NBC of Canada 2015

Equivalent Static Force Procedure

Equivalent Static Force Procedure is implemented according to Section 4.1.8.11, Division B of NBC 2015. You are refereed to Sections 4.1.8.1 - 4.1.8.13 as well Commentary J of User's Guide - NBC 2015 -Structural Commentaries.

Seismic Base Shear

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

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

(4.1.8.11.2)

The above equation is subjected to the following limits:

For walls, coupled walls, and wall-frame systems, $V > \frac{S (4.0) M_{v} I_{E} W}{R_{d} R_{0}}$ (for this equation, M_v must be calculated with T ≥ 4.0 sec according to the Table 4.1.8.11)
For moment-resisting frames, braced frames, and other systems, $V > \frac{S (2.0) M_{v} I_{E} W}{R_{d} R_{0}}$ (for this equation, M_v must be calculated with T ≥ 2.0 sec according to the Table 4.1.8.11)
And if R_d ≥ 1.5, then $V < m a x (\frac{2}{3} \frac{S (0.2) I_{E} W}{R_{d} R_{o}}, \frac{S (0.5) I_{E} W}{R_{d} R_{o}})$

The limits given above are applied in the program.

The Fundamental Period (Ta)

The fundamental period, T_a, is based on one of the following choices (4.1.8.11 - 3):

Clause (a):

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}

(4.1.8.11.3a)

Clause (b):
$T_{a} = 0.025 h_{n} for
braced frames$ (4.1.8.11.3b)
Clause (c):
$T_{a} = 0.050 {(h_{n})}^{3 / 4} for shear walls and other
structures$ (4.1.8.11.3c)

Note that h_n is height of the building (in meters). In the preceding equations, Clauses (a), (b), and (c) are implemented in the program except T_a = 0.1N. You may also enter your own value for T_α directly.

The program can calculate the fundamental period from an Eigenvalue analysis, and, if required, the calculated period can also be used. In this case, the program checks the following limits:

Clause d - i: For moment resisting frames: $T_{a} \leq 1.5 \times that determined in
Clause(a)$
Clause d - ii: For braced frames: $T_{a} \leq 2.0 \times that determined in
Clause(b)$
Clause d - iii: For shear wall structures: $T_{a} \leq 2.0 \times that determined in
Clause(c)$
Clause d - iv: For other structures: $T_{a} \leq that determined in
Clause(c)$

If the load case is created for Drift provisions (i.e., calculating drift/deflections), the above limits are not checked but instead the following limits are enforced (see Clause d-v):

T_a ≤ 2.0 if it is a moment-resisting frame, braced frame, or other system
T_a ≤ 4.0 for all others (i.e., walls, coupled walls, and wall-frame systems)

Also, it is stated in the code that these upper limits specified may not be checked for deflection and period calculations.

Spectral Response Acceleration (Sa)

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

Higher Mode Factor (Mv)

The higher mode factor, M_v, applied to base shear is read from Table 4.1.8.11, based on type of lateral resisting system, T_a and values of Sa(0.2) & Sa(2.0). It is also possible to enter M_v directly.

If engineer directly enters values for M_v, these values are used in base shear equation (4.1.8.11.2). On the other hand, a few supplemental equations still require values of M_v calculated at 2.0 sec. and 4.0 sec. (i.e., the equations impose limits on calculated value of base shear). In this case, the program still refers to the table to calculate M_v values at 2 and 4 seconds.

Importance Factor (Ie)

It is given in Table 4.1.8.5 and you must provide this value.

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) is to account for capability of a structure to dissipate energy through inelastic behavior. Over-strength related force modification factor (R_o) is to account for the dependable portion of reserve strength in a structure. Values for both factors are defined in Table 4.1.8.9 for different types of structural systems. You must provide this value for each direction.

Design Spectral Acceleration Curve

The design spectral acceleration value is determined as follows (linear interpolation is allowed for intermediate values of T):

S (T) = {\begin{matrix} \begin{matrix} F (0.2) S a (0.2) o r F (0.5) S a (0.5) w h i c h e v e r i s l a r g e r T \leq 0.2 s \\ F (0.5) S a (0.5) T = 0.5 s \\ F (1.0) S a (1.0) T = 1.0 s \end{matrix} \\ F (2.0) S a (2.0) T = 2.0 s \\ F (5.0) S a (5.0) T = 5.0 s \\ F (10.0) S a (10.0) T \geq 10.0 s \end{matrix}

(4.1.8.4-9)

he values of F(T) are tabulated with Tables 4.1.8.4.-B to 4.1.8.4.-I. In these tables, a reference value of PGA is needed (i.e., PGA_ref). The section 4.1.8.4. (4) provides the following definition:

{P G A}_{r e f} = {\begin{matrix} 0.8 P G A \frac{S_{a} (0.2)}{P G A} < 2.0 \\ P G A \frac{S_{a} (0.2)}{P G A} \geq 2.0 \end{matrix}

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}}

(4.1.8.11-7)

where

F_t

=

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 (Commentary J, p.J-52):

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

The overturning moment calculation as given in 4.1.8.11.(8) is not implemented in the program.

Torsional Sensitivity

Torsional sensitivity ratio, B_x, as given in 4.1.8.11. (10) is neither calculated nor reported in the program.

Torsional Effects

Torsional effects are accounted for in analysis by applying lateral forces eccentrically at story levels. This is carried out according to the following equation 4.1.8.11. (11):

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	=	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

It should be noted that the above equation is allowed to be used in the equivalent static force procedure if B_x ≤ 1.7. Otherwise, a dynamic analysis procedure is required. The program applies eccentric loading according to the equation without checking B_x. It is engineer’s responsibility to justify the applicability of the equation in the equivalent static force procedure.

The torsional effects related to natural eccentricity (i.e., e_x) is implicitly included in the analysis (by definition, e_x is associated with having center of mass and rigidity at different positions). To this end, the remaining eccentricity (i.e., ± 0.10 D_nx) is the only eccentricity calculated and applied in the equivalent static force procedure. Conveniently, this is exactly the same set of load applications required for the determination of the torsional sensitivity parameter, B_x. Even though B_x is not calculated or reported in the program, it is still possible to compute value of B_x from analysis results in which lateral loads are applied eccentricity according to the given equation.

Stability Factor (θ_x)

The stability factor as given in Commentary J, pg. J-26 is neither calculated nor reported.

Orthogonal Loading (4.1.8.8)

The program optionally creates additional load cases to account for 100/30% orthogonal loadings per 4.1.8.8. (c).

Direction of Loading

For eccentric loading, the following 4 load cases are generated:

Generated Load Cases for NBC of Canada 2015 Seismic

Two additional options are provided:

Consider Orthogonal effects (100/30)
Generate Additional Load Cases for Analysis with Tension-Only Members

If either of these options is selected, additional load cases are generated accordingly.

Response Spectra Analysis

Modal Response Spectrum Analysis according to 4.1.8.12 of Division B of NBC of Canada 2015 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:

S (T) = {\begin{matrix} \begin{matrix} F (0.2) S a (0.2) o r F (0.5) S a (0.5) w h i c h e v e r i s l a r g e r T \leq 0.2 s \\ F (0.5) S a (0.5) T = 0.5 s \\ F (1.0) S a (1.0) T = 1.0 s \end{matrix} \\ F (2.0) S a (2.0) T = 2.0 s \\ F (5.0) S a (5.0) T = 5.0 s \\ F (10.0) S a (10.0) T \geq 10.0 s \end{matrix}

(4.1.8.4-9)

For intermediate values, linear interpolation is used.

Accidental Torsional Eccentricity

Section 4.1.8.12.4 provides two alternative approaches for accidental torsional eccentricity:

4.1.8.12-4a: 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).
4.1.8.12-4b: 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.

Only the method given in 4.1.8.12-4b is implemented in the program. In this case, a total of 4 load cases is generated: two cases in X-directions (i.e., X ± 0.05 D_nx ) and two cases in Y-directions (Y+ ± 0.05 D_nx).

Dynamic Base Shear

The elastic base shear (V_e) can be obtained from a modal response spectrum analysis (i.e., linear dynamic analysis). Sections 4.1.8.12.-6 and 4.1.8.12. -7 provides a method to obtain design base shear, V_d These sections are not implemented in the program.

Story Shears, Member Forces, and Deflections

Section 4.1.8.12.8 states that story shears, member forces and deflections obtained from response spectrum analysis shall be multiplied by $\frac{V_{d}}{V_{e}}$ . This section is not implemented. On the other hand, the following procedure is recommended: the program provides scale factors defined for the response spectra load case. The load case is run twice: the first run is with scale factors set to 1.0. This gives elastic base shear values (V_e). The second run includes the scale factors set to $\frac{V_{d}}{V_{e}}$ . The analysis results from the second run reflects the necessary modifications mentioned in 4.1.8.12.8.

NBC of Canada 2015

Equivalent Static Force Procedure

Seismic Base Shear

The Fundamental Period (Ta)

Spectral Response Acceleration (Sa)

Higher Mode Factor (Mv)

Importance Factor (Ie)

Seismic Weight of the Building (W)

Force Modification Factors (Rd and Ro)

Design Spectral Acceleration Curve

Distribution of Lateral Earthquake Force

Overturning Moments

Torsional Sensitivity

Torsional Effects

Stability Factor (θx)

Orthogonal Loading (4.1.8.8)

Direction of Loading

Generated Load Cases for NBC of Canada 2015 Seismic

Response Spectra Analysis

Design Spectral Acceleration

Accidental Torsional Eccentricity

Dynamic Base Shear

Story Shears, Member Forces, and Deflections

Stability Factor (θ_x)