Formulation and validation of minimum brace stiffness for systems of compression members

Journal of Constructional Steel Research 129 (2017) 263–275

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Journal of Constructional Steel Research

Formulation and validation of minimum brace stiffness for systems of compression members Ronald D. Ziemian a,⁎, Constance W. Ziemian b a b

Bucknell University, Department of Civil & Environmental Engineering, Lewisburg, PA 17837, United States Bucknell University, Department of Mechanical Engineering, Lewisburg, PA 17837, United States

a r t i c l e

i n f o

Article history: Received 24 September 2016 Received in revised form 12 November 2016 Accepted 15 November 2016 Available online xxxx Keywords: Ideal brace stiffness Brace design Parallel compression members Lateral stiffness Buckling

a b s t r a c t Bracing that is used to reduce the effective length of compressive members, and thereby increase their load carrying capacity, must be designed to provide adequate stiffness and strength. In practice, a targeted minimum or ideal brace stiffness is typically defined, and then this value is increased by a factor of two or three in order to ensure brace strength demands can be satisfied efficiently. While straightforward expressions are available to determine the ideal brace stiffness for single members, the brace stiffness for systems of multiple parallel compression members is more complex. This paper presents the derivation and validation of a simple mathematical expression that is useful for obtaining the ideal brace stiffness of structural systems composed of multiple parallel compression members or sub-assemblages. Use and validation of the proposed expression is explored through a variety of examples. Agreement with results obtained using finite element analyses suggests that the developed expression sufficiently determines the minimum bracing stiffness for systems of multiple parallel members with single and multiple brace points. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Bracing is used to reduce the unsupported or effective length of compression members or systems, and thereby increase buckling strength. Brace stiffness and strength requirements, as needed to achieve a specified increase in the buckling strengths of the braced members, are well established in the literature with regard to a single column-brace model [1–5]. Economical solutions for stability bracing further take advantage of the relationship between stiffness and strength, which often includes a portion of the domain in which a relatively small increase in the brace stiffness can dramatically reduce brace forces or strength demands. Far fewer studies have been carried out, however, to determine the stiffness and strength requirements for multiple member bracing systems [6–8]. Stability bracing provided for a system of parallel compression members or sub-assemblages (e.g. trusses) can typically be separated into two key parts; tie bracing and anchor bracing (Fig. 1a). The first represents the stiffness of the components used to tie the parallel members together, and the second embodies the stiffness of the anchorage (e.g. discrete point) or terminating system (e.g. shear panel). In defining the stiffness and strength requirements for systems with parallel elements, design specifications often assume that the stiffness of the tie bracing is significantly greater than the stiffness of the anchor bracing. In such cases, the tie bracing is typically treated as rigid. The subsequent result is that stability bracing design methods are often simplified to ⁎ Corresponding author. E-mail address: [email protected] (R.D. Ziemian).

http://dx.doi.org/10.1016/j.jcsr.2016.11.015 0143-974X/© 2016 Elsevier Ltd. All rights reserved.

only include consideration of the stiffness and strength requirements of the anchor-bracing system; and the demands on the anchor-bracing system can then be defined by the product of the number of parallel compression members being stabilized and the bracing demands of one of these compression members. This paper presents an investigation of the bracing stiffness requirements for structural systems composed of multiple parallel compression members, in which the stiffness of the tie bracing is not necessarily much larger than that of the anchor bracing; thereby removing the suitability of assuming that the tie bracing is rigid. Such cases include buildings or industrial facilities with a significant number of lean-on columns and/or girders, open web steel joists, and studs within cold-formed steel framed walls. For example, the system shown in Fig. 1a can be represented by the parallel column analogy displayed in Fig. 1b. In this case, the stiffness of the anchor bracing, kanchor, would include the net lateral stiffness contributions of the resisting column, the connection clips, and the column base fixity. The stiffness of the tie bracing, ktie, includes the net axial stiffness of the bridging and connections between the joists. This paper will demonstrate that an increase in the flexibility (decrease in stiffness) of the tie bracing can result in significant increases on the anchorage stiffness demands. The stiffness of such systems has been considered by multiple researchers in the literature. Medland and Segedin, for instance, completed significant work on this topic in the mid to late 1970's [9–11]. Although their approach is comprehensive and detailed, its implementation requires a level of mathematics that lends itself more to the subsequent use of design charts, as opposed to the use of concise

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Fig. 1. Multiple-member systems. (a) Key elements of stability bracing for system with multiple members, and (b) Model of system of n-parallel compression members.

expressions that are more suitable for design specifications. More recently, Sputo and Beery [12] and Blum et al. [13,14] investigated this topic through the implementation of curve fitting results to a series of finite element studies. These studies addressed only the stiffness of the braces tying the members together, however, and did not consider the additional impact of non-rigid bracing anchorages. The work presented here focuses on the determination of the “ideal” brace stiffness, or the minimum brace stiffness required to ensure that the full buckling strength of braced members can be achieved. This minimum brace stiffness is often the basis of bracing design, which typically targets a stiffness of two or three times that of the ideal value in order to keep brace forces or strength demands within reason [1]. The work presented herein specifically develops expressions for the minimum stiffnesses of both the tie and anchorage bracing for systems of multiple parallel compression members, such as that shown in Fig. 1.

2. Ideal brace stiffness of single compression member The ideal brace stiffness for a single column such as that shown in Fig. 2 is derived on the basis of Winter's work [1], which is well documented in the literature [15,16]. An interpretation of the mathematical formulation is repeated here to serve as the foundation for the forthcoming derivation for parallel members. The objective is to determine ks, the minimum or ideal stiffness of the brace, such that the column non-sway buckling load Pcr (Fig. 2a) will equal the column side-sway buckling load Pcr (Fig. 2b). The buckled shape diagram and free body diagram in Fig. 2b and c indicate that the brace force Q1 is a function of (Δ1 − Δ0), the axial extension of the

brace, as follows: Q 1 ¼ ks ðΔ1 −Δ0 Þ

ð1Þ

For the case of Δ0 = 0, this becomes: Q 1 −ks Δ1 ¼ 0

ð2Þ

Consideration of the free body diagram in Fig. 2c and application of the condition of equilibrium of the moments about the base, provides: LQ 1 −P cr Δ1 ¼ 0

ð3aÞ

This can be rewritten as: Q 1−


 P cr Δ1 ¼ 0 L

ð3bÞ

Expressing Eqs. (2), and (3b) as a homogeneous system of linear algebraic equations, provides: "

1 1

−ks P cr − L

#

Q1 Δ1



 ¼

0 0

 ð4Þ

The solution to Eq. (4) is either the trivial case, where Q1 = Δ1 = 0, or is based on the requirement that the determinant of the coefficient

Fig. 2. Single column ideal brace stiffness; (a) Deflected shape of non-sway column buckling, (b) Deflected shape of side-sway column buckling, and (c) Free body diagram of side-sway column.

R.D. Ziemian, C.W. Ziemian / Journal of Constructional Steel Research 129 (2017) 263–275

265

matrix is equal to zero:   1 −ks  P cr   1 − L

  P cr  −ð−ks Þ ¼0¼−  L

ð5Þ

From this requirement, the ideal brace stiffness can be expressed as expected:

ks ¼

P cr L

ð6Þ

where Pcr is the column's desired non-sway buckling strength, and L is the unbraced length of the column. 3. Ideal brace stiffness, system of parallel compression members Although an abbreviated description of the following formulation has been presented previously [17], a detailed explanation is provided here for the sake of completeness and as the basis for the subsequent validation study (Section 5). 3.1. System with rigid anchor Using the same general approach as that presented in Section 2, the ideal brace stiffness for a system of parallel columns with a rigid anchorage (Fig. 3) can be obtained. The formulation is based on the assumptions that: (a) each of the n columns are simply supported, (b) each has the same non-sway buckling strength Pcr, and (c) each of the tie braces has the same stiffness, kn, i.e. the ideal brace stiffness for an n-column system. Consideration is first given to the buckled shape of the i-th column in the system, which is displayed in Fig. 4.

Fig. 4. The ith column of an n-column system; (a) Deflection diagram of side-sway column i, and (b) Free body diagram of side-sway column i.

From Fig. 4, it can be seen that the brace force Qi is related to the axial extension of the brace, (Δi − Δi − 1) as follows: Q i ¼ kn ðΔi −Δi−1 Þ

ð7aÞ

Eq. (7a) can be rewritten as: kn Δi−1 þ Q i −kn Δi ¼ 0

ð7bÞ

Consideration of the free body diagram of the ith column, in Fig. 4b, and applying the conditions of moment equilibrium about its base,

Fig. 3. Model of a system of parallel columns with a rigid anchor; (a) Non-sway column buckling mode, and (b) Side-sway column buckling mode.

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ideal brace stiffness for the system of n parallel compression members:

provides: LQ i −P cr Δi −LQ iþ1 ¼ 0

ð8aÞ

This can be rewritten as:
 P cr Δi −Q iþ1 ¼ 0 Q i− L

ð8bÞ

Eqs. (7b) and (8b) together represent a recursive relationship for all n columns in the system: "

kn

1

0

1

−kn 0 P cr −1 L

8 #> Δi−1 < Qi > : Δi Q iþ1

9 > = > ;

 ¼

0 0

 for i ¼ 1; …; n

ð9Þ

for which Δ0 = Qn + 1 = 0. By representing the ideal brace stiffness of each column in the n-column system, kn, as a multiple of the ideal brace stiffness of the single column system, ks, substitutions can be made using kn = aks. The coefficient a is an unknown scale factor. Using this relationship and substituting ks for Pcr/L, Eq. (9) can be expanded to include all n members in the parallel column system as follows:

2 6 6 6 6 6 6 6 6 6 6 4

1 −aks 1 −ks ⋮ ⋮ 0 0 0 0 ⋮ ⋮ 0 0 0 0

0 ⋯ 0 −1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ aks 0 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 0 0 ⋯ 0

0 0 0 0 ⋮ ⋮ 1 −aks 1 −ks ⋮ ⋮ 0 0 0 0

0 0 ⋮ 0 −1 ⋮ 0 0

⋯ ⋯ ⋱ ⋯ ⋯ ⋱ ⋯ ⋯

0 0 ⋮ 0 0 ⋮ aks 0

8 > > > > 3> > > > 0 0 > > > > 0 0 7 7> > > > ⋮ ⋮ 7 > 7> < 0 0 7 7 0 0 7 > 7> > > ⋮ ⋮ 7 > 7> > 5 > 1 −aks > > > > 1 −ks > > > > > > :

Q1 Δ1 Q2 ⋮ Δi−1 Qi Δi Q iþ1 ⋮ Δn−1 Qn Δn

9 > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > ;

¼

8 > > > > > > > > > > > > > > > > > > <

0 0 0 ⋮ 0 0 0 0 ⋮ 0 0 0

> > > > > > > > > > > > > > > > > > :

9 > > > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > > > ;

ð10Þ Eq. (10) represents a homogeneous system of linear algebraic equations that only permits a nonlinear trivial solution when the determinate of the coefficient matrix is equal to zero. Expansion of this determinate yields an equation of the general form n

f ðaÞ∙ðks Þ ¼ 0

ð11Þ

in which n is the number of parallel columns in the system, and f(a) is an nth-order polynomial with constant coefficients. Given that the product shown in Eq. (11) must equal zero, and ks is a nonzero value, the roots of the polynomial f(a) are required. These roots represent the potential values of the scale factor a that satisfy Eq. (11), and consequently Eq. (10). The maximum root, amax, multiplied by the ideal brace stiffness of the single column system, ks, provides an expression for the required

kn ¼ amax ∙ks ¼

amax P cr L

ð12Þ

Using this approach, and accomplishing the root finding task with the symbolic processor feature within MATLAB [18], solutions for the ideal brace stiffness for systems that include n = 1 to 100 parallel columns were obtained. A sample of the resulting polynomials and maximum roots are provided in Table 1. The accuracy of the results provided in Table 1 were verified using the critical load (buckling) analysis option within the finite element program MASTAN2 [19]. Given that this procedure requires significant effort and the use of a symbolic processor, an approximate solution was pursued using least squares regression. The maximum roots, amax, were first obtained for each value of n in the range of [1100] using the aforementioned procedure and MATLAB. These “exact” values, corresponding to systems of n parallel compression members, are plotted (Fig. 5) and analyzed as a data set. A non-linear regression analysis was performed on the data set, and the resulting parabolic function: amax ¼ 0:405285n2 þ 0:405255n þ 0:185440

ð13aÞ

demonstrated a very good fit to the data, with a coefficient of determination of R2 = 0.999. Simplifying this function, for ease of use, provides the following approximation: amax ¼ 0:4n2 þ 0:4n þ 0:2

ð13bÞ

Approximate values for the maximum root, as obtained using Eqs. (13a) and (13b), are displayed in Fig. 5 as well. A numerical comparison of the theoretically exact values for amax, as determined using the aforementioned determinant and root-finding procedure (Table 1), was made with the approximated values obtained using Eq. (13b). The maximum percentage error resulting from this comparison is 1.3% (Table 2), suggesting excellent agreement between exact and approximated values for amax. Using this simplified approximation, the ideal brace stiffness of a system of n parallel and identical compression members can be closely approximated by   kn ¼ 0:4n2 þ 0:4n þ 0:2 ∙ks

ð14Þ

in which ks is the ideal stiffness for a single column. For the specific problem studied in this section, ks = Pcr/L, where Pcr is the column's desired non-sway buckling strength, and L is the column's unbraced length. It should be noted that this equation is comparable, but not the same as that developed by Sputo and Beery [12], in which they

Table 1 Ideal brace stiffness for n compression members, kn, with rigid anchor. n

f(a)

max. root, amax

kn = amax ks

1 2 3 4 5 10 15 25 50 100

a−1 a2 − 3a + 1 a3 − 6a2 + 5a − 1 a4 − 10a3 + 15a2 − 7a + 1 a5 − 15a4 + 35a3 − 28a2 + 9a − 1 10th-order polynomial 15th-order polynomial 25th-order polynomial 50th-order polynomial 100th-order polynomial

1.0000 2.6180 5.0489 8.2909 12.343 44.766 97.453 263.62 1033.7 4093.6

Pcr/L 2.62Pcr/L 5.05Pcr/L 8.29Pcr/L 12.34Pcr/L 44.77Pcr/L 97.45Pcr/L 263.6Pcr/L 1034Pcr/L 4094Pcr/L

Fig. 5. Exact, fitted, and approximated values for the maximum root, amax, of the polynomials of order n, corresponding to systems of n parallel compression members with rigid anchors.

R.D. Ziemian, C.W. Ziemian / Journal of Constructional Steel Research 129 (2017) 263–275 Table 2 Comparison of exact and approximate values for the maximum root, amax. n

Exact amax

Approximate amax (Eq. (13b))

% error

1 2 3 4 5 10 25 50 100

1.0000 2.6180 5.0489 8.2909 12.343 44.766 263.62 1033.7 4093.6

1 2.6 5 8.2 12.2 44.2 260.2 1020.2 4040.2

0 0.6888 0.9689 1.0959 1.1629 1.2645 1.2972 1.3022 1.3035

performed a similar curve fitting approach to finite element results obtained for a smaller range of parallel columns. 3.2. System with flexible anchor The formulation to determine the ideal stiffness of the bracing or springs used to tie together parallel compression members can be extended to include the lateral stiffness of a non-rigid or flexible anchoring system. In order to accomplish this, the model shown in Fig. 3 is modified to include an additional brace or spring with stiffness klat at the upper left support point, as seen in Fig. 6. The net stiffness of the two springs in series at the left support of the system is defined as 1 1 1  ¼
 kn ¼ kn 1 1 kn ð1 þ c Þ þ 1þ klat kn klat




ð15Þ

with c = kn/klat; the ratio of the tie-bracing stiffness kn to the anchorbracing stiffness klat. In order to include this non-rigid anchor within the mathematical model derived in Section 3.1, the stiffness in the first tie brace must be scaled by 1 / (1 + c). Eq. (10) is then modified to replace the term appearing in row 1 and column 2 of the coefficient matrix (i.e. −aks) with − aks / (1 + c). As was the case in the previous formulation, the non-trivial solution is determined by setting the determinant of this now modified coefficient matrix to zero. By once again employing the symbolic processor within MATLAB to formulate the root-finding problem, results in the form of Eq. (11) are obtained. In this case, however, the coefficients of the nth-order polynomial, f(a), are now functions of the stiffness ratio c. For illustrative purposes, a sample of these polynomials is provided in Table 3, for systems including n = 1 to 5 parallel compression members. For a system of n parallel compression members, the theoretically exact solution for the ideal brace stiffness can once again be obtained from the maximum root amax of the system's corresponding nth-order

267

Table 3 Polynomials associated with ideal brace stiffness for n compression members and a flexible anchor. n

f(a)

1

½a−ð1þcÞ ð1þcÞ

2

½a2 −ð3þ2cÞaþð1þcÞ ð1þcÞ

3

½a3 −ð6þ3cÞa2 þð5þ4cÞa−ð1þcÞ ð1þcÞ

4

½a4 −ð10þ4cÞa3 þð15þ10cÞa2 −ð7þ6cÞaþð1þcÞ ð1þcÞ

5

½a5 −ð15þ5cÞa4 þð35þ20cÞa3 −ð28þ21cÞa2 þð9þ8cÞa−ð1þcÞ ð1þcÞ

polynomial. Given that the coefficients in these polynomials are functions of c, the value of this maximum root will vary as this stiffness ratio changes. It is worth noting that these polynomials will reduce to those provided in Table 1 when the ideal stiffness of the anchorage klat is infinitely larger than that of the ideal tie-bracing stiffness kn, i.e. c = 0. Once the polynomial has been established for a given stiffness ratio c, and the maximum root amax has been obtained, the ideal brace stiffness pair is given by tie bracing : kn ¼ amax ∙ks anchor bracing : klat ¼

ð16aÞ

kn amax ¼ ks c c

ð16bÞ

Tables 4a and 4b provide exact values for the scale factor amax and the ratio amax/c for a range of stiffness ratios c and number of parallel columns n, as determined using the aforementioned mathematical approach of finding the maximum roots of the determinant of the coefficient matrix. The accuracy of the results in Tables 4a and 4b were subsequently verified using the critical load (buckling) analysis option within the finite element program MASTAN2. As shown in Table 4b, the ideal stiffness for the anchorage klat approaches the product n·k s as the stiffness of the tie bracing kn becomes significantly greater than the stiffness of the anchor bracing klat, i.e. c N 1000. It is actually with this assumption that many design specifications recommend using a “summation-P” concept in defining bracing requirements, whereby all column axial forces on a level are summed (which for this study is n·Pcr) and then used as the axial force in the ideal brace stiffness expression derived in Section 2 (Eq. (6)). Use of this approach could be quite non-conservative for many combinations of n and c, as shown in Table 4b. Non-linear regression was used to obtain an approximate solution for different values of c, and parabolic curves were again determined to provide a close fit to the data displayed in Table 4a. A summary of best-fit functions is presented in Table 5. Although the coefficient of determination R2 values indicate that the parabolas explain 100% of the variation in the data obtained by the exact solution and provide an excellent fit, the coefficients within the parabolas appear to vary significantly for the range of stiffness

Table 4a Theoretically exact scale factor amax, flexible anchor.

Fig. 6. Model of a system of parallel columns with a flexible anchor.

n

c 0.1

1

10

100

500

1000

1 2 3 4 5 10 25 50 100

1.1 2.8 5.3 8.7 12.8 45.6 265.7 1037.8 4101.7

2.0 4.6 7.9 12.0 16.9 53.5 284.6 1075.0 4175.4

11.0 22.5 34.7 47.6 61.2 140.2 491.1 1471.3 4941.8

101.0 202.5 304.7 407.5 511.0 1038.7 2724.3 5883.1 13,560.0

501.0 1002.5 1504.7 2007.5 2511.0 5038.5 12,722.0 25,864.0 53,426.0

1001.0 2002.5 3004.7 4007.5 5011.0 10,039.0 25,221.0 50,861.0 103,405.0

268

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4. Design procedure

Table 4b Theoretically exact ratio amax/c, flexible anchor. n

c 0.1

1

10

100

500

1000

106

1 2 3 4 5 10 25 50 100

11.0 28.1 53.3 86.5 127.9 456.2 2656.9 10,378.0 41,017.0

2.0 4.6 7.9 12.0 16.9 53.5 284.6 1075.0 4175.4

1.1 2.3 3.5 4.8 6.1 14.0 49.1 147.1 494.2

1.0 2.0 3.1 4.1 5.1 10.4 27.2 58.8 135.6

1.0 2.0 3.0 4.0 5.0 10.1 25.4 51.7 106.9

1.0 2.0 3.0 4.0 5.0 10.0 25.2 50.9 103.4

1.0 2.0 3.0 4.0 5.0 10.0 25.0 50.0 100.0

ratios c that was studied. Using multi-dimensional regression analysis software, it was found that (1) the coefficients of the secondorder n2-term only varied slightly from the 0.4 value employed in Eq. (13b), (2) the coefficient of the first-order n-term has a nearly linear relationship with stiffness ratio c, and (3) the constant terms in the parabolic equations are only significant when n is very small. With this in mind, and in an effort to obtain an equation that would become Eq. (13b) for the case when c = 0, the following approximation was adopted:

amax ¼ 0:4n2 þ ð0:4 þ cÞn þ 0:2

ð17Þ

To assess the accuracy of Eq. (17), the approximate values of amax were divided by the theoretically exact values given in Table 4a. A summary of this assessment is provided in Table 6, which shows a maximum non-conservative error of 1% and a maximum conservative error of approximately 4%. Eq. (17) appears to provide a generally good approximation for amax. In many cases, the system of parallel members can be anchored at both ends (Fig. 7) and the engineer may want to take advantage of this by designing the bracing for both tension and compression forces. Given the symmetry of the buckling mode, the lateral deflections at the column ends result in Δ1 = Δn, Δ2 = Δn − 1, Δ3 = Δn − 2, … With the two centermost columns deflecting by the same amount, Δn / 2 = Δn / 2 + 1, there would be no relative movement between them and, hence, there would be no axial force in the brace tying these two columns together. As a result, a bifurcation analysis of a system of n parallel columns with two anchors of equal stiffness would be equivalent to an analysis of a system of n / 2 columns with only one anchor. This holds true regardless of whether the total number of columns is even or odd. With this in mind, Eq. (17) can then be modified to

amax ¼ 0:4N2 þ ð0:4 þ cÞN þ 0:2

ð18Þ

in which N = n / j, with n equal to the total number of columns in the system and j equal to the number of ends anchored, j = 1 or 2.

For a system of n parallel members, the equations presented in the Section 3 can be effectively used in the following proposed design procedure: a. Based on the number of ends anchored, j, compute N = n / j. b. Compute, assume, or estimate a target value for the brace stiffness ratio c. c. Compute the required ideal brace stiffness ks for a single member braced as in the parallel system under consideration (Section 2). d. Compute the ideal brace stiffness of the tie and the corresponding ideal brace stiffness of the anchorage, using: h i kn ¼ 0:4N2 þ ð0:4 þ cÞN þ 0:2 ∙ks klat ¼

ð19aÞ

kn c

ð19bÞ

e. Ensure that the actual stiffnesses for the tie bracing ktie and anchor bracing kanchor employed in the final design exceed the ideal values kn and klat obtained using Eqs. (19a) and (19b). This procedure requires one to commit to a value of the stiffness ratio c. This step can be avoided if either the actual tie-bracing stiffness ktie or the actual anchor-bracing stiffness kanchor is known beforehand, or can be closely approximated. Algebraic manipulation of Eqs. (19a) and (19b) results in the following equations, which alleviate the need for estimating or assuming a value of c by defining one of these ideal stiffness values in terms of the other: kn ¼

0:4N2 þ 0:4N þ 0:2 ∙ ks ks 1−N kanchor

klat ¼

ð20aÞ

N

k ∙ks s 2 1− 0:4N þ 0:4N þ 0:2 ktie

ð20bÞ

In this form, it is obvious that Eq. (20a) will reduce to Eq. (14) for the case when the anchor can be assumed to be rigid (kanchor ≫ ks), and Eq. (20b) will reduce to the “summation-P” concept when the tie bracing can be assumed rigid (ktie ≫ ks). It is also worth noting the significance of the requirement that the denominator of each equation must be positive. For Eq. (20a), this ensures that the stiffness of the anchorage must always exceed Nks, regardless of the stiffness of the tie-bracing. Similarly, for Eq. (20b), the stiffness of the tie bracing ktie must at least exceed the value given by Eq. (14), regardless of the stiffness of the anchor bracing. In other words, these requirements on the denominators can Table 6 Ratio of approximate to exact amax values, flexible anchor.

Table 5 Approximated scale factor amax, flexible anchor. c

Best-fit parabola, amax(n)

R2

0 0.1 1 10 100 500 1000

0.4053n2 0.4053n2 0.4052n2 0.3989n2 0.3571n2 0.3392n2 0.3363n2

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

+ + + + + + +

0.4053n + 0.1854 0.4868n + 0.2185 1.2315n + 0.6200 9.4568n + 4.7663 99.798n + 4.3514 500.31n + 1.3061 1000.4n + 0.7592

n

c=0

c = 0.1

c=1

c = 10

c = 100

c = 500

c = 1000

1 2 3 4 5 10 25 50 75 100

1.00 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99

1.00 1.00 1.00 0.99 0.99 0.99 0.99 0.99 0.99 0.99

1.00 1.01 1.01 1.02 1.02 1.01 1.00 1.00 0.99 0.99

1.00 1.00 1.01 1.01 1.02 1.03 1.04 1.03 1.03 1.02

1.00 1.00 1.00 1.00 1.00 1.01 1.01 1.02 1.03 1.04

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.01 1.01

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01

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269

Fig. 7. Model of a system of parallel columns with flexible anchors.

provide lower bounds or minimum values on the stiffness of the tie and anchor braces.

5. Results/discussion The proposed formulation and design procedures have been applied in four different types of example problems in order to validate and illustrate the use of the developed equations. The quality of the results is evaluated through comparison with those obtained from finite element analyses using MASTAN2 and the commercially available program Strand7 [21].

5.1. Example 1: system of parallel columns in load bearing wall This example is used to demonstrate the design procedure described in the Section 4. The problem, originally studied by Sputo and Beery [12], involves a system of multiple parallel cold-formed steel columns that represent the studs in a load bearing wall. Here, a system of n = 23 columns is used, with non-rigid lateral bracing anchoring the system at both ends of the wall (Fig. 8). The tie bracing is being conservatively modeled as a series of truss elements (without continuity) located at column mid-height. The columns are spaced at 610 mm (24 in.) on center, each with a length L = 2.4 m (96 in.) and braced at mid-height, i.e. at 1.2 m (48 in.) from the base support. Interior column section properties reflect the use of 250S137-54 studs, including an in-plane buckling moment of inertia of I = 33.299 mm4 (0.08 in.4).

For an assumed value of the stiffness ratio c = 100, the objective is to determine the ideal or minimum stiffnesses required for the flexible anchor columns klat located at each end of the system and the interconnecting straps or ties kn of the 23-member system. Equivalently, the objective can be considered to be the determination of the minimum cross-sectional area An of the straps and the minimum moment of inertia Ilat of the anchor columns. 5.1.1. Analytical technique Using the equations developed in this work and the design procedure outlined in Section 4, the following is determined for this example problem: a. With j = 2 anchored ends, N = n / j = 23 / 2 = 11.5; b. A bracing stiffness ratio of c = 100 is assumed; c. The ideal brace stiffness ks for a single column braced as in Fig. 8 is defined as the stiffness that results in Pcr being the elastic critical load at which the column buckles between the points of lateral restraint, i.e. the brace point and the support (Eq. (21)).

P cr ¼

π2 E I Lc 2

where Lc ¼ L

 2

ð21Þ

where E is the elastic modulus, I is the area moment of inertia, and L is the column length. Using this expression for Pcr, the ideal brace stiffness ks for a single column braced at mid-height is computed, as per Timoshenko and

Fig. 8. Model of a load-bearing wall including 23 parallel cold-formed steel columns.

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Gere [15]:

2 ks ¼

2P cr ¼ Lc

! π2  29; 500  0:08   96 = 2 kip N 2 ¼ 0:4212 ¼ 73:763  96 in: mm

ð22Þ

2

where Pcr is equal to 44.97 kN (10.1 kips) in this example, per Eq. (21). d. Given a bracing stiffness ratio of c = 100, the ideal brace stiffness of the tie is computed using Eq. (19a):

presentation, this factor of two has not been included in the calculations of the above example. In many cases, the details of the strapping and the corresponding connections can be defined up front in order to meet other design constraints. In this situation, the value of the tie stiffness ktie would be known, and the required ideal brace stiffness for the anchor columns would be calculated without the need to assume a value for the stiffness ratio c. As an illustration of this scenario, and using this same example problem, suppose that the tie stiffness ktie was known to be 5341.4 N/mm (30.5 kips/in.). The ideal anchor brace stiffness of the system could then be determined using Eq. (20b): 0

h i kip kn ¼ 0:4ð11:5Þ2 þ ð0:4 þ 100Þ11:5 þ 0:2 ∙ 0:4212 ¼ 508:68 in: N ¼ 89; 083:52 mm If the axial stiffness of the connections at the tie ends is assumed rigid, this value of kn can then be used to determine the associated minimum cross sectional area of the ties:
 EAtie ð29500ÞAtie 24 ≥kn ⟶Atie ≥508:68 ¼ ¼ 0:414 in:2 Ltie 2ð12Þ 29500 ¼ 267 mm2

ktie ¼

The ideal brace stiffness of the anchorage klat is similarly computed using Eq. (19b): klat ¼

kn 508:68 kip N ¼ ¼ 5:0868 ¼ 890:84 c 100 in: mm

The associated minimum moment of inertia of the anchor columns is then determined as: kanchor ¼

48 EIanchor

ðLanchor Þ3 ¼ 132:2 cm4

963 ≥klat ⟶Ianchor ≥5:087 48  29500

! ¼ 3:179 in:4

The bracing stiffness used in the actual design process is typically at least two times the ideal stiffness values, otherwise the corresponding strength demands on the bracing may be excessive. For clarity in

klat

B ¼@

1

C

0:4212 A∙ 0:4212 2 1− 0:4 11:5 þ 0:4ð11:5Þ þ 0:2 30:5 kip N ¼ 4175:0 ¼ 23:84 in: mm 11:5

As demonstrated above, this value could be then used to compute the minimum required moment of inertia of the anchor columns. 5.1.2. Computational results Using the elastic critical load feature in the finite element program MASTAN2, two-dimensional elastic eigenvalue analyses [20] were performed on a model of this system. Each pin-ended column member was modeled with 8 line (6 degree of freedom) elements. The properties reflected 250S137-54 steel studs with L = 2.4 m (96 in) and E = 203.000 MPa (29,500 ksi). Tie bracing was modeled as axial force (truss) elements located at mid-height. For a given tie-bracing stiffness, such as ktie = 89.084 N/mm as determined above, finite element analyses are repeated by varying and refining the amounts of anchorage stiffness kanchor until the critical buckling mode just changes to the onset of columns buckling about the intermediate brace points as if they were fixed points (Fig. 9). The stiffness at which this transition occurs is identified as the ideal brace stiffness for the anchors and is computationally obtained as kanchor = 891 N/mm, which is nearly identical to the klat reported above. It should be noted that the ideal bracing stiffness of the system can be represented by a wide range of pairs of complementing ideal tie- and anchorbracing stiffnesses [kn, klat]. In other words, and as demonstrated above, a tie-bracing stiffness of ktie = 5341.4 N/mm and an anchorage stiffness of kanchor = 4175.0 N/mm could also represent a pair of ideal

Fig. 9. MASTAN2 models for parallel column example; Deflected shapes (buckling modes), elastic critical load with ktie = kn and (a) kanchor b klat (b) kanchor ≥ klat.

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271

Fig. 10. One-half of the symmetric18K3 joist studied.

bracing stiffnesses for the given system. These same values were also obtained using this iterative computational FEA procedure. 5.2. Example 2: system of open web steel joists This example is used to demonstrate the accuracy of the approximate equations in determining the ideal brace stiffness for a system of parallel open web steel joists. Open web steel joists, such as K-series joists, are simply supported uniformly loaded truss-shaped members that are typically used to support floors or roof decks. The system studied here includes a number of parallel K-series (18K3) joists, each spanning 8.61 m (28.25 ft), as seen in Fig. 10. The top chord of each joist includes two back-to-back 38.1 × 38.1 × 3.18 mm (1.5 × 1.5 × 0.125 in.) angles, spaced at 15.875 mm (0.625 in.). The bottom chord is two back-to-back 38.1 × 38.1 × 2.77 mm (1.5 × 1.5 × 0.109 in.) angles, spaced at 15.875 mm (0.625 in). Web members are 15.875 mm (0.625 in) diameter round bar that run between the top and bottom chord double angles. Systems of n parallel joists were studied, with multiple values of n considered between 2 and 32. For each system, the spacing between parallel joists or tie-bracing lengths is 1.22 m (4 ft), and rigid lateral bracing (i.e. infinite stiffness) was assumed to be anchoring the system at one end, i.e. klat → ∞ and c = 0. The tie bracing was conservatively modeled as a series of interconnected truss elements, and each pair of parallel joists was braced with a total of six identical truss elements in locations as shown in Fig. 10. Two different loading cases were considered for each system, including uniformly distributed gravity and uplift loadings on the top chord, thereby resulting in compression in the top chord or bottom chord, respectively. Using computational analysis, the ideal brace stiffness ks for a single joist braced as shown in Fig. 10 was first obtained using an iterative process of varying the stiffness of the six tie braces used to provide stability to the joist. This iteration continued until the point was achieved at which the controlling out of plane buckling mode of the joist occurred at 96% of the load required to buckle the joist if the braces are assumed rigid. The 96% value was used on the basis of the shape of the associated

knuckle curve [22], which is generally described as a plot of system strength versus bracing stiffness. While the knuckle curve asymptotes to the horizontal line that represents the maximum resistance of the rigidly braced structure [23], the shape of the curve as it approaches the asymptote must be considered in defining the appropriate percentage of the theoretical maximum and thereby avoid unreasonably large brace stiffness requirements. In this example, the moderately gradual approach of the knuckle curve justified the use of 96% of theoretical value, a value which also resulted in the desired buckling mode of little to no out-of-plane movement of the chord at the brace points. Applying this method, the value of the ideal brace stiffness for a single joist was determined to be ks = 106.2 N/mm (0.6061 kips/in.) for gravity loading, and ks = 70.5 N/mm (0.4027 kips/in.) for uplift loading. Using this single joist value, the ideal brace stiffness for a system of n joists can now be determined using two different methods that will allow for a comparison of results, including (1) continued use of the lengthy iterative computational FEA-based procedure and (2) the previously derived approximate equations. As an illustration, the ideal brace stiffness for a system of n = 4 joists, modeled as shown in Fig. 11, is considered here. Computationally, the described iterations now include variations of the stiffness ktie of the 24 (i.e. 4 × 6) identical tie braces until the 96% value is achieved. This process results in the buckling modes shown in Fig. 11 and an ideal system brace stiffness of kn = 4 = 880.1 N/mm (5.026 kips/in.) for gravity loading, and kn = 4 = 584.7 N/mm (3.339 kips/in.) for uplift loading. If, instead and without the need to prepare extensive computational models, Eq. (19a) is applied, the kn = 4 value for gravity loading is simply computed as: h i N kip ¼ 4:971 kn¼4 ¼ 0:4ð4Þ2 þ ð0:4Þ4 þ 0:2 ∙106:2 ¼ 870:8 mm in: The percent error for this case (n = 4; gravity loading) is found to be 1.1%, indicating good accuracy of the proposed equation. Similarly, Eq. (19a) yields a value of kn = 4 = 578.1 N/mm (3.302 kips/in.) for uplift loading with the same percent error.

Fig. 11. Controlling buckling modes for case of n = 4 open web joists (Strand7 computational models); Gravity loading with (a) ktie b kn, (b) ktie ≥ kn; Uplift loading with (c) ktie b kn, (d) ktie ≥ kn.

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Table 7 Comparison of ideal brace stiffness ratios, kn/ks. Gravity loading

Uplift loading

# of joists, n

Eq. (19a)

FEA

% error

Eq. (19a)

FEA

% error

2 4 8 16 32

2.60 8.20 29.0 109.0 422.6

2.62 8.29 29.36 110.43 428.09

0.76 1.1 1.2 1.3 1.3

2.60 8.20 29.0 109.0 422.6

2.62 8.29 29.4 110.2 427.7

0.76 1.1 1.2 1.1 1.2

These results can also be compared through consideration of the ratio of kn to ks for the two different approaches (Table 7). The computational results were obtained using Strand7, and further verified using MASTAN2. It can be seen that the percent error between the FEA results and those estimated using the proposed formulation increases only slightly as the number of joists n increases, with error values ranging only from 0.76% to 1.3%.

three brace points and the supports. The corresponding elastic critical load Pcr is determined using Eq. (21), here with Lc = L/4, and the ideal brace stiffness ks for a single column braced in this configuration is given by Timoshenko and Gere [15]: 3:41 3:41P cr ¼ Lc kN ¼ 26:22 mm

ks ¼

! π 2  29; 500  44:1  2 264 = kip ¼ 149:7  4 264 in: 4

ð23Þ

in which Pcr equals to 12,889.5 kN (2897.7 kips). In considering the case of n = 12 compression members, as an illustration, the ideal stiffness of the tie bracing is computed using Eq. (19a): "

#
2
 12 12 þ 0:4 þ 0:2 ∙26:22 ¼ 17∙26:22 2 2
 kN kip ¼ 2; 545:2 ¼ 445:7 mm in:

kn¼12 ¼ 0:4

5.3. Example 3: discrete bracing of system of elastic and inelastic columns The following example is used to demonstrate the use of the proposed approximate equations to determine the bracing demands associated with both elastic and inelastic buckling of columns with multiple brace points. A system of parallel compression members is investigated (Fig. 12), each with multiple tie braces, and with rigid lateral bracing anchoring the system at both ends (i.e., c = 0 and j = 2). The columns are oriented such that the braces resist minor axis buckling of the columns (out-of-plane major axis buckling is assumed not to control), and both elastic and inelastic buckling are considered. The geometry of this system is modeled after one previously studied by Ales and Yura [24]. It is composed of n parallel W310 × 60 (W12 × 40) columns of A992 steel (σy = 345 MPa = 50 ksi; E = 203.395 MPa = 29.500 ksi). Systems with n equal to 3, 6, 9, and 12 columns are presented here. The tie bracing or girts are modeled as discrete truss elements, which are located at ¼, ½, and ¾ of the total column height of L = 6.7 m (22 ft). Column properties reflect the use of W310 × 60 (W12 × 40) sections, including area A = 7.6(103) mm2 (11.7 in.2), and in-plane buckling moment of inertia I = 18.4(106) mm4 (44.1 in.4). 5.3.1. Elastic buckling analysis The ideal brace stiffness ks for a single column braced as shown in Fig. 12 is defined as the minimum stiffness of the ties such that the column buckles between the points of lateral restraint, i.e. between the

5.3.2. Inelastic buckling analysis For the inelastic analysis, ks for a single column that is also braced as shown in Fig. 12 is now defined as the minimum tie stiffness required for the column to experience inelastic buckling between the points of lateral restraint (Eq. (24a)). The inelastic stiffness reduction factor τb employed in this example is from reference [25], although other expressions are also applicable. P cr ¼

π2 Et I ðLc Þ2

with Lc ¼ L

 4

and Et ¼ τb E

8 > > <

ð24aÞ

P cr ≤0:5 Py P cr N0:5 Py

1
 where τb ¼ P P cr > cr > 1− : 4 Py Py

with P y ¼ Aσ y

By first assuming (and then later confirming) that Pcr/Py exceeds 0.5, the above expression for τb is substituted into Eq. (24a), which can then be manipulated to provide an expression for an inelastic buckling load of ! ðLc Þ2 P y Py ¼ P cr ¼ 1− 4EIπ2 ¼ 2471 kN

L  2 1−

4

Py

!

4EIπ2

P y ¼ 555:5 kips ð24bÞ

As with the above elastic buckling study, this value can then be used per reference [15] to obtain the ideal brace stiffness for a single column ks ¼

3:41Pcr 3:41ð555:5Þ kip kN ¼ ¼ 28:7 ¼ 5:03  264 Lc in: mm 4

Table 8 Comparison of ideal brace stiffness ratios, kn/ks. Elastic buckling

Fig. 12. Model of n = 12 parallel W310 × 60 (W12 × 40) columns.

Inelastic buckling

# of columns, n

Eq. (19a)

FEA

% error

Eq. (19a)

FEA

% error

3 6 9 12

1.70 5.00 10.10 17.00

1.71 5.05 10.21 17.20

0.58 0.99 1.2 1.2

1.70 5.00 10.10 17.00

1.71 5.05 10.22 17.21

0.58 0.99 1.2 1.2

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(a)

273

(b)

Fig. 13. Controlling buckling modes for case of n = 12, (a) ktie b kn, (b) ktie ≥ kn.

Again considering the case of n = 12 compression members as an illustration, the ideal stiffness of the tie bracing is computed using Eq. (19a): "

kn¼12

#
2
 12 12 ¼ 0:4 þ 0:4 þ 0:2 ∙5:03 ¼ 17∙5:03 2 2
 kN kip ¼ 487:9 ¼ 85:4 mm in:

5.3.3. Comparison with computational results In Table 8, the values obtained for kn using the proposed equation are compared with those obtained from iterative finite element analyses (using Strand7, and further verified with MASTAN2; Fig. 13). The percentage error resulting from these comparisons ranged from 0.58% for n = 3 to 1.2% for n = 12. In addition to validating that Eq. (19a) can work equally well for elastic and inelastic buckling, this example also illustrates that the ideal brace stiffness for compression members failing in inelastic buckling is significantly less than the stiffness required for elastic buckling. For example, the ideal brace stiffness of the ties for n = 12 columns is reduced from kn = 12 = 445.7 kN/mm for elastic buckling to k n = 12 = 85.4 kN/mm for inelastic buckling, which represents an approximate 80% reduction in required stiffness. 5.4. Example 4: single story industrial building This final example is representative of a typical one-story industrial building in which a single portal frame (or braced frame) provides lateral stability to a large number of bays with leaning columns. The system shown in Fig. 14 is a variation of one used extensively to develop AISC's direct analysis method [26], which has been slightly modified here to illustrate the key features of the proposed equation. The portal frame providing lateral stability is composed of two W250 × 73 (W10 × 49) columns rigidly connected to a W690 × 125 (W27 × 84) beam. A varying number of W310 × 67 (W12 × 45) parallel lean-on columns gain their lateral stability from a system of braces, located at the tops (height L from their base), that is attached to the portal frame (Fig. 14). Systems with n = 1 to 6 leaning columns have been

considered. Dimensions include column heights of 5.49 m (18 ft) and column spacing of 10.67 m (35 ft). All members are A992 steel and are oriented for in-plane major-axis bending. Using Eq. (24b), the inelastic flexural buckling load of each leaning column is determined to be 2690 kN (604.8 kips). With a length of L = 5.49 m (18 ft), Eq. (6) provides the ideal brace stiffness for a single column of ks = 0.49 kN/mm (2.80 kips/in.). The lateral stiffness of the portal frame or anchor system can be shown to be kanchor = 3.01 kN/mm (17.17 kips/in.). For the above system with one leaning column, the ideal brace stiffness ks is represented by the net stiffness of two springs in series as shown in Fig. 6 with n = 1, which include the tie brace and the anchor system (portal frame). By simultaneously considering Eq. (6) and the left side of Eq. (15), the following equality is obtained: ks ¼

P cr 1  ¼
1 1 L þ kanchor kn¼1

ð25Þ

Substituting the inelastic buckling load of the column Pcr and the known lateral stiffness of the portal frame kanchor, Eq. (25) is used to determine the required stiffness of the tie brace for the case of n = 1, i.e. kn = 1 = 0.5854 kN/mm (3.346 kips/in.). It is important to note that Eq. (20a) would provide the same value for kn = 1. In determining the required tie-brace stiffness for a system of multiple (n N 1) leaning columns, however, Eq. (25) is no longer applicable, and Eq. (20a) can be used instead. As an illustration, Eq. (20a) for the case of n = 4 columns provides an estimate of kn = 4 = 11.56 kN/mm (66.02 kips/in.) or nearly 20 times the tie-bracing stiffness required for a single column. Furthermore, employing an iterative computational analysis (Fig. 15), as described in Section 5.2, an exact value of kn = 4 = 10.94 kN/mm (62.49 kips/in.) is obtained; with the approximation given by Eq. (20a) resulting in a percent error of 5.7%. If the stiffness of the lateral frame was increased by an arbitrary factor of say 5 (kanchor = 5 × 3.01 = 15.05 kN/mm), which at a minimum would be more representative of a braced frame than the given portal frame, Eq. (20a) would yield kn = 4 = 4.621 kN/mm (26.40 kips/in.). On the other hand, if the anchor system was considered to be rigid, Eq. (14) would yield kn = 4 = 4.017 kN/mm (22.958 kips/in.). In comparison with the original portal frame, it is clear that the tie-bracing

Fig. 14. Model of n = 4 leaning columns, with tie bracing at their tops and system lateral bracing provided by a single portal frame.

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Fig. 15. Controlling buckling modes for case of n = 4, (a) ktie b kn, (b) ktie ≥ kn.

stiffness demands are significantly reduced as the stiffness of the lateral (anchor) system increases. Eq. (20a) can also be used to determine the maximum number of leaning columns that a given lateral system can support. For example, consider the portal frame shown in Fig. 14. Given that the denominator of Eq. (20a) must be positive, and ks = 0.49 kN/mm (2.80 kips/in.) and kanchor = klat = 3.01 kN/mm (17.17 kips/in.), it can be easily shown that n must be less than or equal to 6.13. In other words, this lateral system can provide stability to a maximum of 6 leaning columns, regardless of how stiff the tie bracing is. Further, Eq. (20a) would show that for n = 6, the stiffness of the tie bracing must equal or exceed kn = 6 = 4.00 kN/mm (22.80 kips/in.), which is a conservative estimate of what the more exact iterative finite element procedure, kn = 6 = 3.44 kN/mm (19.63 kips/in.). Using this same approach to investigate the lateral system with stiffness increased by a factor of 5, it can be shown that stability can be assured for a maximum of 30 columns, but with an approximated required tie-bracing stiffness of kn = 30 = 8463 kN/mm (48,360 kips/in.); with this value being a moderately conservative estimate of that obtained from iterative finite element analyses, kn = 30 = 7153 kN/mm (40,875 kips/in.). It is important to note that an increase in the lateral system's stiffness does increase the number of columns it can support, but at the expense of significantly higher tie-bracing stiffness demands. Eq. (19a) could also be used in a reverse fashion to solve for the maximum number of leaning columns n that can be supported when given the system lateral stiffness kanchor and the tie-bracing stiffness k tie. For example, with given values of k anchor = k lat = 15.05 kN/mm (85.85 kips/in.) and k tie = k n = 210 kN/mm (1200 kips/in.) resulting in c = 13.98, the minimum positive root of Eq. (19a) yields n = 19.36; a maximum of 19 columns can be supported by this specific combination of lateral-system and tie-bracing stiffnesses. This maximum value of n = 19 leaning columns was also confirmed using iterative finite element analyses. It should also be noted that in this discussion of the maximum number of leaning

columns supported, these values would be reduced when the portal frame itself is subject to gravity loading. Returning to the system originally described in Fig. 14, Table 9 provides a summary of results obtained for n = 2 to 6 parallel leaning columns. In addition to further validating the use of the proposed approximate equations, the data presented in Table 9 clearly indicates that as the lateral system becomes stiffer, Eq. (20a) becomes more accurate and the stiffness demands on the tie bracing are significantly reduced. 6. Conclusions The bracing used in order to achieve the full buckling strength of a compression member must be designed to have adequate strength and stiffness. An integral part of this design process is the computation of the minimum or ideal brace stiffness. For systems composed of multiple parallel compression members or sub-assemblages, this ideal stiffness is a nonlinear function of both the anchor-bracing stiffness and the stiffness of the components (ties) interconnecting the compression members. A mathematically exact formulation and a simple approximate expression for determining such an ideal stiffness has been developed and presented herein. Validation of this approximate expression included the comparison of its results with those from an iterative procedure employing linear buckling (eigenvalue) analyses, using finite element software, for a variety of examples. This expression performed well in all cases, achieving good to excellent agreement with the computational results, suggesting it sufficiently determines the minimum bracing stiffness for systems of multiple parallel members with single and multiple brace points. In addition to providing a basis for this validation, the four example problems demonstrate the robustness of the expression for a wide range of applications requiring consideration of bracing stiffness, and illustrate several key concepts related to bracing stiffness.

Table 9 Comparison of ideal brace stiffness ratios, kn/kn = 1 at several values of lateral stiffness klat. kanchor = 15.05 kN/mm kn = 1 = 0.506 kN/mm

kanchor = rigid kn=1 = 0.489 kN/mm

# of columns

kanchor = 3.01 kN/mm kn = 1 = 0.585 kN/mm

n

Eq. (20a)

FEA

% error

Eq. (20a)

FEA

% error

Eq. (20a)

FEA

% error

2 3 4 5 6

3.23 8.19 19.7 55.3 660.2

3.20 7.95 18.7 50.9 586.7

−0.95 −3.1 −5.7 −8.7 −12.5

2.69 5.36 9.12 14.10 20.45

2.70 5.37 9.10 13.99 20.18

0.37 0.16 −0.26 −0.78 −1.3

2.60 5.00 8.20 12.20 17.00

2.62 5.05 8.29 12.34 17.21

0.73 1.0 1.1 1.2 1.2

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It is relevant to note that there are several additional factors related to bracing requirements that this paper and the supporting research did not explore, including but not limited to the impact of initial imperfections of the compression members being braced and the influence of the relative angle between the bracing and the compression members (which in all presented examples is π/2 rad). In general, these specific factors and other destabilizing effects would tend to increase the demands on the strength and stiffness of the bracing. Given that this paper provides a method for obtaining the bracing stiffness for multiple parallel members that is based on proportionally increasing the bracing stiffness requirements of a single compression member, the authors are confident that the proportionality factor presented would still apply as long as the bracing stiffness for the single compression member accounted for all significant destabilizing effects, such as member imperfections, bracing orientation, and inelasticity. References [1] G. Winter, Lateral bracing of columns and beams, Trans. ASCE 125 (Part 1) (1960) 809–825. [2] N.S. Trahair, D.A. Nethercot, Bracing requirements in thin-walled structures, Dev. Thin-Walled Struct. 2 (1984) 93–130. [3] R.H. Plaut, Lateral bracing forces in columns with two unequal spans, J. Struct. Eng. ASCE 119 (10) (1993) 2896–2913. [4] J.A. Yura, Bracing for stability, state-of-the-art, Proc. Struct. Congr. XIII, ASCE, Boston 1995, pp. 88–103. [5] R.D. Ziemian (Ed.), Guide to Stability Design Criteria for Metal Structures, sixth ed. John Wiley & Sons, Inc., Hoboken, NJ, 2010. [6] G.S. Tong, S.F. Chen, Design forces of horizontal inter-column braces, J. Constr. Steel Res. 7 (5) (1987) 363–370. [7] S.F. Chen, G.S. Tong, Design for stability: correct use of braces, Steel Struct. J. Singapore Struct. Steel Soc. 5 (1) (1994) 15–23. [8] Y. Zhang, J. Zhao, W. Zhang, Parametric studies on inter-column brace forces, Adv. Struct. Eng. 11 (3) (2008) 293–303. [9] I.C. Medland, A basis for the design of column bracing, Struct. Eng. 55 (1977) 301–307.

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