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Introduction to structural dynamics m 1 m 2 m 3 m n 2 u u 1 3 u n u ........... n-1 m u n-1 ........... p 2 p 1 p 3 n-1 p n p u 1 1 c k 1 1 p k 2 2 c u 2 p 2 n n c k u p n n 1 m 2 m m n
Introduction tostructural dynamics
m1
m2
m3
mn
2u
u 1
3u
nu
...........
n-1mu n-1
...........
p 2
p 1
p 3
n-1p
np
u 1
1c
k11p
k2
2c
u 2
p2n
nc
ku
p
n
n1m 2m mn
Static vs dynamic analysis
• Static analysis is based on the assumption that external loading is applied very slowly to the structures
• Velocities and accelerations are negligible and considered to be equal to zero (no motion)
• No inertia or resistance/friction forces are developed
• All quantities are constant and not functions of time
• Dynamic analysis
• External excitation leads to the motion of the structure
• Every quantity is a function of time
SDOF systems
The simplest type of structures for dynamic analysis are Single Degree Of Freedom (SDOF) systems
SDOF systems can be idealized as a concentrated or lumped mass m supported by a massless structure with stiffness k in the lateral direction
mu
mu
SDOF systems
u
m
For a cantilever column (or beam) with height h the stiffness k is:
3
E Ik 3
h
b
d
where
E is the modulus of elasticity
I=b·d3/12 for a rectangular column
SDOF systems
Free vibration of SDOF systems
The differential equation governing the lateral displacement u(t) of SDOF systems without any external excitation (and with zero damping) is:
mu ku 0
2
2
u: displacement
duu : velocity
dt
d u duu : acceleration
dt dt
No external excitation means that we just pull the SDOF system once, and then let it oscillate (move) freely without any external force.
Free vibration of SDOF systems
The solution of this differential equation is:
u(0)u(t) u(0)cosωt sinωt
ω
where k
ωm
and is called the natural circular frequency of vibration
u(0) and ú(0) are the initial conditions, i.e. displacement and velocity when t=0sec
Free vibration of SDOF systems
Free vibration of SDOF systems
-1.0
0.0
1.0
0 1 2 3 4 5
t / T
u(t
) / u
(0)
T
Free vibration of SDOF systems
T is the time required to complete a free vibration and is called the natural or fundamental period vibration of the system
2π mT 2π
ω k
• An increase of k leads to a decrease of T
• An increase of m leads to an increase of T
Free vibration of SDOF systems
For the idealized SDOF systems studied before, the vibrations continue forever and the systems would never come to rest. This behaviour is unrealistic, of course.
The process by which free vibration steadily diminishes in amplitude is called damping. The energy of the vibrating system is dissipated by various mechanisms such as:
• the thermal effect of repeated elastic straining of the material
• the internal friction
and/or for real structures
• opening and closing of microcracks in concrete structures
• friction between structural and non-structural elements such as infill walls
• friction of steel connections in steel structures, etc
Free vibration of SDOF systems
mu cu ku 0
The differential equation of the free vibration becomes:
k
c m p
u
simple model of a mass-spring-damper system
for free vibration the external excitation p=0
-1.0
0.0
1.0
0 1 2 3 4 5
t / T
u(t
) / u
(0)
Free vibration of SDOF systems
Through damping, the amplitude of the vibration progressively decreases until the system finally rests
MDOF systems
Most realistic structures are Multi Degree Of Freedom systems and can be discretized as systems with a finite number of degrees of freedom
m1
m2
m3
mn
2u
u 1
3u
nu...........
n-1mu n-1
...........
p 2
p 1
p 3
n-1p
np
u 1
1c
k11p
k2
2c
u 2
p2n
nc
ku
p
n
n1m 2m mn
For usual buildings a very common assumption is that mass is concentrated in each floor level (columns are assumed to be massless) and slabs are idealized as rigid (infinitely stiff) within their plane
MDOF systems
• The motion of MDOF systems can be studied through the synthesis of several simple motions (as many as the degrees of freedom of the system) which are independent of the external excitation of the system.
• These motions (or simpler these shapes of deformation) are called modes and are the basis for the dynamic analysis of structures as used in modern seismic codes (EC8, EAK2000, etc.)
• Modes are not equivalent to one another and participate in a different degree to the final response of the structure (usually the first modes are the more important and govern the final response)
MDOF systems
With the assumption that mass is concentrated in each floor level, a 2D (2-dimensional) six storey building can be considered as a 6DOF system
MDOF systems
Vibration according to the first (fundamental) mode.
The first mode usually governs the response of usual buildings
MDOF systems
Vibration according to the 2nd mode.
MDOF systems
Vibration according to the 3rd mode.
MDOF systems
Vibration according to the 4th mode.
MDOF systems
Vibration according to the 5th mode.
MDOF systems
Vibration according to the 6th mode.
