Graitec Advance Design: How to consider the eccentricity of dynamic masses and theta coefficient calculation

In this article we’ll discover how to consider the eccentricity of dynamic masses on seismic calculation and how to calculate the drift sensitivity coefficient according to Eurocode 8 and NTC 2018, using the tools in Advance Design.

Eccentricity of dynamic masses

Theoretical background

The seismic action is applied in the centre of gravity of the masses of each floor. If the floor has a rectangular plan, the centre of gravity of the masses will be the same as the geometric centre of gravity of the rectangle. But is it a realistic situation?

The mass of a floor is composed of:

–        The self-weight of the structural elements of the floor (for example structural beams, slabs)

–        The dead load coming from the non-structural elements (for example floor screed, partitions, etc.)

–        Live loads (for example the presence of people)

Assuming the centre of gravity of the masses coincides with the geometric centre of gravity of the floor, this means that all live loads are positioned in a way that the centre of mass coincides with the geometric centre of gravity of the floor.

It would be very difficult for this situation to appear. For this reason, the European and Italian standard prescribe a precaution to consider the randomness of the real position of the loads.

Modern codes for seismic design require consideration of the so-called accidental eccentricity, to consider the torsional response caused by several factors not explicitly considered in design. This provision requires that the mass centres in the building floor be moved a certain percentage of the building’s dimension (usually 5%) along both the x and y axes and in both positive and negative directions.

The Italian standard NTC 2018 at chapter 7.2.6 states that:

“To consider the spatial variability of the seismic motion, as well as any uncertainties [in the localization of the masses, ed.], an accidental eccentricity with respect to its position must be attributed to the centre of mass as it derives from the calculation. For buildings only and in the absence of more accurate determinations, the accidental eccentricity in any direction cannot be considered less than 0.05 times the average size of the building measured perpendicular to the direction of application of the seismic action. This eccentricity is assumed to be constant, in size and direction, on all horizons.”

The Eurocode 8 at chapter 4.3.2 states that:

“(1) P In order to account for uncertainties in the location of masses and in the spatial variation of the seismic motion, the calculated centre of mass at each floor i shall be considered as being displaced from its nominal location in each direction by an accidental eccentricity:

eai = ± 0,05⋅ Li

where:

eai is the accidental eccentricity of story mass i from its nominal location, applied in the same direction at all floors.

Li is the floor-dimension perpendicular to the direction of the seismic action.”

The eccentricity of the centre of mass will consequently generate four different positions of the centre of mass. We will therefore have four different “packages” of seismic combinations, one for each position of the centre of mass. In the figure below you can see the four different positions of the masses schematized.

Accidental eccentricity inside seismic combinations

The second factor that generates 32 seismic combinations is the simultaneous presence of the seismic action in the X and Y direction. Together with the seismic action prevailing in one direction, 30% of the seismic action in the orthogonal direction will act simultaneously.

Example: for the dominant earthquake in the X direction, 30% of the seismic action will act in the Y direction. For the dominant earthquake in the Y direction, 30% of the seismic action will act in the X direction.

The simultaneity of the seismic action along two orthogonal directions (X and Y) must be considered for the four different positions of the centre of mass generated by the accidental eccentricity mentioned above. I report below the combinations that are obtained considering the simultaneity of the earthquake in the two directions X and Y.

Position 1 of the centre of mass

Ex + 0.3Ey

0.3Ex + Ey

Position 2 of the centre of mass

Ex + 0.3Ey

0.3Ex + Ey

Position 3 of the centre of mass

Ex + 0.3Ey

0.3Ex + Ey

Position 4 of the centre of mass

Ex + 0.3Ey

0.3Ex + Ey

We have so far reached 8 combinations, i.e., 2 combinations for each position of the centre of mass. There are 24 combinations left to reach the 32 seismic combinations. We just have to consider the last factor that affects the creation of load combinations in the presence of an earthquake.

The consideration to generate 32 seismic combinations is the positive or negative direction of the seismic force, along the two directions (X and Y) in which the earthquake can act. The permutation of the positive and negative signs, together with the eccentricity of the masses and the simultaneity of the earthquake along two orthogonal directions will generate the 32 load combinations to be considered for the analysis of buildings in seismic areas. I list below the 32 seismic combinations grouped for the 4 different positions of the centre of mass.

Center of mass position 1 (Eccentricity +5%, +5%)

Dominant seism on X direction:

1) + EX + 0.3 * EY

2) + EX – 0.3 * EY

3) – EX + 0.3 * EY

4) – EX – 0.3 * EY

Dominant seism on Y direction:

5) + 0.3 * EX + EY

6) + 0.3 * EX – EY

7) – 0.3 * EX + EY

8) – 0.3 * EX – EY

The 8 combinations above should be repeated for the remaining three positions of the centre of mass.

Center of mass position 2 (Eccentricity +5%, -5%)

[The same 8 combinations seen for position 1 of the centre of mass]

Center of mass position 3 (Eccentricity -5%, +5%)

[The same 8 combinations seen for position 1 of the centre of mass]

Center of mass position 4 (Eccentricity -5%, -5%)

[The same 8 combinations seen for position 1 of the centre of mass]

We finally get the total number of 32 seismic combinations: 8 combinations x 4 positions of the centre of mass = 32 total seismic combinations.

How to manage the eccentricity of dynamic masses in Advance Design

Possibility to create more than 1 seismic load case family

With Update 1 to Advance Design 2023, it is possible to define multiple seismic load families in the current model. Within each family, different numbers of seismic load cases can be defined, and each family can have different parameters defined (for example, soil parameters, different spectra, etc.).

The ability to define multiple independent seismic case families makes it possible mainly to perform an analysis using more than one spectrum in the same project, for example, a design spectrum for ULS seismic load case family and an elastic spectrum for SLS load case family.

Possibility to change properties of Modal Analysis between phases

One of the features available in Advance Design is “Calculation by Phases.” It gives the possibility to pause calculation and modify some parameters of certain analyses after each iteration of the calculation process. One example of the use of this function is the ability to get results for some load combinations with a given element stiffness, and other combinations with a different one. With version 2023.1, it is possible to modify modal analysis parameters between phases. This makes it possible to run new calculation scenarios, for example, when you want to run a seismic analysis with different modal mass eccentricities.

In the following example, displacements and efforts will be determined for a concrete building subjected to gravitational and lateral loads, considering two cases of modal mass distribution:

–        Phase 1: modal mass with no eccentricity.

–        Phase 2: modal mass with +X +Y mass eccentricity.

For this purpose, two identical pairs of seismic cases were defined, differing only in name. The modal analysis parameters at the beginning were left without the definition of dynamic masses eccentricity.

Next, the analysis model is created by clicking the Analysis button in the Project Browser, and in the Calculation sequence window only Verify, Mesh, and Evaluate options are checked.

The next step is to prepare the phases definition. Using the command Calculation by Phases available on the Analysis tab, we open a dedicated dialog, and we create two phases – the first one includes load cases 1-4, while the second phase includes load cases 5 and 6. Options to enable calculation by phases and pause calculation after every phase should be selected and be able to make changes between phases.

Now we can run the calculations. Calculations only for the cases in phase one will be made. When they are finished, the results for the first 2 seismic cases and related combinations will be available. We can now check the result, generate reports, and add them to the documentation.

Then we go to the modal analysis properties and change the settings – in this example related to eccentricities.

Now it’s time to run the calculations again. This time only the cases assigned to phase two will be calculated. Once completed, results for these cases and their combinations will additionally be available, for example, results including dynamic mass eccentricities.

To consider the 5% eccentricity in the other combinations (X+ Y-, X- Y+, X- Y-) we should create the other 3 seismic load cases families from the descriptive model and repeat the same operation, changing (after evaluating the model) the direction inside the modal mass property dialog.

Inter-story drift sensitivity coefficient

Theoretical background

In seismic analysis, the second-order effect (P-Δ) can be neglected when the additional moment generated in vertical elements is very small compared to the one created by horizontal seismic forces.

This is formulated with the inter-story drift coefficient (ϑ coefficient) according to chapter 4.4.2.2 (2) from EN 1998-1 and 7.3.1 from NTC 2018 (formula 7.3.3)

Where:

–        𝑃𝑡𝑜𝑡 is the total gravity load at and above the story considered in the seismic design situation.

–        𝑑𝑟 is the inter-story drift, calculated as the difference between the average lateral displacements at the top and bottom of the story under consideration.

–        𝑉𝑡𝑜𝑡 is the total seismic story shear.

–        ℎ is the inter-story height.

For 𝜃 ≤ 0.1 the second-order effect (P-Δ) can be neglected.

Workflow in Advance Design

Advance Design can perform this verification for each seismic combination and at each story level. For this, Advance Design identifies for every story the maximum value of 𝜃 and the corresponding seismic load combination that is causing it.

The verification is carried out after meeting the same conditions required for inter-story drift calculations, namely:

–        levels are defined (by activating the “Level” option of the system).

–        levels have correctly defined heights and story numbers.

–        seismic analysis is defined.

A new table is displayed together with the table summarizing the level drift verification and is available in the category “Seismic results by mode” / “Level drift verification and theta calculation for seismic combinations.”