N. M. Belyaev- Strength of Materials- Mir

N. M. Belyoev I

1

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MIR PUBLlSHERS MOSCOW

H. AA 6Ullle.

C0fV>01l1BnEliHE MATEPIo1AnOB

N. M. BELYAEV

Strength of Materials

frallslafed from [he Russial1 byN. K. Me/l/a

MIR PUBLlSHERS MOSCOW

Frrst publlsM containing dala on materials have beeo dropped from appendkes. A part or the dala on materials hu lransierred lo

• corresponding secHons. The ob3otete sted pruliles grading has bern replaccd by ntw ones. A" in \he previous edlttons lt ""1$ Out endeavour lo pfl:5efVe Belyaev's stylt Ind melhod ol presentalkm oi maleríll/. Therl'rof1! Ihl' author's 161 ha! In general been preserved. 11 Nlkolll Mikhailovich allve loday he would pclS:5ibly write man)' thmgs In a dill"erent way. However, slnce the book WOIl wide popularlly as II,'ritlen by N. M. Bel)'aev, we tried 10 preserve tM original lal as {¡Ir as pos.sible. The work involved In preparing lhe edltion loe publica· tlon '#ias dlstributed amoog lhe group as follows: Olapler 13, § 80 of Olapter 14, Q¡apters 15·19, 24·25-L. A. Belyavskii; Chaplers 6,

8·12, 27-28-Ya./. Kipnls; Chapters \-5, 26and appendices-N. Yu. Kushelev; Chapter 1, § 79 of Chapter 14, 21)·23. 29·32A. K. SinHskiL A. K. Sinitski¡

Contents

"!toJaI Mikhailovldl 5 lo !be FiIlftlllb RuUlln EdItlon 7

PART 1. Inlroduction. rendon .nd Compression Ou.plff l. lnlrodoo11

'6.1.1

I

1OIdlics of resiStallCe 01 malerial, lo faHure ¡tnd plaslic dclormatiom. To e:llSurt the smooth funelioning or lhe structure wilhoul a risk ollailur('. \1,'(musl see lo it lhal Ihe element is unly subjedtd lo stresses whicb 11":$$ Ihan its ullimatl' strenglh.

"

lhe s/rlSS is square bracbls: il is klwing expression:

bv Ihe lo Ihe uit imale

letler bul is pul in P. b)' Ule fol-

[p]-1' whffe k thC:S(lfl'ly {De/o' whkh 5hOl\'5 ho\\' many times lhe stress [5 [ess Ihan Ihl" ultima!e (tenslle) strenglh. 1h(' value 0' Ibis factor varíes lrom 1.1·1.8 lo 8·10 and depends upon IheoperaHng con· dilions al the slrurlure. It will be di5cl.'ssed in grealer detall In §§ 16 and 17. Cenoling b)' P... Ihe muimum stress Ihal in lhe desl¡ntd

eJemenl uoder lhe acllao d external forces. we may lhe bnsic oondilion, \\'hlch lhe ami material oi lhe elenK'nl musl satisry• .J5 (0110\\"'5:

(1.1 )

This is lhe sfreflgtl¡ (XmdilioJl, whlch slales Iha! the' adual slrE'SS mllS! be not grtaler lhan Ihe permi!>Sible. Nov.' we ma)' compile lhe plan rOl" solving lhe problelllS al slrcngth 01 materlals as folkN.'S. (1) Ascerbin lhe magnilude and nalure of all Ihe external forces, includu¡g lhe reattions, IICting on Ihe element IInder cornideution. (2) Selecl an appropriale material Ihat is mosl suilable in lhe "..orking conditions 01 lhe eleml!nt (struclure) and lhe natureof loading; dell'fmine lhe permissible stress. (3) Set the cross-seclional arra oi lhe elenlCnl in numerical ur algebraie iorm, and calculale 1he maximum aclual stress p",,, which maximum load \\'lIieh IhE' spe'Cinllm can sus!ain befare ruplure. Tlle normal slrm due lo this load is

0'.=" p.

and is cal1et11he ultimalr (/ensile) ef lhe material under lension. It ls USUllJly ell.pressed in l\lt units kgl mm' or kgi cm'. As poinled oul earlier in § 4, lhe maximum permissible normal stress 10'1 is Sl'veral times 1m Ihan lile ultimate slrength (l.; lhe pero missible slress is oblained by dividing Ihe ullimale slrength by lhe sarely factor k. The value of k dt'pends upon a lIumb« 01 faclors, whic:h $hall dist::ussed in laltr on (§ 16). Al any rale, lhe value of the saftl}' factor must tnsUfl.' nol on1y Ihe normal "''Ol''king of lhe eltmenl, i.t. wOfkmg witlW)lJ1 failufl.'. bul also prevenl Ihe formaHon of plaslic deformations whkh nlaY affte! lhe ....-orking 01 lhe Illac:hilW or slruc:ture. Tbe salely htctor depends upon lile mattrial al' lhe eleml'l'll. natllrt- 01 lhe lorces ac:ling on the e1emml, economic. l:OOdiHoos ami a number 01 oiher laclors_ [n \'ie"" of lhe importance of properly st"ltdin¡ Ihe salely faclor and Ihe pennissiblc str<

"

¡PO'I I

othn- hand. upon i1s c:ross·seclional area A. Somelimes il is more C'Onvenitrlllo thl' lerro ulatifJt rigidity E/l, Le. Ihe ratio of rigidity lo lhe length al lhe bu. Formulas (2.5) lmd t2.6) enable us lo lhe elongaUOll or in liJe bar o( a structure uoder Imsion or rompression. koowing lhe elongalion, dirncR$ions and lhe material 01 the bar onc ma)' cakulale lhe normal stres.ses acling in ¡l. Thus. normal stress can be deierminrd by 1"'0 methods. If the !eosHl' or compressive lortt P is knoWII, o IS calculated from ¡he formula {2.l):

,

o"""A H lhl' externa! force is nol lmown bul Ihe elongation 01 the bar can be n1l'usurctl, a is de-terminl."d frum formula (2.7):

o=tE The relalive t'longaliOll ma)' be eakulated according lo lhe follo\\'ing formula ¡¡ lhe {olal elona:!Iion M lar a lenglh 1 aflhe bar can be mea· surl'd:

A' t=T

We shall show lalt'r lhat lhe s«ond mdho5 rinl again slarts resisling lurther !cns¡le slrain and !o clongate it by a kngth tJ.l lhe force should b(' ¡n(reased. Poiol D of the diagram (orresponds to lhe maxiUlllln load.

P JI 21

Xl

Al

..!.. ----

.........

-T-/7l----

7t""'

,f-----cof,

K.

i1

iI

I I I

I I I

F,!( 12

• fig l;¡

Al this instan! lhere is again a sharp change in ¡he behaviour o[ malerial. Up lo Ihis poinl, Ihe whole bar was being drforl1Jl'd; ('nth unil length of lhe specimen elongated nlmast f'qually. Similarly lllere was a uniform decrease in lhe cross·sectional area ollhe speci melL. From lhe inslant Ihe load achieves Ihe value 00" Ihe, dC'formalion gels eoneenlraled in a certain parl of thE' speciml.'n. A small porlion oi lhe specimen around Ihisspol ¡s from no\\."on subjecled to Ihe m;lximum stress. This resulls in a loealized redudion 01 Ihe ere$ we get a slightly curved line instead or the strnitiht Jine on the diagram, Le. a slríclly linear preparo tionalíty bel\\'l'C'n the lorce or slress Fig. 22 and lhe eorresponding delormation is absenl. Therelore, lhe modulus 01 elasticit)· which is equal lo lhe slope or lhc ,liagram (see § 11) cannot be eonsidercd a lily lor britllc il changcs dependillg lIpOtl (he strl.'!>S lar

,

r.

54

IPwt f

whích lhe deiormalion is to be calculated. As lhe stress iocreases, lhe

modulus

or e-Iastkíly increases

Q(

decreases clepending upon Ihe direc-

lion in which lhe curve 15 bulging-upwards or dO'l\Tlwards. HO'o\1!:vtf, Ihe devialion from Hooke's law i.s insigni6canl rOl lhe stres.s tange in which lhe malerial! groerally function in slruclurn. Thereiore. in procHce. lhe curved that a companliively srnall ¡ncrease in {he ductilily 01 a britlle malerial (even up to 2% relalive elongalion before breakenables lis use In a number el cases \\'hich are olherwi5e preclud· ed lor briltle malerials (in machine parlsl. Thert'!"ore, re.search \l:ork on improving Ihe ductili!)' al brittle malerials such as concrete and casi iron demands lhe maximum poMible a!lention. § 16. Considuations .in Selec.tion 01 Saltty Faclor

A. In the preceding sedions, \Iooe discussed the 01 computing 01 malerials under ItIlslresses, delermining lhe mechanical slon 2nd compre 600kgf¡cm'

The assumed value 01

A

is nol enough; It should be

N.

lUlO L 85 600 =. cm '

To maintain Ihe conclition A,=A. which lormed the basls 01 our calculation, we musi lake A,=A.= 1.&5 cm' instead 01 the rrquired value 01 1.67 cm' oblained fram the first l:ondilion. In this way we shall have an additional reserve in Iho:! side ban. lf we wish to avoid lhis extra reserve and lake

A,=A,=1.67 cm',

A.=I.83 cm'

(4.9)

Ihen forces N" N,. and N, will change immellialcly; Ihe ratio A,/A, will no longer equal 1, as assumed earlier. bu! wil1 beO.9. In formu13 (4.5) the denominator becomes less and N, ¡ncreases; in formula (4.6) the decrease in Ihe value o[ the denomínalor will be less as compare side bal'5 by 4l,-A8._AC,_ Orawing Ihrough poinls C. and 8, lo lhe lnilial posHions oi Ihe s¡de bars as explalntd in § 18, we get lhe point 01 junction A, of the ends of al] lhe lhree bars. From (he figure we may wrile dmrm the c:ondilion of joint defOlmation:

"

(4.13)

Sin«' lhere are no external forces and N, is a rompressive force whertas N. is tensile, the equilibrium condition (4.1) takes Ihe form N.-2N,cosa=O

(4.14)

Replacing in (4.13) tr.l, and 6f. by Ihelr values

M,

E,A,

M _ N.t.

and

N,I, cosa

and solving equations (4.13) and (4.14), we have N

.- [ l.

OC.A,

1+ 2E.A,

fc¡¡ J 7iJ

N_2L , 2COSll

(4.15)

The plus sign befare Ihe values 01 N, and N. signifies that our assump· lions abaot Ihe dlf('CIIOI1s 01 these lorces are ccrrt'C1. II should be poinledoot lha! in formula (4.3) ttK-length of Ihe middle b;¡r can be replllced b)·I. and not b)' l. -ó, b«ause 6 is an infinitesimal as fomp and p._ From (4.18) \lie

'o,

A,E. P,= PtA;!;

Subslituling this value 01 p. in (4.17), \Ve gel P1>

(1

A b El>

'"' (4.19)

('.20) The dislribulion al rClrC'tS the demenls 01 slalically inde· ttrminale struclUre5 depends upon the ralios or their cross·sectiona[

----- -----------W

+\-._._._._._._._.

-

Fil_ 39

areas and moduli 01 elaslicily. From

(4.18), taking into

consideration (hat

'"'

p. A, =0,

,,·e find that the ralio of Ihe stresses in brome and steel depends only upon the ralio ol their moduli el elasticity: If.

E,

E,

"

o. 41

are directly proporlional lo Ihe moduJi 01 elaslkily. and Ihe Aswming Ihal under E.-2XI0' kgl!cm' aod E.=lx X 10' ktf/cm" (Table 1), it is obvious that slresst'S in steel bar will alwl)" be 1\00 limes higher Ihan lhe slresses in brome jacket. rhe permissible slresscs for sl("el are U!ually Ihft.'(' llmes grealer than permissibIl" stresscs lar brome. TherefoR', if !he stres.ses in lhe brome jacket ¡;qual lo lhe pcrmissible slress Inr bronU', Ihe stre:sses in lhe sltel IJar will be smaller Ihan Ihe permissible stress lor sleel. Hence lhe dimcnsions of Ihe colurnn are oblained (rom slrenglh condition of bronte jacket under compresslon: p

E -

A.+A.'t A.

(

p A

t. )

1+ A: i''

'-;.[0.1

(4.21)

Let P=2S If. rhe ratio A,IAb of Ihe cross-secllonal areas is usually seleded ¡rom designconsiderations. Let AJA.-=2, and lhe permisslble stress [ObJ-SOO kgf/cm'. Equalion (4.21) will lhe" be wrilten as 2>000

A.(l+2X2)'¡;;;;SOO. \loMe/mm

''''''

A.=2xlO_20em t

and

TIte diameter of tbe sleel bar is calculaled from lhe coodition ;> A., wherelrom d. -

=

-V

S.05an z 51 mm

TIIe dimensions of thl" bronu jacket scetion can be round ir we assume a particular value 01 waH 1 lrom design coosidentions. Lel I-S mm-O.S cm. No\\', applying lhe approximale formula lar a nng. Wl" have A.:ro;; 11 d.t,

wherelrom

ni

3,,10 ti' _

_ 6.48 cm

65 mm

lhe ddormalions of such structures are calculllled according to the gl"neral principies. Sinct Ihe sleel and bronze porlions 01 Iht' colurnn $horlen by the same amount, it is imm31erilll ....hich al the formula.s in equation (4.18) is employed for calculaling al.

§ 21. Slresscs Duc to Temperalure Change In $talically indtltrminale $yslems. withoul any external looding occur nol only due lo the inaccuracy oC manulacluring and assembling, but lllso due lo a change: in lernperalure. Considerable siresses 01 Ihis lype may arise: in rails weldl"d into a llne. Tht rails are subjecled lo tensile or cornpressh·t slres-

.. S('IS

Ip"r¡ 11 whef1 Ihe lempenture changes wilh respect lo Ihal al whÍl.:h lIK'Y

...."re "",,[ded. lile problem may be ¡chemalicaJly exprC'Ssed as 1011(N:s: \\'e ha ...e a bar Ilo'ho.sl! both ends have bten rigidl'l Tixedat a tempcrlllure t,; ftnú lhe stresses ¡rising \\'ntn thetempeulurechanges lo t. lFig. 40). Tite length al ¡he biU is 1, cross-setUonal area is A and modulus oí elaslkity is E. Lel us ilSCertain Ihe forces which \\'m ad on the bar 9.'heo 1M lem· perl'ltureri.ses 'rom tI lo ' •. The bar will tl'nd lo eJangate and push apart

/A--- fl

.-í-

,

ti

Fil(. 40

lhe supporls A and B. lhe supporls will resist Ihis wilh rt'adions ,Iirecled asshown in Ihe figure. These l()("ces will cause lhe bar lo be rompressed. These lorces cannol be round from static condiUons, becausc 1111 Iha! come lo know Irom Ihe single equilibrium condmoo is lhal the reactions al points A Ind B 3rt equal in magniludt and opposile lO each olher. value 01 the reactiol'l P rernllins unklJO\l,T1, and ht':fIQ lhe! may be oonsidered slalically indeterminale. The! addiliooal equalion can be writ\en 100m lhE' oonsideration lh81 lenglh f 01 Ihe reslrained bar remains unchanged in spiteol ¡!'le change in ltmperalure. This implres that shortening lJ.{p due lo fo.n P is tqual in ils absoluh.' \'alue lo {he temperalure elongation t'!.l, whkh Ihe bar \lo'ould have experienced had lhe end A been fixed and end 8 Iree lo loove. Hence (4.22)

This is Ihe oondilion 01 joinl deTorrnation; il shows thal Ihe lenglh 01 [h... bar remains constan! despite lhe tcrnpcrllture change, since JI nol tear away frOll1 (he fixOO supports.

Sioc\!

"

ó/p-n.

and

lJ.1,_4/(I,_I,)

where is fhe linear thnticallllan-

.

[P"rl 11

ner. lei us consider Ihe lo11owing example lo compare the eil"ec:tiveness 01 using uniform·slrength bars. slep bal"$ and han 01 COf'Gtant seclJon. A support ",,¡lh height h_42 ro is SUbjKted to eompression by axial force. If . Assumin¡ the uni! wei¡ht ollhe laying as 2.2 tf!m", and Ihe permissible stress under compression as 12 kgf/cm", compare

lbe volume el lario¡¡: lO!" support ol constanl sectlon; support made 01 three prlsmatic parb el equal IMglh; support of uníJorm slren¡th under compression.

We $hall (aff)' out lhe caltulalions in tans (force.) and ml'tres. For Ihe firsl case Ihe cross·sedional area is

[(It

A_

lile volume

hy -120

v_ Ah -

14.5x42 ¡::: 610 m"

In fhe second C8SC. Ihe area 01 the upper portion I.s A,"'"

P le Illl-

a'

The crnss·sectiooal are.a ol lhe second portioo is

•

A.=

P+'l'A,'i 10

'fOO+2.2X4.48XI4 -

10 1-'31'

120

14)(2.2

6.04 m"

rhe cross·secllonaJ area or lhe thircf portion Is

A l -

•

•

P+yA,3'+l',"'s h (1" its lellglh, but oyer lile horizontal projecllon 01 ils length, Le. aloog Ihf' -"pan J. We Shllll sludy only Ihis Iype of cables. Let liS aSSlllRe lhat lhe oí Ihe load uniformly dislribuled the cllble span is q. This 10llíl. having dimensionality 1l0rcelllJenglhl. may be nol only dUe to the weighl 01 the cable pt'f uníl span but also the wcighl of ice or 30)' olher lood also distribult'd unilormJy. This assumption about lhe la\\.' 01 load dislribu!ion considerably simplifies lhc cakuliltlons, bul si· multaneously renders [hem approximllle. In exacl calculajions (load p

uishibulion aloog Ihl' ("urvt') lhe sag curve a c:alft\arr. when..as in approxim.:Jle clllculalions il .s lound lo be a parabola.. ul us lakt the lowtsl poinl 01 O as Hit" ol"lgln of coordmllltt' make 1\\'0 assump· smallt'l' quanlily as Iions: Ihe lenglh of the cable is equal lo Hs span. and lensile forte is constanl and equal to H. Thesc givc ulI1all error in genll}' sloping cables. In this case {he cable due lo lhe ¡ncrease in lernperaturc will be (5.16) whcre Ct Is lhe linear lheflllal expansion coeflicient of Ihe cable male· nal. The cable dongales when lhe lemperalure ¡ncreases. Thi! wiU I't". sult in 3n increase in lIs sag and, con.sequenlly. in v.'ith formula 15 10) decrease in ils tenslon. On Ihe olher hand. [rom the $lime formula (5.10) il is evidenl that lensile force will iocrease due lo in load. L('l us lI$Sume Ihe final ded is lhe tightenh¡g 01 the cable. Then. according lo Hooke's la\\". lhe 1'Iongalion of Ihe cable due lo iocrease in It'I\Slon -.-i1I he ,

_

(5.11)

98

Clmlplicaltd

CI1S!5

of Ten.slon lUId ComPrt!3l'on

¡Par! If

If H,n (rom fhe upper portion (1) lo lile lower portian (11), \l.·e imagine Ihe upper portion lo be removed and ils action on the lower porlian replaced by slresses Pa' To majnlaln lhe cquilibrium afilie lower portion, S!ces5C5 must compensa le flX" force P and musl be diredtd parallel lo lhe axis 01 Ihe bar. I t is n>ident that theslresses are nol pcfl)endicular lo Ihe plane on which Ihe)' are actlng. Their value will abo dUler !rom that in .seclion mk.

IPaf/ 11

100

Assuming lha! al a suflidenl dislal1«' (mm lhe poin! 01 applkalion or ulffilal fon:cs P slressf$ p", arE' uniformlr dislribultd over SC'ttion mil, "''i! find p

P",--

A.

· A Bul slnce

A

P.. - A;- =

(J.

cosa

00-;' is Ihe normal stress in section mk perpendicular lo lhe a.::>a. wiIJ be satisfied. lhus, we dislinguish thrl"e kinds or stressed 1. triullll slrm, when alllhe three principal suesses are not equal to UfO «(ex- eumple, lension Of compr!SSiotI in lhree mulually pt'Tpeldlcular dirte:Uons); 2. biaxial slress, when one principal stress is equal lo uro (tenslon or oompression in two directions); 3. uniaxial stress, when two principal slresses are equal lo zero (Iension or compressioo io one directlon). In § 27 we sludied the stress dislribution in unlaxial slressed slale; below we give examples 01 planar and volumélrlc slressed slstes uplaining hO\\' stresses are distributed in dilferent planes in lhese cases. § 79. Examples of Bluial Ind Trlula' Slresse:s. Deslin of I Cyllndflcal

A. As an example of a composite stressed state .... shall con.sid« the in lhe malerial 01 a thio-walled cylindrical ms('fvoir whkh is filie«qDa. The area of lhe rHamelrical seclion (t\Vo \\'aJls) which bears this pressllre in A,=2/a, and slresses in lhe walls lIre

"

•

'iDa

'10

These 5tresscs lite two limcs greater than stres.scs rr' ading in lhe ring se;¡ring slreues in Ihe! gh-en seclion: 1...

[o, sin2a,

2a..J ={ro,

,... -"';"·Sin':ht

+o,s]n1

(a, +901] (6.6)

In Ihese formulas angle (1, has bet'n measured Iroro ¡he d lIxis I \slresli o,) up lo Ihe normal lo Ihe J!iven $l'Clion by rol"Unl! We shal! 10110110' lhe rules laid dO'V1\ earlier in § 27

in choosing proper signs fO(

(J"

and

T..

33 well as for llngles a, ¡¡oo

el••

FI,. III

In lulure. in formulas ¡iving lhe values or o" and T" we $hall denote a, by 11., 1I1ways meuuring il from lhe muilllum principal slre:ss in lhe anliclockwise direclion. Employing formulas (6.5) ¡nd (6.6) whictl ¡¡ive Ihe stresses in .se::tion a-o (Fig. 61). we can easily Ihe stressf'S in secUon b·b which has norlnal forming angle with lhe

n.

61

'"

Co".po.",4 St"j& Sltt'" """ SIIfIIIt

dirtction of lhe muimum principal stress:

=a, l'OS' P+a.sin'f' -o, cos' {a + a, _a, sin' a +a,.,;o",4 et

(et

+90')

}

(65')

16.6')

formulas derivl!d aoove c1:lriiy lhein lIl\ltually perpl.'ndicular planes. FuI normal

cJ slresses aclin¡:: we llave

a.. =o,cos'a+a,sin'u = o, sin' a + 0". COSO et Summing up, .... e get

0"

=0, +0, = consl

16.7)

j.c. lhe sum oi normal s\resses in 1\\'1) mutual1y perprndicular plane5 js conslanl an:lring slre'SM'5 in 1Il"O muluall)' perpendicular plan s)'slem wHh all"es a and 1. Tht>I1-axls dir('{'tt>o'lIll"is we plol

,

segllM."nls DA and 08 reprl'Senling in a enlajn seale Ihenum{1'jul \ alul'S do, and a.(i1ls convenienllo draw Iheo'Uls parallcllo Ihe olall"imUlIl principal slro!SS oJ.

'"

OJmpou.ruJ SI.tU- Sfrttl lJIII1 SIra,,,

CA "

In Fig. G2 botll slresses are cor.sidered and are Ilid 011' on lhe a-axis in the posili\'(' Had ene oc bolh 01 Ihe slresses compressivl', wt' would have laid thelD 011' in lbe opposill' directlon. Taking segment AB as the diamE'ter, we draw 11 circle with the -- -,

1',

Fil. 64

(J.

SIres.ses and are by D,K, and OK•• res· pedively, It is clear lrom Ihe diagram Iha! and 0",+0,=0, +0, =cons! The stresses acling on the lace; 01 the element cul by planes a Ind b are shown in Fig. 64 00 lhe rrght. By bringing in line the direclion 01" lhe maxlmum (1I1gebrai zero. For eumple, far Ihe case shown in Pig_ 60, we have

1

i

(1,

(1,

(1,

(6.218)

".

Lel u... calculale the chaogl.' in Ihe volume of a redangular parallelepiped having edges. of a, b and e, ir It is I,;nder Iriaxia! stress. lis volulrJe bdore defoTlllalion is V.=abc. Alter deformalion, due lo e1ongallon of its edges its volume be,olume (§ 34). rhe stress fFlg. 73 on lhe r¡gh!) i5 onlr re¡ponsible lor Ihe change oF Sh3JK'. \Ve shall rtturn to the problem of Ihe change In volurnc amlshllfle later whilc discussJng pro[)lems or strength 01 materials in compound stressed state (Chapler 7J.

§ 35.

Ener¡y 01 Elaslic Ddormalion in Compound Stress

Polellllal energy 01 defocmaUon is lhe energy acc:umulaled by Ihe malerial as a resull oS elastic deformalion c.aused by ulernal forces. To calculale lhe polmtiat energy by 01'1 ela.slic S)'slem, may use Ihe lal'o' or e:onservalion ol energy. Let us first consider Ihe case 01 simple tensioo (Fig. 74). Ii we load a bar statically by gnadually suspending small 1000ds !!P, lhen aHer each addiUon lhe suspended load C'O!IteS dou." aOO ils polential enern decreases. whereas lhe potential ener-gy 01 delormalion of lhe slrelched bar ¡neceases.

o, Gl

e_pe",,,J SIr=. SfrtSJ

S/,o"',

12S

Whcn the load increases slowly and gradually, lhe velocily of displ._ bar is ver)' sm.U. Thtreiort', mar «-ment of Ihe iree end 01 glect ¡he inertia 01 Ihe moving mass ando consequenlly. assume Ihat thfo deformalion IS 001 accompanied by change in lhe kinelic enero gr of Ihe 5)·skm. Unoo thl'SC.' rondHioN liJe polential tflf'l'gy 01 Ihe loU'ering load 15 lransfofnted iolo lhe poh'ntial energy of elastic dclorlMlion o{ Ihe bar

(",e neglect lhe dissipation of eneriY due lo lhcrmal and eleclromag· 1'K'lic accompao)'ing (he c[aslic deformation). Thus an t'lastic system undcf slalic ¡oadiog may be

consicterNl as a machine transformo

ing cne form 01 polenlial energy jolo

analher.

As Ihe potenlial energy losl by Ihe load is equal to Ihe work accomp· Iished by il in IO\I:ering, lhe problem

of determining lhe pottnlial energy 01 deformationcoOll'$lo calculal· in¡ Ihe \\'Ol"k done by Ihe externa! fortei. In § 10 ....-e obtained expres-

dP FIl:. 74

sm (3.1) rOl' Ihf wock done b)'the external lorces in simple Icmion:

IV

-,

_ plJ,/

This implies that the polential energy 01 lension is siso P'I ,,'Al

. o,

U_w ... 2EA

=v

(6.26)

PI

=EA'

rhe potential encrgy accumulaled by a unit volume 01 mulerial 15 00

(6,21)

Lel us now passover lo tht detCf"minalion 01 potenlial energ}' accumu·

lated in a unU volume el a mat!'1'iat \I.'hich 15 in a compound (planar or \'olumelric) stress.• U5e 01 lhe principIe of superposilion of lorces and assuming that the princip"l stresses inttease gradually, "':1' can determine lbe potentiall'lle1fY as Ihe 5UUl of the mergies accumulalee! by a unit volume olthe material under the acliOll 01 each of lhe principal stres.se:s a" a•. and u. aa:ording lo (6.27)

'"

Qf

Complitllffd e"..,

Tmst"" ""d ComprusiOll

[Parl JI

artes- multiplication

u

=it. [0'1 +a: +0:- 2J1 (0,0. +(1',0", +o,o.lJ

(6.28)

Hence, Ihe tutal /'lIerr¡JI (JI de[lNma/ioll accumulaled in a uuif volum/' of material (8 eube wilh e0,>0. are caku1ll1ed; (2) lhe material is seleded;

",

01.71

(3) Ihe crilica! slre0. This [heor,' is confirmed by lensile tests of brittlc ma{crials such as slonc, brick, concrele, glass, and porcelain. In Ihe case of compound stress thl' theory oflen comes inlo conAict with experimental data be· cause il does not take iolo account the olher l\Vo principal stresses upon which Ihe str(!nglh of material depends in many B. The idea Ihat briltle laiture is conneclerl nol wilh Ihe rnaximum lensile slress butwilh maximmn slrain was lirst expressed by FrE'nch scientisls Ed. Mariot!elin 1686) and C. M. L. Na"ier (in [826) 3nd tater sllpported by other French scientists, J. V. Poncele! (1839) and

(;1,.

71

S'rtlllllh

o/

11\ COmpOUM SrrtSJ

139

B. Saml·Venafll (IB37). The strength Iheory based upon Ihis supposiHun is klli)\vn as Ih" fheCJryCJ! maximum strailt, or lhe seconrl )/renglh l!loor". According lo 11m Ihl'Ory raHure occurs irrespedive of tile slllle 01 stress \vhl:n maximulIl elaslic sira in t m" equal lo a rerlain valuc l';,up which is conslanl for Ihl! given malerial. in general E",. . ""'1';,

--rrI

whereas In simple lension e=oIE; it is obvious that t,"p=a,u.jE. In Ihe compound slress, faiture \Vil! Ol:h 111 C ['1,1

We see ¡haj although slrength condilion is satislied, lhe shaft (]iameter sh()uld be incrl.'¡jsed lo 111lprOVe rigidity amI il shouJd be: calculatcd from cxpn:ssion (9.19):

J whffe!rom, by

in radj;¡m

MI!

P#GI'Í'1

Ihe value of permissible angle of lorsion we obtain ItlX10'XlíJO 1 5 d ;;. -1/ OlXtlXIO'XU:¡= 4.6cm

Hence, lhe shalt's dillmeler should be taken d=H.6 cm lo ensure required rigidity.

§ 51. Slresses Under Torslon in a Seclion Indined lo lhe Shait Axis Wtlile s\udying Ihe stresses in aued by a force couplt which ('reales a Ulomenl Ihal gradually inaeases from U'fO lo a linHe value M. TIK" iocrease in M \\'i11 resull in a corrC'SJKlnding iocrease or lJ' which is relllled lo M I by equalien 19.18): Ip ....

MIl

H W(' piel Ihe llngle 01 l\Visl along lhi.-' x-axis anu Ihe corrcsponding values or Ihe Iwisting momenl atOlle: the y·axis, lhen Ihe relation between Ihe 1\\'0 will be represenleu b)' an im:1ined slraighl line OA (Fig. 112). By Ihe samc reasoning as employed in calculatini lhe work donf.> by a lensile force p. we lind Ihat the work rlonr by lhe fon:e couple M may be upressed Ihrough Ihe are. or lr¡angle 0/18:

(9.20) conslant 1:'2 in formula (9.20) is dLH! to the f.eI ¡ha! fhe moment

M las !Jot bem applied in i1$ full Ulagmlude al once bul il1Cl?Jsed

grarlually. "stalically" fmm tero lo il, hile vatlJ('. Replacing

3.'

O.'"

O.'" O."" O.'"

S.,

..O.'""" 0.a18 O.a93 0.645

0.76íl

,•••.. •••

10.0

1.123 1••55 I.'R'} 2.4St> 3.123

• 0.1Il1 lo 12l!

..... 1.455

3.123

, 0.753 0.7.5 0.744 0.7&:1 0.7a2 0.142

SMor and Tars/on

192

[Parl 111

10

Oda lln Tursilln 01 Nlln-cjrcular Scctlons .'10",..1 ol 1.,. 0.5 t,....

:If,

-w;-

"f

:\1 Ihe

lile

'"

O'

J,-Cll'ir

(l

M, t",,, ""lt'"¡""

and 11 from tlle lable dcp.ndlnll tlpon lht

fOI]"

dlD.

dlD

0.0

0.05

0.10

0.20

0.40

O.GO

0.80

LO

"

1.51

lUlO

11.81

IUl2

lJ.i6

O.m

0.52

O.:3S

1.57

1.5',

1.5(,

0.':1'1

0.63

0.:18

,

0." , 22

'"

'".9)

Attording lo Table 9. lor narrow recf3ngular SfClions (};;t10) roeffidenls IX and alld a, and are approximalely equal t01-ffrom 0.312100.333). In accordance wilh formula (9.37'\ lar suth rectangular seclions we oblain 1

J'='jIllJ'.

1

W'-:rhb'

(9.38)

Table 10 O ir Iheir resultllnt is direcll'd dO\\1lwards. Acc:ording lo this convention lhe direclion of Q eoincicles wilh Ihe dirl'Clion 01 shl'aring slresst's t whieh eonstitule Ihe shearing force (Fig. 134). The bending mOlnenl will be considcred positive 11 fhe algebraic sum 01 momenf.s of forres lo lile lelt ollhe Sl.'Ction givtS ¡¡ resulting momenlllCting in lhe clod:\\'ise direclion; or ir lorces applied lo Ihe right of Ihe secUon give an anlkJockwise Il'SUlling momenl ¡Fig. 135). HellCl'. for ihe Idl culoR porlion iM- bending moTI)!'nl due lo individual force is considereUing Q. and M·diali'"ams or" lllvell 1" p",bl.m In Slr{fJgtll (lf MaleriO. Le. tan a>O. the momenl increases; (b) Q iI works no! as a beam bul as a plale and il musl be analrud in 8 mannl'!". • In genf'l"al. assumplions llIade aboYe lIre only lIl"proximatd)' lrue. HOl\'evl'f. Ihe lilroreliclll trror is so slnall (exccpl in special cases) that it can be ignored.

§ 63. Delermination 01 Normal 5tresses in Bendlllll. ' Hooke's Law alld PolenUal Energy of Bendlng A. Le! us consider a beaul subjected lo pur!.' brndlng by 11 momrnt M (Fig. 151). Let us cullne bealU In l\\'o parts by secfion n, and using Ihe Illefhod of sed ions cooslder Ihe equilibrium ti 0f'K' 01 the portlOrd, Sily. lhe leH. sh ¡he z·axis \\'uds); lhe neutral axis of secfion has been laken as Ihe y·axis, ils loca{¡OIl along Ihe height al" the beam being nol )'1'1 koo\\'n. Tht:

Ch. JI)

NQrln(l/ StreS$1s in IJCnlJillg. Strcngtll

"f

Oeams

""

has been taken along lhe neulral layer perpendicular lo Ihe -,md z·axis. Every poi ni in Ihl' cross secl ion is acll'd upon by a normal stress (J. Let us isolate 3n elemenlary area dA aboul an arbitrary poin! hadng coordinales y and z, and denote (he force acling on iI by dN="dA, The culoff portion of Ihe beam mainlains its equilibrium under lhe action or exlernal rorces constiluting a couple of rnoment M and

Fig. ISI

Ihc normal rorce dN which r$ iscalled the axial sectlon modulus and is denote secUan is symmelrkal aboot the n("ulral axis, rar e.umple, a rectangular seclion. Ihe outer strelched and oompressed 6bres are )f

Ch. 1/1

of

Normal Sfrt&l!tJ

Btom&

IOCllled al equal distance from lhe neutral axis, and such a secUon has a singll' liefinite value ol Ihe .serljon modulus aooul the y-axis. Thus, ir we considC'r a seh" = Bll"-b/l' ----,.."..-

(12.2')

,,' /Illrtia 01

The scclion

Is W _ - 1M = BH'

12HI2

z'ou

011

Nole Ihal lhc section modullls cannot be caklllaled in !he form oi dirference or W=BH'¡6-bIl'/6, because lhis runs couoter to the ver)" concept 01 secHon modulus as Ihe ralio

r O

8

•

Y

¡ddz

bu}

,1 Frg. 158

Fig. "9

While determlning lhe momeni 01 inerlia 01 a circle of radius r \Ve similarly dIvide lis lolal atea into elementary slrips 01 thickness dz along the axis Dl; lhe widlh al Ihe strips b=b(z) also vuries ¡¡long the hcight al thc section. The elementary area is

(Fig. 159)

dA=b(z)dz

Tbe moment 01 inertia is

J=

, z'b(z) dz

As Ihe upper and lower halves of the scction are idenlical, it 18 suflicienl lo calcula!e the momenl 01 inertia for one hali and double lhe resulto liJe lirnJts in whlch z varies are irom O lo r:

Wc introduce nO\\' n new variable af integralion, angle IX (Fig. 159):

z=rcos";',

b(z)=2rsin%

'"

IParl fV

.

The limils of integration are a=tt al z=O and a=O al z=r, lherefore

" (12.4)

(12.5) For a cirde soy axis passing through lhe centre of gravity is the axis of syrnmelry. Therefore lormulas (12.4) aod (IZ.5) are valid for aJl 5uch axes.

11.

Hg. 160

Fil[. 161

Substiluting r={, we shaJl now diallleter;

J and W Ihrough Ihecirde's "d'

The llJomelt!

I =1J4

(IVI')

"d'

(12.5')

01 ¡nerita 01 a triangle (Fig.

160) abou! AB is:

JAB=r'b.dz, J"B=

J•bz' (l-i )dz _

-

-

Laler (§§ 66·68) we shall explain how lo ca!cutate Ihe momenl of

inerlia af a sct:!ion of sny complex shape abou! .cd in solving problclIIs. We shall sho\\' in § 70 Ih:ll one 01 lhe principal mOIll('nls 01 [nerUa is J,., and Ihe olher Is 1"'1.' Formulas (12.IS) can be lIIodiñed into a form whiclt does nol contalo a., ExpressillJ!" cos'a. and sin'a. in Itrms 01 ros 2«., pulllng lheir valucs in Ihe first formula in (1218) and simultaneously suhslltul1ng value of J., from formula (12.17), \Ve gel

J!Io

/,+/,

Jy-J,.

,,_

----r- +-,- cOS ""'. +-,- 00SZii; /9+/' J -,- +-,- coa 2a.

From formula (12.17), repladng lhe froclion

,

.(t>'i

2"'. by

± V'+I an ' 1400 kgficm'

strenglh condilion according lo lhe folltflJ

of slrt'ngth

(13.17) is

1510 kgl,cm' ,.1400

k,l:!f,cm'

lhe reduccd strcs:> according 10 lhe iourth thcory is 8°0 gre¡¡!cr lhan Ihe pCrluissible SITCSI;, 1111.' dllllcnsions 1)1 lhe shouhi be increased by takmg no [-bE'alll No, 22a. Mlcr com¡wtalions v.'e !lel lor lhis seoclion 0"'.... -1260 kRI'cm;. t111d for 1=10.11 l'm. o... 1158 kgf CIII' ilod T=442 kgl l'm'. Tlle rt>dlll'ed slress accordin¡: lo lhe hst thNry 1329 kgl cm', and llccording lo (he fourlh Iheor)' of slrength, 1423

§ 78. Directions 01 Ihe Prindpill Slresses Jo lhe precedmg secllon \Ie dl'termlOl'd onl} thf:' magniludf:' o/ the prlOc!pill stresses lor ao arbltranl}" sclecled eJemeo! II'Hhouf con· QCming oursdn',1-0

0,

,

--,,

,

r._r

-.,

,

L

.:c! q

r;.-,y

•

-;;:-

0, Fil. 197

-

Ch. 131

SMaring and Prlndpni Slrtw.

'"

enough lor maleriaJs which have eju¡¡1 rC'Sistance- lo lension and compression. Fur lTIalerials like rein orc('¡] concreh.', howen'r, it is exlremely importan! lo kno\\' Ihe direction 01 leMUe in ever)' point so {hat \'\'C can place {he rein[orccl1Il'nl rous in Ihis UlfKtion.

Flg. 1911

The 01 the principal stresses rnay b\' delermined \vith lhe help 01 lhe stress circle (Fig. 197). 5IJppose 0" ancl To, llc:ling in a plane perpendicular lo the .uis 01 Ihe beoam, are posilive:

,w, a.. = + a=7

JiUldlng. SlrUlgII¡ af lJtams

lPar/ lV

and QS' 't'«=+t=-¡{

Arter ploll(ng ihe stress circle we see Iha! Ihe relative posllion of lhe Unes 01 adion of slrm a" and Ihe ma:dmum prin. clpal stress a, is Ihe same as thl' relalive posiliof! of Jlne BD" lInd lhe x-axJs; lhe lal'ter lwo make an angle (J; in lhestress cirde (fig. 197). To mark Ihe direclion of a, on Ihe drawing we mus! lay off angle a from the lliredion or G" dockwlse. The principal siresses change Ihelr direclioo ",¡¡hin lhe limils 01 lhe sectíon. Near the edges oí lhe beam Orle 01 the principal siresses 4"IIIiIIIllI!!lIlI!!!11l11fh!!!III!!1II1II!!!!lIlt

774fl FIg. 199

is uoro, "",hereas the other is dlrect.ed parallello {he axis 01 Ihe beam; al the neutral layer principal slresses lIIake al! angle 01 45" with {he axis 01 Ihe b('am. Figure 198 shows lhe slress circles and 01 tiJe principal slrcsses in varlous poinls of theseclion. 11 is assumed thal Ihe bending Inomenl and shearin¡,: for'::l' in the M'clion are posilive. Having oblained lhe direclions of lhe princ,ipal stresses in an aro bilrary poiot 01 Ihe given srelion, '.1'lnlilles, thC5e shearing 011

TI' and in attordance wilh the law al

T,

,

'1

• ,I

"

" " " "

1,

"

'e,

,

,

,

" (,)

('I

dislribuled lhe :lrea 01 fare ABeD. ConseqUffilly. lhe sum of elemenlar)' ¡nternal shearing lOITl'S artine htre ""ill be

dT--TfJdx The condlUon of ¡he isolaled wrillen as follo\\'s: N,+dT_N,=dT_dN_O

CM be

..,..hetd"rom I

I

'*

(14.2)

Thu5, Zhuravskii's formula (13.3) can also employed loc shear· in¡ stresses parallel lo ¡ht' neutral ;u:is in Ihin-wall«l seclions íf quanllly b in {he deoomin.loc is takm u lhe w¡dlh of Iht' I.)'tr in whlCh wuin¡:¡ stt'l'U b calculaled, ¡'respectlvt' of the thinnlled 5eClion is lo be cut p3ralJd oc perpendicular lo Ihe neutral axis.

Ch.

141

,,,

S/lmr

In our case (wHh the assumption Iha! N,>N,) shearing slresses TI in the leH hall 01 the top lIange ael in Ihe eross sectiOIl ¡rom lerl to righ!. lt can be easily seen lhat in Ihe 1erl hall 01 lhe lower flange, where lhe normal slresses are eompressive and as belore 10.1>10>1. shearing slresses 'tI ad in Ihe opposile direclion (Fig. 2Oó(d)); in Ihe righl hall of Ihe top ftange Ihey aet ¡rom right to leH (Fig. 200{f»), whcreas in lhe righl hall of lhe lower flange ¡rom lel! lo righ!. 'I-diaj...", ,

lIIO.I::..

,-, ," 1

Y-;...

-dll'l""'"

The shearing slresses in the flnnges and \\'eh 01 Iht Ihin-\\'a1Jed section form lhe so-callcd shearing str5S "s!re;Jm1.IIU;·', Ihe slrcnmlines lor nn l-section are depictcd in Fig. 201. Lel us \\'rile ¡he expression for sh"llTlnf/ T,_ One 01 lhe qullnlilies in formula (14.2) is Ihe tl,ornl'I\t o( lhe llange area hatched in Fig_ 201:

Therelore (14.3)

Le. shenring str01 Fig 24(1

ql'

['\="32/=(; OrdinalC' ..o 01

f!..:

ctnlre 01 gravHy of (he parabola is

,

'.

o

..,tI

61

..!.. /' fI q

J

1 l'

li q

Tlle distante belween Ihe centre 01 01 tll(' para bota :md the of mallimull1 bending mo!nenl (rigidly end) is I'ljual lo one-{ourlh of the beam splln. Let us salve the following exa mples usíng lhe ttliltions ucrived htre.

Offorma1iQrl

lo 8Mdlng

[Por! V

Find lbe deftecliOIl al {be middle 01 lbe beam shol.:n in Fig. 238. lhe area 01 the bending·momeni diagram is laken as the ficlitious load. The diagrarn is posilive, therelore lhe ficlilious load is direcled up· wards. The ficlilious support reactions are '" 9/' A,=8,="'2"""R"

These reactions are direcled downwards (Fig. 238). The tictiUous bending momenl in Ihe middle of lhe span is equal lo Ihe sum of statie momenls al lhe tictitious forees loeated on one side of lhe middle seclion, sar. on the lefl·hand side. The forees localed lo tbe leH of ¡he seclion are A / and Ihe lefl balf 01 the parabola. The arm of loree A I is equal lo half of Ihe span; arm al the hall parabola is Therefore, Ihe f1ctHious bending momen{ in lhe middle of ¡he span is

and the delleclion in Ihe middle 01 lhe span is 5 q/'

11"--38477 Consider a beam rigidly tixe.1....

II "1:

3P

b.,. - 2h Itl lhe correded shape oF the! beam ¡s shown in F'ig. 2013.

§ 95, Praclica' Examplt.s af 8eanl$ 01 Unifotm Slrenglh lhe txal1lple distussed 6nds practical applkalion in design or springs. 11 we ils small curllature, a spting mar be looked upon as a simpl)' supported beam (Fig. 2014(17) l03ded ..lIh a larte P in lhe of lts span and havin¡ .1 iis ends, We design luch a bar b)' Ihe UlIDe principies as a beam al unilorm

'"

lo BeNfing

[Pad V

slrenglh 01 constant height 1,. and variable widlh b(x); as lhe looding is symmelric il is sulficil:n! lo studr jusi ane half af Ihe span. The section rnoduli W(x) and W. can be expressed by ¡he 33me foro f. p P mul1l5 as in ¡he prlXeding example. The maximum bending moment in IJ) I Ihe middle 01 Ihe span i5:

it Ji.,

lb}

1

A,

r--7

Itl ti, , ,

-+--l---.,: ;;:>\

I

I (,)

,

I

,

,

-1,

mu

=!:..!.. 4

Bending mamenl in any arbilrary seclion Is:

,

I

1/

,,

1='-

3

I

M

"

M(x)=T Solving, as in ¡he prcceding exam· pie. we gel:

,I ,1, ,

b(x)-=b. 2: ,

I

It}

: : :

"

:

:

:

I

1

: :

: : : :

I ' I , ',' IJI,'

I I I I

I ' I

The maximum widlh required lo resist ¡he shearing force

I

1 kC«CDDSD: I

1'1

,

can be delermined from Ihe fol-

lowing formula:

I I I

' •

!

(17.5)

I

J

;,

°mln =

'T

, h. (T)

lhe front and top views 01 the spring are sho\\-'T1 in Fig, 244(b) and (e). Iiowever, such a shape of the spring is highly inconvenient ¡rom the practical point of view; therefore Ihe shape is slightl)' modilied without allecting the performance 01 the spring. 1magine that Ihe spring is divided into Ihin slrips when seen from Ihe top, as sho\\'n in fig. 244(d). JI we place these strips nol adjacenl lo earh olher but one over lhe oLher and neglecl the friction belween lhem, then wllhout allccling Its working the spring may be given a shape lhe lop and front views of whkh are shown in Fig. 244(e) and (j), respectivel,.. Obviously, in actual praclice each spring plate, Ihe 1st, 2nd. ele., is manulac1ured in one piece and not in two halves. Non-unilorm beams are olten used in mechanical engineering For example, sharts are oflen designed as bealTl5 01 unHorm slrenglh. Flg. 2H

§ 96. Displacements in Non-uniform Beams When delermining lhe defledion and angle of rotal ion 01 a nonuniform heam. il should be borne in mind thal {hE' rigidity 01 sud a bealll is a lunction 01 x. Therelore. Ihe differenlial equalion 01 lhe de·

Ch. 17)

NOII·ulllfnrm

'"

Beams

lIecled axis may be writien as

d',

EJ (x) dx' ... M where J (x) is the variable momen! 01 inertla in dilferent beam sec· lions. Before inlegrating lhis equation we must express J (x) in lerms of J, Le. the momen! of ¡nerlia of the section in whleh the maximum bend· ing moment ac\s. Having done this, we can carry out !he computa· lions in the same manner as lar a beam 01 uniform seclion (§ 82). Lel us show Ihis through the example discussed earlier. We shall delermine lhe dellection In a beam of uniform slrength (Fig. 242), which is lixed al orre end, loadetl at the other by a force P and has a fixed heighl. Lel the free end or lhe beam be the origin or coordina tes. Then a ... 12 1

J(,)= b(x)h' = bGh

M(,)=-Px,

.

12

J!...1

(17.6)

• d',

EJ T"Tx7=-Px !he di!ieren!ial cquation may be writlen as EJ!!:J!..=- Pxl=_PI

'"

Integrating twice,

= - Plx+C,

El

(17.7)

,

.'

EJy=-Pl T+Cx+D

We have the lollowing cOllditions ror determining Ihe constanls 01 inlegration: ¡JI polnt A (x""l) dellection y=O and angle of rotation =0. Therefore pp O__ PI'+C and O=-T+ Cl + D

.jf

wherefrom C=Pl'

The expressions for y and d'l

y=-

PI PI

2f1J

'"'

PI' D=--,-

a may be writlen as PI'

PI" x x'+ET -

Pl' (

foJlows:

• ')

I-T

PI'

PI'

2EJ = - 2EJ

(

"')

1-2¡+""'i'

Ma:dmum dellcction al Ihe free end is oblained by putling x=O: pp

1m.. =-'ii!7

'"JI 11:(' had a bum el umrorln so:lioo wilh

:l

Ihera lhe muimum tkRC\:lion . . .,ouhl be.-

lllomenl' of inl>rtia J,

("1'

or two-thirds gn:a\cr. Henc:e, non-umiQl'IlliK.>:llLlS are more flexible Ihan ofllRiform sectioo of the $.!Ime stre:ngth. It of lhis properly ltnd not

due lo saving 01 melal Ihat n:>o-unilorm slrength bcams are used In Ihe manufacture or e1ements sucll as sptings. Equation (17.7) indicales Ibat in Inis ('unlp;le lhe curvaturt 0\ t/K! beam is constant, Le. lhe bea.Ul axis dl:'llech along Ji cirele. Bul lIpon intcgralion lbe equation obtained was Iba! a para bola. U 15 sug·

or

gested Ihat Ihe reader should Clplain lhe reason lar Ihis..

Whcn lhe graph·analrtic RII..'ltlOd is used lor determining lh", de· lornl$lion af non-uniform heams. it does nol present any difñcultil:$. Insltad of dividing lhe bentling momml and shearing force in the rKlilious bearo by El lo compule! alld 8. oblain Ihe load by dividing lhe ordlnAtes of Ihe bending momenl d¡agram of the real beam by rigidily El. 1hen •

,\t ¡..l

qf""'""'""'FJ'

!"",:U r.

artd

e-Qí

Whc:n applying Ihis IlIethod to oon-umform bean\,S. \l'e assume Ihal •

.\1

'''1

qf= EJC") Then ""e load tlle fietilious beam by Ihis fOlce and oblain Ihe required defleelloo and angle of rolallon Ihe bending moment and shearlng force in secllons 01 the ftclilious beam.

q; -- ffi.. . -fT,

In Ihe exa1l1ple above Le. Ihe ficllliollS beam should be 1000ded nol by 11 triangular force bul by 11 unifoflnly dislribuled force (Fig. 245). lhe d",ntt:lion 01 seclion 8, whkh is equal (o IhE' bcnding momenl in the Iixed end of the ftclitlous be'.1I11, can be expu'Mt'd by lhe formulll • 1_.\11,_

qjl' 2

PC' --2FT

We could have oblalnt'd lhe salDe re'>ull by a.qoming thal lhe heam has constanl ngidilr El antl lis bertdmg momenl diagram is oblaint'd by llIulliplying elll:h ordinale by Ihe ralio !lendmg momenl dlagr.m lhus oblained are

:

Ihe onlirules 01 lite

-Px-'-=P.x!..!=Pi J ¡.) J..

Ch. l7j

329

NOII-WI;'orm BMI/IJ

(Fig 246) lhen llcQording lo the gencral lY!11: method I

PI"

MI =-Plxl"2=-T

ol lt1e graph-ana.

and

9= -

PI"

nI

lhus, ddormatiOI1 of non-unilorm beams can be cal¡;ulated by the sallJe melhod as Iha! lor beams 01 unilorm rigidily. lhe only difieren.:... is tha( ¡he bending momenl diagram Ilscd in (his case is obtained by mulliplying wllh (he ratio J:X)'

1',

'.

1:.-..-, 'Fig.

rlg.

AF,6 . , "

PI

b:rnm:rrrrJl rig. 246

F,g. 248

tion is J in lhe ¡eH hlllf ami 0.51 in IhX¡. Ju

+ s;n'i'XYo =0 J.

(20.3)

This is Ihe equation ollhe neulral axis. lt represents a siraight Hne passing through the cenlre 01 gravify 01 Ihe seclion tal Yo=O snd zo=O). Figure 299 shows Iwo beam sections; Ihe y. and Ihe z-axis are lhe principal axes of inerlia. Assurning lha! Ihe beams are loaded as in Fig. 297, Ihe ptojedion of force P has been SROWn in bolh Ihe sections and Ihe proper signs 01 Ihe nornlal stresses have been given lor each quadranl; the signs above and below the seclion are for stresses dUi

CA. ID)

to IlJOI:K'J1t M" whereM the signs lo the right and leH d the secliOl1 are ror moment M,. Fot a beaRl which 15 loaded and conslrained in. dilfercnl way (Fig. 3(0), Ihe signs 01 lhe slresses will also change ac· cordingly.

fl(.

m

Approximatt lacalion 01 lhe n('ulral axis ls .sho..." , in Fig. 299. Po! lhe neutral axis passes lhrough Ihe cenlre 01 gravity. U 15 sullicient lo know angle which il makes wHh the y.axis in orde!" lo lacate il fully. ) I is evident fronl Fig. 299 tha! the lanIml 01 Ihis angle is equal lo the absolule value of lhe ratio 01 l. lo Y.:

o:

tanct=I;:1 From equalion (20.3) v'"e oblain

tano:_I"I=bn, '" Y.

J.

p

(20.4)

Heoce, lhe loc:alion 01 lhe netJlral axis does not depend upon Ihe magni· tude 01 lorce P, bul onl)' llpon angle which Ihe plane al applicaliOfl of external rorces makes with Ihe z-axis and llpon Ihe shape of lhe secliol1. Mler calculaling angle from foro mula (20.4), we plol the neulral lllfis 300 on Ihe diagram, and by drawing lan· gents lo lhe sectiOfl parallel to lhe neulral axis .....efind lhe muimum slre5sed points. which ar. Ihe polnts larthesl rrom lhe neutral axis (points 1 and 2 in Fig. 299).

o:

(Po,! VII

Subslituling the coordinales oí lhese poinlJ {vi. 1" or Yr, lJ wilh their proper signs in formula (20.2). "lOe calculate lhe m.u:imum lensile or compres&i\'e sbes,sts, lhe strmgth eondition loe lhe beam l1lay be. wrillen as (ro S)

where y, and 1, (orJ' and l j art' Ihe coordinalf'S 01 Ihe poinl (in Ihe coordinale system rrinclpal axes passing Ihl'Ough lIJe centroid) lar· IIlesl frOfil lhe neutra u is.

For s«lion with corners in which bolh lhe principal ¡¡¡es of inedia are lhe aXe:! 01 syrnmetry tredangle. I-beam). i.e.

fy.I-ly.I-Iy... 1

aud

11,1-lz,I""'ll...,1

formula (2O.S) ma}' be simplilied and the expression for cr,,, l' ma)' be wriHen as (ollo\\'1:

'1n,)

"(

,11",.)

.,(....lSt, cr",.,=±I> -Y;-'""Il";"" .... ± W; coslpTiV;smlp

lhe slrE'nath condilion lar such

10••• /=- "{:-;' (roSIjI

1'06) ,

is as Jol\owl:

+;;

¡in ..),.;;;: [aj

• i.

(20.7)

While St'leding Ihe secllon \\1! sel Ihe yalue: o! 'lOd kno""ing 10J, /ti... , ¡nd angle"we 6nd by trilll and error (he v.lues of IV, and IV, whifh satlsry lhe strtnglh candilion /2lI.7). In unJymmetric leclloru .....llhoul comers. Le. when we US(' slrE'ngth condilion (20.5), the loca·

CII. MI

UI1s¡¡mlMlrlt BtI1dil1f

'"

lion of neuiral llxis and Ihe coordinales 01 the rarlhest poinl (PI' ll) musL be delermil1ed cVl'ry bdorehand. For a rectangular seclion,

-F

Therefore assuming Ihe ralio known \Ve can easily [¡nd W' ¡¡nd the dimensions of Ihe secUon from condilion (20.7). the diagrams showing lhe dislribulion 01 slresses in a rectangular sedion are given in Flg. 301. [1 is e1ear (rom equalion (20.4) Iha! angles a and lJ' are nol equal, Le. {he neutral axis is nol perpendicular to Ihe plane 01 application or exlernalloras as was lhe case in uni-planar bending. lhe perpendicularit)' can be achieved onl)' jf Jv=J.

(20.8)

bul ,hen aH alies become the principal axes, and unsymmelric bending becomes impossible; o[ Ihe plane of Joading we shall have uni·planar bending. lhis will be Irue for square, circular and aH otber sedions whlch satisry cquation (20.8). lhe shearing may also be calculated by a melhod similar lo the 0111' Bdopted ror determining lhe normal slresses: Ihe total shearing stress wi11 be equal lo lhe geometric sum of the stressti due lo benrling in eaeh 01 tht' principal Usually the value 01 the shearing slresses has no praclical importance.

§ 121. Delermining Displaeemenls in Unsymmetrlc Bendlnll We shall agajn appl)' the principle or sup{'rposilion 01 [orces loclr· lh 'f=O.OB75x 11 "" 0.963,

•

from Ihe z·axis,

a;:::::;

Dencctions in the diredron of the y-axis wi[] al mast be equal lo the in Ihe direclion 01 the z-axis: fu = f# tanu.=0.963' < ,'.\oreover, the deviation or the plane of applicalion of Sorces from the plane oi maximum rigidlty is accompanied by a considerable in· crease in Ihe normal siresses. In Ihe example discussed above Ihe maximum normal strcsses (as compared lo uni·planar bending whenlfl=O) incrrase in (he ratio (see formula (20.6)) M W

( co,q:+WSIn'l' "".)

Al'

h

2(1

-(I+ll(antp)costp;::;;:(I+ao.0875 )1=,¡.29

"

fig. 303 shows Ihe relative lacalion of lhe neutral surlace, ¡he plane or bending and lhe plane 01 lo' from anolher, work satisfaC!orily if bending occurs

304

in the plane 01 maximum rigidity (high rectangular seclions, l·beams, channel bars). They, however, fail under unsyrnmetric bending. Tllerelore in sHualions where (he designer is no! very sure 01 a sullicJently accurate! coincidence 01 the plane of loading wilh Ihe principal plane, he shoutd avoid using sucll seclions or make! additional provisions (by pulting constrainls) lo prevenl laleoral deforma \ion, which might occur during lmsymmeLric bending. However, carelm reinforcern(!I\t of {he existing slruclures may be llxtremely harmlul. We know a case when a beam 01 channel secUon

Re¡;islaJII;¡: Utuk, CQm.pO/md l.ooding

[1'(lfl VII

consisting 01 S pIste. and two angles (Fig. 304(a», ......o rking under s lolld acting in plane xOz was reinlorced by welding lo il an e.:drll angle (Fig.304(b)). This resutted in of {he principal axes from Ihe plane 01 londlng and gave rise lo de-Iermallan in Ihe lale-ral cliredion, which was comp!elely unforeseen. bample. Selecl Ihe secUon lor a wooden 18th 01 heighl h andll'ldlh b and dele-rmine the denection of its midd!epeint. AS5ume Ihal -} =2; Ihe lenglh 01 Ihe beaO! (dislance between fhe Iwo supporUng lrusses) is 1=4 m and the rool is inclined at 23" te lhe horironlal; Ihe load due lo lhe letll's weight and Ihe weight of snowon Ihe roo Ola)" beconsid· ered as unilorrnly dislribuled and ha.... ing the inlensity q=400 kgr1m. rhe lath is simply supporled. Permissíble stress is 100 kgl¡cm", and Ih(' modlllus of elasticily ls [=10j\\aximum bending momenl will OCCllr al lhe middle 01 Ihe span; ii will be '11' 41l1.1xJ6 M u ,n=g=-g-=800kgl.m As angle lJ' is ('qual lo Ihe angl" oí slant d lhe roof, Le. 25", H 101· lo\\'s ¡rom formula (20.7) and condilion -2 t11at blr1h'M(

h)

i

Coscp+¡-sinq> 00000

= 1402 cm'

wherefrom 12x 1402·....25.6::::::26 cm and b= 13 cm. Maximurn del1eclion 01 Ihe bearn occurs al the rniddle of Ihe span. Moments of inerlia of the 5ecllon are: bIP' 13x'!h' Ir" Iv =T2 ""---¡r= 19050cm', J. =12- ---¡z- ... 4760cm' The angle ol inclinalion, a, of the neutral axis can be determined as follows;

lana =tan

= tan 25" 1;/:...

1.865

wherelrom a=--6IoSO', and lhe angle made by lhe plane of bendlng wilh Ihe plane of loodlng is:

a-tt = 61"50' _25" .... 36°50' The dellection in Ihe ptane 01 maximum rigidity is: , _6qI'coso¡>=5X4X4'XO.905XIO' -064 • 3::s4xJ!}ü59xlí)( - . cm

'"

Tolal deilection ls:

¡_.1L._ , Le. the neulral axis will be located on (he other side 01 thecentre of gravity ¡han poinl A (fig. 312). The neutr.l uis divida lhe secUon inlo two parts, IInd siretdied. In Fig. Jl2 lhe slretch!d parl has been sh.ded. Drt"'ing lo the conlour of lhe sec:tion, IlIngenls parallel lo neulral axis, "':1' oblain n:o polnts D, and D. whkh m subjecttd to the rtlallimum compressive and tensile slresses. ¡\\easuring the t'OOrdinales!l and l al lhrse points and Ihem in formula 121.4), Wf' eakulatc Ihe maximum stresses al poinls D, 3nd Do by the formula

a

-

P (\

S

+1 ; - +---¡;Z":(I.I) )

(21.9)

1f Iht bar's malerial has equal resistance to lensLn and compress.ion, then Ihe slren¡lh condilion mil)" be \\'rilten as

..

p(

zf,' ) ,(a)

(21.10)

'"

CI¡. 2/J

For sections wilh comers in which both principal axes oF inertia are also Ihe axes oF s)'mmelry (rectangle. [·beam, etc.), Y'=Yn... and z,= =zm..' Therefore formula (21.10) may be simplified and wriiten as lollows:

IOmul=P(:r+

+::J

[a]

(21.11)

H lhe material oF lhe bar has unequal resistance lo tension amJ compression, lhen Hs slrenglh must be checked in lhe slrelched as well as compressed zone. However, in sorne cases one check may sulnce for Ihese malerials a15O. lt is eviden! from lormulas (21.7) and (21.8) Ihal Ihe localion 01 poinl A 01 appliclllion oF Force and thalof Ihe neulral axis ar{' inlerrelaled; lhe ncarer poinl A is to Ihe cenlre 01 gravily lhe smaller Ihe coordinates YP and Zp and Ihe grealer Ihe segments 0v and O,. Thus, as polnl A approaches Ihe cenlre 01 gravily 01 Ihe secUon, the neutral fIlOves away ¡rom il, and vice nrS3. Therefore, in certain positions of poinl A the neulral axis \ViII pass outside Ihe secl ion and Ihe whole secUon will experhmce eilhe.r tensile or compressive slress. Ob· viously, in such cases il is always sufncient lo check the slrength oF Ihe malerial at poinl DI only. fij!. 313 Let us analyz.e a case of practica] imporlance, when a bar of recian· guiar secUon (Fig. 313) is eecenlrically loaded by force P al point A on lhe principal axis Dy. The eccenlricily DA is l.'qu(0) and (e). Lel os lirst sludy lhe case when enly normal appear in the bar ¡«Iion. 1I can be mily seen Ihat ¡his is a particular case el como pound loading-lension or compressian wllh pure bending In 1\\011 princlpal inerlja planes paSlling !hrough lhe cenlroid.

§ 128. Determination of NOfmal Slresses Lel us assume Iha! Ihe lorctS acting on lhe removed parl 01 Ihe bar can be reduce(b». Surnming up lhese component.s of normal stress \\'e rel the lollowing expressiDn far calculaling ¡he lolal normal stress al poinl A: ,)

,

J,

(23.1)

CA, DI

'"

For ca1culating total normal stres.s at any olher point of ¡he bar's cross :s«lion, il is suflicienl lo subslilule in formula (23.1) the and M, and coordlnates !I and 2 wilb lhe proper values of N, signs; IbIs ¡Ives us tbe total normal stress witb lhe proper signo It is obvious from formula (23.1) Ihat lhe normal stresses are linear lunctions 01 coordinales II and 2: lbey mus! attain maximum al lhose points of tbe section which are lartbut Irom tbe neutral axis (at lbe

M,.

'f

'

,.

,

r,)

(1)

Fi,. 330

neulral axis tlle normal slresses lre uro). Figure 331 (a) depicts a 5«. tion ollbe bar; in all lbe quadrants lhesigns 01 normal stresses, (t) (J', (2) (J" and (3) are shown in Ibe assumplion Ihat N, M, and M. are positive. It is ob\'k!us Ihat lbe neutral axis will intersed Ibe qUlId· rants witb normal slresses of diflerent signs and in the given case will nol pass Ihrough Ihe centre of gravlty and the top leH quadran!. Assuming that in formula (23.1) stress (J is eqllal to lera and dcnoting we get Ihe the coordinates of a poin! on lhe neutral nis by !In and lollo\\,jng equalioo of the neutral axis:

.

N M, M 11.",,0 5+7'.-7 '

Equating lo UfO first 2. and then /l., we find Ihe intercepls cul by lhe neutral axis on lhe axes of !I and '. respectively (Fig. 331 (&)): NI, a,_ SM,

,"d

NI a,=--iit:

•

(23.2)

As lhe presente or absence of factor N/S in.formula (23.1) does not all'ecl lhe Inclination 01 Ihe neulral axis wlth respect lo lbe coordinate axes, tM tnclinalian may be delermined [rom lhe equation

M.

M

7; 1.-7';11. =

O

IPOfl VII

wherelrom 11 ensues that

j

By ion mil in planes

'J. M, lanu'""--r"""" ,. • "'r

(23.3)

summing the moments M, and M, ading and xy, we obtain the resultant bending mamen!

(23.4) Angle

bebA."ffn ¡he plane in u'hkh MI 8ch and Ihe ver! ical piune

n may be lound (rom Ihe eJlpressioo tan

•

(23.5)

This expression tnables liS to wrile formula (23.3) in lhe follO\l,jng for",:

(236)

Angles lX and 'JI wlll be considerellIn !ji and a:>lf, Le. Ihe centre aS gravity geb displaced in a diredion whlch is indined lo lhe pllne o( acUon of bending momenl M_ and lends towards Ihe I/·axis. It can be easily notked Ihal Ihe eenlre of gravily always deOecu lrom Ihe plal\(' or lhe resullan! bending moment lowuds Ihe axis lIboul'whic.h !he momelu 01 inerlia is maximum. lt lollows Irom the aboye that lhe delltcle()n aecount of rela_ tive rolalion of these sedions. Lel us delermine the normal stresFlg.3-lli ses in these sections al points A, and A, which lie al a distancez {rolO their respective neutral BXes. We seled the posilive direction 01 the z·axis towards the ouler fibres. Fibre A,A, elongates by A,D.; Ihe corresponding stress is (1=tE

where t is the relative elongation 01 fibrf' A,A,. lt is equal to lhe ratio of absolule elongation AtO. to the ini/ial length 01 the libre A,A,;

e= A,D,

A,A,

Denoting the radius of curvature 01 libre A ,A, by p, we oblain

A,O.=z6dq¡, e=.!.6d'f p

,

A,A,=pdq¡ P d'J'

(24.7)

Formula (24.7) gives the dislribulion of normal slresses due fo bending momenl Mover Ihe heighl of the seclion. As and E are oonstants for each seclion, (J depends only upon the z-(:oordlnale and the radius of curvature 01 libre A,A, (p=r+z, where r is the radius of curvature of Ihe neulral1ayer). For a straight beam we had oblained a linear law 01 distribulion 01 oormal slresses: in a curved bar o varies according to a hyperbol ic law (Fig. 347). It is also evidenl from formula (24.7) that in libres whlch are on the outside w.r.t. {he neulral layer the increase of stresses

Or.

241

subleml.etl at Ihe cenlr(!. It is clear rrom the diagram that p ... R.+{sinq¡,

dA=b¡1¡>

cutl

,

Bol b.=dr051p llnd

dj:!=2'"COS9dtp

which implies lhat (24.9) may be written

The denominator in equation

Arter Integral ion we gel el'

.". S

..-:t(2RI-V4R: d')

-"12

Putting this yalue in equalion (24.9) and 5ubslHuting :td'/4 fOf A. Wt oblain 4'

r=

4

(24.20)

bR.-V 4R:- ....)

FO! a lrapezoid (Fig. 351) we agaln use equalion (24.9). lhe area 01 the Irapezoid is

The width of lhe Irapezoid al a dislal1C(! p from {he centre of curvalure b

b(pl-b,+(b.-b,){l-RP , The integral

f':

,- o

dA_b(p)dp

has lhe iollowing value (dropping lhe inlermediale

operalions):

id: -( b1+ R¡ Now irom equation (24.9) r=

v,'e

In

ha)

gel

l.)

• R, ( H R' - , - In'l;_(6'._b,>

(24.21)

When b,=-b" l.e. when lhe frapez.old beoome I rectangle. lbe above formula becomes ldenlical to formula (24.16).

,,.

IP C2U5), ror poinl 1, 1, ami R, libres) {or poml 2, - l t ami R. llnsidc libres)

in

o. 141

Ol/Vtd Safl

." Table 11

t.«at11l1 ¡he /rfeulfll Llytf Irolll billa

I

,.•• ].2

t."¡

I.G

,..... '.8

'.0

0.229 O.lr..s 11.128 0.102

0.116

0.000 0.082

0.«0 0.052

r

10.0

O.Ola

0._ 0._

0._

'.0010

'.-1

•

w. . '.

,,.... ,.,

L8 2.0 .

2.8 3.0 3.5 •.0 6.0 '.0

10.0

0.418

O.'"

0.285

fl.l83 0.149 0.125 0.106 0.091

O.IGO

O.'"

O."'" 0._ O.,.,

0.0rable wall thkkness as compared lo Iheir rlldius is sure to result in large {'rrors. lhe 01 weh cyllnders was worked out by .....o rldwide G. Lamé and A. V. Gallolln in 1852·4. lhe latlc:>r lame lhanks lo his lVorks on una!ysis li curvel1l, L.. ...b

457 CHArTflt l '

Oesign for Loads. Olesign for Limlllng States § 147. Drsign lor Pmniuible Loads. Appliu.lion lo SlllIcIUy Determinlte Systcms

In Ihe methods c.xplained aboYe designing tensioo or compression s1atically delerminate as well as ¡ndeterminate we proceeded from lhe lundamen!al slrenglh condition (§§ 4 and 18). to this condition, lhe dimensions ol lhe slruclure should ensure that lhe maximum in tht' critical ,celion dOl'$ no! tlreeed !he pCrlnissiblc value. "c Let us view Ihe problem from a diffcrffil Q,. Tht material d al lhe lhree rods is lully emplo}'ed at the load q:. Hence. the melhod 01 permwible loads hdps us lo discovcr lhe l.· lent sources 01 reducll18 the salety fllC10l of st.tically slructures. increasing their design Iiftingcapadly and achieving great· ('f uniformily oi strenglh of .11 their parts. Wilhout any diFficulty Ihe atlsNedional .reas the mel!lod can be applied to the case .... oF lhe mlddJe ami slde rods are not equal. The Ihl'{Jrelical consideratlons lliscussed .bove were uperimenlally verilied a number of times, and Ihe calculatc.d and uperimelltal v¡¡lues ollhe ultimate load were found lo be in good lIgreement with each oth· ('(. lhis assurcs tha! lhe Iheoretical premises on whlch the method al permlssible loods is wrrect.

.re

§ 1"9. Dr:terminalion 01 limitlng UfUng Capacity 01 I Twlsled R;od lile method al desi¡ning fOl" permiSliible loads may also be applied lo lorsion. As aIread)' exp!ained in § 148 Ihe result obl.inl'd by this melhod in tension J.nd compression dilJlfi ftOI:l Ihe one obtained by designing lor perrnissible stresses only rOl' a slatically indelerminale s)'stem oi bars, because lhe slresses are dislribuled uniformly over lhe cross sections of each har. lile situalion is dilJerffit in lorsion: the slresses are oot dislributed lIniformly over Ihe cross sa:tions. In § 49 we delermlned lhe required dimensions oi a twisll.'d shQft from lhe condition tha! (he maxinlUm shearing stress al points on Ihe con tour 01 the cross section should not e)(ceed Ihe permissible shl'aring analysis on (he hasis al permissihle slresslress [TI. \Ve conducl:, , 1

l'!,

T

In

i-¡i¡"i¡'i:,...,(1111'

/1

• ,

.".,

r..

1.1.

*1,

,

"

,

,

'I 1.\Fl

,-,-;...--;...: 1 Ir

1

I

,,

1

•

,

Fil1.. 371

Lel coosidrr a I....n-span ronlinuous beam or unifNm ' hiTge5. n, h(f,\'ever. lhe bar bends in lhe pl.line ol llJies, Ihet1 Ihe ends should be considered rilidly fixed (wilh the reserva· tions. discussed below lor rigidly fixed ends). In slruetures U"e oflen lind compresstd bars, ...:hich are riveled Of" welded al the t'f1ds lo other elemenl5 quite oftt'n wilh ¡he help al cover plales. Stx:h a cornlrainl cartl1Ot, however. be considered rigid. !he elemenls lo v!hich Ihe compressed bars are- secured are nol absolutely ri'lid. IncidentaJl}', a sUght rolalion al Ihe liJed md in the fLl3lion is enough lo render it more c10se lo a hinged con• PtOfrtdü1ts o{ /JI(I Confmna Rl3Slan).

af R,,¡t;w, EflfÍ1U't'T,

St. Pttenbur¡, 1892 (In

o.

271

e

,••

"

•

.;

•".. e o

o

" •• • • .,• <

,,

i¡ o

,

" 1

•

•j

<

"••

e

"

•

•

o

eo

8

• •e

" •, e

j

O

:; l ;S ...

;

;

."

Slllbilif" 01 Btm Udtr CDmplUJiOll

.

-,'1,:- • ,

:

•I

.,- '1. t's formula conlirms Ihat the formula ¡ives proper results only wilhin cero tain limlts. For example, ir v.'C ulculale the critical slress according lo formula (27.12) rOl" a sleel bar (E-2x 10" kgllclll:'j al IIcxibilHy ;'.-50, 1""e obtllin ,.'/: 3·14'x2xlO" 8000 k I1 • O-V"" 5@ g cm

lhis is almosl Iwice Ihe uUimate strength of sleel; Ihl' bar \VIII ccase to \\'Otk even belare the critical stress is achie\·ed. We Ihus set Ihat ter 10\\' lIexibility baTS Euler's IOfmula ¡ives exaggeralt."d of critica! slresscs and fOtt'e$. Whal are lhe reasons roc ihis? Figure 390 shov.s lhe relalion belweetl and l. lhe curve is 2 hypt'toolll. whkh is known as -Eul"'s hypubola-, While using this curvt. it shouht be ktpl In mina lormula (27.12), which il reprcsenls• ..,,·as oblain a'!.'\.IJnlfllion lhnl Ihe ),,1....... 100 (on lhe righl). hori1.Olltal Hne lor AIii;40 when (00 the leH) and an inclilU'd

...

o,

Slabl/lfll 01

slraight Une (27,20) when

SI"'e/u.es

IParl VII/

(joining poinls

f/l

and

nI,

In

otder to avoid sharp inilections in ¡he (fe versus), curve al poinls m Ilnd n. "''e may use ¡he empirlcal formula due lo Johnson which recomo mends a parabo1ic vadalion 01 critical stress in lhe inelastic wne O'c=a,,-cxl.' (27.20')

"="11,"'

Here, when :1.=0 then (1 [a]

a,

The .section does not salisfy lhe slabilily condilion. Thrrel

d

.\1,4•

Ket'ping in rnind Ihat Mt"""GJ, ili'"

and

d far a rectangular ,9l'(:lion as l¡h is inecease:1p, Ihen ¡he formulas dl'f'ived ahove give {'xaggerated values of lhe l'ritlcal as Euler's formula gh'es overslated '"alues lor compressed baN 01 ow DexibilHy. Dala collccte0,,_

Lel us consider lhe rollowing example. A simply supported I·beam No. 60 l.see Appendix) witb a span 1-6 m is loaded by a unilormly

".

SÚlbJl//1I

of ElunmlJ of Slruct..rn

IP..rl VIII

disttibuted force of inlensity q=-9 U/m. Check Ihe strength and sta· bilHy ol plane 01 lhe beam ir the permissible stress 101_ -1600 kgf/cm' and lile safely factor ro.. slrenglh and stability aTe both to 11-1.7. The dimensions of Ihe sedion (Fig. 401) and ils g«¡metrical charaeterisUcs, as obUinM from the sp«"ilicalioos. 1ft': h-60 cm, b= 19 cm. 6,=1.2cm, 6.=1=17.8 mlD",=,I.8cm, h l -60--2x 1.8-56'" cm, 1.-76 S(X) cm', J,-1725011', U7 -2560 cm'. Tlle torsionar momen! 01 ine!"tia has been calculalt'd from formula (9.38'): J wherC' '1-1.2 for an ¡·bcam. 5ubstituting ¡he numerkal values. we obtain ¡

-

1

-, h,

- 1&

'qualion lhen

i

Here " =10-1 is lhe main permissibJe stress undcr The strength condilion may be writtcn as lolJows: I

p

,-P,

(28.22)

lhe elfed 01 axial fornl P,.

l

e

P

(28.23)

P, We no\\' impose a reslriction on defledion by writing lhe rigidity condilion:

1m..

1- f

'"

p

(28.24) <

where ¡II is Ihe pernüssíbJe deflecllon and k f is ¡ht' safety lactor against defleclíon. Bcsídes checking the strenl':th and rigidity of lhe bar in lhe plane 01 bcnding il is necessary to check i1s stalJilily in plane 01 mini·

517

Ch. 1.51

mum rigitlily when it is subjected onl}' lo cempressive ¡orces P (§ 158) ;lnd al$o check lhe stabilily lor plane surillcc in bending (§ 161). Lf'1 U$ oon.sider onf' more f'Jtample. that a simply supported beam is subjected (o bending br a P. acling al the nliddle ol .Is span and is compressed by an axiallorce P. From formulas 128.(7) and (:18.18) the deílecUOI1 al the middle oJ span is;

f-f¡;,

f_ PtP

or

I 4m7"""P -1';

(28.25)

The maxhnum absolule normal str(';S5eS in the crilical seclion are:

(28.2G)

12EJ

II'''J

p ..

--¡z .. ii]'2'f¡i = [¡¡rz \Ve obtain (28.27) Aftcr Iransl'ormations,

Wl'

get

_.!:.+p.lP,

".,,_ S

4lf'

O.17SP

JS.. P

(28.28)

p +"·C a·"=5 lF '

H«e

e.

is slightly les than C. Foc example, wben P"""O.5P.., C=

"'p:.!:.p_2 ..,.-nile C,-I.822. Calculalions

lhe same as in the

example.

§ 1&3. Effed DI' Eccentric Compressive Force and Initlal Curvalure 01 Bar

A. In case 01 eccmlric application the axial compressive force lealb lo ecctnlric (."Omprc:ssion. which, as shown ea.rlier (Chapler 21) results in axial compression and bending (§ 162)- By using lite resuHs ebtained in § 162 ""'e can ta.ke inlo accounl lhe elIecl ei initial eccen· lticity, of lhe axial IXJmpressive force P (Fig. 404 (o).

Stability 01

[Parl VIII

el S/fuclufa

'"According to formula (28.17) lhe maximum defieclion in lhis case. is f=!.C where

f.

¡s Ihe defledion due to bending caused by momen! M... Pe

and is equal lo (Chapler 15)

M,i'

PeI'

t· ... w"""W and coeHicienl e ¡hal accounls lar the e!red 01 axial lorces on tienedium vl'locll)' v.. may be rOl.md ¡rolO lheorem: Q

O+aQ. v",

(30.29)

Here Q and are Ihe weight and mass 01 Ihe striking body, Q. and .'11. are Iheweigbl and massollhe body subjecled lo Impact (beam), and I"J. is Ihe 1I1ass reduction coerliciellt (Iess Ihan unily) which has to be inlroduced to accOl.llll for Ihe lacl Ihal nol aH parls of Ihe bod)' su!It'ring Impad move alter impact wHh {ht' same velocily, v... (see, lar example, Fig. 422). For lenslon llnd compre.ssion I"J.""+: ir Ihe beam is slIbjected lo bending as shown in Fig. 422{a), th" >'" >" >" >'0 >8 >8

Nktel Slcel: e 0.20. NI 3.0

...58

28·'"

7"

>7

7" >'" >" >" >• >. >3

"

24,18

>'" 20.18

makt 11 ciar lhal 11 reductlon in lemperature ma)' cause a sharp reduclion in impad strenglh and Ihus brlttle fracture 01 parls oi struclure. Tbis phenomenon has been oflen observed in practice; eteltlents used in railway cold brlttleness al rans, rims and ¡ransport has 0111'" been Ihe 01 11 number 01 problems. A very importanl point in this contexl is Ihat For quile a few ma· terlals (Flg. 431. curve al lhe transHion Irom ptaslk failure, havlne high impact slrength, lo brittle Failure takes plac:e In a smalllemperature intervalo For Instance. I malerlal havlng iood impact strength at room or nearly room temperalure may experlence britth: !aHure even w1th a smafl reductlon in lemperature. Then>fore, resullS oIlhe usual impact tests It room lemperalurt -, The rt'duetion may be explained by the fael Ihat In tenslon and compression all seetions of (he sped men experience equal stresses, and in bendlng mulmum stresses occur only al lhe oull'r fibres (Ihe rernalning malerial remains underloaded and thus somewhal Impedes Ihl' l.'rnergence of fatigue eracks): besides, Ihere is alw8Ys bound lo be sanie l'Ccentricity in Ihe Ilppliclltlon 01 axial Joalls, Plnally. the torsional endurance IIm\t llnder a syrnmelrielll eycle oompnse5 0.55 01 the bending endurance limit. Thus. under a s)'mml!tfleal cycle we ¡:d the lollowlng values lor sieel: _ 0.400.

0'_, ... O

)

_0.280. = 0.220.

(31 3}

ThesE.' relat inns can be lor obtllining formulas for the slrength check. In the case of nonk'rrous ml'tals I'."e gel a more lIexlble relallon betwN.·n endurance 11mil ami ultimale strength; the empirical for• lbld.

'"

Ch. .tll mula b

a!., "'" (O 24-0.50) o.

While usin¡ (31.31, tl VIollld be borne in mind Ihal !he endllrance Ilmit of 11 material dt'penlh upon a largl: nllmbef o, larlors I§ Ih", rel.lions given In I:Jlprl"UIOO (31.3) ....ere oblair\ed on :spcdrnens 01 s,",1I diarnetcr (7-10 mm) having 11 pohIDed surlJM.:t' and no sharp changcs of 5hape .Iont lhe lenglh

§ 185. Endurance Umlt In an UMymmclrlcal Cyde The equlpmenl required loc delermimng l:Odurance slrtnRth Ilnder IIn un:symnltlrical cycle b much man oomplicall.'d lhan 1M t>qUlP' menl llSe'd lar symmelrical cycle.. A spcclal spring capable 01 strelching Ilnd compress1ng Ihe :spt:elmm Mlould be added to lhe simple Ie.ting discUSkd e¡;rJier, in which the spedrnen only rolales. Quite oHm v.re have lo t"mpJo)" t."Vel1 more complicaled machines, which are c:apable el ulal load on Ihe specimen (h:nsion, compression) under difTerent extn:lIlt- values of Ihe variable slres.ses. HOlI/eVd, lile now have al our dlsposal sulficlenl exp;.>rllllt"nral dat:; to obtaln I graphicel or analylical rellllon belween Ihe endllrllncl' limil and

r,

0"',,

L... t IlS remind Ihe reader abaIJl the nalallons used here: Pv renrl:' S'.'llts (lit ulJlmlte strength of the material, the yleld p, lhe endurance limil correspondlng lo a cyde 01 ch3!aclenstl(' r, p_, Ihl' endurance limil in a syrnmetrical cycle, P.... aod P..,. Ihe IlPpL'f Ilnd 10lller exlreme \'aIIJes 01 lhe tyde, P.. -P",•.;P'''' the mean stress in lhe cyele,

P.=O...,.P,.l.

the amplitude 01 osctlla

tion! ol Ihe c)'Cle, 2p. the double IImplilude of the C)'cle. aod "p,,;./p_ (he charllderistic of Ihe cyele. The value:s al P_., P.l..., P. IInd P. which rorrespond lo

r_

01 the matl!rial at lhe endurance limil will be d{'OOled by a subsc!lpl r; P, ....

p, .-

P••

Uh.. oluimum abSO!ute valued Pr ... or p{Il!' musl coincide wilh p,l. Tite re:sull.s 01 upedments ror endUrJOL'l: 1Ir.l11 under dtllerent cyeJes ..re coO\'enierltly represenled 10 Ihe larm of dlagrlllll§. The ,Implest InlOng these diagrlllll'> is Ihe diagram in lhe P.- and II.-coorctinates IHay's dlagram) shown ;n F'1t "36. On Ihis diagram

Ihe "alul'S al P.. are laid off on lhe x-axis lO a cerlalO seale llnd Ihe "afues of P. are laid 01( 00 Ihe y-axis in lhe .sIHIo! scale. Curve AOCO has b«n plotlC'd on lhe ba:sis 01 experlmenls ror dl'll'rJmning the endurance limil under dUJerent c)'des 01 variable slresses. Foc deld"-

Dynam/c Adl"", of Fora.s

IPMI IX

rnining wllh Ihe help 01 lhis diagram Ihe endurance limil p, far a cycle having coeJfklent 01 asymmetry f lile draw Irom the cenlre of coordlnales O a straight llne OS al 3n angle P. so that (see (31.1) ami 131.3))

lan/l=&= o,.

(31.4)

1-'

1+'

and it until il inlersecls curve ABCD al polnl C; lrom this poinl IVe lhen drop a normal CE on lhe absdssa. The sum of segmenls CE and EO, whieh are resptdively equal lo pe(; and PmC. gives Ihe enduranee limil (31.5)

Thns. polnl A havlng ordinale OA =Po=P_f and absclssa Pm=O represents lhe endurante limit under a symmetricsl cyde: r--l, rJ,.lgf!u/

" "

V

-"

•

"''l/lt t

"-

/f--':' .'

-<

2IJ ·,9os'!'!/iII!! Fig. 437

p=n/2, whiJe potn! D having ordinale Pa=O Bnd abscissa OD=p,.= =P ...

represents the enduranee limll under stalie loadlng (r=+ 1,

p=O), which is equal to lhe ultlmate slrength of the material. 11 we

draw a slraighl Jine

-

or al 4S ine]jnatioo lhrough lhe cenlre of coor.

•" 6IJ

-b/J

lO Fig. 438

O

+

'.

dinales, poinl 8, where lhis line inlerseds eurve ABCD, represenls lhe enduranee limil under 6uelualing (zero base) load, b('C3Use oro dinale BF=Pas of poinl 8 Is equal lo abscissa Of=Pms. Le. from (31.1) and (31.2) l+r= l-r, and r=O.

'"

437 c1epicls H3y'S diagram lor comrnOl1 10000I, In.1 Fig. 438 shows the ume dlltgram lor ¡rey iron h.vlng tile follOllo'Illl properties: (1...... 78 kgl/mm', 0.,=22 kgi/mm', 0.,_7.3 kgf/mm", and 11 ••-46 kgf'mm" (o•• is lhe enduTilnce Ilnlll under cyde in compressm). This dillgram proves lila! undel' variable loadir.g loo casi iron has a mcch strenglh in conlpresslon lo Ihat in Imsion. lile endurance tim.l should looked upon as a critical stress sImilar lo ullimale under slalic loading, because a !1«'SS

sllRhll)' greater Ihan lhe enduro

i1fl1:e slrenglh can cause ISI¡ure

wilhin a practicall)' feasible numo ber 01 varialions of the load. curve ABCD in Fill. 4311 represents a curve of eril·

k .. t (1Irnlting) stresses lor mate-

rials Iha\ do 001 have lO yield

,

tOllt. H the m,krial Is ductile amI liJe critical s1re$S ror iI un· der sl.tic loodillg is yield slress .1 p .Ihen il can bt' e3llily estab· Ifshtd IromFil!.439lha! lineAGH should be takM as lhe Ilne el crili· cal slresse5. In lhe eodun.nce 11m· il dlagram ABD (Figs. 436 and 439), iI we Cut eegmenLS OH anlJ ON equal to yield stre.u P. on lhe;r· and y-axis. respecllYl'ly. and jOtn poln!s N and H by a slraigbl1t1JE' NH, Ihen Ihe surn of lhl' absds.sa and ordinale Di any poin! on Ihis Hne is alwa)'$ equal to p. (IOT in· larpoinl K.OL+LK-p"'A·+P.II -OL+LH=p . as LK=LH). Hl'oce. slraight Hne NH determines the crilicallimH o, slresses under sllltic loading, while curve ABD dl."termille5 I! under variable loading. In the region in which slraight Ilne NH Hes aboye curve ABO (from Ihe axis of ordinales lo poiot G, the poinl 01 inters«t!on 01 these Jine.o.) the critical liml1 01 stresses is lJelermined by curve AG. Bul whffi' Jine NH I¡es btlow curve ABO. the crilical IirnU al stresses IS deterrnlned by Ihe lilll." NH. In Pigs. 436 and 439 Ihe crllical stress Jines hatched. On .ccounl of Ihe fael tha! usually \\'1." delermine experlmenlally Pr' P•• lInd P_. whill' !he endurance Iimits lor olher valul'S Di , are generally nol delermirK'd, straighl tines AD (Fig.436) Of AH (Fig. 439) have often lo lX" acttpted as Ihe critlcal slrtsS lioes in Ihe lIbsence Di experimental dala. The critical stress curve is somelimes replacf'd by a slraight Jine in creer lo simplify caJculations; [el us note thal soch an approximalion adds to ¡he existing safety fllClot. A fair[y largt" number 01 formulas .....ere proposed estllblishing the analytical between endurance Ilmit. ultimale slrength

, ,'"

o/ '" ami lile cys ami recesses in parls subjected to leosion or bl'nding.

IPul IX

The lollowing melhods ate emplo)'t'd lar deitrrninin¡ Ihe coeIficiffil of stress concenlralion. In I number of cases (roc example, in tension antl bltndlng of bars wilh holes and necks) iI is possible lo c!eletnline lhe local stresses by lhe thoory 01 elullclly. popular metho. as a rulco, about :JlOO stress c)Ocles per EXpl"rimenlal sludie; >hoIl llul nrialion 01 lhe 01 cydes behH'fll 500 ,,"d 10 (XX) does not have any "pprKilIble ellect on e1ldurantt slrenglh_ Thereforl', Ilhik' dynamK: designing p"rts subjecled lo variable 1000dmg lhe stress roe!'ficie11t KD should be ustd .... hnt lhe C)c1co frequenC) is less than 500 or grealer lh.1n 10 (XX) alld "Iso Il-hen lhe variable load is simultancou.sly an impacl load.

§ 188. Practical bamples 01 Failure Undtr Variable loading. Causes 01 EmugMel' and Dr"rlopmenl of Faligue Cr;KU Having eslablished a1l aspecls or railure unucor variable looding. lel us sllld} a rell- praClical cases or such railllre:s. 01"6 anll 441 )110", Ihe JI.,le or ¡l \\a¡;!OI1 eOlllpal1il'lI by 101'5iol1), in whieh failllre occurred dul' lo sharu chan!!l'

fmm a Ihick porlion lo a Ihin portion; insl{'ad oí a lllloolh Ih(' lransilion \las sharp, II.-ilh rough nolciml 011 lht- wrlacco lcoft ti", cut\tnJ( 1001. The hlligllC crack alll)fl'arl'd al tlM.' outl'r surfact' ami ,J¡'H'lopetl aloni! a rmJ( lhaped path, Thl' tllah'flal or lhl' a\lf' 'lOlILSfaclor): IhlS is bofllt' out by Ihe I',llreultl} stllall llrt';l oí lIIU'" llM'nlar)' rUlllurl'o Fii!ure 4018 lhO'A'S lhe ¡ractuTe or a notHolalmg ",ll(" Ilhich bemls in Ihe H'rllcal platICo The is shaH .... llh an "ppro,litllale

...

iP"r/ IX

ultimale strenglh of 50 kgf mm", The crack and developtd due lo sharp transition (al f¡ght angles) from a square shape lo a circular shape. 449 shO'A"'li lhe longitudinal secUon of lhe olher end of lhe faligue crillCks oo'elopin¡ Illmt. shaft. "..hich. has )'d 001

Fig 44A

lrom lhe ouler libres IO\l!ards !hc interior are cltarly visible in lhe region 01 loharp \ramilion. file rolle 01 faligue cracles and the zone of ultimate failure are bolh cJearl)' visible in Fig. 448. Par altenlioo to Ihe series of curvtd slrips and lines on Ihe surface of lhe faligue cracks. These are lile

Fig. 1.50

¡,acts oi gradual dM'dopmenl el lhe cracks; Ihe lailure orcUT:!i approximall'lr along lhe OOTmal lo lhese linu. Hrnce, by stud)'ing 1'lirll"S. \\:e can aJways poinl oul Ihe origin of Ihe crack; as a Tull'. Ihis i5 Ihe poin! where lhe source of stress concenlration is tIlOSt e!fI'Clive,

Slrcnglh Clwk for Variable l.IXlding

Ch. 3/1

59'

resulting In Ihe fatigue crack. The develollment 01 Ihe can be explaincd by Ihe lad lhal high local slresSlormation: tension. rompression. lorsion and bt'nding. 1111' question that now arises is: How to apply mtolhods in of oompound stressed stale-? FroUl a practical poml 01 view lhe masl impo:lant situalion is thal 01 rornhinfd bendiog and lorsion. As explained earlier in § 125. slrtngth rlle tepresenlcd by a

COlllloon equalioll

(31.23) 5ince the latigue crack is rallSf.'d by lhe same physiral pr0cess6 ol l!eformalion of lhe material \\ilith result in lallure und('f sllltic loading, equation (31.23) nlal' also be. empfoyed for clWcking Ihe strength 01 materials under variable loading. Stresses a and' 1 may be broken inlu the components 1)'"" ami T",.

Here 101 and Irl represent lhe permissible slresst$ under bendinR IInd ¡orsion. ami h:1. res.p«:tivell', oblained lr"OUl Ihe sirnplilietl Ip.I./p.1 diagram (Fig. 455) by laking inlo ronsideration lhe stres,s concenlrallon roefñcienl 01 lne plIrUcular Iype d dtionnation and lhe cyde rharacleri5Ilt.',

or (I. .

I..

",2

Dgnam/c ilclilJJl of

[Pon IX

§ 191. Practlcal Measures tor Preventing Fatigue Fallure

Tlle resulls derived in {he preS , A accompaoying nller-eifecl in the firsl nnd second stages has not yel !let.·lJ slll(lied io sullkient deLails. There sti!1 very fe\\' good experimental sct-ups for studyio,l! aftcr-{'iTect, antl this rnakes il dilfleult lo compare Ihe resulls of experin1t'nls 00 aflerefree! wílh lhose on creep. lo majority 01 Ihe machines which have beeo used unlil now lor invesligaliog aiter-eifecl it has been impossible lo achieve pure aller-eifecl. "L --" lt is generally assumed thal growlh or p!astic deformalion in Flg. 464 after-efrect is similar lo ils growlh in creep and therefore Ihe rale 01 alter· eifecl may be calculated Irom lhe creep velocily. fr Ihis assurnplion were Irue, there would be no need for sludying ailer·effecl.separalely_ ¡'¡owever, lhere is anollll.'r view which holds fhat creep rate caonot be taken as Ihe Tale 01 after.elfecl, because Ihese processe.s 3re eally differ('n!. the mechanism 01 origln and growfh of prasUc delormation in alter-elfecl is somewhal differen! from that in crC(!p. In aiter·effecl {he reduction of stressc.' in lhe eremen! eaused by delormalion, the growlh of plastie deformaUon allhe rosl 01 and lhe lenglh 01 Ihe eremenl remalns const:l1lt. In the gro\\'lh 01 plasLie deforrnalion is exc1usively due lo e-Iongalion 01 Ihe elemen\. The tolal deformation in rreep i5 grea!er than in afterelft' Iirsl ar curve of fig. 463 whm creep defOl"malion no! appear al allor disappears soon afler lhe part is loadl'd. Howt'ver. lhe- corresponding slresscs a, equal lo lile creep ¡imil llnd slrC!Ms a. 3rt' USu¡tI)' SO small in Ihat il lhey wert' lo bi.> acceplcd as lhe upper strl'ss limil, Ihis \\'ould lt'ad lo an unjustilied incrt>ase in Ihe dinlt'n· '23 \'.1 .... 01

U••, 1=01 ......'

Annc.Jlilló': 844"C

AnnnUlIt; tl«"c

Annullne &14"C

J\nne.liru: 8ol4"C

Annulln¡: 8·U"C Hardt-nint 1093"C

Hlldl'rlll1t Norm.1I1l1110ll 8.'iO"C

.'>

.'"'", ....'", m

'" '" '", '" ''''

..

'427"

'"'"..,

.."".. '" '" "6

In '·,",nf.

"'l .... , . . .

4.

¡q1 ... ')

• 110-251)

....,."

211)-g)

6.0 3.'

l410-mO

"".""

:11-90 1(lf,O. 1&90 1W).1llO 910_l4l0 :;('o.HIGO 210.420 00-120 970014(0 28Q....'O

3.IU

2." , .7

,.'" 4.r.o

3."

3.10 8.35

.. " r..90

140·280 1410.2110 ;no. 1060

6."

8M-I340 51;0.1000 350·8«) 110.2m

••• '.3 ,., '.7

ro.'"

...... «10-1500

1500-2500 :?ol).5(lO

3.25

3." 2.95

3.16

...

2.9

3.'

.(,... O.17XIO·" O.12xIO-l> O.2t.xl()-U O.2xIO- U O.I4Xlo-" 0.12xI0-1.2xI0-·' 0.6xI0- 1O O.23xlO-" 0.2>.0 419.0 516.0 667.0

....

IOC.O 1356.0 1725.0

K', 1.22'

>."

l. 70

>'88

""...... ,'.32... '.31 ,." .., ,.'"

........ . "' .. "'.. 114.0 131.0 143.0 163.0

,,

4l.SO

49.90 foO,1()

oo." 71,10

1KI.1O

101.00 123.00 151.00 182.00

2.12 2.01

2.27

2.03

,." '.19 2.69

'.59 3.ro

3.09 3.23 3.39 3."

_.

0Ianne1 5«liN .11.. SI",I..,

••

,.,..

•

•

Flan,"

_.,..

,.., •••••• ..•••, ...

,..,

_ c . l l ..

....

".," .. " • " ". ,., ' " "" ... ,". ••• .., . " ". ,. ,". ,,., ,". ,.,,... "" ..,....,, ,., ..,

7.0

27,

10.2

26.70 211.80

JO.O

30.00

"..,

ll.O

35.20 40.:-.0

27.70 :'ll.80

".SO

".SO

GI.SO

".30

00

10

14

140

'.0

20 20,

" " ". 27 30 33

m m

.. '" '" 270

".. '"... 330

."

U

7.' 7.' 7.'

5.90

,..,

7.05

10 . .0

'.7

•••

17.00 18.10

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