Model Analysis of Steel Frame Structures Considering Interactions between Racks and the Frame

Abstract

The steel racks on the floor are seen as live loads in the current design process, ignoring the interaction with the supporting frames. In this paper, multiple steel racks with different masses and stiffnesses are placed on the first floor of a two-story main structure to form different real structures (RS). The corresponding simplified structures (SS) are frames with the mass of steel racks concentrated on the first floor of the main structure. Modal analysis is performed to analyze the relationship between the periods of RS and SS in the cross-aisle direction. Firstly, the beams on the first floor are assumed to be infinitely rigid. The relationship between the periods of the rack T_Rk, the simplified structure T_SS, and the real structure T_RS under different mass ratios α is established, and an accurate equation relating T_RS with T_Rk and T_SS is proposed. Moreover, by considering the influence of finite beam stiffness, the interaction between racks and the main structure is studied by constructing different analysis models. The effect of the main structure on the racks is reflected by a combined system consisting of beams and racks. A modified model, distinguished from SS by considering the effect of no-mass racks, is constructed to study the strengthening effect of the racks on the first-floor beams. The effect of the top connecting bars is also analyzed.

1. Introduction

In recent years, a large number of two-story warehouses have been built in China, as shown in Figure 1. The warehouse height can reach 20–25 m, and the story height is about 10–12 m. The cold-formed steel storage racks are installed on the ground and the first floor to make effective use of precious spaces.

Currently, the steel racks and main structures are designed, respectively, by rack manufacturers [1] and authorized institutes for two-story warehouses where racks are installed both on the ground and first floors. The steel racks are regarded as nonstructural components fixed to the ground [2,3,4,5]. Since the arrangement of steel racks is usually uncertain in the early design stage, the steel racks are simplified as additional live loads and the interaction between the main structures and the racks is ignored.

These two-story warehouses with racks installed on the first floor can be seen as primary-secondary combined systems, where the interaction between the primary and secondary structures can be called the Primary-Secondary Structure Interaction (PSSI), ensuring that the structure performs as a whole system. By considering the influences of PSSI, the floor response spectrum (FRS) method is applied to check the strength of non-structural members and the connections between non-structural members and supporting structures. For the calculation of primary-secondary combined systems, Sackman and Der Kiureghian [6,7] adopted the perturbation theory based on the dynamic properties of the primary and secondary structures. However, this method, based on the available literature, is only suitable when the secondary system is relatively light (usually below 1%) compared with the total mass of the whole system [8,9,10,11,12,13].

The FRS approach has attached much attention recently, however, these research works [14,15,16,17,18,19,20,21,22] concentrated on the simplified single-degree-of-freedom (SDOF) secondary system and elastic and inelastic multi-degree-of-freedom (MDOF) primary structure systems. It should be noted that secondary constructions may be connected to more than two locations in practical engineering, but the FRS method only allows for the construction of one or two points of attachment.

For the large two-story warehouse described in Figure 1, the racks on the first floor enlarge the demand on the stiffness of the floor, so the floor slabs of the main structures are generally made of reinforced concrete with thicknesses of 200–250 mm. On the first floors of the main structures, the dead load is 5–8 kN/m² and the live load is usually as high as 20 kN/m². Under these loading conditions, the mass of goods on the steel racks can account for more than 70% of the total mass of the structures. The current FRS approach is mainly for structures where the mass from the secondary equipment is very small and is not applicable for frames investigated in this paper where most of the mass is from secondary structures.

Meanwhile, the rack is a multistory structure with a height of 8 m–12 m and should be treated as an MDOF secondary system. Thus, the racks on the first floor should be considered as multiple identical MDOF secondary systems interconnected at the top. The interaction between multiple MDOF secondary systems is not considered in the current FRS approach. Therefore, a new analysis must be carried out so that the FRS approach is modified for possible future applications.

In the current design specifications involving multistory racks, no specific design requirements and methods are given for the racks installed on the first floor. It is specified in ASCE 7 [2] that when the weight of a nonstructural element is not less than 25% of the effective seismic weight of the structure, the nonstructural element is regarded as a nonbuilding structure. In addition, when the nonbuilding structure weight is not less than 25% of the combined effective seismic weights of the whole system, the different fundamental period T of the nonbuilding structure leads to different calculating methods. When T is less than 0.06 s, the nonbuilding structure should be treated as a rigid element with appropriate distribution of its effective seismic weight, and it can be simplified as an additional mass when designing the supporting structures. This method is similar to the method used when the mass of the supported nonbuilding structure is less than 25% of the mass of the combined system. However, when T is not less than 0.06 s, the supporting structure and nonbuilding structure should be regarded as a combined system with the appropriate stiffness and effective seismic weight distributions. Moreover, Chinese Codes GB50011 [23] and JGJ339 [24] only specify the seismic design method for the non-structural component of buildings when its mass is less than 10% of the floor.

A two-story frame was constructed by Zhang and Tong [25], where the racks were installed on the first floor. Dynamic history analysis on the overall structures was conducted. The elastic shear force spectra of the frames and racks were studied to verify the applicability of the simplified design method where the racks are regarded as live loads. Simplified design methods for racks and frames were proposed and the period relationship for three types of models was established. The interaction between steel racks and the main structure is considered but the influence of the floor elasticity was neglected, and the mass of racks was constant during the research, so the applicability of the period relationship needs to be further verified.

Therefore, for the two-story main structures with racks installed on the first floor, both steel racks and main structures with different first vibration periods are selected to be combined into different real structures in this paper. The following analysis will be performed in the cross-aisle (CA) direction:

(1)

Modal analysis will be conducted to study the dynamic characteristics of the real structures and the simplified structures. The mass of racks is concentrated on the first floor of the main structures in the simplified structures. The relationship between the periods of the racks, the main structure, simplified structures, and real structures will be studied.

(2)

The influence of the mass ratio of the rack mass to the total mass of the first floor will be involved.

(3)

The influence of the floor beam stiffness will be considered.

(4)

The interaction between the main structures and the racks considering the stiffness of the floor will be discussed.

The spine-bracing system is always used in the down-aisle (DA) direction to bear the weight of seismic actions [2,4], representing a different aseismic structural system. Thus, it will be examined in future studies.

2. Calculation Models and Process

2.1. Calculation Models

In this paper, a two-story structure (Figure 2d) with multiple six-story racks installed on the first floor is studied. Based on this original structure, the following four models are analyzed.

• The first model is the steel rack. As shown in Figure 2a, the steel rack height is 9.0 m with each story possessing a height of 1.5 m. There are four rack pieces in the DA direction, and the span between rack pieces is 4.0 m. The spacing between two rack columns is 1.0 m in the CA direction, and the effective mass of each story is denoted by m. The elasticity modulus of elements in the rack is denoted by E_Rk, and the natural period of the steel rack in the CA direction is denoted by T_Rk. The value of T_Rk is changed by varying E_Rk.
• A two-story steel frame is taken as the main structure, which is the second model and is described in Figure 2b. The spans of the main structure in two directions are both 12.0 m, and the height of each story is 10.0 m. The mass of the first floor is M₁, and the mass of the roof is M₂. The elastic modulus of the column of the main structure is denoted by E_C, and the elastic modulus of the beam is denoted as E_B. The natural period of the main structure in the CA direction is denoted by T_SS0 (without including the masses of racks). Based on the given sections of main structural components, the period T_SS0 will vary in the analysis through changing E_C and E_B.
• As presented in Figure 2c, the simplified structure is the third model analyzed in this paper, which is widely used in current seismic design. The total mass of racks is concentrated on the first floor of the main structure, and the interaction between the main structure and racks is ignored. Thus, the mass of the first floor in the simplified structure is denoted as $M_1+n\cdot 6m$, in which n represents the number of steel racks. The natural period of the simplified structure in the CA direction is denoted by T_SS, which varies by changing the values of E_C and E_B. The simplified structure is abbreviated as SS in the following analysis for convenience.
• The last model is the real structure shown in Figure 2d, which is composed of the main structure and n regularly spaced steel racks with the same stiffness. As a result, the mass of the first floor is M₁, while that of the roof is M₂. The effective mass of each story m is considered as live loads subjected to the main structure, but the active points of the live loads are at their original positions, which is different from the SS. All the racks are interconnected at their tops. The natural period of the real structure in the CA direction is denoted by T_RS, and the real structure is abbreviated as RS.

For the convenience of calculation, the sections of calculation models stay the same during the calculation.

Table 1. The Sections of Calculation Models.

Model	Element	Section	Size (mm)
			Depth	Flange Width	Flange Thickness	Web Thickness
Main structure	column	W14 × 211	399	401	39.6	24.9
	beam	W16 × 89	427	264	22.2	13.3
Rack	upright	W8 × 31	203	203	11	7.24
	Pallet beam	W8 × 15	206	102	8	6.22
	Upright bracing	W4 × 13	106	103	8.76	7.11

summarizes the details of the calculation models in OpenSees, and the initial elasticity modulus is set as $2\times {10}^6\mathrm{N}/{\mathrm{mm}}^2$.

2.2. Calculation Methods and Processes

The software OpenSees [26] is used for structural modal analysis. Since a large amount of modal analysis is carried out in this paper, MATLAB is used for data analysis. The steel members of calculation models are linearly elastic since the study is intended to serve the practical design. The beams and columns are simulated by elastic beam-column elements, and the elastic truss elements are used to model the braces of the racks. For the convenience of calculation, the sections of calculation models stay the same during the calculation. In addition, the masses of the main structures in OpenSees are M₁ = 8.155 × 10⁴ kg and M₂ = 2.5 × 10⁴ kg. The seismic masses are assumed to be combined on the nodes in terms of numerical modeling. Furthermore, because concrete slabs are usually used on the floor, a rigid diaphragm is applied for all floors [27]. The connections between beams and columns and between columns and foundations are set to be rigid. For the real structures with racks on the first floor, the uprights of racks are simply supported on the floor beams in the models, which means that only translational restraints are given to the bottom nodes of the rack uprights.

Two types of modal analysis are firstly defined as follows:

Modal Analysis 1 (MA1): The period T_Rk of the steel rack remains constant, and the period T_SS0 of the main structure changes continuously.

Modal Analysis 2 (MA2): The period T_SS0 of the main structure remains constant, and the period T_Rk of the steel rack changes continuously.

The number of steel racks n and the mass of each rack story m varies during the calculation. For the convenience of calculation, the sections of calculation models stay the same during the calculation, and the change of the periods of different calculation models will be realized by changing the elastic modulus of the structural members of models.

Based on the four calculation models mentioned above, the following aspects are studied respectively:

(1)

The beam stiffness of the first floor is infinite ($E_{\mathrm{B}}=\infty$), which means the influence of the elasticity of the first floor of the main structure is not considered.

Modal analysis is performed on both RSs and SSs to calculate periods T_RS and T_SS, and the relationship between T_SS, T_Rk, and T_RS is studied in MA1 and MA2, respectively. In these cases, the period T_SS0 will be varied by changing the elastic modulus E_C of the columns of the main structure.

(2)

When the beam stiffness of the first floor is finite, the relationship between T_SS, T_Rk, and T_RS is also studied in MA1 and MA2. In these cases, the period T_SS0 will be varied by changing the elastic modulus E_C (column) and E_B (beam) of the main structure simultaneously.

(3)

The interaction between the regularly spaced steel racks and the main structure is then studied, considering the influence of the floor elasticity of the main structure.

3. Period Relationship with Infinite Beam Stiffness

3.1. Rack Periods

As shown in Figure 2a, the period of an independent rack fixed on the ground is T_Rk. When this rack is installed on the first floor of the main structure (Figure 3a), the beams become the elastic support of the racks. As a result, the period of the rack installed on the first floor will change. Therefore, a combined system consisting of floor beams and racks can be constructed with the beams being fixed, and the period of this beams-racks combined system is denoted by $T_{\mathrm{Rk}n}^{\prime }$, where the subscript n represents the number of racks. The combined system with one rack is shown in Figure 3b and the period is $T_{\mathrm{Rk}1}^{\prime }$.

In this section, the beams of the first floor are assumed to be infinitely rigid ($E_{\mathrm{B}}=\infty$). Given the mass m of each rack floor, the elasticity modulus of the rack E_Rk is modified to obtain an independent rack with the period T_Rk = 1.0 s. The period of the beams-racks combined system with one rack can be obtained as $T_{\mathrm{Rk}1}^{\prime }=T_{\mathrm{Rk}}=1.0 \mathrm{s}$, indicating that T_Rk does not change if the first-floor beams are infinitely rigid.

As the number of racks increases from 1 to 6, different beams-racks combined systems are shown in Figure 4. The connections between the racks at their tops ensure that all racks in the combined system vibrate in the same first vibration mode, as shown in Figure 5. It can be seen that without considering the influence of floor beam stiffness ($E_{\mathrm{B}}=\infty$), the number and the arrangement of racks do not affect the modal shape of beams-racks combined systems, and the periods of the six combined systems are the same, i.e., $T_{\mathrm{Rk}n}^{\prime }=T_{\mathrm{Rk}}=1.0 \mathrm{s}$. Therefore, the vibration characteristics of the combined systems with n racks are consistent with those of the combined system with one rack whose masses and stiffness are multiplied by n (Figure 6).

Based on the simplified equivalent model shown in Figure 6, modal analysis will be conducted to study the relationship between the periods of the rack T_Rk, the simplified structure T_SS, and the real structure T_RS under two different cases:

(1)

The total mass of the steel rack is constant, and the value of T_Rk is changed by varying the elasticity modulus E_Rk of the racks.

(2)

The stiffness of the rack is constant, and the period T_Rk of the rack varies through changing the total mass of the steel rack.

3.2. Period Relationship with Constant Rack Mass

Firstly, the mass ratio α of the rack mass to the total mass of the first floor in a real structure is calculated as:

$$\alpha =\frac{6nm}{M_1+6nm}$$

(1)

The mass M₁ of the first floor of the main structure is 8.1646 × 10⁴ kg. The mass m is firstly taken as 1.3607 × 10⁴ kg, with which the mass ratio is $\alpha =6nm/(M_1+6nm)$ = 0.5, i.e., the mass of the first floor of the main structure is equal to the total mass of the racks.

Modal analysis is conducted for different real structures composed of different racks and main structures. The period T_Rk varies by changing the E_Rk of racks, and the period T_SS0 varies by changing the elastic modulus E_C of columns. It should be emphasized that varying the period T_SS0 will consequently change the period T_SS of the simplified structure in Figure 2c. Since the simplified structure is used in the current aseismic design, T_SS can be used directly instead of T_SS0 in MA1 and MA2 in this section.

In MA1, T_SS changes continuously (0.02, 0.04, ···, 2.98, 3.0 s) while T_Rk remains constant (T_Rk = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 s, respectively). Under this condition, modal analysis is conducted on a total of 1050 real structures. Taking the simplified structure period T_SS as the abscissa, seven T_RS–T_SS curves corresponding to the seven T_Rk are obtained in Figure 7a.

In MA2, T_Rk changes continuously (0.02, 0.04, ···, 2.98, 3.0 s) while T_SS remains constant (T_SS = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 s, respectively). Under this condition, modal analysis is conducted on a total of 1050 real structures. Taking the period T_Rk as the abscissa, seven T_RS –T_Rk curves corresponding to the seven T_SS are obtained in Figure 7b.

From the curves in Figure 7a, it can be seen that for different T_Rk, each T_RS–T_SS curve starts at T_RS = T_Rk and gradually increases to the straight line of T_RS = T_SS with the increase of T_SS. For $T_{\mathrm{Rk}}$ = 0.1 s, the $T_{\mathrm{RS}}$ curve is basically coincident with this straight line. The curves in Figure 7b obtained in MA2 assume a similar pattern: each T_RS curve starts at T_RS = T_SS and gradually increases to the straight line of T_RS = T_Rk with the increase of T_Rk.

As the effects of T_SS and T_Rk on T_RS are mathematically the same, the relationship between the three periods can be accurately expressed by Equation (2), which is obtained according to the least squares method. The comparison between the numerical results and Equation (2) is also presented in Figure 7. The comparison shows that the maximum deviation is only 1.0%, so Equation (2) has excellent accuracy for both MA1 and MA2.

$$T_{\mathrm{RS}}^{3.2}=T_{\mathrm{Rk}}^{3.2}+T_{\mathrm{SS}}^{3.2}$$

(2)

The mass ratio α is constantly equal to 0.5 in this analysis procedure, and α is not considered as a variable that may affect the relation between the three natural periods. Therefore, the role of α will be addressed through parametric studies in the rest of this section.

Firstly, the mass m is set to be m = 5.4431 × 10⁴ kg, giving the mass ratio α = 0.8 and indicating that the mass of the rack is four times the mass M₁. Similarly, the modal analysis in MA1 and MA2 is conducted to calculate the T_RS–T_SS and T_RS–T_Rk curves, and the results are given in Figure 8. The curves in Figure 8 have similar shapes to those in Figure 7, but the fitting equation according to the least squares method is Equation (3), which is slightly different from Equation (2). The comparison between numerical results and Equation (3) is also given in Figure 8 and the maximum deviation is only 0.8%.

$$T_{\mathrm{RS}}^{2.5}=T_{\mathrm{Rk}}^{2.5}+T_{\mathrm{SS}}^{2.5}$$

(3)

It can be concluded from Equations (2) and (3) that the relationship between T_Rk, T_SS, and T_RS for any value of α can be expressed in the form of:

$$T_{\mathrm{RS}}^{\xi }=T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\xi }$$

(4)

where the power exponent ξ is dependent on the mass ratio α.

To further determine the expression of ξ as a function of α, α is set to 0.1, 0.2, 0.3, 0.4, 0.6, 0.7, and 0.9 in MA1 and MA2. Due to the symmetry of the results in MA1 and MA2, only the T_RS–T_Rk curves obtained in MA2 with different values of α are plotted in Figure 9. Meanwhile, Equation (4) is also plotted in all the subfigures of Figure 9 using the best fitting values of ξ (shown in the notation of each subfigure). The T_RS–T_Rk curves with α = 0.9 are no longer given and the power exponent is ξ = 2.35. The maximum deviations between Equation (4) and the numerical results in Figure 9a–f are 0.94%, 1.07%, 1.08%, 0.92%, 0.96%, and 0.97%, respectively.

According to the results shown in Figure 7, Figure 8 and Figure 9, the relationship between the power exponent ξ and the mass ratio α is concluded as plotted in Figure 10, and the empirical expression obtained by the least squares method is proposed as Equation (5). The maximum deviation is only 1.2%.

$$\xi =\frac{2.25}{\sqrt{\alpha }}$$

(5)

Therefore, in the case of infinitely rigid first-floor beams ($E_{\mathrm{B}}=\infty$), if the total mass of racks on the first floor of the main structure is given, the mass ratio α can be determined and subsequently the power exponent ξ is available in Equation (5). Next, the period T_RS of the real structure can be determined by Equation (4). Therefore, when the simplified structure is used for analysis in design practice, Equations (4) and (5) can be used to calculate T_RS to consider the difference between T_RS and T_SS in the simplified method.

3.3. Period Relationship with Constant Rack Stiffness

When the stiffness of the steel rack is determined (i.e., the elastic modulus E_Rk is constant), but the total mass of the rack is uncertain, the period T_Rk of the rack will vary through changing the mass of the rack. The constant elastic modulus E_Rk of the rack is determined following the principle that T_Rk = 3.0 s when the mass ratio α is 0.95. With the known E_Rk, T_Rk is 0.158 s when α = 0.05. Similarly, as T_Rk changes continuously (0.02, 0.04, ···, 2.98, 3.0 s), the corresponding relationship between α and T_Rk can be shown in Figure 11. It can be seen that the constant E_Rk obtained by this principle can make the mass ratio α reasonably distributed in the range of 0~0.95.

For the calculation models in Figure 2, the period T_SS of the simplified structure changes with the change in mass of the racks. The modal analysis is conducted on both the simplified structures and the real structures according to MA2, in which the T_Rk changes continuously (0.02, 0.04, ···, 2.98, 3.0 s) while T_SS0 remains constant (T_SS0 = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 s, respectively). The simplified equivalent models for real structures in Figure 6 are still used in modal analysis. Therefore, T_RS–T_Rk and T_SS–T_Rk curves can be obtained corresponding to the T_SS0 of the seven periods of the main structures, as shown in Figure 12a. Meanwhile, according to the relationship between T_Rk and the mass ratio α (Figure 11), T_RS–α and T_SS–α curves are shown in Figure 12b.

Since the rigidity of the rack is constant, Figure 12 shows that:

(1)

For the main structure with maximum rigidity (T_SS0 = 0.1 s), T_RS is basically the same as T_Rk, and T_SS hardly changes with T_Rk and α;

(2)

For relatively flexible main structures (T_SS0 ≥ 2.0 s), T_RS is close to T_SS and the maximum deviation is only 6.0%, which means the simplified structure can be used in place of the real structure in modal analysis;

(3)

With the decrease in T_SS0, (i.e., the rigidity of the main structure is gradually increased relative to the rack), the difference between T_RS and T_SS becomes more obvious;

(4)

With the increase of T_Rk, the T_RS–T_Rk and T_SS–T_Rk curves gradually become straight lines, which means T_RS and T_SS are linearly related to T_Rk. Moreover, the difference between T_RS and T_SS gradually increases.

The numerical results of T_RS and T_SS in Figure 12 can also be used to validate Equation (4). Based on the corresponding relationship between T_Rk and α in Figure 11, the power exponent ξ in Equation (4) can be obtained by Equation (5), as shown in Figure 13a. Therefore, the periods T_RS of real structures can be calculated using Equation (4) ($T_{\mathrm{RS}}^{\xi }=T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\xi }$) and plotted in Figure 13b. The T_RS–T_Rk curves obtained from Equation (4) are in good agreement with the numerical results, with a maximum deviation of 0.6%.

To conclude, the T_Rk–T_SS–T_RS relationship can be accurately expressed using Equations (4) and (5) when the influence of floor beam stiffness is not considered and the first-floor beams are assumed to be infinitely rigid ($E_{\mathrm{B}}=\infty$).

4. The Period Relationship with Finite Beam Stiffness

The T_Rk–T_SS–T_RS relationship, Equation (4), is based on the assumption that the first-floor beams are assumed to be infinitely rigid ($E_{\mathrm{B}}=\infty$). However, the floor beam stiffness is finite in practice. Therefore, the influence of floor beam stiffness on the T_Rk–T_SS–T_RS relationship will be taken into account in this section.

The real structures with 1–6 steel racks installed on the first floor, as shown in Figure 14, are used in the modal analysis. For the main structure shown in Figure 2b, because the frame beam is normally heavy in such warehouses, the period T_SS0 is mainly controlled by the elastic modulus of columns (E_C). The elastic modulus of beams (E_B) is always equal to E_C. The variation of T_SS0 is realized by changing E_B and E_C simultaneously, and the ratio of beam-to-column stiffness $i_b/i_{\mathrm{c}}$ is constant at 0.41 according to the given sections of the main structural components in Table 1.

Considering that Equation (4) is related to the mass ratio α, in order to more directly show the influence of the number of racks, the mass of racks can be prescribed as follows: The total mass of racks is prescribed (the mass ratio α is constant), and it is uniformly distributed to different numbers (n = 1, 2, 3, 4, 5, and 6) of racks on the first floor, ensuring the same period for each rack. Since the total mass of racks in engineering practices would not be very small, the mass ratio α is assumed to be in the range of 0.4–0.9.

For each constant mass ratio α, modal analysis is carried out for the six real structures in Figure 14 and simplified structures according to MA2, in which T_Rk changes continuously (0.02, 0.04, ···, 2.98, 3.0 s) while T_SS0 remains constant (T_SS0 = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 s, respectively).

It should be noted that when the T_SS0 remains constant, the changing α will consequently change the period T_SS of the simplified structure in Figure 2c. With different values of α, the corresponding T_SS values are listed in

Table 2. Period T_SS of simplified structures with different values of α.

Periods of the Main Structure TSS0 (s)	Periods of Simplified Structures TSS (s)
	α = 0.4	α = 0.5	α = 0.6	α = 0.7	α = 0.8	α = 0.9
0.1	0.114	0.121	0.131	0.146	0.172	0.235
0.5	0.572	0.606	0.655	0.730	0.862	1.175
1.0	1.144	1.213	1.310	1.460	1.724	2.351
1.5	1.717	1.819	1.965	2.190	2.586	3.526
2.0	2.289	2.425	2.620	2.919	3.448	4.701
2.5	2.861	3.032	3.275	3.649	4.311	5.877
3.0	3.433	3.638	3.930	4.379	5.173	7.053

.

Firstly, modal analysis on the six real structures (Figure 14) with different T_SS0 and T_Rk is conducted to obtain the corresponding T_RS when the mass ratio α is 0.4, as shown in Figure 15. For comparison, the T_RS can be calculated using Equation (4) ($T_{\mathrm{RS}}=(T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\xi }{)}^{\frac{1}{\xi }}$), in which the period T_SS can be determined in Table 2 with the given mass ratio α and T_SS0. The power exponent ξ is calculated by Equation (5) ($\xi =2.25/\sqrt{\alpha }=3.558$). Therefore, the calculation results of Equation (4) are the same for the six real structures in Figure 14, as shown in Figure 15. It can be seen that: when T_Rk > 1.0 s, the maximum deviation of T_RS between the numerical results and Equation (4) is only 3.4%. When the racks are nearly rigid (T_Rk = 0.02 s) and T_SS0 = 3.0 s, the maximum deviation is 18.9%.

Similarly, for α = 0.6 and 0.8, the numerical results of T_RS are plotted in Figure 16 and Figure 17, and T_RS–T_Rk curves obtained from Equation (4) with $\xi =2.25/\sqrt{\alpha }=2.905$ and $\xi =2.25/\sqrt{\alpha }=2.516$ are also plotted in Figure 16 and Figure 17 for comparison, respectively. It can be seen from Figure 16 that when T_Rk > 1.0 s, the maximum deviation of T_RS between the numerical results and Equation (4) is only 4.4%. When the racks are nearly rigid and T_SS0 = 3.0 s, the maximum deviation is 18.2%. For Figure 17, the maximum deviation is 6.6% when T_Rk > 1.0 s, while it increases to 17.0 when the racks are nearly rigid and T_SS0 = 3.0 s.

Based on Figure 15, Figure 16 and Figure 17 illustrating the influence of finite floor beam stiffness, the following conclusions can be drawn:

(1)

When T_Rk > 1.0 s, the numerical and theoretical (Equation (4)) results of T_RS are in good agreement. Therefore, Equation (4) is still applicable.

(2)

When T_Rk < 1.0 s, the numerical results of T_RS are shorter than those obtained from Equation (4).

(3)

With the increase of the rack number n, the difference between the numerical results and Equation (4) becomes more obvious, especially in the cases of longer T_SS0. Maximum differences occur when n is 6, T_SS0 = 3.0 s, and T_Rk < 1.0 s, in which case the racks are relatively rigid and the floor beams are flexible.

For practical engineering, the warehouse height can reach 20–25 m, and the story height is about 10–12 m. The steel racks are essentially very tall and flexible structures, and many racks supporting heavy pallet units have periods of 1.5–2.5 s in the CA direction. When multiple racks with periods of 1.5–2.5 s are placed on the first floor of the two-story real structure, most of the mass of the real structure comes from the racks and the period of the real structure will be longer than 1.0 s. For these cases, Equations (4) and (5) proposed in this article are applicable in the design practice.

Although these differences are almost acceptable in practical engineering, the reasons for the differences should be further analyzed in the following part to include them in practice.

5. The Interaction between Racks and the Main Structure

Based on the above analysis, the reason for the difference between Equation (4) and the numerical results is that the interaction between racks and the main structure (mainly floor beams) has been neglected in Equation (4). In the following, the influence of the racks on the main structure and the influence of the main structure on the racks are studied by constructing different analysis models.

5.1. Influence of the Main Structure on Racks

The influence of the main structure on racks on the first floor is mainly reflected in the influence of the rigidity of the first floor beams as the elastic supports of the racks. In the case of infinite beam stiffness ($E_{\mathrm{B}}=\infty$), all racks in the six combined systems (Figure 4) have the same first vibration mode in the CA direction, i.e., $T_{\mathrm{Rk}n}^{\prime }=T_{\mathrm{Rk}}$, as seen in Figure 5

When the floor beam stiffness is finite, however, the period $T_{\mathrm{Rk}n}^{\prime }\ne T_{\mathrm{Rk}}$. Therefore, the different beams-racks combined systems given in Figure 4 are selected to study the influence of the main structure on the racks. The elastic modulus E_B of the beams will be taken as the elastic modulus of the members when the period T_SS0 of the main structure is 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 s, respectively, ensuring that T_SS0 basically covers the range of 0.02 s–3.0 s. For example, when T_SS0 is 1.0 s, the elastic modulus of beams is denoted as E_B (T_SS0 = 1.0 s).

The real structures with 1–6 steel racks installed on the first floor, as shown in Figure 14, are used in the modal analysis. For the main structure shown in Figure 2b, the period T_SS0 is mainly controlled by the elastic modulus of columns (E_C). The elastic modulus of beams (E_B) is always equal to E_C, and thus, the variation of T_SS0 is realized by changing E_B and E_C. The ratio of beam-to-column stiffness i_b/i_c is constant at 0.41, according to the given sections of main structural components in Table 1.

Therefore, the mass ratio is set as α = 0.6 and the T_Rk changes continuously (0.02, 0.04, ···, 2.98, 3.0 s) for each elastic modulus E_B (T_SS0 = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 s). The modal analysis is conducted for the six different beams-racks combined systems to obtain corresponding $T_{\mathrm{Rk}n}^{\prime }$, and the relationship between $T_{\mathrm{Rk}n}^{\prime }$ and T_Rk is shown in Figure 18.

It can be shown in Figure 18 that when the elastic modulus E_B of the beams results in a main structure period of T_SS0 = 0.1 s, meaning that the first-floor beams are nearly rigid, $T_{\mathrm{Rk}n}^{\prime }$ is closed to T_Rk.

For the case of the rack number n = 1, the difference between $T_{\mathrm{Rk}1}^{\prime }$ and T_Rk increases gradually with the decrease in E_B, which means that the restraint of the beams to the racks decreases. However, as n increases, the differences become insignificant, e.g., when n = 6, Figure 18f shows that $T_{\mathrm{Rk}n}^{\prime }$ and T_Rk are almost the same, which means the restraint to the racks is nearly rigid and the effect of beam stiffness E_B is small.

For the calculation of $T_{\mathrm{Rk}n}^{\prime }$ of the combined system shown in Figure 19a, a simplified equivalent method (Figure 19b) can be introduced: the connecting bars between racks at their tops ensure that all racks in the combined system vibrate in the same shape, and the floor beams act as the elastic rotational restraint for the racks under horizontal vibration. The period is written out by considering the original system as the two sub-systems set up in series:

$$T_{\mathrm{Rk}n}^{\prime 2}=T_0^2+T_{\mathrm{Rk}}^2$$

(6)

in which T₀ is the period when the racks are integrated as one infinitely rigid rack which is supported by a rotational spring with a stiffness of K_Z. Therefore, T₀ physically represents the rotational restraint of the floor beams to the racks. T_Rk is the period of the rack fixed on the ground.

Based on this simplified equivalent method, referring to the values of $T_{\mathrm{Rk}n}^{\prime }$ in Figure 18 (α = 0.6), the period T₀ for different numbers of racks can be determined using Equation (6) ($T_0=\sqrt{T_{\mathrm{Rk}n}^{\prime 2}-T_{\mathrm{Rk}}^2}$) and plotted in Figure 20.

T₀ decreases with the increase in floor beam stiffness (E_B), as shown in Figure 20, and T_RS becomes closer to that obtained using Equation (4), which can also be concluded from Figure 16.

By comparing the curves with T_SS0 = 3.0 s in Figure 20a–f, it is found that:

(1)

In the case of n = 1, T₀ approaches the maximum value of 1.6 s, which means that the restraint of the beams to the racks is relatively minimal. The difference between $T_{\mathrm{Rk}1}^{\prime }$ and T_Rk also reaches the maximum. However, as illustrated in Figure 16a, the numerical results of T_Rk are approximated to those obtained from Equations (4) and (5) with good accuracy.

(2)

In the case of n = 6, the maximum value of T₀ is only 0.5 s and T₀ decreases further in the short period range of T_Rk < 1.0 s, which means the difference between $T_{\mathrm{Rk}6}^{\prime }$ and T_Rk is small and the restraint of the beams to the racks is nearly rigid. However, as depicted in Figure 16a, the maximum difference of T_RS between the numerical results and the prediction of Equation (4) occurs exactly within this range.

Therefore, in the period range of long T_SS0 and short T_Rk in Figure 16, the differences between numerical results and Equation (4) are not due to the influence of the floor beams on the racks.

Comparing Figure 18 and Figure 20, it can be seen that such differences are basically consistent with the trend of T₀ curves. Based on the simplified equivalent method for $T_{\mathrm{Rk}n}^{\prime }$ in Figure 19b, the decrease in T₀ is due to the increase in K_Z, which means the reversed rotational constraint of racks on the beams also increases gradually. Therefore, the differences of T_RS between numerical results and Equation (4) are probably caused by the strengthening effect of the racks on the floor beam stiffness, which will be discussed in detail in the following.

5.2. Influence of Racks on the Main Structure

The racks may increase the stiffness of the first floor, and this can be studied using the new model shown in Figure 21. This modified model reserves the racks but integrates the total mass of the racks into the first floor of the real structure. This modified model is distinguished from the simplified structure in Figure 2c by taking into account the effect of the no-mass racks, so the period of this modified model is denoted by $T_{\mathrm{SS}}^{\prime }$. Therefore, the analysis of the influence of the racks on the main structure can be obtained by comparing the periods of the simplified structure (T_SS) with the periods of the modified model ($T_{\mathrm{SS}}^{\prime }$).

The analysis started with the case leading to the maximum difference in T_RS between numerical results and Equation (4) in Figure 16, in which T_SS0 = 3.0 s, n = 6, and α = 0.6. In this case, the period T_SS is 3.93 s. The T_RS obtained from Equation (4) ($T_{\mathrm{RS}}^{\xi }=T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\xi }$) is larger than the numerical result, especially when T_Rk < 1.0 s.

For the new modified model with no-mass racks in Figure 21, the influence of the different racks is included by setting different values of E_Rk corresponding to T_Rk = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 s. Modal analysis on modified models with these different racks is conducted. The obtained $T_{\mathrm{SS}}^{\prime }$ are listed in

Table 3. Period $T_{\mathrm{SS}}^{\prime }$ of the modified model with no-load racks (n = 6, α = 0.6).

TSS0 (s)	TRk (s)	TSS (s)	${\mathit{T}}_{\mathbf{SS}}^{\prime }$ (s)	Frame Beam Amplitude of Simplified Structure (mm)	Frame Beam Amplitude of Modified Model (mm)
3.0	0.1	3.93	3.29	6.16	0.49
	0.5		3.53		2.08
	1.0		3.70		3.77
	1.5		3.79		4.71
	2.0		3.84		5.22
	2.5		3.87		5.51
	3.0		3.88		5.69

, and corresponding modal shapes are plotted in Figure 22a–f. For comparison, the modal shape of the simplified structure without racks (T_SS = 3.93 s) is also given in Figure 22g.

It is seen in Table 3 that $T_{\mathrm{SS}}^{\prime }$ is always shorter than T_SS and the difference increases with the increase in the rack stiffness. Combined with Figure 22 for the first modal shape, it is found that with the increase in rack stiffness, the deflection amplitude of the floor beam in the vibrational mode becomes smaller. The specific deflection values of the first vibration amplitude are given in Table 3.

When T_Rk = 3.0 s, the racks have a small effect on the stiffness of the frame beam. The vibration amplitude of the frame beam is 92.3% of that in the simplified structure shown in Figure 22h with the corresponding $T_{\mathrm{SS}}^{\prime }$ = 3.88 and s = 0.99T_SS. Thus, the vibration characteristics of the two models are almost identical. In this case, the value of T_RS obtained from numerical results is basically equal to that obtained from Equation (4), as shown in Figure 16f.

On the other hand, when T_Rk = 0.1 s, the frame beam of the first floor is almost rigid, and the vibration amplitude is only 8% of the amplitude in the simplified structure, and $T_{\mathrm{SS}}^{\prime }$ = 3.29 and s = 0.837T_SS. Therefore, the racks with high rigidity increase the stiffness of the floor beams and contribute to reducing the period of the simplified structure. However, in Equation (4) which represents the simplified design method, this strengthening effect has not been included in the simplified structure, thus a long T_SS leads to a longer T_RS obtained from Equation (4) than the numerical results, as shown in Figure 16f.

The role of the connecting bars between racks in this strengthening effect should also be addressed in this section. Modal analysis is conducted on the modified model in Figure 21, without connecting bars between the top of racks. Similarly, given T_SS0 = 3.0 s and different values of E_Rk corresponding to T_Rk = 0.1, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 s, the $T_{\mathrm{SS}}^{\prime }$ and the vibration amplitudes of frame beams are listed in

Table 4. Vibration characteristics of the modified model with/without connecting bars (n = 6, α = 0.6).

TSS0 (s)	TRk (s)	TSS (s)	TRS (s)	${\mathit{T}}_{\mathbf{SS}}^{\prime }$ (s)		Frame Beam Amplitude of Modified Model (mm)
				With Bars	Without Bars	With Bars	Without Bars
3.0	0.1	3.93	3.28	3.29	3.50	0.49	3.27
	0.5		3.54	3.53	3.73	2.08	4.32
	1.0		3.75	3.70	3.80	3.77	4.92
	1.5		3.90	3.79	3.85	4.71	5.33
	2.0		4.04	3.84	3.87	5.22	5.59
	2.5		4.21	3.87	3.89	5.51	5.76
	3.0		4.41	3.88	3.90	5.69	5.86

. Meanwhile, the periods T_RS of the seven real structures in Figure 16f are also listed. The first modal shape of the six modified models without connecting bars are shown in Figure 23.

It is found in Table 4 that, when T_Rk > 1.0 s, the $T_{\mathrm{SS}}^{\prime }$ of the modified model with no-mass racks is not significantly affected by the top connecting bars. The modal shapes corresponding to different $T_{\mathrm{SS}}^{\prime }$ are quite similar, as shown in Figure 23. However, when T_Rk < 1.0 s, the strengthening effect of the connecting bars on the frame beams cannot be ignored.

To further analyze the effect of connecting bars on the real structure, the real structure with T_SS0 = 3.0 s and T_Rk = 0.1 s is selected, and modal analysis is conducted on real structures with and without the top connecting bars, respectively. The modal shapes of this real structure without connecting bars are plotted in Figure 24, and those of real structures with connecting bars are plotted in Figure 25.

It can be seen that the first modal shapes are similar with or without bars, and the connecting bars reduce the first natural period of the real structure by 6.6%. The top connecting bars change the second and higher-order vibration modes.

Therefore, the strengthening effect of the racks on the beams is partly due to the high rigidity of the racks and connecting bars. The connecting bars eliminate the local vibration modes between racks and ensure that the modes of all racks are laterally consistent. As a result, the real structure will vibrate as an integrity.

Thus, Equation (4) can be modified by replacing T_SS with $T_{\mathrm{SS}}^{\prime }$:

$$T_{\mathrm{RS}}^{\xi }=T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\prime \xi }$$

(7)

in which the power exponent ξ corresponding to the mass ratio α can be obtained by Equation (5).

In order to validate Equation (7), modal analysis is conducted on the modified model shown in Figure 21 according to MA2, in which T_Rk changes continuously while T_SS0 remains constant. The obtained $T_{\mathrm{SS}}^{\prime }$ are plotted in Figure 26a, and the corresponding T_SS are also given. Therefore, the T_RS can be calculated by Equation (7) to be compared with the numerical results, as shown in Figure 26b. It can be seen that they are in good agreement and the maximum deviation is only 1.1%.

In addition, it is also indicated in Figure 26a that the difference between $T_{\mathrm{SS}}^{\prime }$ and T_SS due to the strengthening effect of racks on floor beams is significant only in the cases where the racks are relatively rigid compared with the first floor of the main structure. With an increase in floor stiffness, the difference becomes smaller. For T_SS0 = 3.0 s, when T_Rk = 0.5 s, $T_{\mathrm{SS}}^{\prime }$ = 3.53 and s = 0.9$T_{\mathrm{SS}}$, and the deviation is 10% while it was only 5% when T_Rk = 1.06 s. When T_Rk continues to increase, the influence of steel racks on the main structure floor can be neglected.

For convenience of calculation, this type of strengthening effect can be considered by multiplying the obtained period T_RS by a reduction factor of 0.8–0.95, similar to the effect of infilled interior walls on the period of a frame. When T_Rk > 1.0 s, a reduction factor of 0.95 can be used. A reduction factor of 0.8 should be used when the racks are nearly rigid and T_SS0 is approaching 3.0 s. However, these cases rarely happen in practice, as the rigidity of the first floor is generally large as they are required to support multiple fully distributed and fully loaded racks. As the steel racks are essentially very tall and flexible structures, many racks supporting heavy pallet units have periods T_RK ≥ 1.5 s in the CA direction.

Therefore, in the design practice of a real structure with racks on the first floor, the simplified structure period T_SS can be obtained by concentrating the mass of all racks into the first floor of the main frame. Combined with the characteristics of racks, the period T_RS of the real structure can be accurately predicted by Equations (4) and (5), with a suitable account of the stiffening effect of racks on beam stiffness by a period reduction factor of 0.8–0.95.

6. Summary and Conclusions

In this paper, a two-story steel frame is selected as the main structure. Steel racks with different masses, stiffnesses, and quantities are placed on the first floor of the main structure to form different real structures (RSs). The corresponding simplified structures (SS) are frames with the mass of steel racks concentrated on the first floor of the main structure. Modal analysis is performed to analyze the dynamic characteristics of RS and SS in the CA direction. The periods T_RS of the RSs and T_SS of the SSs are calculated, and the interaction between regularly spaced steel racks and the main structure is studied. The following conclusions can be drawn based on the results:

(1)

When the first-floor beams are assumed to be infinitely rigid ($E_{\mathrm{B}}=\infty$), the connecting bars at the top of the racks ensure that all racks vibrate in the same first vibration mode. The number and arrangement of racks have no influence on the modal shape of the real structure.

(2)

When $E_{\mathrm{B}}=\infty$, the relationship between the period T_Rk, T_SS, and T_RS can be accurately expressed by Equation (4): $T_{\mathrm{RS}}^{\xi }=T_{\mathrm{Rk}}^{\xi }+T_{\mathrm{SS}}^{\xi }$, where the power exponent $\xi =2.25/\sqrt{\alpha }$. The influence of the mass ratio α is considered, so that the relationship is applicable for the mass ratio α changing in the range of 0~1.0.

(3)

The influence of the finite floor beam stiffness of the main structure on the relationship between the period T_Rk, T_SS, and T_RS is taken into account. When T_Rk > 1.0 s, T_RS predicted by Equation (4) are in good agreement with the numerical results. However, with the increase of the rack number n, the difference becomes more obvious, especially for the cases of long T_SS0 with relatively rigid racks and flexible floor beams.

(4)

With finite beam stiffness, the influence of the main structure on the racks is reflected in the influence of the rigidity of the first-floor beams as the elastic supports of the racks. Different beams-racks combined systems are selected to study the influence of the main structure on the racks. The period $T_{\mathrm{Rk}n}^{\prime }$ of the beams-racks combined system is always greater than T_Rk, and the difference increases gradually with the decrease in beam stiffness, but with the increase in the number of racks (which is always the case in practice), the difference is becoming non-significant and negligible.

(5)

With finite beam stiffness, the strengthening effect of the racks on the stiffness of the floor beams of the main structure is studied by constructing a new modified model, which is distinguished from the simplified structure by taking into account the effect of the no-mass racks. The racks with high rigidity and the continuous inter-connecting bars at the tops of the racks increase the stiffness of the floor beams and contribute to reducing the period of the simplified structure. The inter-connecting bars eliminate the local vibration modes between racks and ensure the modes of all racks are laterally consistent. As a result, the real structure will vibrate as an integrity.

(6)

This strengthening effect leads to a shorter period T_RS than that predicted by Equations (4) and (5), especially in the cases where the racks are relatively rigid compared with the first floor of the main structure. This effect can be considered by multiplying the obtained period T_RS by a reduction factor of 0.8–0.95, similar to the effect of infilled interior walls on the period of a frame, but these cases rarely happen in practice.

Therefore, in the design practice of a real structure with racks on the first floor, the simplified structure period T_SS can be obtained by concentrating the mass of all racks on the first floor of the main structure. Combined with the characteristics of racks, the period T_RS of the real structure can be accurately predicted by Equations (4) and (5), with the mass of steel racks concentrated on the first floor of the main structure.