# Design Forces of Horizontal Braces Unlocated at Middle of Columns considering Random Initial Geometric Imperfections

Authors: Jinyou Zhao; Junming Wei; Jun Wang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

The horizontal bracing forces of column-bracing systems derived from past studies and current design codes were considered only located at middle of columns. Actually, the horizontal braces used to reduce the out-of-plane effective column lengths are frequently designed not to locate at middle of columns. In this paper, a large number of column-bracing systems with the horizontal braces unlocated at middle of columns were modelled and analyzed using the finite element method, in which the random initial geometric imperfections of both the columns and the horizontal braces unlocated at middle of columns were well considered by the Monte Carlo method. Based on the numerical calculations, parametric analysis, and probability statistics, the probability density function of the horizontal bracing forces was found, so that the corresponding design forces of horizontal braces unlocated at middle of columns were proposed which were compared with the design mid-height horizontal bracing forces in the previous study and the relevant codes. The results indicate that the design forces of the horizontal braces located at 0.6 column height are smaller than the mid-height horizontal bracing forces in the previous study while the design forces of horizontal braces located at 0.7 column height are larger than the mid-height horizontal bracing forces in the previous study. The proposed design forces of the horizontal braces located and unlocated at middle of columns are both smaller than the mid-height horizontal bracing forces stipulated in GB50017-2017, Eurocode 3-1992, and AS4100-1998. The above conclusions provide references for the engineering applications and further related code revisions.


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## Body

## 1. Introduction

One of the most important functions of bracing members in structural systems is to provide intermediate lateral support to compression members for enhancing their stability [1]. The brace strength requirements to achieve a specified increase in the buckling strengths of the braced columns have generally been developed from a single column-brace model [2–4] to column-bracing systems [5–8]. Among the above studies, the horizontal braces used to reduce the out-of-plane effective column lengths were assumed to be located at middle of columns. In engineering practice, the above horizontal braces are frequently designed to be located a bit higher than middle of columns, but no significant definitive research has been performed for the design forces of the horizontal braces unlocated at middle of columns.In past studies on the mid-height horizontal bracing forces for column-bracing systems [7, 8], the random initial geometric imperfections of both the columns and the horizontal braces were well considered by the Monte Carlo method. The research results showed that the random combination of the initial geometric imperfections between the columns and the horizontal braces has a favourable effect on the bracing forces as it leads to the randomness of the mid-height horizontal bracing forces in compression or tension when the ultimate load of the braced columns is reached. Therefore, in order to estimate and to ensure the accuracy of the design forces of the horizontal braces unlocated at middle of columns, it is still necessary to consider the effects of uncertainties of initial geometric imperfections [9].Bach et al. utilized the Bayesian optimization method to solve multi-objective robust design optimization problems of steel frames under aleatory uncertainty in external loads and material properties [10]. Reza et al. applied the uniform deformation method to obtain the optimal distribution of dampers for reducing the seismic response of steel frames with multi-degree-of-freedom [11]. Bergami and Nuti employed the capacity spectrum method to establish a displacement-based procedure applied to design dissipative bracings for the seismic protection of frame structures [12]. In this paper, a large number of column-bracing systems with the horizontal braces unlocated at middle of columns have been carried out by second-order analysis using ANSYS, in which the initial geometric imperfections of both the columns and the horizontal braces were randomly sampled according to the Monte Carlo method. Moreover, the probability density function of the design forces of the horizontal braces unlocated at middle of columns was found by probability statistics, so that the corresponding design forces of the horizontal braces unlocated at middle of columns were also proposed which were compared with the design mid-height horizontal bracing forces in the previous study and the relevant codes. Based on the research results, recommendations for the revision of relevant codes were provided.


## 2. Analytical Model and Parametric Design

The analytical models of column-bracing systems with the horizontal braces unlocated at middle of columns are shown in Figure1. The n-columns in the row which are equally spaced and simply supported have the same vertical axial load P at the top of the columns, and the horizontal braces used to reduce the out-of-plane effective column lengths are located at 0.6 column height or 0.7 column height, respectively. The cross-sectional areas of the columns are the same and are denoted by Ac; the cross-sectional areas of the horizontal braces at 0.6 column height or 0.7 column height are the same and are denoted by Am, the cross-sectional areas of the horizontal braces at the top of the columns are the same and are denoted by At, and Am is equal to At; the cross-sectional areas of the diagonal bracings are the same and are denoted by Ad.Figure 1
Analytical models of column-bracing system with the horizontal braces unlocated at middle of columns. (a) Horizontal braces located at 0.6L. (b) Horizontal braces located at 0.7L.
(a)(b)The parameter selection of the analytical models is mainly based on reference [8]: the half slenderness ratio of the column about the minor axis λc, the slenderness ratio of the horizontal brace λb, and the ratio of the horizontal brace length over the column height b/L. In engineering practice, the commonly used parameters are b = 6 m, 50≤ λc ≤100, 100≤ λb ≤200, and 0.4≤ b/L ≤0.7.A biaxially symmetric I-section is used for the column section whose flexural buckling along the longitudinal direction of the building occurs about the minor axis of the column. The cross-sectional areas of the columns are equal to 250 cm2, and the geometric dimensions of the columns are shown in Table 1.Table 1
Geometric dimensions of column sections.
L (m)b/LλcB (mm)H (mm)tf (mm)tw (mm)Αd (cm2)8.570.750392.4580018.231412.58.570.775284.2380021.331711.88.570.7100226.9880023.561910.5100.650446.6580016.831313.5100.675322.2880019.921612.5100.6100256.4980022.221811.2120.550523.5480015.051214.5120.575376.5880017.981513.5120.5100298.8180020.221712.0150.450639.9380012.881115.5150.475446.6580016.831314.5150.4100352.5480019.261514.0Note. H, B—height and width of I-section, respectively; tw, tf—thickness of web and flange, respectively; Ad—the cross-sectional area of the diagonal bracings.A circular tube section is used for the horizontal brace section. According to the ultimate loads of the braced columns and the different numbern of the braced columns, the horizontal braces are designed and the geometric dimensions of the horizontal braces for n = 6 are shown in Table 2.Table 2
Geometric dimensions of horizontal brace sections.
λcλbAm/At (cm2)Db (mm)tb (mm)5010028169.625.2512535135.528.2215044112.4512.461755595.2218.392006680.7726.017510027169.635.0712534135.537.9915043112.4812.171755495.2818.042006580.9125.5710010024169.654.5012532135.567.5115042112.5111.881755395.3517.692006481.0125.14Note. Am, At—area of brace at mid-height and top of column, respectively; Db, tb—diameter and thickness of circular tube section, respectively.The circular solid bars are used for the diagonal bracings. According to the horizontal bracing forces, the diagonal bracings are designed against tension only and the cross-sectional areas of the diagonal bracingsAd for n = 6 are shown in Table 1.


## 3. Random Combination of Initial Geometric Imperfections

### 3.1. The Monte Carlo Method

The Monte Carlo method is also called the statistical testing method which can be used to solve complex engineering problems containing random variables by numerical simulation and probability statistics [13]. According to the Monte Carlo method, the probability models of the initial imperfections of the columns and the braces are set up, all of which are normally distributed. The probability distributions of the important parameters in the problems can be established based on the results of numerical simulation [14]. The Monte Carlo method can resolve large sample problems with small sample sizes sampled randomly, and it is considered that the results of the Monte Carlo method are credible as long as a sufficient number of sample sums are sampled and the simulation tests are accurate. The specific process of the Monte Carlo method applied to the research in this paper is shown in Figure 2.Figure 2
Specific process of the Monte Carlo method.


### 3.2. Probability Distribution Model of Initial Geometric Imperfections

The randomness of the initial geometric imperfections in a structure is caused by the sum of many independent random variables, and each random variable can be represented by a normal distribution according to the Monte Carlo method [15]. Given that the number of the braced columns is n, δ0i is the initial bow imperfection of the ith column, and its probability distribution can be written as(1)Piδ0i=12πσ1e−δ0i−μ12/2σ12,i=1,2,…,n.Given thatΔ0i is the initial sway imperfection of the ith column, its probability distribution can be written as(2)PiΔ0i=12πσ2e−Δ0i−μ22/2σ22,i=1,2,…,n.Given thatu0i is the initial imperfection of the ith horizontal brace unlocated at middle of columns, its probability distribution can be written as(3)Piu0i=12πσ3e−u0i−μ32/2σ32,i=1,2,…,n−1.Given thatv0i is the initial imperfection of the ith top horizontal brace, its probability distribution can be written as(4)Piv0i=12πσ4e−v0i−μ42/2σ42,i=1,2,…,n−1,where μ1=μ2=μ3=μ4=0, σ12=σ22=L/980, and σ32=σ42=b/980 [16]. The meanings of L and b are shown in Figure 1.


### 3.3. Random Initial Geometric Imperfection Equation

The initial bow imperfection of theith column is given as a half-sinusoid curve:(5)δi=δ0isinπxL.The initial sway imperfection of theith column is expressed as a straight line with the same slope:(6)Δi=Δ0ixL.The initial bow imperfection of theith horizontal braces unlocated at middle of columns is given as a half-sinusoid curve:(7)ui=u0isinπyb.The initial bow imperfection of theith horizontal braces at the top of the columns is also given as a half-sinusoid curve:(8)vi=v0isinπyb,where δ0i and Δ0i are the initial bow and sway imperfections of the ith column sampled by the Monte Carlo method, respectively, and u0i and v0i are the initial bow imperfections of the ith horizontal braces unlocated at middle of columns or at the top of the columns sampled by the Monte Carlo method, respectively, and the meanings of x and y are shown in Figure 1.


## 3.1. The Monte Carlo Method

The Monte Carlo method is also called the statistical testing method which can be used to solve complex engineering problems containing random variables by numerical simulation and probability statistics [13]. According to the Monte Carlo method, the probability models of the initial imperfections of the columns and the braces are set up, all of which are normally distributed. The probability distributions of the important parameters in the problems can be established based on the results of numerical simulation [14]. The Monte Carlo method can resolve large sample problems with small sample sizes sampled randomly, and it is considered that the results of the Monte Carlo method are credible as long as a sufficient number of sample sums are sampled and the simulation tests are accurate. The specific process of the Monte Carlo method applied to the research in this paper is shown in Figure 2.Figure 2
Specific process of the Monte Carlo method.


## 3.2. Probability Distribution Model of Initial Geometric Imperfections

The randomness of the initial geometric imperfections in a structure is caused by the sum of many independent random variables, and each random variable can be represented by a normal distribution according to the Monte Carlo method [15]. Given that the number of the braced columns is n, δ0i is the initial bow imperfection of the ith column, and its probability distribution can be written as(1)Piδ0i=12πσ1e−δ0i−μ12/2σ12,i=1,2,…,n.Given thatΔ0i is the initial sway imperfection of the ith column, its probability distribution can be written as(2)PiΔ0i=12πσ2e−Δ0i−μ22/2σ22,i=1,2,…,n.Given thatu0i is the initial imperfection of the ith horizontal brace unlocated at middle of columns, its probability distribution can be written as(3)Piu0i=12πσ3e−u0i−μ32/2σ32,i=1,2,…,n−1.Given thatv0i is the initial imperfection of the ith top horizontal brace, its probability distribution can be written as(4)Piv0i=12πσ4e−v0i−μ42/2σ42,i=1,2,…,n−1,where μ1=μ2=μ3=μ4=0, σ12=σ22=L/980, and σ32=σ42=b/980 [16]. The meanings of L and b are shown in Figure 1.


## 3.3. Random Initial Geometric Imperfection Equation

The initial bow imperfection of theith column is given as a half-sinusoid curve:(5)δi=δ0isinπxL.The initial sway imperfection of theith column is expressed as a straight line with the same slope:(6)Δi=Δ0ixL.The initial bow imperfection of theith horizontal braces unlocated at middle of columns is given as a half-sinusoid curve:(7)ui=u0isinπyb.The initial bow imperfection of theith horizontal braces at the top of the columns is also given as a half-sinusoid curve:(8)vi=v0isinπyb,where δ0i and Δ0i are the initial bow and sway imperfections of the ith column sampled by the Monte Carlo method, respectively, and u0i and v0i are the initial bow imperfections of the ith horizontal braces unlocated at middle of columns or at the top of the columns sampled by the Monte Carlo method, respectively, and the meanings of x and y are shown in Figure 1.


## 4. Finite Element Analysis considering Random Initial Geometric Imperfections

### 4.1. Finite Element Model

The finite element models were built using the ANSYS program, in which the Beam188 element was used to simulate the horizontal braces and the columns while the Link10 element was used to simulate the diagonal bracings. Hinged joints between the horizontal braces and the columns were formed by coupling their translational displacements at their nodes. There were 6 elements of each horizontal brace member and 12 elements per column member, and each diagonal bracing member used only one element. Based on equations (5)–(8), the coordinates of these elements were established which indicated the magnitudes and the directions of the random initial geometric imperfections.


### 4.2. Random Sampling of Initial Geometric Imperfections

The tolerance of initial geometric imperfections is also mainly based on reference [8]. According to the “Code for Acceptance of Construction Quality of Steel Structures” in China (GB50205-2001) [17], the allowable fabrication deviations of columns for one-story steel structures are given as follows: the maximum value of initial bow imperfection should be less than 1/1200 of the column height and the maximum value of initial sway imperfection should be less than 1/1000 of the column height. In addition, the allowable fabrication deviations of horizontal braces should be less than 1/750 of the horizontal brace length.According to the mean and the variance of the initial geometric imperfections, the initial geometric imperfections of the columns and the horizontal braces were randomly sampled by the function “Normrnd()” in the software MATLAB. In this study, 1.05 times the allowable fabrication deviations in GB50205-2001 were used for the initial geometric imperfection sample limits, and the sampled initial geometric imperfections must all be less than the limits.


### 4.3. Finite Element Analysis

In the finite element analysis, the constitutive relationship for steel was a bilinear strengthened elastic-plastic model, yield strength was 235 N/mm2, Young’s modulus was 2.06e5 N/mm2, tangent modulus was 2.06e3 N/mm2, and Poisson’s ratio was 0.3. A non-linear large displacement analysis was carried out using the arch-length method to find the maximum loads of the column-bracing systems which were taken as the strengths of the braced columns.The column-bracing systems with the random initial geometric imperfections were analyzed forn = 4, 6, 8, and 10, in which the horizontal braces used to reduce the out-of-plane effective column lengths were located at 0.6 or 0.7 column height. When n, λc, and the horizontal brace location (at 0.6L or 0.7L) were certain, there were twenty combinations for various values of b/L and λb. For every combination, fifty groups of the initial geometric imperfections of the columns and the horizontal braces were randomly sampled according to the Monte Carlo method. Therefore, there were one thousand combinations with random initial geometric imperfections for every certain constitution. The instability deformation modes could be obtained for every combination analyzed using the ANSYS program.Figure3 shows four kinds of instability modes for column-bracing systems with the horizontal braces located at 0.6 column height. For non-sway instability mode as shown in Figures 3(a) and 3(b), the maximum force of the horizontal braces located at 0.6L is in compression and in tension, respectively; for sway instability modes as shown in Figures 3(c) and 3(d), the maximum force of the horizontal braces located at 0.6L may be in compression or in tension but much more in compression. Similarly, column-bracing systems with the horizontal braces located at 0.7 column height also have four kinds of instability modes as shown in Figure 3, and the maximum force of the horizontal braces may be in compression or tension. The analysis results were found to be sensitive to both the magnitude and the direction of the random initial geometric imperfections of the columns and the horizontal braces.Figure 3
Instability deformation modes for horizontal braces located at 0.6L. (a) Non-sway instability (horizontal braces in compression). (b) Non-sway instability (horizontal braces in tension). (c) Sway instability (horizontal braces in compression or tension). (d) Sway instability (horizontal braces in compression or tension).
(a)(b)(c)(d)


## 4.1. Finite Element Model

The finite element models were built using the ANSYS program, in which the Beam188 element was used to simulate the horizontal braces and the columns while the Link10 element was used to simulate the diagonal bracings. Hinged joints between the horizontal braces and the columns were formed by coupling their translational displacements at their nodes. There were 6 elements of each horizontal brace member and 12 elements per column member, and each diagonal bracing member used only one element. Based on equations (5)–(8), the coordinates of these elements were established which indicated the magnitudes and the directions of the random initial geometric imperfections.


## 4.2. Random Sampling of Initial Geometric Imperfections

The tolerance of initial geometric imperfections is also mainly based on reference [8]. According to the “Code for Acceptance of Construction Quality of Steel Structures” in China (GB50205-2001) [17], the allowable fabrication deviations of columns for one-story steel structures are given as follows: the maximum value of initial bow imperfection should be less than 1/1200 of the column height and the maximum value of initial sway imperfection should be less than 1/1000 of the column height. In addition, the allowable fabrication deviations of horizontal braces should be less than 1/750 of the horizontal brace length.According to the mean and the variance of the initial geometric imperfections, the initial geometric imperfections of the columns and the horizontal braces were randomly sampled by the function “Normrnd()” in the software MATLAB. In this study, 1.05 times the allowable fabrication deviations in GB50205-2001 were used for the initial geometric imperfection sample limits, and the sampled initial geometric imperfections must all be less than the limits.


## 4.3. Finite Element Analysis

In the finite element analysis, the constitutive relationship for steel was a bilinear strengthened elastic-plastic model, yield strength was 235 N/mm2, Young’s modulus was 2.06e5 N/mm2, tangent modulus was 2.06e3 N/mm2, and Poisson’s ratio was 0.3. A non-linear large displacement analysis was carried out using the arch-length method to find the maximum loads of the column-bracing systems which were taken as the strengths of the braced columns.The column-bracing systems with the random initial geometric imperfections were analyzed forn = 4, 6, 8, and 10, in which the horizontal braces used to reduce the out-of-plane effective column lengths were located at 0.6 or 0.7 column height. When n, λc, and the horizontal brace location (at 0.6L or 0.7L) were certain, there were twenty combinations for various values of b/L and λb. For every combination, fifty groups of the initial geometric imperfections of the columns and the horizontal braces were randomly sampled according to the Monte Carlo method. Therefore, there were one thousand combinations with random initial geometric imperfections for every certain constitution. The instability deformation modes could be obtained for every combination analyzed using the ANSYS program.Figure3 shows four kinds of instability modes for column-bracing systems with the horizontal braces located at 0.6 column height. For non-sway instability mode as shown in Figures 3(a) and 3(b), the maximum force of the horizontal braces located at 0.6L is in compression and in tension, respectively; for sway instability modes as shown in Figures 3(c) and 3(d), the maximum force of the horizontal braces located at 0.6L may be in compression or in tension but much more in compression. Similarly, column-bracing systems with the horizontal braces located at 0.7 column height also have four kinds of instability modes as shown in Figure 3, and the maximum force of the horizontal braces may be in compression or tension. The analysis results were found to be sensitive to both the magnitude and the direction of the random initial geometric imperfections of the columns and the horizontal braces.Figure 3
Instability deformation modes for horizontal braces located at 0.6L. (a) Non-sway instability (horizontal braces in compression). (b) Non-sway instability (horizontal braces in tension). (c) Sway instability (horizontal braces in compression or tension). (d) Sway instability (horizontal braces in compression or tension).
(a)(b)(c)(d)


## 5. Probability Statistics of Numerical Results

### 5.1. Probability Density Figures ofF/P

F was defined as the maximum force of the horizontal braces unlocated at middle of columns induced by the ultimate load P of the braced columns, and then the ratio F/P was obtained for the horizontal braces located at 0.6L or 0.7L, respectively. Considering the combination of n = 4, 6, 8, 10 and λc = 50, 75, 100, twelve groups of F/P ratios were obtained, respectively, when the horizontal braces are located at 0.6L or 0.7L, respectively. As above mentioned, there are one thousand numerical results for every certain combination between n and λc, which are directly affected by the random initial geometric imperfections of both the columns and the horizontal braces. Because of the randomness of the initial geometric imperfections of both the columns and the horizontal braces, the numerical results of F/P obviously have randomness. Hence, numerical results were analyzed with probability statistics.As an illustration, the results of twelve groups’F/P ratios for the horizontal braces located at 0.6L and 0.7L are presented in Figures 4 and 5, respectively. In these figures, the sign of abscissas is defined as follows: F/P is positive when F is in compression while it is negative when F is in tension. The ordinates of these figures show the relative frequency of various (F/P) × 100 (shown as abscissas). The area of each single blue rectangle represents the probability of that abscissa value being calculated by the finite element analysis, and the sum of all the rectangular areas is equal to 1. The solid lines represent the theoretical probability density functions that are chosen to fit the data.Figure 4
Probability density figures ofF/P for horizontal braces located at 0.6L. (a) λc = 50 and n = 4. (b) λc = 50 and n = 6. (c) λc = 50 and n = 8. (d) λc = 50 and n = 10. (e) λc = 75 and n = 4. (f) λc = 75 and n = 6. (g) λc = 75 and n = 8. (h) λc = 75 and n = 10. (i) λc = 100 and n = 4. (j) λc = 100 and n = 6. (k) λc = 100 and n = 8. (l) λc = 100 and n = 10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 5
Probability density figures ofF/P for horizontal braces located at 0.7L. (a) λc = 50 and n = 4. (b) λc = 50 and n = 6. (c) λc = 50 and n = 8. (d) λc = 50 and n = 10. (e) λc = 75 and n = 4. (f) λc = 75 and n = 6. (g) λc = 75 and n = 8. (h) λc = 75 and n = 10. (i) λc = 100 and n = 4. (j) λc = 100 and n = 6. (k) λc = 100 and n = 8. (l) λc = 100 and n = 10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


### 5.2. Probability Density Figures ofF/P

The probability density function ofF/P is given by(9)fx,a1,μ,σ,μin,σin=a12πσexp−12x−μσ2,x≤0,1−a12πσinxexp−12Inx−μinσin2,x>0,and the distribution function of F/P is(10)Fx,a1,μ,σ,μin,σin=a12πσ∫−∞xe−1/2t−μ/σ2dt,x≤0,a12πσ∫−∞0e−1/2t−μ/σ2dt+1−a12πσin∫0+x1te−1/2lnt−μin/σin2dt,x>0,where a1 is the ratio of the number of negative value data over the number of the total data; μin and σin are the mean value and the mean square deviation of the logarithm of positive value data, respectively; and μ and σ are the mean value and the mean square deviation of negative value data, respectively. The statistical results of various (F/P) × 100 for the horizontal braces located at 0.6L and 0.7L are shown in Table 3, respectively.Table 3
Statistical results ofF/P×100andFn/P∗.
Horizontal braces located at 0.6LHorizontal braces located at 0.7LλcnStatistical characterFn/P (%)1.2 ×Fn/P (%)λcnStatistical characterFn/P (%)1.2 ×Fn/P (%)504μ = −2.0375; σ = 1.07021.631.96504μ = −0.7927; σ = 0.41242.032.44μΙn = 0.6923; σΙn = 0.3946μΙn = −0.1401; σΙn = 0.5870a1 = 0.2300a1 = 0.2980506μ = −1.1544; σ = 0.59582.372.84506μ = −1.1693; σ = 0.62793.053.66μΙn = 0.2089; σΙn = 0.4510μΙn = 0.1563; σΙn = 0.6624a1 = 0.2810a1 = 0.3720508μ = −1.1907; σ = 0.56623.163.79508μ = −1.4877; σ = 0.69063.964.75μΙn = 0.2647; σΙn = 0.5743μΙn = 0.4434; σΙn = 0.5127a1 = 0.3850a1 = 0.33605010μ = −1.5193; σ = 0.73324.034.845010μ = −1.7635; σ = 0.84335.136.15μΙn = 0.4805; σΙn = 0.5294μΙn = 0.6485; σΙn = 0.4866a1 = 0.3760a1 = 0.3520754μ = −1.0674; σ = 0.51761.872.24754μ = −1.3492; σ = 0.75662.613.13μΙn = 0.0317; σΙn = 0.6390μΙn = 0.2587; σΙn = 0.6496a1 = 0.3530a1 = 0.3660756μ = −1.1134; σ = 0.77182.883.46756μ = −1.8843;σ = 0.83564.215.05μΙn = −0.0331; σΙn = 0.8046μΙn = 0.5768; σΙn = 0.5375a1 = 0.3060a1 = 0.3110758μ = −1.9497; σ = 1.11043.854.62758μ = −2.4379; σ = 1.17735.786.94μΙn = 0.6112; σΙn = 0.6170μΙn = 0.7872; σΙn = 0.5883a1 = 0.3910a1 = 0.42607510μ = −2.3161; σ = 1.32395.486.587510μ = −3.2291; σ = 1.59067.669.19μΙn = 0.8144; σΙn = 0.5938μΙn = 1.0570; σΙn = 0.5564a1 = 0.3810a1 = 0.46101004μ = −0.9750; σ = 0.42612.332.791004μ = −1.4045; σ = 0.64923.033.63μΙn = −0.1504; σΙn = 0.5849μΙn = 0.2742; σΙn = 0.6987a1 = 0.3720a1 = 0.38801006μ = −1.7184; σ = 0.74603.33.961006μ = −2.5231; σ = 1.19844.885.85μΙn = 0.4839; σΙn = 0.5801μΙn = 0.7677; σΙn = 0.7717a1 = 0.4270a1 = 0.44801008μ = −2.3836; σ = 1.17154.825.781008μ = −3.3931; σ = 1.83796.978.36μΙn = 0.7768; σΙn = 0.6407μΙn = 1.0623; σΙn = 0.7615a1 = 0.4240a1 = 0.458010010μ = −2.8440; σ = 1.44116.758.110010μ = −4.3562; σ = 2.50719.3911.27μΙn = 0.9566; σΙn = 0.6333μΙn = 1.2740; σΙn = 0.7612a1 = 0.4340a1 = 0.4760∗Fn/P: refer to Figure 5 for the definition of the characteristic value.


### 5.3. Design Forces of Horizontal Braces Unlocated at Middle of Columns

According to the distribution functionsF (x), the F/P ratios can be predicted to a certain significance level, say, α. The significance level was selected as α = 0.05 in this study. Then, the characteristic values Fn/P are taken as the horizontal bracing force with a 95% guarantee rate (see Figure 6).Figure 6
The characteristic values atα = 0.05.In order to provide full support for longitudinal columns in the row,Fn/P is multiplied by a coefficient of 1.2, and 1.2 × Fn/P is taken as the design horizontal bracing forces. Table 3 summarizes Fn/P and 1.2 × Fn/P for the horizontal braces located at 0.6L and 0.7L.


## 5.1. Probability Density Figures ofF/P

F was defined as the maximum force of the horizontal braces unlocated at middle of columns induced by the ultimate load P of the braced columns, and then the ratio F/P was obtained for the horizontal braces located at 0.6L or 0.7L, respectively. Considering the combination of n = 4, 6, 8, 10 and λc = 50, 75, 100, twelve groups of F/P ratios were obtained, respectively, when the horizontal braces are located at 0.6L or 0.7L, respectively. As above mentioned, there are one thousand numerical results for every certain combination between n and λc, which are directly affected by the random initial geometric imperfections of both the columns and the horizontal braces. Because of the randomness of the initial geometric imperfections of both the columns and the horizontal braces, the numerical results of F/P obviously have randomness. Hence, numerical results were analyzed with probability statistics.As an illustration, the results of twelve groups’F/P ratios for the horizontal braces located at 0.6L and 0.7L are presented in Figures 4 and 5, respectively. In these figures, the sign of abscissas is defined as follows: F/P is positive when F is in compression while it is negative when F is in tension. The ordinates of these figures show the relative frequency of various (F/P) × 100 (shown as abscissas). The area of each single blue rectangle represents the probability of that abscissa value being calculated by the finite element analysis, and the sum of all the rectangular areas is equal to 1. The solid lines represent the theoretical probability density functions that are chosen to fit the data.Figure 4
Probability density figures ofF/P for horizontal braces located at 0.6L. (a) λc = 50 and n = 4. (b) λc = 50 and n = 6. (c) λc = 50 and n = 8. (d) λc = 50 and n = 10. (e) λc = 75 and n = 4. (f) λc = 75 and n = 6. (g) λc = 75 and n = 8. (h) λc = 75 and n = 10. (i) λc = 100 and n = 4. (j) λc = 100 and n = 6. (k) λc = 100 and n = 8. (l) λc = 100 and n = 10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Figure 5
Probability density figures ofF/P for horizontal braces located at 0.7L. (a) λc = 50 and n = 4. (b) λc = 50 and n = 6. (c) λc = 50 and n = 8. (d) λc = 50 and n = 10. (e) λc = 75 and n = 4. (f) λc = 75 and n = 6. (g) λc = 75 and n = 8. (h) λc = 75 and n = 10. (i) λc = 100 and n = 4. (j) λc = 100 and n = 6. (k) λc = 100 and n = 8. (l) λc = 100 and n = 10.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)


## 5.2. Probability Density Figures ofF/P

The probability density function ofF/P is given by(9)fx,a1,μ,σ,μin,σin=a12πσexp−12x−μσ2,x≤0,1−a12πσinxexp−12Inx−μinσin2,x>0,and the distribution function of F/P is(10)Fx,a1,μ,σ,μin,σin=a12πσ∫−∞xe−1/2t−μ/σ2dt,x≤0,a12πσ∫−∞0e−1/2t−μ/σ2dt+1−a12πσin∫0+x1te−1/2lnt−μin/σin2dt,x>0,where a1 is the ratio of the number of negative value data over the number of the total data; μin and σin are the mean value and the mean square deviation of the logarithm of positive value data, respectively; and μ and σ are the mean value and the mean square deviation of negative value data, respectively. The statistical results of various (F/P) × 100 for the horizontal braces located at 0.6L and 0.7L are shown in Table 3, respectively.Table 3
Statistical results ofF/P×100andFn/P∗.
Horizontal braces located at 0.6LHorizontal braces located at 0.7LλcnStatistical characterFn/P (%)1.2 ×Fn/P (%)λcnStatistical characterFn/P (%)1.2 ×Fn/P (%)504μ = −2.0375; σ = 1.07021.631.96504μ = −0.7927; σ = 0.41242.032.44μΙn = 0.6923; σΙn = 0.3946μΙn = −0.1401; σΙn = 0.5870a1 = 0.2300a1 = 0.2980506μ = −1.1544; σ = 0.59582.372.84506μ = −1.1693; σ = 0.62793.053.66μΙn = 0.2089; σΙn = 0.4510μΙn = 0.1563; σΙn = 0.6624a1 = 0.2810a1 = 0.3720508μ = −1.1907; σ = 0.56623.163.79508μ = −1.4877; σ = 0.69063.964.75μΙn = 0.2647; σΙn = 0.5743μΙn = 0.4434; σΙn = 0.5127a1 = 0.3850a1 = 0.33605010μ = −1.5193; σ = 0.73324.034.845010μ = −1.7635; σ = 0.84335.136.15μΙn = 0.4805; σΙn = 0.5294μΙn = 0.6485; σΙn = 0.4866a1 = 0.3760a1 = 0.3520754μ = −1.0674; σ = 0.51761.872.24754μ = −1.3492; σ = 0.75662.613.13μΙn = 0.0317; σΙn = 0.6390μΙn = 0.2587; σΙn = 0.6496a1 = 0.3530a1 = 0.3660756μ = −1.1134; σ = 0.77182.883.46756μ = −1.8843;σ = 0.83564.215.05μΙn = −0.0331; σΙn = 0.8046μΙn = 0.5768; σΙn = 0.5375a1 = 0.3060a1 = 0.3110758μ = −1.9497; σ = 1.11043.854.62758μ = −2.4379; σ = 1.17735.786.94μΙn = 0.6112; σΙn = 0.6170μΙn = 0.7872; σΙn = 0.5883a1 = 0.3910a1 = 0.42607510μ = −2.3161; σ = 1.32395.486.587510μ = −3.2291; σ = 1.59067.669.19μΙn = 0.8144; σΙn = 0.5938μΙn = 1.0570; σΙn = 0.5564a1 = 0.3810a1 = 0.46101004μ = −0.9750; σ = 0.42612.332.791004μ = −1.4045; σ = 0.64923.033.63μΙn = −0.1504; σΙn = 0.5849μΙn = 0.2742; σΙn = 0.6987a1 = 0.3720a1 = 0.38801006μ = −1.7184; σ = 0.74603.33.961006μ = −2.5231; σ = 1.19844.885.85μΙn = 0.4839; σΙn = 0.5801μΙn = 0.7677; σΙn = 0.7717a1 = 0.4270a1 = 0.44801008μ = −2.3836; σ = 1.17154.825.781008μ = −3.3931; σ = 1.83796.978.36μΙn = 0.7768; σΙn = 0.6407μΙn = 1.0623; σΙn = 0.7615a1 = 0.4240a1 = 0.458010010μ = −2.8440; σ = 1.44116.758.110010μ = −4.3562; σ = 2.50719.3911.27μΙn = 0.9566; σΙn = 0.6333μΙn = 1.2740; σΙn = 0.7612a1 = 0.4340a1 = 0.4760∗Fn/P: refer to Figure 5 for the definition of the characteristic value.


## 5.3. Design Forces of Horizontal Braces Unlocated at Middle of Columns

According to the distribution functionsF (x), the F/P ratios can be predicted to a certain significance level, say, α. The significance level was selected as α = 0.05 in this study. Then, the characteristic values Fn/P are taken as the horizontal bracing force with a 95% guarantee rate (see Figure 6).Figure 6
The characteristic values atα = 0.05.In order to provide full support for longitudinal columns in the row,Fn/P is multiplied by a coefficient of 1.2, and 1.2 × Fn/P is taken as the design horizontal bracing forces. Table 3 summarizes Fn/P and 1.2 × Fn/P for the horizontal braces located at 0.6L and 0.7L.


## 6. Design Recommendations

Table3 summarizes 1.2 × Fn/P for the horizontal braces located at 0.6L and 0.7L with n = 4, 6, 8, 10 and λc = 50, 75, 100, respectively. It should be noted that 1.2 × Fn/P increases with an increase of λc, and the same result was obtained in reference [8]. Furthermore, according to different λc, 1.2 × Fn/P may be approximated as follows:(1)
The design formulas of horizontal braces located at 0.6L are(11)FnP=n1000.48+0.001n,λc=50,n1000.45+0.020n,λc=75,n1000.52+0.028n,λc=100.(2)
The design formulas of horizontal braces located at 0.7L are(12)FnP=n1000.6+0.0015n,λc=50,n1000.71+0.021n,λc=75,n1000.75+0.037n,λc=100.Figures7 and 8 show the comparisons between equations (11) and (12), and the results are given in Table 3. It is suggested that equations (11) and (12) may be used in engineering practice for the horizontal braces located at 0.6L and 0.7L, respectively.Figure 7
Variation in 1.2 ×Fn/(P) for the horizontal braces located at 0.6L against n.Figure 8
Variation in 1.2 ×Fn/P for the horizontal braces located at 0.7L against n.


## 7. Comparison with Mid-Height Horizontal Bracing Forces

### 7.1. Comparison with Previous Study

In reference [8], the mid-height horizontal bracing forces were proposed according to the same research method adopted in this study. Comparison among various 1.2 × Fn/P is shown in Table 4, and the variation of the ratios is also plotted in Figure 9. It is shown that the ratios of 1.2 × Fn/P for the horizontal braces at 0.7L are the largest, followed by the ratios of 1.2 × Fn/P for the horizontal braces at 0.5L, and the ratios of 1.2 × Fn/P for the horizontal braces at 0.6L are the smallest. The reasons are summarized as follows: when the horizontal braces are located at 0.6L, the effective length of lower columns can be increased, so that the ultimate load P and horizontal bracing forces F are lower than those with the horizontal braces located at 0.5L, and the relatively low ultimate load P of the former more significantly decreases its horizontal bracing force F, which eventually results in the ratios of 1.2 × Fn/P with the horizontal braces at 0.6L being lower than the ratios of 1.2 × Fn/P with the horizontal braces at 0.5L; when the horizontal braces are located at 0.7L, the effective length of lower columns can be further increased, so that the ultimate load P decreases much more, but horizontal bracing force F which is needed to maintain the longitudinal stability of the braced columns decreases much less, which eventually results in the ratios of 1.2 × Fn/P with the horizontal braces at 0.7L being higher than the ratios of 1.2×Fn/P with the horizontal braces at 0.5L.Table 4
Comparison of 1.2 ×Fn/P values (%).
Horizontal brace locationλcn = 4n = 6n = 8n = 100.5L502.283.184.185.16752.714.446.128.161003.025.107.269.72GB500174.676.678.67—Eurocode 35.818.0610.2712.46AS41006.258.7511.25—0.6L501.962.843.794.84752.243.464.626.581002.793.965.788.100.7L502.443.664.756.15753.135.056.949.191003.635.858.3611.27Figure 9
Comparison of 1.2 ×Fn/P among various horizontal brace locations. (a) λc = 50. (b) λc = 75. (c) λc = 100.
(a)(b)(c)


### 7.2. Comparison with Relevant Codes

GB50017-2017 [18], Eurocode 3-1992 [19], and AS4100-1998 [20] specify the mid-height horizontal bracing forces of the column-bracing system which are shown in Table 4. It was found in reference [17] that the mid-height horizontal bracing forces proposed by considering the random initial geometric imperfections were smaller than those given in GB50017-2017, Eurocode 3, and AS4100-1998. Although the design forces of horizontal braces located at 0.7L from the proposed method are larger than the mid-height horizontal bracing forces from the same proposed method, they are still smaller than the mid-height horizontal bracing forces stipulated in GB50017-2017, Eurocode 3-1992, and AS4100-1998. As the horizontal bracing forces increase with an increase of λc, the comparison of horizontal bracing forces between the proposed method and the relevant codes is plotted in Figure 10 when λc is equal to 100.Figure 10
Comparison ofFn/P values for λc = 100 with relevant codes.


## 7.1. Comparison with Previous Study

In reference [8], the mid-height horizontal bracing forces were proposed according to the same research method adopted in this study. Comparison among various 1.2 × Fn/P is shown in Table 4, and the variation of the ratios is also plotted in Figure 9. It is shown that the ratios of 1.2 × Fn/P for the horizontal braces at 0.7L are the largest, followed by the ratios of 1.2 × Fn/P for the horizontal braces at 0.5L, and the ratios of 1.2 × Fn/P for the horizontal braces at 0.6L are the smallest. The reasons are summarized as follows: when the horizontal braces are located at 0.6L, the effective length of lower columns can be increased, so that the ultimate load P and horizontal bracing forces F are lower than those with the horizontal braces located at 0.5L, and the relatively low ultimate load P of the former more significantly decreases its horizontal bracing force F, which eventually results in the ratios of 1.2 × Fn/P with the horizontal braces at 0.6L being lower than the ratios of 1.2 × Fn/P with the horizontal braces at 0.5L; when the horizontal braces are located at 0.7L, the effective length of lower columns can be further increased, so that the ultimate load P decreases much more, but horizontal bracing force F which is needed to maintain the longitudinal stability of the braced columns decreases much less, which eventually results in the ratios of 1.2 × Fn/P with the horizontal braces at 0.7L being higher than the ratios of 1.2×Fn/P with the horizontal braces at 0.5L.Table 4
Comparison of 1.2 ×Fn/P values (%).
Horizontal brace locationλcn = 4n = 6n = 8n = 100.5L502.283.184.185.16752.714.446.128.161003.025.107.269.72GB500174.676.678.67—Eurocode 35.818.0610.2712.46AS41006.258.7511.25—0.6L501.962.843.794.84752.243.464.626.581002.793.965.788.100.7L502.443.664.756.15753.135.056.949.191003.635.858.3611.27Figure 9
Comparison of 1.2 ×Fn/P among various horizontal brace locations. (a) λc = 50. (b) λc = 75. (c) λc = 100.
(a)(b)(c)


## 7.2. Comparison with Relevant Codes

GB50017-2017 [18], Eurocode 3-1992 [19], and AS4100-1998 [20] specify the mid-height horizontal bracing forces of the column-bracing system which are shown in Table 4. It was found in reference [17] that the mid-height horizontal bracing forces proposed by considering the random initial geometric imperfections were smaller than those given in GB50017-2017, Eurocode 3, and AS4100-1998. Although the design forces of horizontal braces located at 0.7L from the proposed method are larger than the mid-height horizontal bracing forces from the same proposed method, they are still smaller than the mid-height horizontal bracing forces stipulated in GB50017-2017, Eurocode 3-1992, and AS4100-1998. As the horizontal bracing forces increase with an increase of λc, the comparison of horizontal bracing forces between the proposed method and the relevant codes is plotted in Figure 10 when λc is equal to 100.Figure 10
Comparison ofFn/P values for λc = 100 with relevant codes.


## 8. Conclusions

The design forces of the horizontal braces unlocated at middle of columns were determined by Monte Carlo simulation, finite element analysis, and probability statistics, in which the random initial geometric imperfections of both the columns and the horizontal braces were well considered. The conclusions are summarized as follows:(1)
The structural performance of the column-bracing system resulting from real random initial imperfections was well predicted by the Monte Carlo method. When the ultimate loads of the braced columns are reached, the horizontal braces used to reduce the out-of-plane effective column lengths may be in compression or in tension, i.e., the horizontal bracing forces have randomness.(2)
The probability density function of horizontal bracing forces unlocated at middle of columns was established by probability statistics, the practical design formulas for horizontal bracing forces unlocated at middle of columns were also proposed, and it was found that the design forces of horizontal braces located at 0.6 column height are smaller than the mid-height horizontal bracing forces in the previous study while the design forces of horizontal braces located at 0.7 column height are larger than the mid-height horizontal bracing forces in the previous study.(3)
As the random initial geometric imperfections of the columns and the horizontal braces were well considered, the proposed design forces of the horizontal braces located and unlocated at middle of columns were both decreased to more reasonable values which were smaller than the mid-height horizontal bracing forces stipulated in GB50017-2017, Eurocode 3-1992, and AS4100-1998.



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