Structural dynamics

1)

<> Static analysis

• variation of load with time is negelected in this analysis.
• Motion of structure can be disregarded.
• only equivalent lateral force is considered in equilibrium equation.
• we don't need to consider the effect of inertia and damping in the equation of motion.

<> Dynamic analysis

• variation of load with time is considered.
• displacement of structure is regarded.
• due to displacement of structure, two addational force inertial and damping force are considered in the equation of motion.

2)

<> Elastic behavior

• when SDOF system is subjected to external static force,there is deformation in the system u, if the system is linear elastic, the deformation is not large.
• so external force is directly proptional to deformation,

Fs=Ku , where K= lateral stiffness.

<> Inelastic behavior

• stiffness value is not linear.
• It makes the equation is very complex.

<> Plastic behavior

• when external static force is reached beyound the yield stree of material, deformation is not linear. Then the structure may undergo heavy damage but not totally collapse.

<> Non- linear inelastic behavior

• force-displacement relatiionship becomes non-linear and inelastic at large deformation.
• force to deformation is not single valued and depends on history of deformations and whether deformation is incresing or decreasing.

Fs = fs(u,v)

3) The differential equation governing the displacement of an idealized SDOF of one storey frame subjected to an externally applied force p(t) can be derived using Newton's second law of motion.

m u ¨ + c u + f s (u)= p(t)

It is combination of three pure independent systems:-

• Stiffness component: The frame without damping or mass. The external force on the stiffness component related to displacement 'u'.

Fs=Ku

• Damping component: The frame without stiffness or mass. The external force on the damping component related to velocity 'v'.

Fd=cv

• Mass component: The frame without damping or stiffness. The external force on the mass component related to acceleration 'a'.

Fi=m u ¨

4) Relationship between Tn and f is:

f=1/Tn

• Natural period (Tn): The time required to complete its one cycle oscillatory motion is called natural period.
• Natural frequency(f): The number of cycles completed by system in 1 second is called natural frequency.

5) Response spectrum is a plot of the peak value of a response quantity(u,v and a) to the natural vibration period Tn of the system.

• since we intrested in only peak value of all possible SDF syytem to a particular component of ground motion, so this response spectrum method provides us. It makes calculation easy.
• In the response spectrum, each plot for SDOF systems having fixed damping ratio and we plot several plots for different values of damping ratio.

<> It can be three types:

• Deformation response spectrum
• Relative response spectrum
• Acceleration response spectrum

Graph:

• This is the recorded ground motion
• After that we analyze the deformation with different natural period with fixed damping ratio.

• we get different peak value of deformation in different time period and we put the value and plot the graph and similary get another graph of different damping ratio.

• and if you plot graph between accleration and time period, we get opposite of the graph of deformation vs time period. That means the stiffer the material , less deformation and get higher psuedo acceleration.