** Cov ntsiab lus kws tshaj lij Archival: ** cov ntawv nyeem thiab cov qauv qhia txog keeb kwm theoretical kev nthuav qhia thiab tsis yog ib qhov project, static laculation, load-bearing capacity check, assessment of the current element, or instruction for construction or rehabilitation. Cov qauv tsim, kev xav thiab cov qauv kwv yees yuav tsum raug kuaj xyuas raws li cov txheej txheem tshwj xeeb, cov khoom siv, geometry, ciam teb thiab cov qauv siv tau. Kev suav ntawm torsion, kev ruaj ntseg, qaug zog, pob qij txha thiab kev ntxhov siab vim kev tiv thaiv kev khoov yuav tsum tau ua los ntawm tus kws tshaj lij uas muaj ntawv tso cai, nrog kev tswj hwm ywj pheej thaum xav tau.
Hnub no, Savo Kusić tau tsom mus rau qhov rais ntoo, wood-aluminium Qhov rais, kev cai Qhov rais, qhov rooj thiab thov lus hais. Kab lus no tseem yog keeb kwm kev tshaj lij archive thiab tsis sawv cev rau kev pabcuam tsim qauv.
Saint-Venant txoj kev xav ntawm torsion (dawb torsion)
Sib npaug sib txawv
Sl.1
Cia tus pas nrig ncaj nraim ntawm ib qho ntawm nws qhov kawg x=0thiab cia nws thauj mus rau lwm qhov kawg nrog lub sijhawm torsion M (fig.1). Tom qab ntawd txhua qhov ntawm nws cov seem tig los ntawm ib lub kaum sab xis α(x) nyob ib ncig ntawm ib qho kab uas tig mus rau lub axis ntawm tus pas, uas peb yuav tau txais raws li x-axis. Yog hais tias tag nrho cov seem yog sib npaug zos rau ib leeg, txhua txoj kab ncaj nraim mus rau lub x-axis yuav dhau mus rau hauv ib lub kauj, thiab lub kaum sab xis α yuav proportional rau qhov kev ncua deb ntawm qhov kawg: α = ϑx, los ntawm cov khoom tawg nyob rau hauv lub dav hlau ntawm seem tshwm sim (Fig.2):
v = - r a sin__ψ = - ϑ x z,
w = r a cos__ψ = ϑ x y. (1a)
Qhov ntau ϑ hu ua lub torsion lub kaum sab xis (dimension: length-1, rotation ib unit ntev). Lub torsion lub kaum sab xis yog proportional rau lub sij hawm torsion M:
ϑ = M/GJT. (2)
Daim duab. 2
Lub proportionality factor GJT, lub torsional stiffness, yog sib npaug rau cov khoom ntawm lub shear modulus G thiab qhov JT, uas sawv cev rau cov nyhuv ntawm cov duab thiab qhov loj ntawm seem.
Qhov kev sib hloov ntawm ntu no tsis yog tsuas yog deformation ntawm tus pas nrig. Tsis tas li ntawd, kuj tseem muaj kev khoov ntawm ntu ntu, piv txwv li mus rau qhov hloov pauv u nyob rau hauv cov kev taw qhia ntawm x-axis, uas muaj tib qhov loj rau cov ntsiab lus sib thooj ntawm txhua ntu thiab yog li ntawd tsuas yog ib qho kev ua haujlwm ntawm kev sib koom ua ke hauv lub dav hlau ntawm ntu, thiab yog lwm yam proportional rau lub load M ϑ:
u = u(y,z) = ϑ ∙ φ(y,z), dφ/dx=0. (1b)
Yog hais tias cov kab lus saum toj no tau nkag mus rau hauv qhov sib npaug sib npaug, nws tau txais tias qhov kev ntxhov siab tseem ceeb yog sib npaug rau xoom, tsuas yog ob qho kev ntxhov siab τxy thiab τxz:
Ib.3
Qhov sib npaug sib npaug ntawm qhov teeb meem torsion yog tau raws li qhov sib npaug (3) tso:
Ib.4thiab i4b
thiab daws cov kab zauv sib txawv rau qhov tsis paub τxy thiab τxz los ntawm kev qhia txog kev ntxhov siab F(y,z) uas:
τxy = - dF/dz, τxz = + dF/dy. (5)
Nrog rau qhov no, qhov sib npaug (4a) txaus siab zoo ib yam, thiab los ntawm qhov sib npaug (4b) qhov kev sib npaug sib txawv yuav tsum tau txais
d2F/dy2+ d2F/dz2=2G__ϑ (6)
Lub ciam teb qhov xwm txheej yuav tsum tau rau cov kev daws teeb meem tshwj xeeb ntawm qhov sib npaug no tau txais los ntawm qhov yuav tsum tau ua kom tsis txhob muaj kev ntxhov siab ntawm lub hnab ntawv qws, los ntawm qhov uas, vim yog qhov sib txuas ntawm cov kev ntxhov siab, nws ua raws li qhov kev ntxhov siab nyob rau hauv seem dav hlau yuav tsum tsis txhob muaj qhov cuam tshuam rau nws raws qhov contour. Los ntawm no, fig.2:
dy/dz = _τxy/_τxz = - (dF/dz) / (dF/dy)
yog li ntawd
dF/dy dy + dF/dz dz =0,
i.e. tag nrho qhov sib txawv ntawm txoj haujlwm F yuav tsum yog xoom, uas yog, nws yuav tsum yog F=const. raws lub contour.
Yog tias seem tsuas muaj ib qho contour, nws tuaj yeem muab tso rau F= rau nws0, vim hais tias nrog lub zog muaj nuj nqi, ntawm qhov uas peb tsuas yog xav nyob rau hauv derivatives, lub arbitrary additive tas li tsis ua lub luag hauj lwm. Rau cov seem uas muaj ntau tshaj ib qho contour, qhov teeb meem no nyuaj dua.
Raws li pom los ntawm qhov sib npaug (5) qhov tshwm sim ntawm qhov kev ntxhov siab τxy thiab τxz ntawm qhov kev sib tshuam ntawm qhov sib txawv yog sib npaug rau qhov loj ntawm qhov kev ntxhov siab F, tab sis yog perpendicular rau nws, i.e. muaj kev taw qhia tangent rau kab F=const.
Los ntawm kev daws qhov sib npaug sib txawv (6)) cov voltages τ raws li kev ua haujlwm ntawm y, z thiab ϑ tau txais. Lub luag haujlwm ntawm kev suav yog, tau kawg, kev txiav txim siab ntawm lub kaum sab xis torsion ϑ thiab qhov kev ntxhov siab, tshwj xeeb tshaj yog qhov kev ntxhov siab siab tshaj plaws raws li kev ua haujlwm ntawm lub sijhawm torsional, yog li ntawd, kev txiav txim siab ntawm qhov nruj ntawm ntu JT los ntawm qhov sib npaug (2) and the resistance torque WT = Mmaxτ. Yuav kom tau txais cov no, nws tsuas yog tsim nyog los qhia M raws li kev ua haujlwm ntawm F:
Ob lub ntsiab lus ntawm qhov tseem ceeb no, uas yuav tsum tau coj mus rau tag nrho ntu, tuaj yeem hloov pauv los ntawm kev sib koom ua ke ib nrab. Yog hais tias, piv txwv li, thawj ntawm lawv integrates thawj zaug raws kab z=const. los ntawm qhov taw tes ntawm contour y = y1 to the point y = y2 (fig.3) thiab coj mus rau hauv tus account tias hauv ob lub ntsiab lus no F=0, yog tau
Tib tus nqi yog tau los ntawm qhov thib ob integral, yog li ntawd
Ib.7
Sl.3
Lwm qhov xwm txheej ntawm tag nrho ntu
Qhov sib npaug sib npaug (6) yog inhomogeneous equation ntawm tej zaum txoj kev xav. Yog li ntawd, rau kev daws teeb meem torsion nyob rau hauv cov ntaub ntawv ntawm ib tug arbitrary cross-section, cov txheej txheem ntawm lub peev xwm txoj kev xav muaj, uas pab kom nws mus ib txwm nrhiav tau. Cov txheej txheem no, txawm li cas los xij, feem ntau nyuaj dhau los siv rau tus neeg mob uas tshwm sim hauv kev xyaum. Yog li ntawd, cov lus hauv qab no qhia txog cov txiaj ntsig ntawm cov kev suav no rau ntau ntu.
Rau ntu ntu ze rau ncig, JT=F tuaj yeem muab tso4/40_Ip_, qhov twg F yog thaj chaw thiab Ip yog lub caij nyoog polar ntawm inertia ntawm ntu.
** Ceebtoom ntawm cov qauv thiab cov rooj:** cov lus qhia thiab qhov tseem ceeb hauv cov duab scanned yog khaws cia ua keeb kwm. OCR cov ntawv nyeem, cov cim ntsuas thiab luam theej duab tsis txhim khu kev qha txaus rau kev suav. Ua ntej siv, txhua tus qauv, chav tsev, coefficient thiab thaj tsam ntawm kev siv tau yuav tsum tau kuaj xyuas hauv ib qho kev tso cai tshaj lij thiab cov qauv siv tau.
Analogy nrog daim nyias nyias
Ib qho kev pab zoo rau kev kwv yees torsional stresses, thiab tib lub sij hawm ib tug txheej txheem rau lawv qhov kev txiav txim siab yog muab los ntawm kev sib piv ntawm qhov kev ntxhov siab muaj nuj nqi F ntawm qhov teeb meem torsional thiab deflection transverse load ntawm daim nyias nyias. Cia muaj qhov nruab nrab qhib rau ntawm phab ntsa tiaj tus ntawm lub nkoj uas coincides nrog tus ntoo khaub lig-section ntawm ib tug pas nrig loaded nyob rau hauv torsion. Thaum lub membrane yog stretched tshaj qhov qhib no thiab ib tug me me tshaj siab p yog tshwm sim nyob rau hauv lub nkoj, lub membrane bulges sab nraud, kom nws deflections _ξ(y,z) _ txaus siab qhov sib npaug sib npaug.
qhov twg N yog qhov nro nyob rau hauv daim nyias nyias los ntawm capillary rog (qhov ntev: quab yuam ib chav tsev ntev). Qhov kev sib npaug no coincides nrog qhov sib npaug (6), thiab ciam teb yam mob ξ=0yog tib yam li qhov kev ua haujlwm voltage. Nyob rau hauv txoj kev no, lub zog muaj nuj nqi _F (y, z) _ tuaj yeem tau txais los ntawm daim iav no nrog daim nyias nyias thiab yog li daws qhov teeb meem torsion. Qhov kev ntxhov siab τ sib raug rau txoj kab nqes ntawm daim nyias nyias nyob rau hauv kev sib raug zoo rau y-z-plane, yog li qhov chaw steepest yog qhov chaw uas tus pas nrig raug torsion muaj siab tshaj shear stress.
Sl.4
Sl.5
Cov txheej txheem membrane no tshwj xeeb tshaj yog tsim los kom tau txais cov qauv kwv yees rau nyias-walled seem. Thaum, piv txwv li, ntu I tau muab faib raws li daim duab.4ntawm peb lub duab plaub thiab rau txhua tus ntawm lawv cov kev sib piv nrog cov membrane yog nqa tawm cais, cov txiaj ntsig ntawm qhov ntim ntawm qhov tshwm sim protrusions yuav tsuas yog me ntsis me dua qhov ntim uas ua tau sib raug rau qhov kev ua haujlwm voltage. Yog li ntawd, thaum lub thicknesses yog b1 thiab b2 me, kwv yees
JT =2JT1+JT2,
qhov JT1 thiab JT2 cov yam ntxwv nruj ntawm cov duab plaub tus kheej uas tuaj yeem xam raws li lub rooj1. Ib txoj hauv kev zoo dua yog tau txais thaum JT muab tso rau ntawm tav2 = _ ua2b32/3_, yog li ntawd, thaum tus tav raug suav hais tias yog ib feem ntawm cov duab plaub elongated heev. Tus nqi ntau dua yog tau raws li Trayer thiab March\ los ntawm
JT =2JT1+1/3ib2b32+2__A D4,
qhov twg D yog txoj kab uas hla ntawm lub voj voog loj tshaj plaws sau raws li qhia hauv daim duab.4, thiab α tuaj yeem coj los ntawm daim duab.5. Rau T-section nws tau txais zoo sib xws
_JT = JT1+JT2+_A D4,
qhov JT2 ib nrab ntawm tus nqi xam raws li lub rooj1rau a=2_ ua2_, b=b2. Rau L-sections (Fig.6) zoo ib yam li ntawd
JT = JT1+JT2+ _ IB D4_
qhov JT1 rau lub caj npab tuab yog coj ncaj qha los ntawm lub rooj1, thaum JT2 rau cov ceg thinner yog xam raws li T-section, thiab β yog muab los ntawm daim duab.7.
Daim duab. 6
Sl.7
Thaum xam qhov kev ntxhov siab, lub sij hawm torsional muab faib ua cov duab plaub: qhov uas ntog rau ntawm flanges M1=M JT2/JT; lub shear kev nyuaj siab yog ces xam nyob rau hauv nruab nrab point ntawm sab ntev ntawm txhua daim duab plaub raws li lub rooj1. Qhov loj tshaj ntawm cov kev ntxhov siab no yuav tsum tsis yog qhov loj tshaj plaws shear stresses tshwm nyob rau hauv seem. Namely, yog tias lub kaum sab xis ntawm qhov xwm txheej tsis sib npaug, τ=∞ tau txais ntawm qhov chaw ntawd, yog li ntawd rau lub vojvoog me me ntawm kev sib tw, qhov siab tshaj plaws shear stress tuaj yeem xav tau ntawm no.
Engineering Note: tus lej singularity ntawm lub kaum sab xis zoo tagnrho tsis yog daim ntawv tso cai los kwv yees qhov tseeb nthuav dav los ntawm qhov kwv yees tus qauv no ib leeg. Qhov tseeb rounding geometry, tolerances, plasticity, welds, residual stresses, lub zos stability thiab qaug zog yuav tsum tsim nyog qauv thiab cov kws tshaj lij pov thawj.
Rau ib lub kaum sab xis uas nws sab tuab yog t raws li Trefftz\ yog nyob rau ntawm lub voj voog ntawm txoj kab uas hla r:
qhov twg nrog τ0 cim qhov siab tshaj qhov hluav taws xob suav rau txhua tus ceg raws li cov txheej txheem tau piav qhia ntawm no. Rau qhov loj rounding radii, τ= tuaj yeem txais yuav2__t0.
Thin Walled Pipes (Bradt Patterns)
Nyob rau hauv nyias-walled kav, lub bulging membrane muaj cov duab qhia nyob rau hauv daim duab.8, uas muaj ib lub dav hlau saum toj no hollow qhov chaw nyob rau hauv cov yeeb nkab thiab ib tug sloping nto uas nyob ib ncig ntawm nws thiab lies saum toj no qhov zoo ntawm cov yeeb nkab phab ntsa; qhov chaw inclined no yuav tsum tau steeper yog tias cov yeeb nkab phab ntsa yog thinner nyob rau hauv qhov chaw tsim nyog. Nyob rau hauv cov ntaub ntawv no, lub shear stresses yog muab faib ncaj ncaj nyob rau hauv lub thickness ntawm cov phab ntsa thiab yog inversely proportional rau lub thickness ntawm phab ntsa, piv txwv li shear force T = τt tsis hloov raws lub circumference ntawm seem.
Sl. 8
Cov khoom nto t_∙ds_ tau txais ib feem ntawm lub sijhawm torsional
dM = t_∙ds∙τ∙h,_
yog li lub sijhawm tag nrho yog
Ib.8
qhov twg FR nto yog npog los ntawm kab nruab nrab cim los ntawm kab thiab dots hauv daim duab.8. Nws tau los ntawm no
Ib.9- Thawj Bredt\ qauv
Torsional tawv yog tau txais feem ntau tsuas yog thaum lub deformation ua hauj lwm qhia ntawm ib sab los ntawm lub sij hawm torsional thiab torsion lub kaum sab xis thiab ntawm qhov tod tes los ntawm shear stresses. Nyob rau hauv no txoj kev, nws yog tau rau ib tug pas nrig ntawm ntev l
yog li, qhia tus nqi rau τ raws li qhov sib npaug (8),
uas yog, los ntawm kev sib piv nrog cov kab zauv (2):
Ib.10
Yog hais tias lub thickness ntawm phab ntsa yog tib yam nyob ib ncig ntawm, ces nws yog yooj yim (2. Bredt tus qauv):
Ib.11
Ib txwm muaj kev ntxhov siab nyob rau hauv torsion vim yog ntu khoov
Cov ntsiab lus tseem ceeb thiab kev sib npaug sib txawv
Component displacements u, equation (1b), txiav txim siab curvature ntawm cov seem ntawm lawv cov dav hlau. Thaum lub sijhawm torsion tas li, qhov kev khoov no zoo ib yam hauv txhua ntu thiab qhov kev ntxhov siab ib txwm σx tsis tshwm sim.
Hauv ntau qhov xwm txheej, qhov curvature ntawm ntu ntu, uas yuav tsum tshwm sim los ntawm _St. Venant txoj kev xav tsis tau. Yog hais tias lub sij hawm torsional hloov tam sim ntawd nyob rau hauv ib tug ntoo khaub lig-section, dabtsi yog khoov ntawm txawv magnitudes yuav tshwm sim ntawm ob sab ntawm tus ntoo khaub lig-section, thiab qhov txuas ntxiv ntawm deformation yuav cuam tshuam. Ib qho kawg ntawm tus pas nrig tuaj yeem raug clamped uas khoov ntawm ntu yog cais tag nrho ntawm qhov ntawd. Nyob rau hauv tas li ntawd, yuav tsum tau hloov kho _St. Venant txoj kev xav. Supplementary stresses σx nyob rau hauv cov kev taw qhia ntawm lub axis ntawm tus pas nrig thiab ntxiv shear force T (shearing stresses T / t) tshwm nyob rau hauv tus pas nrig, uas yog lub siab tshaj plaws nyob rau hauv seem uas lub continuity ntawm deformation yog tawg thiab uas txo exponentially nrog deb ntawm seem no. Nyob rau hauv cov ntaub ntawv ntawm cov khoom rods thiab hlab, qhov kev xav no yog complex, thiab cov ntxiv voltages txo sai heev uas lawv txiav txim siab tsis tsim nyog nyob rau hauv feem ntau. Rau nyias-walled open-profile rods, qhov kev kho no yog qhov tseem ceeb heev thiab tuaj yeem suav tau yooj yim.
Sl.9
Sl.9qhia cov kab uas nyob nruab nrab ntawm phab ntsa thiab ntu dav hlau sib tshuam (txoj kab nruab nrab ntawm ntu). O yog qhov chaw nruab nrab ntawm kev sib hloov ntawm ob seem uas nyob ib sab. Yog hais tias lawv qhov deb yog dx, lub torsion lub kaum sab xis yog ϑ dx, yog li qhov chaw nruab nrab ntawm qhov nruab nrab nkag mus rau qhov hloov pauv u ϑ dx. Qhov kev hloov pauv no muaj cov khoom tivthaiv rt ϑ dx nyob rau hauv cov kev taw qhia tangent mus rau ntu, thiab yog lub plam ntawm lub caij ntawm nruab nrab ntawm phab ntsa (siv rau hauv tus account tias _u = _φϑ):
γ=__ϑ (rt + d__φ/ds).
Txij li thaum lub shear stress nyob rau hauv nruab nrab ntawm phab ntsa yog xoom, nws kuj yuav tsum yog γ\ =0, yog li ntawd yog vim li cas
Cov qauv no muaj peb qhov tsis txaus ntseeg: lub hauv paus chiv keeb los ntawm qhov ntev arc s suav thiab cov kev tswj hwm ntawm qhov chaw ntawm kev txheeb ze kev sib hloov O, ntawm qhov twg rt nyob ntawm. Qhov pib taw tes los ntawm qhov uas qhov ntev arc s suav tau tuaj yeem xaiv kom tus nqi curvature nruab nrab yog xoom:
thiab cov kev tswj hwm ntawm qhov chaw O tuaj yeem txiav txim siab hauv txoj hauv kev uas lub sijhawm vim qhov khoov ntawm ntu yog xoom:
Nws tuaj yeem pom tau tias center ntawm kev sib hloov yog li txiav txim siab coincides nrog lub chaw shear uas yog ntawm no piav thiab txhais ua lwm yam.
Yog hais tias txoj kev khoov ntawm ntu uas tau piav qhia los ntawm cov qauv no yog tag nrho lossis ib feem tiv thaiv, kev ntxhov siab ib txwm σx nyob rau hauv kev coj ntawm lub axis ntawm tus pas nrig yuav tsum tshwm sim. Kev soj ntsuam ze dua qhia tau hais tias cov kev ntxhov siab no tuaj yeem coj los ua piv txwv rau qhov curvature muaj nuj nqi φ: σx=k__φ, qhov twg φ nyob ntawm s thiab k nkaus xwb ntawm x. Raws li Hooke txoj cai:
σx = E_∂u/∂x = Eφ dϑ/dx_, (12)
yuav tsum yog k=E d__ϑ/dx. Cov kev ntxhov siab li qub σx yog hu ua ntu bending stresses. Xav txog cov kev sib npaug saum toj no, lawv tsim ib qho kev sib npaug ntawm cov rog hauv txhua ntu, ib qho system ntawm lub zog vim qhov khoov ntawm ntu.
Tag nrho torque muaj ob ntu: ib qho ntawm lawv M1=GJT__ϑ tshwm sim vim torsional shear stresses muab los ntawm St. Venant's theory, and the other M2 tshwm sim vim khoov ntawm ntu:
Ib.13
Txij li thaum lub sijhawm M tau muab ua haujlwm ntawm x, qhov kev qhia no sawv cev rau qhov sib npaug sib txawv rau lub kaum sab xis torsion ϑ. Thaum qhov sib npaug no raug daws, M tuaj yeem suav tau1 thiab M2, thiab yog li txhua qhov kev ntxhov siab thiab hom kab mob.
Kev daws teeb meem thiab kev siv
Thaum lub sij hawm torsion M tas li, qhov kev daws teeb meem dav dav ntawm kab zauv (13) yog:
Integration constants C1, C2 yuav tsum tau txiav txim los ntawm ob tug ciam teb.
a) Rigidly clamped kawg x=0(fig.1)
Thaum kawg no yog u≡0, raws li qhov sib npaug (1b) ϑ\ =0. Yog tus pas nrig ntev heev (αl>>1), ces C1= ib0thiab C2= -M/GJT. Nyob rau hauv cov ntaub ntawv ntawm cov pas nrig luv luv, cov tsos ntawm kev ntxhov siab vim yog khoov ntawm ntu yog cuam tshuam los ntawm tus mob ntawm txoj cai kawg ntawm tus pas nrig. Yog tias qhov khoov ntawm ntu tsis tiv thaiv ntawm qhov chaw ntawd, yuav muaj σx≡0thiab dϑ/dx=0. Nws tau los ntawm no
b) Symmetrical torsion load raws li daim duab.10
Ntawm sab laug ib nrab ntawm tus pas nrig, lub sijhawm torsional yog -M/2, thiab ntawm sab xis +M/2. Nyob rau hauv cov ntaub ntawv ntawm dawb torsion, seem rau sab laug thiab sab xis ntawm nruab nrab ntawm tus pas nrig yuav twisted nyob rau hauv lub opposite kev taw qhia, yog li ntawd lub continuation ntawm lub deformation yuav tsis tau khaws cia. Yog li ntawd, ntawm qhov chaw x=0tag nrho lub sij hawm torsional yuav tsum tau pauv los ntawm cov kev ntxhov siab uas tshwm sim vim kev khoov ntawm ntu ntu, thiab yog li hauv ntu ntu no M1= ib0thiab yog li ntawd ϑ=0. Lub sijhawm no, lub zog ntawm lub zog ua haujlwm los xyuas kom meej tias seem nyob ncaj, txawm tias cov seem uas nyob deb ntawm qhov nruab nrab ntawm tus pas nrig nkhaus. Yog tias, ntxiv rau, nws yuav tsum tau hais tias qhov kawg ntawm tus pas nrig muaj kev ntxhov siab σx≡0, ib qho kev ciam teb ntxiv yog tau txais rau txhua ib nrab ntawm tus pas nrig, yog li nws yog rau ib nrab ntawm txoj cai
Daim duab. 10
c) Sab nraud lub sijhawm M ntawm qhov chaw arbitrary (daim duab.11)
Hauv qhov no, qhov kev daws teeb meem ϑ rau ob qho tib si ntawm tus pas nrig yuav tsum tau sau cais, yog li tag nrho ntawm plaub qhov C tshwm sim.1, C2, C3, C4. Tsis tas li ntawd, lub torsion moments Ma, Mb kuj tsis paub. Rau qhov sib npaug yog muaj: ib qho kev txuas ntxiv rau ϑ (qhia los ntawm u) thiab rau dϑ/dx (qhia los ntawm σx) ntawm qhov chaw thau khoom, ib qho xwm txheej ntawm txhua qhov kawg ntawm tus pas nrig (piv txwv li nruj clamping ϑ=0, los yog undisturbed dabtsi yog khoov, dϑ/dx=0), qhov xwm txheej ntawm qhov sib npaug ntawm lub sijhawm -Ma+Mb=M thiab thaum kawg qhov xwm txheej uas cov ntu kawg tsis tig nrog kev hwm rau ib leeg: ∫ ϑdx =0.
Sl.11
** Daim ntawv kawg: ** qhov kev xaiv ntawm tus qauv torsion nyob ntawm qhov qhib lossis kaw ntu, slenderness, txoj kev txhawb nqa, qhov chaw ntawm lub sijhawm tawm tswv yim thiab qhov muaj peev xwm ntawm kev khoov dawb. Rau ib lub ntsiab lus tiag tiag, lub load-bearing thiab serviceability txwv lub xeev, cov ntsiab lus hauv zos, cov pob qij txha, qaug zog thiab ruaj khov yuav tsum tau kuaj xyuas, tsis yog ib qho kev cais tawm ntawm tsab xov xwm.