4. Eccentric axial loading, cross-section core

Transcription

1 . Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we cn replce the orgnl lodng b the sttcll equvlent eccentrc longtudnl lodng, Fg.., nd vce vers. M M Fg.. Eccentrc xl lodng Due to superposton prncple, we cn dd the solutons of the smple cses, tenson nd bxl bendng, nd obtn the norml stress equton for the eccentrc xl lodng: M M x. J J The relton obtned shows tht the dstrbuton of stresses cross the secton s lner. s before, we need to know the neutrl xs poston. The most dstnt fbers from the neutrl xs re the most exerted secton prt, where the norml stress s mxml. The neutrl xs equton cn be wrtten n the form: J M x 0. M J The neutrl xs s the strght lne, but ths tme t doesn t pss through the secton centrod. From the ultmte lmt stte we get the desgn condton: mx x R, where the mxmum vlue of the norml stress s ttned t the most dstnt ponts from the neutrl xs. Tp: Mssve members n eccentrc compresson cn be clculted s n tenson wth the chnge of the stress sgn. Exmples Exmple. Determne the mxmum vlue of the norml stress t the cross-secton tensoned b the force = 50 k, Fg.. (the dmensons n cm):

2 9 6 Fg.. Eccentrcll loded secton Soluton. The cross-secton forces: = 50 k, M = = -.5 km, M = = - km. The geometrc chrcterstcs of the cross-secton: = 96 cm, J = 5 cm, J = 5 cm. The norml stress dstrbuton: M M x MP J J The equton of the neutrl xs: The mxmum stress wll occur t the rght lower corner: mx x 5.7 9( 0.06) MP Exmple. Determne the cross-secton prmeter, Fg.., knowng tht the tenson force s = 50 k nd the bendng moment s M = 75 km. ssume R = 50 MP. M Fg.. Box secton loded Soluton:. The cross-secton chrcterstcs: = (56-9) = 7, c =.9, J = 89. The norml stress dstrbuton: M J x The equton of the neutrl xs:.. The dstrbuton of norml stress shows tht the gretest norml stress occurs t bottom fbers (the most dstnt). We wrte the desgn condton n the form: mx x 90 0 x (.9) R

3 The bove neqult cn be solved numercll or s long s we don t need exct soluton b the trl nd error method. s the frst pproxmton, we cn use seprte solutons for the tenson nd the bendng, obtnng: = m nd = 0.06 m, respectvel. 5. ssumng = m, we get the mxmum norml stress: 7 MP < R. Exmple. Determne the mxmum vlue of the norml stress n the secton loded b the bendng moment M = 65 m nd tensoned b xl force = 0. k, Fg.. (the secton dmensons n cm): M Fg.. Cross-secton wth lodng Soluton. The cross-secton chrcterstcs: = 9 cm, due to the oblque smmetr xs we strt strght w from the prncpl centrl xes (s dfference of two squres): the centrod poston: c =.6 cm, 5 the nert moment bout the smmetr xs = cm = J the second nert moment: cm = J. The cross-secton forces: = 00, M = - M = 5.96 m. The norml stress dstrbuton:. The neutrl xs equton: M J M J x The bove equton shows, tht the most dstnt pont from the neutrl xs s p. ; ts coordntes n the prncpl centrl coordnte set re:.6.8,. 8 (n cm) 6. The extreme norml stress t the pont s: x = 7. MP. Cross-secton core In the cse of eccentrc xl lodng, the norml stress dstrbuton cn be wrtten b mens of the eccentrctes, Fg..5:

4 (, ) x J J The neutrl xs equton becomes: x 0 0. Puttng: Fg..5 Eccentrc xl lodng v b, c, we get the neutrl xs equton n the two ntercept form:, b c where b nd c re the -ntercept nd -ntercept, respectvel, Fg..6.. c b Fg..6 Two ntercept form There re three possbltes of the neutrl xs locton wth respect to the cross-secton: the neutrl xs psses through the cross-secton, the neutrl xs s tngent to the cross-secton, the neutrl xs doesn t pss through the cross-secton. In the frst cse, the neutrl xs dvdes the cross-secton nto two regons wth dfferent sgn of the norml stress: the compressed one nd the tensoned one. The cse of the neutrl xs touchng the crosssecton s the most nterestng s the lmt cse between both prevous cses. The cse of one sgn of the norml stress wthn the cross-secton s of prtculr nterest to the cvl engneers: the mterl ner the neutrl xs doesn t work; prt of the cross-secton s not used, mn mterls hve dfferent compresson strength towrds tenson strength; the cermc mterls work n compresson onl. cross secton core s locus where ppled xl force cuses the norml stress of one sgn wthn the whole secton. In ths cse the neutrl xs doesn t ntersect the cross-secton.

5 The cross-secton core s lmted b the convex core curve. The curve cn be determned on the bss of the neutrl xs equton: 0. The bove equton hs double nterpretton: when nd re unknown, t s equton of the neutrl xs for gven poston of the force (, ), when nd re unknown, t s equton of the force poston for gven poston of the neutrl xs. Moreover, for both nterprettons, for the gven pont we get the lne equton nd vce vers, for the gven lne equton we get the pont. We determne the core curve b cotng the cross-secton wth the possble postons of the neutrl xs, fndng for ech poston the correspondng force poston. The equton of the neutrl xs pssng through two gven ponts hs the prmetrc form, Fg..7: t> c t= t=0 t<0 b Fg..7 Prmetrc form of equton t, t hence, for 0, b nd 0, c, we get respectvel: b, c. The dstnce of the pont P( P, P ) from the neutrl xs s: bp c P bc d. b c From the neutrl xs equton follows the poston of the force nd correspondng neutrl xs, se Tb... force poston neutrl xs t centrod t nfnt wthn the core outsde the cross-secton on the core curve tngent to the cross-secton wthn the cross-secton, outsde the core crosses the secton outsde the core on the lne tngent to the cross-secton tngent to the core outsde the cross-secton crosses the core t nfnt crosses the centrod Tb.. Force nd neutrl xs reltve postons The lgorthm s s follows:. We cot the cross-secton b the such lnes tht the cross-secton s from one sde (mgne tht we use ver long plnk nd there re protrudng nls t ech convex corner); we get the smllest convex fgure wth the cross-secton nsde,. For ech lne we get the force poston, usng the pproprte formule.

6 Exmples Exmple. Determne the core for the rectngulr nd box cross-sectons, Fg..8. Soluton For the rectngulr cross-secton h, nd the neutrl xs postons re: h c h h Fg..8 Rectngulr nd box cross-sectons h we hve the nert rd squred: h 6 b 6 (we cn use the smmetr propert now). For the box cross-secton wth the wll wdth of h/0 nd /0, the nert rd squred re: 0.6h, 0.6 nd the coordntes of core curve re: 0.8,0. 8h nd the core s pproxmtel two tmes greter. Exmple.5 Construct the core for the cross-secton n Fg Soluton. the centrod poston. the prncpl centrl nert moments Fg..9 Cross-secton nd the core , 0 0 8

7 . the nert rd squred J (.58.5) cm J J. the core curve: -: b, c. 0 0, : b, c , 0 0 -: b, c , 0. 0 (we use the smmetr propert) Exmple.6 Determne the core for the secton, Fg..0..6, J 0.7cm.56 5 Soluton. the centrod poston. the prncpl centrl nert moments Fg..0 T-secton nd ts core , J 5(.5) 5(5.5 ) cm 5 5 J cm. the nert rd squred J J.,.08. the core curve: -: b, c 0 0, : b., c , : b, c 0 0., 0 0 -: b c 0,. 67, 0 0

8 Exmple.7 Determne the core for the ngle profle, Fg,.. 5 E D C B Fg.. ngle profle nd ts core Soluton We use the results from the prevous secton: J =.98, J = 6.6, =.5 The corners coordntes n the prncpl centrl coordnte sstem s before. The results re presented n Tb. 7.. Lne b c B BC CD DE E Revew problems Problem. Determne the prmeter, Fg.., ssumng P = M nd R = 0 MP. (ns.: =.0 cm) cm 5cm 5 5 Fg.. Secton loded t constnt eccentrctes Problem. Determne the prmeter, Fg.., ssumng P = M nd R = 0 MP. (ns.: =. cm) 5 5

9 Fg.. Secton loded t vrble eccentrctes Problem. Determne the norml stress dstrbuton cross the secton n Fg.., = cm, loded b force = 50 k. The force s ppled to rgd plte, welded t the member end secton. Drw the dgrm of the norml stress. (ns.: mx σ x =59. MP t lower rght corner) Fg.. Secton wth lodng pont Problem. The spot footng, Fg..5, s loded b moment M =. Mm nd xl force =.5 M. Determne the d dmenson so tht the norml stress dstrbuton t the bottom of footng would be unform. (ns.: d =.96 m) M m d m m Fg.5 Spot footng Problem.5 Determne the core for the cross-sectons gven n Fg..6. ssume the geometr dt. (Check our clcultons wth the prekroj.exe progrm)

10 Fg..6 Core - revew problems ddendum Glossr eccentrct mmośród eccentrc mmośrodow constnt/vrble eccentrctes stłe/menne mmośrod spot footng stop fundmentow core rdeń two ntercept form równne odcnkowe core curve krw rdenow