# How to derive Equivalent Static Load for irregular structure modeled with FEM?

Building codes such as the Uniform Building Code (UBC) or International Building Code (IBC) require the determination of Equivalent Static Lateral Forces in order to compute $P-\Delta$ effect and to perform checks on the structural design.

However, all these overturning moment computations and structural checks are done (in the building code) assuming a lumped mass model (a cantilever column). But what if the structure is irregular? How can we compute the Equivalent Static Lateral Forces in an irregular building/structure discretized by finite element?

These quantities have clear physical meaning in a cantilever column, but may not even necessarily have clear meaning/clear equivalent in an irregular building.

I am looking for the FEM way of computing equivalent static forces that act on global level. Any ideas?

To illustrate why I need FEM version of Equivalent Static Load: consider that the overturning moment is defined as $M_o=F_xh$, where $F_x$ is the lateral load at floor level, and $h$ is the distance between the top floor to the point where overturning moment is calculated.

$\frac{P_x\Delta}{V_xh_s}$

where

$P_x$ is the total unfactored gravity load at and above level $x$

$\Delta$ is the seismic story drift by seismic forces

$V_x$ seismic shear between level $x$ and $x-1$

$h_s$ sotry height below level $x$

Consider also accidental torsional effects ( EC8 example, section 2.5.3, pp 39)

$M_t=F_xe_y$

Where

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

$e_y$ is the accidental eccentricities

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

But if we do FEM seismic dynamic analysis, then all I get is the forces for each node, not at each floor. So we need to be able to resolve FEM seismic nodal forces into floor forces.

How can we compute the Equivalent Static Lateral Forces, the overturning moment and other quantities in an irregular building/structural discretized by finite element?

is, simply, you can't. ASCE7-10 speaks directly to this in Section 12.6 when it describes the conditions under which Equivalent Lateral Force (ELF) can and cannot be used. If your structure is located in an area with moderate to high seismic activity (SDC C-F), you cannot use ELF if your structure has "structural irregularities." See Table 12.6-1 in ASCE7-10 for details, I've simplified a bit. There are certain structural irregularities that do not preclude ELF.

That said, there are a couple different ways you can go about calculating equivalent static seismic forces for highly irregular structures using FEM. These forces are not directly analogous to the forces you'd get using ELF method from ASCE 7, but they are forces obtained via rational analysis and commonly used/accepted in order to simplify the seismic analysis of irregular structures. They are:

1. Perform a (fixed base) time history analysis on your structure. Use the results to generate acceleration response spectra at each level of your structure. Use the accelerations from all of your RS to design the structure. (Most conservative)
2. Perform a (fixed base) time history analysis on your structure. Pull the peak acceleration in all 3 directions across all your time steps for each node. Use these accelerations to design the structure.
3. Perform a (fixed base) time history analysis on your structure. Use the accelerations at each time step to design your structure (at each time step). The most conservative design controls. (Least conservative)

## Time History to RS Method (#1)

This method is the most conservative of the three. The general procedure would be to apply the peak acceleration from your response spectra to the structural elements at each level. In my industry, when using this method, we even multiply the peak acceleration by 1.5 to ensure that we envelope the actual acceleration including any multi-modal effects. This is generally only done for designing components within the structure or modifications to an old structure when time & budget do not allow a more accurate (\) analysis. It is considered less conservative for component design because you, theoretically, know the natural frequency of the piece of equipment you are working with, and therefore can pull a more accurate acceleration off the response spectra.

## Max Time History Acceleration Method (#2)

This method is fairly straightforward. After performing your time history analysis, you can extract acceleration time histories for every node in your FE model. For each node in your model, pull the highest acceleration in the X, Y, and Z directions. Multiply these accelerations with your tributary nodal mass to obtain an equivalent static seismic nodal force in all three directions. This is still considered (highly) conservative because you are pulling accelerations from different time steps and combining them.

There are several variations of this method related to exactly which accelerations/time steps you design to. The tricky part is in convincing yourself and/or the body having jurisdiction over your design that you are enveloping reality.

## Time step Time History Acceleration Method (#3)

This method is similar to #2 except you will design your structure/run your analysis for the seismic accelerations at each time step. At each time step, for each node in your model, pull the acceleration in the X, Y, and Z directions. Multiply these accelerations with your tributary nodal mass to obtain the equivalent static seismic nodal force in all three directions.

This method is the least conservative but, by far, most computationally expensive approach and is mostly unrealistic. Your time histories will likely have upwards of 10,000-100,000 time steps. It is incredibly expensive to chug through the analysis.

I've personally used all 3 methods at one time or another in my career. #2 seems to hit the sweet spot with respect to the accuracy:expense ratio.

• In method 2: Multiply these accelerations with your tributary nodal mass to obtain an equivalent static seismic nodal force in all three directions-- but these nodal forces are individual node force and not floor force ( as in equivalent static load). I need the floor force in order to calculate overturning moment, p delta effect and to compare the limits of base shear as specified in the UBC/IBC code. So how to get that if my structure is modelled in FEM? – Graviton Mar 1 '16 at 4:06
• If I can't get equivalent static load in FEM version, this means that I can't compute overturning moment according to the UBC/IBC. Then does that mean that FEM is not actually code compliant? – Graviton Mar 1 '16 at 4:16
• You can use nodal forces to calculate overturning moment. You'd multiply each nodal force by its elevation above the base of the structure. There's nothing special about ELF method and using it to calculate OTM per the building code. The code described process just a simplification/abstraction of basic statics. – William S. Godfrey- S.E. Mar 1 '16 at 15:12
• in that case, I will have a lot of OTM in a floor, and according to the building code, I will have to add these OTM to the internal member moment, because the member must be strong enough to resist them all. Is my interpretation correct? – Graviton Mar 2 '16 at 0:16
• You're interpretation, as I am understanding it, is incorrect. You typically only calculate OTM to verify stability. Assuming that you have loaded (with gravity and lateral loads), run, and extracted design stresses from your FE model, there's no need to design for an OTM. Remember, OTM is just a (very rough) calculation of a rigid, global response of a structural system. Your FE has already captured the local response of your structure to the forces you'd use to calculate an OTM. . – William S. Godfrey- S.E. Mar 2 '16 at 1:19