODE

49
Modal Combination Rules
The maximum or peak of the desired response quantity is first obtained for each mode and then
these modal maxima are combined according to a modal combination rule.
1.Square Root of Sum of Squares (SRSS) Method:-
Assumed that the modes achieve peak responses at randomly distributed time instants.
 It provides a very good approximation of the peak response for modes with well-separated
natural frequencies but fails for closely spaced modes
2.Complete Quadratic Combination (CQC) Method:-
Based on the use of cross-modal coefficients, it is an improvement over the SRSS
method and applicable to a wide class of structures.
In this method all possible quadratic combinations are incorporated.

60
 Ductility is the ability of a member to deform beyond its
elastic limit without failure.
Ductility
 Concrete and masonry
are brittle.
 Steel is ductile.

61
Ductile and Brittle Failure

62
Ductile Chain design
 As more and more force is applied, the chain will eventually
break when the weakest link in it breaks.
 If the ductile link is the weak one (i.e., its capacity to take load is
less), then the chain will show large final elongation.
 Instead, if the brittle link is the
weak one, then the chain will fail
suddenly and show small final
elongation.
 Thus, to design a ductile chain, the
ductile link has to be made the
weakest link.

63
Strong-Column Weak-Beam Design
 The beams should be made the ductile weak link in the chain
and not the columns because the failure of a column affects the
stability of the whole building while the failure of a beam has a
localized effect.
Strong-Column Weak-Beam
Design Method

64
Design Provisions for Ductility
 IS codes (such as IS 13920 : 1993) enforce ductility
specifications with the following objectives:
 Provide large capacity for inelastic deformations.
 Prescribe relative strengths of different members to control
failure mechanism at joints.
 Permit structure to undergo large inelastic deformations
before collapse – “fail-safe” design philosophy.

Clause 7.4.5 :-Special confining reinforcement shall be provided over the full height of column
which has significant variation in stiffness along its height.
This clause deals with short column effect.

6 d ( ! 65 mm )
6 d ( ! 65 mm )

Earthquake resisting materials1.Masonry
Masonry is made up of burnt clay bricks and cement or mud mortar.
Masonry can carry loads that cause compression (i.e. pressing together) but
can hardly take load that causes tension (i.e. pulling apart). Masonry is a brittle
material, these walls develop cracks once their ability to carry horizontal load is
exceeded.
2.Concrete
Concrete is another material that has been popularly used in building
construction particularly over the last four decades. Cement concrete is made
of crushed stone pieces (called aggregate), sand, cement and water mixed in
appropriate proportions. Concrete is much stronger than masonry under
compressive loads, but again its behavior in tension is poor.
3. Steel
Steel is used in masonry and concrete buildings as reinforcement bars
of diameter ranging from 6mm to 40mm. reinforcing steel can carry both tensile
and compressive loads. Moreover steel is a ductile material.

Approaches for Earthquake-Resistant Design of
Structures
 First Approach :
Design the structure with sufficient strength, stiffness and inelastic deformation
capacity.
 Second Approach :
Use of control devices to reduce the forces acting on the structure.
Control devices may be defined as external structural protective systems that
reduce the energy dissipation demand on primary structural members when the
structure is subjected to external input energy.

Classification of Control Systems

1.Passive Control Systems:-
 These do not require power to operate – hence termed “passive”
 Systems in this category are very reliable since they are unaffected by power outrages, which are
common during earthquakes.
2.Active Control System:-
 These require considerable amount of external power, in the order of tens of kilowatts
 They are more effective than passive devices because of their ability to adapt to different loading
conditions and to control different modes of vibration
 Since the large amount of power required for their operation may not always be available during seismic
events, they are not very reliable.
3. Semi-Active Control Systems:-
 cannot inject energy into the controlled system, but their mechanical properties can be adjusted to
improve these performance.
 These are often viewed as controllable passive devices.
4. Hybrid Control Systems:-
 Combined passive and active control systems – less power and resources are required than active
control systems

COMPARISION OF DIFFERENT CATEGORIES OF CONTROL SYSTEMS

Types of Passive Control Systems
 Base Isolation Systems
 Metallic Dampers
 Friction Dampers
 Viscoelastic Dampers
 Fluid Viscous Dampers
 Tuned Mass Dampers
 Tuned Liquid Dampers
 Shape Memory Alloy Dampers

Base Isolation Systems
• One of the most powerful tools of earthquake engineering, applicable for
new structures as well as for the retrofit of existing structures.
• It is the only practical way of reducing simultaneously inter-story drift and
floor acceleration

84
Structural Response with and without Base Isolation
Model of a one story building without
base isolation – SDOF system
Model of a one story building with
base isolation – 2-DOF system
c1
c2

Base isolation drastically reduces the fundamental
frequency of the system
Fixed Base
Base Isolated
Period

86
Base Isolation Systems
Elastomeric-type
Lead
Rubber
Bearing
Sliding-type
Elastomeric
Bearing with
steel shims
(Laminated
Rubber
Bearing)
High
Damping
Rubber
Bearing
Super-
high
Damping
Rubber
Bearing
Friction
Pendulum
System

87
Elastomeric Rubber Bearing
Made from natural rubber or neoprene.
Improved by vulcanization bonding of sheets of rubber and thin steel
reinforcing plates or steel shims laminated rubber bearing.
The steel shims reduce the vertical deformation of the bearing and lateral
bulging of the rubber layer.
• Critical damping is of the order of 2% -3% low damping bearing
Easily manufactured at comparatively low cost, are unaffected by time and
resistant to environmental degradation.
.

88
Used as bridge bearing and vibration isolator for buildings
Elastomeric Rubber Bearing….contd.

89
Lead Plug Bearing
Disadvantage of elastomeric bearing’s low damping property can be
overcome by plugging a lead core into the bearing.
A hole is introduced in the center of the elastomeric bearing and a lead
plug is tightly fitted into that hole.
The bearing is designed so that it is very stiff and strong in the vertical
direction but flexible enough in the horizontal direction.
Top and bottom of the bearing are fitted with steel plates which are
used to attach the bearing to building through its foundation.
Critical damping 15% - 35%

90
Lead Plug Bearing….contd.

91
High Damping Rubber Bearing (HDRB)
Another way to increase the damping is to modify the rubber
compounds. The modification is to add carbon black or other types of
filler material with rubber.
Damping obtained – 10% - 15% of critical.
Super high damping rubber bearing (HDRB-S) has damping 20%
higher than that of conventional HDRB and very stable against cyclic
deformation during a large-scale earthquake.

92
Sliding Systems
This works by limiting the transfer of shear across the isolation
interface.
In China there are at least three buildings on sliding systems that use a
specially selected sand at the sliding interface.
A type of isolation containing a lead-bronze plate sliding on stainless
steel with an elastomeric bearing has been used for a nuclear power
plant in South Africa.

CONCLUSION
 Earthquake is still unpredictable. So we should
design our structures with proper Earthquake
Codal provisions.Strict laws should be enforced
by the Government for following the codes.
Codes should be revised on the basis of
probabilistic approaches.
So, by using these we can cope up
with earthquake effects.
93

References
1. IS: 1893-2002
2. IS:13920-1993
3. IS: 875(Part I,Part II)
94

Elastic and Dynamic analysis of a multistorey frame

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