The reference pipe rack structure adopted in this study is reproduced from the steel pipe rack described by Puttatt (2020) [3], which was adapted from an actual petrochemical complex located in a region of moderate seismicity. The structure is a multi-level steel frame measuring 42.0 m in the longitudinal direction and 7.5 m in the transverse direction, with seven bays of 6.0 m each. The total height is 25.0 m, with piping supported at four elevations (EL 5.0, 10.0, 15.0, and 20.0 m) and a cable tray level at EL 25.0 m. The lateral force-resisting system consists of moment-resisting frames (MRF) in the transverse direction and diagonal bracing in the longitudinal direction. Secondary beams are provided between the main frames to support small-diameter pipes and are laterally restrained by struts and plan bracing. Large-diameter pipes are supported directly on the main frames. All column bases are modelled as pinned.

The structural members consist of European HE-series wide-flange sections of steel grade S275 (f_y = 275 MPa). The member sizes vary along the height of the structure, with heavier sections assigned to the lower levels where gravity and seismic demands are greatest. The member assignments are summarised in Table 1. A schematic of the structural configuration showing the three-dimensional view, the transverse elevation, and the longitudinal elevation is presented in Figure 1.

Figure 1. Reference pipe rack structure: (a) three-dimensional view; (b) transverse elevation (MRF direction); (c) longitudinal elevation (braced direction). Adapted from Puttatt (2020) [3].

Table 1. Structural member assignments for the reference pipe rack model (after Puttatt (2020) [3]).

All structural members are modelled using S275 European structural steel. A bilinear stress-strain relationship with kinematic hardening is adopted, consistent with the material model used by Puttatt (2020) [3]. The key material parameters are: yield strength f_y = 275 MPa, elastic modulus E = 210,000 MPa, Poisson’s ratio ν = 0.3, strain-hardening ratio b = 0.01 (post-yield stiffness equal to 1% of the elastic stiffness), and steel density ρ = 7,850 kg/m³. Geometric nonlinearity is accounted for through P-Δ effects in the analysis.

The nonlinear finite element model is developed in SAP2000 [11]. Beam and column members are subdivided along their length into several sub-elements, with fiber-section hinges assigned at relative locations of 0.1 L and 0.9 L within each sub-element, where L denotes the sub-element length. This discretisation scheme approximates distributed plasticity by capturing the spread of yielding along the member length through multiple discrete fiber sections.

This approach is adopted to maintain consistency with the distributed inelasticity modelling strategy employed by Puttatt (2020) [3], who used force-based (FB) distributed inelasticity elements with five integration points in SeismoStruct [12]. Although the software platforms differ, the underlying modelling philosophy is equivalent: both approaches capture nonlinear material behaviour at multiple sections along the member length through fiber discretisation, rather than relying on pre-defined lumped moment-rotation relationships.

For diagonal bracing members, the simulation of global buckling in compression is addressed through multi-linear link elements, following the approach of Puttatt (2020) [3]. The link elements are defined at coincident joints at the bracing ends and are characterised by an asymmetric bilinear force-displacement relationship that distinguishes between the tensile yield capacity and the reduced compressive buckling capacity of each brace. A link element with elastic stiffness of 17,298 kN/m, corresponding to an L50 × 50 × 5 angle section, is used. This formulation captures the essential hysteretic behaviour of bracing members under cyclic loading, including strength degradation following buckling.

The gravity loads are computed in accordance with the owner’s engineering design specification as reported by Puttatt (2020) [3]. The dead load (DL) is taken as 1.20 kPa and the corresponding product load (PL) as 0.80 kPa, yielding a total operating load of 2.00 kPa at each pipe level. Since no live loads are anticipated during seismic events, the full values of gravity loads are considered for the computation of seismic masses. Masses are lumped at beam-column joints and divided equally among all joints at each elevation. The total pipe mass across the four pipe levels is 256.8 tonnes, with the cable tray level at EL 25.0 m carrying a fixed mass of 12.8 tonnes (see Section 3.1 for the distribution cases). The self-weight of the structural members is included automatically by the software. The distribution of dead and product loads per level is summarised in Table 2.

Table 2. Load distribution per level for pipe racks and cable trays.

To verify that the SAP2000 baseline model reproduces the dynamic characteristics of the reference rack defined by Puttatt (2020) [3], an eigenvalue analysis was performed on each mass-distribution case and compared against the periods reported in the reference study. Puttatt reports first-mode periods of T₁ = 1.402 s in the transverse (MRF) direction and T₁ = 0.716 s in the longitudinal (braced) direction, with cumulative mass-participation ratios exceeding 90% within the first three modes in both directions [3].

In the transverse direction, the four cases of the present model bracket Puttatt’s reference value. The uniform case yields T₁, Y = 2.790 s, while the non-uniform cases yield T₁, Y = 1.041 - 1.200 s. The transverse direction is governed by the bending stiffness of the moment-resisting frames and is therefore sensitive to the total seismic mass and to its vertical distribution: the uniform case maximises the generalised mass associated with the first mode, lengthening the period, while the non-uniform cases relieve mass from the upper elevations and produce periods close to Puttatt’s value. The bracketing of the reference period by the four cases is taken as evidence that the transverse-direction modelling—section properties, frame geometry, gravity-load to mass conversion, and connection assumptions—is consistent with the reference rack.

In the longitudinal direction, the present model produces first-mode periods longer than Puttatt’s reported 0.716 s for all four cases. This discrepancy is attributable to the modelling of the diagonal bracing. Puttatt (2020) [3] illustrates the bracing idealisation through a generic asymmetric bilinear force-displacement schematic for the link element but does not specify the angle section adopted for the brace, the elastic stiffness of the link, the tensile yield force, or the post-buckling residual capacity. In the absence of these numerical specifications, the link element in the present study was independently defined with an elastic stiffness of 17,298 kN/m derived from the axial stiffness of an L50 × 50 × 5 angle, as reported in Section 2.3. This independently derived stiffness is evidently softer than the value implicitly used by Puttatt, lengthening the longitudinal-direction periods relative to the reference. Because Puttatt’s link properties are not retrievable from the published material, exact agreement in the longitudinal direction is not achievable and is not pursued here.

Notwithstanding the longitudinal-direction period offset, the modal analysis of the present model reproduces the first-mode dominance reported by Puttatt, with more than 90% of the total mass captured within the first three modes in both directions, and the qualitative nonlinear response observed in Section 4—progressive brace buckling controlling the longitudinal direction and plastic-hinge formation controlling the transverse direction—is consistent with the mechanisms identified by Puttatt for the same reference rack. The baseline model is therefore considered to be dynamically consistent with the reference rack in those aspects that are fully specified in Puttatt (2020). The parametric comparisons in Section 4 are made between the four cases of the present model, not against the absolute response values of the reference, so the period offset in the longitudinal direction does not affect the conclusions of this study.

Four vertical pipe mass distribution cases are defined to investigate the sensitivity of the seismic response to this parameter. In all cases, the total pipe mass is held constant at M_total = 256.8 tonnes, distributed across the four pipe levels (L1 = EL 5.0 m, L2 = EL 10.0 m, L3 = EL 15.0 m, L4 = EL 20.0 m). The cable tray mass at EL 25.0 m is fixed at 12.8 tonnes and is not redistributed. Lumped masses are assigned at beam-column joints, divided equally among all joints at each elevation. The four cases are defined in Table 3, and the corresponding per-joint mass values are reported in Table 4.

Case 1 (Uniform) represents the standard design assumption in which pipe mass is distributed equally across all pipe levels and serves as the baseline against which all other cases are compared. Case 2 (Top-Heavy) simulates a configuration in which heavy process lines are routed at the upper elevations, a scenario expected to amplify P-Δ effects and upper-story drift demands. Case 3 (Bottom-Heavy) represents the routing of heavy lines at lower levels, typically associated with grade-level equipment connections, which may increase base shear demands. Case 4 (Mid-Concentrated) models a major pipe corridor concentrated at mid-height, representative of a pipe bridge scenario, and tests the sensitivity of the response to mass concentration at a single elevation zone.

Table 3. Pipe mass distribution cases (percentage of M_total = 256.8 tonnes per level).

Table 4. Pipe mass distribution per joint (16 joints per level).

Eleven pairs of bidirectional ground motion records are selected from the PEER NGA-West2 database [13]. The metadata of the selected records, station name, rupture distance Rrup, and shear-wave velocity Vs30 are reported in Table 5. The records were chosen according to the following criteria to ensure compatibility with the seismotectonic context and the characteristics of the reference structure. Records were restricted to shallow crustal earthquakes with moment magnitudes M_w in the range 6.5 - 7.5 and rupture distances R_rup not exceeding 30 km, consistent with the dominant seismic hazard scenario for moderate-to-high seismicity regions in which petrochemical facilities are typically sited [14]. The magnitude range was selected to capture ground motions with sufficient energy content in the period range of interest (0.2T₁ to 2.0T₁, spanning approximately 0.2 s to 5.6 s for the uniform case), while the distance constraint ensures that near-source directivity effects are not excluded [15].

The target design spectrum is constructed in accordance with ASCE 7-22 [7] for South Fork Long Canyon. The structure is assigned to Risk Category III, as pipe rack structures in petrochemical facilities support piping systems carrying hazardous materials whose failure could pose a substantial risk to human life and the environment, satisfying the criteria of ASCE 7-22. The site is classified as Site Class D (default soil type D recommended by ASCE), and the mapped spectral acceleration parameters are S_S = 0.78 g and S₁ = 0.273 g, consistent with the moderate seismic region described in Puttatt (2020) [3].

Table 5. Selected ground motion records.

Spectral matching was performed using SeismoMatch [16], which employs the wavelet-based algorithm proposed by Al Atik and Abrahamson (2010) [17]. This algorithm adjusts the frequency content of each record to match the target spectrum while preserving the non-stationary characteristics of the original ground motion, including the phasing, duration, and Arias intensity distribution. The matching was applied over the period range 0.2T₁ to 2.0T₁ of the uniform mass distribution case (Case 1), which encompasses the fundamental periods of all four cases. A matching tolerance of ±10% relative to the target spectral ordinate was enforced across the matching period range. Both horizontal components of each record were matched independently to the target spectrum, yielding 22 matched accelerograms (11 pairs). The quality of the spectral match is verified in Figure 2, which shows that the mean spectrum of the matched records closely follows the target spectrum across the full period range.

Figure 2. Response spectra of the eleven matched ground motion pairs compared with the ASCE 7-22 target design spectrum (5% damping): (a) X-direction; (b) Y-direction.

The use of spectral matching rather than amplitude scaling was adopted to minimise record-to-record variability in the spectral domain, thereby reducing the number of ground motions required for stable mean estimates [18][19]. This approach is consistent with the recommendations of ASCE 41-17 [20] and has been shown to produce unbiased estimates of mean nonlinear structural response when applied over an appropriate period range [18]. The number of records is consistent with the minimum requirement of ASCE 7-22 [7] Section 16.2.2, which stipulates that when eleven or more ground motions are used in nonlinear response history analysis, the mean value of the response parameter of interest may be used for design, rather than the more conservative maximum of the mean and the maximum response. The use of mean response is adopted in this study, as it provides a statistically stable estimate while maintaining computational tractability (4 cases × 11 pairs × 2 directions = 88 analysis runs).

Nonlinear response history analyses are performed for all four mass distribution cases by applying each horizontal component of every ground-motion pair separately to the three-dimensional model. For each of the eleven pairs, the first horizontal component (H1) is applied along the longitudinal (X, braced) direction with no transverse excitation, and the second horizontal component (H2) is applied along the transverse (Y, MRF) direction with no longitudinal excitation. This yields 4 cases × 11 record pairs × 2 loading directions = 88 nonlinear time-history runs in total. The unidirectional excitation protocol was adopted to isolate the effect of pipe mass vertical distribution on the response in each principal direction without compounding contributions from cross-axis coupling, and is consistent with the parametric scope of the study. Simultaneous bidirectional excitation, while permitted by ASCE 7-22 [7] Section 16.2.4 for design verification of three-dimensional models, is not required for the comparative parametric assessment performed here and is identified as a topic for future work in Section 5.

Viscous damping is modelled using Rayleigh damping (mass- and stiffness-proportional) with a damping ratio of 5% of critical, specified at two control periods: T₁, taken equal to the fundamental period of the case under analysis, and T₂, taken equal to the period of the higher mode at which the cumulative mass-participation ratio first exceeds 90% for that case. Anchoring T₁ at the fundamental period suppresses spurious damping of the first mode, while anchoring T₂ at the 90%-MPR mode prevents over-damping of the higher modes that contribute to the seismic response. The mass-proportional coefficient α and stiffness-proportional coefficient β are computed automatically by SAP2000 from these two control periods at 5% critical damping, in accordance with Equation (1) and Equation (2), where ξ = 0.05, ω₁ = 2π/T₁, and ω₂ = 2π/T₂.

The stiffness-proportional component of Rayleigh damping is formulated on the initial elastic stiffness matrix, consistent with the SAP2000 default for Direct Integration nonlinear time-history load cases. No supplementary control was applied to suppress artificial damping after yielding or brace buckling; the Rayleigh formulation defined above was the only viscous-damping source throughout the analysis. The 5% damping ratio is consistent with the conventional assumption for steel structures and with the damping level implicit in the ASCE 7-22 design spectrum [7]. Time integration was performed using the HHT-α method as implemented in SAP2000, which provides controlled high-frequency numerical dissipation without affecting the response in the period range of interest.

The analysis procedure is automated using the SAP2000 Open Application Programming Interface (OAPI) through Python scripting, which enables batch execution of all 88 runs and systematic extraction of the following engineering demand parameters (EDPs) from each analysis: i) peak interstory drift ratio (IDR) at each level; ii) maximum interstory drift ratio (MIDR), defined as the maximum of the IDR profile; iii) peak roof displacement; and iv) peak base shear, normalised by the seismic weight (V/W). For each mass distribution case, the mean value of each EDP across the eleven ground motions is computed for each loading direction and used as the representative response quantity.

The modal properties obtained from eigenvalue analysis for each mass distribution case are summarised in Table 6. For each case, the table reports the fundamental period and mass participation ratio (MPR) corresponding to the dominant translational mode in the longitudinal (X, braced) and transverse (Y, MRF) directions, together with the first three modal periods.

Table 6. Fundamental modal properties for each mass distribution case.

The results reveal a substantial influence of the pipe mass distribution on the dynamic characteristics of the structure. The uniform distribution (Case 1) yields the longest fundamental periods in both directions (T1, Y = 2.790 s, T1, X = 2.593 s), while the non-uniform distributions produce significantly shorter periods, ranging from 0.951 s to 1.200 s. This difference is attributable to the manner in which the equal distribution of mass across all pipe levels in Case 1 results in a higher effective mass at upper elevations relative to the non-uniform cases, thereby increasing the generalised mass associated with the first mode and elongating the period. In contrast, the non-uniform cases concentrate mass at specific levels, shifting the effective modal height and reducing the contribution of upper-level mass to the first mode.

The mass participation ratios are relatively consistent across all cases, with the first mode capturing approximately 79% - 81% of the total mass in the transverse direction and 92% - 94% in the longitudinal direction. The structural response is therefore governed by the first mode in both directions, with more than 90% of the total mass captured within the first three modes, consistent with the findings of Puttatt (2020) [3] for the reference configuration. This first-mode dominance is a characteristic feature of pipe rack structures due to the absence of floor diaphragms and the relatively uniform stiffness distribution along the height [1].

The mean interstory drift ratio (IDR) height profiles for all four mass distribution cases are presented in Figure 3, with subplots showing the longitudinal (X) and transverse (Y) directions separately. The corresponding maximum interstory drift ratio (MIDR) values are compared in Figure 4 and summarised in Table 7.

Figure 3. Mean interstory drift ratio height profiles: (a) longitudinal direction (X); (b) transverse direction (Y).

Figure 4. Mean maximum interstory drift ratio (MIDR): (a) longitudinal direction (X); (b) transverse direction (Y).

Table 7. Summary of mean peak response quantities for each mass distribution case.

In the longitudinal direction (Figure 3(a)), the uniform distribution (Case 1) exhibits the highest IDR values at all elevations, with the peak drift occurring at the top level (EL 25 m), reaching 7.19%. This pronounced concentration of drift at the upper story is consistent with the softening behaviour expected from progressive brace buckling under strong ground shaking, as observed by Puttatt (2020) [3] in the same reference structure. The non-uniform cases (Cases 2 - 4) produce substantially lower IDR profiles, with MIDR values ranging from 3.35% (Case 3, Bottom-Heavy) to 3.95% (Case 2, Top-Heavy), representing reductions of 45.1% to 53.4% relative to the uniform baseline.

In the transverse direction (Figure 3(b)), a similar trend is observed, though at lower absolute magnitudes owing to the greater flexibility and energy dissipation capacity of the moment-resisting frames. The uniform case produces an MIDR of 3.10% at EL 15 m, while the non-uniform cases yield MIDR values between 1.27% and 1.42%, corresponding to reductions of 54.1% to 58.9%. The drift profiles for the non-uniform cases are notably more compact and shifted toward the lower elevations compared to the uniform case.

The observation that the uniform distribution consistently produces the highest drift demands in both directions is a counterintuitive but physically explicable result. The substantially longer fundamental period of the uniform case (Table 6) positions the structure in a more displacement-sensitive region of the response spectrum, resulting in larger spectral displacement demands. Furthermore, the uniform distribution of mass across all levels maximises the effective modal height, thereby amplifying the displacement response at upper elevations. This finding has direct design implications, as discussed in Section 4.5.

The peak roof displacement and normalised base shear (V/W) for each mass distribution case are shown in Figure 5 and Figure 6, respectively, and are summarised alongside the MIDR values in Table 7. The percentage changes relative to the uniform baseline are presented in Table 8.

Figure 5. Mean peak roof displacement: (a) longitudinal direction (X); (b) transverse direction (Y).

Figure 6. Mean normalised peak base shear (V/W): (a) longitudinal direction (X); (b) transverse direction (Y).

Table 8. Percentage change in mean peak response quantities relative to the uniform baseline (Case 1).

To verify that the ranking of the four mass-distribution cases is stable across the eleven records and is not an artefact of the mean operation, the record-to-record dispersion of each engineering demand parameter was computed. Standard deviation (SD) and coefficient of variation (CoV) across the eleven pairs are reported in Table 9 together with the minimum and maximum response over the suite. The CoV ranges from 6% to 18% in the transverse direction and from 6% to 33% in the longitudinal direction, the higher dispersion in the longitudinal direction reflecting the well-known sensitivity of braced-frame response to the post-buckling sequence under individual records. The mean ranking—Case 1 producing the largest demands and Case 3 producing the smallest—is preserved record-by-record in the transverse direction for all eleven pairs, and in nine of the eleven pairs in the longitudinal direction. The dispersion, therefore, confirms that the demand reductions reported in Table 8 are stable and not driven by outlier records.

Table 9. Record-to-record dispersion of engineering demand parameters across the eleven ground-motion pairs (n = 11).

The peak roof displacement follows the same trend as the MIDR. In the longitudinal direction, the uniform case produces a mean roof displacement of 1022.1 mm, which reduces to 580.9 - 677.4 mm for the non-uniform cases (reductions of 33.7% to 43.2%). In the transverse direction, the corresponding reduction is from 493.7 mm to 205.0 - 231.7 mm (reductions of 53.1% to 58.5%). The bottom-heavy distribution (Case 3) consistently yields the lowest displacement demands in both directions, attributable to the concentration of mass at lower elevations, which reduces the effective modal height and, consequently, the displacement amplification along the height of the structure [9].

The normalised base shear (V/W) exhibits a pronounced sensitivity to the mass distribution. The uniform case produces V/W values of 16.49% and 39.01% in the longitudinal and transverse directions, respectively, while the non-uniform cases yield values ranging from 4.84% to 5.25% (X) and 12.80% to 14.15% (Y), representing reductions of 68% - 71% (X) and 64% - 67% (Y) relative to the uniform baseline. To distinguish the contribution of elastic spectral effects from that of nonlinear yielding and buckling, the 5%-damped ASCE 7-22 design-spectrum ordinates at T₁ are reported for each case in Table 10.

Table 10. ASCE 7-22 design-spectrum ordinates Sa (T₁, 5%) at the fundamental period of each case.

Table 11 reveals that the uniform case sits on the descending velocity-sensitive branch of the spectrum (T₁ ≈ 2.6 - 2.8 s, Sa ≈ 0.11g), while the non-uniform cases sit near the plateau (T₁ ≈ 1.0 - 1.2 s, Sa ≈ 0.25 g - 0.31 g). The elastic spectral demand at the fundamental period is therefore 2.4 - 2.7 times larger for the non-uniform cases than for the uniform case—that is, the period shift on its own would predict an increase in base shear for Cases 2 - 4 relative to Case 1, not the observed reduction of 64% - 71%.

Table 11. Comparison of mean MIDR values against FEMA 356 indicative performance drift limits.

FEMA 356 indicative drift limits for reference: Braced frame (long.): IO ≈ 0.5%, LS ≈ 1.5%, CP ≈ 2.0%; MRF (trans.): IO ≈ 0.7%, LS ≈ 2.0%, CP ≈ 5.0%.

The observed reduction must therefore be attributed predominantly to nonlinear effects. The uniform case enters a strongly nonlinear regime under all eleven records: extensive brace buckling in the longitudinal direction and significant plastic-hinge formation in the transverse direction MRF beams. The base shear of Case 1 reflects the post-buckling residual capacity of the structure (i.e., a capacity-limited response), and the very large MIDR of 7.19% (longitudinal) is a direct consequence of this softening combined with the long fundamental period that places the structure in a displacement-sensitive region. In contrast, the non-uniform cases experience comparatively limited yielding and only localised buckling, so their base shear approaches the elastic-spectrum demand reduced by moderate ductility. The 64% - 71% reduction in V/W is therefore best understood as the combined outcome of i) a long-period uniform-case response that is capacity-limited by widespread brace buckling and plastic hinging, and ii) shorter-period non-uniform cases that respond near the elastic plateau without comparable softening. Period-dependent elastic spectral effects [7], [21] are present but, on their own, would act in the opposite direction; the dominant mechanism is the difference in nonlinear inelastic engagement between the cases.

Among the non-uniform distributions, the mid-concentrated case (Case 4) produces slightly higher base shear in the transverse direction (V/W = 14.15%) compared to Cases 2 and 3. This is consistent with the observation that concentrating mass at mid-height elevations generates a higher effective seismic force for the MRF system, where the mode shape amplitude is maximum near mid-height [21].

To contextualise the computed drift demands within a performance-based framework, the mean MIDR values obtained for each mass distribution case are compared against the interstory drift acceptance criteria prescribed by ASCE 41-17 [20] and FEMA 356 [22] for steel structures. Although these standards were developed primarily for building structures, they are commonly applied to building-like non-building structures such as pipe racks in the absence of dedicated performance criteria [1][3]. The applicable drift limits depend on the structural system type and the target performance level: Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP). To place the computed drift demands in context, the mean MIDR values obtained for each mass distribution case are compared with the indicative interstory drift thresholds reported in FEMA 356 [22] for steel structures. The FEMA 356 thresholds were developed for steel buildings; they are not formal acceptance criteria for industrial pipe racks, and the present comparison is therefore made strictly for comparative purposes between the four mass-distribution cases rather than as a performance verification of the rack. The comparison highlights the relative shift in performance classification that the assumed pipe mass distribution can induce; it does not constitute a substitute for a component-level performance assessment.

For steel moment-resisting frames (transverse direction), FEMA 356 [22] Table C1-3 reports indicative interstory drift values of approximately 0.7% (IO), 2.5% (LS), and 5.0% (CP). For steel concentrically braced frames (longitudinal direction), the corresponding indicative values are approximately 0.5%, 1.5%, and 2.0%. ASCE 41-17 [20], which superseded FEMA 356, prescribes acceptance criteria at the component level (plastic rotation of beams and columns, axial deformation of bracing members, and so on) rather than global interstory-drift limits. A formal performance-based assessment of the rack in accordance with ASCE 41-17 would require these component-level checks to be evaluated for the most-stressed members under each case; such checks were not performed in the present parametric study and are identified as future work. The FEMA 356 global drift thresholds are therefore retained here only as approximate global indicators against which the relative performance of the four mass-distribution cases can be compared.

Table 11 summarises the assessment. In the longitudinal direction, the uniform case (Case 1) produces an MIDR of 7.19%, which substantially exceeds the CP threshold of 2.0% for braced frames, indicating that the structure under this mass configuration would be classified as experiencing collapse-level demands. The non-uniform cases yield MIDR values of 3.35% - 3.95%, which also exceed the CP limit but by a considerably smaller margin. In the transverse direction, the uniform case produces an MIDR of 3.10%, which exceeds the LS limit of 2.0% but remains below the CP threshold of 5.0%, placing it in the LS-CP range. The non-uniform cases yield MIDR values of 1.27% - 1.42%, placing them within the IO-LS range—a markedly more favourable performance classification.

These results underscore two important points. First, the longitudinal (braced) direction is the critical direction for all mass distribution cases, with all cases exceeding the CP drift limit. This is consistent with the progressive brace buckling mechanism observed in the IDR profiles (Figure 3(a)) and with the findings of Puttatt (2020) [3], who similarly reported that the CP limit state could not be verified in the braced direction. Second, the difference in performance classification between the uniform and non-uniform cases is pronounced in the transverse direction: the uniform case falls in the LS-CP range, while the non-uniform cases remain within the IO-LS range. This demonstrates that the assumed pipe mass distribution can shift the performance assessment outcome by one or more performance levels, reinforcing the practical significance of the parametric variable investigated in this study. It is reiterated that the performance classifications presented in Table 9 are indicative only, derived from global drift thresholds developed for steel buildings and applied here for comparative purposes between cases. A formal performance-based assessment of the rack would require component-level acceptance checks per ASCE 41-17 [20] (plastic rotations of beam and column elements, axial deformation demands on the bracing link elements, and member-by-member chord rotations), which were not performed in this parametric study and are identified as a topic for future work.

The results of this parametric study yield several findings with direct relevance to the seismic design practice of industrial pipe rack structures.

First, the uniform mass distribution assumption, commonly adopted in design practice for simplicity, is shown to be the most conservative case across all engineering demand parameters and both structural directions. The non-uniform distributions consistently produce lower MIDR, roof displacement, and base shear demands by margins ranging from 34% to 71%. This finding suggests that the uniform assumption provides an inherent safety margin that is not explicitly recognised in current design codes [7][1]. From a design verification perspective, this is a favourable outcome: engineers who design pipe racks under the uniform mass assumption are unlikely to underestimate the seismic demands for any realistic operational configuration.

Second, the sensitivity of the structural response to the mass distribution is primarily mediated through the fundamental period. The uniform distribution, by spreading mass equally across all levels, maximises the generalised mass associated with the first mode and elongates the fundamental period, placing the structure in a more displacement- and acceleration-sensitive region of the response spectrum. This mechanism is analogous to the effect of vertical mass irregularity in building structures, where floors with concentrated mass attract higher seismic demands [9][10]. However, the effect observed here operates in the opposite direction: the uniform distribution, by virtue of its higher effective modal mass, produces the most severe response.

Third, the bottom-heavy distribution (Case 3) yields the lowest demands across nearly all EDPs, suggesting that configurations in which heavy process lines are routed at lower elevations are inherently beneficial from a seismic perspective. This finding is consistent with the general principle that lowering the centre of mass reduces overturning demands and the effective modal height [21].

Fourth, the magnitude of variation observed—up to 71% reduction in base shear and 59% reduction in MIDR—is sufficiently large to warrant consideration in design optimisation. While the uniform assumption remains conservative, the substantial over-prediction of demands may lead to unnecessarily heavy member sections, particularly in the longitudinal (braced) direction where the baseline MIDR exceeds 7%. This over-design could be mitigated by accounting for the actual pipe mass distribution when it is known at the detailed engineering stage, as recommended by Di Sarno and Karagiannakis (2020) [1] for the general case of pipe rack design.

It should be noted that the present study considers the pipe mass as lumped at beam-column joints without modelling the dynamic interaction between the piping system and the supporting structure. As demonstrated by Di Sarno and Karagiannakis (2020) [5] and Karagiannakis and Di Sarno (2021) [8], the coupling between the piping system and the rack can modify the seismic demands, particularly at elevations where heavy pipes are concentrated. The trends identified in this study may therefore be modulated when pipe-structure interaction is explicitly considered, and this aspect is recommended for future investigation.