NBC of Canada 2005

Equivalent Static Force Procedure

Equivalent Static Force Procedure is implemented according to Section 4.1.8.11, Division B of NBC 2005. The user is refereed to Sections 4.1.8.1 - 4.1.8.13 as well Commentary J of User's Guide - NBC 2005 -Structural Commentaries.

Seismic Base Shear

The minimum lateral earthquake force (seismic base shear) is calculated according to the following equation (see 4.1.8.11.2):

V = \frac{S (T_{a}) M_{v} I_{E} W}{R_{d} R_{0}}

The above equation is subjected to the following limits:

$V > \frac{S (2.0) M_{v} I_{E} W}{R_{d} R_{0}}$ (Note that M_v must be calculated with T ≥ 2.0)
And if R_d ≥ 1.5, then $V < \frac{2}{3} \frac{S (2.0) I_{E} W}{R_{d} R_{0}}$

The Fundamental Period (Ta)

The fundamental period (Ta) is based on one of the following choices:

Clause (a): Use 4.1.8.11.3a :
$T_{a} = {\begin{matrix} 0.085 {(h_{n})}^{3 / 4} & for steel moment frames \\ 0.075 {(h_{n})}^{3 / 4} & for concrete moment frames \\ 0.1 N & for other frames \end{matrix}$
Clause (b): Use 4.1.8.11.3b : T_α = 0.025 h_n for braced frames
Clause (c): Use 4.1.8.11.3c : T_α = 0.050 (h_n)^3/4 for shear wall and other structures

Note that h_n is height of the building (in meters). In above equations, all options are implemented in the program except 0.1N. Instead, you are always allowed to defined your own value for T_α.

In addition, if required, the program calculates fundamental period from an Eigen analysis, and you are also provided a method to use this as well. The above equation is subjected to the following limits:

For moment resisting frames: T_α ≤ 1.5 * that determined in Clause (a)
For braced frames: T_α ≤ 2.0 * that determined in Clause (b)
For Shear Wall Structures: T_α ≤ 2.0 * that determined in Clause (c)

Note that these upper limits specified above may not be checked for deflection/drift calculations.

Spectral Response Acceleration (Sa)

It is the acceleration read from Design Spectral Acceleration Curve for the value of Ta.

Higher Mode Factor (Mv)

This factor is read from Table 4.1.8.11, based on type of lateral resisting systems, Sa(0.2), Sa(2.0) and calculated value of Ta. You are required to choose lateral resisting system for each direction.

Importance Factor (Ie)

It is given in Table 4.1.8.5

Seismic Weight of the Building (W)

It is calculated by the program.

Force Modification Factors (Rd and Ro)

Ductility related force modification factor, R_d, reflecting the capability of a structure to dissipate energy through inelastic behavior is required from the user for each direction. This factor is provided in Table 4.1.8.9.

Overstrength related force modification factor, R_o, accounting for the dependable portion of reserve strength in a structure is required from the user for each direction. This factor is provided in Table 4.1.8.9.

Design Spectral Acceleration Curve

Based on Site Class and values of SA(0.2) and Sa(1.0), the design spectral acceleration curve is generated as follows:

\begin{matrix} S (T) = \end{matrix} {\begin{matrix} F a S a (0.2) & T \leq 0.2 s \\ F v S a (0.5) or F a S a (0.2), whichever is smaller & T = 0.5 s \\ F v S a (1.0) & T = 1.0 s \\ F v S a (2.0) & T = 2.0 s \\ F v \frac{S a (2.0)}{2} & T \geq 4.0 s \end{matrix}

where Fα and Fv are determined according to Tables 4.1.8.4.B and 4.1.8.4.C, respectively. For intermediate values, linear interpolation is used.

Distribution of Lateral Earthquake Force

Calculated base shear (V) is distributed over the height of the building based on the following equation:

F_{x} = (V - F_{t}) \frac{W_{x} h_{x}}{\sum_{i = 1}^{n} W_{i} h_{i}}

Where F_t is concentrated force applied at the top of the building and it accounts for effects of higher order modes. It is calculated according to the following equation:

F_{t} = {\begin{matrix} 0 & T_{a} \leq 0.7 s \\ 0.07 T_{a} V & 0.7 s < T_{a} < 3.6 s \\ 0.25 V & 3.6 s \leq T_{a} \end{matrix}

Overturning Moments

Overturning moment calculation as given in 4.1.8.11 and it is not implemented in the program.

Torsional Sensitivity (B)

Determination of torsional sensitivity requires static analysis using 3D elastic model with static lateral loads at each floor level applied at distances ±D_nx (see 4.1.8.11.9). It is only applicable for rigid diaphragms. This is not implemented in the program.

Torsional Effects

Torsional effects are accounted for according to the following equation (it applies for both X- and Y-directions):

T_x = F_x(e_x ± 0.10D_nx)

where

e_x	=	natural eccentricity due to center of rigidity and mass being at different positions
0.10 D_nx	=	The portion ± 0.05 D_nx of ± 0.10 D_nx represents accidental torsion, and the remainder takes into account natural torsion, including dynamic amplification.
D_nx	=	plan dimension of the building at a level x perpendicular to the direction of seismic loading being considered

Note that in a 3D model, the effects of e_x is already included in 3D analysis (if mass center and rigidity center are at different locations, this is already reflected in analysis results). So, there is no need to include it explicitly. Hence, the only eccentricity considered is ± 0.10 D_nx. Conveniently, this is exactly the same set of load applications required for the determination of the torsional sensitivity parameter, B. The eccentricity of ± 0.10 D_nx is implemented in the program.

Stability Factor (θ_x)

Stability factor as given in Commentary J, p. J-26 is not implemented.

Orthogonal Loading (4.1.8.8)

100%/30% orthogonal loading is implemented in the software as referenced in 4.1.8.8.

Direction of Loading

Four load cases are generated as follows:

Generated Load Cases for NBC of Canada 2005 Seismic

100%/30% orthogonal loading is implemented in the software as referenced in 4.1.8.8.

Response Spectra Analysis

Modal Response Spectrum Analysis according to 4.1.8.12 of Division B of NBC of Canada 2005 is implemented. Other methods (Numerical Integration Linear Time History Method and Nonlinear Dynamic Analysis) are not covered.

Design Spectral Acceleration

Based on Site properties, the design spectral acceleration curve is generated as follows:

\begin{matrix} S (T) = \end{matrix} {\begin{matrix} F a S a (0.2) & T \leq 0.2 s \\ F v S a (0.5) or F a S a (0.2), whichever is smaller & T = 0.5 s \\ F v S a (1.0) & T = 1.0 s \\ F v S a (2.0) & T = 2.0 s \\ F v \frac{S a (2.0)}{2} & T \geq 4.0 s \end{matrix}

where Fa and Fv are determined according to Tables 4.1.8.4.B and 4.1.8.4.C, respectively. For intermediate values, linear interpolation is used.

Accidental Torsional Eccentricity

Section 4.1.8.12.4 provides two alternative approaches for accidental torsional eccentricity:

In this approach, which can be used for any value of B (torsional sensitivity) but is intended primarily for torsionally sensitive structures, the effects of static torsional moments, ( ± 0.10 D_nx ) F_x, at each level "x" are calculated and then combined with the effects determined from a dynamic analysis that includes the actual eccentricities (i.e., eccentricities due to mass center and center of rigidity of floors. In a 3D analysis, this is already covered).

The second approach is only for permissible for structures that are not torsionally sensitive (B < 1.7). This approach allows the effects of accidental eccentricity to be included by shifting the center of mass by ± 0.05 D_nx.

The second approach is implemented in this version. Thus, a maximum of 4 load cases is generated due to these eccentricities.

Dynamic Base Shear

The elastic base shear, V_e, , obtained from Linear Dynamic Analysis (i.e., Modal Response Spectrum Analysis or also called as Response Spectra Analysis) is multiplied by the Importance factor, I_e, and divided by R_dR_o to obtained dynamic base shear V_d:

V_{d} = \frac{I_{e}}{R_{d} R_{0}} V_{e}

The above equation is not implemented. Thus, reported base shear is the base shear obtained from Response Spectra Analysis (i.e., V_e), and it is the engineer's responsibility to make this adjustment.

In addition, V_d is subjected to further modifications:

V_d = 0.8V if V_d < 0.80 V, where V is the lateral earthquake design base shear that can be obtained from Equivalent Static Force Procedure.
V_d > V, then V_d can be used as the design base shear.
V_d = max( V_d, V) for irregular structures. In other words, reducing V_d is not permitted for irregular structures.

Again, this set of adjustment is not enforced in the program. It is the engineer's responsibility to include this adjustment.

Story Shears, Member Forces, and Deflections

Section 4.1.8.12.8 states that story shears, story forces, member forces, and deflections obtained from Linear Dynamic Analysis shall be multiplied by $\frac{V_{d}}{V_{e}}$ . This can be carried out in the program by setting X Scale Factor and Y Scale Factor in the load case dialog. Thus, the engineer is required to run the Response Spectra Analysis twice, once with 1.0 for these scale factors to obtain V_e and then once again with setting the scale factors to $\frac{V_{d}}{V_{e}}$ (the engineer needs to calculate V_d). Finally at the end of this process, analysis results (displacements, story shears, and member forces) reflect this adjustment.