Accidental Torsion and Equivalent Static Load for an irregular structure

Important note: I am writing a structural FEM software from scratch, so I won't able to use existing commercial FEM package. I would need to know how all the important quantities are calculated from the first principles.

I would need to determine the accidental torsional effect ( UBC97, 1631.5.6, pp-27, this accidental torsion effect is needed for both response spectrum and linear time history analysis) for seismic dynamic analysis.

$M_t=F_xe_y$

Where

$F_x$ is the equivalent lateral static force in $x$ direction

$e_y$ is the accidental eccentricities in $y$ direction

Take note that all these involved quantities are evaluated on floor basis.

My structures are quite irregular and general, such as this: For a cantilever column or an one dimensional lumped mass model, the term on floor basis is easy to interpret because there is only 1 node per floor, and the $F_x$ can only be applied on this node itself. Compare this lumped mass model with the above 2D structure FEM model and you can see the difference: But when you have such a 2 dimensional irregular structure, analyzed using FEM, calculating $F_x$ and $M_t$ is no longer easy:

1. How should we obtain $F_x$, given that it is lateral static force, but we are doing dynamic analysis?
2. There are so many nodes on a floor, so on which node(s) should the $M_t$ be applied?
3. In FEM, each beam, column, wall and slab are multiple-node elements. And each node has a internal force value resulted from FEM analysis, what should I do with all these forces at different nodes? The comment here says that I should sum up all of the forces on each related node in order to obtain the correct $F_x$ ( I'm not sure whether I interpret it correctly or not). But I am highly skeptical of this approach, because won't that make $F_x$ dependent on the number of nodes, an artifact of how we mesh an element? How can a real physical quantity increases with the number of nodes we decide to mesh?
4. If 3) is true, then what about the $F_x$ on the wall above a particular floor? Should we sum them in? What about the internal slab and beam? Do they have this force also?
5. How should we intepret $M_t$? Is it internal member force to be resisted by beam/slab? If yes, if there are many slabs and beams into $x$ direction, then how should this $M_t$ be distributed? Or is it an external static force; so one would have to apply this external force on the structure in order to calculate the individual internal member force induced by it ( via the FEM equation $ku=F$)?

So the question is, how to calculate and interpret accidental torsion in a FEM model? I've tried to obtain a general way to reduce a 2 Dimensional general FEM structure into 1 Dimensional lumped mass model, so that I can apply the above formulation easily, but I was told that this wasn't possible.

1. How should we obtain $F_x$, given that it is lateral static force, but we are doing dynamic analysis?

$F_x$ is the sum of the lateral force for a story. If you've performed a dynamic analysis, this is computed (usually) by multiplying your nodal masses by their corresponding nodal accelerations. You should be able to easily output both these from your analysis results.

1. There are so many nodes on a floor, so on which node(s) should the $F_x$ be applied?

This question is ambiguous, but if you are asking which node $F_x$ would be applied to in order to capture accidental torsional effects (let's assume that you are using ELF since, as I mentioned above, you don't need to worry about accidental torsion if you are doing a dynamic analysis), the answer is, you don't apply $F_x$ to any node, really.

The equation, $$M_t = F_x e_y \tag{1}$$

is only intended to be used to determine the magnitude of the torsional moment you need to account for in your design. It is not meant to be used to calculate a literal load you would directly apply to a model. In the past, we have accounted for torsional moments on structures (accidental and inherent) by calculating the torsional effect and then, via a rational process, applying loads to our model that would produce that torsional effect.

For example, lets say you have a very simple structure represented in your finite element model as a single 4 node, thin shell element. This structure sees a small torsional moment of 1 N/m. Your "structure" would look something like this:

By decoupling the moment into a force applied at the 4 nodes, you might load your model like this:

[Edit: Oops, drew arrows in wrong direction. Pretend the applied moment is rotating clockwise instead.]

There is no single correct way to account for any type of load in your models. An accidental torsional moment is no different. It is up to the engineer to use a rational process to determine how his/her mathematical models and methods will account for the loading applied to the structure being designed (in this case, accidental torsion).

1. In FEM, each beam, column, wall and slab are multiple-node elements. And each node has a internal force value resulted from FEM analysis, what should I do with all these forces at different nodes? The comment here says that I should sum up all of the forces on each related node in order to obtain the correct $F_x$ ( I'm not sure whether I interpret it correctly or not). But I am highly skeptical of this approach, because won't that make $F_x$ dependent on the number of nodes, an artifact of how we mesh an element? How can a real physical quantity increases with the number of nodes we decide to mesh?

Yes, proceed as I suggested the last time you asked this question. No, this doesn't make $F_x$ dependent on the number of nodes in your model. $F_x$, in this context, is dependent on the tributary mass to each node. As you increase the number of nodes in your model, the amount of mass tributary to each node decreases (linearly), therefore, the force does not increase if you change the scale of your mesh discretization.

1. If 3) is true, then what about the $F_x$ on the wall above a particular floor? Should we sum them in? What about the internal slab and beam? Do they have this force also?

This question is a little unclear and indicates a certain level of misunderstanding that I think is way beyond the scope of this answer. Refer to my answer to #2. It is up to the engineer to determine how to account for loading.

1. How should we intepret $M_t$? Is it internal member force to be resisted by beam/slab? If yes, if there are many slabs and beams into xx direction, then how should this $M_t$ be distributed? Or is it an external static force; so one would have to apply this external force on the structure in order to calculate the individual internal member force induced by it ( via the FEM equation $ku=F$)?

Please refer to my answer to #2. It is up to the engineer to determine how to account for loading. The torsional moment is resisted by the building structure as a whole. $M_t$ is not meant to simply be "plugged in" to your model/analysis.

I don't mean to sound mean spirited, but these are questions related to some very fundamental aspects of structural/stress analysis and code interpretation. My short answer here is in no way everything you need to know to understand the full implication and answers to your questions. I strongly urge you to consult with an experienced professional if these questions are the result of a real world analysis of something that will eventually be constructed.

Also, a quick glance at your included images suggests that you have a whole lot of inherent torsion in this structure as well. Make sure that your analysis accounts for that as well.

• According to UBC 97 1631.5.6, pp 2-17, both response spectrum and linear time history requires special handling for accidental torsion, because it is not accounted for in FEM analysis. Apr 1 '16 at 6:25
• Also, according to the book Dynamics of structure, 4th, pp 555: the structural response associated with accidental torsion is not amendable to structural design calculation... therefore building codes include a simple design provision... Apr 1 '16 at 6:26
• I think it is safe to say that accidental torsion is needed, for response spectrum and linear time history, and not just ELF Apr 1 '16 at 6:27
• On question 2, I am simply asking how can I account for $M_t$ in my FEM model. I've clarified the question. Apr 1 '16 at 6:37
• My question is simple: given a FEM model, how can I include accidental torsion in my analysis? Apr 1 '16 at 6:38

There is a paper on how to account for accidental eccentricity in dynamic modal analysis of buildings:

The methodology is based on dynamic modal superposition technique by applying of accidental torsion to global force vectors related to each modal shape. For this purpose, the global displacement vectors are computed for each modal shape at the first step. Then, corresponding global force vectors are computed by using the global mass matrix, the eigenvalues of the system and the global displacement vectors. For each mode shape, the accidental torsional moments of the global force vector are updated by the required amount of eccentricity in either direction. Static analysis is carried out to find the modified global modal displacement vectors and the internal forces for each member for each modal shape. The nodal displacements and the internal force resultants can be combined by using standard modal combination techniques.