Economic efficiency of glued joints in timber truss systems

\ PERGAMON

Building and Environment 23 "0888# 090Ð096

Economic e.ciency of glued joints in timber truss systems J[ O[ Afolayan Civil Engineering Department\ Ahmadu Bello University\ Zaria\ Nigeria Received 17 November 0885^ revised 00 June 0886^ accepted 08 January 0887

Abstract The potential losses associated with glue as a jointing material in timber construction is examined when inherent uncertainties in the basic design variables are quantitatively appraised[ The Mean First!Order Second Moment Method "MFOSMM# and Value Analysis "VA# are developed to estimate the reliability indices and the associated risk costs of the joints of an example roof truss[ The variables which describe the loading\ grade stress and geometrical properties applicable to joint analysis are treated as random variables and they are assumed to be normally distributed[ The results show indicative trends of the dependence of joint reliability and cost of failure loss on the degree of workmanship[ Þ 0887 Elsevier Science Ltd[ All rights reserved[

0[ Introduction Timber as a low density\ cellular\ polymeric composite does not fall into any one class of materials ^ rather\ it tends to overlap a number of classes[ Because of its high strength performance and low cost\ timber is found to be the world|s most successful _bre composite ð0Ł[ One factor that can a}ect the e}ective use of wood as a structural component is the limitation associated with the con! nection of wood members to each other or to other materials[ The relatively low bearing strength of wood\ combined with the possibility of cracks or splits develop! ing at mechanical fasteners\ can result in connections being considered as the {weak link| in a wood structure[ However\ recently\ the use of adhesives has provided an e}ective means of connecting a variety of wood com! ponents ð1Ł[ Glued joints in timber are very e.cient in load dis! tribution and as such\ concentrated local stresses are not so pronounced as for nails\ screws and bolts[ Desch ð2Ł said {{gluing does for timber what welding does for steel^ it enables joints to be made without cutting any material out of the members joined and makes it possible for the designer to take advantage of all the economies associ! ated with monolithic construction||[ He also remarked that glued joints are noted to be governed by the dimen! sions and physical properties of the wood used\ and the

Corresponding author[ Tel[] 99123 958 49470 ex 58^ e!mail] josefolÝ abu[edu[ng[

shape of the joint\ rather than by the qualities of the particular glues selected[ Moisture has also been noted to be the most important factor in glue bond durability ð3Ł\ although _re\ living organisms and mechanical force can weaken or destroy glued joints[ Currently\ there are no standards for the design of glued joints\ but there are e}orts underway in the U[S[A[ and in Europe to develop design method and standards for structural glued assembly joints ð4Ł[ Tyler ð5Ł noted that the proper use of glue in the fabrication of structures would lead to joints of high strength[ However\ their use should be con_ned to high quality work where adequate supervision could be given[ For instance\ for bolted joints it is possible after fabrication to observe that bolts and connectors are in position\ while it is not possible to ascertain whether a glue has been mixed properly and applied in a manner that will ensure a good joint[ This paper makes an attempt to describe the probable loss associated with the use of glue as a jointing material\ and the dependence of its reliability on the degree of quality control in timber construction[ The analysis simu! lates the possibility of varying workmanship by using arbitrary coe.cients of variation for the relevant design variables[ A typical glue!jointed timber truss system designed in accordance with BS 4154^ Part 1 and 2 ð6\ 7Ł and NCP 1^ 0861 ð8Ł\ is examined[ The material strength\ geometric properties and loading are considered as ran! dom variables[ The Mean First Order Reliability Method ð09Ð01Ł and the concept of value analysis ð02Ł are implemented to compute the inherent failure probability levels and the associated quiescent losses[

9259Ð0212:88:,*see front matter Þ 0887 Elsevier Science Ltd[ All rights reserved PII] S 9 2 5 9 Ð 0 2 1 2 " 8 7 # 9 9 9 9 2 Ð 0

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J[O[ Afolayan:Buildin` and Environment 23 "0888# 090Ð096

1[ Basic assumptions for glued joint design

z ¹ `"x#¦S"xi−xi#

Most references in the literature address adhesive design in a qualitative manner ð1Ł[ Descriptions usually concentrate on schemes for minimizing stress con! centrations\ tension bond stresses\ or joint eccentricity[ Discussion of allowable design parameters for adhesives are not common in literature[ The general conventional approach to designing adhesive joints has basically been empirical and strength tests are often recommended to supplement theoretical analysis ð1Ł[ In the design of adhesive bonded timber\ the engineer has traditionally taken one of two approaches[ In the _rst case\ the adhesive selected for use is stronger and more durable than the adherents\ under all loads and end use conditions[ However\ the process of selecting such an adhesive may be quite di.cult and requires extensive testing and evaluation[ The alternative approach is to develop a set of design stress criteria which enable the engineer to both rationally analyse and design the com! ponent[ This approach requires not only analytical pro! cedures to predict the actual stress conditions at critical points of the bonding surface\ but also a set of allowable stresses for comparison[ In this paper the assumptions are based on the BS 4154 ð6Ł and NCP 1 ð8Ł requirements\ viz]

$ % 1` 1xi

"1# x

where x is the vector representing the linearization point at which the partial derivative in eqn "1# is evaluated[ When x corresponds to the mean value vector "as pre! supposed in this study#\ the procedure is called the Mean First Order Second Moment "MFOSM# method[ A sur! vival indicator\ b\ "also commonly referred to as the safety or reliability index#\ is obtained by dividing the mean value of z by its standard deviation[ Although the computation strategy of MFOSM is simple\ its exactness depends on the magnitudes of the coe.cients of variation of elements of x and the e}ect of the higher order terms[ As long as `"x# is linear and the variables are independent\ the computed b is also accurate[ Hence\ the probability of failure\ Pf can be related to b in the form] Pf  0−F"b#

"2#

where F"b# is the cumulative distribution of the variate b[

3[ Potential loss model

"0# Joints should be designed to resist\ in tension\ com! pression and shear\ the axial forces in abutting mem! bers subjected to the service loads[ In addition\ all joints should be able to sustain the handling loads^ "1# The arrangement of members in a joint should be such that the longitudinal axis of each member passes through or close to a common point[ When this is not the case\ the resulting eccentricity should be allowed for\ in the design of both the fasteners and the jointed members[

Codes of practice should be able to provide or indicate tolerable levels of risk which achieve equitable balance in bene_ts "failure prevented# and costs[ Conventionally there is no deterministic based design philosophy that accedes this expectation in an explicit manner[ Naturally\ if one examines the basic economics of cost value analysis "see ð02Ł#\ it becomes reasonable to assume that the initial expected cost ECI of any structure is directly proportional to its estimated resistance[ On the other hand\ the risk cost ECP decreases with increasing design resistance[ It then follows that the expected total cost ETC can be expre! ssed as

2[ Method of rating joint ef_ciency

ETC  ECI¦ECP

The satisfaction of operational and economic require! ments with a speci_ed measure of reliability is a key design objective[ But in practical design\ phenomena which are non!orderable sets\ are often being dealt with because structural problems are practically non! deterministic[ Thus\ a probabilistic analysis is required when the actual scatter of each variable involved in deter! mining the response of a structural system to uncertain loading is to be incorporated in the safety assessments[ The probabilistic failure analysis used in this study fol! lows the _rst order second moment procedures ð09Ð01Ł[ The performance equation relating the interaction of the resistance and loading variables

If the life!time and the annual probability of failure are related and accounting for economic ~uctuations\ eqn "3# can be expanded to give]

`"x#  9 is linearized to yield

"0#

"3#

N

ETC  ECI¦ECVlF s ECPiPfi

"4#

i0

in which ECV 

0−e−"R−I#T T"R−I#

is a function accounting for economic ~uctuations and dependent on the interest rate R\ in~ation rate I\ and the design life T^ and lF is the life!time failure probability related to the probability\ P_ associated with the event due to cause i out of N resulting into failure and ECPi

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J[O[ Afolayan:Buildin` and Environment 23 "0888# 090Ð096

denotes the potential loss in respect of failure cause i[ According to ð03Ł\ lF  0−e−Pf T

"5#

Since our concern is a check on the probability of any glued joint being loaded beyond its recommended capacity under uncertainties inherent in loading\ and material and geometrical properties\ eqn "4# becomes ETC  ECI¦

ECPlF"0−e−"R−I#T# Pf T"R−I#

"6#

"0−e−A# ECP A ECI

"8#

G"X#  R−S

in which R  joint strength and S  load e}ect in member[ If N is the design glue stress and A being the glueline area required\ then "09#

R  N[A[

From eqn "6# the risk cost index is the second term and can be expressed as] PL  lF

These loads are transformed to point loads applied at the node points as given in Section 3\ clause 05\ on the truss system given in Fig[ 0[ De_ning a failure function G"X# for failure of any joint as required in Section 2\ we have]

"7#

For the purpose of illustration the joint detail in Fig[ 1 is considered[ Thus\ the glueline area required in case of Fig[ 1"a# is A  L0L1z0−cos1 u

so that A  "R−I#T[ Thus\ using any plausible com! bination of interest and in~ation rates\ and design life\ a range of tolerable risk levels can be de_ned[ As intended in this investigation\ failure is de_ned as the violation of the allowable stress levels speci_ed for a glued joint in respect of quanti_ed uncertainties in material\ loading and geometrical properties[ 4[ Estimate of joint failure The truss system shown in Fig[ 0 is analysed and the design loads used are taken to be considered for long term loading] "0# the dead load  9[574 kN:m1 uniformly distributed on the full length of the rafter along the slope^ and "1# the ceiling tie loads uniformly distributed on the full length of the rafter consisting of the dead load  9[14 kN:m1 and the imposed load  9[14 kN:m1[

"00#

For all the internal members inclined at an angle u\ N in eqn "09# is modi_ed by the Hankinson formula given in NCP 1 ð8Ł\ Clause 6[7[1 as N

PQ

"01#

1

P sin u¦Q cos1 u

where P  grade stress for wood species for shear parallel to grain and Q  grade stress for wood species for shear perpendicular to grain[ Relating Q and P as suggested in NCP 1 ð8Ł and substituting for N and A in eqn "09#\ we have\ from eqn "8#] G"X# 

Ph10 sin u"0¦2 sin1 u#

−S[

"02#

Equations "2#\ "6# and "02# essentially lead to the com! putation of the probability of failure levels and the associ!

Fig[ 0[ Schematic of example truss[

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J[O[ Afolayan:Buildin` and Environment 23 "0888# 090Ð096

Fig[ 1[ Details of typical joints[

ated losses[ Simulating the degree of quality control by the coe.cients of variation of the relevant decision vari! ables\ the results for some selected joints for the di}erent details are shown in Figs 2Ð5[ 5[ Signi_cance of results In the example problem\ it is assumed that R−I  09)\ T  49 years and the ratio ECP:ECI is 49 in eqn "5#[ Also\ an arbitrary but practical range for the coe.cients of variation of the decision variables is assumed\ in the estimate of the probabilities of failure and the risk cost parameters for the selected joints[ Figs 2Ð5 show the indicative trends of the consequence of quality control on the reliabilities and the implied failure costs of the joints[ These results show that for a range of coe.cients of variation in the decision variables\ joint e.ciency and the associated failure cost can be predicted[ In all cases\ a and a0 respectively\ represent the safety index and the implied failure cost parameter in relation to the grade stress[ Similarly b and b0 are for the e}ects associated with the angle of inclination of the member\ while c and c0 stand for the e}ects related to the pro! portion of the member length within the glueline area[ Figs 2 and 3 represent the results associated with the joint detail given in Fig[ 1"a#\ while Figs 4 and 5 are for Fig[ 1"b#[ It is observed from this example problem that the reliability and failure cost parameter vary from joint to joint[ This reveals one of the drawbacks of the design methodology that is non!probabilistic just as the BS 4157] Parts 1 and 2 ð6\ 7Ł and NCP 1 ð8Ł design requirements connote[ It is also possible from these results to advise on the degree of quality control that is commensurate with a desired reliability level and minimum risk cost[ The risk cost parameter develops rapidly after a certain

range of the coe.cients of variation\ which signi_es a drop in joint e.ciency[ It is to be noted that when there is a high quality control\ the proportion of the member length\ h\ in the glueline area\ contributes more sig! ni_cantly to joint e.ciency than the angle of inclination of a member or the grade stress[ However\ this obser! vation is limited to a range of coe.cients of variation[ As workmanship becomes poorer "i[e[\ the coe.cient of variation becomes large#\ joint reliability drops abruptly in relation to this variable with an accompanying startling cost of failure[ Meanwhile\ from these results\ it can be suggested that to maintain a reasonable e.ciency and minimise failure cost of glued joints\ more attention should be given to ensure adequate member length within the glueline area[ It is also evident that the empirical strategy employed in designing glued joints cannot provide a uniform reliability level nor predict a consistent expected potential loss[

6[ Conclusions The implied reliability levels and the associated failure costs of glued joints in a truss system have been estimated[ All the variables relevant to a joint performance are tre! ated as normally distributed random variates[ The results have highlighted the relative dependence of a joint e.ciency and its potential loss on the quanti_ed quality control measure[ It is also observed that higher control e.ciencies are required for the member geometrical properties connected with the glueline area than the grade stress in order to maintain high reliability and minimum failure cost[ Also\ the conventional design methods invoked cannot provide uniform level of reliability for all the joints\ as noted in the paper[ It is therefore suggested

J[O[ Afolayan:Buildin` and Environment 23 "0888# 090Ð096

Fig[ 2[ Relationship between joint e.ciency\ potential loss and quality control measure for member BH for joint detail "a#[

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Fig[ 3[ Relationship between joint e.ciency\ potential loss and quality control measure for member DG for joint detail "a#[

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Fig[ 4[ Relationship between joint e.ciency\ potential loss and quality control measure for member BH for joint detail "b#[

Fig[ 5[ Relationship between joint e.ciency\ potential loss and quality control measure for member DG for joint detail "b#[

J[O[ Afolayan:Buildin` and Environment 23 "0888# 090Ð096

that probability!based design criteria be adopted in glued joint design\ if safety and economy must be traded!o}[

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References ð0Ł Illston JM\ Smith AA\ Smith JM[ Concrete\ Timber and Metals[ In] The Nature and Behaviour of Structural Materials[ New York\ U[S[A[ Van Nostrand Reinhold Company\ 0868[ ð1Ł Faherty KF\ Williamson TG[ Wood Engineering and Con! struction Handbook[ New York\ U[S[A[] McGraw!Hill Publishing Company\ 0878[ ð2Ł Desch HE[ Timber\ its Structure and Properties[ London] Mac! Millan and Co Ltd\ 0845[ ð3Ł Selbo ML[ Adhesives Bonding of Wood[ Technical Bulletin\ No[ 0401\ U[S[ Department of Agriculture\ Forest Service\ Washington DC\ U[S[A[\ 0864[ ð4Ł River BH[ Gluing of Wood and Use of Adhesive for Structural Purpose[ Personal Communication\ Madison\ U[S[A[\ 0881[ ð5Ł Tyler RG[ The Structural Properties of West African Timbers[

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Note N[ 2\ West African Building Research Institute\ Accra\ Ghana\ p[ 01\ 0851[ British Standard 4157] Part 1[ Code of Practice for Permissible Stress Design\ Materials and Workmanship[ British Standards Institution\ London\ 0873[ British Standard 4157] Part 2[ Code of Practice for Trussed Rafter Roofs[ British Standards Institution\ London\ 0874[ Nigerian Code of Practice "NCP# 1[ Structural use of Timber\ Nigerian Standards Organisation Publication\ Lagos\ Nigeria\ 0862[ FORM\ User|s Manual\ RCP GmbH\ Munich\ 0877[ Rackwitz R\ Fiessler B[ Structural Reliability Under Combined Random Load Sequences[ Computers and Structures\ Vol[ 8[ pp[ 378Ð83\ 0867[ Ang AH!S\ Tang WH[ Probability Concepts in Engineering Plan! ning and Design\ Vol[ II] Decision\ Risk and Reliability[ New York] John Wiley and Sons\ Inc[\ 0873[ Afolayan JO[ Management Value Analysis for E}ective Quality Control[ Accepted for Publication in Quality Review "An Inter! national Journal of Quality Management Practice#[ Dover AR\ Bea RG[ Application of Reliability Methods to the Design of Coastal Structures[ Paper presented at the Coastal Struc! tures |68 Conference\ Alexandria\ Virginia\ March 03Ð05\ 0868[