Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
DYNAMIC LOADS
There are two types of forces/loads that may act
on structures, namely static and dynamic forces.
Static forces are those that are gradually applied
and remain in place for longer duration of time.
These forces are either not dependent on time or
have less dependence on time.
Live load acting on a structure is considered as a
static load because it usually varies gradually in
magnitude and position.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Similarly moving loads may also be considered
as statically applied forces.
Dynamic forces are those that are very much
time dependent and these either act for small
interval of time or quickly change in magnitude or
direction.
Earthquake forces, machinery vibrations and blast
loadings are examples of dynamic forces.
Structural response is the deformation behavior
of a structure associated with a particular loading.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Similarly, dynamic response is the deformation
pattern related with the application of dynamic
forces.
In case of dynamic load, response of the
structure is also time-dependent and hence
varies with time.
Dynamic response is usually measured in terms
of deformations (displacements or rotations),
velocity and acceleration.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Dynamic force, F(t), is defined as a force that
changes in magnitude, direction or sense in much
lesser time interval or it has continuous variation
with time.
Impact load is the other extreme where the load is
applied only for an infinitesimal interval of time with
some momentum and is considered separate from
the dynamic loads.
The variation of a dynamic force with time is called
history of loading.
Prescribed dynamic loading is regularly varying
loading in which well-defined cycles of loading are
repeated after equal intervals of time.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
F(t)
t
Fig. A Typical Dynamic Force.
Example of prescribed loading is a regular
vibration of machinery with a certain amplitude
and frequency.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Amplitude of vibration is the maximum
structural displacement during one complete
cycle of load.
Frequency is the number of loading cycles in a
unit time (usually one second).
Types Of Prescribed Loading
a)
Periodic loading
i) Sinusoidal Loading:
ii) Stepped Loading:
iii) Complex Variation Loading:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
F(t)
Typical Sinusoidal Loading.
t
F(t)
Typical Stepped Loading.
t
F(t)
Typical Complex
Variation Loading.
t
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
b)
Non-periodic loading
F(t)
Typical Impulsive Loading.
t
F(t)
Typical Earthquake Loading.
t
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
EARTHQUAKES
An earthquake is the vibration of earth produced
by rapid release of energy from within itself.
This extra energy may be stored in earth and
released at intervals due to many different
phenomena, some of which are as under:
1.
Plate tectonics.
2.
Volcanic eruptions.
3.
Atomic explosions.
4.
Collision of massive meteorites with the
surface of earth.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Plate Tectonics
Crust: Crust is the outermost layer of earth
consisting of solid material varying in temperature
from surface temperature to a maximum
temperature of 1000° C. Its thickness under deep
oceans is between 4 to 6 km and the thickness
under continents is approximately 30 to 40 km.
Mantle: This layer has an approximate thickness
of 3000 km and consists of semi-solid to plastic
material. The temperature ranges from 1000 to
3500° C.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Outer Core: Outer core is a thickness of
approximately 2250 km and consists of liquid at a
temperature of 3500 to 4000° C.
Inner Core: The inner core has a radius of
approximately 1200 km and is a layer of solid
material at temperature higher than 4000° C.
The outer layer of earth having a thickness of 100
km is relatively rigid and is called lithosphere.
The layer of earth below lithosphere having a
thickness of 400 km is softer and is called
asthenosphere.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
The lithosphere acts as rigid plate that moves over
partly molten asthenosphere.
According to this theory, lithosphere is cracked in
places or broken in to smaller pieces or plates.
This may have happened during initial drying of
the earth from a molten state.
There are seven large and several small plates.
The largest plates are the Pacific plate, the North
American plate, the Eurasian plate, the Antarctic
plate, the Indo-Australian plate and the African
plate.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Plate boundaries
a)
Mid-Oceanic Ridge:
b)
Subduction Zone:
Further, there are three types of plate boundaries
depending on the relative movement between the
two adjoining plates.
i)
Convergent Plate Boundary:
ii)
Divergent Plate Boundary:
iii)
Transform Plate Boundary:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
According to continental drift theory, all the
continents were once part of a huge landmass,
which have slowly moved apart.
The Indian sub-continent was not a part of Asia.
It drifted over millions of years from Australia to
Asia and the collision produced Himalayas.
Modern techniques such as GIS and GPS prove
such movements of continents.
Faults are cracks which are developed within the
main plates.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Focus And Epicenter
The point within the earth along the rupturing
geological faults where an earthquake originates
is called the focus or hypocenter.
The point on the earth’s surface directly above
the focus is called the epicenter. Earthquake
waves radiate out from the focus.
The focal depth is the depth of the hypocenter
below the epicenter.
Focal distance is the distance from the
hypocenter to a given reference point.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Earthquake / Seismic Waves
The waves originated at the rupture zone are
called body waves and are of the following two
types:
(1) P-Waves or Primary Waves or Dilation
Waves:
These waves involve particle movement parallel to
the direction of propagation of the wave, as shown
in Fig.
The speed of travel of these waves is appr. 1.73
times greater than the other waves.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
These waves are felt earlier in an earthquake
and cause relatively less damage.
There is usually an after-shock at an interval
during which the other more damaging waves
approach the area.
Wave Direction
Wave Direction
Particle Movement
Particle Movement
(a) P-Waves
(b) S-Waves
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
(2) S-Waves or Secondary Waves or Shear
Waves:
These waves involve particle movement
perpendicular to the direction of propagation of the
wave (refer the Fig.).
When body waves reach the ground surface, part
of these is reflected back while other part
produces surface waves.
Surface waves are the waves produced on the
earth’s surface due to an earthquake and are of
following two types:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
(1) R-Waves or Rayleigh Waves:
These waves produce a circular motion
analogous to the motion of ocean waves.
Hence, rotation along with vertical movements
takes place in case of Rayleigh waves (Fig.).
Movement of Particles in Rayleigh Waves.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
(2) L-Waves or Love Waves:
These waves produce horizontal motion along the
ground surface transverse to the direction of
propagation.
Earthquake magnitude and Richter scale
Earthquake magnitude is a measure of the energy
released during an earthquake.
It defines the size of the seismic event but is not
related with damage or effect of earthquake at a
given location.
The magnitude of earthquake is usually measured
on Richter scale, which is a log scale.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
A magnitude of M5 Richter scale is ten-times
greater than a magnitude of M4 and is
associated with an increase in energy release of
31.6 times.
A magnitude of M5 is 100 times greater than a
magnitude of M3 scale.
Earthquake intensity and Mercalli scale
Intensity is an assessment of the effect of the
earthquake at a given location and is not directly
related to the earthquake magnitude.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
This is determined not by reading instruments but
by observing the effects on structures, human life
and disturbance to the ground surface.
Modified Mercalli index is based on the observed
effects of an earthquake at a specific site.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Mercalli
Scale
Effect
I.
Felt by almost no one.
II.
Felt by very few people.
III.
Tremor noticed by many, but they often do not realize it as an earthquake.
IV.
Felt indoor by many. Feels like a truck has struck the building.
V.
Felt by nearly everyone; many people awakened. Swaying trees and poles
may be observed.
VI.
Felt by all; many people run outdoors.
occurs.
VII.
Everyone runs outdoors. Poorly built structures considerably damaged; slight
damage elsewhere.
VIII.
Specially designed structures damaged slightly, others collapse.
IX.
All buildings considerably damaged, many shift off at foundations.
Noticeable cracks in ground.
X.
Many structures damaged. Ground is badly cracked.
XI.
Almost all structures fall. Bridges wrecked. Very wide cracks in ground.
XII.
Total destruction. Waves seen on ground.
Furniture moved, slight damage
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Main Considerations For Seismic Design
Design of structures to withstand the maximum
intensity earthquake is highly expensive and may
not even be possible due to the following factors:
1.
The magnitude, intensity and other
characteristics of future earthquakes are not
precisely known.
2.
Stiffer structures attract more earthquake
loads. These structures cannot dissipate
energy and all the energy is stored in them
making them unstable.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
3.
Heavier design means more mass of the
structure. Due to larger mass, more
inertial forces are produced during the
ground excitation.
The most common method to design
earthquake resistant structures is to design for
mild earthquakes of expected common
occurrence in the elastic range or in the
inelastic range with less or no permanent
deformations.
Ductility is then provided for maximum expected
intensity of earthquakes.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Ductility is a measure of inelastic deformations
that may be produced in a structure before its
collapse.
Inelastic deformations release energy in the form
of heat and make the structure stable.
Permanent deformations may be produced in the
structure with considerable cracking and structure
may not be useable after a severe earthquake.
However, the life is saved as the people may
escape out of the building.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
In essence the main aim of earthquake resistant
design is to avoid loss of life and then less loss to
property is the second criterion.
It may be tried that the damage is repairable for
moderate earthquakes.
Methods Of Analysis For Earthquake Loading
1 Free Vibration Analysis
2 Response History Analysis (RHA)
3 Response Spectrum Analysis (RSA)
4 Equivalent Static or
Pseudo-Static Load Method
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Related Methods Of Dealing With Earthquakes
1 Base Isolation Method
2 Use Of Special Energy Dissipating Devices
DAMPING
Damping means the presence of frictional forces
in the structure, which transforms the mechanical
energy of system in to other forms of energy,
such as, heat.
If damping is completely absent in an ideal
system, a structure once excited will oscillate
indefinitely with constant amplitude at its natural
frequency.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
The viscous damping forces produced are
proportional to the velocity of the piston.
However, the actual damping in a structure may
result from looseness of joints, dry friction between
components (called Coulomb damping), material
damping (or internal damping found by examining
the area within the hysteresis loop between
stresses and acceleration), structural damping
(general term for all types of damping in a
structure) and many other complex causes that
would lead to nonlinear behavior of the structure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Critical Damping (ccr): It is defined as that
amount of damping due to which a freely excited
system does not oscillate but returns to its
original position in the shortest possible time.
Damping Ratio Of System (ξ):
Damping ratio of a system is defined as the ratio
of damping present in a system to its critical
damping. ξ = c / ccr.
Citical damping coefficient usually ranges
between 2 to 10% of ccr (ξ = 0.02 to 0.10) for
actual structures.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
ξ=1 ⇒
ξ>1 ⇒
ξ<1 ⇒
critically damped response
over-damped response
under- damped response
EQUIVALENT STATIC LOAD METHOD
The parameters discussed in the following
sub-sections are required to be evaluated to
get the values of equivalent static loads
according to UBC-97.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Seismic Zone Factor (Z)
The zone factor (Z) is given as a factor of peak
acceleration with respect to acceleration due to
gravity (g) and it varies from 0.075 to 0.40.
The suggested values correspond to recurrence
interval of 475 years giving a 10 percent
probability of being exceeded in a 50 years
period.
Table. Seismic Zones and Effective Peak Ground Accelerations.
Zone
Effective Peak Ground
Acceleration (EPA)
4
3
2B
2A
1
0.40
0.30
0.20
0.15
0.075
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Soil Profile Types
The ground vibrations traveling through the soil
may be amplified or reduced depending upon the
fundamental period and type of strata.
UBC classifies soils into six profile types, as
given in Table.
This classification depends on the average shear
wave velocity in the top 30m of material.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Table. Soil Profile Types.
Soil Profile Type
Description of Soil
Shear Wave Velocity
(m/s)
SA
Hard rock
> 1500
SB
Rock
760 to 1500
SC
Soft rock
360 to 760
SD
Stiff soil
180 to 360
SE
Soft soil
< 180
SF
Very soft clayey soil
Detailed investigations required
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Seismic Source Types
The seismic source types are decided based on
the maximum moment magnitude potential of a
fault and its slip rate per year.
Type C represents almost an inactive fault in
Table.
Table. Seismic Source Characteristics.
Seismic Source
Type
Maximum Moment Potential
Slip Rate (mm/year)
A
≥ 7.0
≥ 5.0
B
C
Source Characteristics
The fault which is not A or C
≤ 6.5
≤ 2.0
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Near-Source Factors
Two factors, Na and Nv, are used to consider
increased ground motions near a fault.
The factor Na is the acceleration-based factor
that is important for short-period structures and
velocity-based factor Nv that is important for
periods exceeding one second.
Table. Near Source Factors (Na and Nv) for Various Seismic Source Types.
Seismic
Source
Type
Distance From Fault
≤ 2 km
5 km
10 km
15 km
Na
Nv
Na
Nv
Na
Nv
Na
Nv
A
1.5
2.0
1.2
1.6
1.0
1.2
1.0
1.0
B
1.3
1.6
1.0
1.2
1.0
1.0
1.0
1.0
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Ground Response Coefficients
The two ground response coefficients, Ca and Cv,
give indication of the vibration amplification
capacity of a soil depending on zone factor (Z), soil
profile factor (S) and the near-source factors (Na
and Nv).
The fundamental period of a structure determines
whether Ca or Cv is more important for design of a
structure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Table. Ground Response Coefficients, Ca and Cv.
Soil
Profile
Zone 1
Zone 2A
Zone 2B
Ca
Cv
Cv
Zone 3
Ca
Cv
Zone 4
Ca
Cv
Ca
Ca
Cv
SA
0.06
0.06
0.12
0.12 0.16 0.16 0.24 0.24 0.32 Na 0.32 Nv
SB
0.08
0.08
0.15
0.15 0.20 0.20 0.30 0.30 0.40 Na 0.40 Nv
SC
0.09
0.13
0.18
0.25 0.24 0.32 0.33 0.45 0.40 Na 0.56 Nv
SD
0.12
0.18
0.22
0.32 0.28 0.40 0.36 0.54 0.44 Na 0.64 Nv
SE
0.19
0.26
0.30
0.50 0.34 0.64 0.36 0.84 0.36 Na 0.96 Nv
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
When soil parameters are unknown, soil profile
type SD may be assumed in seismic zones 3 and 4
and profile SE may be assumed in other zones.
For a regular structure, the near source factor
needs not exceed 1.3.
Fundamental Time Period Of A Structure
The time period of a structure may exactly be
calculated by performing free vibration analysis of
the structure, which involves lengthy calculations.
Following empirical methods are also available to
reasonably guess the fundamental time period of
a structure:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Approximate method
Fundamental time period, T =
number of stories
sec
10
Method A of UBC
TA
= Ct (hn )
3
4
where
= height of the roof above the base in
hn
meters, not including the height of parapets.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Ct
= 0.085 for steel moment resisting frames
= 0.073 for reinforced concrete moment
resisting frames and eccentric braced steel
frames
= 0.050 for all other buildings
Method B of UBC
TB
= 2π
∑W δ
g∑ f δ
i
2
i
i
i
≤ 1.4TA for Zones 1,2 and 3
≤ 1.3 TA for Zones 4
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
where
δi
=
=
ki =
fi =
wi =
and
g =
=
static elastic deflection at level “i” due
to the forces applied at all levels,
increasing in a linear way with height.
The value of deflection must be with
respect to the base in mm.
total lateral force at i - th floor
+ δ i −1
ki
shear stiffness of columns under floor “i”
lateral force at level “i”, N
dead load located at level “i”, N
acceleration due to gravity
9810 mm/sec
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Ductility
Ductility of an element shows its capacity to
deform in the inelastic range without collapse.
Due to these inelastic deformations, the energy
is dissipated making the structure relatively
stable against earthquake forces.
If these deformations successfully occur in the
two opposite directions causing reversal of
stresses in the members, hysteresis loops are
produced dissipating energy in each cycle of
loading, unloading and loading in the opposite
direction.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Response Modification Factor (R)
The response modification factor of a structure (R)
is ratio of the seismic base shear of an elastic
system to a reduced design base shear depending
upon ductility, energy absorbing capacity, increase
in natural period due to yielding and increase in
damping ratio of the structure.
If shear walls or braced frames provide support to
gravity loads and all the lateral loads, the structural
system is a Bearing Wall System (BWS). In other
words, the gravity loads are resting on walls.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
If separate systems are provided to resist lateral and
gravity loads, the structural system is called Building
Frame System (BFS).
No special detailing is required for gravity load
supporting frames.
Special Moment Resisting Frames (SMRF) are
frames specially detailed to provide high ductility and
support for lateral and gravity loads by flexural action.
Moment Resisting Frames With Masonry Shear
Walls are called MRWF systems.
Dual Systems are those in which more than one
systems are used together.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Table. Response Modification Factor, R.
Structural System
R
Height Limit
(m)
1. BWS with concrete or masonry shear walls
4.5
49
2. BWS with steel braced frames
4.4
49
3. BFS with steel eccentrically braced frames
7.0
73
4. BFS with concrete shear walls
5.5
73
5. BFS with masonry shear walls
5.5
49
6. BFS with steel ordinary braced frames
6.4
73
7. Steel or concrete SMRF
8.5
None
8. Masonry MRWF
6.5
49
9. Concrete shear walls with SMRF
8.5
None
Many other types given in UBC
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
The value of the response modification factor (R) is
determined from consideration of a structure’s
over-strength capacity beyond the point at which
the elastic response of the structure is exceeded.
The value of R always exceeds unity, which
indicates that all structures are designed for forces
less than would be produced in a completely
elastic structure.
This reduced force level is made possible by the
energy absorption and dissipation capacity of the
structure at displacements in excess of initial yield.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Seismic Importance Factor (I)
The factor is equal to 1.25 for essential and
hazardous facilities and 1.00 for special
occupancy, standard occupancy and
miscellaneous structures.
Seismic Response Coefficient (Cs)
The seismic response coefficient (Cs) is the
fraction of total dead load of the structure that is
acting as base shear on the structure.
This means that the base shear (V) is: V = Cs W.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
This factor depends upon velocity of acceleration
based ground response coefficient (Cv or Ca),
importance factor (I), response modification factor
(R) and time period (T).
Cv
and Ta = 0.2Ts
Response time Ts =
2.5C a
Cv I
(if T > Ts) subjected to maximum
Cs
=
RT and minimum values
Max. value = 2.5 Ca I / R
(Controls when T = Ta to Ts)
Min. value = 0.11 Ca I
(OR) 0.8ZNvI / R for zone-4
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Seismic Dead Load (W)
The seismic dead load (W) consists of the
following:
i)Dead load of the structure.
ii)25 percent of the floor live load for storage
warehouses.
and
iii)A minimum allowance of 50 kg/m2 for movable
partitions.
iv)The total weight of permanent equipment and
fittings.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Magnitude Of Base Shear (V)
1 UBC refined formula
Base shear V = Cs W
Maximum inelastic displacement ∆m = 0.7 R ∆s
Where ∆s = the displacement corresponding to
the shear V, given above.
2 UBC simplified formula
Base shearV = (3.0 Ca / R)W
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
This is a conservative formula having the
following restrictions:
i)Ordinary occupancy type.
ii)Light-frame construction not exceeding 3
stories.
iii)Any construction, except bearing wall
systems, but not exceeding two stories.
Distribution Of Base Shear
At Various Story Levels
V ′W x a x
Shear at a particular story, FX =
∑ Wi a i
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Where
level-i
level-x
and
level-x
V′ = modal base shear
Wi= seismic dead load at level-i
ai= mode shape component at
for the given mode
wx= seismic dead load located at
ax= mode shape component at
for the given mode
For uniform distribution of mass over height and
for first linear mode, the distribution of base
shear may be simplified as follows:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
W x hx
FX = V1
∑ Wi hi
Where V1 = base shear corresponding to first
mode
hi = height above the base to level-i
and
hx = height above the base to level-x
In order to account for higher mode effects in the
above expression for long period buildings, an
additional force Ft is added at the top of the
structure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Ft = 0.07 T V when T > 0.7 sec
Where
V= total base shear
= Ft + Σ Fx
In such cases: FX
W x hx
= (V − Ft )
∑ Wi hi
RESPONSE SPECTRUM ANALYSIS (RSA)
Spectral response means the maximum
displacement, velocity or acceleration response.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Pseudo-Acceleration is the maximum
displacement of the structure multiplied with
square of natural circular frequency.
Pseudo-Velocity is the maximum displacement
of the structure during an earthquake multiplied
with the natural circular frequency.
Sd
Sv
= spectral displacement
= spectral velocity
Sa
= ω Sd
= spectral acceleration
= ω Sv
= ω 2 Sd
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Because of this inter-relationship, all three spectra
may be plotted on the same graph using tripartite
axes and logarithmic scales.
A response curve or response spectrum is a
graph of the spectral (or maximum) response of a
range of single-degree-of-freedom oscillators to a
specified ground motion, plotted against the
frequency or period of the oscillators.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
1,000
500
200
Spectral Velocity
100
(Sv), mm/sec (on
log scale)
50
ξ=0%
ξ=2%
ξ=5%
ξ = 10 %
20
10
50
0.05
0.10
0.20
0.50
1
2
Time Period, sec (on log scale)
5
A Typical Out-of-Scale Response Spectrum.
10
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Procedure To Use Response
Spectra For SDOF Systems
The procedure to use response spectrum to calculate
the earthquake lateral forces for single degree of
freedom systems is summarized as under:
i)
Calculate angular speed ω and time period T
for the structure.
ii)
Estimate the damping ratio, ξ.
iii)
Use applicable response spectrum for a
particular area and find Sd or Sv or Sa (the
values are inter-convertible) against the time
period.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
iv)
Find shear force in each column as:
S.F
v)
= Sd × k
Find the total lateral force by adding shear
forces in all the columns.
Procedure To Use Response
Spectra For MDOF Systems
The procedure to use response spectrum for the
calculation of the earthquake lateral forces in
case of multiple degrees of freedom systems is
summarized as under:
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
1. Calculate angular speed ω and time period T
for the desired mode of vibration of the structure.
2.
Find the mode shape ai.
3.
Estimate the damping ratio ξ.
4. Find Sd, Sv and Sa from the response
spectrum or calculate others after knowing one
out of these.
5.
Calculate the effective weight as follows:
(∑W a )
(∑W a )
2
WE
=
i
i
i
2
i
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
6. Calculate the base shear as follows:
V = WESa / g
7. Find the lateral force at each level as follows:
Wi ai
Fi = V
∑Wi ai
Example : Determine the base shear for the
fundamental mode with ξ = 0.05. Also determine
the lateral load at each level for the fundamental
mode.
1.00
ω= 15.1 rad/sec




ai = 1.95 
2.86


Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
3
2
1
880 kg/m2
590 kg/m2
880 kg/m2
30m × 30m
Frame For Example.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Solution:
Dead Loads:
Level 1 & 3 = (880)(30)(30) = 7770 kN
Level 2
= (590)(30)(30) = 5209 kN
ω = 15.1 rad/sec
T = 2π / ω = 0.416 sec
From the response spectrum, Sv = 350 mm/sec
Sa = ω Sv = 5285 mm/sec2 = 5.285 m/sec2
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
The calculations are made in Table using the
following expressions:
(∑W a )
(∑W a )
2
WE
=
V =
i
i
i
2
i
WESa
Wi ai
/ g and Fi = V
∑Wi ai
Table. Calculation Sheet For Example.
Wi
ai
Wiai
Wiai2
Fi
kN
mm
kN-mm
kN-mm2
kN
1
7770
1.00
7770
7770
1844
2
5209
1.95
10158
19807
2411
3
7770
2.86
22222
63555
5275
Total
20749
40150
91132
9530
Level
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
=
(40150)
(91132)
=
17689 kN
2
WE
V =
17689 × 5.285
= 9530 kN
9.81
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
UBC RESPONSE
SPECTRUM METHOD
UBC Design Response Spectrum is an elastic
response spectrum for 5 percent equivalent
viscous damping used to represent the dynamic
effects of the Design Basis Ground Motion for
the design of structures.
This response spectrum may be developed for a
site-specific spectrum based on geologic,
tectonic, seismological and soil characteristics
associated with a specific site.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Alternatively, the spectrum constructed in
accordance with the spectral shape given in UBC
using the site-specific values of Ca and Cv and
multiplied by the acceleration of gravity, 9.815
m/sec.2.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
According to UBC 1633, the following
requirements must be satisfied:
1.
Only the elements of the designated
seismic-force-resisting system need to be
used to resist design forces.
2.
The individual components should be
designed to resist the prescribed design
seismic forces acting on them.
3.
All building components in Seismic Zones
2, 3 and 4 shall be designed to resist the
effects of the calculated seismic forces and
the effects of gravity loadings from dead,
floor live and snow loads.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
4.
Consideration shall be given to design for
uplift effects caused by seismic loads.
5.
In Seismic Zones 2, 3 and 4, provision should
be made for the effects of earthquake forces
acting in a direction other than principal axes.
6.
If the axial load in the column due to seismic
forces acting in either direction is less than 20 %
of the column capacity, the orthogonal effects
may simply be considered by designing for 100
% of the design seismic forces in one direction
plus 30 % of the design seismic forces in the
perpendicular direction.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
7.
The combination requiring the greater
component strength should be used for design.
8.
Alternatively, the effects of the two orthogonal
directions may be combined on a square root of
the sum of the squares (SRSS) basis.
9.
When the SRSS method of combining
directional effects is used, each term computed
shall be assigned the sign that will result in the
most conservative result.
10.
The strength and stiffness of the framing
between the base and the foundation shall not
be less than that of the superstructure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
According to UBC 1633.2.7, concrete frames
that are part of the lateral-force-resisting
system should conform to the following:
1.
Should be special moment-resisting
frames in Seismic Zones 3 and 4.
2.
Should be a minimum of intermediate
moment-resisting frames in Seismic
Zone 2.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
According to UBC 1631.4.1, response spectrum
analysis is defined as an elastic dynamic analysis
of a structure utilizing the peak dynamic response
of all modes having a significant contribution to
total structural response.
Peak modal responses are calculated using the
ordinates of the appropriate response spectrum
curve which correspond to the modal periods.
Maximum modal contributions are combined in a
statistical manner to obtain an approximate total
structural response.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
This condition is considered satisfied by
demonstrating that for the modes considered, at
least 90 percent of the participating mass of the
structure is included in the calculation of response
for each principal horizontal direction.
The peak member forces, displacements, story
forces, story shears and base reactions for each
mode should be combined by recognized methods.
When three-dimensional models are used for
analysis, modal interaction effects shall be
considered when combining modal maxima.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
Reduction of Loads Based On Ductility
According to UBC 1631.5.4, the Elastic
Response Parameters may be reduced for
purposes of design in accordance with the
following items, with the limitation that that the
corresponding design base-shear should not be
less than the Elastic Response Base Shear
divided by the value of R.
The corresponding reduced design seismic
forces shall be used for design.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
1.
For all regular structures where the ground
motion representation complies with Ground
Motion definition of UBC using design
spectrum, Elastic Response Parameters
may be reduced such that the corresponding
design base shear is not less than 90
percent of the base shear determined in
accordance with Static Force Procedure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
2. For all regular structures where the ground
motion representation complies with Ground
Motion definition of UBC using site-specific design
spectrum, Elastic Response Parameters may be
reduced such that the corresponding design base
shear is not less than 80 percent of the base
shear determined in accordance with Static Force
Procedure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
3. For all irregular structures, regardless of the
ground motion representation, Elastic
Response Parameters may be reduced such
that the corresponding design base shear is
not less than 100 percent of the base shear
determined in accordance with Static Force
Procedure.
Prof. Dr. Zahid A. Siddiqi, UET, Lahore.
4. The vertical component of ground motion may
be defined by scaling corresponding horizontal
accelerations by a factor of two-thirds.
Alternative factors may be used when
substantiated by site specific data. Where the
Near Source Factor, Na, is greater than 1.0,
site-specific vertical response spectra should be
used in place of the factor of two-thirds.