fundamentals of dynamic analysis

1. 1
CE 412: Introduction to Structural Dynamics and
Earthquake Engineering
MODULE 1
Fundamentals of Structural Dynamics

2. Static Vs Dynamic Force
Static Force
A Static force is one which does not produces
acceleration in the acting body . A Static force may
or may not vary with time
2
P(t)
u(t)
Force, P , does not change with time

3. Static Vs Dynamic Force
Static Force
3
P(t)
u(t)
Force, P , change with time but does not produce acceleration in the beam

4. Static Vs Dynamic Force
A Dynamic force not only varies with time but
also produces acceleration in the acting body
i.e a = dv/dt= d2u/dt2 ≠ 0.
4
Why inertial force in the
beam is maximum at the
point of application of
dynamic force, P(t) ?
P(t)
t
2o curve
1o curve
t1 t2
u(t)
t
2o curve
1o curve
t1 t2
Dynamic Force

5. Effect of Dynamics Force on structural response
5
(Po)static
-
BMD due to static force, Po
(Po)dynamic
pI = ma
-
+
BMD due to dynamic force, Po

6.  Consider a model building mounted on a
truck with most of mass lumped at the roof
level.
 Inertia force , FI , in the model building are
produced in leftward direction when the
truck move in the right ward direction with
certain acceleration.
 Resultant FI act at the roof level since
greater portion of mass is lumped there.
6
How Inertial force are produced
Force exerted by
truck’s engine
FI
a≠0
 Model will overturn , if the destabilizing
moment due to FI at the bottom of model
exceeds stabilizing moment due to resultant
weight of model

7. The response of structures to static load is different than its response to dynamic load.
Dynamic loading may cause large displacement and severe stresses, especially in cases
where the frequency of loading is close to the natural frequency of structures.
Fluctuating stress, even of moderate intensity, may cause material failure through fatigue.
Vibratons may at times cause wearing and malfunction of machinery.
The vibration from one machine may transferred to a delicate instrument through support
structures.
Vibration cause discomfort to the occupants
Why to Study Dynamics of Structures?

8. Classification of Dynamic loads
Dynamic loads may be broadly classified as ‘Deterministic’ and ‘Non-deterministic’.
If the magnitude, point of application of the load and the variation of the load with
respect to time are known, the loading is said to be Deterministic and the analysis of a
system to such loads is defined as Deterministic analysis.
On the other hand, if the variation of load with respect to time is not known, the
loading is referred to as Random or Stochastic loading and the corresponding analysis is
termed as Non-deterministic analysis.

9. Classification of Dynamic loads
Dynamic loadings can be also divided into periodic loadings and non-periodic loadings.
Below given table summarizes the different types of dynamic loadings that are encountered
in civil engineering. Permanent and live loads that are applied slowly compared to the period
of vibration of structures are generally considered static loadings, such as dead loads.

10. Periodic Loading
Harmonic loadings
The simplest periodic loading varies as a sinusoid and is called simple harmonic loading .
This type of loading is generated by rotating machines video1and exciters with unbalanced
masses and it gives rise to the resonance phenomenon when the excitation period matches the
structure’s natural period of vibration.
A Periodic loading repeats itself after a regular time interval, T , called the period.
 Periodic loadings can be divided into Simple Harmonic loadings and Arbitrary Periodic
loadings.

11. Periodic Loading
Arbitrary periodic loadings
Arbitrary periodic loadings repeat themselves at regular interval of time.
P
t
This type of loading is generated by reciprocating machines, by walking or jogging by one or
many persons crossing a pedestrian bridge (Figure on left), by rhythmic jumping and dancing
by one or many persons on a floor, by hydrodynamic forces generated by the propeller of a
boat (Figure on the right), by waves, etc.

12. Non-periodic loadings
Non-periodic loadings vary arbitrarily in time without periodicity.
Non-periodic loadings can be divided into Impulsive short-duration loadings and
Arbitrary long duration transient loadings.
Impulse loadings
Impulse loads have a very short duration with respect to the vibration period of
the structures and are caused by explosions (see below figure), shock, failure of structural
elements, support failure, etc.

13. 13
Non-periodic loadings(Impulse loadings)
Variation of blast loading w.r.t time and its effect
1
1
2
1
3
1
4
1
5
1
1
1
2
1
3
1
4
1
5
1

14. 14
Non-periodic loadings(Arbitrary loadings )
Arbitrary loads are of long duration and are caused by earthquakes, wind, waves, etc.

15. Dynamic response
The strains, stresses, internal forces, and reactions are determined once the
displacement time history is known.
The end result of the deterministic analysis of a structure excited by a given
dynamic loading is the dynamic response expressing the displacements of the
structure with time, which is also called the displacement time history.
Dynamic response varies with time. However, for design or verification, all that is
required is the maximum dynamic response which, for a linear system, can be added
to the maximum static response to yield the maximum total response.

16. 16
Various types of waves video 2
What happens during an earthquake?

17. What happens to the structures?
Inertia force and relative motion within a building
The upper part of the structure
however ‘would prefer’ to remain
where it is because of its mass of
inertia.
If the ground moves rapidly back and forth, then the foundations of the
structures are forced to follow these movements due to the friction at the base.
17
Resultant Inertia force
Foundation movement

18. 18
Variation of horizontal acceleration at various story levels in San Francisco’s
Transamerica Pyramid during to 1989 Loma Prieta Equake
1. Why accelerations increases along the height? Video 3
Amplification of accelerations along height
2. Will structure fail at story with maximum inertial
force (top floor) or ground story?

19. The Structural Response* during an earthquake mainly depends:
19
Factors influencing Structural Response
* Response is the structural system reaction to a
dynamic force.
Thus a response quantity may be structural
displacement, velocity, acceleration, internal shear,
bending moment, axial force etc.
1. Natural time period video 4
2. Configuration, material, structural system, age, or quality of construction. video 5

20. In comparison with rock, softer soils are particularly prone to
substantial local amplification of the seismic waves.
20
Note the amplification of
ground displacement with
decrease in soil stiffness
Influence of local soil conditions on structures

21. 21
The 1.6 mile long cypress freeway structure in Oakland, USA, was built in the 1950s.
Part of the structure standing on soft mud (dashed red line) collapsed in the 1989
magnitude 6.9 Loma Prieta earthquake.
Influence of local soil conditions on structures
Adjacent parts of the structure (solid
red) that were built on firmer ground
remained standing.
Seismograms (upper right) show that
the shaking was especially severe in
the soft mud.video 5a

22. The dynamic response of structural systems, facilities and soil is very
sensitive to the frequency content of the ground motions.
The frequency content describes how the amplitude of a ground motion is
distributed among different frequencies.
Using Fourier transformation (mathematical technique) we can find the
frequency content of seismic waves by shifting from time domain to
frequency domain
Frequency content parameter
22

23. Frequency content parameter
23
The plot of Fourier amplitude
versus frequency is known as a
Fourier amplitude spectrum

24. Frequency content parameter
24
Fourier amplitude spectrum of a
strong ground motion expresses the
frequency content of a motion very
clearly.

25. Fundamentals of Dynamics Analysis
25

26. 26
Structural Degrees of Freedom
Degrees of freedom (DOF) of a system is defined as the number of
independent variables required to completely determine the positions
of all parts of a system at any instant of time.

27. Continuous and Discrete systems
Some systems, specially large system involving continuous elastic members, have
an infinite number of DOF. These are referred to as Continues or Distributed mass
system. An example of this is a cantilever beam with self weight only .
27
𝑢̈
𝑢̈
𝑢̈
The beam have infinite DOF because its Dynamic analysis requires determination of
displacement, u, inertia forces, FI=ma for each point on the beam. Since distributed
mass is split into infinite number of small masses hence determination of acceleration
associated with each each mass mean calculation of infinite accelerations.

28. Discrete/ Lumped mass systems
4 DOF Lumped mass system corresponding to the above given cantilever beam with distributed
mass. ρ = Mass per unit length.
(How? there are 5 lumped masses.) 28
In most cases, for practical reasons, continuous systems are approximated as discrete systems
with finite number lumped masses. Higher number of lumped masses leads to better accuracy
Distributed mass system
Transformed Lumped
mass system

29. Structural DOF for Static Analysis
Usually beam and connected slabs have axial stiffness thus u4=u1. Moreover, for low-
rise structures column longitudinal deformations can be neglected with very little
impact on accuracy thus u2=u5=0.
Actual DOF Reduced DOF
Thus DOF = 3 in this case. i.e. 1 translation, u21 and 2 rotations (u2 & u3 )

30. Structural DOF for Dynamic Analysis
Sine no mass exist on joints, therefore, translational moment of inertia in vertical direction is
neglected (i.e. u3 & u6 as shown in figure =0).
Actual DOF Reduced DOF
This assumption also implies that u2=u5=0. Similarly as mentioned on previous slide ,u4=u1.
DOF = 1 in this case

31. 31
In a single degree of freedom system, the deformation of the entire structure can be
described by a single number equal to the displacement of a point from an at-rest position.
True single DOF systems are extremely rare in practice and are most often idealizations
resulting from simplifications of the distribution of the essential properties (mass, stiffness
and damping ) of a mechanical or structural system.
The study of SDOF systems is justified by the fact that dynamic analysis results obtained
from such simplified systems are often very close to the exact solution.
Another very important reason for studying SDOF systems (demonstrated in Module 8)
is that the response of complex linear elastic systems can often be obtained by superposing
individual SDOF system responses.
Single Degree-of-Freedom (SDOF) System

32. Idealization of a structural system as SDOF system
The structural system of water tank
may be simplified by assuming that
the column has negligible mass along
its length*. This means that we can
consider that the tank is a point mass
32
* This is reasonable, assuming that the tube is hollow and that the
mass of the tube is insignificant when compared with the mass of the
water tank and water at the top.
SDOF model of water tank
This 3-dimensional water tower may
be considered as a single degree of
freedom system when one considers
vibration in one horizontal direction
only.

33. 33
ρ = Mass per unit height, H= total height, y = Any distance along height
and k = lateral stiffness of cantilever member = EI/H3
0.227 ρ .H
Lumped mass idealization (on the right side) corresponding to
vertical distributed mass system (on the left side) is made by
using a shape function (?), Ψ(y) = [1-Cos (πy/2H)].
SDOF Lumped mass idealization of a cantilever wall with
uniformly distributed mass
H
Ψ(y)

34. An overhead water tank is supported on RC tower with hollow x-section. Total weight of tower
is 0.1W. Where W is the weight of water tank when full of water . Using the shape function
given on previous slide, determine the corresponding SDOF lumped mass model.
Comment on the result
 Mass of tower= 0.1m
 According to shape function, Ψ(y) = [1-Cos (πy/2H)], the mass of tower lumped at the
top is 0.227* total mass.
Total mass lumped at the top end = m+0.227*0.1m = 1.023m. Conclusion ??
k=3EI/H3
H
1.023m
H

35. In a Multi degree of freedom system, the deformation of the entire structure cannot be
described by a single displacement. More than one displacement coordinates are
required to completely specify the displaced shape.
Multiple Degree-of-Freedom (MDOF) System
Considering all DOFs : DOF = 6
m2
m1
u1(t)
u2(t)
DOF = 2 for dynamic analysis
m2
u1(t)
u2(t)
m1

36. Multiple Degree-of-Freedom (MDOF) System
What is the DOF for this system…?
36
DOF is 2 when we have a flexible beam
m
u1
u2
DOF can be taken 1 when flexural stiffness of
beam is very high as compared to column
m
u1
u2 =0

37. Exercise No. 3.1
Determine the DOF of systems shown in given figures.
(a)
(b)
(c)
(d)
(e)
37

38. 38
Vibrations
Vibration is “the rapid to and fro motion of an elastic / inelastic system
whose equilibrium is disturbed”
0
Activity….
Graphically represent the position of mass for positions 0,1,2, 3

39. Physical Explanation of Vibration video 6
1 2
3
39

40. 40
Periodic and Random vibrations
Just like type of dynamic loading, Vibrations can be Periodic (cyclic) or Random (arbitrary).
If the motion is repeated after equal intervals of time, it is called Periodic motion.
The simplest type of periodic motion is Harmonic motion.
Periodic vibration (Harmonic vibration)
Periodic vibration (Non-harmonic vibration)
u

41. 41
Random vibration
u
t
Periodic and Random vibrations
Such type of vibrations are produced in a system due to wind, earthquake, traffic etc

42. 42
Free vibrations vs. Forced vibrations
When a structure vibrates without any externally applied forces, such as
when it is pulled out of position, and then released.
Free vibration of a SDOF lumped mass
system when released after being stretched by
a displacement u(0) at the top end .
The vibration of strings on a musical
instrument after they are struck is a common
example of free vibration.
SDOF system always exhibits Simple
Harmonic motion during free vibration video 7

43. 43
Vibration of a system during the presence of an external force is known is known as
Forced vibration.
The vibration that arises in machine such as diesel engines occur due to force exerted
by piston and cylinder arrangement is an example of Forced vibration.
As stated above vibration of a system in the absence of external force is known is
known as Free vibration.
In order for Free vibration to occur, its equilibrium shall be initially disturbed by an
external force. e.g., vibration of rotating machines continues to occur for some time
after power supply is switched off. Similarly, a structure subjected to earthquake
continues to vibrate video 8 for some time after there are no seismic waves to impart
energy to structure
Free vibrations vs. Forced vibrations

44. 44
Undamped vibrations
If no energy is lost or dissipated in friction or other resistance during vibration, the
vibration is known as Undamped vibration video 9
Undamped vibration is a hypothetical phenomena which help in providing an
understanding of the Damped vibration.
In actual system the energy is always lost due to a number of mechanisms. Such
type of vibration is known as Damped vibrations video 10
Damped vibrations

45. 45
Damping
Any energy that is dissipated during motion will reduce the kinetic and potential (or
strain) energy available in the system and eventually bring the system to rest unless
additional energy is supplied by external sources.
The term Damping is used to describe all types of energy dissipating mechanisms.
In structures many mechanism contributes to the damping. In a vibrating building
these include friction at steel connections, opening and closing of microcracks in
concrete, and friction between the structures itself and nonstructural elements such as
partition walls.

46. 46
Damping
Since there is considerable uncertainty regarding the exact nature and magnitude of
energy dissipating mechanisms in most structural systems, the simple model of a
dashpot is often used to quantify damping.
The Dashpot or viscous damper is a ‘device’ that limit or retard vibrations.
Simplified diagram of linear dashpot
Dashpot can be imagined as a cylinder filled with a viscous
fluid and a piston with holes or other passages by which the
liquid can flow from one side of the piston to the other. video 11

47. 47
Energy dissipation in buildings is graphically shown
by diagonally installed (imaginary) dash pots.
Pistons in dash pots move back and forth and cause
viscous friction during the building’s vibration
Damping
Viscous fluid dampers video 12 are
provided in building structures to
dissipate the energy imparted by the
earthquake

48. 48
Damping
 Simple dashpots as shown schematically in figure a exert a force fD whose magnitude is
proportional to the velocity of the vibrating mass.
 The damping force fD is related to the velocity across the linear viscous damper by:
Where the constant c is the viscous damping coefficient
u
c
f D


Dash Pot