Derivation of the Bi-axial Bending, Compression and Shear Strengths of Timber Beams

Transcription

1 Send Orders of Reprints at The Open Mehanis Jornal Open Aess Deriation of the i-axial ending Copression and Shear Strengths of Tiber eas T A C M Van Der Pt * Falt of Ciil Engineering and Geosienes Tiber Strtres and wood tehnolog TU Delft PO ox GA Delft The Netherlands Abstrat: The deriation is gien of the obined bi-axial bending opression and shear strength of tiber beas As for other aterials the elasti fll plasti liit design approah applies whih is known to preisel explain and predit niaxial bending strength behaior The deriation is based on hoosing the loation of the netral line This proides the stress distribtion in the bea ross setion in the ltiate state for that ase aking it possible to allate the assoiated ltiate bending oents in both ain diretions and ltiate noral- and shear fore The deried general eqations are siplified to possible eleentar design eqations appliable for bilding reglation Kewords: Tiber beas liit analsis bi-axial bending-opression-shear strength INTRODUCTION As known tiber beas behaes qasi isotropi for the loading ase of bending opression with shear p to the ltiate state [] and the oon bea theor an be applied The fitie bending strength f based on the linearied bending stress in the failre state gien in [] onl applies for retanglar ross-setions and for the ost eleentar loading ase For obined loading ases and to explain easreents the elasti-fll plasti diagra has to be sed as shown eg in [] where the deriation is gien of the niaxial bending opression and shear strength of tiber beas For profiles this elasti-plasti approah has to be applied to obtain the neessar profile fators on the fitie linear bending strength f The elasti-fll plasti approah is the basis for liit design and applies for all aterials and it is extensiel shown also for other aterials as steel and onrete [4] to be sffiient for the real strength predition For wood this neessar design ethod was alread generall known and widel applied sine 90 see [5] Althogh neessar as basis for stabilit design and for the presribed allable reliabilit bi-axial bending strength obined with opression and shear loading is neer deterined atheatiall before and therefore is gien here to orret this oission The elasti-plasti stress diagra with a negligible plasti range for tension applied in the figres below represents an adissible eqilibri sste satisfing eqilibri and bondar onditions iolating nowhere the ield riterion and ths is lower bond soltion The highest lower bond soltion is eqal to the real strength and this is reahed in this *Address orrespondene to this athor at the Falt of Ciil Engineering and Geosienes Tiber Strtres and wood tehnolog TU Delft PO ox GA Delft The Netherlands; Tel: ; Fax: ; E-ail: anderp@xs4allnl ase when the netral line is a straight line and when nliited flow in pre opression is possible ths when the shear stress is arried in the elasti part of the ross setion Ths as onfired in [] the niaxial ltiate obined bending-opression strength is deterined b the ltiate tensile stress f t and b nliited flow in opression at the flow opression stress f ending failre ths alwas is an ltiate tension failre at f t This therefore is the starting point for the deriations in Setion and is an iproeent with respet to the old odel applied in [6] whih was based on a liited ltiate opression strain and therefore did not fit preisel to the data The deriations of Setion are in priniple based on hoosing the loation of the netral line and allate the assoiated ltiate bending oents and noral and shear fores There are three ases to regard for the loation of the netral line The netral line a go throgh two opposite planes of the ross setion as gien in Fig or the netral line goes throgh two adjaent planes at the tension side or at the opression side as gien b respetiel Fig 6 and 7 Matheatiall sipler is not to hoose the loation of the netral line bt of the parallel border line of the fll plasti opression area of the ross setion as done in the following I-AXIAL ENDING STRENGTH CASES Doinating ending in the Stiff Diretion! b ; Z! h in Fig The ltiate state of the deterining ross-setion of a bea loaded nder biaxial bending is gien in Figs 6 and 7 The line EF in Fig is the bondar of the fll plasti ltiate opression strength area of the ross-setion of a bea In Fig an eqilibri state is gien of a bea with diension b and h loaded in doble bending For the analsis the bending stresses of the ltiate state are re / 0 entha Open

2 Deriation of the i-axial ending The Open Mehanis Jornal 009 Vole 5 N! T! f t bz! b + b = f! * * Z h! b + b where s = ft / f Ths with the axial possible ale of N is with N = f : Z h! b + b =! N 8 N Fig Copression with bi-axial bending garded to be a sperposition of opression fore N = f of the nifor opression stress f oer the entire ross-setion and a tension fore b the linear inreasing tensile stresses in the plane AEF with a axial tensile stress ft + f in point A In Fig is:! = line C! = line AD! = line E! 4 = line AG Then:! =! b! =! b! Z =! b Noral Fores =! b! 4 Z = b The tensile stress at point is sing eq: = b! =!! f t f + f t The total tensile fore T onsists of the s of stresses within the stress praid of T in the enter of grait of this stress praid aboe plane AG with axial stress at A of ft + f! and the part T aboe plane AG below the stress praid T with onstant stress! and finall of T aboe plane EFG with a linear inreasing stress to! at points and G Then is sing eq and : T = f t! b! 4 = f + f t b Z 6 T =! b! 4 = f + f t b Z b 4 T =! b! = f + f t bz b 5 b T = T + T + T = Z f t! 6 + b Z! b + bz! b + + * - Ths the ltiate ale of T is: T = f t bz! b + b 6 The ltiate noral fore N with for oneniene a positie sign for opression is: For niaxial bending in the stiff diretion! or: b /! 0 this eqation agrees with Eq of [] where it was shown that the theor preisel fits the data ease of the bondar onditions of the eqations of Setion :! b ; Z! h is:! N / N! s / 9 with a tension liit negatie sign when s > for the eqations of Setion ending Moents The bending oent b T = T + T + T aording to eq to 6 with respet to the resltant opression fore N = f ths with respet to the enter of the rosssetion of the bea is: in the stiff diretion: /! / 4 /! / /! /! / M = T h! + T h! + T h!! = = T! h / T! / 4 T! / T!! / +! / and in the weak diretion: M = T b /! b / 4 + T b /! b / + 0 = T b / 4 + T b / 6 Ths sing eqs to 6: = f t bzh 4! b + b! f + f t bz! b + b! b or b sbstittion of Z aording to Eq8 this eqation beoes:! N +!! N / N 6 N + * + 4! 6b / + 4b /! b /! b / + b / - = f + f t b Z! b 4 sbstittion of Z aording to Eq8 Eq4 beoes: hb 6! b / b / b / + b /! N 5 N For! M = 0 as follows fro Eq 4 or 5 and ths niaxial bending ors and Eq 8 then beoes: Z / h =! N / N When this is sbstitted in Eq for! the niaxial bending strength beoes:! = f t bzh 4 f + f t bz 6 = f 6 + s + 4N / N as fond before in [] based on the data of [6] N 6 N

3 6 The Open Mehanis Jornal 0 Vole 6 T A C M Van Der Pt Fig ending and shear stress Fig 4 s = - 50th strength perentile Fig s = - 95th perentile of the bending opression strength The fitie linear elasti design bending stress applied in the ilding Codes ths is:! = 6 + s + 4N / N N * 7 N whih is eqal to the niaxial bending strength N = 0 Ths when: f 6M! s = = f f when 8 gien in Fig In [] the ale of s = was fond for the ean strength while s = for the 95th perentile and s = 077 for the 5th perentile of the niaxial obined bending - opression strength gien in Figs 4 and 5 of [] where = 6 M / f and n = N / f These ales are based on the data of [6] and appl for standard liate onditions Iportant is that these ales of s are independent of the load-obination showing that there is no ole effet de to tensile stress distribtion bt onl for ole alone This is explained b a derease of qalit with ole inrease This also explains wh b not brittle opression failre a ole effet is possible as reported in literatre Aording to the bondar onditions the eqations of Setion appl for: f / 4! / Fig 5 s = 077-5th strength perentile and 0! / 4 9 Shear Fore in The total ltiate reslting shear fore V = V x + V the elasti region of the ross setion is: V = f b! + b Z!! = f b bz! 0 based on the paraboli shear stress distribtion in the elasti region The possible range of V is: 0! V! f / for all ases of Setion Sbstittion of Z of Eq8 into Eq0 gies: V = f b!! N / N! b / + b / or with niaxial V 0! aording to Eq: V = b N / N b / + b / The shear strength V 0! is deterined at ltiate niaxial bending for N = 0 and! and is gien for design as a fitie linear elasti paraboli stress distribtion oer the total depth h aording to Fig Ths

4 Deriation of the i-axial ending The Open Mehanis Jornal 009 Vole 7 Fig 6 Doinating opression with bi-axial bending for Z h and b Fig 7 Doinating tension with bi-axial bending for Z! h and = f = f f For! Eq represents the niaxial loading ase see [] giing: V! = N N 4 Doinating ending in the Weak Diretion Z! h and! b All eqations ths far appl for the ase of Fig ths for Z! h and! b For Z! h and! b the sae eqations of Setion appl with interhange of and ; Z and b and h as follows b eqs 5 to 7: hb! N +!! N / N 6 N + * + 4! 6h / Z + 4h / Z! h / Z! h / Z + h / Z N 5 -! h / Z 6!+ Z / h + h / Z! N 6 V h / Z = V 0! h / Z + h / Z N 7 N Again one oponent of the biaxial oent shows the linear relation with - N/N When Eq7 is deterining also re Eq5 is t off to this relation see Setion 4 Doinating loading in opression! b and Z! h in Fig 6 For doinating opression the ondition! b and Z! h a appl aording to Fig 6 Noral Fores N! Z f + f t! 6 Z or: 6! Z = N N 8 Aording to the bondar onditions is 5! s / 6 N / N 9 Shear Fore For the ltiate shear fore applies: V = f Z = f Z = f! N N De to the bondar onditions Z here axial: V = f 0! h and! b is V Fro Eq 0 follows when shear strength is deterining: Z V = f = f! N f N bease aording to Eq 9: N / N! 5 s / 6 ending Moents As before the resltant fore of the tensile stress praid ties the distane to the resltant opression fore in the enter of the ross setion deterines the bending oent For bending applies sing Eq 8: = f t Z = f t Z h! Z 4 f h 4! N N b! 4 f b 4! N N Knowing M N and M and Z are known and the fond prodt Z shold be saller for bending failre than the ale of Z for shear failre aording to Eq0 ths Z! V / f 4 bending There is no adantage to first eliinate h/z and to end with an expression in b/ as done before for the ases of Setions and 4 Doinating Loading in Tension For doinating tension the ondition! b and Z! h a appl aording to Fig 7 In Fig 7 is:! Z = K KA =! b LF = Z! h Z 5 In the liit ase for the here applied eqations when! = h and line LF = b is: b + h Z = 6

5 8 The Open Mehanis Jornal 0 Vole 6 T A C M Van Der Pt 4 Noral Fores The tension fore T follows fro the ontent of the tension praid AHKA with height f t +f ins the ontents of praids EK and FHLF with heights! and! F with:! F = GF AD f + f t! = C AD f + f t = HF f + f t = Z h f t 7 HA Z = K f + f t = h f t 8 KA Ths the total tension fore is: T! T F! T 9 with: T A = f t T F =! F Z LF FH! h Z = f t 6 In the sae wa is: T = f t 6 Ths: Z! b! b T!! b!! h Z h Z h Z Z h = f t The applied ltiate noral opression fore is: N! T! Z 6!! b!! h Z 6 40 Z h Z To replae Z in other eqations this an be written with N = f : Z 6!! b!! h Z =! N 45 N In the liit ase of Eq45 is for: h = Z! or b /! 0 : 6 b!! b = 6 b!! b + b! b =! N 46 N The sae applies for b 4 Shear Fore = Z! The ltiate shear fore is: h!! V = f! b! FL = f! b! Z! h Z h!! b Z = = f!!! h Z b!! b Z h +! h Z! b Z = = f b + Z h! h bz! bz h! Z 47 In the liit ase is for: h = Z! or for b = Z! : V = f / 4 ending Moents The ltiate bending oent is: h! Z 4! T h!! 4! T h F! h + Z! h 4 = T h! T Z A 4 + T! 4 + T h + Z! h F 4 =!! b!! h h Z! T Z A 4 + T! b! A 4 + T! h A Z h + Z! h 4 = h!! b h!! h h Z! Z 4 + Z 4! b 4 +! h Z h!! b!! h Z! Z h + Z h! b 4 +! h Z Ths: h + Z! h 4 = + Z h 48! Zh 6! b h Z Z h + Z h b 4 + h Z + Z h 49 For the liit ase in aordane with Eq6: h = Z! applies for M giing: 6 50 The liit ase: b = Z! applies to M Z leading to! hb 6 5 In general is M : b! 4! T b! b +! b 4! T b F! 4! h Z = = T b! T A 4 + T b! 4 + T F! h Z 4 =!! b!! h b Z! 4 +! b b + +! h 4 Z Ths: 4 Zb!! b!! h Z! b +! b + b +! h 4 Z 5 b 44 High Tensional Loading At higher tensile loading the behaior is linear elasti and deterining for failre is the axial tension stress leading to: 4

6 Deriation of the i-axial ending The Open Mehanis Jornal 009 Vole 9 f t = hb + N or: = + + N N t 5 In this eqation tension has the positie sign SIMPLIFICATION FOR PRACTICAL DESIGN For pratial design at doinating bending the expressions in ariables b / and h / Z in the eqations an be siplified withot loss of ara Doinating Copression For doinating opression Z! h and! b Eqs 8 to 4 are siple enogh as design eqations Doinating ending in the Stiff Diretion Eq for doinating bending in the stiff diretion Z! h and! b is:! N *!! N / N b - 6 N / + / with:! b / = 4 6b / + 4b / b / b / + b / 54 / In the appling range Z! h and b/ between 0 to! b is a bend re whih preisel an be approxiated b the parabola: a power for: b 8 b b b! = + + or b / 4 / 4 /! b / = / 55 Ths! N!! N / N b + 6 N + * and aording to Eq and 6 is: 8-56 = N / N 5b / 8 57! s + 4N / N and b/ is diretl known fro: M M! and N where M is the oponent in the stiff diretion of the ltiate biaxial oent and M! the diret allable niaxial strength in the stiff diretion ths! or b / = 0 in Eq 56 For the oponent of the ltiate bi-axial bending strength in the weak diretion M is aording to Eq 5 in the range of 0! b /! : hb 6! N hb! N b / 6! N b! N + 05 b bease! b / = b / + / b + b / b / b / 044b / or b a power for: 8 58! b = b + b 8 59 / / 05 / Aording to Eq is: V = N b / with: N b /! b / = b / + b / =! b / b / Ths: V = N N + 05 b Doinating ending in the Weak Diretion Stiff The eqations of Setion appl with interhange of b with h and with Z and with Ths for doinating bending in the weak diretion when! b and Z! h : hb! N!! N / N h + 6 N + Z * = N / N 8 5h / Z 6! s + 4N / N 6! N h! N Z + 05 h Z V = N N + 05 h Z 4 Doinating Tension The foregoing eqations also appl for tension with bending and will oer the ases in pratie For doinating tension the allation aording to Setion 4 to 4 of Fig 7 shold be followed These eqations an not be siplified and shold be tablated for different ales of h / Z and b / or soled b a nerial ethod for a gien loading ease for high tension and for lower qalities and large strtral sies the long ter tensile strength will be lower than the opression strength and the behaior is linear elasti based on the ltiate tensile stress aording to Setion 44:! N / N = / M + / M 65 4 DESIGN EQUATIONS The gien eqations for biaxial bending are eas prograable for nerial soltions Howeer it alwas is neessar to proide siple Code rles The bondar onditions of appliation of the eqations are deterined b the niaxial bending ases and therefore the onditions:! N / N! s / 9 = f = f f V! / = N / N 4

7 0 The Open Mehanis Jornal 0 Vole 6 T A C M Van Der Pt appl for doinated bending in the stiff and weak diretion and for doinated tension aording to Setions and 4 For doinating high opression and therefore low shear loading and b N reded bending oents aording to Eqs and of Setion applies: 5! s / 6 N / N 9 V = f! N / N / = f f f / / 4 + s / 4 N / N / 4 hb + s / 4 hb N / N The bondar onditions for appliation of the other bending oent eqations are for: Doinating bending in the stiff diretion Setion : f / 4! / and 0! / 4 9 Doinating bending in the weak diretion Setion : / 4 and / 4! hb / 0! hb f hb The sae onditions appl for M and M of doinating tension aording to Setions 4 to 4 Howeer as Code rle these eqations are too extended and shold be replaed for design for doinating tension b the sae Eq5 Ths for doinating opression when Z! h and! b Eqs8 to 4 appl Else design an be based on the eqations of Setion Eq54 an be written sing Eq8: f / 6 =! N s +!! N / N b / + N * + s! - whih will be written: s +!! n =! n s! = s + s!! n Siilarl Eq58 an be written:!! n s! 66 =! n s! 67 This is a linear relation between and n the sae as applies for the shear strength Eq 4 Aording to the -point bending test is: V a = M where a = L/ is the distane of the load in the iddle of the bea to the spport Ths a / h = M / V h is the shearslenderness with ritial ale: a / h = f / 6 / / f f = f / 4 f f This ritial ale of axial bending and axial shear failre at the sae tie is a / h! aording to the test-bea diensions of the shear strength test for ean qalit Eropean softwoods Eq 4 ths a be written: Fig 8 Interation re t off b the dashed bending- opression strength shear line or no t off b the drawn ltiate shear line V! = a! a = h! a = N N or: n =! h / a 68 For a = h the bondar is reahed where below the axial possible bending oent will be reded b the axial possible shear fore and Eq68 then beoes: n =! This linear relation was the basis of the Dth Code [7] and shold appl for all Codes as long as the shear allation aording to Eq 68 is absent Eq 68 shows the paraboli Eroode line to be a fator too nsafe when a = h see Fig 8 and [] ease sh a linear relation also is deterining for one oponent of the biaxial strength as gien b Eq 67 The red line Eq 66 the best also an be approxiated b straight lines throgh the end points = ; n = 0 and = 0; n = and the point on the re for n = 05 see [] where aording to Eq 66: = 05 s +! / s! for n =05 giing: =! s! 5 + / n when n! 05 s! s! n =! s +! / when n! The sae eqations appl for doinating bending in the weak diretion after interhange of b with h and with Z and with 5 CONCLUSIONS A deriation is gien of the biaxial bending strength in aordane with the liit analsis ethod and ths based on elasti-fll-plasti behaior Therefore with the restrition of negleting hardening stages after initial flow the anal-

8 Deriation of the i-axial ending The Open Mehanis Jornal 009 Vole sis is rigoros and the strength predition realisti and the reslt has to be applied in the ilding Codes to proide the presribed sffiient preise reliabilit allation For the highest lower bond soltion of biaxial bending strength is neessar that the netral axis is a straight line and that nliited flow in pre opression ors ths when there is bending-tension failre and when the shear stress is arried in the elasti part of the ross setion This is an iproeent with respet to the ths far applied old odel [6] restriting the ltiate plasti opression strain The deried general expressions in oordinates of the bondar line of the fll opression area are siplified to eleentar design eqations whih also has to be in the Reglations Chosen is for sipliit of design for separate ltiate shear strength and ltiate bending-opression strength eqations The eqations ontain also the soltion for niaxial bending ases whih are alread shown to preisel explain and fit data b the elasti fll plasti liit approah The ale of s = ft / f appears to be abot onstant for all load obinations of bending with opression indiating again that there alwas is failre b the ltiate tensile strength A ole effet b stress distribtion ths needs not to be regarded as follows fro the niaxial data The ole effet ths now is ased b the ole alone de to dereasing qalit b ole inrease CONFLICT OF INTEREST The athors onfir that this artile ontent has no onflits of interest ACKNOWLEDGEMENT Delared none REFERENCES [] an der Pt TACM A ontin failre riterion appliable to wood J Wood Si 009; 55: 5- [] Design of Tiber Strtres Part -: General Coon rles and rles for bildings Erode 5 EN 995: 004 [] an der Pt TACM Failre riterion for tiber beas loaded in bending opression and shear Wood Mat Si Eng 00; : 4-9 [4] Chen WF Atsta T Theor of bea olns Spae behaior and design USA: J Ross Pblishing 007 ol [5] Kollann F Tehnologie des Holes nd der Holwerkstoffe erlin: Springer Verlag 95 [6] Johns KC hanan AH Strength of tiber ebers in obined bending and axial loading CI/IUFRO Meeting oras Sweden 98 see also CI W8 985 Israel [7] Tiber strtres - asi reqireents and deterination Methods NEN TG 990 Reeied: Jl 07 0 Reised: Agst 0 0 Aepted: Agst 0 T A C M Van Der Pt; Liensee entha Open This is an open aess artile liensed nder the ters of the Creatie Coons Attribtion Non-Coerial Liense whih perits nrestrited non-oerial se distribtion and reprodtion in an edi proided the work is properl ited