Dynamic of structures software progress report No 9.

Dynamic of structures software progress report No 9.

In order to find the response of a structure under an external dynamic force (an earthquake) we solved the eigenvalue problem, whose solutions gives the natural frequencies and modes of a system, u(t)=qn(t)Øn

Where the deflected shape Øn does not vary with time. The time variation of the displacements is described by the simple harmonic function:

qn(t) = AnCosWnt +BnSinWnt

Therefore, u(t)= Øn (AnCosWnt+BnSinWnt)

Where Wn and Øn are unkown.

Taking into account the equation of motion: mü +ku= 0

We could determine:

Therefore K Øn = Øn

The problem is to determine the scalar and the vector Øn.

We obtain the frequency equation: [K-m] Øn =0

Which has a Non trivial solution: Det[K-m] Øn =0

When the determinant is expanded, a polynomial of order N in Wn is obtained.

The N roots of the frequency equation determine the N natural frequencies Wn (n=1,2,3…) of vibration. When a natural frequency is Known Wn, The equation can be solved for the corresponding vector Øn to within a multiplicative constant.

There are N independent vectors Øn, which are also known as natural modes of vibration, or natural mode shapes of vibration.

To commence, we used the elastic bending theory to determine the deformations of a three story structure subjected to unitary load.

Then we obtained the 6x6 stiffness matrix of the structure:

Then, we used the Goyan method to condense the 6x6 stiffness matrix in order to find the lateral stiffness matrix of the structure.

{Ktt}

Lateral stiffness matrix:

With the lateral stiffness matrix {Ktt} of the structure, we solved the frequency equation:

Det[K-m] Øn =0

We also solved the polynomial equation to find the four frequencies of vibration Wn of the structure.

Then we normalized the frequency equation to obtain the natural modes of vibration Øn.

So the latest version of the dynamics of structures software DynRFRv9.0 could calculate the frequencies, periods of vibration and the internal forces of any given three story structure.

Since structural design is based on the peak values of forces and deformations over the duration of the earthquake-induced response.

Therefore, we used the Peak Modal Response analysis to determine the peak value Rno of the nth-mode contribution Rn(t)to the response r(t) using the earthquake response spectrum or design spectrum.

Rno= rn*An

Where An is the pseudo-aceleration.

The modal static response rn is calculated by static analysis of the building subjected to lateral forces Sn.

With the new update, the software is able to calculate the base shear Vb and the base overturning moment for a three story building in each mode of vibration.