# Numerical Study on Seismic Response of Steel Storage Racks with Roller Type Isolator

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. General Description of Numerical Model

_{p}) were analyzed, shown in Figure 1. The internal damping of the structure was considered 3% of the critical damping, according to NCh2369 [21]. The ASTM A36 material was used in the numerical model because it is commonly used in Chile for steel storage racks.

_{I}= 3, 3.5, 4, 4.5 and 5 s. To evaluate the effect of the dissipation capacity of the isolation device on the response of the rack, three friction coefficients were considered on the device’s ball joint (Figure 2 and Figure 3), where µ = 0.2, 0.4 and 0.6. The range for this parameter was chosen considering as a reference that the friction coefficient between steel-steel is µ = 0.4. Each rack with baseline isolation was analyzed considering all possible combinations between the three values of friction coefficient µ, and the five values of the isolation period T

_{I}mentioned above, with a total of 15 cases with baseline isolation per rack.

#### 2.2. Description of Roller Type Isolator

_{0}. As a restriction, it was established for this stress that σ

_{0}≤ σ

_{adm}= σ

_{y}/SF, where σ

_{y}= 31.0 MPa is the yield strength of Teflon at 23 °C [30] and SF is the safety factor. The latter was considered as SF = 2, taking into account that the maximum vertical seismic overload can at most equal the static load. The resistance to compression of the tension was considered as a reference in order that said material can be used in the coating of the support housing of the ball joint. However, the design of the ball joint and its support housing is not restricted to this material; other materials of equal or greater strength may be used.

_{0}(Figure 3b), considering a height between load levels equal to 1.5 m for calculating the rack height.

_{b}, due to a displacement “u” with velocity $\u201c\dot{u}\u201d$ and dissipation by friction in the ball joint of the basal support of the device is defined by:

_{n}is the normal reaction to the surface of the ball joint at the base of the isolator.

_{n}, in all the isolators is approximately equal to the weight of the structure above the isolation level. Therefore, given isolator height H, and total weight structure W, the dissipation capacity of the isolation system as a whole is approximately proportional to the product d∙μ. This implies that, if in practice it is desirable to modify the dissipation capacity of the isolation system, the materiality of the surfaces in contact in the ball joint must be changed—change μ—, or the diameter of ball-joint d, or both.

_{0}; (2) it can withstand the elongation imposed by the maximum possible lateral displacement, in addition to the pre-tension elongation; and (3) its stiffness is not very high so that T(u) does not differ substantially from T

_{0}. The purpose of the latter is to avoid an excessive increase in the lateral stiffness of the isolator. Maureira-Carsalade et al. [25] propose a feasible alternative: using a compressed air piston as an elastic element (Figure 2).

_{max}, and the ball joint in the basal bearing.

_{0}, allow the three requirements described above to be met and provide the desired lateral stiffness to the isolator. Maureira-Carsalade et al. [25] describe the tension T(u) of the cable that connects the isolator to the base of the rack (Figure 2b) by means of the real gas law as T(u) = p(u)A

_{cs}, where A

_{cs}is the cross section of the lower air chamber. However, if the working pressure p(u) is less than or approximately equal to 2 MPa, the well-known ideal gas law—p

_{1}V

_{1}= p

_{2}V

_{2}—allows obtaining a suitable approximation for the expression T(u), that is:

_{0}. This is achieved when ${V}_{ac}^{\left(2\right)}\gg {V}_{ac}^{\left(1\right)}$, which is why in this investigation it is considered that the body of the isolator is formed by two cylindrical chambers of greater diameter in the upper portion than in the lower portion. (${D}_{ac}^{\left(2\right)}>{D}_{ac}^{\left(1\right)}$ in Figure 2).

_{0}and the initial pressure p

_{0}are related to each other through the cross-sectional area of the lower portion of the pneumatic piston (A

_{cs}in Figure 2). These parameters are in turn related to the isolation period T

_{I}, the height of the isolator H, the total mass of the superstructure m

_{t}, and the number of isolators n

_{I}, using Equation (6), adapted from [25]:

_{I}, given the initial pressure and the other structural parameters.

_{0}, inside the pneumatic piston that causes the piston inside the isolator to provide the necessary lateral stiffness so that each rack analyzed has the desired isolation period. In the calculations, an inner diameter of the lower and upper air chambers was considered ${D}_{ac}^{\left(1\right)}=$ 90 mm y ${D}_{ac}^{\left(2\right)}=$ 120 mm, respectively.

_{0}, of the cable connecting the piston plunger to the rack base (Figure 2b), was determined for each rack and each isolation period, using Equation (6) with the data from Table 2. The calculated values are shown in Table 3 for each rack analyzed according to its number of load levels and T

_{I}isolation periods.

_{max}= 300 mm. This displacement is the reference value established as the demand for the NCh2745 design displacement spectrum in seismic zone 3 and type of soil III, for a structure without mass eccentricity and considering an effective damping ratio of the isolation system equal to 10%.

## 3. Nonlinear Analysis

#### 3.1. Nonlinear Racks Modeling

_{p}in parallel with an elastic element of stiffness (1 − α)·k

_{1}, which works as long as the capacity of the frictional element is not equalized. This pair of in-series elements is in turn parallel to an elastic element of stiffness α∙k

_{1}, which works independently. The hysterical characteristic curve is shown in Figure 5b, in which a yield point (θ

_{f}, M

_{p}), is defined, in which the stiffness changes from k

_{1}to k

_{2}= α∙k

_{1}. This behavior is the sum of the behavior of the elastic branch (Figure 5c) where only the stiffness spring k

_{2}= α∙k

_{1}works, plus the plastic branch. In the latter, only the stiffness spring (1 − α)∙k

_{1}works, as long as the stress is less than M

_{p}= (1 − α)∙k

_{1}∙θ

_{f}= (1 − α)∙M

_{f}. Once this capacity is reached, all the displacement concentrates in the frictional element. The capacity being constant and equal to M

_{p}, the displacement is concentrated in the frictional element (Figure 5d).

_{1}in Figure 5b). If −1 < z < 1, then M

_{P}< M

_{p}= (1 − α)∙M

_{f}(Figure 5d), if |z| = 1, then |M

_{P}| = (1 − α)∙M

_{f}. According to the above, the expression that defines the total moment M (Figure 5b) is:

#### 3.2. Seismic Input

## 4. Results and Discussion

_{abs}); and maximum in time for basal shear load relative to the seismic weight (Q

_{0}/P

_{s}). The maximum values of ü

_{abs}and Q

_{0}/P

_{s}were obtained by imposing the seismic force perpendicularly to the direction of the aisles of racks. This is consistent with the fact that in this direction, the structure is more rigid due to stiffer braced axes that provide lateral resistance. The maximum drift was obtained by imposing the seismic force parallel to the direction of the aisles of racks since it is the most flexible direction of the structure where the resistant axes are flexible frames.

_{0}was divided by the seismic weight P

_{s}to be able to compare this response in racks of different heights and weights. To determine the average maximum absolute floor acceleration ü

_{abs}, the average value of the absolute floor accelerations was first determined for all levels, obtaining a single value for each analysis instance. Then, the maximum of said average acceleration was determined throughout the duration of the analysis in order to soften the maximum response obtained, eliminating specific peak values of very short duration, since the effect of absolute acceleration is reflected in practice in the overturning of stored pallets. However, acceleration peaks of very short duration fail to generate this effect, since, by rapidly reversing its direction, the pallet stabilizes again on the rack. This effect is shown graphically relative to the acceleration of gravity in Figure 11.

_{I}) and load levels. In order to verify the effectiveness of the use of the base isolation and show the decrease of the target responses, the corresponding maximum response of the fixed base rack is included.

_{I}= 3 s, friction coefficient in the ball joint μ = 0.2 and its diameter d = 10 cm, according to Table 1.

_{I}, the response decreases consistently, independent of the value of the friction coefficient μ and the load levels of the rack. The effect of modifying the friction coefficient μ shows a clear trend, the higher it, the lower the basal shear load, regardless of the number of rack levels. However, the reduction in this response is greater in racks with higher load levels. As shown in Table 1, the more load levels the rack has, the greater the diameter d assigned to the ball joint of the base support of the isolators. According to the analysis of Equation (3) in Section 2.2, the energy dissipation provided by the isolation system as a whole is proportional to the product d∙μ. This explains why the maximum basal shear load in racks with base isolation is lower in racks with more load levels, for the same T

_{I}and μ values; because the greater the number of levels, the greater the diameter d of the ball joint at the base (Table 1).

_{I}. Table 4 shows that the natural periods of the rack fixed at its base are longer when it has more load levels. The closer the period of isolation is to the fundamental period of the fixed rack at its base, the less the effect of the basal isolation in decoupling the movement of the structure from the ground. This phenomenon explains why the maximum drifts are larger in racks with more load levels with basal isolation of the same T

_{I}period and the same friction coefficient μ on the isolator ball joint. However, in all cases with base isolation, the maximum drift is below 1%, complying with the requirement of NCh2369, which establishes 1.5% as the desired limit.

_{I}isolation period values that differ depending on the number of rack levels. For low isolation periods, an increase in μ leads to a reduction in the maximum drift. On the other hand, for high T

_{I}values, the trend is the opposite. The T

_{I}value for which the trend is reversed is lower the more load levels the rack has. In general, either the higher the rack, the longer its isolation period T

_{I}, or both, the more convenient it is to have a lower value of the friction coefficient μ to reduce the rack’s maximum floor drift.

_{I}= 4, 4.5 and 5 s and friction coefficient in the ball joint μ = 0.6, where the critical acceleration is slightly exceeded. This value is consistent, since the friction dissipation acts as a brake, increasing the absolute accelerations, although the basal shear load is reduced. However, this problem is solved by reducing the dissipation capacity in the isolation system by decreasing the friction coefficient μ or the diameter of the ball joint at the base of the isolation devices d.

_{p}= 1.35 m and width between support beams B = 0.85 m. This figure illustrates the horizontal acceleration condition considered critical for overturning effects. This condition is defined by the state in which the projection of the force resulting from the seismic action and the acceleration of gravity cuts the basal edge of the pallet (pivot rolling in Figure 11). This effect corresponds to an unstable equilibrium condition, since a horizontal acceleration greater than that illustrated in Figure 11 and defined analytically by Equation (12), will cause the stored pallet to overturn. In the illustrated condition, if the absolute horizontal acceleration is greater than 0.63∙g, the pallets will overturn.

_{I}= 3, 4 and 5 s. It is observed that there is a reduction in the lateral displacements of the base isolated rack (models 3) with respect to the models (1) and (2) of racks fixed at its base. In storage level 8, a reduction of 46.2 cm is observed between rack (3) with T

_{I}= 5 s and rack (1); such reduction is 26.9 cm when compared to the rack (2). Furthermore, the lateral displacements in level 8 are 9.6 and 7.9 cm, for the racks with an isolation period of 3 and 5 s, respectively. The displacement of the racks with isolation periods of 4 s and 5 s are practically the same. It is inferred that the longer the period of isolation the structure will have a greater reduction in the target response of floor drift. It is also observed that the maximum floor displacements of the rack with semi-rigid connections are less than those of the rack with rigid connections. Although the former is more flexible than the latter, the energy dissipation capacity provided by the semi-rigid connections contributed to reducing lateral displacement, achieving better performance in terms of maximum drift.

_{I}. It is observed that the greater the magnitude of μ, the maximum displacement in the base becomes less and less, due to the greater dissipation of energy. It is also observed that the less slender the structure—fewer load levels—the greater the maximum displacement achieved at the base, compared to more slender structures. This relation makes sense since the structure becomes more flexible as it is more slender, that is, its fundamental period increases, having a lower acceleration in the design spectrum (Figure 6).

_{I}= 2π/T

_{I}is the isolation frequency (rad/s). The energy dissipated by a linear dynamic system in an imposed displacement cycle is given by [25]:

_{n}= m

_{t}∙g and performing the calculations, the following is obtained:

_{I}= 3 s and a rack with 2 load levels (Figure 13a). This value is due to the fact that the equivalent damping is very low, ${\xi}^{eq}=$ 1.2% in this case, as shown in Figure 14. However, when the displacement is less than 30 cm, there is a greater ${\xi}^{eq}$. This is exemplified when μ = 0.2, a rack with 4 load levels and T

_{I}= 4, 4.5 and 5 s is considered, obtaining ${\xi}^{eq}$ = 8.5, 11.4 and 14.5%, respectively (Figure 14).

_{I}= 4 s. It is observed that the greater the magnitude μ, the band of the hysterical cycle becomes wider, so there is a greater dissipation of energy; the same is observed in the equivalent damping ratio (Figure 14). The effectiveness of energy dissipation can be seen when the magnitudes of the maximum displacement of the isolation level (Figure 15) decrease with increasing μ for racks with the same number of levels.

_{0}), considering the seismic input acting in cross the aisles direction of the rack. While in Figure 17, the seismic input is considered to be acting in the longitudinal direction. If the quotient is greater than 1 there is lifting in the isolator, otherwise, the isolator is compressed and works stably.

## 5. Summary

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${N}_{p}$ | Number of pallet load levels |

${T}_{I}$ | Isolation period. |

T_{0} | Initial stress of the elastic element inside the isolator, T_{0} > 0 in tension. |

T(u) | Tension of the cable in series with the elastic element as a function of $u$. |

d | Ball diameter at the insulator basal patella (d = 2r, Figure 3b). |

H | Isolator height (Figure 2a). |

R_{n} | Normal reaction in the basal ball patella (Equation (3)), R_{n} > 0 in compression. |

µ | Coefficient of friction between materials in contact in the basal ball joint. |

${V}_{ac}^{\left(1\right)}$ | Volume of the upper air chamber. |

${V}_{ac}^{\left(2\right)}$ | Lower air chamber volume. |

${L}_{ac}$ | Length of the compressed air chamber, ${L}_{ac}^{\left(1\right)}$ and ${L}_{ac}^{\left(2\right)}$ is the length of the upper and lower cylinder, respectively. |

${p}_{0}$ | Initial air chamber pressure. |

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**Figure 2.**Scheme of the isolator used with a compressed air piston as an elastic element (

**a**) static or undeformed equilibrium position, (

**b**) deformed position with large lateral displacement.

**Figure 3.**Pressure distribution in the ball joint of the isolator base. (

**a**) Ball joint inside its bracket at the base of the isolator, (

**b**) free-body diagram (F.B.D.) of the ball joint.

**Figure 5.**(

**a**) Bouc–Wen bilinear conceptual model. (

**b**) Hysteretic characteristic curve of the model, (

**c**) linear elastic component of the model, (

**d**) plastic component of the model.

**Figure 6.**Design spectrum of NCh2745 for type soil III, seismic zone 3 and 5% critical damping. The response spectra of the original and scaled frequency domain recording are overlaid to fit the design spectrum.

**Figure 7.**Llolleo seismic record of 3 March 1985, component N10E: (

**a**) original record, (

**b**) record scaled in frequency domain to fit with design spectrum of NCh2745.

**Figure 8.**Maximum basal shear load in racks with base isolation with different load levels, friction coefficient (μ) and isolation period (${T}_{I}$). The black dotted line is the response for fixed base racks.

**Figure 9.**Maximum drift in racks with basal isolation with different load levels, friction coefficient (μ) and isolation period (${T}_{I}$). The black dotted line is the response for fixed base racks.

**Figure 10.**Maximum acceleration in racks with base isolation with different load levels, friction coefficient (μ) and isolation period (${T}_{I}$). The black dotted line is the response for fixed base racks.

**Figure 12.**Lateral displacement of the load levels of 8-level racks with fixed support at the base and with base isolation.

**Figure 14.**Equivalent damping ratio (${\xi}^{eq})$ of the isolation system considering different friction coefficients on the insulator ball joint (μ) and maximum basal displacement $({u}_{b}^{max}$ ).

**Figure 16.**Ratio between the maximum axial load (traction > 0) or minimum axial load (compression < 0) on the isolator and the initial tension of the elastic element inside the insulator (T

_{0}). Orange segmented line (value 1) delimits the occurrence of the survey (points above said line). The seismic action is acting in cross the aisles direction of the rack.

**Figure 17.**Ratio between the maximum axial load (traction > 0) or minimum axial load (compression < 0) on the isolator and the initial tension of the elastic element inside the insulator (T

_{0}). Orange segmented line refers to the value of 1. Input in the longitudinal direction of the rack.

Level | Total Height (m) | Weight (kN) | d (mm) | σ_{o} (MPa) |
---|---|---|---|---|

2 | 3 | 41.1 | 50 | 7.85 |

4 | 6 | 82.9 | 70 | 8.07 |

6 | 9 | 124.6 | 85 | 8.24 |

8 | 12 | 166.4 | 100 | 7.95 |

**Table 2.**Initial pressures inside the pneumatic piston (MPa) according to the rack levels and T

_{I}.

Level | 2 | 4 | 6 | 8 | |
---|---|---|---|---|---|

T_{I} | |||||

3.0 | 0.433 | 0.874 | 1.314 | 1.754 | |

3.5 | 0.318 | 0.642 | 0.965 | 1.289 | |

4.0 | 0.244 | 0.491 | 0.739 | 0.987 | |

4.5 | 0.193 | 0.388 | 0.584 | 0.780 | |

5.0 | 0.156 | 0.314 | 0.473 | 0.632 |

**Table 3.**Initial tension of the elastic element or cable that joins the isolator with the rack base, T

_{0}(kN).

Level | 2 | 4 | 6 | 8 | |
---|---|---|---|---|---|

T_{I} | |||||

3.0 | 2.76 | 5.56 | 8.36 | 11.16 | |

3.5 | 2.02 | 4.08 | 6.14 | 8.20 | |

4.0 | 1.55 | 3.13 | 4.70 | 6.28 | |

4.5 | 1.22 | 2.47 | 3.72 | 4.96 | |

5.0 | 0.99 | 2.00 | 3.01 | 4.02 |

Mode (N°) | 2 Level | 4 Level | 6 Level | 8 Level |
---|---|---|---|---|

1 | 0.329 | 0.640 | 0.956 | 1.276 |

2 | 0.137 | 0.304 | 0.562 | 0.858 |

3 | 0.097 | 0.199 | 0.305 | 0.413 |

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## Share and Cite

**MDPI and ACS Style**

Álvarez, O.; Maureira, N.; Nuñez, E.; Sanhueza, F.; Roco-Videla, Á.
Numerical Study on Seismic Response of Steel Storage Racks with Roller Type Isolator. *Metals* **2021**, *11*, 158.
https://doi.org/10.3390/met11010158

**AMA Style**

Álvarez O, Maureira N, Nuñez E, Sanhueza F, Roco-Videla Á.
Numerical Study on Seismic Response of Steel Storage Racks with Roller Type Isolator. *Metals*. 2021; 11(1):158.
https://doi.org/10.3390/met11010158

**Chicago/Turabian Style**

Álvarez, Oscar, Nelson Maureira, Eduardo Nuñez, Frank Sanhueza, and Ángel Roco-Videla.
2021. "Numerical Study on Seismic Response of Steel Storage Racks with Roller Type Isolator" *Metals* 11, no. 1: 158.
https://doi.org/10.3390/met11010158