Guyan Reduction for Modal/Eigen Analysis of Buildings

[Ronankeating]

Hi all,

Before posting here I've read couple of articles, some books and through google search for this topic.

I've basic perception of what's the reduction, what's the main goal and how we achieve it. But still got a dangling questions in my mind.

For sake of simplicity let's assume that I've 3 storey building (comprised from columns, beams, and slabs) and I want perform the Guyan reduction for modal anlysis or eigenvalue solution on that.

My point of interest is 2 translational(deltaX,deltaY) and 1 rotational(theta-Z) degreee of
freedom per storey basis. The thing that I dont' understand is: normally columns or beams have 6 DOF per node but in order to conform with overall building DOF (3 ) I should reduce / neglect those DOF's which doesn't conform with storey DOFs. (That is, erasing the columns/rows of corresponding local stiffness/mass matrices)

If the procedure is correct, I'd like to know how we actually keep the the statical consistency?
Those stiffness and mass matrices are derived from statical equations and IMHO can not produce statically correct results by simply omitting them from global equation.

Regards,

 

[AlephZero]

You can think of Guyan reduction and other reduction methods in a general way, as a process of applying constraints to the model.

If you have one varable $x_s$ that you want to eliminate, you can write an constraint equation $x_s = L x_m$ which defines how that variable moves relative to the other variables $x_m$ in the model. Then, you can reduce the size of the stiffness and mass matrices by one, by applying the constraint equation.

For Guyan reduction, $L$ represents the static response of the structure. In other words, applying $L$ to the stiffness matrix is exactly the same as eliminating $x_s$ by Gaussian elimination when solving the equation $Kx= F$.

For a dyamics problem, this is equivalent to assuming there are no inertia forces applied to the variable $x_s$. The inertia forces that were applied to $x_s$ it are redistributed to the other variables in the model. Mathematically, you do that by operations on the corresponding row and colum of the mass matrix, using the constraint equation $L$.

For a frame modeled with beam elements, the "rotary inertia" terms on the rotation variables usually represent a small amount of the structure's kinetic energy compared with the "translational inertia" terms, so representing that small amount of KE approximately doesn't introduce significant errors in the lowest frequency vibration modes of the structure.

The Guyan reduction process does not introduce any approximations into the stiffness matrix. You can think of it as being like doing part of the process of Gaussian elimination to solve the equations, then stopping and remembering the "reduced stiffness" corresponding to the variables you haven't eliminated yet, then finishing the solution process later.

The statical consistency is preserved, because when you reduce out a variable in the stiffness matrix, you introduce new off-diagonal terms in the matrix that represent the same coupling using the smaller set of variables. If you try it on a numerical example, you will find that the sparse K and M matrices for the original structure usually become fully populated in the reduced model.

Destroying the sparsity of K and M may seem like a disadvantage of Guyan reduction today, but when it was first developed there were no good methods known for finding eigenvalues of large sparse matrices, so reducing a large sparse matrix to a smaller full one was more or less the only game in town for vibration analysis.

 

[Ronankeating]

First of all, thank you very much indeed for commenting.

If you have one varable xs that you want to eliminate, you can write an constraint equation xs=Lxm which defines how that variable moves relative to the other variables xm in the model. Then, you can reduce the size of the stiffness and mass matrices by one, by applying the constraint equation.

Probably I didn't fully digest the reduction concept, but why we try to get the constrained equation of slaves, IMHO it should be X_m = L * X_s because we try to express the motion of masters as functions of slaves and later on slaves can be omitted before the comprison of Global matrix.

For the rest of your post, I deduce that elimination of unwanted degrees of freedom is performed simply by deleting the corresponding row / columns and doesn't affect the overall solution process. (which similar to Gauss elimination ).


Another question arises, suppose that I have the structure that I mentioned in my previous post, for space frame the well - known beam/column stiffness and mass matrices are 12x12, in order to integrate the slab as lumped mass in dynamic/eigen anlaysis I should derive the mass and stiffness matrices of slab. To solve the slab I actually use the DKT element which has 3DOFS (1 vertical displacement and 2 rotations) .

Based on those fact, what is the mass matrix of slab element how am I supposed to derive it and how to integrate the stiffness and mass matrices of slab to global matrix assembly?

Regards,

 

[Ronankeating]

Your comment will be appreciated.

 

[AlephZero]

Probably I didn't fully digest the reduction concept, but why we try to get the constrained equation of slaves, IMHO it should be $X_m = L X_s$ because we try to express the motion of masters as functions of slaves and later on slaves can be omitted before the comprison of Global matrix.

No, that's the wrong way round. For Guyan, the L matrix is based on the static response of the structure.

Suppose you want to eliminate just one variable from the model. That's not very practical, but the math still has to work right - and with Guyan, if you take N steps of reducing the model size by one variable at a time, in any order, you will get the same final reduced K and M as if you reduced out all the slave variables at once.

Suppose you applied arbitrary independent exterrnal loads to each of the $X_m$ variables, but no load to the $X_s$ variable. Then the single displacement of $X_s$ is defined by the displacement of all the $X_m$'s, not the other way round. The "independent loads" are actually the inertia loads generated by the structure's mass, but for Guyan L is only a function of the stiffness and independent of the mass, so L is the same for any possible mass distribution that the structure might have.

For the rest of your post, I deduce that elimination of unwanted degrees of freedom is performed simply by deleting the corresponding row / columns and doesn't affect the overall solution process. (which similar to Gauss elimination ).

I'm not quite sure what you mean there, but just deleting a row/column is equivalent to fixing that variable to zero displacement, not eliminating it with Gauss elimination.

Another question arises, suppose that I have the structure that I mentioned in my previous post, for space frame the well - known beam/column stiffness and mass matrices are 12x12, in order to integrate the slab as lumped mass in dynamic/eigen anlaysis I should derive the mass and stiffness matrices of slab. To solve the slab I actually use the DKT element which has 3DOFS (1 vertical displacement and 2 rotations) .

Based on those fact, what is the mass matrix of slab element how am I supposed to derive it and how to integrate the stiffness and mass matrices of slab to global matrix assembly?

You can formulate the mass matrix for the slab elements based on their geometry and material properties, similarly to any other element type.

If your slab only has a vertical displacement at a node, and two rotations, presumably you need to restrain the other two displacements and "twisting" rotation of the frame to zero where it joins to the slab, otherwise you have modeled the frame sitting on a sheet of ice, not fixed to its foundation. The three varables that are retained assemble together like any other elements.

I'm not a civil engineer, so I have one question I can't answer - assuming the slab is connected to the earth, then how much of the "mass of the earth" do you actually include in your slab model? If you take the slab mass as very large, that is equivalent to assuming it won't move at all in vibration modes, compared with the motion of the frame - so you might as well not model the slab, and just constrain all 6 DOFs at the base of the frame to zero. I think you need a CE to say what the "standard practice" is for this.

 

[AlephZero]

It might help to write down the bare bones of Guyan using my "L" notation.
Partition the full stiffness, mass, and variables into slaves and masters.

$K = \begin{bmatrix}K_{ss} & K_{sm} \\ K_{ms} & K_{mm} \end{bmatrix} \quad M = \begin{bmatrix}M_{ss} & M_{sm} \\ M_{ms} & M_{mm} \end{bmatrix} \quad X = \begin{bmatrix} X_s \\ X_m \end{bmatrix}$

Guyan reduction makes the approximation that there are no external loads on the slave variables, so the equation for statics is

$\begin{bmatrix}K_{ss} & K_{sm} \\ K_{ms} & K_{mm} \end{bmatrix} \begin{bmatrix} X_s \\ X_m \end{bmatrix} = \begin{bmatrix} 0 \\ F_m \end{bmatrix}$

From the first row we have

$K_{ss}X_s + K_{sm}X_m = 0$ or $X_s = -K_{ss}^{-1}K_{sm}X_m = L X_m$

So we have the equation connecting the full and reduced variables:

$\begin{bmatrix} X_s \\ X_m \end{bmatrix} = \begin{bmatrix} L \\ I \end{bmatrix} \begin{bmatrix} X_m \end{bmatrix}$

The reduced stiffness and mass matrices are therefore

$K' = \begin{bmatrix} L^T & I \end{bmatrix} \begin{bmatrix}K_{ss} & K_{sm} \\ K_{ms} & K_{mm} \end{bmatrix} \begin{bmatrix} L \\ I \end{bmatrix} \quad M' = \begin{bmatrix} L^T & I \end{bmatrix} \begin{bmatrix}M_{ss} & M_{sm} \\ M_{ms} & M_{mm} \end{bmatrix} \begin{bmatrix} L \\ I \end{bmatrix}$

Multiplying out the reduced stiffness and using the definition of L, some of the terms cancel out and the result is the same as for Gaussian elimination:

$K' = K_{ms}K_{ss}^{-1}K_{sm} + K_{mm}$

Nothing "special" happens with the reduced mass matrix, which is just

$M' = L^T M_{ss}L + L^T M_{sm} + M_{ms} L + M_{mm}$

The reduced matrices are of course symmetric, since

$K_{ms} = K_{sm}^T$ and $M_{ms} = M_{sm}^T$

 

[Ronankeating]

Ok, Guyan reduction concept is more clear for me now, it basically focuses on the stiffness relation (ignoring the inertial contributions) writing the unknown variables(X_s) in the form of known variables X_m with the help of constraint matrix L .
For frames element we have, U_x, U_y,U_z, R_x, R_y, R_z per nodes bolded DoFs are to be kept and others should be eliminated.

You can formulate the mass matrix for the slab elements based on their geometry and material properties, similarly to any other element type.

If your slab only has a vertical displacement at a node, and two rotations, presumably you need to restrain the other two displacements and "twisting" rotation of the frame to zero where it joins to the slab, otherwise you have modeled the frame sitting on a sheet of ice, not fixed to its foundation. The three varables that are retained assemble together like any other elements.

I'm getting confused especially with this part, I couldn't realize how the one FEM element (frame) can be integrated with another FEM element(slab) which has different DoFs (triangle). So those DoFs for triangle are exactly that I'm trying to eliminate, so what remains for the rest? If it was a triangular shell element 6 DoFs per nodes, than the case would be different and elimination of unwanted DOFs will result in automatically as DoFs that are to be kept.

As a result, I couldn 't discretize slabs in the building, presuming the slabs as a DKT triangle FEM element, because they don't have enough Dofs , instead shell element should be used.
If my reasoning is correct, could you just approve it?

The issue has got more to do with sub-structuring of building and specific to CE rather than elimination process, elimination is just part of it. Probably there are more convenient or expedient ways out there to do that especially for slabs.

In books also it's been mentioned that, remaining forces, which can not be expressed or transformed to master DOfs, in slave nodes(e.g. U_z, θ_x, θ_y ) should be applied to the building as external forces. If it's so, how that can be achieved because I'll have only U_x, U_y and R_zafter the reduction ?

Thanks for helping,

 

[AlephZero]

The DKT triangle only represents the bending stiffness, right?

If you want to join something (e.g. the building) to the slab, you also need to model the membrane stiffness. For a flat shell the membrane and bending are not coupled, so you could superimpose a 3-node membrane element on the same nodes as the DKT.

If your membraine element only had translation degrees of freedom, that would give you 5 degrees of freedom not 6 at the nodes in the slab. The missing rotation about the vertical direction would correspond to the "torsion" variable in a vertical beam element, so it might not matter much if that was not connected. If the bulding as a whole had a twisting mode about its vertial axis, that will be restrained mainly by the horizontal shear forces where the frame connects to the foundation, not by the "torsiional" rotation of each beam about its own axis.