Reliability-based design of mechanically connected built-up wood columns

Reliability-based design of mechanically connected built-up wood columns

S. K. MALHOTRA A N D A. P. SUKUMAR Departtnetzt of Civil Etigitzeering, Techt~ical University of Nova Scoria, Halifar, N.S., Canada B3J 2x4

Received February 27, 1990

Revised manuscript accepted July 13, 1990

Built-up wood structural components and systems have been in use in the building industry for decades. Though easy to construct, they are rather complex to analyze because of numerous factors affecting their strength and behaviour. This paper explains the development of a reliability-based design formulation in limit states for built-up columns, using a rational mathematical model for the analysis and design. Also, a brief discussion is given on a set of simplified design rules for layered columns which have been incorporated into the Canadian Standards Association Standard CAN3-086. ILM89 "Code for engineering design in wood (limit states design)." The rational model as well as the simplified rules are subjected to reliability analysis.

The Canadian specifications for the design of wood structures in limit states format have been available since December 1984. These specifications are essentially a soft conversion of the previous working stress design code. The 1989 edition of the code is mostly based on the principles of reliability-based design. Some portions of the code are still not fully based on reliability approach, as additional information is needed for the establishment of a reliable data base upon which the "uncertainty factors" could be redefined in a more rational and logical manner. This paper provides that type of information and the details of reliability analysis as applied to mechanically connected built-up timber columns. The reliability analyses are performed based on first-order second-moment methods, using a data base of column strengths. The data base consists of experimental as well as simulated results. Two reliability analysis programs, BUCREL (built-up column reliability analysis) and POINT (reliability analysis using design point algorithm), are developed. Design recommendations are derived in a reliability-based design format.

Key words: buckling, columns, design specifications, efficiency, joints, layered columns, rational model, reliability, resistance factor, slip, spaced columns, timber, wood.

Les ensembles et les ClCments de charpente en bois compose sont utilisCs depuis des annCes. Bien qu'ils soient faciles B construire, ils sont plut8t complexes a analyser en raison des nombreux facteurs qui influent sur leur resistance et leur comportement. Cet article traite de I'Claboration, B I'aide d'un modkle mathematique rationnel d'analyse et de calcul, d'une formule de calcul aux Ctats limites bas& sur la fiabilitC, pour les poteaux composes. En outre, une brkve discussion est proposee sur un ensemble de rkgles de calcul simplifiCes pour les poteaux en couches, qui ont CtC intCgrCes B la norme CAN3-086. I-M89, ~Rkgles de calcul aux Ctats limites des charpentes en b o b . Le modkle rationnel ainsi que les rkgles simpli- fiCes ont CtC soumis B une analyse de la fiabilitC.

Les spCcifications canadiennes pour le calcul aux Ctats limites des charpentes en bois sont disponibles depuis dCcembre 1984. Ces specifications decoulent en majeure partie d'une conversion du Code de calcul du taux de contrainte admissible. L'Cdition 1989 du code s'inspire surtout des principles de calcul bases sur la fiabilitt. Certaines portions du code ne sont pas encore basCes sur la mtthode de fiabilitC car des renseignements additionnels sont ntcessaires afin d'etablir une base de donnCes fiable B partir de laquelle les ((facteurs d'incertituden pourraient Ctre redCfinis d'une manikre plus logique et ration- nelle. Cet article prCsente ce type de renseignements ainsi que les details de I'analyse de fiabilitC appliquCe B des poteaux de bois composCs, assemblCs B I'aide de fixations mCcaniques. Les analyses de fiabiliti sont effectuCes i I'aide d'une base de donnCes des resistances des poteaux et tiennent compte des mCthodes de premier ordre, second moment. La base de donnCes est constituie d'informations obtenues B la suite d'experiences et de simulations. Deux programmes d'analyse de la fiabilitC, BUCREL et POINT, ont CtC dCveloppCs. Des recommandations de calcul sont obtenues dans une forme baste sur la fiabilitC.

Mots clis : flambement, poteaux, spCcifications de calcul, efficacitC, joints, poteaux en couches, modkle rationnel, fiabilitk, coefficient de rCsistance, glissement, poteaux espacCs, bois.

[Traduit par la rCdaction]

Can. I. Civ. Eng. 18, 171 - 181 (1991)

Introduction layered columns; and the ones with spacer blocks, spaced

Built-up wood components and systems have been in use in the structural wood construction for many decades. Among them, two types of built-up wood columns are commonly used, namely layered columns and spaced columns. Built-up compression members are primarily formed by joining two or more members together with bolts, nails, spikes, or split-ring connectors and bolts, with or without spacer blocks between members. Built-up columns without spacer blocks are called

columns. The strength of a built-up column depends on various

factors, like the characteristic material properties, column configuration, and connector data. Though easy to construct, the exact analysis of built-up wood systems is rather complex because of the large number of factors affecting their strength. Design specifications (CSA 1984, 1989; NFPA 1986) provide very limited information for the design of such structural members. In the last few years, the concept of interlayer slip and its significance on the behaviour of built-up columns were

NOTE: Written discussion of this paper is welcomed and will be subjected to detailed experimental and theoretical investiga- received by the Editor until August 3 1, 1991 (address inside front tions by Rassam and Goodman (1970), Malhotra and cover). Van Dyer (1977), and Van Dyer and Malhotra (1985). The Pr~ntcd in Canada / Imprime ;IU Canada

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172 CAN. J. CIV. ENG. VOL. 18, 1991

mathematical model developed by Malhotra and Van Dyer (1977) for the buckling strength of built-up columns takes into account the various variables involved and is quite versatile in its application. This theory has been further verified by con- ducting laboratory tests and used to develop simplified design rules (Malhotra and Sukumar 1989).

Reliability-based design methods are rational and consistent techniques of balancing the "capacity -demand1' situations in structural design. A number of building and structural design codes in various countries have recently been changed or are in the process of changing from the traditional design philoso- phy to reliability-based limit states methods. Whereas the 1984 edition of the Canadian standard for engineering design in wood (CSA 1984) had a limit states version based on a "soft conversion" of the previous working stress design code, in the 1989 edition many sections have been developed using principles of reliability-based design (Foschi 1989). For the design specifications that are still not fully implemented on reliability concepts, additional information is needed for the establishment of a reliable data base upon which the "uncertainty factors" could be redefined in a more rational and logical manner.

The objectives of this paper are twofold. The first is to dis- cuss the reliability levels of a set of simplified design specifications for layered columns as adopted in the Canadian standard (CSA 1989) and the second is to obtain suitable resistance factors for the design of built-up columns in general, based on the model developed by Malhotra and Van Dyer (1977). These objectives are achieved by performing reliability analysis for layered and spaced columns. A data base of column strength (obtained from laboratory tests and Monte- Carlo simulations) has been developed (Malhotra and Sukumar 1990), which can be extended further to suit almost all types of built-up wood columns. Two FORTRAN programs, BUCREL

(built-up column reliability analysis) and POINT (reliability analysis using design point algorithm), were developed for creating data base and reliability analysis. These programs along with their algorithms are explained in this paper. Details of an experimental program, which provided the necessary data for this study, are also given.

Design of built-up compression members Design standards (CSA 1984, 1989; NFPA 1986) do not

provide detailed specifications for the design of built-up wood columns. For layered columns, the specifications are very limited in the design options and for spaced columns, they are mainly based on empirical formulations. A design approach that takes into account the interaction between the component members and the role of connector type, size, and configuration is required to replace the current design provisions. The design procedure developed by Malhotra and Van Dyer (1977) for built-up wood compression members takes into account various parameters affecting the strength of layered members (referred to as Malhotra and Van Dyer model, hereafter). The theory has been verified by a series of experimental investi- gations.

In the present paper, the above-referred theory is adopted as such for developing a reliability-based design procedure for built-up wood columns. All analytical and experimental values specified in this paper apply to columns subjected to axial loads, which are effectively held in position at both ends, but are not restrained against rotation.

Malhotra and Van Dyer model - a review The model takes into account various parameters affecting

the strength of built-up columns. Some of the salient points in the theory for the mathematical model to describe the behavior of built-up timber columns are listed below: -An ideal, centrally loaded compression member is assumed. -The non-rigidity due to the use of mechanical fasteners and

the resulting interlayer slip between individual components are considered in the theory.

-To study the system in elastic as well as inelastic ranges of stress, the concept of tangent modulus theory is employed.

-The buckling occurs in the direction of smaller dimension of the individual members of the column. Detailed explanation and formulation of the theory can be

found in Van Dyer (1976) and Malhotra and Van Dyer (1977). Based on the theory, the buckling stress for a layered or a spaced column is given by the following equation:

where Fcr is the column buckling stress (MPa);

The parameter p equals 0 for layered columns; Fc is the compressive stress (MPa); E is the modulus of elasticity (MPa); c is a constant depending on the shape of the stress -strain curve and is taken as 0.9;

in which I; is the moment of inertia of the ith layer about its own centroidal axis (mm4) and I is the moment of inertia of the cross section of the column based on the overall dimensions (mm4); m is the number of interfaces in the column;

[6] a = m X L

Total number of connectors

in which L is the column length (mm);

in which A; is the cross-sectional areas of the ith layer (mm2) and equals 0 for blockings in spaced columns; zi is the distance of the centroid of the cross section of the ith layer, G;, to the centroid, G , of the entire cross section (mm) (Fig. 1); r; is the distance from the centroid of the cross section of the ith layer to the centroid of the cross section (mm) (Fig. 1); 1, and 1, are defined in Fig. 1 (mm); K is the slip modulus, which is the slope of the load-deformation curve of a connection at the instant of column buckling (Nlmm); Lb is the buck-

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MALHOTRA AND SUKUMAR

ling length of the column (mm), which is the distance between the centers of the two pinned-end supports; X = Lblr is the slenderness ratio, where r is the radius of gyration of the cross section, based on the overall dimensions.

Figure 1 shows typical layered and spaced columns, with a cross section of rectangular configuration. Equation [ l ] may be applied to any configuration of built-up compression members. With the value of p = 0, the column stress predictions correspond to layered columns. By substituting the value of B = 1 in [ l ] , the stress predictions for equivalent solid columns can be obtained. An equivalent solid column refers to a solid column having the same length and cross-section dimensions as the composite column. The values for slip modulus, K, used in [ l ] are obtained in accordance with Malhotra and Van Dyer (1977) and Van Dyer and Malhotra (1985). Approximate value of slip modulus can be taken as given in [8].

where K is the slip modulus (Nlmm); t is the depth of connector penetration in a component member (mm); d is the connector diameter (mm); E is the modulus of elasticity of wood (MPa); and x = 200 for split-ring connectors, 250 for bolts, and 300 for nails.

Equation [8] is an approximation of the theory based on the concept of beam-on-elastic-foundation applied to nailed and bolted timber joints under lateral load developed by Kuenzi (1955).

Efficiency of layered columns Design of layered wood compression members can be sim-

plified if it can be related to equivalent solid members. This simplification is achieved by developing efficiency graphs for nailed and bolted layered compression members. The efficiency is defined as the ratio of the strength of a built-up compression member to the strength of an equivalent solid compression member of the same overall dimensions as those of the composite member. A computer program has been developed, which gives the efficiency of a layered column as a percentage of the buckling strength of the equivalent solid column (Sukumar and Malhotra 1989). Two typical efficiency graphs for nailed and bolted layered columns are given in Figs. 2 and 3.

It should be mentioned at this juncture that the efficiency of built-up columns does not change considerably with the type and the grade of column material. For bolted columns fabricated with various species and grades of wood as given in CSA (1989), the efficiency falls within 80- 100% range.

Based on the model and the efficiency analysis as mentioned above, a set of simplified design rules for layered columns have been formulated (Malhotra and Sukumar 1989). These rules have been Incorporated into the 1989 edition of the CSA standard (CSA 1989). The model was applied to a number of possible configurations of layered columns made with nails, or bolts, or split-ring connectors in conjunction with bolts. To use these rules, particular sets of requirements pertaining to column configurations and fastenings are to be satisfied in each case as listed in CSA (1989). For each type of connectors, the essence of the design rules are briefly given as follows:

FIG. 1. Spaced and layered columns.

* b=114, 2 nails

90 - b=184, 2 nails - - b=286. 3 nails s - 7 b=387, 4 nails h 80 - C - 0 70 - .- - Ci . LID

-4-J"' 1 0 20 3 0 40 5,O 6 0 7 0

50 . I . ' ~ - ~ : ~ ~ I ' ~ I . ; ~ I ~ I . I ' . I . I ~ ~ ~ I . ~

0 20 4 0 60 8 0 100 120 140 160 180 200 220 240 260 280 300 Llr

FIG. 2. Efficiency graph - nailed layered columns. Three 38 mm layers with nails of 5.38 mm in diameter.

A LID

b= l l4 . 2 bolts - b=184, 3 bolts - b=286, 4 bolts --a- - b=387, 6 bolts

1 ip 2 0 3 0 40 5.0 6 0 7 0 5 0 1 . , . 1 . 1 ! , . I ' . , . ' I . t . , . I ' . I . ; . , . , . ~

0 20 4 0 6 0 8 0 100 120 140 160 180 200 220 240 260 280 300

Llr

FIG. 3. Efficiency graph - bolted layered columns. Three 38 mm layers with bolts of 6.4 mm in diameter. The compressive resistance of a built-up (layered) compres-

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174 CAN. J . CIV. ENG.

1 3 8 1 6 4 1 3 8 1 T T i 1 (all dlmenslons In mm.)

Connectors : 6.4 mm dia. bolts Wood: Eastern Spruce

COnstrucIlon Grade No. 1

FIG. 4. Test specimens - layered columns.

sion member fastened together with nails or spikes may be taken as 60% of the factored compressive strength of a solid member having a slenderness ratio based on the least dimension of the composite member. The corresponding values for members fabricated with bolts and split-ring connectors in conjunction with bolts are 75 % and 80 % , respectively.

Experimental program - built-up columns The experimental work was designed and undertaken to

accumulate data on the strength of built-up columns of two types, namely layered columns and spaced columns. For each type, a total of 48 columns were tested in four slenderness ratios. Details of the test columns are given in Figs. 4 and 5. All columns were made of 38 mm x 140 mm construction grade No. 1 eastern spruce lumber obtained from a mill in Nova Scotia. The lumber pieces were not discriminated for knots, but severally warped pieces were avoided. The lumber as received from the mill was used to fabricate the columns without adjusting for moisture content, as the intention was to produce a data base of column strength in the field conditions. From the lumber pieces forming built-up columns, samples were taken to assess the average moisture content of the wood material at the time of testing. Compression specimens of dimensions 38 mm x 38 mm x 150 mm were also taken to find the characteristic material properties of the wood, that is, the ultimate compressive strength and the modulus of elasticity. Some 269 clear compression samples were tested to get these values for the column material. These values were then adjusted to a standard moisture content of 12% in order to unify the basis from which the property statistics are obtained. Since the average moisture content of each column at the time of testing was recorded, individual values in the distributions of E and F, were adjusted to the moisture content of the corresponding columns.

The columns were fabricated using steel bolts of 6.4 mm in diameter for the layered and spaced columns in different connector configurations as detailed in Figs. 4 and 5. For layered columns, the bolts were placed in two longitudinal rows, with

• Connectors : 16 - 6.4 mm dla. bolts . Wood : Eastern S p ~ c e Consl~ction Grade No. I

Spacer blocks: 140 X 38 mm

Slenderness Speclmen para. ~ a l l o Leng lh LS L C

L I ~ L/r (mm) (mm)(mm)

s 1 3 0 3 3 . 6 3 1 1 8 0 3 4 6 2 8 2

5 2 4 5 4 8 . 2 9 1 7 6 0 5 4 0 4 7 6

5 3 6 0 6 2 . 4 5 2 2 8 0 7 2 7 6 6 3

5 4 9 0 9 2 . 2 9 3 5 0 0 1 1 2 0 1 0 5 6

FIG. 5. Test specimens - spaced columns.

spacing of the bolts along the column length being 6 times the thickness of the individual piece of the column. For spaced columns, 16 steel bolts of 6.4 mm in diameter were used in each of the two spacer blocks and two end blocks in each column, as shown in Fig. 5. In total, 48 layered columns and 48 spaced columns were fabricated and tested. Test columns were of slenderness parameters LID = 10, 15, 20, and 30 in layered column series (C1 to C4) and Lld = 30, 45, 60, and 90 in spaced column series (S1 to S4).

Test results Layered columns

The results of layered column tests along with the model predictions are shown in Fig. 6. The figure shows the average test results, 95% confidence limit values, and the prediction mean values. The column strength predictions are found to be in close agreement with the test values. Table 1 gives the results of layered column tests.

Spaced columns The results of spaced column tests along with the model pre-

dictions are shown in Fig. 7. The figure shows the average test results, 95 % confidence limit values, and the prediction mean values. The column strength predictions are obtained using effective number of connectors in the calculation (see discussion below). Table 2 gives the results of spaced column tests.

Effective number of connectors The design model assumes that the slip is uniformly dis-

tributed throughout the column length. Thus, it is implied that each connector transfers the same magnitude of shear, irrespective of its position along the length. Actually, as bending occurs, the end connectors will be subjected to maximum slip and shear transfer, varying to a minimum in the middle length. This behaviour is more evident in the case of spaced columns, which have connectors grouped at the end blocks and spacer blocks. The average test results of the spaced columns are smaller than the predictions, when all the connectors are

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MALHOTRA AND SUKUMAR 175

500 - 500 - - 5 400 - - 5 5 4. 2 300 - 2 300 - b '.,. Prediction Mean E 95 %Confidence Limits !! 95 %Confidence Limits z

200 - z 200 -

E 5 - - 0 100- 0

0 1 0 0 - 0

FIG. 6. Test results and model predictions (layered columns).

considered equally effective. However, upon considering end connectors alone to be effective, the test results matched closely with the predictions. The predictions and the test results matched almost exactly in the case of long columns. Similar views about the effectiveness of end connectors were expressed by Bryant (1987). The current design specifications also recognize this effect, requiring a designer to provide con- nections for end blocks to transfer a minimum specified strength which is dependent on Lld of the member. The spacer blocks along the column length ensure the composite column action in bending and buckling.

In the case of layered columns, there is no heavy concentra- tion of shear on end connectors, as in the case of end block connectors of spaced columns. The assumption that all connectors are effective is more realistic in the case of layered columns than for spaced columns. An investigation to study, in detail, the effect of connector configuration along the built- up column length on the column strength is in progress.

Development of the strength data base

From the compression samples taken from the lumber used to fabricate the test columns, the distributions of a fairly large population of ultimate compressive strength and modulus of elasticity are obtained. Similarly, from the knot data of the lumber pieces used to fabricate 96 built-up columns, the distribution of knot ratios in the test material is calculated. Test data of E and Fu are fitted to normal, lognormal, and three- parameter Weibull types of distributions (Figs. 8 and 9). The types of distributions that gave the best fit were selected for simulation purpose. The knot ratio followed approximately the normal distribution and was taken as such for further computations. The ultimate compressive strength, Fu, and the modulus of elasticity, E, were close to the three-parameter Weibull type of distribution as seen in Figs. 8 and 9, respectively. Table 3 gives the details of material property data obtained from compression tests, along with the Weibull parameters.

The column strength data base was then expanded by a Monte-Carlo simulation by "picking the wood" from the data base of material properties and knot ratio, and by drawing a set of these values at random. The column strength values so calculated were added to the column test results. For the results presented in this paper, 2000 simulations were done and any additional simulations did not show any significant change in the final outcome of reliability analysis. The procedure was repeated for layered and spaced columns for various slenderness ratios.

A FORTRAN program called BUCREL (built-up column relia-

FIG. 7. Test results and model predictions (spaced columns).

bility analysis) was developed in order to create strength data base for built-up wood columns of any geometric configuration and type of wood material. Details of this program are given in Sukumar (1990). The distribution of ultimate strength of the test columns was compared with the normal and Weibull distributions as shown in Figs. 10 and 11 for layered columns. Similar plots were also created for spaced columns.

Reliability analysis using BUCREL

The load and resistance factor design equation, applied in limit states design format, states that the factored resistance should equal or exceed the effects of factored loads.

where 4 is the resistance factor; R,, the mean resistance; y, the load effects factor; and S,, the mean load effects.

Considering a, and as to be the standard deviations of R and S, respectively, a function Znl could be defined which has a standard deviation, az, as follows:

The reliability index, P , can be expressed as

The reliability index values are calculated using the program BUCREL. AS explained earlier, BUCREL creates an expanded data base and then performs the reliability analysis. The reliability index values corresponding to various combinations of load ratios and resistance factors are calculated using the first- order second-moment expression, [13] :

where VR and Vs are coefficients of variation of resistance and load effect, respectively.

Typical results of reliability analysis of layered columns of L/D = 20 and spaced columns of L/d = 60 are given in Figs. 12- 14. Details of the results for other slenderness ratios are available in Sukumar (1990). The inputs for BUCREI. are column configuration, connector data, and values of column strength as obtained from laboratory tests of built-up columns along with the material property distributions. Twelve

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CAN. J . CIV. ENG. VOL. 18. 1991

TABLE 1. Test results and model predictions of layered columns

95% C.L . t (kN) Test Fu* (MPa) E* (MPa) Test value Model

column average average mean (kN) prediction (kN) Low High

*Adjusted for knots and moisture content in columns. tC.L. refers to confidence limits.

TABLE 2. Test results and model predictions of spaced columns

95% C.L.$ (kN) Test Fu* (MPa) E* (MPa) Test value Model

column average average mean (kN) prediction? (kN) Low High

*Adjusted for knots and moisture content in columns. tModel predictions are based on the effective number of connectors. K . L . refers to confidence limits.

t Weibuli -u- Normal

. . . . . . . 1 2 ' ' 1 4 ' '16 ' ' 18 ' 20 ' ' 2 2

H Test t Weibull I

> -u- Normal 0.20

a 0 D L

u 0.10

0.00 la ' 2 0 ' 2 4 ' 2 8 ~ 3 2 ' 3 i j ~ 4 0 ' 4 4 ' 4 8 ' 5 2 ' 56' 6 0 ' 6 4 '

E (x1000 MPa) F, (MPa)

FIG. 8. Modulus of elasticity (E) values obtained from compression FIG. 9. Ultimate compressive strength (F,) values obtained from tests on clear samples. (Mean E = 11798.31 MPa; standard devia- compression tests on clear samples. (Mean E = 39.57 MPa; standard tion = 2471.68 MPa.) deviation = 5.477 MPa.)

columns were tested in each category of slenderness ratio, both in layered and spaced columns, and these values were used as inputs in performing the reliability analysis. Figure 12 shows the variation of reliability index for layered columns (test results combined with simulated values) corresponding to different resistance factors. Figure 14 shows a similar graph for spaced columns. BUCREL also calculates the reliability index values based on the simplified design rules for the layered columns (Fig. 13).

Reliability analysis using design point method Reliability analysis is also performed by taking into account

most of the important random variables that influence the judgement of survival or failure of a structure, using the design point method originally proposed by Rackwitz and reported in a CIRIA report (CIRIA 1977). According to Rack- witz, for reliability analysis in the advanced second-moment methods, the limit state equation in the generalized form can be written as

[14] Z = [g(XI, X2, . . . , Xi . . . , X,,) - load effect] > 0

where XI, X2, . . . are the basic variables affecting the design. In the case of built-up wood columns, Z is a function

representing the load effect and the column strength computed from the model used with the following variables: XI = E, modulus of elasticity of the column material; X2 = F,, ultimate compressive strength of the column material; X3 = K, slip modulus of the connection; and X4 = A, slenderness ratio of-the column.

For a particular set of values of random variables, the buckling strength is given by g(XI, . . . , X4). The value of load effect (S,) is a constant for a set of particular snow load to dead load ratio (SLIDL) and resistance factor (4). With these four major variables (XI to X4) and a constant load effect, a design point, X*, can be established for a particular reliability level. The calculation of the load effect, S,, is discussed later in this paper.

The design point method of reliability analysis is suitable

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MALHOTRA A N D SUKUMAR 177

TABLE 3. Strength properties of column material

Compressive strength, Modulus of elasticity, F, (MPa) E (MPa)

For c l e a r t e s t s a m p l e s * Average = 39.57 Std. deviation = 5.48 Weibull parameters:

Location, E = 24.3 Scale, w = 17.19 Shape, K = 3.08

For k n o t t e d s a m p l e s t Average = 28.56 Std. deviation = 3.95

Average = 11 798 Std. deviation = 2472 Weibull parameters:

Location, E = 5541 Scale, w = 7033 Shape, K = 2.742

Average = 10 156 Std. deviation = 2128

*Average o f 269 samples. ?Average o f 2000 simulat ions.

-"" I I ~ 4 Simulation I

Bar Charts I C3 Simulation 1 C2 Simulation I C1 Simulation

+- C4 Normal I -n- C3 Normal + C2 Normal -A- C1 Normal

Column Strength (kN)

FIG. 10. Simulations and normal distributions of column strengths - layered columns.

400 - U) c

300 m -

;1 200

so 8 := 100

0

Column Strength (kN)

FIG. 11. Simulations and Weibull distributions of column strengths - layered columns.

when the random variables are normally distributed. So, if the distribution of a variable Xi is not normal, an equivalent and mg are to be deduced from the actual distributions of this variable. In the present study, the variables XI and X2 (modulus of elasticity and compressive strength) are considered to follow Weibull distribution as explained in a previous section. The variables X3 and X4 (slip modulus and slenderness ratio) are taken as normally distributed.

For reliability analysis using the design point method as applied to built-up timber columns, an iterative procedure is used in the development of a FORTRAN program called POINT, the details of which are shown in a flow chart (Fig. 15).

Load Ratio, SLIDL

FIG. 12. Variation of reliability index values computed by BUCREL

- layered columns (C3) with L / D = 20. -

Load Ratio, SLIDL


for simplified design rules - layered columns (C3) with LID = 20.

l ! . I . l . I . I . l . l . l . i

0 1 2 3 4 5 6 7 8

Load Ratio, SLIDL


- spaced columns (S3) with L / d = 60.

Program POINT

The strength space of built-up wood column is repre- sented by

1151 z = g(A, E, F,, X, K ) - S,

where Sm (mean dead load + mean snow load) is the factored load effect derived on the basis of recommendations from the National Building Code of Canada (NBCC 1985), for a particular dead load to snow load ratio.

[16] Z = A(Fcr) - Sm

in which Fcr (buckling stress), a function of E, F,, X, and K, is given by [I]; A is the cross-sectional area of the column; and

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I Assume lnltlal X' with mean values of the varlables

Compute Z=g(X*)-SM I Compute

V) g'l = aglaxl (numerically) c for all variables 0 .- - m L Q, - .- Compute the sensltlvlty factor,

0 N

C g i ' (X)* o i2

+ y e s

I Print, O and X' I FIG. 15. Flow chart for program POINT.

S, is the factored load effect, calculated assuming a resistance factor for each cycle of iteration for which the corresponding reliability indices are obtained.

For a given column configuration, material properties, connector details, and load ratios, POINT gives more or less exact reliability levels in terms of reliability index values, from which a suitable resistance factor can be chosen for possible adoption in a reliability-based design procedure. The input requirements of the program are geometric configuration of the column, connector data, and mean and standard deviation values for each of E, F,, X, and K . For E and F,, which follow Weibull distributions. minimum values are also to be provided. From the specified strength and modulus of elasticity values for design, the program computes the factored compressive strength and the mean load effect. Usually, a value of 3.0 for the reliability index, 0, is taken as an initial value for the analysis.

Compute equivalent

normal parameters for the variables that are not

normal

Figure 16 shows a set of graphs corresponding to the test layered columns of slenderness parameters L/D = 20; and Fig. 17 for spaced columns corresponding to slenderness parameters L/d = 60. To get a complete spectrum of reliability levels associated with the rational design model, the reliability index values are calculated for different resistance factors and for a wide range of load ratios.

Results of the reliability analysis For the reliability analysis, using BUCREL and POINT, the fac-

tored compressive resistance of the columns is calculated using specified compressive strength and 5th percentile value of modulus of elasticity as given in the CSA standard (CSA 1989). In computations, the Malhotra and Van Dyer model is adopted and the load effect (S,) is taken as the sum of mean dead load and mean snow load. Mirza and MacGregor (1982)

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MALHOTRA A h ID SUKUMAR 179

l l . l ' l . l . l ' 1 . 1 . 1 . 1

0 1 2 3 4 5 6 7 8

Load Ratio, SLlDL

FIG. 16. Variation of reliability index values computed by POINT -

layered columns (C3) with L/D = 20.

l l ' l . l . l . l ' 1 . 1 ' 1 . 1

0 1 2 3 4 5 6 7 8

Load Ratio, SLlDL

FIG. 17. Variation of reliability index values computed by POINT -

spaced columns (S3) with L/d = 60.

Resistance Factor, 0

FIG. 18. Variation of reliability index with respect to resistance fac tor - layered columns (C3) with L/d = 20.

suggested the ratios of mean-to-nominal dead load and mean- to-nominal live load as DL,/DL = 1.05 and SL,/SL = 0.80. Similar values have been used by many other researchers such as Lind (1971) and Sexsmith and Fox (1978). In this research, no specific study has been done on the statistical distribution of loads. The nominal dead load and snow load are calculated from the factored compressive strength (PC,), corresponding to particular load ratio and resistance factor using [17].

[17] q5Pc, = 1.25(DL) + 1.5(SL)

Layered columns Reliability analysis for layered columns was performed

using BUCREL and POINT for various combinations of load ratios and resistance factors as explained earlier. Figure 18

Resistance Factor, 0

FIG. 19. Variation of reliability index with respect to resistance factor for simplified design rules - layered columns (C3) with L/D = 20.

1 : . I . , ' , . 1 . 1 . 1 ' 1 ' 1

0 1 2 3 4 5 6 7 8

Load Ratio, SUDL

FIG. 20. Plots of reliability index versus load ratio for Layered columns (resistance factor = 0.80).

Load Ratio, SLlDL

FIG. 21. Plots of reliability index versus load ratio for layered columns - simplified design rules (resistance factor = 0.80).

shows the plots of reliability indices calculated using design point method, and Fig. 19 is based on the simplified design rules. The plots show the variation of reliability index against resistance factor for columns with L/D of 20. The reliability levels obtained by the simplified design rule and the model for a resistance factor, 4 , of 0.8 are shown in Figs. 20 and 21. The /3 values calculated by the basic first-order second- moment analysis (as in BUCREL) and the design point method (as in POINT) are found to be in close agreement. This comparison is illustrated in Fig. 22. Taking a target reliability index of about 3.0, the resistance factor could be safely taken as 0.8 as seen in the examples presented here. For a higher reliability index, the resistance factor may accordingly be reduced, depending on the design situation in hand. It can be observed

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180 CAN. J. C N . ENG. VOL. 18, 1991

VD=2O, (BUCREL) 1.1 , , , , a , , , , , , - VD=20, (POINT) tc

1 0 1 2 3 4 5 6 7 8

Load Ratio, SUDL

FIG. 22. Comparison of reliability index values calculated by using BUCREL and POINT - layered columns (C3) with L/D = 20.

Resistance Factor, o

FIG. 23. Variation of reliability index with respect to resistance factor - spaced columns (S3) with L/d = 60.

that the reliability index values are fairly consistent throughout the entire range of load ratios for columns of different slenderness parameters.

Spaced columns Reliability analysis similar to that for bolted layered

columns is carried out for spaced columns as well. Figure 23 shows the variation of reliability index with respect to resistance factor for various load ratios for columns with L/d of 60. The reliability index values are consistent throughout the entire range of load ratios. For a resistance factor of 0.8, Fig. 24 gives the comparison of 0 values as obtained by the two programs (EUCREL and POINT).

Application to design procedure

The factored compressive resistance of built-up columns can be calculated in the same format as that for solid columns as given in the CSA standard (CSA 1989).

where P, is the factored compressive resistance (N); 6 is the resistance factor obtained from reliability analysis; A is the cross-sectional area of the composite column (mm2); and Fi is given by

in which F,, is the column buckling stress (MPa) calculated by [I], in which the value of F,, the specified compressive strength, is to be modified with a size factor, K,,, for visually graded lumber using the column dimensions. For spaced

Vd=6O. (POINT) - Vd=6O, (BUCREL)

Load Ratio, SLIDL

FIG. 24. Comparison of reliability index values calculated by using BUCREL and POINT - spaced columns (S3) with L/d = 60.

columns, only the connectors provided at the end blocks are to be considered effective for using [I] ,

in which d is the actual dimension of an individual piece in the direction of buckling (mm) and L is the actual length associated with the member dimensions (mm). KD is the load dura- tion factor; KH, the system factor; Ksc, the service condition factor; and KT, the treatment factor.

The values of Ks for different end-use conditions are to be taken according to the CSA standard (CSA 1989).

The value of the resistance factor, 6, can be safely taken as 0.8, as in almost all cases it gives a reliability index of around 3.0. By adopting this value of the resistance factor, a safety level consistent with that obtained for other materials of construction can be achieved. Moreover, this value of 6 equal to 0.8 is in agreement with the recommendations for solid columns in CSA (1989). Equation [18] can be used as a general expression in reliability design format for compression members. Using either EUCREL or POINT, a set of values of the reliability indices (P) and the corresponding resistance factors are obtained, from which 6 may be suitably selected for a particular reliability level desired. So, in [18], one would need to bring in the value of 6 by performing a reliability analysis. In the case of layered columns conforming to certain guidelines regarding column and connector configurations, the simplified design rules can be used, as it is easier to use them in day-to- day designs. For exact results and for other configurations, the column buckling equation [ l ] is to be applied. The simplified design rules are based on the lowest efficiency for a particular configuration of intermediate columns and connector data. For short and long columns, simplified design rules give conserva- tive results, as the efficiency of such columns could be much more than the specified values (that is, 75% for bolted columns and 60% for nailed columns, etc.). In the case of spaced columns, the formulations based on the model in conjunction with the design equation [18] is found to be quite versatile, as it takes into account almost all the variables and the uncertainties involved. These formulations can be extended to other types of built-up columns as well, although a detailed reliability analysis is needed for obtaining suitable resistance factors.

Conclusions

1. The mathematical model predicts closely the column buckling strength in the case of layered and spaced built-up

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MALHOTRA A N D SUKUMAR 181

columns. The model, based on a rational theory, is further verified by laboratory experiments. A procedure for the for- mation of a column strength database based on laboratory tests and simulations is presented, which forms the basis of reliability analysis.

2. A detailed reliability analysis is performed to assess the reliability levels attainable using the rational mathematical model. For this purpose, two reliability analysis computer programs, BUCREL and POINT, are developed. For a large number of cases tested, the reliability index values obtained by both programs are quite consistent and the variation is found to fall within a small range.

3. For built-up wood columns, a design formulation consistent with that of solid columns is recommended with resistance factor 6 = 0.8. This factor corresponds to an acceptable reliability index of about 3.0. Also, the reliability index values are observed to be consistent throughout a wide range of load ratios.

4. A set of simplified design rules based on the efficiency of built-up (layered type) columns have been formulated (CSA 1989). While these rules make the designs slightly conserva- tive, they are easy to apply in common design situations. Use of these rules will result in designs that are more economical than those provided by the 1984 version of the standard (CSA 1984). The simplified rules provide a reasonable and consistent reliability index throughout a wide range of snow load to dead load ratios.

5 . For other types of built-up timber columns, the design model may be extended, and reliability analysis performed, for developing a suitable reliability-based design.

Acknowledgements The authors would like to acknowledge the financial support

provided by the Natural Sciences and Engineering Research Council of Canada for this research.

BRYANT, A. H. 1987. Built-up wood columns. ASCE Journal of the Structural Division, 113(ST1): 107- 121.

CIRIA. 1977. Rationalization of safety and serviceability factors in structural codes. Construction Industry Research and Information Association, London, England, pp. 87- 113.

CSA. 1984. Engineering design in wood (limit state design). Stan- dard CAN-086.1-M84, Canadian Standards Association, Rexdale, Ont.

1989. Engineering design in wood (limit state design). Stan- dard CAN-086.1-M89, Canadian Standards Association, Rexdale, Ont.

FOSCHI, R. 0 . 1989. Reliability-based design: the Canadian experience. Proceedings, Second Pacific Timber Engineering Con- ference, Auckland, New Zealand, pp. 309 - 3 14.

KUENZI, E. W. 1955. Theoretical design of a nailed or bolted joint under lateral load. Report No. D 1951, U.S. Forest Products Laboratory, Madison, WI.

LIND, N. C. 1971. Consistent partial safety factors. ASCE Journal of the Structural Division, 97(ST5): 165 1 - 1669.

MALHOTRA, S. K., and SUKUMAR, A. P. 1989. A simplified design procedure for built-up wood compression members. Proceedings, Annual Conference of the Canadian Society for Civil Engineering, Vol. lA, St. John's, Nfld., pp. 1-18.

1990. Formulation of data base for reliability analysis of built-up wood columns. Proceedings, 1990 International Timber Engineering Conference, Tokyo, Japan, pp. 295 -301.

MALHOTRA, S. K., and VAN DYER, D. B. 1977. Rational approach to the design of built-up timber columns. Wood Science, 9(4): 174- 186.

MIRZA, S. A., and MACGREGOR, J. G. 1982. Probabilistic study of strength of reinforced concrete members. Canadian Journal of Civil Engineering, 9(1): 43 1-448.

NBCC. 1985. National building code of Canada. Associate Commit- tee on National Building Code of Canada, National Research Council of Canada, Ottawa, Ont.

NFPA. 1986. National design specification for wood construction. National Forest Products Association, Washington, DC.

RASSAM, H. Y., and GOODMAN, J. R. 1970. Buckling behaviour of layered wood columns. Wood Science, 2(4): 238-246.

SEXSMITH, R. G., and Fox, S. P. 1978. Limit states design concepts for timber engineering. Forest Products Journal, 28(5): 49-54.

SUKUMAR, A. P. 1990. Reliability-based design of mechanically connected built-up wood columns. Ph.D. dissertation, Technical University of Nova Scotia, Halifax, N.S.

SUKUMAR, A. P., and MALHOTRA, S. K. 1989. Efficiency of layered wood columns fabricated with nails, bolts, and split ring connectors. Proceedings, Twelfth Canadian Congress of Applied Mechanics, CANCAM '89, Ottawa, Ont., Vol. 1. pp. 366-367.

VAN DYER, D. B. 1976. Strength of built-up timber columns. Ph.D. dissertation, Technical University of Nova Scotia, Halifax, N.S.

VAN DYER, D. B., and MALHOTRA, S. K. 1985. Strength of spaced timber columns. Canadian Journal of Civil Engineering, 12(1): 36-42.

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Reliability-based design of mechanically connected built-up wood columns