ADRS curve generation functions for the New Zealand seismic code (Part 4)

Following on from the previous blog posts in this series that provided functions for calculating the NZS1170.5 seismic coefficient. I’ve added some further functions to the GitHub repository that utilise the previous functions to generate an ADRS curve in accordance with the provisions of NZS1170.5 and the draft Seismic Assessment of Existing Buildings guidelines produced by MBIE in NZ.

What is an ADRS curve you might ask, well it’s an Acceleration Displacement Response Spectrum, and they look like the below image (the pink line) if you’ve generated them correctly (fingers crossed). While at first glance it looks suspiciously like the “normal” period vs acceleration plot, don’t be fooled!

It’s essentially another way of visualising the design response spectrum, but in terms of displacement and acceleration.

Usually the “normal” response spectrum is plotted in terms of period on the horizontal axis, and acceleration on the vertical axis. The ADRS curve is simply plotting the same thing after some conversions but as a displacement on the horizontal axis, and acceleration on the vertical axis.

A straight line drawn from the origin to the curve represents a specific period.

This representation of the design spectrum is used in displacement-based design. In NZ when assessing existing buildings, we are “encouraged” to use a displacement-based approach. The idea being that force-based approaches don’t work too well for accurately assessing the strength and more importantly the deformation capacity available in existing structures.

What the ADRS curve allows us to do is evaluate the collective response of the bracing elements in the lateral load resisting system for a structure even if they have different ductilities and or deformation capacities, and determine the overall capacity for a structure.

This is achieved usually via the use of a simplified hand-based pushover analysis to inform the designer of the overall available deformation capacity. The particular analysis method we use in New Zealand is called SLAMA (Simple Lateral Mechanism Analysis).

Enough of the background (if you must know more, check out the guidelines I linked to above), onto the VBA code: –

The code

Function Loading_ADRS(T_1 As Variant, Site_subsoil_class As String, Hazard_factor As Double, Return_period_factor As Double, _
                      Fault_distance As Variant, S_p As Double, zeta_sys As Double, _
                      Optional D_subsoil_interpolate As Boolean = False, Optional T_site As Double = 1.5) As Variant
'function to calculate the spectral displacement and spectral acceleration for plotting an ADRS Curve (Acceleration Displacement Response Spectrum)

'________________________________________________________________________________________________________________
'USAGE
'________________________________________________________________________________________________________________
'=Loading_ADRS(T_1,Site_subsoil_class,Hazard_factor,Return_period_factor,Fault_distance,S_p,zeta_sys,D_subsoil_interpolate,T_site)
'T_1 = FIRST MODE PERIOD
'Site_subsoil_class = SITE SUBSOIL CLASS, i.e. A/B/C/D/E (ENTERED AS A STRING)
'Hazard_factor = HAZARD FACTOR Z
'Return_period_factor = RETURN PERIOD FACTOR Ru OR Rs
'Fault_distance = THE SHORTEST DISTANCE (IN kM's) FROM THE SITE TO THE NEAREST FAULT LISTED IN TABLE 3.6 OF NZS1170.5
'                 IF NOT RELEVANT USE "N/A" OR A NUMBER >= 20
'S_p = STRUCTURAL PERFORMANCE FACTOR
'zeta_sys = SYSTEM DAMPING (AS A PERCENTAGE)
'OPTIONAL - D_subsoil_interpolation is optional - TRUE/FALSE - consider interpolation for class D soils
'OPTIONAL - T_site is optional - Site period when required for considering interpolation for class D soils
'           (note default value of 1.5 seconds equates to no interpolation)
'________________________________________________________________________________________________________________

    Dim results As Variant
    Dim C_d_T As Variant
    Dim K_zeta As Double
    Dim i

    'determine the spectral damping reduction factor
    K_zeta = Loading_K_zeta(zeta_sys)

    'determine the 5% damping Response Spectrum design spectrum with ductility of 1.0 (so k_mu = 1.0)
    C_d_T = Loading_C_d_T(T_1, Site_subsoil_class, Hazard_factor, Return_period_factor, Fault_distance, 1, S_p, False, D_subsoil_interpolate, T_site)

    ReDim results(LBound(T_1) To UBound(T_1), 1 To 2)

    For i = LBound(T_1) To UBound(T_1)

        'determine the spectral acceleration (S_a)
        results(i, 2) = K_zeta * C_d_T(i, 1)
        'determine the spectral displacement (S_d)
        results(i, 1) = Loading_K_delta_T(T_1(i, 1)) * results(i, 2)

    Next i

    'return results
    Loading_ADRS = results

End Function

Function Loading_K_delta_T(T_1 As Variant) As Variant
'function to calculate the Displacement spectral scaling factor

'________________________________________________________________________________________________________________
'USAGE
'________________________________________________________________________________________________________________
'=Loading_K_delta_T(T_1)
'T_1 = FIRST MODE PERIOD
'________________________________________________________________________________________________________________

    Dim pi As Double
    pi = 3.14159265358979

    Loading_K_delta_T = 9810 * T_1 ^ 2 / (4 * pi ^ 2)

End Function

Function Loading_K_zeta(zeta_sys As Double)
'function to calculate the spectral damping reduction factor

'________________________________________________________________________________________________________________
'USAGE
'________________________________________________________________________________________________________________
'=Loading_K_zeta(zeta_sys)
'zeta_sys = SYSTEM DAMPING (AS A PERCENTAGE)
'________________________________________________________________________________________________________________

    Loading_K_zeta = (7 / (2 + zeta_sys)) ^ 0.5

End Function

Usage example

Let’s create the ADRS curve for the same Welington site we had in the example in part 3.

Firstly, we would generate our period range in the same manner as we did in part 3 using the Loading_generate_period_range  function, so we won’t cover that here again. As noted in part 3, it’s important to use a period range if plotting the curve that accounts for all the transition points between all the various factors.

Next, we need to know the equivalent viscous damping of the system, this scales the spectrum to account for ductility. If our curve is going to be in accordance with the Seismic Assessment of Existing Buildings guidelines, then we can use appendix C2D in section 2 of the guidelines to determine this.

Let’s say for the sake of the example that our structure was a Reinforced Concrete (RC) frame, then we can assess the system damping to be 5% inherent damping + 10% hysteretic damping for a system ductility of 2.0. Therefore, the total equivalent viscous damping of the system = 5 + 10 = 15%.

To generate our curve assuming our range of periods is in a sequence referred to as being a variable T_1: –

=Loading_ADRS(T_1, Site_subsoil_class, Hazard_factor, Return_period_factor, Fault_distance, S_p, zeta_sys, D_subsoil_interpolate, T_site)

For our particular problem we would enter the following in a cell to generate the ADRS curve coordinates. It’s worth noting here that the ADRS curve requires the use of the Response Spectrum derived spectral shape factors.

=Loading_ADRS(T_1, "D", 0.4, 1.8, 2, 0.7, 15, TRUE, 0.87)

If we enter this in a cell with T_1 referring to the range of the periods we get the following result: –

The blue column (1st column of the functions results) is the spectral displacement and the red column (2nd column of the functions results) is the spectral acceleration: –

Plot these on an XY scatter chart and you get the final ADRS curve: –

Now that we have our ADRS curve, we can show that for any points we plot that it describes the stiffness (or in other words the period).

The stiffness of a structure can be represented by radiating lines from the origin
of the acceleration-displacement response spectrum. These lines, for periods with 0.5 second increments are plotted on the same curve below to show the concept: –

Code Update…

Note I’ve made a few small tweaks to the code on GitHub to address some inconsistencies I noted while writing this post. Part 2 has been updated, so if you downloaded the code from GitHub previously, make sure you download an updated version.

Part 5 can be found here, explores how to get the functions working in non-Excel 365 versions of Excel.