# PERMANENT BRACING DESIGN FOR MPC WOOD ROOF TRUSS WEBS AND CHORDS.

CATHERINE R. UNDERWOOD [*]FRANK E. WOESTE [*]J. DANIEL DOLAN [*]

ABSTRACT

Permanent bracing of metal-plate-connected trusses is required to stabilize specific members of each truss throughout the life of the roof structure. The objectives of this research were to: 1) determine the required net lateral restraining force to brace multiple webs or chords in a row braced by one or more continuous lateral braces (CLBs); and 2) develop a methodology for permanent bracing design using a combination of lateral and diagonal braces. Three system analogs used to model multiple truss chords braced by n-CLBs and one or two diagonal braces were analyzed with a finite element analysis program. Single member analogs were analyzed that represented web members braced by one and two CLBs and chord members braced by n-CLBs. For analysis and design purposes, a ratio R was defined as the net lateral restraining force per web or chord divided by the axial compressive load in the web or chord. For both 2 by 4 and 2 by 6 webs braced with one CLB, the R-value was 2.3 percent for all web lengths studied, an d for webs braced with two CLBs, the R-value was 2.8 percent for all web lengths studied. Calculated R-values for truss chords braced by n-CLBs, assumed to be spaced 2 feet on center, ranged from 2.3 to 3.1 percent. A design procedure was offered for determining the net lateral restraining force required for bracing multiple webs or chords in a row based on the results of the single member analogs studied.

Consideration of permanent and temporary bracing of metal-plate-connected (MPC) wood trusses is critical for safety during erection and for reliable performance of a roof structure in service. Temporary bracing is used to position and stabilize trusses until permanent bracing or other building components can be installed. Permanent bracing, the subject of this paper, is required to stabilize specific members of each truss throughout the life of the roof structure. Based on truss design assumptions, various chords and webs require lateral support. The chords that require sheathing or the members that require lateral support at a specified interval are indicated on the truss design drawing [5]. Bracing, whether it is temporary or permanent, is required to help prevent the trusses from deflecting laterally and potentially causing the trusses to topple over or collapse [3,15, 16, 19].

Permanent truss bracing can include several different components, but are typically designed using one of the following options: 1) continuous lateral braces (CLBs) at required brace locations in conjunction with diagonals; or 2) properly nailed sheathing, either oriented strandboard (OSB) or plywood. If carefully designed, some elements of temporary bracing [11] can serve dual roles and become permanent bracing also.

LITERATURE REVIEW

The building designer is responsible for permanent bracing design of MPG wood trusses [12]. The Commentary for Permanent Bracing of Metal Plate Connected Wood Trusses [5] provides typical bracing design strategies. The Commentary does not provide a specific design procedure, but it depicts the various locations in roof truss constructions that typically require a permanent bracing design and installation. The Commentary addresses common truss configurations and provided various permanent bracing options. Although some permanent lateral bracing information may be shown on truss drawings, the building designer is responsible for the complete bracing design and details needed by the contractor to properly install the permanent bracing system. For the case where a roof truss has a long span or a high pitch and a piggyback truss is needed for shipping purposes, the building designer must provide a permanent bracing design and specifications for the supporting trusses. The Commentary provides several alternatives for bracing the flat compression chord of the supporting truss that includes the use of CLBs in conjunction with diagonal braces. No design guidelines are given for determining the necessary capacity of the diagonals needed to stabilize the CLBs.

Currently, the 2 percent Rule is the most common and accepted design practice used for designing permanent bracing for a compression web that utilizes one CLB with diagonals spaced at some interval. The 2 percent Rule is a strength model based on a percentage of the axial load in a column needed to stabilize the column [17]. Designers primarily use strength models during design due to the simplicity of the calculations. In the derivation of the 2 percent Rule, the column was assumed to be pinned at each end and at the center brace location. The column is assumed to be 1 inch out of plumb for an assumed story height of 100 inches [9]. A compression web or chord braced at the center of its span will have an initial column slope of 1/100 above and below the brace [9,17]. A force balance for a freebody diagram at the brace and chord connection is the basis for the 2 percent Rule [6].

Waltz [17] studied the design requirements for bracing a 2 by 4 compression web member at one brace support point at the center of the web. Waltz tested Select Structural and Standard Douglas Fir-Larch (DFL) specimens axially loaded and braced by his computer-controlled testing apparatus to determine the required brace force. Using finite element (FE) analysis, Waltz estimated the stiffness of the support provided by a lateral and diagonal bracing system for a number of braced 2 by 4 truss webs in a row. His research objective was to determine if one of four existing models could be used to estimate the required brace strength and stiffness. The four existing bracing models considered included Plaut's method [7,80], Winter's method [18], the 2 percent Rule [9,10,11], and Tsien's method [13]. Waltz [17] concluded that either Plaut's or Winter's method could be used for web bracing design.

OBJECTIVES

For permanent bracing design purposes, a simple "hand calculation" method was desired to approximate the brace force required to stabilize numerous compression webs braced by one or two CLBs or chords braced by CLBs at multiple locations. The objectives of this research were to: 1) determine the required net lateral restraining force to brace multiple webs or multiple chords in a row braced by one or more CLBs; and 2) to develop a methodology for permanent bracing design using a combination of lateral and diagonal braces. In addition to the primary objectives of this research, the impact of lumber specific gravity, modulus of elasticity, and chord lumber size on required bracing forces were investigated.

ANALYSIS TOOLS AND ASSUMPTIONS

SAP2000 [2] is a finite element analysis program used for analyzing different types of problems, including structural analysis problems. In this research, SAP2000 was used with the assumption that beam elements were linear elastic. A minimum of three beam elements were used to represent the compression members.

To create the structural analogs used in this research, the following assumptions were made:

* The CLBs and diagonal braces were dry 2 by 4 nominal Stud grade sprucepine-fir (SPF) lumber;

* The ends of the braced webs or chords were pin connected; therefore, pin or roller reactions were used to support the member ends;

* 2-16d Common nails were used for all wood-to-wood connections;

* Truss chords in a row and multiple CLBs used for bracing the chords were spaced 2 feet on center;

* Truss webs were braced with one or two CLBs at the center or third points of the length, respectively.

The load-displacement relationship for a single nail connection made with SPF lumber was determined by using Equation [1] from Mack [4]:

[F.sub.n] = [52d.sup.1.75][k.sub.s] (3.20[omega] + 0.68) [(1 - [e.sup.-75[omega]]).sup.0.7] [1]

where [F.sub.n] = the load per nail (lb.) applied to a single shear nail joint; [omega] = the slip between the two members of the nail joint (in.); d = the nail diameter (in.); [k.sub.s] = the species constant from Mack [4].

Since Mack's [4] paper did not provide a species factor for SPF, a linear regression was performed to determine the equation suitable for that particular species using available data. Equation [2] gives the variables and results from a linear regression of 10 pairs of [k.sub.s] and specific gravity (SG) data reported in Mack [4]:

[k.sub.s] = 265 x SG + 3.5 [2]

where [k.sub.s] = the species factor, and for use in Equation [1]; SG = specific gravity for the species.

From Equation [2], [k.sub.s] equals 115 for SPF that has a published SG value of 0.42 [1]. The right-hand side of Equation [1] must be doubled because the stiffness of a two-nail joint is assumed to be twice as stiff as a single-nail joint. And finally, substituting d equals 0.162 inches for a 16d Common nail into Equation [1] and by applying a factor of 1.25 for dry material as referenced in Mack, Equation [3] results:

[F.sub.2n] = 618 x (3.20 [omega] + 0.68) [(1 - [e.sup.-75[omega]]).sup.0.7] [3] where [F.sub.2n] = the load applied to a 2-16d Common nail joint (lb.); [omega] = the joint slip between the SPF members (in.).

Equation [3] was implemented in the bracing analysis by using the secant modulus at different joint load levels. To find the secant modulus, a slip was calculated using SAP2000 and then the corresponding joint force was calculated using Equation [3]. The joint force g(x), where x is slip, was determined by a line drawn from the origin to the point (x, g(x)). The slope of the line, called the secant modulus, was the stiffness for the specified joint force and slip.

A linear spring stiffness was estimated for each joint of the structural analog, and then the structural analog was re-analyzed. Calculated spring forces were then compared to the specific force and displacement used to input the linear spring (secant modulus) constants. A new stiffness value for each spring was determined based on the new deflections and entered into SAP2000 [2]. The structure was analyzed again. This procedure was repeated until the force in the springs matched the assumed force and displacement used to calculate the secant modulus spring stiffness within a tolerance of 1 percent. This iterative procedure was necessary to accommodate a nonlinear spring (nail connection) in the SAP analysis of the braced system.

For the purpose of discussion and comparison to the 2 percent Rule, the net lateral restraining force from the web and chord analyses were divided by the axial compression load in the web or chord, respectively. This ratio will be referred to as R and is defined by Equation [4]:

R = Net lateral restraining force (lb.)/Axial load level in web or chord (lb.) [4]

where R = the variable of interest for both webs and chords.

WEBS BRACED WITH ONE CLB

The structural analog used to analyze a multiple number of webs in a row (j) braced with one CLB is depicted in Figure 1. To simplify the structural analog for the case of multiple truss webs in a row, the diagonal brace was neglected and a single truss web was analyzed. The response of a multiple web case encountered in actual construction can be obtained by multiplying j times the response from the single web case. The structural analog was represented in SAP2000 [2] as depicted in Figure 2, where the length of the web was varied from 3 to 12 feet. The lumber was assumed to be 2 by 4 Stud grade SPF with a modulus of elasticity (MOE), of 1,200,000 psi. The load applied to the web (P) varied from 10 to 100 percent of the allowable load based on the assumption that the webs are pin-connected to the chords and that effective web buckling length is 80 percent of the actual web length [1,12].

A second analog was created to determine the effects of web size, if any, on the required bracing force. The lumber was assumed to be 2 by 6 nominal Stud grade SPF with an MOE of 1,200,000 psi. The allowable load (P) applied to the web was recalculated based on the lumber size and the procedures presented in the NDS for column design [1,12].

A third structural analog was created for the one web braced by one CLB case to determine if lumber species (as characterized by MOE and SG) affected the required brace force. Using the analog shown in Figure 2, 2 by 4 nominal No. 2 DFL was assumed for the web and CLB. DFL has a 17 percent higher MOE value than SPF. The published SG value for DFL is 0.50 versus 0.42 for SPF. DFL was chosen as the species to compare to SPF because the nail slip data were Figure 2. -- Structural analog representing one web braced by one CLB. The nailed connection is represented by a spring and an applied load (P) is a compression force determined using design equations given in the NDS 97 [1].

available for a DFL joint and because of the higher SG value.

ANSI/TPI 1-1995 [12] installation limits were assumed for all web lengths studied. All web members were assumed to have equal initial curvature in the same direction (for example, all webs bowed left). The assumption for initial curvature of all webs is conservative because in reality truss webs would have initial deflections in both directions (bowed right and bowed left). Initial curvature in both directions (with a CLB installed) would help provide support to the overall structure. Web members were assumed to have an initial maximum deflection of L/200 and have initial curvature matching a half sine wave defined by Equation [5]:

[[delta].sub.i] = L/200 sin ([pi] X x/L} [5]

where [[delta].sub.i] = assumed initial deflection of the truss webs; L = length of the compression chord, and x = distance from the member end in inches; [pi] is in radians.

A deflected web member, by nature, is a smooth curve as opposed to a series of straight lines. Therefore, the half sine configuration was assumed and used in all web and chord analyses. A summary of the cases studied for one web braced by one CLB is given in Table 1.

RESULTS (WEB WITH ONE CLB)

R-values for the case of one web braced by one CLB were 0.023 (2.3%) for all web lengths and load levels (ranged from 10% to 100% of the allowable compression for the assumed lumber grade). R-values were not affected by lumber species (higher MOE and SG) or by using a 2 by 6 versus 2 by 4 web (14). The difference in the 2 percent Rule versus 2.3 percent results from this study, is due to the fact that a flatwise 2 by 4 is very flexible, and thus not dramatically affected by member continuity. The computer analog constructed for this research did not have a pin joint at the center of the web as is assumed for the derivation of the 2 percent Rule.

WEBS BRACED WITH TWO CLBs

For the case of one web braced by two CLBs shown in Figure 3, the structural analog included the same modeling assumptions as the structural analogs representing one web braced by one CLB. The structural analog was represented in SAP2000 [2] as depicted in Figure 4, where the length of the web varied from Figure 4. -- Structural analog representing one web braced by two CLBs. The nailed connections are represented by springs and an applied load (P) is a compression force determined using design equations provided in the NDS [1].

5 to 12 feet. The lumber was assumed to be 2 by 4 nominal Stud grade SPF with an MOE of 1,200,000 psi. Table 2 contains a summary of the cases studied for one web braced by two CLBs; the study objectives and analysis methods for each case were identical to the case of one web braced by one CLB.

RESULTS (WEB WITH TWO CLBs)

This case, consisting of one web and two CLBs, produced an R-value of particular interest. In the past, one option for design purposes was to assume the bracing force was equal to 2 percent of the applied load, times the number of CLB connections per web, which yields 4 percent of the axial load as the required bracing force per web. The R-value was determined to be 0.028 (2.8%) for all lengths and axial load levels studied. The 2.8 percent R-value is substantially less than the 4 percent calculated by the assumption that the brace force increases in proportion to the number of CLBs. The R-values were not affected by increasing the web lumber MOE and SG values or increasing the web size to a 2 by 6 (14).

CHORDS BRACED BY N-CLBs

Figure 5 is a line representation of an actual set of j truss chords braced by n-CLBs and one diagonal brace. In this research, the cases of one and two diagonal braces were investigated. A total of 10 structural analogs were developed based on the varying lengths of lumber. The diagonal braces were neglected in the first phase of the investigation. Figure 6 depicts the structural analog models as they were analyzed using SAP2000 [2]. Structural analogs were constructed using the same boundary conditions and assumptions for connections between each CLB and chord as were made for the cases of one web with one and two CLBs.

The center panel (or two panels) of a symmetric truss with a flat-top chord under symmetric loading will have the maximum stress interaction according to the NDS [1] when the truss panel lengths are equal. Equation [6] gives the stress interaction criterion for a chord subjected to bending and compression load.

[([f.sub.c]/[F'.sub.c]).sup.2] + ([f.sub.b]/[F'.sub.b](1 - [F.sub.cE])) [less than or equal to] 1 [6]

where [f.sub.c] = actual compression stress parallel to grain (psi); [F'.sub.c] = allowable compression design value parallel to grain (psi); [f.sub.b] = actual bending stress (psi); [F'.sub.b] = allowable bending design value (psi); [F.sub.cE] = critical buckling design value for a compression member for the applicable [l.sub.e]/d ratio (psi).

From a permanent bracing designer standpoint, one needs to determine the maximum axial force in all the panels of the chord. When an unbraced truss chord is assumed to be continuous in the structural analysis, bending moments will exist in all panels. The amount of bending moments will vary in magnitude from one design to the next. A conservative assumption with respect to permanent bracing design is that the bending moment is zero in all panels. Since the bracing is designed to resist a percentage of the axial load in the web, the higher axial load due to this assumption results in a higher design load in the brace. If the stress interaction is at the maximum equal to 1.0, Equation [6] reduces to Equation [7]:

[f.sub.c] = [F'.sub.c] [7]

Equation [7] applies to the center panel (or two symmetrical panels). It is conservative because assuming zero bending moment allows for the maximum applied axial compression load to be present in the assumed chord. The design compression load (C) in the center panel (or two symmetrical panels) is therefore given by Equation [8]:

C = A X [F'.sub.c] [8]

where A = chord area ([in.sup.2); [F'.sub.c] = allowable compression design value parallel to grain (psi).

Axial load in the outer panels under the symmetry assumptions stated previously will be lower than the axial load in the center panels. A conservative assumption for permanent bracing design is to assume all panels have the same axial load equal to the maximum value of the center panel as determined from Equation [8].

The allowable compression parallel-to-grain design value ([F'.sub.c]) was calculated using NDS (1) procedures for column design. Chords can buckle about both axes depending on the [l.sub.e]/d (effective column length-depth ratio) of each axis. When CLBs are installed at 24 inches on center, the effective column length-depth ratio about the weak axis was determined using Equation [9] as follows:

le/d = Ke X la/d = 1.0 X 24"/1.5 = 16 [9]

If the strong axis [l.sub.e]/d is greater than 16, the truss designer uses the larger [l.sub.e]/d. For example, the strong axis [l.sub.e]/d controls the design of a 2 by 4 truss panel 8 feet long. If the strong axis [l.sub.e]/d is less than 16, 16 is used. A situation such as this occurs when determining [l.sub.e]/d for a 2 by 12 member that is 10 feet in length. For permanent bracing design, it is therefore conservative to assume [l.sub.e]/d equals 16 and this assumption will always yield the maximum possible axial load in the top chord [14].

The final assumptions used in creating the structural analogs included the lumber type, chord lengths, and the duration of load factor. Snow load plus dead load was assumed and thus a duration of load factor of 1.15 was used. The lumber was assumed to be No. 2 Southern Pine for the truss chords and Stud grade SPF for the CLBs and diagonal braces. The truss chord size was varied from 2 by 4 to 2 by 12. The truss chord length was varied from 4 to 40 feet by increments of 4 feet, but also included a length of 6 feet. Allowable loads were determined based on the size of the members and the aforementioned [l.sub.e]/d value and the NDS (1) design equations for column design.

The assumed initial deflected shape of the chords was determined from Equation [5] and the same assumptions as for the cases of one web braced by one and two CLBs. If the length of the chord exceeded 400 inches, Equation [10] was used to meet the installation guidelines provided in ANSI/TPI 1-1995 [12]:

[[delta].sub.i] = 2 X sin ([pi] X x/L) [10]

ANSI/TPI 1-1995 (12) states that the maximum initial deflection allowed in a truss chord is the lesser of L/200 or 2 inches. In cases where compression chord length (L) is greater than 400 inches, the 2-inch maximum allowance was observed. Table 3 summarizes the truss chords modeled for the investigation of j truss chords braced by n-CLBs.

The same analysis and procedures as were used for the case of a braced web were used to analyze chords with n-CLBs, except the chords were assumed to be No. 2 grade Southern Pine lumber. For calculations of the nail slip of the 2-16d Common nail connections, it was assumed that both the chord and the CLB were SPF because nail slip data were not available for a joint having mixed species. This assumption will result in lower stiffness values for the connection, which is conservative.

RESULTS (CHORDS BRACED BY N-CLBs)

The chord sizes and spans studied for this case are given in Table 3. The CLBs for all cases were assumed to be 2 feet on center. Chord size did not affect the R-value for assumed chord lengths of 4 to 40 feet.

For the 2 by 4 chord size, only spans up to 24 feet were studied. As reported in Table 4 for 2 by 4 truss chords, all R-values for the 4-foot 2 by 4 No. 2 Southern Pine chord were equal to 0.023 (2.3%) and were the same as the R-value determined for the case of an SPF web braced with one CLB. The R-value was the same because the same number of bracing locations were present for both cases (one at the center) and it was determined that member length did not affect the R-value for webs. The results in Table 4 for the 2 by 4 chord size are similar to the results for 2 by 6 through 2 by 12, thus they are the only results presented in this paper [14].

R-values (all equal to 0.028 or 2.8%) for the 6-foot Southern Pine chords (two CLB bracing locations) were the same as the R-values for the case of a 2 by 4 SPF web braced with two CLBs. Again, the results were the same due to the bracing locations being the same (at the 1/3 points). A-values for the 8-foot chords (all equal to 2.8%) were the same as for the 6-foot chords using two significant figures. R-values for chords between 12 feet and 32 feet had a peak value of 0.031 (3.1%) [14]. R-values for all lumber sizes (2 by 4 to 2 by 12) for 36- and 40-foot chords were less than R-values for the shorter lengths. R was 0.029 (2.9%) for the 36-foot Southern Pine chord with n-CLBs spaced 24 inches on center, and R was independent of lumber size. R was equal to 0.026 (2.6%) for the 40-foot Southern pine truss chord with n-CLBs spaced 24 inches on center, and R was independent of lumber size. The values for R for chord lengths (L) greater than 400 inches were different due to the maximum initial member deflection o f 2 inches [14].

In addition, the chord load level as a percent of [F'.sub.c] did not affect R for any size or length. The SAP2000 analysis was based on linear elastic beam elements with nonlinear springs representing the nail connections, and thus one would expect the system to behave in a non-linear manner. However, the nail connections at low load levels are apparently so stiff that the calculated value of R is not substantially affected by the load level [14].

SYSTEM ANALOG: EFFECT OF INCLUDING DIAGONALS WITH MULTIPLE CHORDS

"System" structural analogs were composed of j chords, n-CLBs, and diagonal braces. The first system structural analog analyzed with SAP2000 [2] represented five 8-foot roof truss chords spaced 24 inches on center, three continuous lateral braces (CLBs) spaced 24 inches on center, and one diagonal (Fig. 7). For this case and the remaining two cases, the lumber used in the construction of the structural analog consisted of 2 by 4 Stud grade SPF, with an MOE of 1,200,000 psi for the CLBs, and 2 by 4 nominal No. 2 Southern Pine, with an MOE of 1,600,000 psi for the truss chords. The truss chords were assumed to be columns with pin supports at one end and roller reactions supports at the other [14].

An axial chord load was applied to all chords in the analysis. The maximum allowable design load (A x [F'.sub.c]) was determined to be 6,842 pounds based on NDS procedures for column design [1]. However, 50 percent of the allowable compressive load based on an [l.sub.e]/d ratio of 16 is a typical load level in a wood truss chord. For the analyses of the required bracing forces, the chord axial load level was increased from 10 to 50 percent of [F'.sub.c]. Maximum allowable axial load was based on the grade, species combination, size of lumber, and the duration of load factor of 1.15 for snow plus dead loading.

The second system structural analog analyzed represented six 20-foot roof trusses spaced 24 inches on center, nine continuous lateral braces (CLBs) spaced 24 inches on center, and two diagonals in a V shape with an angle of 45 degrees (Fig. 8). The truss chords were assumed to be columns with roller supports, free to translate in the vertical or Z-direction, on both ends. Roller supports free to translate in the horizontal or X-direction were used on the chords where the middle CLB crossed the chords to stabilize the structure, but still allowed the chords to deflect.

The third system structural analog analyzed represented 11 20-foot roof truss chords spaced 24 inches on center, 9 continuous lateral braces (CLBs) spaced 24 inches on center, and 2 diagonals in a V shape with an angle of 45 degrees (Fig. 9). Truss chords were assumed to be columns with roller reactions as described for the previous study case.

RESULTS (OF SYSTEM MODELS)

To calculate the required net lateral restraining force (NLRF) using the SAP2000 analysis results for multiple truss systems braced by n-CLBs and one or two diagonals, the X-components of the joint force between each diagonal and truss chord (tabulated in Table 5) were summed taking into account the direction of the force. For the case of five 8-foot trusses braced by three CLBs and one diagonal brace, the single member analog estimate of the required net lateral bracing forces was approximately 5 to 6 percent greater than the estimate of the required net lateral bracing forces predicted by the system analog analysis. For the case of six 20-foot trusses braced by nine CLBs and two diagonal braces, the single member analog estimate of the required NLRF was 2 percent or more greater than the estimate of the required NLRF predicted by the system analog analysis. For the case of 11 20-foot trusses braced by 9 CLBs and 2 diagonal braces, the single member analog estimate of the system required NLRF was 5 percent or more greater than the bracing force from the system analog analysis (14). However, it should be noted that the 40 and 50 percent load level cases theoretically caused a 2-16d Common nail connection to fail based on the 0.1-inch slip criterion from Mack (4). The practical solution to prevent an overload of a nail connection is to reduce the number of trusses in a row to be braced by the two diagonals in a V shape.

In summary, for the three cases studied with chord loads from 10 to 50 percent of the allowable [[F.sup.'].sub.c], the predicted net lateral bracing force by the single member analysis was greater than the bracing force predicted by the system analog analysis.

CONCLUSIONS

For design purposes, R is the ratio between the NLRF (lb.) and the axial load level in the web or chord (lb.). R-values of 2.3 percent for the case of one web braced by one CLB, 2.8 percent for one web braced by two CLBs, and a conservative value of 3.1 percent for j truss chords braced by n-CLBs were determined using structural analyses for single-member analogs. Based on the results of this study, it was concluded that the length of the web member did not affect the R-values; however, the chord length had a significant effect on R-values. The chord length affected the R-values because the numbers of brace locations (24 in. on center) along the chord were based on the length of the chord, whereas the web brace locations were fixed at the center and 1/3 points of the web member.

Based on the three cases studied involving 2 by 4 chords braced as a unit (and believed to be representative of typical truss construction), the bracing force from the single member analog analysis was a conservative estimate for bracing design purposes. Based on other single member analog studies in this research [14] that showed chord size and chord lumber (characterized by MOE and SG) did not affect bracing force ratio, it was concluded that the single member analysis analog will yield approximate bracing forces for chords greater than 2 by 4 and for typical constructions beyond the three cases studied in this research. It is believed that the presence of a diagonal brace(s) stiffens the braced set of j chords and thereby reduces the net lateral force required to brace the j chords compared to the required bracing force from the single member analysis. It is not practical to attempt to analyze all possible combinations of truss lumber and bracing scenarios (j chords braced either by a V diagonal or a singl e diagonal, and all possible spans and chord load levels). The R-values reported for webs and chords can be used by permanent bracing designers since the R-values are based on a rational engineering analysis.

DESIGN CONSIDERATIONS

Based on the results of this study [14], the following design rules may be used for calculating the required NLRF for webs and chords when utilizing CLBs in combination with one or two diagonal braces. R is the ratio between the net lateral restraining force (lb.) and the axial load in the web or chord (lb.).

When designing braces for j webs in a row, the required NLRF for j webs braced by one CLB can be calculated as follows:

NLRF = j x 2.3% X maximum axial force in web from all load combinations where j is the number of identical webs in a row to be braced.

When designing braces for j webs in a row, the required NLRF for j webs braced by two CLBs can be calculated as follows:

NLRF = j X 2.8% X maximum axial force in web for all load combinations

where j is the number of identical webs in a row to be braced.

When designing permanent bracing for j chords in a row, the required NLRF for j chords braced by n-CLBs can be approximated by:

NLRF = j X R X maximum axial force in chord from all load combinations

Where j = number of identical truss chords in a row to be braced. An R-value of 3.1 percent is conservative with respect to the variable chord length since for chord lengths between 4 and 40 feet evaluated using the single member analog, 3.1 percent was the maximum R-value obtained [14].

The authors are, respectively, former Graduate Research Assistant and Professor, Biological Systems Engineering, Virginia Tech, Blacksburg, VA 24061; Professor, Wood Science, Virginia Tech, and Alumni Distinguished Professor, Civil and Environmental Engineering, Virginia Tech. This paper was received for publication in May 2000. Reprint No. 9128.

* Forest Products Society Member.

[C] Forest Products Society 2001.

Forest Prod. J. 51(7/8):73-81.

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(7.) Plaut, R.H. 1993. Requirements for lateral bracing of columns with two spans. ASCE J. of Structural Engineering 119(10):2913-2931.

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(9.) Throop, C.M. 1947. Suggestions for safe lateral bracing design. Engineering News Record, February 6. pp. 90-91.

(10.) Truss Plate Institute. 1989. Recommended design specification for temporary bracing of metal plate connected wood trusses. DSB-89. TPI, Madison, WI.

(11.) _____. 1991. Commentary and recommendations for handling, installing and bracing metal plate connected wood trusses. HIB-91 Pocketbook. TPI, Madison, WI.

(12.) _____. 1995. National design standard for metal plate connected wood truss construction. ANSI/TPI 1-1995. TPI, Madison, WI.

(13.) Tsien, H.S. 1942. Buckling of a column with nonlinear lateral supports. J. of the Aeronautical Sciences 9(4):119-32.

(14.) Underwood, C.R. 2000. Permanent bracing design for MPC wood roof truss webs and chords. M.S. thesis. Virginia Polytechnic Inst. and State Univ., Blacksburg, VA.

(15.) _____ and F.E. Woeste. 2000. Conceptual model for temporary bracing of MPC wood trusses. ASCE Practice Periodical on Structural Design & Construction 5(1):36-40.

(16.) Vogt. J. and R. Smith. 1999. Wood truss installation considerations: Permanent bracing. Wood Design & Building 6:40-42.

(17.) Waltz, M.E. 1998. Discrete compression web bracing design for light-frame wood trusses. M.S. thesis. Oregon State Univ., Corvallis, OR.

(18.) Winter, G. 1960. Lateral bracing of columns and beams. Transactions of the ASCE Vol. 125:807-845.

(19.) Woeste, F.E. 1998. Permanent bracing for piggy-back trusses. J. of Light Construction 16(6):79-83.

ABSTRACT

Permanent bracing of metal-plate-connected trusses is required to stabilize specific members of each truss throughout the life of the roof structure. The objectives of this research were to: 1) determine the required net lateral restraining force to brace multiple webs or chords in a row braced by one or more continuous lateral braces (CLBs); and 2) develop a methodology for permanent bracing design using a combination of lateral and diagonal braces. Three system analogs used to model multiple truss chords braced by n-CLBs and one or two diagonal braces were analyzed with a finite element analysis program. Single member analogs were analyzed that represented web members braced by one and two CLBs and chord members braced by n-CLBs. For analysis and design purposes, a ratio R was defined as the net lateral restraining force per web or chord divided by the axial compressive load in the web or chord. For both 2 by 4 and 2 by 6 webs braced with one CLB, the R-value was 2.3 percent for all web lengths studied, an d for webs braced with two CLBs, the R-value was 2.8 percent for all web lengths studied. Calculated R-values for truss chords braced by n-CLBs, assumed to be spaced 2 feet on center, ranged from 2.3 to 3.1 percent. A design procedure was offered for determining the net lateral restraining force required for bracing multiple webs or chords in a row based on the results of the single member analogs studied.

Consideration of permanent and temporary bracing of metal-plate-connected (MPC) wood trusses is critical for safety during erection and for reliable performance of a roof structure in service. Temporary bracing is used to position and stabilize trusses until permanent bracing or other building components can be installed. Permanent bracing, the subject of this paper, is required to stabilize specific members of each truss throughout the life of the roof structure. Based on truss design assumptions, various chords and webs require lateral support. The chords that require sheathing or the members that require lateral support at a specified interval are indicated on the truss design drawing [5]. Bracing, whether it is temporary or permanent, is required to help prevent the trusses from deflecting laterally and potentially causing the trusses to topple over or collapse [3,15, 16, 19].

Permanent truss bracing can include several different components, but are typically designed using one of the following options: 1) continuous lateral braces (CLBs) at required brace locations in conjunction with diagonals; or 2) properly nailed sheathing, either oriented strandboard (OSB) or plywood. If carefully designed, some elements of temporary bracing [11] can serve dual roles and become permanent bracing also.

LITERATURE REVIEW

The building designer is responsible for permanent bracing design of MPG wood trusses [12]. The Commentary for Permanent Bracing of Metal Plate Connected Wood Trusses [5] provides typical bracing design strategies. The Commentary does not provide a specific design procedure, but it depicts the various locations in roof truss constructions that typically require a permanent bracing design and installation. The Commentary addresses common truss configurations and provided various permanent bracing options. Although some permanent lateral bracing information may be shown on truss drawings, the building designer is responsible for the complete bracing design and details needed by the contractor to properly install the permanent bracing system. For the case where a roof truss has a long span or a high pitch and a piggyback truss is needed for shipping purposes, the building designer must provide a permanent bracing design and specifications for the supporting trusses. The Commentary provides several alternatives for bracing the flat compression chord of the supporting truss that includes the use of CLBs in conjunction with diagonal braces. No design guidelines are given for determining the necessary capacity of the diagonals needed to stabilize the CLBs.

Currently, the 2 percent Rule is the most common and accepted design practice used for designing permanent bracing for a compression web that utilizes one CLB with diagonals spaced at some interval. The 2 percent Rule is a strength model based on a percentage of the axial load in a column needed to stabilize the column [17]. Designers primarily use strength models during design due to the simplicity of the calculations. In the derivation of the 2 percent Rule, the column was assumed to be pinned at each end and at the center brace location. The column is assumed to be 1 inch out of plumb for an assumed story height of 100 inches [9]. A compression web or chord braced at the center of its span will have an initial column slope of 1/100 above and below the brace [9,17]. A force balance for a freebody diagram at the brace and chord connection is the basis for the 2 percent Rule [6].

Waltz [17] studied the design requirements for bracing a 2 by 4 compression web member at one brace support point at the center of the web. Waltz tested Select Structural and Standard Douglas Fir-Larch (DFL) specimens axially loaded and braced by his computer-controlled testing apparatus to determine the required brace force. Using finite element (FE) analysis, Waltz estimated the stiffness of the support provided by a lateral and diagonal bracing system for a number of braced 2 by 4 truss webs in a row. His research objective was to determine if one of four existing models could be used to estimate the required brace strength and stiffness. The four existing bracing models considered included Plaut's method [7,80], Winter's method [18], the 2 percent Rule [9,10,11], and Tsien's method [13]. Waltz [17] concluded that either Plaut's or Winter's method could be used for web bracing design.

OBJECTIVES

For permanent bracing design purposes, a simple "hand calculation" method was desired to approximate the brace force required to stabilize numerous compression webs braced by one or two CLBs or chords braced by CLBs at multiple locations. The objectives of this research were to: 1) determine the required net lateral restraining force to brace multiple webs or multiple chords in a row braced by one or more CLBs; and 2) to develop a methodology for permanent bracing design using a combination of lateral and diagonal braces. In addition to the primary objectives of this research, the impact of lumber specific gravity, modulus of elasticity, and chord lumber size on required bracing forces were investigated.

ANALYSIS TOOLS AND ASSUMPTIONS

SAP2000 [2] is a finite element analysis program used for analyzing different types of problems, including structural analysis problems. In this research, SAP2000 was used with the assumption that beam elements were linear elastic. A minimum of three beam elements were used to represent the compression members.

To create the structural analogs used in this research, the following assumptions were made:

* The CLBs and diagonal braces were dry 2 by 4 nominal Stud grade sprucepine-fir (SPF) lumber;

* The ends of the braced webs or chords were pin connected; therefore, pin or roller reactions were used to support the member ends;

* 2-16d Common nails were used for all wood-to-wood connections;

* Truss chords in a row and multiple CLBs used for bracing the chords were spaced 2 feet on center;

* Truss webs were braced with one or two CLBs at the center or third points of the length, respectively.

The load-displacement relationship for a single nail connection made with SPF lumber was determined by using Equation [1] from Mack [4]:

[F.sub.n] = [52d.sup.1.75][k.sub.s] (3.20[omega] + 0.68) [(1 - [e.sup.-75[omega]]).sup.0.7] [1]

where [F.sub.n] = the load per nail (lb.) applied to a single shear nail joint; [omega] = the slip between the two members of the nail joint (in.); d = the nail diameter (in.); [k.sub.s] = the species constant from Mack [4].

Since Mack's [4] paper did not provide a species factor for SPF, a linear regression was performed to determine the equation suitable for that particular species using available data. Equation [2] gives the variables and results from a linear regression of 10 pairs of [k.sub.s] and specific gravity (SG) data reported in Mack [4]:

[k.sub.s] = 265 x SG + 3.5 [2]

where [k.sub.s] = the species factor, and for use in Equation [1]; SG = specific gravity for the species.

From Equation [2], [k.sub.s] equals 115 for SPF that has a published SG value of 0.42 [1]. The right-hand side of Equation [1] must be doubled because the stiffness of a two-nail joint is assumed to be twice as stiff as a single-nail joint. And finally, substituting d equals 0.162 inches for a 16d Common nail into Equation [1] and by applying a factor of 1.25 for dry material as referenced in Mack, Equation [3] results:

[F.sub.2n] = 618 x (3.20 [omega] + 0.68) [(1 - [e.sup.-75[omega]]).sup.0.7] [3] where [F.sub.2n] = the load applied to a 2-16d Common nail joint (lb.); [omega] = the joint slip between the SPF members (in.).

Equation [3] was implemented in the bracing analysis by using the secant modulus at different joint load levels. To find the secant modulus, a slip was calculated using SAP2000 and then the corresponding joint force was calculated using Equation [3]. The joint force g(x), where x is slip, was determined by a line drawn from the origin to the point (x, g(x)). The slope of the line, called the secant modulus, was the stiffness for the specified joint force and slip.

A linear spring stiffness was estimated for each joint of the structural analog, and then the structural analog was re-analyzed. Calculated spring forces were then compared to the specific force and displacement used to input the linear spring (secant modulus) constants. A new stiffness value for each spring was determined based on the new deflections and entered into SAP2000 [2]. The structure was analyzed again. This procedure was repeated until the force in the springs matched the assumed force and displacement used to calculate the secant modulus spring stiffness within a tolerance of 1 percent. This iterative procedure was necessary to accommodate a nonlinear spring (nail connection) in the SAP analysis of the braced system.

For the purpose of discussion and comparison to the 2 percent Rule, the net lateral restraining force from the web and chord analyses were divided by the axial compression load in the web or chord, respectively. This ratio will be referred to as R and is defined by Equation [4]:

R = Net lateral restraining force (lb.)/Axial load level in web or chord (lb.) [4]

where R = the variable of interest for both webs and chords.

WEBS BRACED WITH ONE CLB

The structural analog used to analyze a multiple number of webs in a row (j) braced with one CLB is depicted in Figure 1. To simplify the structural analog for the case of multiple truss webs in a row, the diagonal brace was neglected and a single truss web was analyzed. The response of a multiple web case encountered in actual construction can be obtained by multiplying j times the response from the single web case. The structural analog was represented in SAP2000 [2] as depicted in Figure 2, where the length of the web was varied from 3 to 12 feet. The lumber was assumed to be 2 by 4 Stud grade SPF with a modulus of elasticity (MOE), of 1,200,000 psi. The load applied to the web (P) varied from 10 to 100 percent of the allowable load based on the assumption that the webs are pin-connected to the chords and that effective web buckling length is 80 percent of the actual web length [1,12].

A second analog was created to determine the effects of web size, if any, on the required bracing force. The lumber was assumed to be 2 by 6 nominal Stud grade SPF with an MOE of 1,200,000 psi. The allowable load (P) applied to the web was recalculated based on the lumber size and the procedures presented in the NDS for column design [1,12].

A third structural analog was created for the one web braced by one CLB case to determine if lumber species (as characterized by MOE and SG) affected the required brace force. Using the analog shown in Figure 2, 2 by 4 nominal No. 2 DFL was assumed for the web and CLB. DFL has a 17 percent higher MOE value than SPF. The published SG value for DFL is 0.50 versus 0.42 for SPF. DFL was chosen as the species to compare to SPF because the nail slip data were Figure 2. -- Structural analog representing one web braced by one CLB. The nailed connection is represented by a spring and an applied load (P) is a compression force determined using design equations given in the NDS 97 [1].

available for a DFL joint and because of the higher SG value.

ANSI/TPI 1-1995 [12] installation limits were assumed for all web lengths studied. All web members were assumed to have equal initial curvature in the same direction (for example, all webs bowed left). The assumption for initial curvature of all webs is conservative because in reality truss webs would have initial deflections in both directions (bowed right and bowed left). Initial curvature in both directions (with a CLB installed) would help provide support to the overall structure. Web members were assumed to have an initial maximum deflection of L/200 and have initial curvature matching a half sine wave defined by Equation [5]:

[[delta].sub.i] = L/200 sin ([pi] X x/L} [5]

where [[delta].sub.i] = assumed initial deflection of the truss webs; L = length of the compression chord, and x = distance from the member end in inches; [pi] is in radians.

A deflected web member, by nature, is a smooth curve as opposed to a series of straight lines. Therefore, the half sine configuration was assumed and used in all web and chord analyses. A summary of the cases studied for one web braced by one CLB is given in Table 1.

RESULTS (WEB WITH ONE CLB)

R-values for the case of one web braced by one CLB were 0.023 (2.3%) for all web lengths and load levels (ranged from 10% to 100% of the allowable compression for the assumed lumber grade). R-values were not affected by lumber species (higher MOE and SG) or by using a 2 by 6 versus 2 by 4 web (14). The difference in the 2 percent Rule versus 2.3 percent results from this study, is due to the fact that a flatwise 2 by 4 is very flexible, and thus not dramatically affected by member continuity. The computer analog constructed for this research did not have a pin joint at the center of the web as is assumed for the derivation of the 2 percent Rule.

WEBS BRACED WITH TWO CLBs

For the case of one web braced by two CLBs shown in Figure 3, the structural analog included the same modeling assumptions as the structural analogs representing one web braced by one CLB. The structural analog was represented in SAP2000 [2] as depicted in Figure 4, where the length of the web varied from Figure 4. -- Structural analog representing one web braced by two CLBs. The nailed connections are represented by springs and an applied load (P) is a compression force determined using design equations provided in the NDS [1].

5 to 12 feet. The lumber was assumed to be 2 by 4 nominal Stud grade SPF with an MOE of 1,200,000 psi. Table 2 contains a summary of the cases studied for one web braced by two CLBs; the study objectives and analysis methods for each case were identical to the case of one web braced by one CLB.

RESULTS (WEB WITH TWO CLBs)

This case, consisting of one web and two CLBs, produced an R-value of particular interest. In the past, one option for design purposes was to assume the bracing force was equal to 2 percent of the applied load, times the number of CLB connections per web, which yields 4 percent of the axial load as the required bracing force per web. The R-value was determined to be 0.028 (2.8%) for all lengths and axial load levels studied. The 2.8 percent R-value is substantially less than the 4 percent calculated by the assumption that the brace force increases in proportion to the number of CLBs. The R-values were not affected by increasing the web lumber MOE and SG values or increasing the web size to a 2 by 6 (14).

CHORDS BRACED BY N-CLBs

Figure 5 is a line representation of an actual set of j truss chords braced by n-CLBs and one diagonal brace. In this research, the cases of one and two diagonal braces were investigated. A total of 10 structural analogs were developed based on the varying lengths of lumber. The diagonal braces were neglected in the first phase of the investigation. Figure 6 depicts the structural analog models as they were analyzed using SAP2000 [2]. Structural analogs were constructed using the same boundary conditions and assumptions for connections between each CLB and chord as were made for the cases of one web with one and two CLBs.

The center panel (or two panels) of a symmetric truss with a flat-top chord under symmetric loading will have the maximum stress interaction according to the NDS [1] when the truss panel lengths are equal. Equation [6] gives the stress interaction criterion for a chord subjected to bending and compression load.

[([f.sub.c]/[F'.sub.c]).sup.2] + ([f.sub.b]/[F'.sub.b](1 - [F.sub.cE])) [less than or equal to] 1 [6]

where [f.sub.c] = actual compression stress parallel to grain (psi); [F'.sub.c] = allowable compression design value parallel to grain (psi); [f.sub.b] = actual bending stress (psi); [F'.sub.b] = allowable bending design value (psi); [F.sub.cE] = critical buckling design value for a compression member for the applicable [l.sub.e]/d ratio (psi).

From a permanent bracing designer standpoint, one needs to determine the maximum axial force in all the panels of the chord. When an unbraced truss chord is assumed to be continuous in the structural analysis, bending moments will exist in all panels. The amount of bending moments will vary in magnitude from one design to the next. A conservative assumption with respect to permanent bracing design is that the bending moment is zero in all panels. Since the bracing is designed to resist a percentage of the axial load in the web, the higher axial load due to this assumption results in a higher design load in the brace. If the stress interaction is at the maximum equal to 1.0, Equation [6] reduces to Equation [7]:

[f.sub.c] = [F'.sub.c] [7]

Equation [7] applies to the center panel (or two symmetrical panels). It is conservative because assuming zero bending moment allows for the maximum applied axial compression load to be present in the assumed chord. The design compression load (C) in the center panel (or two symmetrical panels) is therefore given by Equation [8]:

C = A X [F'.sub.c] [8]

where A = chord area ([in.sup.2); [F'.sub.c] = allowable compression design value parallel to grain (psi).

Axial load in the outer panels under the symmetry assumptions stated previously will be lower than the axial load in the center panels. A conservative assumption for permanent bracing design is to assume all panels have the same axial load equal to the maximum value of the center panel as determined from Equation [8].

The allowable compression parallel-to-grain design value ([F'.sub.c]) was calculated using NDS (1) procedures for column design. Chords can buckle about both axes depending on the [l.sub.e]/d (effective column length-depth ratio) of each axis. When CLBs are installed at 24 inches on center, the effective column length-depth ratio about the weak axis was determined using Equation [9] as follows:

le/d = Ke X la/d = 1.0 X 24"/1.5 = 16 [9]

If the strong axis [l.sub.e]/d is greater than 16, the truss designer uses the larger [l.sub.e]/d. For example, the strong axis [l.sub.e]/d controls the design of a 2 by 4 truss panel 8 feet long. If the strong axis [l.sub.e]/d is less than 16, 16 is used. A situation such as this occurs when determining [l.sub.e]/d for a 2 by 12 member that is 10 feet in length. For permanent bracing design, it is therefore conservative to assume [l.sub.e]/d equals 16 and this assumption will always yield the maximum possible axial load in the top chord [14].

The final assumptions used in creating the structural analogs included the lumber type, chord lengths, and the duration of load factor. Snow load plus dead load was assumed and thus a duration of load factor of 1.15 was used. The lumber was assumed to be No. 2 Southern Pine for the truss chords and Stud grade SPF for the CLBs and diagonal braces. The truss chord size was varied from 2 by 4 to 2 by 12. The truss chord length was varied from 4 to 40 feet by increments of 4 feet, but also included a length of 6 feet. Allowable loads were determined based on the size of the members and the aforementioned [l.sub.e]/d value and the NDS (1) design equations for column design.

The assumed initial deflected shape of the chords was determined from Equation [5] and the same assumptions as for the cases of one web braced by one and two CLBs. If the length of the chord exceeded 400 inches, Equation [10] was used to meet the installation guidelines provided in ANSI/TPI 1-1995 [12]:

[[delta].sub.i] = 2 X sin ([pi] X x/L) [10]

ANSI/TPI 1-1995 (12) states that the maximum initial deflection allowed in a truss chord is the lesser of L/200 or 2 inches. In cases where compression chord length (L) is greater than 400 inches, the 2-inch maximum allowance was observed. Table 3 summarizes the truss chords modeled for the investigation of j truss chords braced by n-CLBs.

The same analysis and procedures as were used for the case of a braced web were used to analyze chords with n-CLBs, except the chords were assumed to be No. 2 grade Southern Pine lumber. For calculations of the nail slip of the 2-16d Common nail connections, it was assumed that both the chord and the CLB were SPF because nail slip data were not available for a joint having mixed species. This assumption will result in lower stiffness values for the connection, which is conservative.

RESULTS (CHORDS BRACED BY N-CLBs)

The chord sizes and spans studied for this case are given in Table 3. The CLBs for all cases were assumed to be 2 feet on center. Chord size did not affect the R-value for assumed chord lengths of 4 to 40 feet.

For the 2 by 4 chord size, only spans up to 24 feet were studied. As reported in Table 4 for 2 by 4 truss chords, all R-values for the 4-foot 2 by 4 No. 2 Southern Pine chord were equal to 0.023 (2.3%) and were the same as the R-value determined for the case of an SPF web braced with one CLB. The R-value was the same because the same number of bracing locations were present for both cases (one at the center) and it was determined that member length did not affect the R-value for webs. The results in Table 4 for the 2 by 4 chord size are similar to the results for 2 by 6 through 2 by 12, thus they are the only results presented in this paper [14].

R-values (all equal to 0.028 or 2.8%) for the 6-foot Southern Pine chords (two CLB bracing locations) were the same as the R-values for the case of a 2 by 4 SPF web braced with two CLBs. Again, the results were the same due to the bracing locations being the same (at the 1/3 points). A-values for the 8-foot chords (all equal to 2.8%) were the same as for the 6-foot chords using two significant figures. R-values for chords between 12 feet and 32 feet had a peak value of 0.031 (3.1%) [14]. R-values for all lumber sizes (2 by 4 to 2 by 12) for 36- and 40-foot chords were less than R-values for the shorter lengths. R was 0.029 (2.9%) for the 36-foot Southern Pine chord with n-CLBs spaced 24 inches on center, and R was independent of lumber size. R was equal to 0.026 (2.6%) for the 40-foot Southern pine truss chord with n-CLBs spaced 24 inches on center, and R was independent of lumber size. The values for R for chord lengths (L) greater than 400 inches were different due to the maximum initial member deflection o f 2 inches [14].

In addition, the chord load level as a percent of [F'.sub.c] did not affect R for any size or length. The SAP2000 analysis was based on linear elastic beam elements with nonlinear springs representing the nail connections, and thus one would expect the system to behave in a non-linear manner. However, the nail connections at low load levels are apparently so stiff that the calculated value of R is not substantially affected by the load level [14].

SYSTEM ANALOG: EFFECT OF INCLUDING DIAGONALS WITH MULTIPLE CHORDS

"System" structural analogs were composed of j chords, n-CLBs, and diagonal braces. The first system structural analog analyzed with SAP2000 [2] represented five 8-foot roof truss chords spaced 24 inches on center, three continuous lateral braces (CLBs) spaced 24 inches on center, and one diagonal (Fig. 7). For this case and the remaining two cases, the lumber used in the construction of the structural analog consisted of 2 by 4 Stud grade SPF, with an MOE of 1,200,000 psi for the CLBs, and 2 by 4 nominal No. 2 Southern Pine, with an MOE of 1,600,000 psi for the truss chords. The truss chords were assumed to be columns with pin supports at one end and roller reactions supports at the other [14].

An axial chord load was applied to all chords in the analysis. The maximum allowable design load (A x [F'.sub.c]) was determined to be 6,842 pounds based on NDS procedures for column design [1]. However, 50 percent of the allowable compressive load based on an [l.sub.e]/d ratio of 16 is a typical load level in a wood truss chord. For the analyses of the required bracing forces, the chord axial load level was increased from 10 to 50 percent of [F'.sub.c]. Maximum allowable axial load was based on the grade, species combination, size of lumber, and the duration of load factor of 1.15 for snow plus dead loading.

The second system structural analog analyzed represented six 20-foot roof trusses spaced 24 inches on center, nine continuous lateral braces (CLBs) spaced 24 inches on center, and two diagonals in a V shape with an angle of 45 degrees (Fig. 8). The truss chords were assumed to be columns with roller supports, free to translate in the vertical or Z-direction, on both ends. Roller supports free to translate in the horizontal or X-direction were used on the chords where the middle CLB crossed the chords to stabilize the structure, but still allowed the chords to deflect.

The third system structural analog analyzed represented 11 20-foot roof truss chords spaced 24 inches on center, 9 continuous lateral braces (CLBs) spaced 24 inches on center, and 2 diagonals in a V shape with an angle of 45 degrees (Fig. 9). Truss chords were assumed to be columns with roller reactions as described for the previous study case.

RESULTS (OF SYSTEM MODELS)

To calculate the required net lateral restraining force (NLRF) using the SAP2000 analysis results for multiple truss systems braced by n-CLBs and one or two diagonals, the X-components of the joint force between each diagonal and truss chord (tabulated in Table 5) were summed taking into account the direction of the force. For the case of five 8-foot trusses braced by three CLBs and one diagonal brace, the single member analog estimate of the required net lateral bracing forces was approximately 5 to 6 percent greater than the estimate of the required net lateral bracing forces predicted by the system analog analysis. For the case of six 20-foot trusses braced by nine CLBs and two diagonal braces, the single member analog estimate of the required NLRF was 2 percent or more greater than the estimate of the required NLRF predicted by the system analog analysis. For the case of 11 20-foot trusses braced by 9 CLBs and 2 diagonal braces, the single member analog estimate of the system required NLRF was 5 percent or more greater than the bracing force from the system analog analysis (14). However, it should be noted that the 40 and 50 percent load level cases theoretically caused a 2-16d Common nail connection to fail based on the 0.1-inch slip criterion from Mack (4). The practical solution to prevent an overload of a nail connection is to reduce the number of trusses in a row to be braced by the two diagonals in a V shape.

In summary, for the three cases studied with chord loads from 10 to 50 percent of the allowable [[F.sup.'].sub.c], the predicted net lateral bracing force by the single member analysis was greater than the bracing force predicted by the system analog analysis.

CONCLUSIONS

For design purposes, R is the ratio between the NLRF (lb.) and the axial load level in the web or chord (lb.). R-values of 2.3 percent for the case of one web braced by one CLB, 2.8 percent for one web braced by two CLBs, and a conservative value of 3.1 percent for j truss chords braced by n-CLBs were determined using structural analyses for single-member analogs. Based on the results of this study, it was concluded that the length of the web member did not affect the R-values; however, the chord length had a significant effect on R-values. The chord length affected the R-values because the numbers of brace locations (24 in. on center) along the chord were based on the length of the chord, whereas the web brace locations were fixed at the center and 1/3 points of the web member.

Based on the three cases studied involving 2 by 4 chords braced as a unit (and believed to be representative of typical truss construction), the bracing force from the single member analog analysis was a conservative estimate for bracing design purposes. Based on other single member analog studies in this research [14] that showed chord size and chord lumber (characterized by MOE and SG) did not affect bracing force ratio, it was concluded that the single member analysis analog will yield approximate bracing forces for chords greater than 2 by 4 and for typical constructions beyond the three cases studied in this research. It is believed that the presence of a diagonal brace(s) stiffens the braced set of j chords and thereby reduces the net lateral force required to brace the j chords compared to the required bracing force from the single member analysis. It is not practical to attempt to analyze all possible combinations of truss lumber and bracing scenarios (j chords braced either by a V diagonal or a singl e diagonal, and all possible spans and chord load levels). The R-values reported for webs and chords can be used by permanent bracing designers since the R-values are based on a rational engineering analysis.

DESIGN CONSIDERATIONS

Based on the results of this study [14], the following design rules may be used for calculating the required NLRF for webs and chords when utilizing CLBs in combination with one or two diagonal braces. R is the ratio between the net lateral restraining force (lb.) and the axial load in the web or chord (lb.).

When designing braces for j webs in a row, the required NLRF for j webs braced by one CLB can be calculated as follows:

NLRF = j x 2.3% X maximum axial force in web from all load combinations where j is the number of identical webs in a row to be braced.

When designing braces for j webs in a row, the required NLRF for j webs braced by two CLBs can be calculated as follows:

NLRF = j X 2.8% X maximum axial force in web for all load combinations

where j is the number of identical webs in a row to be braced.

When designing permanent bracing for j chords in a row, the required NLRF for j chords braced by n-CLBs can be approximated by:

NLRF = j X R X maximum axial force in chord from all load combinations

Where j = number of identical truss chords in a row to be braced. An R-value of 3.1 percent is conservative with respect to the variable chord length since for chord lengths between 4 and 40 feet evaluated using the single member analog, 3.1 percent was the maximum R-value obtained [14].

The authors are, respectively, former Graduate Research Assistant and Professor, Biological Systems Engineering, Virginia Tech, Blacksburg, VA 24061; Professor, Wood Science, Virginia Tech, and Alumni Distinguished Professor, Civil and Environmental Engineering, Virginia Tech. This paper was received for publication in May 2000. Reprint No. 9128.

* Forest Products Society Member.

[C] Forest Products Society 2001.

Forest Prod. J. 51(7/8):73-81.

LITERATURE CITED

(1.) American Forest & Paper Association. 1997. National design specification for wood construction. ANSI/AF&PA NDS-1997. AF&PA, Washington, DC.

(2.) Computer and Structures, Inc. 1995. SAP 2000 Analysis Ref. Vol. 1. CSI, Berkeley, CA.

(3.) Kagan, H.A. 1993. Common causes of collapse of metal-plate-connected wood roof trusses. J. of Performance of Constructed Facilities 7(4):225-234.

(4.) Mack, J.J. 1966. The strength and stiffness of nailed joints under short-duration loading. Forest Prod. Pap. No. 40:28. CSIRO, Australia.

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TABLE 1. -- Summary of trusts web cases studied involving one web braced by one CLB. a Nominal Range of web Web lumber CLB lumber lumber size lengths investigated Increments species species 2 by 4 3 ft. to 12 ft. 1 ft. SPF SPF 2 by 6 3 ft. and 12 ft. N/A SPF SPF 2 by 4 3 ft. and 12 ft. N/A DFL DFL (a)SPF = spruce-pine-fir; DFL = Douglas-fit.larch. TABLE 2. -- Summary of truss web cases studied involving one wbe braced by two CLBs. a Nominal Range of web Web lumber CLB lumber lumber size lengths investigated Increments species species 2 by 4 5 ft. to 12 ft. 1 ft. SPF SPF 2 by 6 5 ft. and 12 ft. N/A SPF SPF 2 by 4 5 ft. and 12 ft. N/A DFL DFL (a)SPF = spruce-pine-fir; DFL = Douglas-fir-larch. TABLE 3.-- Summary of truss chord cases studied involving j chords braced by n-CLBs. a Nominal Range of chord Chord lumber lumber size lengths investigated Increments species 2 by 4 4 ft. to 24 ft. 4 ft. SP including 6 ft. 2 by 6 4 ft. to 36 ft. 4 ft. SP including 6 ft. 2 by 8 4 ft. to 36 ft. 4 ft. SP including 6 ft. 2 by l0 4 ft. to 40 ft. 4 ft. SP including 6 ft. 2 by 12 4 ft. to 40 ft. 4 ft. SP including 6 ft. Nominal CLB lumber lumber size species 2 by 4 SPF 2 by 6 SPF 2 by 8 SPF 2 by l0 SPF 2 by 12 SPF (a)SP = southern pine; SPF = spruce-pine-fir. TABLE 4.-- Net lateral bracing force (lb.) divided by the axial load (lb.) for comparison to the 2 percent Rule for 2 by 4 Southern Pine truss chords. Lateral force produced in the n-web-CLB connection (lb.) Axial load level in chord (lb.) Chord No. of braces length (n + 2) a 10 b 20 b 30 b (ft.) 4 3 0.023 0.023 0.023 6 4 0.028 0.028 0.028 8 5 0.028 0.028 0.028 12 7 0.03 0.03 0.03 16 9 0.031 0.031 0.031 20 11 0.031 0.031 0.031 24 13 0.031 0.031 0.031 Chord length 40 b 50 b 60 b 70 b 80 b 90 b 100 b (ft.) 4 0.023 0.023 0.023 0.023 0.023 0.023 0.023 6 0.028 0.028 0.028 0.028 0.028 0.028 0.028 8 0.028 0.028 0.028 0.028 0.028 0.028 0.028 12 0.03 0.03 0.03 0.03 0.03 0.03 0.03 16 0.031 0.031 0.031 0.031 0.031 0.031 0.031 20 0.031 0.031 0.031 0.031 0.031 0.031 0.031 24 0.031 0.031 0.031 0.031 0.031 0.031 0.031 (a)When n-CLBs are used, one additional brace is typically installed on each end of the chord. (b)Percent of maximum allowable axial compressive load in the truss chords. TABLES.5--Net lateral forces (lb.) produced by Southern Pine truss n-chords braced by multiple Spruce-Pine-Fir (SPF) CLBs and one or two SPF diagonal(s). Applied axial compressive load from 10% to 50% of allowable load (lb.) a Length No. of No. of No. of of chords trusses CLBs diagnol braces 684 (ft.) 8 5 3 1 91 20 6 9 2 124 20 11 9 3 221 Length of chords 1368 2053 2737 3421 (ft.) 8 182 272 363 453 20 247 367 486 602 20 434 637 824 b 983 b (a)Based on [l.sub.e]/d equals 16 and 2 by 4 No. 2 Southern Pine truss chords (b)Joint slip was greater than the theoretical failure limit for a 2-16d Common nail connection.

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Title Annotation: | metal-plate-connected trusses |
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Author: | UNDERWOOD, CATHERINE R.; WOESTE, FRANK E.; DOLAN, J. DANIEL; HOLZER, SIEGFRIED M. |

Publication: | Forest Products Journal |

Article Type: | Statistical Data Included |

Geographic Code: | 1USA |

Date: | Jul 1, 2001 |

Words: | 6745 |

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