Joining Sheets Perpendicular To One . - Técnico Lisboa
Joining sheets perpendicular to one other bysheet-bulk metal formingI. M. F. Bragança, C. M. A. Silva,L. M. Alves & P. A. F. MartinsThe International Journal ofAdvanced Manufacturing TechnologyISSN 0268-3768Int J Adv Manuf TechnolDOI 10.1007/s00170-016-9083-51 23
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Author's personal copyInt J Adv Manuf TechnolDOI 10.1007/s00170-016-9083-5ORIGINAL ARTICLEJoining sheets perpendicular to one other by sheet-bulkmetal formingI. M. F. Bragança 1 & C. M. A. Silva 2 & L. M. Alves 2 & P. A. F. Martins 2Received: 25 April 2016 / Accepted: 16 June 2016# Springer-Verlag London 2016Abstract This paper presents a variant of the traditional ‘mortise-and-tenon’ joint, which has been used for thousands ofyears by carpenters and blacksmiths to connect wood or metalparts. The new proposed joint is utilized to fix longitudinallyin position two metal sheets (or plates) perpendicular to oneother by sheet-bulk metal forming, at room temperature. Thedevelopment is performed by means of a combined finite element and experimental investigation focused on the identification of the major process parameters and on the understanding of their influence on the overall joining feasibility.Destructive testing is carried out to characterize the performance of the new proposed joint, and an analytical expressionis provided to determine the maximum tensile force that thejoint can safely withstand.Keywords Joining by plastic deformation . Joining of sheets .Sheet-bulk metal forming . Experimentation . Finite elementmodelling* P. A. F. Martinspmartins@ist.utl.ptI. M. F. Bragançaibraganca@dem.isel.ptC. M. A. Silvacarlos.alves.silva@tecnico.ulisboa.ptL. M. Alvesluisalves@tecnico.ulisboa.pt1ISEL, Instituto Superior de Engenharia de Lisboa, InstitutoPolitécnico de Lisboa, Rua Conselheiro Emídio Navarro,1959-007 Lisboa, Portugal2IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av.Rovisco Pais, 1049-001 Lisboa, Portugal1 IntroductionRecent years saw the development of sheet-bulk metal forming(SBMF) processes (also designated as ‘plate forging processes’) for producing sheet metal parts with local thickening, thinning or functional features such as teeth, ribs and solid bossespositioned outside the plane of the sheets from which they areshaped [1]. The state-of-the-art reviews by Merklein et al. [1]and by Mori and Nakano [2] include several examples of functional components produced by SBMF as well as details on theassociated tooling and tribological systems.From a ‘plasticity and friction only’ point of view, the potentialities offered by SBMF processes led Merklein et al. [3]to propose its classification into a separate standalone metalforming category placed in between the two well-establishedsheet and bulk metal forming categories, which encompassplastic flow under plane-stress or three-dimensional stressconditions, respectively. This classification into a newstandalone metal forming category was recently strengthenedby Isik et al. [4] who showed that failure by cracking in SBMFmay be triggered by tension and in-plane shearing like in sheetmetal forming but also by out-of-plane shearing, which is themain separation mode in bulk metal forming.From a production engineering point of view, the fabrication of complicated sheet metal parts by SBMF requires singleor combined application of loading by tools across and/orperpendicular to sheet thickness [5]. The directions acrossand perpendicular to sheet thickness are designated as ‘S’(across), ‘T’ (transverse-perpendicular) and ‘L’ (longitudinalperpendicular) in Fig. 1. The kinematics of the tools (punchesor rolls) and workpieces along these directions allowed Moriand Nakano [2] to characterize SBMF as a new category ofmetal forming processes that include compressive bulkforming operations such as upsetting, extrusion and ironingin conventional sheet metal forming sequences involving
Author's personal copyInt J Adv Manuf TechnolFig. 1 Loading directions in sheet-bulk metal forming (SBMF)processesstamping, shearing, bending and deep drawing with the intention of producing complex three-dimensional plastic flows insheets (or plates).Taking a broader view on the published research work inSBMF, it is also possible to conclude that research and development efforts have been exclusively placed on the fabricationof sheet metal parts, despite the potential of SBMF for joiningsheets (or plates). In fact, to authors’ knowledge, the potentialof SBMF in applications of joining by forming [6] has not yetbeen addressed in the literature.Under these circumstances, the aim and objective of thispaper is to present a simple, flexible and low-cost solutionbased on a variant of the traditional ‘mortise-and-tenon’ jointto connect two metal sheets (or plates) perpendicular to oneother by SBMF, at room temperature (Fig. 2e). The traditionalmortise-and-tenon joint is glued, pinned or wedged to lock itin place in case of woodcraft and secured by localized heatingfollowed by upsetting of the tenon in case of metalworking.The new proposed joint allows fixing longitudinally in position two metal sheets perpendicular to one other and is analternative to existing joining solutions based on welding(Fig. 2a), adhesive bonding (Fig. 2b), mechanical fasteningor riveting (Fig. 2c) and mechanical folding (Fig. 2d).The solutions based on welding (Fig. 2a) are limited by theheat-affected metallurgical zones, the distortions caused bythe heating-cooling cycles in wide longitudinal joints, the difficulties in joining dissimilar materials and the costly andtime-consuming quality inspection procedures. In addition,the welding joints are permanent joints, meaning that theycannot be reversed without causing damage to the sheets.The solutions based on structural adhesives (Fig. 2b) circumvent the problem of joining dissimilar materials but arelimited by the highest service temperature, by the careful preparation of the mating surfaces and by the utilization of extrajigs and fixtures during the curing time in order to ensure auniform pressure distribution across the entire longitudinalbonded area.Fasteners and rivets (Fig. 2c) provide low-cost, semipermanent joints that can be easily assembled anddisassembled without damaging the sheets. However, thesetypes of joints are often limited by aesthetic and corrosionproblems and by the maximum load that fasteners and rivetscan support safely. The utilization of extra accessories such asstiffeners, brackets and clips may be necessary in case of joining thin sheets.Seamed joints (Fig. 2d) are semi-permanent joints obtainedby mechanical folding and locking (by hemming or clinching)of the periphery of sheets in a press. Seamed joints are easy toobtain in straight longitudinal sections, appropriate for end-oflife and recycling applications and can be applied to sheets (orplates) of different thicknesses made from dissimilar materials. Pre-painted sheets can easily be employed.This paper introduces a variant of the mortise-and-tenonjoint (Fig. 2e) that was specifically developed to fix two perpendicular metal sheets longitudinally in position. Locking isensured by compressing through the tenon at room temperature in a direction perpendicular to the sheet thickness (refer tothe directions ‘T’ and ‘L’ of SBMF in Fig. 1). The presentationidentifies the major process parameters and investigates theirinfluence on the overall deformation mechanics and joiningfeasibility by means of experimental and finite element simulative work. The performance of the new proposed joint isassessed by means of destructive tests aimed at determiningthe force to detach the two sheets.2 Experimentation2.1 Mechanical characterization of the materialThe development of the new mortise-and-tenon joint forconnecting two metal sheets at 90 was performed onFig. 2 Different types of joints for connecting two sheets (or plates) perpendicular to one other. a Welded joint, (b) adhesive bonded joint, (c) mechanicalfastened or riveted joint without and with accessories, (d) seamed joint and (e) the new proposed ‘mortise-and-tenon’ joint produced by SBMF
Author's personal copyInt J Adv Manuf Technolcommercial aluminium alloy EN AW 5754 H111 sheets with1- and 5-mm thickness.The mechanical characterization of the material had beenpreviously performed by the authors in a universal testingmachine by means of standard tensile and stack compressiontests [7]. Figure 3 shows the stress–strain curves obtainedfrom tensile tests performed in specimens cut out from thesupplied sheets at 0 , 45 , and 90 with respect to the rollingdirection and from stack compression test specimens assembled by piling up three discs also cut out from the suppliedsheets.The average stress–strain curve resulting from the entire setof experimental data may be approximated by the followingLudwik–Hollomon’s equation,σ ¼ 325 ε0:18 ðMPaÞð1Þ2.2 Mortise-and-tenon joint locked by SBMFThe new proposed mortise-and-tenon joint is characterized bya rectangular cavity (‘mortise’) cutout in one sheet and by atenon cutout in the edge of the other sheet that passes entirelythrough the mortise of the first sheet (Fig. 4a). The tenon islonger than wider and is compressed perpendicular to thethickness direction in order to plastically deform its free lengthand ensure a mechanical lock between the two sheets to bejoined. Both the mortise and tenon were prepared by blanking,but they could also have been obtained by laser or water jetcutting, among other processes.A close observation of a typical ‘unit cell’ of the new proposed mortise-and-tenon joint for connecting two metal sheetsat 90 (refer to the detail in Fig. 4a) allows concluding that thetenon acts like a rivet. The smooth head end of the rivet isreplaced by the connection of the tenon to the surroundingmaterial of the sheet, and the opposite free end of the tenon(hereafter designated as ‘the tail’) is upset by compression inorder to produce a flat-shaped surface head.The major process parameters are identified as (i) thelength-to-width ratio lf /w0, where lf is the free length and w0is the width of the tenon, (ii) the thickness-to-width ratio t0/w0where t0 is the thickness and w0 is the width of the tenon, (iii)the mechanical strength of the two sheets to be joined and (iv)the thickness of the two sheets to be joined.Figure 4b presents a schematic representation of the laboratory tool in which the tests were performed. The main activecomponents of the tool are (i) the compression punch, (ii) thedie segments and (iii) the blank holder with suitable screws toclamp the lower sheets firmly in position during the tests. Thetool was installed in the universal testing machine where material characterization had been performed, and the tests wereperformed in displacement control under a constant verticalvelocity equal to 5 mm/min.The work plan is summarized in Table 1. The first part ofthe work plan (first row of Table 1) only takes the variation ofthe length-to-width ratio lf /w0 into account because the widthw0 of the tenon (and of the mortise) was set as twice theoriginal sheet thickness w0 2t0. The two sheets to be joinedare made from the same material and have equal thicknesses.The connection of sheets made from dissimilar materials withdifferent thicknesses was left out of the first part of the investigation because its aim and objectives were focused on thepresentation of the new proposed joining concept and on theevaluation of its overall performance. The latter was accomplished by means of destructive tests aimed at determining theforce to detach the two sheets.The second part of the work plan (second row of Table 1)takes the variation of the thickness-to-width ratio t0/w0 intoconsideration and allows tests to be carried out with two different sheet thicknesses.3 Numerical modellingThe development of the new SBMF process for joining sheetsperpendicular to one other was supported by finite elementmodelling with the in-house computer program I-form, whichhas been extensively validated against experimental results ofmetal forming processes since the end of the 1980s. I-form isbased on the irreducible finite element flow formulation whichis built upon the following variational statement,ZZZ:1:ð2ÞΠ¼σ ε dV þ Kε2v dV T i ui dS2VZþFig. 3 Stress–strain evolutions obtained from standard tensile testsperformed in specimens cut out from the supplied sheets at 0 (L), 45 (LT) and 90 (T) with respect to the rolling direction and from stackcompression tests built upon three circular discs piled up to formcylindrical test specimensSfVZjur j!STτ f dur dS0where the symbol σ denotes the effective stress, ε: is the effective strain rate, εV is the volumetric strain rate, K is a largepositive constant imposing the incompressibility of volume V,
Author's personal copyInt J Adv Manuf TechnolFig. 4 Schematic representationand terminology of the newprocess for joining two sheetsperpendicular to one other. a Thenew proposed ‘mortise-andtenon’ joint with a detail showinga typical unit cell. b Laboratorytool for joining a unit cellTi and ui are the surface tractions and velocities on surface STand τf and ur are the friction shear stress and the relative velocity on the contact interface Sf between the compressionpunch and the tail of the tenon. Further details on the computerimplementation of the finite element flow formulation withspecial emphasis to contact and frictional sliding between rigid and deformable objects are provided by Nielsen et al. [8].Finite element modelling was firstly employed to investigate the critical length-to-width ratio lf/w0 to trigger plasticinstability in the form of buckling out of the sheet plane whenthe tenon is subjected to compression loading perpendicular toits thickness. The numerical simulations made use of threedimensional and simplified two-dimensional finite elementmodels under plane strain deformation conditions. In case ofthree-dimensional models, the specimens were discretized bymeans of hexahedral elements and the punch and blank holderwere discretized by means of contact-friction spatial lineartriangular elements. Figure 5 shows the initial and final geometries for two different test specimens with lf/w0 ratios equal to1 and 2, which give rise to symmetrical (Fig. 5a) and nonsymmetrical (Fig. 5b) upset deformations. The nonsymmetrical deformation is due to failure by buckling.As will be seen later in the presentation, the simplified twodimensional models not only provide results closer to thoseobtained by the full three-dimensional models as they are capable of finishing a complete simulation with remeshing inless than 3 min in a personal computer equipped with oneIntel Xeon E5-2620 CPU (2.10 GHz) processor.Finite element modelling was subsequently employed tosimulate the joining of the two sheets perpendicular to oneother by means of the new proposed mortise-and-tenon jointproduced by SBMF. The experienced gain in the upsetTable 1The plan of experiments (nomenclature according to Fig. 4)Test case t0 (mm) w0 (mm) l0 (mm) lf (mm) lf/w01–56–951, .50.25–0.5 0.1, 0.5compression of the tenons allowed authors to discretize typical unit cells of the new proposed mortise-and-tenon joint bymeans of two-dimensional models. These were built by halving the two sheets lengthwise and discretizing the resultingcross section by means of quadrilateral elements under planestrain deformation conditions.4 Results and discussion4.1 Upset compression of the tenonThe first task in the development of the new proposed joiningprocess was the determination of the critical length-to-widthratio lf/w0 that gives rise to plastic instability in the form ofbuckling out of the sheet plane by upset compression of thetenon. Different test specimens with various lf/w0 ratios wereemployed (refer to the first row of Table 1), and the corresponding experimental and finite element predicted evolutionsof the force with displacement are summarized in Fig. 6.As seen, the force-displacement evolution is characterizedby two different trends. The specimens with ratios lf/w0 2experience a steep rise in the force followed by a monotonicincrease at a lower rate without signs of plastic instability andfailure by out-of-plane buckling. The ratios lf/w0 2 give riseto symmetric modes of deformation (refer also to Fig. 5a) andare suitable for the design of the new proposed mortise-andtenon joints locked by SBMF.The specimens with ratios lf/w0 2 behave differently dueto the occurrence of out-of-plane buckling. In fact, after a firststeep increase of the force with displacement up to the onset ofplastic instability, they experience bifurcation into a secondaryloading path during which the compression force remains approximately constant or may even decrease (e.g. lf/w0 3).The resulting deformation is non-symmetrical (refer also toFig. 5b), and therefore, the ratios lf/w0 2 are unsuitable tofix longitudinally in position two metal sheets perpendicularto one other by means of the new proposed mortise-and-tenonjoints.
Author's personal copyInt J Adv Manuf TechnolFig. 5 Finite element modellingof the upset compression of atenon with critical length-to-widthratios equal to a lf/w0 1 and b lf/w0 2. The thickness-to-widthratio t0/w0 0.5 is constant for alltest specimensA comparison of three-dimensional and two-dimensionalfinite element predicted evolutions of the force with displacement for test specimens with lf/w0 1 and lf/w0 2 allows concluding that two-dimensional models can also be successfullyutilized for modelling the upset compression of the tenon under plane strain deformation conditions.To conclude, it is worth noting that the last part of the forcedisplacement evolutions shown in Fig. 6 may be influenced bythe contact of the tenon with the tool. This means that in caseof the test specimen with lf/w0 2, for example, the increase incompression force beyond 7.5 mm displacement is not due toplastic instability but to the contact of the deformed tenon withthe upper surface of the blank holders (refer to Fig. 4b).4.2 Mortise-and-tenon joints produced by SBMFFigure 7a shows the experimental and finite element predictedevolutions of the force with displacement for a unit cell of thenew proposed mortise-and-tenon joint. Five different length-towidth ratios lf/w0 of the tenon were utilized (refer to the first rowof Table 1), and the overall results show good agreement withthose obtained in the previous section on upset compression(Sect. 4.1).In fact, when the length-to-width ratio lf/w0 2, there are nosigns of out-of-plane buckling and the force-displacement evolution allows disclosing three different regions labelled as ‘A’,Fig. 6 Experimental and finiteelement predicted evolution of theforce with displacement forvarious tenons with differentlength-to-width ratios lf/w0 loadedin uniaxial compressionperpendicular to the sheetthickness (t0/w0 0.5)‘B’ and ‘C’ (refer to lf/w0 1, in Fig. 7a). Region A correspondsto a very short initial period of time when the force experiencesa steep rise and is immediately followed by region B in whichthe force grows monotonically at a lower rate as the free lengthof the tenon is progressively upset by compression along thedirection per
1 ISEL, Instituto Superior de Engenharia de Lisboa, Instituto Politécnico de Lisboa, Rua Conselheiro Emídio Navarro, 1959-007 Lisboa, Portugal 2 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal Int J Adv Manuf Technol DOI 10.1007/s00170-016-9083-5 Author's personal copy
A) Rotating perpendicular lines result in parallel lines. B) The lines remain perpendicular only if rotated 180 . C) The lines remain perpendicular only if rotated 360 . D) Rotated perpendicular lines always remain perpendicular lines. Explanation: Rotated perpendicular lines always remain
In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line. If h k and j h, then j k. Proof Example 2, p. 150; Question 2, p. 150 Theorem 3.12 Lines Perpendicular to a Transversal Theorem In a plane, if two lines are perpendicular to the same line, then they are parallel to .
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Any two vertical lines are parallel. Postulate 18 Slopes of Perpendicular Lines In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1. The slopes of the two lines that are perpendicular are negative reciprocals of each other. Horizontal lines are perpendicular to vertical lines
indicate parallel lines. Perpendicular lines are lines in the same plane that intersect at a right angle, which measures 90 .You draw a small box in one of the angles to show that the lines are perpendicular. Investigation: Slopes The opposite sides of a rectangle are parallel, and the adjacent sides are perpendicular
Theorem 10.7 Perpendicular Chord Bisector Theorem If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc. Proof Ex. 22, p. 550 Theorem 10.8 Perpendicular Chord Bisector Converse If one chord of a circle is a perpendicular bisector
If two parallel lines are cut by a transversal, then the Pairs of alternate exterior angles are congruent. Theorem 3.7 Perpendicular Transversal If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other. Ex. 1 Proving the A
299 MMaintaining Mathematical Proficiencyaintaining Mathematical Proficiency Writing an Equation of a Perpendicular Line Example 1 Write the equation of a line passing through the point ( 2, 0) that is perpendicular to the line y 2x 8. Step 1 Find the slope m of the perpendicular line. The line y 2x 8 has a slope of 2. Use the Slopes of Perpendicular Lines Theorem (Theorem 3.14).
