SHEARING IN WELDED T-JOINTS

Transcription

1 Proceedings of he 7h Inernaional Conference on echanics and aerials in Design Albufeira/Porugal June 17 Ediors JF Silva Gomes and SA eguid Publ INEGI/FEUP 17) PAPER REF: 64 SHEARING IN WELDED T-JOINTS Alexandre de acêdo Wahrhafig 1*), Bruno Felix dos Sanos Coelho 1 Deparmen of Consrucion and Srucures DCE), Federal Universiy of Bahia, Salvador, Brail Deparmen of echanical Engineering DE), Federal Universiy of Bahia, Salvador, Brail *) alixa@ufbabr ABSTRACT Sudies of welded joins using classical mechanics and calculus mus be employed by welding engineers o esablish he dimensions of bead-welds and o se load hresholds ha can be applied for connecing srucural elemens In analyical models, one normally considers an accepable kinemaics hypohesis for a problem and obains he formulaion ha permis he calculaion of sress for he given case If necessary, one hen analyses he sress sae esablished a he poin of ineres, normally where he value is maximal This paper develops a mechanical- and calculus-based formulaion for he design and verificaion of weld lines on joins of perpendicular plaes subjeced o shear sresses due o orque The resuls obained are compared wih mahemaical modelling by he finie elemen mehod FE) o evaluae he disribuion of shear sress In comparison wih FE, i was possible o conclude ha he analyical soluion developed in his work is a formulaion appropriae for he design and verificaion of bead-weld in T-joins of a recangular plae Keywords: T-join, shearing, bead-weld, srucural design, compuaional modelling INTRODUCTION Usually, he developmen of a mahemaical soluion for pracical engineering problems sars wih he esablishmen of an admissible kinemaic for he problem Alhough hese soluions inrinsically uilie simplifying assumpions, hey are sill quie suiable for a relaively large range of applicaions In his conex, he design and verificaion of welded T-joins composed of perpendicular plaes subjeced o a orsional momen can be accomplished by an undersanding of he shear sresses on he bead-weld weld See Chen 5) for an example According o Ivan e al 8) of he American Insiue of Seel Consrucion AISC), in many cases, eccenrically loaded join configuraions are unavoidable, creaing more complex sress condiions in he join han concenrically loaded joins, where he welds are generally subjec o shear in only one direcion Considering his, a suiable mehod was developed in his work for calculaing he properies of he joins of plaes welded perpendicularly, providing a formulaion ha direcly considers he shear sress induced by orque a he join The necessary relaionships for calculaing he shear sresses relaed o orque acing wih he polar momen of ineria are defined, aking ino accoun he basic dimension of he weld line and he hickness of he plaes o be unied Taking ino accoun he saemens made by AISC, as menioned by Cynhia 4), and in order o evaluae he formulaion developmen in his work, compuer modelling was performed, seeking o address he problems encounered when joins and plaes are welded perpendicularly nd subjeced o orque This was accomplished using he deformable solids heory, assessing he sress disribuion in he base secion of he weld line -861-

2 Topic-I: Civil Engineering Applicaions ANALYTICAL DEVELOPENT Take he union shown in Fig 1 consising of a base plae, B, and he verical plae o be welded, A The dimension of he weld line is designaed in he plan by a and he hickness of he verical plae by is he orsional momen ha requires a connecion A y Weld Weld L B B A y x x Fig 1 - T-join references When he connecion is required by he momen acing on is own plane, i is possible o consider ha he disribuion of he shearing sress obeys he known law arising from he resisance of maerials given in Eq 1) There, is he orsional momen acing on he sysem, ρ) is he generic disance in relaion o he cenre of he join, dependen on variable, and J is he polar momen of ineria o be deermined as described below This implies ha i should be considered ha he weld is circumscripive in a circle of differen radius, according he polar disance varies, which is a good approximaion because he sresses induced are shearing sress, given by each posiion in he weld line given by ordinae ρ ) ) 1) J Take an elemenary area da, as shown in Figure, given by da a`d, where a` is equal o a, which is he widh of he bead weld This assumpion is a formalism ha jus arises because heoreically he sress obained is perpendicular o he lever arm in relaion o he cenre The sress decomposiion in he weld direcion can be found for he elemen of area a` as poseriorly for he vecor shear sress Fifure ) The disance from he cenre of he connecion is obained by he hypoenuse of he righ riangle, as indicaed in Figure and Eq 1)1) ) ρ ) + 1) a a -86-

3 Proceedings of he 7h Inernaional Conference on echanics and aerials in Design a / d ρ a` x sinθ) cosθ) ρ θ x Fig - T-join references for analyical developmen Concepually, he polar momen of ineria is he inegral over he area of he polar disance o he square, obained in accordance wih Eq3): J ρ ) da 3) A Subsiuing Eq ) ino Eq3), wih he inegraion limis appropriae o he problem, one finds ) a + L/ J + dad 4) L/ Solving he above inegral, one finds Eq5), which provides a way o calculae he polar momen of ineria o ake ino accoun he dimension of he weld and he hickness of he verical plae: J a L+ al 5) Replacing he polar momen of ineria obained from Eq 5) in Eq1), Eq 6) is found, which is he equaion for he shear variaion sress on he weld bead ) ) ρ 1 1 a L+ al To consider jus he componen aligned o he weld line, i is necessary o decompose he previous resulan, as indicaed in Eq7): where ) ) ρ sin θ ) ), a L+ al 4 1 θ ) arcan 6) 7) 8) -863-

4 Topic-I: Civil Engineering Applicaions Knowing ha he disribuion of sresses requires ha he maximum sress occur a he end of he weld, a value for i can be approximaed, assuming an iniial simplificaion o ρ max equal o L/, and he equaion of maximum shear sress can be wrien in he form of Eq9): max ) + L ) 9) a L+ al 4 1 In he direcion of he bead weld, i is necessary o decompose he shear sress vecor according o he sine of he opening for ha posiion, so he previous equaion becomes Eq19): max ) + L ) sin θ L / ) ) 1) a L+ al 4 1 I is imporan o keep in mind ha in he case of dimensioning, he maximum shear sress should be calculaed relaive o he lower weld line area, which lies on he bisecor of he righ angle, so Eq9) and Eq 1) mus ake ino accoun he cosine of he 45 angle; hen: max ) + L ) and a L+ al cos π / 4) 4 1 max ) + L ) sin θ L / ) ) cos / 4) 3 a L+ al π 11) In addiion o checking he horional shear sress, Expressions 9) and 1) can be used for he design of he base a of he bead weld, aking i in funcion of he hickness of he verical plae To do his, Eqs 11) mus be pu in he polynomial form shown in Eqs1 1/ + L ) adm 3 + L ) ) 3 and a ) a L 1 1/ + L ) sin θ L / ) ) L adm 3 + L ) 3 1) where adm is he admissible ension ha can be applied o he weld, which can be provided by consuling, for example, he Srucural Welding Code - Seel [American Welding Sociey AWS) Specificaions] or recoendaions from Eurocode 3 Srucural Seel Design) COSNTRUCTION OF THE ODELLING BY THE FINITE ELEENT ETHOD I is ineresing iniially o menion ha he finie elemen mehod FE) includes he classical mechanics designaed for coninuum mechanics and follows he behaviour of bodies according o he heory of deformable solids I should be noed ha he FE is an original conribuion of engineering srucures, which reassembles he findings of sudies by Argyris, -864-

5 Proceedings of he 7h Inernaional Conference on echanics and aerials in Design in 1954; Turner, Clough, arin and Topp, in 1956; and Clough, in 196, among ohers ha used he principle of virual work FE is a discreiaion echnique used o inerpre coninuous sysems and heir numerical approximaions from differenial equaions FEs have roos in he rial funcions used in he variaional mehods of Rayleigh 187) and Ri 199) and he residual weighed mehods of Galerkin 1915), affirms Brasil 1995) Tradiional references o FE are provided by Klaus 1996), Rober 4), and iguel 11) For he area of welding, modelling by FE has been used o analyse problems relaed o several aspecs Khiabani and Sadrnejad 9) used i o sudy residual sresses in he cold bending and welding of hick plaes Chen, Hashemadeh, Garbaov, e al 15) employed finie elemen analysis o deermine he residual sress disribuion and he disorion field in bu and fille plaes os recenly, Zain-ul-abdein and Nélias 16) used FE o calculae and compare he bulk maerial properies wih hose of experimenal findings and o characerie he effec of grain sie on he sress sae of maerial in a welded aluminium join A mahemaical model based on FE was buil in Ansys Academic Version 17, a coercial sofware package The model was compuaionally elaboraed reproducing he dimensions of he connecion described in his sudy, allowing o analysis he variaion of shear sress in he weld secion For his purpose, he sudy used daa employed in mahemaical simulaions, considering he base of he weld o be 1 and is lengh 1 and he hickness of he verical plae o be 1, represened by 3D solid elemens The compuaional model and principal references as well he discreiaion of he model are included in Fig 3 a) 3D solid elemen b) Discreiaion of he model Fig 3 - odelling of he connecion by FE The orsional effor was applied in he form disribued o minimie he effecs of he concenraion of effor a he upper posiion of he verical plae Figure 4- in red) and he horional plae was resrained in he whole of he lower plane Figure 4- in blue) -865-

6 Topic-I: Civil Engineering Applicaions a) Effor applicaion plane b) odel resrained posiion Fig 4 - omen posiion applicaion and resrain The deformaion of he welded join, wih he configuraion of he sysem observed afer applicaion of loading, can be seen in Fifure 5 Afer compuer processing, he resuls reveal he appearance of shear sresses in he weld line secion The shear sresses are presen in he horional direcion, wih he maximum values occurring a he farhes poin in relaion o he cenre However, a meiculous sudy of he sae of sress induced in he weld join as well he influence of modelling parameers can be performed in a fuure analysis a) Fronal view b) Top view Fig 5 - Deformaion of he model by FE RESULTS AND DISCUSSION Considering a weld lengh L 1 and hickness of verical plae 1, in Figure 6 i is possible o see he rend of he variaion of he polar disance of ineria given by Eq) The shear sress behaviours given by Eqs 6) and 7), all of hem along he lengh of he bead weld, are shown infigure

7 Proceedings of he 7h Inernaional Conference on echanics and aerials in Design 6 ρ ) Fig 6 - Polar momen of ineria along he weld for L 1 Wih Fig 7 i, i is possible o observe ha since Eq 6) depends on he polar disance, which is always posiive, he values obained from ha equaion are always posiive oo black line) Oherwise, when he sine of he angle is inroduced in he same equaion, giving Eq 7), he resuls obained assume as many posiive as negaive values blue line) However, if he modulus of Eq 7) is considered, only posiive values are found red line) A special poin of ineres is he case of he minimum value in comparison wih hese wo equaions The minimum value ha appears using Eq 6) is no ero he black line), since he hickness of he verical plae is considered in he soluion, which is in accordance wih he resul obained by FE When using Eq 7), his does no occur because he angle formed a he posiion of he minimum value is ero, and herefore he sress a he referred poin is ero oo blue and red lines) For more clariy, below, on he righ side, he curves of Eqs 6) and 7) are ploed Τ ) Pa T ) Pa T ) Pa Eq 6) Eq 7) Eq 7) Τ ) 13 Pa T ) Fig 7 - Variaion of he shear sress along he weld Pa 65 Eq 6) Eq 7) Wih he aim of improving he evaluaion, i is imporan o emphasie ha Eqs 6) and 7) make i possible o perform some numerical simulaions of imporance for engineering welding, allowing he influence of he hickness of he verical plae and orsional momen on -867-

8 Topic-I: Civil Engineering Applicaions he dimensions of he weld o be sudied In he firs case, he dimension a of he weld base is obained in funcion of he hickness of he verical plae o be welded [Eqs 1)-a1, a)] The relaion beween a and can be seen in he graph of Fig 8, which shows he reducion of he minimum dimension of he weld line wih increasing hickness of he elemen o be conneced In his simulaion, an arbirary and consan orsional momen equal o 5 knm, which was used for he compuaional modelling oo, was considered The resul was produced aking ino accoun he admissible sress of Pa, calculaed in accordance wih he AISC Specificaion A second simulaion made i possible o evaluae he variaion of he basic dimension a of he hread weld wih he demanded effor For ha, a was formulaed in funcion o he orsional momen, keeping he plae hickness unchanged, admied as 1 The resuls can be seen in he graph in Fig 9, in which is possible o observe a ypical linear increase in he dimension a of he weld wih he orque I is eviden ha when he momen is bigger, a verical plae of greaer hickness will be necessary and hen he dimension a will increase oo 6 a 1 ) 4 a ) Fig 8 - Variaion of he bead weld wih he hickness of he verical plae - Eq1) 1 9 a ) Fig 9 - Dimension a of he hread weld wih he demanded effor kn m -868-

9 Proceedings of he 7h Inernaional Conference on echanics and aerials in Design Boh of he previous graphs can be used ogeher forming an Abacus in order o perform weld design, supporing pracical problems in engineering The model devised o compuaionally analyse he join described in his sudy akes ino accoun he variaion of shear sress in he weld secion, and was elaborae wih he goal of faihfully reproducing he dimensions of he connecion For his purpose, daa employed in mahemaical simulaions were used, considering he base of he weld and he hickness of he verical plae where he orsional momen occurs The resul found by FE can be seen in Figure 1 Fig 1 - Resul found by FE Table 1 compares he resuls found by he analyical soluions proposed in his work and he modelling by FE ehod Table 1 - Tension resuls of maximum shear aximum values 1 8 Pa) inimum values 1 7 Pa) Difference in maximum values %) Analyical Eq 9) Eqs 9) and 1) 717 Analyical Eq 1) 396 Eq 9) and FE 5386 FE Eq 1) and FE 4636 CONCLUSION In comparison wih he compuaional modelling, i was possible o conclude ha he analyical soluion developed in his work is an appropriae formulaion for he design and verificaion of bead-welded joins beween perpendicular plaes subjeced o orsional momen, allowing an evaluaion of he horional shear sresses -869-

10 Topic-I: Civil Engineering Applicaions REFERENCES [1]-Brasil LRFR, The phenomenon locaion modes in srucural dynamics and sabiliy modulaed linear behavior or nonlinear, Thesis Free Teaching) - Polyechnic School, São Paulo Universiy São Paulo, 1995 []-Chen BQ, Hashemadeh, Garbaov Y, e al In J ech aer Des, 15, 11, p 439 doi:117/s [3]-Chen P, Li Y Effec of weld on design of seel momen-resising connecion reinforced wih seel plaes, J Cen Souh Univ Technol, 5, , S-74-4 [4]-Cynhia JD Connecion design in he 5 AISC Specificaion, Connecions in Seel Srucures V, Behaviour, Srengh & Design, Proc Fifh In Workshop, Amserdam, The Neherlands, 4 [5]-Ivan G, Ami K, Yu KK, Gilber G Srengh and duciliy of welded joins subjeced o ou-of-plane bending, American Insiue of Seel Consrucion, July 8 [6]-Khiabani, AC, Sadrnejad, SA, In J ech aer Des, 9, 5, p 53 doi:117/s [7]-Klaus JB Finie elemen procedures Prenice Hall, Englewood Cliffs, NJ, 1996 [8]-iguel LB, Klaus JB The mechanics of solids and srucures - hierarchical modeling and he finie elemen soluion, Springer, Germany, 11 [9]-Rober DC, Conceps and applicaions of elemen analysis John Wiley and Sons, NJ, USA, 4 [1]-Zain-ul-abdein, Nélias D In J ech aer Des 16, 1, p 55 doi:117/s x -87-