Dynamic of structures software progress report No 7.





Dynamic of structures software progress report No 7.

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)PHIn.

Where the deflected shape PHIn 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)= PHIn(AnCosWnt+BnSinWnt)

Where Wn and PHIn are unkown.

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

We could determine: [ -W**2mPHIn+KPHIn]qn(t) =0

Therefore KPHIn=W**2mPHIn

The problem is to determine the scalar W**2 and the vector PHIn.

We obtain the frequency equation: [K-W**2m]PHIn =0

Which has a Non trivial solution: Det[K-W**2n]=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 PHIn to within a multiplicative constant.

There are N independent vectors PHIn, 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.

We obtained the 6x6 stiffness matrix of the structure:


Stiffness Matrix















K11

K12

K13

K14

K15

K16



39573,33333

26382,22222

13191,11111

197866666,7

-2968000

-2968000



K21

K22

K23

K24

K25

K26



26382,22222

39573,33333

26382,22222

-2968000

197866666,7

-989333,333



K31

K32

K33

K34

K35

K36



2968000

2968000

39573,33333

-989333,3333

-989333,3333

197866667



K41

K42

K43

K44

K45

K46



1978666,667

2968000

297789333,3

593600000

989333,3333

989333,333



K51

K52

K53

K54

K55

K56



13191,11111

989333,3333

989333,3333

989333,3333

593600000

989333,333



K61

K62

K63

K64

K65

K66



989333,3333

989333,3333

989333,3333

989333,3333

989333,3333

6595,55556










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


ktt11

Ktt12



6004666,522

6641050,804

0

0

Ktt21

Ktt22



37111,5029

-594374,4346

0

0

Ktt31

Ktt32



6580,470696

23668,25634

0

0

Ktt41

Ktt42



-1963756,504

-2941792,867

0

0

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

Det[Ktt-W**2m]=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 PHIn.

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 three story structure.

The next step is to scale up the software in order to solve bigger structures.

The big picture: we would like to use a Maglev Train type of system for buildings, therefore we need to find the size of the magnetic field located at the building foundation in order to reduce the pseudoaceleration effect produced by any earthquake.