BEAMS SUBJECTED TO TORSION AND BENDING - II

Transcription

1 BAS SUBJCTD TO TORSION & BNDING-II 8 BAS SUBJCTD TO TORSION AND BNDING - II.0 INTRODUCTION In the previous chpter, the bsic theor governing the behviour of bems subjected to torsion ws discussed. A member subjected to torsionl moments would twist bout longitudinl is through the sher centre of the cross section. It ws lso pointed out tht when the resultnt of pplied forces pssed through the longitudinl sher centre is no torsion would occur. In generl, torsionl moments would cuse twisting nd wrping of the cross sections. When the torsionl rigidit (GJ) is ver lrge compred with its wrping rigidit (Γ), the section would effectivel be in uniform torsion nd wrping moment would unlikel to be significnt from the designer's perspective. mples of this behviour re closed hot-rolled sections (e.g. rectngulr or squre hollow sections) nd rolled ngles nd Tees. Note tht wrping moment is developed onl if wrping deformtion is restrined. Wrping deformtion in ngle nd T-sections re not smll, onl wrping moment would be smll. On the other hnd, most thin wlled open sections hve much smller torsionl rigidit (GJ) compred with wrping rigidit (Γ) vlues nd these sections will be ehibiting significnt wrping moment. Hot rolled I sections nd H sections would ehibit torsionl behviour in-between these two etremes nd the pplied loding is resisted b combintion of uniform torsion nd wrping torsion..0 DSIGNING FOR TORSION IN PRACTIC An structurl rrngement in which the lods re trnsferred to n I bem b torsion is not n efficient one for resisting lods. The messge for the designers is "Avoid Torsion - if ou cn ". In ver lrge number of prcticl designs, the lods re usull pplied in such mnner tht their resultnt psses through the centroid. If the section is doubl smmetric (such s I or H sections) this utomticll elimintes torsion, s the sher centre nd centroid of the smmetric cross section coincide. ven otherwise lod trnsfer through connections m - in mn cses - be regrded s ensuring tht the lods re effectivel pplied through the sher centre, thus eliminting the need for designing for torsion. Furthermore, in situtions where the floor slbs re supported on top flnges of chnnel sections, the lods m effectivel be regrded s being pplied through the sher centre since the fleurl stiffness of the ttched slb prevents torsion of the chnnel. Where significnt eccentricit of loding (which would cuse torsion) is unvoidble, lterntive methods of resisting torsion efficientl should be investigted. These include Copright reserved Version II 8 -

2 BAS SUBJCTD TO TORSION & BNDING-II design using bo sections, tubulr (hollow) sections or lttice bo girders which re full tringulted on ll fces. All these re more efficient mens of resisting torsionl moments compred with I or H sections. Unless it is essentil to utilise the torsionl resistnce of n I section, it is not necessr to tke ccount of it. The likel torsionl effects due to prticulr structurl rrngement chosen should be considered in the erl stges of design, rther thn left to the finl stges, when perhps n inpproprite member hs lred been chosen..0 PUR TORSION AND WARPING In the previous chpter, the concepts of uniform torsion nd wrping torsion were eplined nd the relevnt equtions derived. When torque is pplied onl t the ends of member such tht the ends re free to wrp, then the member would develop onl pure torsion. The totl ngle of twist (φ ) over length of z is given b T q z φ G J where T q pplied torque GJ Torsionl Rigidit () When member is in non-uniform torsion, the rte of chnge of ngle of twist will vr long the length of the member. The wrping sher stress ( w ) t point is given b w S wms φ t () where odulus of elsticit S wms Wrping stticl moment t prticulr point S chosen. The wrping norml stress (σ w ) due to bending moment in-plne of flnges (bi-moment) is given b σ w -.W nwfs. φ'' where W nwfs Normlised wrping function t the chosen point S..0 COBIND BNDING AND TORSION There will be some interction between the torsionl nd fleurl effects, when lod produces both bending nd torsion. The ngle of twist φ cused b torsion would be mplified b bending moment, inducing dditionl wrping moments nd torsionl shers. The following nlsis ws proposed b Nethercot, Slter nd lik in reference (). Version II 8 -

3 BAS SUBJCTD TO TORSION & BNDING-II. imum Stress Check or "Cpcit check" The mimum stress t the most highl stressed cross section is limited to the design strength (f /γ m ). Assuming elstic behviour nd ssuming tht the lods produce bending bout the mjor is in ddition to torsion, the longitudinl direct stresses will be due to three cuses. σ σ b bt σ. W w Z Z t nwfs ''. φ () σ bt is dependent on t, which itself is dependent on the mjor is moment nd the twist φ. t φ () Thus the "cpcit check" for mjor is bending becomes: σ b σ bt σ w f /γ m. (5) ethods of evluting φ, φ, φ nd φ for vrious conditions of loding nd boundr conditions re given in reference ().. Buckling Check Whenever lterl torsionl buckling governs the design (i.e. when p b is less thn f ) the vlues of σ w nd σ bt will be mplified. Nethercot, Slter nd lik hve suggested simple "buckling check" long lines similr to BS 5950, prt b ( σ bt σ w ) ( f / γ ) m 0.5 b () where, equivlent uniform moment m nd b, the buckling resistnce moment φ B ( ) φ B p p Version II 8 -

4 in which φ B p ( η ) P, the plstic moment cpcit f. Z p / γ m LT BAS SUBJCTD TO TORSION & BNDING-II Z p the plstic section modulus, the elstic criticl moment λ where λ LT is the equivlent slenderness. p LT. Applied loding hving both jor is nd inor is moments When the pplied loding produces both mjor is nd minor is moments, the "cpcit checks" nd the "buckling checks" re modified s follows: Cpcit check: σ b σ bt σ w σ b f /γ m (7) π f γ m Buckling check: b f Z / γ m ( σ bt σ w ) ( f / γ ) m 0.5 b (8) where σ bt m / Z. Torsionl Sher Stress Torsionl sher stresses nd wrping sher stresses should lso be mplified in similr mnner: ( ) vt t w 0.5 (9) b This sher stress should be dded to the sher stresses due to bending in checking the dequc of the section. 5.0 DSIGN THOD FOR LATRAL TORSIONAL BUCKLING The nlsis for the lterl torsionl buckling is ver comple becuse of the different tpes of structurl ctions involved. Also the bsic theor of elstic lterl stbilit cnnot be directl used for the design purpose becuse Version II 8 -

5 BAS SUBJCTD TO TORSION & BNDING-II the formule for elstic criticl moment re too comple for routine use nd there re limittions to their etension in the ultimte rnge A simple method of computing the buckling resistnce of bems is given below. In mnner nlogous to the Perr-Robertson ethod for columns, the buckling resistnce moment, b, is obtined s the smller root of the eqution ( - b ) ( p - b ) η LT. b (0) As eplined in pge, b is given b, b φ φ where φ B B p ( ) B ( η ) LT p [ As defined bove, lstic critcl moment p f. Z p / γ m η LT Perr coefficient, similr to column buckling coefficient Plstic section modulus] Z p In order to simplif the nlsis, BS5950: Prt uses curve bsed on the bove concept (Fig. ) (similr to column curves) in which the bending strength of the bem is epressed s function of its slenderness (λ LT ). The design method is eplined below. The buckling resistnce moment b is given b p b p b.z p () where p b bending strength llowing for susceptibilit to lterl -torsionl buckling. Z p plstic section modulus. It should be noted tht p b f for low vlues of slenderness of bems nd the vlue of p b drops, s the bem becomes longer nd the bem slenderness, clculted s given below, increses. This behviour is nlogous to columns. The bem slenderness (λ LT ) is given b, λ LT π f λ LT () where λ LT p Version II 8-5

6 BAS SUBJCTD TO TORSION & BNDING-II 00 Bem fils b ield 00 p b N/mm Bem buckling λ LT Fig. Bending strength for rolled sections of design strength 75 N/mm ccording to BS 5950 Fig. is plotted in non-dimensionl form compring the observed test dt with the two theoreticl vlues of upper bounds, viz. p nd. The test dt were obtined from tpicl set of lterl torsionl buckling dt, using hot-rolled sections. In Fig. three distinct regions of behviour cn be observed:- stock bems which re ble to ttin the plstic moment p, for vlues of λ LT below bout 0.. Slender bems which fil t moments close to, for vlues of bove bout. λ LT bems of intermedite slenderness which fil to rech either p or. In this cse 0. < <. λ LT Bems hving short spns usull fil b ielding. So lterl stbilit does not influence their design. Bems hving long spns would fil b lterl buckling nd these re termed "slender". For the prcticl bems which re in the intermedite rnge without lterl restrint, design must be bsed on considertions of inelstic buckling. In the bsence of instbilit, eqn. permits tht the vlue of f cn be dopted for the full plstic moment cpcit p b for λ LT < 0.. This corresponds to λ LT vlues of round 7 (for steels hving f 75 N/mm ) below which the lterl instbilit is NOT of concern. Version II 8 -

7 BAS SUBJCTD TO TORSION & BNDING-II Plstic ield / P 0. / p stock intermedite slender λ LT P Fig. Comprison of test dt (mostl I sections) with theoreticl elstic criticl moments For more slender bems, p b is function of λ LT which is given b, λ LT uv λ r u is clled the buckling prmeter nd, the torsionl inde. () For flnged sections smmetricl bout the minor is, ( ) p u Z γ nd 0.5 h A s A h J s For flnged sections smmetricl bout the mjor is u I Z p γ A Γ nd. I AΓ J In the bove Z p plstic modulus bout the mjor is I γ I A cross sectionl re of the member Version II 8-7

8 We cn ssume BAS SUBJCTD TO TORSION & BNDING-II Γ torsionl wrping constnt h s t t b b ( t b t b ) J the torsion constnt h s the distnce between the sher centres of the flnges t, t flnge thicknesses b, b flnge widths u 0.9 for rolled UBs, UCs, RSJs nd chnnels.0 for ll other sections. λ v function of is given in Tble of BS5950: Prt I r, (for preliminr ssessment v ) D/T providing the bove vlues of u re used. 5. Unequl flnged sections For unequl flnged sections, eqn. is used for finding the buckling moment of resistnce. The vlue of λ LT is determined b eqn. using the pproprite section properties. In tht eqution u m be tken s.0 nd v includes n llownce for the degree of monosmmetr through the prmeter N I c / (I c I t ). Tble of BS5950: Prt must now be entered with (λ /r )/ nd N. 5. vlution of differentil equtions For member subjected to concentrted torque with torsion fied nd wrping free condition t the ends ( torque pplied t vring vlues of αl), the vlues of φ nd its differentils re given b T q α λ (- For 0 z α λ, φ T q GJ. ( α ) z αλ sinh λ tnh α λ cosh z sinh Version II 8-8

9 BAS SUBJCTD TO TORSION & BNDING-II φ T q GJ ( α ) αλ sinh λ tnh α λ cosh z cosh φ T q G J αλ sinh λ tnh αλ cosh z sinh T q φ G J αλ sinh λ tnh αλ cosh z cosh Similr equtions re vilble for different loding cses nd for different vlues of αλ. Reders m wish to refer Ref. () for more detils. We re unble to reproduce these on ccount of copright restrictions..0 SUARY This chpter is imed t eplining simple method of evluting torsionl effects nd to verif the dequc of chosen cross section when subjected to torsionl moments. The method recommended is consistent with BS 5950: Prt. 7.0 RFRNCS () British Stndrds Institution, BS 5950: Prt : 985. Structurl use of steelwork in Building prt : Code of Prctice for design in simple nd continuous construction: hot rolled sections. BSI, 985. () Nethercot, D. A., Slter, P. R., nd lik, A. S. Design of embers Subject to Combined Bending nd Torsion, The Steel construction Institute, 989. () Steelwork design guide to BS 5950: Prt 985, Volume Section properties nd member cpcities. The Steel Construction Institute, 985. () Introduction to Steelwork Design to BS 5950: Prt, The Steel Construction Institute, 988. Version II 8-9

10 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel CALCULATION SHT Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 Checked b RN Dte Jn. 000 mple The bem shown below is unrestrined long its length. An eccentric lod is pplied to the bottom flnge t the centre of the spn in such w tht it does not provide n lterl restrint to the member. The end conditions re ssumed to be simpl supported for bending nd fied ginst torsion but free for wrping. For the fctored lods shown, check the dequc of the tril section. A B W 00 kn 000mm λ 000 mm e 75 mm Stiffener to prevent flnge nd web buckling W 00 kn Replce the ctul loding b n equivlent rrngement, comprising verticl lod pplied through the sher centre nd torsionl moment s shown below. φ T q W.e e W W negtive ngle of twist due to T q Version II 8-0

11 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 Lodings due to plne bending nd torsion re shown below. W λ (i) Plne Loding (Note: These re fctored lods nd re not to be multiplied b γ f ) Point lod, W 00 kn Distributed lod (self weight), w kn/m (s) ccentricit, e 75 mm Bending effects ( t U.L.S) z α λ T q (ii) Torsionl oment t B, Sher t A, Sher t B, B 0 knm F va 5 kn F vb 50 kn Torsionl effects ( t U.L.S) Torsionl moment, T q W.e T q knm This cts in negtive sense, T q -7.5 knm Generll wide flnge sections re preferble to del with significnt torsion. In this emple, however, n ISWB section will be tried. Tr ISWB kg/m Section properties from steel tbles. Depth of section Width of section D 500 mm B 50 mm Version II 8 -

12 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel CALCULATION SHT Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 Checked b RN Dte Jn. 000 B 50 mm Web thickness t 9.9 mm Flnge thickness T.7 mm oment of inerti I 59 cm oment of inerti I 988 cm Rdius of grtion r 9. mm lstic modulus Z 09 cm lstic modulus Z 9 cm Cross sectionl re A. cm 7mm 9.9 mm D 500 mm Additionl properties Torsionl constnt, J [ BT ( D T ) t ] [ ( ) ] mm I h Wrping constnt, Γ ( 500.7).7 0 mm Sher modulus, G 0 5 ( 0.) 7.9 ( υ ) kn / mm Torsionl bending constnt, Γ G J mm Version II 8 -

13 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 Normlized wrping function, W nwfs h B ( 500.7) 50 Wrping stticl moment, S wms 0 mm h B T mm Stticl moment for flnge, Q f A f. f ( ) mm Stticl moment for web, Q w (A/) w w mm Q w mm Version II 8 -

14 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 5 of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn teril Properties.5 Sher modulus, G 7.9 kn/mm Design strength, p 50 / γ m 50 /.5 7 N/mm Check for Combined bending nd torsion 507kNm (i) Buckling check ( t Ultimte Limit Stte) b m ( σ σ ) bt f γ m w.0 m 0.5 B.0 B b 0 knm ffective length λ.0 L λ 000 mm The buckling resistnce moment, where elstic criticl moment p plstic moment cpcit f.z p / γ m b φ B φ B p ( ) φ B p ( η ) LT p BS 5950: Prt I App.B. Version II 8 -

15 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 lstic criticl moment, λ LT p π λlt p the equivlent slenderness nuv λ BS 5950: Prt I App.B.. λ the minor is slenderness λ / r 000 / n 0.8, u 0.9 v slenderness fctor (ccording to N nd λ/) Icf N 0.5 ( for equl flnged sections) I I cf tf AΓ. I.J BS 5950: Prt I Tble BS 5950: Prt App.B..5 λ / 80.7 /.. v 0.98 λ LT nuvλ knm π Version II 8-5

16 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 7 of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 φ B p ( η ) LT BS 5950: Prt App.B.. The Perr coefficient, η LT α b ( λ LT - λ LO ) Limiting equivlent slenderness, To clculte φ λ LO t. φ π 0. p 5 π η LT ( ) 0.5 λ / 000 / 59.5 z α λ, α α λ / 0.77 ( 0.5 ) 507 φ B 9 knm p b φ ( ) B φ B p ( 9 507) 8. knm Version II 8 -

17 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 8 of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 φ Tq. ( α ) GJ rds z 59 α λ sinh λ tnh α λ cosh ( 0.5) 0.77 z sinh sinh0.77 tnh.5 cosh0.77 sinh0.77 Ref..0 App. B t knm σ bt t Z 9.89 N / mm σ w. W nwfs.φ To clculte φ φ T q G J α λ sinh λ tnh α λ cosh 59 z sinh sinh0.77 tnh.5 cosh 0.77 sinh0.77 Ref..0 App. B σ w N / mm Version II 8-7

18 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 9 of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 b 0 0 ( σ σ ) 0 bt f m Buckling is O. K γ w 0.5 ( ) ( 50 ).5 b < (i) Locl "cpcit" check σ b σ bt σ w f / γ m σ b / Z 0 0 / N / mm N / mm < 7 N / mm O. K Strictl the sher stresses due to combined bending nd torsion should be checked, lthough these will seldom be criticl. Sher stresses due to bending (t Ultimte Limit stte) At support:- In web, bw FVA. Q I. t w N / mm Version II 8-8

19 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 0 of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 In flnge, bf F VA I. Q. T f N / mm At midspn :- In web, bw. N / mm In flnge, bf.8 N / mm Sher stresses due to torsion ( t Ultimte Limit stte ) Stress due to pure torsion, t G.t.φ Stress due to wrping, w. S t wms. φ To clculte φ nd φ T q φ ( α ) G J α λ sinh λ tnh α λ cosh z cosh Ref..0 App.B φ At α 0.5, α λ α λ sinh T q G J α λ sinh λ tnh α λ 0.85, cosh α λ cosh 0.77., z cosh λ tnh 0.9 Version II 8-9

20 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 At support, z 0 At midspn, z z cosh 000 z cosh cosh(0) cosh(0.77).0. At support φ φ φ ( 0.5) Stresses due to pure torsion. In web, tw G.t.φ tw ( ) -.95 N / mm In flnge, tf G. T. φ tf ( ) - 9. N / mm Version II 8-0

21 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 Stresses due to wrping in flnge,. Swms. φ wf T wf N / mm At midspn φ 0 φ Stresses due to pure torsion, In web, tw G.t.φ 0 In flnge, tf G.T.φ 0 Stresses due to wrping in flnge,. Swms. φ wf T wf N / mm B inspection the mimum combined sher stresses occur t the support. Version II 8 -

22 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn At support vt In web t, ( ) t tw vt w b.95 N / mm 0. N /mm This must be dded to the sher stresses due to plne bending. bw vt ± N / mm ( cting downwrds) In the top flnge t, tf - 9. N / mm wf -. N / mm vt 0 ( 9..) N / mm bf vt N / mm ( cting left to right) Version II 8 -

23 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple. Fleurl member de b RSP Dte Jn. 000 CALCULATION SHT Checked b RN Dte Jn. 000 Sher strength, f v 0. f / γ m /.5 0 N / mm Since < f v 7.9 < 0 N / mm Section is dequte for sher Referring bck to the determintion of the mimum ngle of twist φ, in order to obtin the vlue t working lod it is sufficient to replce the vlue of torque T q with the working lod vlue s φ is linerl dependent on T q. Since T q is due to solel the imposed point lod W, dividing b the pproprite vlue of γ f will give :- Working lod vlue of T q is 7.5. knm 0.0 the corresponding vlue of φ 0.0 rds On the ssumption tht mimum twist of is cceptble t working lod, in this instnce the bem is stisfctor. Version II 8 -

24 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 mple Redesign the member shown in emple, using rectngulr hollow section. Tr kg / m R. H. S Section properties. Depth of section D 00 mm Width of section B 00 mm Web thickness t 8 mm Flnge thickness T 8 mm Are of section A 77. cm oment of inerti I 9798 cm Rdius of grtion r 8. cm lstic modulus Z 5 cm lstic modulus Z 5 cm Plstic modulus Z p 785 cm 8mm 00 8 mm D 00 Additionl properties t h Torsionl constnt J K Ah Are enclosed b the men perimeter of the section, A h (B - t ) (D - T) (neglecting the corner rdii) ( 00-8 )(00-8) The men perimeter, h [(B - t) ( D - T)] 50 mm [( 00-8) ( 00-8)] 98 mm Version II 8 -

25 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 Ah t 50 8 K h mm Torsionl constnt, J mm Torsionl modulus constnt, C t J K t mm teril properties Sher modulus, G 0 ( υ) ( 0.) kn / mm Design strength, p 50 / γ m 50 /.5 7 N / mm Check for combined bending nd torsion (i) Buckling check b ( σ σ ) bt f γ m w 0.5 b Since slenderness rtio (λ / r 000 / 8. 8.) is less thn the limiting vlue given in BS 5950 Prt, tble 8, lterl torsionl 50 f buckling need not be considered.. Version II 8-5

26 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 Hence b c Sher cpcit P v 0. f / γ m. A v D 00 Sher re A v A 77.. cm D B P v 0. (50 /.5) kn Since F VB < 0. P v 50 < c f. Z p / γ m. f / γ m. Z ( for plstic sections) c. (50 /.5) knm m m B knm To clculte φ The 00 kn eccentric lod gives vlue of T q knm BS 5950: Prt..5 BS 5950: Prt..7. tble 00 kn 75 mm 00 kn L T 0 T q / T 0 T q / Version II 8 -

27 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 φ T 0 T0 z G J T q knm At centre of spn, z λ / 000 mm φ rdins t φ. B knm σ t bt Z 0.95 N / mm Wrping stresses ( σ w ) re insignificnt due to the tpe of section emploed. Check becomes b 0 70 σ f bt γ m b < O. K (ii ) Locl cpcit check σ b σ bt σ w f /γ m σ b B / Z Version II 8-7

28 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet 5 of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 σ b < 7 N / mm O. K N / mm Sher stresses due to bending ( t Ultimte Limit stte) imum vlue occurs in the web t the support. bw FVA. Q I. t w 00 Q w A bw A A Q w A A A N / mm cm A cm Sher stresses due to torsion ( t Ultimte limit Stte) T T0 q t.5 N / C C 87 0 mm Version II 8-8

29 BAS SUBJCTD TO TORSION & BNDING-II Structurl Steel Job No. Sheet of Rev. Design of members subjected to bending nd torsion Worked mple.fleurl member de b RSP Dte Jn 000 CALCULATION SHT Checked b RN Dte Jn 000 Totl sher stress ( t Ultimte Limit Stte ) vt bw ( ) t (.5 0) N / mm N / mm w vt 0.5 b 0 70 Sher strength p v 0. f / γ m /.5 0 N /mm Since < p v 9 < 0 N / mm BS 5950: Prt.. the section is dequte for sher. Version II 8-9