Dynamic of structures software progress report No.26

Dynamic of structures software progress report No 23

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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

The problem is to determine the scalar Wn 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 Wn is Known, 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 and to obtain the 6 x 6 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[Ktt-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 DynRFREKv1.0 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.

With the latest update, the software is able to calculate the total modal static response of the building subjected to an earthquake: the story drift, the floor displacement, the base shear and the moments.

We created a simplified version for the dynamic analysis, using the general equation for the beam to column stiffness ratio and we considered a rigid diafragm to perform the algorithm.

We compared the two versions in order to have a better safety factor when performing bigger and more complex structures.

At the same time we are studying the electromagnetic field fundamental concepts in order to assembly the Electromagnetic matrix with the mass matrix and the stiffness matrix as part of our research goal.

We completed the electrostatic field review. We are now able to calculate the electric field intensity using the Gauss' laws, also we can calculate the potential, the electric flux as well as the potential energy of the field.

Now, we are heading to the electrodynamics field, in particular to review the numerical methods to solve the required equations we will need to incorporate the electromagnetic matrix into the mass matrix.

We completed the chapter for the general idea of numerical solutions to find the matrix of potentials V.

We dare to formulate the first attemp to couple the matrix the mass with the matrix of potentials V in the equation of motion of the structure subjected to an earthquake in the following way:

mü*Ep +ku*Ep= 0

Where m is the matrix of mass, K the matrix of stiffness of the structure and Ep the matrix of potentials.

The Electrostatic elemental matrix could be considered as equivalent to the stiffness matrix.

Solving the potentials at the nodes in the mesh created for the elemental matrix, could be the equivalent to solving the displacements at the nodes for the stiffness matrix!

Now we are planning to assembly the electrostatic elemental matrix with the stiffness matrix to solve the static problem before to procede with the dynamic problem.

Now we completed the chapter XI Maxwell's Equations.

We redefined the equation of motion, having a different approach to the problem, using the magnetic forces instead of the magnetic field.

mü* +ku*-IB= -müg(t)

Where B is the magnetic flux density due to the current element dI' at a distance R in space.

We commence to develop the magnetic flux denisty matrix for each element:

We are expecting to have a similar approach than Maglevs for the soil-structure interaction while developing a novel concept for the upper part of the structure.

We used a single steel frame with two circuits to calculate the electromagnetic force at the columns, the Electromagnetic force vector will oppose the base shear force on the frame.

We performed a new iteration using a higher permetivity material to calculate the Electromagnetic force in the base of the steel frame.

The total Electromagnetic force at the base of the steel frame is enough to counterbalance the acting base shear force.

We reviewed different classical cases using different values for the RSA (Response Spectrum Analysis) method and compared the results to the NEW RSA and Electromagnetism method.

We calculated the total response of the building, including the electromagnetic forces generated by the capacitors to counterbalance the basal shear forces generated by the earthquake. With the total forces we now calculated the internal forces of the structure and the support reactions, both considering the RSA case and the RSAEK case.

We completed the calculation of the total internal forces and the total response of the frame, both considering the RSA clasic case and the new RSAEK.

We are solving different cases with different input values in order to determine the main factors which affect the structure.

The preliminar configuration for the electromagnetic field is based in placing capacitors in each floor of the structure, to generate a current of 4,000 amperes with a magnetic flux density of 40 tesla and a force of 160,000 Newtons to counterbalance the horizontal forces in each floor due an earthquake.

We updated the algorithm by enhancing the matrix to 8x8 to calculate the total response of a fourth story building subjected to an earthquake.

We calculated the Adjugate sub-matrix Koo and inverted it. Dealing with the 8x8 matrix will determine our path to scale up the algorithm.

We condensed the 8x8 matrix to obtain the lateral stiffness matrix Ktt which was used to solve the equation of frequencies for the four story building.

The algorithm is now able to calculated the total response for the four story building with six modes of vibration.

The equation of inertial reduction in function on time was solved for the four story building.

mr=Fn-Fek/(TT*Phi*An)

Several topological examples were performed for the new 8x8 matrix.

We already commenced to develop the 10x10 matrix for the five story building.