COMPUTATIONAL MODEL OF ADHESIVE SCARF JOINTS IN TIMBER BEAMS

Transcription

1 Drewno 04, Vo. 57, No. 9 DOI: 0.84/wood RSARCH PAPRS Piotr Rapp COMPUTATIONAL MODL OF ADHSIV SCARF JOINTS IN TIMBR BAMS The sbject of this paper is the enera formation of a mode for scarf adhesive joints in timber beams within the framework of pane inear easticit. It is assmed that wood is orthotropic. The joint can be sbjected to a compe oadin state incdin an aia force, a bendin moment and a shear force. The joint mode is iven in dispacements b means of a set of for partia differentia eqations of the second order. Bondar conditions cater for sharp edes in the adherends. Compete sotions to theor of easticit eqations are presented and discssed. The manner in which the joint transmits the aia force, the bendin moment and the shear force is presented. It is shown that the scarf joint does not featre stress concentrations and that there eists an approimate eqivaence of dispacements and stress states in scarf jointed and continos eements. Kewords: timber beam, adhesive scarf joint, anatica mode, two dimensiona dispacement-stress anasis, orthotrop, inear easticit Introdction The repair of timber strctres incdin beams is of practica importance in civi enineerin. An eampe of this wod be the repacement of a biooica-corroded, damaed end section of a foor beam or a rafter. Simpe methods, sch as when a new frament is joined with the od beam b varios tpes of wooden pates or stee profies, have been in se for some time. The are effective bt sa ead to a chane in the eterna ook of the eements, chanes in the static scheme and appication of the materias which differ from the oriina ones. Sch sotions are not acceptabe in historica strctres, where not on are the Piotr Rapp (piotr.rapp@pt.poznan.p, piotrrapp@mai.com), Poznan Universit of Technoo, Poznan, Poand

2 6 Piotr Rapp architectre and decoration of reat vae, bt aso the strctre itsef. In sch cases, the shapes, the appearance of the strctra eements, the static schemes, materias, etc. mst be preserved. One possibe sotion aowin for the repair of a damaed eement, whie preservin its oriina shape, appearance, materia, strenth, and static scheme, is the se of an adhesive scarf joint (fi. ). Fi.. Adhesive scarf joints in wooden beams: a) scarf joint and its oadin, b) scarf joints in a wide beam It is important that in the considered joint the adhesive pane is parae to the Y ais in in the beam bendin pane 0XY (fi. a). In a wide beam, a sine scarf joint wod be too on, therefore mtipe joints can be sed to keep them sma see fi. b. In sch a case, it is assmed that the wide beam consists of severa narrower sements with sine scarf joints and each sement transmits an appropriate part of oadin. These tpes of joints are rare sed in wooden beam strctres and, as technooica sotions, the do not et possess compete theoretica or eperimenta docmentation. In other cases, differin from the ones described in this paper, scarf joints are freqent sed nder aia tension [rdoan, Ratawani 97; Redd, Sinha 975] or in beams nder bendin as joints in the form of micro-dovetais (finer joints), where the joint is perpendicar to the bendin pane [Tomasik 988; Smardzewski 996]. In this paper, the enera rests previos presented b Rapp are sed [00a, 00b].

3 Comptationa mode of adhesive scarf joints in timber beams 7 The adhesive scarf joint mode A timber beam with a rectanar cross-section consistin of two eements with thickness vaes and is considered. In enera, varios wood tpes ma be sed. The adherends are connected b an adhesive scarf joint. A set of co-ordinates 0XYZ is associated with the beam. The pane 0XY represents the bendin pane, accordin to fi.. The adhesive is a pane rectane formin the ane φ with the bendin pane 0XY. The adhesive projection on the pane 0XY is the rectane ABCD with dimensions (fi. ). Fi.. The adhesive scarf joint mode It is assmed that stresses across the adherend thickness are constant and form pane stress states parae to the pane 0XY. Hence, the adherends are considered as pane stress eements parae to the pane 0XY. The adherend thickness is measred in the direction norma to the pane 0XY. In the joint zone, the adherend thickness vaes () and () var inear from zero to and aon the X ais and are constant aon the Y ais. ), ( ) () ( The adhesive thickness t is measred in the direction norma to its pane. The adhesive is modeed as an isotropic inear-eastic medim with Yon s mods s, the shear mods s and Poisson s ratio s, where s ( s ) s.

4 8 Piotr Rapp Stresses in the adhesive are defined as interactions between adherend and the adhesive. There are shear stresses τ τ (, ) and τ τ (, ) in the adhesive pane and the stress σ N σ N (, ) norma to the adhesive pane. The stress τ is parae to the pane 0XZ, and the stress τ is parae to the Y ais (fi. ). Fi.. Stresses in the adhesive It is assmed that the stresses in the adhesive are constant across its thickness. The stresses presented in fi. are positive. De to the action of the shear stresses τ and τ, a shear deformation in the adhesive occrs. This eads to reative dispacements of aers in the adhesive in the direction parae to the adhesive pane. The stress σ N eads to aia strains norma to the adhesive pane. The dispacements in adherends and of the scarf joint are defined b the fnctions (, ) and (, ) aon the X ais and the fnctions (, ) and (, ) aon the Y ais. The dispacements,,, and are positive when the coincide with the positive orientation of the X and Y aes. It is assmed that the fnctions,,, and are C continos with respect to variabes and. In the foowin, the fnctions of the dispacements,,, and are considered as nknowns and a the qantities reated to the adhesive joint are epressed in their terms. The oadin of the adhesive joint can be represented b forces parae to the pane 0XY distribted aon the edes of the adherends. The oadin is positive when its orientation coincides with the positive orientation of the X and Y aes. There are aia forces, bendin moments and shear forces in the beam cross- -sections. The ead to the aia stresses p and p distribted inear, and the shear stresses p and p with a paraboic distribtion. Hence, it is assmed that the edes of the adherends can be oaded b aia forces, bendin moments and shear forces. The positive orientation of these forces and the correspondin stress distribtions are presented in fi. 4. The joint oadin mst remain in eqiibrim.

5 Comptationa mode of adhesive scarf joints in timber beams 9 Fi. 4. Loadin of the scarf joint adherends Constittive eqations for the adherends It is assmed that enera the adherends are made of different wood tpes, both orthotropic, and that the main aes of orthotrop coincide with the X and Y aes. In the pane stress state, the constittive reations for adherends and are iven b: ε k k σ k σ k () k k ε k k σ k σ k () k k γ where: k for adherend and k for adherend. k k τ (4) k An orthotropic materia in the pane stress state is described b five materia constants: two coefficients of onitdina easticit k and k, one shear mods of the set of eqations () () is smmetric, i.e. k, and two Poisson s ratios k, and. It is assmed that the matri of coefficients: k

6 0 Piotr Rapp k k k (5) hods. Ths, for ot of five materia parameters are independent. Havin soved the set of eqations () (4) with respect to stresses, one ets: k σ k k k k kk ε k k k ε k (6) σ k kk k k k ε k k k ε k (7) τ (8) k kγ k where: k for adherend and k for adherend. The materia parameters, k, k k, k, and shod be assmed to be k dependent on the annar rin orientation in the beam cross-section. In the tree trnk mode consistin of concentric annar rins, three anatomic directions are defined: parae to the rains L, radia R and transverse T (fi. 5a). The mechanica properties of wood differ in these anatomic directions. At ever point of the trnk mode, the L, R, T aes are mta orthoona and can be treated as the oca set of principa aes of orthotrop. Fi. 5. Anisotrop of wood: a) the trnk mode, b) particar cases of beam cross- -section ocation in the trnk mode At an arbitrar point in the trnk mode, the constittive reations take the form: ε LR LT L σ L σ R σ T (9) L R T

7 Comptationa mode of adhesive scarf joints in timber beams ε R RL L σ L R σ R RT T σ T (0) ε T TL TR σ L σ R σ T () L R T γ τ LR LR () LR γ τ LT LT () LT RT RT τ (4) RT γ ampes of materia constants for some tpes of wood determined eperimenta are presented in [Kewerth 95; oodman, Bodi 970; Nehas 994]. If the beam bendin pane coincides with the radia direction (cross-section in fi. 5b) then it ma be assmed that the beam orthotrop in the bendin pane is defined b the L and R aes. In this case, the materia constants in the eqations () () are k L, k R, k LR, k LR, and k RL. If the coefficient matri of the set of eqations () () is not smmetric, then it can be smmetrised b an approimate assmption that k RL ( R / L ). If the beam bendin pane coincides with the transverse direction ( crosssection in fi. 5b), then it ma be assmed that the beam orthotrop in the bendin pane is defined b the L and T aes. In this case, the materia constants in the eqations () () are k L, k T, k LT, k LT, and. If the k TL coefficient matri of the set of eqations () () is not smmetric, then it can be smmetrised b an approimate assmption that k TL ( T / L ). If the beam bendin pane is not parae to either of the R or T aes (crosssection in fi. 5b), then it can be assmed that the wood can be modeed b a composite with a transverse isotrop. Then the direction L parae to the rain is assmed, and the directions in the cross-sections perpendicar to the L ais are nified. In reait, this is the most freqent case. Transverse isotropic eements sbjected to a pane stress state can be described sin the easticit modi,, and introdced in rocode 5, where 0, the mean easticit mean 90, mean mean 0, mean mods aon the rain, 90, mean the mean easticit mods across the rain and the mean shear mods. It is assmed that,, and mean k k,0, mean k k,90, in the eqations () (4). mean k k, mean

8 Piotr Rapp For coniferos wood, TL RL k 0.45, approimate. From the reation k,90, mean k,0, mean /0 and the smmetr condition (5) k 0.05 can be determined (k for adherend and k for adherend ). Takin into accont the Cach eometric reations: ε k k k k k k ε γ k (5),, the constittive eqations (6) (8) for adherends and can be iven in the form: σ k k k k k kk k k k (6) σ k kk k k k k k k k (7) k k τ k k (8) where: k for adherend and k for adherend. Constittive eqations for the adhesive Stresses in the adhesive are de to differences between dispacements of adherends and. From the enera considerations presented b Rapp [00a, 00b], reations between the stresses in the adhesive and the dispacements of the adherends for a scarf joint can be written in the form: τ t( s ss ( t ϕ )cosϕ s ) (9) τ s ( ) (0) t σ N τ tϕ () Dispacement eqations and bondar conditions A enera formation of the dispacement eqations and the bondar conditions for two-dimensiona adhesive joints was presented b Rapp [00a, 00b]. For an adhesive scarf joint the take the form:

9 Comptationa mode of adhesive scarf joints in timber beams 0 ) ( ) ( γ β β () 0 ) ( γ β () 0 ) ( ) ( γ β β (4) 0 ) ( γ β (5) where: ) ( k k k k k, ) ( k k k k k (6) k k k k k k k k β ) (, k k k k k k k k β ) ( (7) s s k s s k t ϕ ϕ γ )cos t (, k s k t γ cosϕ (8) where: k for adherend and k for adherend. The eqations () (5) represent a set of for eiptic partia differentia eqations of the second order, with varin coefficients where the dispacement fnctions,,, and for adherends and are nknown. From condition (5) and formae (7) one ets β k β k. There is a norma oadin p and a shear oadin p actin on the ede of adherend, whie on the ede of adherend a norma oadin p and a shear oadin p are fond (fi. 4). The edes: for adherend and for adherend have zero thickness and are caed sharp edes (fi. ). The remainin edes are non-sharp. The bondar conditions for non-sharp edes read: adherend, the ede : p ) ( β, p (9)

10 4 Piotr Rapp adherend, the ede ± : 0 ) ( β, 0 (0) adherend, the ede : p ) ( β, p () adherend, the ede ± : 0 ) ( β, 0 () For the sharp edes, we et: adherend, the ede : 0 ) ( ) ( γ β () 0 ) ( γ (4) adherend, the ede : 0 ) ( ) ( γ β (5) 0 ) ( γ (6) In order to ensre the niqeness of the sotion to the bondar vae probem, one shod define in the dispacement formation, besides the static conditions, the kinematic bondar conditions to constrain movement of the joint. The bondar vae probem () (5), (9) (6) is soved sin the finite difference method. The finite difference mesh within the rectane coverin the projection of the adhesive joint on the pane 0XY is shown in fi. 6.

11 Comptationa mode of adhesive scarf joints in timber beams 5 Fi. 6. The finite difference mesh on the projection of the adhesive joint The finite difference mesh has a rear rectanar shape with the side enths Δ and Δ. There are m nodes aon the X ais ( j,,..., m), and n nodes aon the Y ais (i,,..., n), whie n, m 5. It is assmed that n and m are odd nmbers. The nknown parameters in the finite difference method are the dispacements ki,j and ki,j at the nodes of the mesh. Havin determined the dispacements,,, and, one can cacate the stresses in the adherends sin the formae (6) (8). The stresses τ, τ and σ N in the adhesive foow from (9) (). A scarf joint oaded aia Let s now consider an adhesive scarf joint between the adherends with dimensions.5 cm, 0.5 cm, 4.5 cm made of identica wood with the foowin properties:. 0 6 N/cm, N/cm N/cm, 0.0, 0.45 The adhesive has the thickness t 0.05 cm and the materia constants: s N/cm, s N/cm, s 0.5 The joint is sbjected to the tensie forces N N N. Ths, the edes: for adherend and for adherend are nder the action of the niform distribted stresses: p 0.(0084) N/cm and p 0.(0084) N/cm, respective. The probem is soved sin the finite difference method with the mesh n m 45 (Δ.0(7) cm, Δ.05 cm). The kinematic bondar conditions introdced for adherend as the constraint of the node (i, j) (, ) in the direction of the X ais and the nodes (i, j) (, ), (, ) in the direction of the Y ais.

12 6 Piotr Rapp The dispacements of the adherends at seected nodes of the finite difference mesh are iven in tabe. Tabe. The dispacements of the adherends in a scarf joint oaded aia [cm] Dispacements of adherend Dispacements of adherend i\j The distribtions of dispacement fnctions, and, for the adherends are iven in fi. 7. The dispacements for both adherends are depicted in sine drawins, becase with the adopted scae, the differences between the appropriate dispacement vaes are indistinishabe. Fi. 7. The dispacements in the adherends of an aia oaded scarf joint: a) dispacements,, b) dispacements, The remainin rests of the cacations are: the shear stresses τ and τ in the adhesive: τ N/cm const, τ 0 N/cm the norma stress σ N in the adhesive (fi. 0): σ N N/cm const the norma stresses σ, σ and the shear stress τ in adherend : σ N/cm const, σ 0 N/cm, τ 0 N/cm

13 Comptationa mode of adhesive scarf joints in timber beams 7 the norma stresses σ, σ and the shear stress τ in adherend : σ N/cm const, σ 0 N/cm, τ 0 N/cm The sotion presented to the two-dimensiona probem in the theor of easticit ma be verified sin a mode of a one-dimensiona aia oaded rod. Usin the semi-inverse method of the theor of easticit, one can define the dispacements,,, and of the adherends with the foowin formae: (, ) p t( s st ϕ)cos ϕ p A B s s (8) p (, ) A B (9) p (, ) (, ) A C (40) where: A, B and C are arbitrar constants to be derived from kinematic bondar conditions. A simpe sbstittion makes it possibe to check that the fnctions (8) (40) ffi the eqations () (5) and the bondar conditions (9) (6) with the constants,,,, and for both adherends, as we as the eqiibrim condition for the oadin p p 0. Havin sbstitted the reations (8) (40) to the formae (6) (8), the stresses in the adherends can be fond as: σ const (4) (, ) σ (, ) p σ, ) σ (, ) 0 (4) ( τ (, ) τ (, ) 0 (4) Knowin the dispacement fnctions (8) (40), one can determine the stresses in the adhesive τ, τ and σ N sin (9) (): τ (, ) p sinϕ cosϕ const (44) τ (, ) 0 (45) N sin σ (, ) p ϕ const (46)

14 8 Piotr Rapp The kinematic bondar conditions ied the zero constants A and C present in the eqations (8) (40), whie the constant B is iven b: t( s st ϕ)cos ϕ B p ss (47) The dispacement fnctions for the adherends are iven b: p (, ) (48) p t( s st ϕ ) cos ϕ (, ) p ss (49) p (, ) (, ) (50) The fnctions (48) (50) ffi the eqations () (5), the static bondar conditions (9) (6) and the kinematic bondar conditions. The niqeness of the sotion for the theor of easticit probem makes it possibe to concde that the fnctions (48) (50) are sotions to the two-dimensiona probem of the scarf joint. The stress state in the adherends and the adhesive, in the case of a scarf joint oaded aia with two adherends made of an identica materia, does not depend on the adhesive thickness and its materia parameters. Additiona, the stresses are identica to those in a skew section at the ane ö in a one-dimensiona continos eement, nder a niaia stress state. The adhesive parameters infence on the difference between the dispacements of adherends and, which is evident in formae (48) and (49). The eqivaence of the one- and two-dimensiona modes, and the independence of vaes and distribtions of stresses with respect to the adhesive parameters, do not occr in the case of a joint between adherends made from different materias, for a different oadin than a niform aia one or when. For instance, et s ook at a scarf joint oaded aia bt ocated between two adherends made from different materias with the foowin parameters:. 0 6 N/cm, N/cm, N/cm 0.0, N/cm, N/cm, N/cm 0.08, 0.6

15 Comptationa mode of adhesive scarf joints in timber beams 9 A the other data remain the same as in the case considered previos. Two-dimensiona stress and dispacement states are present in the adhesive and the adherends. To istrate the probem, fis. 8 0 present the distribtions of dispacements and stresses in the adherends and in the adhesive. Note, that fis. 9 c f and 0 have a different scae to fis. 9 a, b. The dispacements, and, in fi. 8 for both adherends are depicted in sine drawins, becase with the adopted scae, the differences between the appropriate dispacement vaes are indistinishabe. Fi. 8. The dispacements in the adherends of a scarf joint oaded aia in the case of different materias: a) dispacements,, b) dispacements, Fi. 9. The stresses in the adherends of a scarf joint oaded aia in the case of different materias: a) stress σ (vaes ), b) stress σ (vaes ), c) stress σ (vaes 00), d) stress σ (vaes 00), e) stress τ (vaes 000), f) stress τ (vaes 000)

16 0 Piotr Rapp Fi. 0. The stresses in the adhesive of a scarf joint oaded aia in the case of different materias: a) stress τ (vaes 0), b) stress τ (vaes 0), c) stress σ N (vaes 0) A scarf joint oaded b a bendin moment Let s now consider a scarf joint oaded b the bendin moments M M M N cm restin from the inear distribted stresses on the ede of adherend and on the ede of adherend (fi. ). The dimensions, the materia parameters and the finite difference mesh are assmed identica to the previos considered case of a scarf joint oaded aia. The kinematic bondar conditions are imposed on adherend b constrainin its node (i, j) (, ) in the direction of the X ais and the nodes (i, j) (, ), (, ) in the direction of the Y ais. Now a compete sotion to the eqations of the theor of easticit wi be presented. It wi contain dispacements and stresses in the adherends and in the adhesive. Fi.. Loadin actin on the adherends of a scarf joint sbjected to a bendin moment In the foowin, in the captions of the tabes and fires presentin the compete sotion, the feibiit of the adhesive is emphasized, to differentiate cear from the case with an ndeformabe adhesive. The distribtions of dispacement fnctions, and, for the adherends are iven in fi.. The dispacements for both adherends are depicted in sine drawins, becase with the adopted scae, the differences between the appropriate dispacement vaes are indistinishabe.

17 Comptationa mode of adhesive scarf joints in timber beams Fi.. The dispacements in adherends of a joint with a feibe adhesive oaded b the bendin moment: a) dispacements,, b) dispacements, The vaes of the adherend dispacements at seected nodes of the finite difference mesh are iven in tabe. The vaes of stresses at seected points in adherends and of a joint with a feibe adhesive oaded b the moment are iven in tabes and 4. Tabe. The dispacements in adherends of a joint with a feibe adhesive oaded b the bendin moment [cm] Dispacements of adherend Dispacements in adherend i\j 45 i\j Tabe. The stresses in adherend of a scarf joint with a feibe adhesive oaded b the bendin moment [N/cm ] Stresses i\j σ σ τ σ σ τ σ σ τ

18 Piotr Rapp Tabe. Contined i\j σ σ τ σ σ τ Tabe 4. The stresses in adherend of a scarf joint with a feibe adhesive oaded b the bendin moment [N/cm ] Stresses i\j σ σ τ σ σ τ σ σ τ σ σ τ σ σ τ Tabe 5. The stresses in the feibe adhesive of a joint oaded b the bendin moment [N/cm ] Stresses i\j 45 τ τ σ N τ τ σ N τ τ σ N

19 Comptationa mode of adhesive scarf joints in timber beams The stress distribtions in adherends and of a joint with a feibe adhesive are presented in fi.. Fi.. The stresses in the adherends of a joint with a feibe adhesive oaded b the bendin moment: a) stresses σ, σ, b) stresses σ, σ (vaes 500), c) stress τ (vaes 500), d) stress τ (vaes 500) The vaes of stresses in the feibe adhesive of a joint oaded b the moment are iven in tabe 5, and the correspondin distribtions in fi. 4. Fi. 4. The stresses in the feibe adhesive of a joint oaded b the bendin moment: a) stress τ, b) stress τ (vaes 500), c) stress σ N (vaes 0) In the adherends of the considered scarf joint, the norma stresses σ, σ dominate. The are eqa to one another, constant aon and inear aon with an accrac of 0.8%. The stress state in the adhesive itsef is dominated b the shear

20 4 Piotr Rapp stresses τ and σ N, which are aso approimate constant aon and inear aon. De to the adhesive feibiit. In this case, there are non-zero norma σ k and shear τ k stresses (fi. b d) in the adherends, and non-zero shear stress τ in the adhesive (fi. 4b). However, the stresses σ k and τ k in the adherends are appro. 000 times smaer than the stress σ k, and the stress τ in the adhesive is appro. 500 times smaer than the stress τ. Hence, it can be assmed with sfficient accrac that the stress states in the adherends of a joint with a feibe adhesive oaded b the bendin moment M, are identica to the one in a continos eement sbjected to the moment M. For M N cm the otermost norma stress in the cross-section of the continos beam is σ ± N/cm. These vaes differ b appro. 0.8% from the otermost stresses σ, σ in the adherends of a joint with a feibe adhesive (tabes and 4). If the adhesive feibiit decreases, then the difference between the adherend dispacements decreases, too. In the imitin case, for the ndeformabe adhesive, a phsica interpretation ensres the eqait of these dispacements. This concsion can be drawn from eqations () (5), too. Indeed, when one divides these eqations b s and ets s, then the identities and rest. Ths, it ma be assmed, that in the imitin case, the vaes and the distribtions of stresses in the adherends of a scarf joint are identica, as in the continos beam oaded b the bendin moment M. If we denote these stresses b σ, σ, τ, then we ma write down: M σ (, ) (5) σ (, ) 0 (5) τ 0 (5) Ths, the continos eement ma serve as an approimate mode of a scarf joint with a feibe adhesive. Indeed, it can be considered as a scarf joint with an ndeformabe adhesive. The ndeformabe adhesive is represented be a skew section in the continos eement. That skew section is oriented in the continos eement in the same wa as the feibe adhesive in the scarf joint. The dispacements and of the continos eement with an ndeformabe adhesive ffi the foowin set of eqations: M (54)

21 Comptationa mode of adhesive scarf joints in timber beams 5 0 (55) 0 (56) which rests from sbstittion of the formae (5) (5) to the constittive eqations (6) (8) in the case of the materia properties,,,,. Sovin the eqations (54) and (56) with respect to and, one ets: (, ) (, ) M (, ) (, ) M ( (, ) (, ) 4 A B ) A C (57) (58) where: A, B, and C are arbitrar constants. In the case of s, the fnctions,,, presented above ffi the eqations () (5) and the bondar conditions (9) (6). The constants A, B, and C foow from the kinematic bondar conditions, which can be epressed in co-ordinates as: node (i, j) (,), co-ordinates 0 and 0 (0, 0) 0 node (i, j) (,), co-ordinates 0 and 0 ( 0, 0) 0 node (i, j) (,), co-ordinates 0 and 0 ( 0, 0) 0, where: 0 0.(7) cm. These conditions ead to A 0, B 0 and 0 M C (59) 4 The dispacements in the continos eement oaded b the bendin moment, cacated from (57) (59) are iven in tabe 6. Comparison of the vaes presented in Tabes and 6 makes it possibe to assess that the maima dispacements, of the adherends of the joint with the feibe adhesive and the maima dispacement of the continos eement are eqa, with an accrac of appro. 0.5%. Simiar, it can be assessed that the maima dispacements, of the adherends of the joint with the feibe adhesive and the maima dispacement of the continos eement are eqa with an accrac of appro. 0.85%.

22 6 Piotr Rapp Tabe 6. The dispacements of the continos eement oaded b the bendin moment [cm] Dispacements i\j Norma and shear stresses are present in the continos eement in the skew section defined b the adhesive pane. However, in the imitin case s, the shear stresses in the skew section (ndeformabe adhesive) cannot be determined from the formae (9) (0) becase in the view of 0, 0 and s, indeterminate smbos of the form 0 are fond. The shear stresses in the ndeformabe adhesive can be cacated from the eqations () (5), which do not epicit incde the differences of the dispacements, and the vae s. If we determine, from the formae (9) (0) and sbstitte them to () (5), then for the parameters,,,, the foowin reations foow: 0 cos ) ( ϕ τ β β (60) 0 cos ϕ τ β (6) 0 cos ) ( ϕ τ β β (6) 0 cos ϕ τ β (6) where: ) (, ) ( (64)

23 Comptationa mode of adhesive scarf joints in timber beams 7 β ( ), β ( ) (65) Sbstittion of the fnctions,,, and iven in (57) (59) to the eqations (60) and (6) or (6) and (6) eads to the reations definin the stresses in the ndeformabe adhesive in the continos eement: where: σ (, ) is iven b (5). τ (, ) σ (, )sinϕ cosϕ (66) τ (, ) 0 (67) The formae () and (66) ied the reation for the norma stress in the ndeformabe adhesive: σ (, ) σ (, ) sin φ (68) N The formae (66) (68) can aso be derived in an eementar manner, considerin the eqiibrim conditions of the adherend in the vicinit of the adhesive. The stresses vaes for the ndeformabe adhesive in the continos eement foowin from (66) (68) are iven in tabe 7. Tabe 7. The stresses for the ndeformabe adhesive in the continos eement oaded b the bendin moment [N/cm ] Stresses i\j 45 τ τ σ N τ τ σ N τ τ σ N Comparison of the vaes iven in tabes 5 and 7 makes it possibe to assess with an accrac of appro. 0.% that the maima stresses in the feibe adhesive and in the ndeformabe adhesive in the continos eement are eqa. It can be concded that the vaes and distribtions of the dispacements and stresses in the considered case of the scarf joint with the feibe adhesive are

24 8 Piotr Rapp simiar to those in the case of the continos pane stress eement (in the scarf joint with the ndeformabe adhesive) sbjected to the bendin moment in the pane 0XY. The smaer is the adhesive feibiit, the better the approimation. A scarf joint oaded b a shear force The wa in which a shear force is transmitted throh a scarf joint can be istrated, when the force is constant aon the joint. Let s assme that adherend is sbjected to the shear force T at the ede, and adherend to the shear force T at the ede. The joint oaded in this wa has to be eqiibrated, e.. b additiona bendin moments M T, accordin to fi. 5a. Fi. 5. A scarf joint sbjected to a shear force and a bendin moment In the foowin, the oad of T.0 N is assmed in the form of a paraboic distribted shear stress, and the bendin moment M T in the form of a inear distribted norma stress (fi. 5b). A the remainin data are the same as in the previos case of the joint oaded b the bendin moment. The distribtions and vaes of the dispacements,,, and for a scarf joint with a feibe adhesive are presented in fi. 6 and in tabe 8. Fi. 6. The dispacements in adherends of a joint with a feibe adhesive oaded b the shear force and the bendin moment: a) dispacements,, b) dispacements,

25 Comptationa mode of adhesive scarf joints in timber beams 9 Tabe 8. The dispacements in adherends of a joint with a feibe adhesive oaded b the shear force and the bendin moment [cm] Dispacements in adherend Dispacements in adherend i\j 45 i\j The distribtions of stresses in the adherends of a scarf joint with a feibe adhesive oaded b the shear force and the bendin moment are presented in fi. 7. The particar stresses are represented in sine fires, becase the differences between their vaes for both adherends are neiibe and indistinishabe with the adopted scae. Fi. 7. The stresses in the adherends of a scarf joint with a feibe adhesive oaded b the shear force and the bendin moment: a) stresses σ, σ, b) stresses σ, σ (vaes 00), c) stresses τ, τ The vaes of the stresses in the adherends of a scarf joint with a feibe adhesive oaded b the shear force and the bendin moment are iven in tabe 9. Tabe 9. The stresses in the adherends of a scarf joint with a feibe adhesive oaded b the shear force and the bendin moment [N/cm ] Adherend Adherend i\j σ σ σ σ τ τ 0 0 0

26 0 Piotr Rapp Tabe 9. Contined i\j σ σ σ σ τ τ σ σ σ σ τ τ σ σ σ σ τ τ σ σ σ σ τ τ The vaes of the stresses in the feibe adhesive of a scarf joint oaded b the shear force and the bendin moment are presented in tabe 0, and the correspondin distribtions in fi. 8. Tabe 0. The stresses in the feibe adhesive of a scarf joint oaded b the shear force and the bendin moment [N/cm ] Stresses i\j 45 τ τ σ N τ τ σ N τ τ σ N Fi. 8. The stresses in the feibe adhesive of a scarf joint oaded b the shear force and the bendin moment: a) stress τ, b) stress τ, c) stress σ N (vaes 0)

27 Comptationa mode of adhesive scarf joints in timber beams Comparison of fis. 7a, b and fis. a, b eads to the concsion that the norma stresses σ k and σ k shown in fis. 7a and b rest from the bendin moment M T, whie the shear stresses τ k are de to the action of the shear force T on. Ths, the action of the shear force can be separated, and in this wa, how the joint transmits the shear force can be assessed. As in the case of the joint oaded b the bendin moment, in the scarf joint with the feibe adhesive oaded b the shear force and the bendin moment, the distribtion of the stresses in the adherends are simiar to those in the case of the continos pane stress eement (in the joint with the ndeformabe adhesive). The stresses σ, σ, and τ in the adherends of the joint with the ndeformabe adhesive can be iven b the foowin formae: T σ (, ) σ k (, ) (69) σ (, ) σ k (, ) 0 (70) T ( ) τ τ k (, ) (7) 4 The dispacements and of the continos eement with the ndeformabe adhesive ffi the set of eqations: T (7) 0 (7) T ( 4 ) (74) which rests from sbstittion of the reations (69) (7) to the constittive eqations (6) (8) for the materia parameters,,,,. qations (7) (74) can be soved in a simiar wa to eqations (54) (56) to et: (, ) T T (, ) (, ) 4 4 T 4 T 4 A B (75)

28 Piotr Rapp (, ) T (, ) (, ) 4 T 4 A C (76) where: A, B, C are arbitrar constants. For the ndeformabe adhesive ( s ) and the dispacements,,, and iven b (75) and (76), eqations () (5) and the bondar conditions (9) (6) are ffied as identities. The kinematic bondar conditions epressed in the co-ordinates take the same form as in the previos eampe: (0, 0) 0, ( 0, 0) 0, ( 0, 0) 0, where 0 0.(7) cm. The ied B 0, C 0 and T0 4 A (77) The dispacements of the continos eement oaded b the shear force and the bendin moment restin from (75) (77) are iven in tabe. Comparison of the vaes iven in tabes 8 and makes it possibe to assess that the maima dispacements and of the adherends of the joint with the feibe adhesive and the maima dispacement of the continos eement coincide with an accrac of appro. 0.8%. Simiar, it can be assessed that the maima dispacements, of the adherends of the joint with the feibe adhesive and the maima dispacement of the continos eement, coincide with an accrac of appro. 7.%. It shod be noted, that the dispacements,, and are abot 5 times smaer than the dispacements,, and. Tabe. The dispacements in the continos eement oaded b the shear force and the bendin moment [cm] Dispacements i\j Sbstittion of the fnctions iven b (75) and (76) to the eqations (60) and (6) or (6) and (6) eads the formae for the shear stresses in the ndefor-

29 Comptationa mode of adhesive scarf joints in timber beams mabe adhesive for the continos eement oaded b the shear force and the moment: τ (, ) σ (, )sinϕ cosϕ (78) τ (, ) τ (, ) sinϕ (79) The reations () and (78) ead to the epression for the norma stress in the ndeformabe adhesive: σ (, ) σ (, ) sin φ (80) N The stresses in the ndeformabe adhesive of the continos eement oaded b the shear force and the bendin moment are presented in tabe. Comparison of the vaes in tabes 0 and makes it possibe to assess that the maima stresses in the feibe adhesive of the scarf joint oaded b the shear force and the bendin moment, and the maima stresses in the ndeformabe adhesive in the continos eement oaded ikewise, coincide with an accrac of appro. % in the case of τ and σ N,and appro. 0.5% in the case of τ. Tabe. The stresses in the ndeformabe adhesive of the continos eement oaded b the shear force and the bendin moment [N/cm ] Stresses i\j 45 τ τ σ N τ τ σ N τ τ σ N The cacations carried ot ead to the concsion that the considered scarf joint with the feibe adhesive oaded b the shear force and the bendin moment, featres the vaes of the dispacements and the distribtions of stresses in the adhesive and the adherends simiar to those in the continos pane stress eement (in the joint with the ndeformabe adhesive) oaded ikewise. In particar, the distribtions of the stresses τ k in the adherends and the stress τ in the adhesive de to the action of the shear force are paraboic with an accrac of appro. 0.5%. The smaer is the adhesive feibiit, the better the approimation.

30 4 Piotr Rapp Concsions In a scarf joint oaded aia, where both adherends are made of the same materia, the stress states in the adherends and the adhesive do not depend on the adhesive thickness or its materia parameters, and are identica to those in the continos eement nder a niaia stress state. The adhesive parameters on infence the difference between the adherend dispacements. The eqivaence of the one-dimensiona and two-dimensiona modes, and the independence of vaes and distribtions of stresses with respect to the adhesive parameters, do not eist when: a scarf joint is made of two different materias, adherends are not oaded niform and aia or in the case of adherends with differin thicknesses. A enera concsion ma be formated that a scarf joint with an adhesive of itte feibiit between two adherends made of the same materia and of the same thickness, transmits aia forces, bendin moments and shear forces in the same or in a simiar wa to a continos eement considered as the scarf joint with an imainar ndeformabe adhesive. In enera, the smaer is the adhesive feibiit, the smaer the difference between the sotion to the two-dimensiona probem of the scarf joint with the feibe adhesive and the anatica sotion to the joint with the ndeformabe adhesive. Ths, an approimate eqivaence of dispacements and stress states between an eement made of two adherends with a scarf joint and the continos eement occrs. This fina statement is not obvios, becase in a scarf joint oaded in the wa discssed previos, the decreasin adhesive feibiit eads to a chane in the noninear dispacements and stress distribtions in the adhesive to inear ones, correspondin to the continos eement considered as the scarf joint with an imainar ndeformabe adhesive. On the other hand, in the joint with non-sharp edes, this behavior is reversed de to a decrease in the adhesive feibiit in the vicinit of the oaded edes, the stress concentrations are more prononced and the dispacement and stress distribtions become increasin non-inear. References rdoan F., Ratawani M. [97]: Stress Distribtion in Bonded Joints. Jorna of Composite Materias 5 (): DOI: 0.77/ oodman J.R., Bodi J. [970]: Orthotropic astic Properties of Wood. Jorna of the Strctra Division 96 (): 0 9 Kewerth R. [95]: Die anisotrope astzität des Hozes nd der Laerhözer (Anisotropic easticit of wood and wooden beams). VDI Forschnsheft 40, Asabe, Band 7, Dether Inenir, Dssedorf Nehas H. [994]: Lehrbch des Inenierhozbas (Tetbook on wood strctres). Tebner, Stttart

31 Comptationa mode of adhesive scarf joints in timber beams 5 Rapp P. [00a]: Mechanika połączeń kejowch jako płaskie zadanie teorii sprężstości (Mechanics of adhesive joints as a pane probem of the theor of easticit). Rozpraw Nr 44, Wdawnictwo Poitechniki Poznańskiej, Poznań Rapp P. [00b]: Mechanics of adhesive joints as a pane probem of the theor of easticit. Part I: enera formation. Archives of Civi and Mechanica nineerin 0 (): 8 08 Redd M.N., Sinha P.K. [975]: Stresses in Adhesive-Bonded Joints for Composites. Fibre Science and Technoo 8 (): 47. DOI: 0.06/ (75)900-5 Smardzewski J. [996]: Distribtion of stresses in finer joints. Wood Science and Technoo 0 (6): DOI: 0.007/BF Tomasik A. [988]: Anase der Spannnen in Kebfen bieebeasteter Keizinkenverbindnen (Anasis of stresses in scarf adhesive joints nder bendin). Hoztechnooie Leipzi 9 (): 5 6

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