# NBC 2015 Modal Response Spectrum Analysis and Base Shear Considerations

## Technical Article on the Topic Structural Analysis Using Dlubal Software

### Technical Article

The National Building Code of Canada (NBC) 2015 Article 4.1.8.7 provides a clear procedure for earthquake methods of analysis. The more advanced method, the Dynamic Analysis Procedure in Article 4.1.8.12, should be used for all structure types except those that meet the criteria set forth in 4.1.8.7. The more simplistic method, the Equivalent Static Force Procedure (ESFP) in Article 4.1.8.11, can be used for all other structures.

For structures which qualify for the Dynamic Analysis Procedure, there are two further sub methods defined in 4.1.8.12 [1] including a Linear Dynamic Analysis, with either a Modal Response Spectrum Method or the Numerical Integration Linear Time History Method, or a Nonlinear Dynamic Analysis. This article will focus on a Modal Response Spectrum Method (RSA) and the further base shear requirements defined in the NBC 2015.

#### NBC 2015 Base Shear Variations

Assuming an RSA is calculated according to the NBC which is beyond the scope of this article, the base shear can be determined at the bottom story of the structure in each of the orthogonal directions the lateral seismic load is applied. A series of steps in Article 4.1.8.12 [1] defines the various base shear variations needed to ultimately scale the story shears, story forces, member forces, and deflection if necessary.

Step 1) In Sentence (5) [1], the elastic base shear, Ve, should be determined from a linear dynamic analysis which is the first step in understanding the NBC base shear requirements. Ve should not include any additional factors or scaling applied to the response spectrum.

Step 2) Determine the design elastic base shear, Ved, also known as the "lateral earthquake design elastic force", according to Sentence (6) [1] using Ve determined from Sentence (5) [1]. For all structures located on sites other than Class F and have a seismic force resisting system (SFRS) with a ductility-related force modification factor, Rd, equal to or greater than 1.5, then Ve should be multiplied by the larger of the following two factors:

NBC 2015 Design Elastic Base Shear, Ve, Factor

$$Max 2 S(0.2)3 S(Ta) ≤ 1.0 , S(0.5)S(Ta) ≤ 1.0$$

 S(0.2) Design spectral response acceleration for a period of T = 0.2 sec S(0.5) Design spectral response acceleration for a period of T = 0.5 sec S(Ta) Design spectral response acceleration for the fundamental lateral period of vibration of the structure in the direction under consideration

Step 3) The previously determined Ved should be multiplied by the importance factor, Ie, given in Article 4.1.8.5 [1] and divided by the ductility-related force modification factor multiplied by the overstrength-related force modification factor, RdRo, found in Table 4.1.8.9 [1] to determine the design base shear, Vd.

Step 4) Determine the lateral earthquake design force, V, given in Article 4.1.8.11 [1] according to the simplified ESFP.

#### NBC 2015 Base Shear Scaling Requirements

With the information above in collaboration with 4.1.8.12(8) [1], if Vd from Step 3 is less than 80% of V from Step 4, then Vd should be taken as 0.8V for all regular structures and irregular structures permitted to be designed with the ESFP.

For all irregular structures which require a dynamic analysis specified in 4.1.8.7 [1] as well as wood structures greater than 4 stories which meet the criteria set forth in 4.1.8.12(12) [1], then Vd shall be taken as the larger of Vd or V stated in 4.1.8.12(9) [1].

In addition to these minimums, the scale factor Vd/Ve should be applied to story shears, story forces, member forces and deflections including the effects of accidental torsion to determine the design values from 4.1.8.12(10) [1].

#### NBC 2015 Base Shear Application in RFEM

With the multiple variations for base shear discussed in the NBC 2015, it may be difficult to determine which base shear variant is provided when running a response spectra analysis according to the standard in the structural analysis program RFEM.

Within the RF-DYNAM Pro - Equivalent Loads module, there are two spectrum types available after selecting the NBC 2015 including the "Elastic response spectrum" and the "Design spectrum for linear calculation".

The "Elastic response spectrum" will provide the structure’s elastic base shear, Ve, from 4.1.8.12(5) [1] or Step 1 above. This value does not include the consideration of Ie/RdRo or any other factors.

The "Design spectrum for linear calculation" will provide the structure’s design base shear, Vd, from 4.1.8.12(7) [1] or Step 3 above. However, it should be noted that the additional factor specified in 4.1.8.12(6) or Step 2 above is not included in this calculation. The calculation does include Ie/RdRo as these variables are listed in the RF-DYNAM Pro - Equivalent Loads Code Parameters table whereas they are not listed for the "Elastic response spectrum".

For either spectrum type, the user will have the ability to scale the equivalent loads to account for scale factor requirements within the add-on module. For example, the "Elastic response spectrum", which is needed to analyze the structure deflections as it does not include the effects of Ie/RdRo, may need to be scaled by Vd/Ve as required in 4.1.8.12(10). For the "Design Spectrum for Linear Calculation" type, the Vd/Ve scale factor may also apply in addition to the factor specified in 4.1.8.12(6) or Step 2 above which is not included in the calculation. RF-DYNAM Pro - Equivalent Loads provides a scaling option in each of global directions X, Y, and Z in which the seismic force can be applied.

In summary, the linear dynamic analysis requirement and in particular the modal response spectrum analysis set forth by the NBC 2015 are possible to consider utilizing the FEA program RFEM in addition to the RF-DYNAM Pro - Equivalent Loads add-on module. The many base shear variations in Article 4.1.8.12 are possible to calculate utilizing the program with either the spectrum type designation or the application of the scaling factor entry option within the module.

#### Amy Heilig, PE

CEO - USA Office
Sales & Technical Support Engineer

Amy Heilig is the CEO of the USA office located in Philadelphia, PA. In addition, she provides sales and technical support and continues to aid in the development of Dlubal Software programs for the North American market.

#### Reference

 [1] NBC 2015, National Building Code of Canada 2015

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