Lehigh UniversityLehigh Preserve

Fritz Laboratory Reports Civil and Environmental Engineering

1948

Stresses in reinforcing rings of a reinforced cylinderdue to internal hydrostatic pressure, 1948K. E. Knudsen

Follow this and additional works at: http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports

This Technical Report is brought to you for free and open access by the Civil and Environmental Engineering at Lehigh Preserve. It has been acceptedfor inclusion in Fritz Laboratory Reports by an authorized administrator of Lehigh Preserve. For more information, please contactpreserve@lehigh.edu.

Recommended CitationKnudsen, K. E., "Stresses in reinforcing rings of a reinforced cylinder due to internal hydrostatic pressure, 1948" (1948). FritzLaboratory Reports. Paper 1260.http://preserve.lehigh.edu/engr-civil-environmental-fritz-lab-reports/1260

,II

------------ - --------

--- -----------

J -,!

~--~-------------

--------- --- ----- ._----------

-------..---------

--'~---

-----------

-_._) -- ---------- ------

.,l". • " r

•

I II

I

1I III

1I.,

I

I Ii I I

II i "! 'I I

II II I I I

I.! I -I I II I

III ,

I!I

II

1~-'-~

..,- ,

I

Ii I

II

I!I

I II

II

I ' I

II I .

I

II

i II

!. I I ',1 I

III

I I

II

II

I

II• I I

III

. I,

I' I

I II

I II

I II

·1I

--------------

--1 -(

-----------£~~:.r~z~ ~ ~f~J ------------~--~~--~-------------=

------~--=--=-~ -r_~L CUJfj.~=--~_P_ ,,). e = ?__:_t:~ ~-~- .. ' . _

- [ e e -(>" ~----------J--------~-------- - --~~-~-- ---- - ------ ------------ ------------ - --- --- ---

----------- ---------------- ------------------(]-\-----. ':' ~

~ --:/,-h1------===:.....:..--------- 1 --- --- ------------------ I --'-------- -------

_ ~/ L __ ~ t:!-~:~:h~ -- . Ci,~Z: '~ . --'-----

1------.---ts:-~---r3.- C_~ ~ e ~~__

.0. ~ -= r: G.r -:. ~h1-------- ----------- -------------- -~----=--..fP-=-e.h£----------

_/l-Pv~__.~£\)' Izci-:A.:--=-4--J 3([-,,9 -:_A~_-_r_

L

_

I--- +[~_ ': _J-4<-L£h • jL_'(...:=..~ 'Z. _ _

_________-_~~~~J(~~~J;): - I. N4113-.r >ddt ----

1----------------- ------ -------------- ----'----

----------

-1..-

~ -

~ ~~ %cp( ~~)

'PilJ n.:: 4~ '0/ ~~x)

-~) "~ - ~ e(/?>.><)

-3-

1

"I I\L-o.£ 1. ~~ M P = f 'f(ra) Me = ~ r'Cfb)

~'ti~'Jl(;~ ~ern

Vp+ ::-± eeta,)I

VB 1:: - 2 e ( h)

mA.. C4MMJJ ~ ~I A- -::: yY)~ Me ~ 0

NAui - "t'f1JZ.~

~.fZtWL YA ~ 0 Ye ~o

G<fl-)

mAo 1:: - )

~~"C_ ~1((~Io)

-4-

- - - -L

-&- ~ Zk IP 0.<)

if ~ k 1fCr3aJ[e(f"")-f (f"')]

if c li '0/ ~i b) [e<r".) -j ~3>'- b)l3d '= 'f< Br-) ~",.)

.if C - Ek er b) e\~"b)

rJd&L JfG.~

CP) ( C li.[ fy~") + yyoafe~;"a) - j(.i"..J]

.... VCr~b)[e\ ;><0) -.) G ~b)J

+ e. e1~)eO'<:4)

L. ~C~ b) (~" J

- - -- - --

wfwt"1

t 6tT'\ - J

- -l'- l'.-

..-L

)(. )(.b-#-

.------------------------------

- s-

II ...

3- -

•7

- 4-

iN~d-t, tr I f~) -----,_f_"l

~ ?~i DUi I

-b - .

. ,

~-~it-f-.~ ~ ··teCf~-<)i-;0-~.JJ+La~.y ... ... +1I!~ctca~4H-~Jj -1- 2.a~-k\J-B-~~-J--

-~~~~fkJV-LW .+ ~ .. + ~~~-J=-M-f)J ... ZG(~~~_~,.·--

-l~-i2r:~~ ~(f-lli,~C~~YJ1.-1-3-0~ij--- 2e~'J :.----=----~-. ... + I¥~f)_ra~-Il~)~f-{t00~ + ~~-.tJ-ac_f11-~----'-

+ ¥(rJC-L] + 'Z. e<ft)-·J-

-

l v

(. _.

,t )

-

-

- --, .~ , -

..... J

~ ;

_~}-----.6'I-,~ t-/{J-~-01 .j~~l-=--3-'f-~JJ+ 2.g~ n1.J . . .. .. -1- 1(L~f-)-~-~P-'t}-=-5-0~tt~~j + ZQ.~e,)~'J~---=-----

-40j~,~f~~~e-0iJ4~ ~ 2 e~~~---

-7-

4- ¥~_p_,_) _e~~h-I-)~ -((6-n-~}- f_n-t--.£.'

5c~frJ~Fu~&9~~~-Cfol.JJ + 2G(~ ~~~~~). .

-

((3).... ...

. . ",,¥~n.j.[e.!ft>-0 -5(p--n-~_J~le _l'fLtJli(f-n.J-J_.

~~-~~4J~J~r#1fLJtft-fj----J?~-J <-. 2{?~--'!L~.

. .' ~n~)-bl .J + 2e~J' . ~I.

I

-

-

-

I-~ .' 6:1-~~I-~~~)~;t~~~~~k~J_.'~)Jl~~~ia~~

-~ b~ ~~tl ~ ¥ffm')f~~_mJ -~tf-m_'~i :f~~ltr ii'

-~ -

\' '.

I--------------~---_·

'- q-

-

~1Q)-i~4~~J-:f!t£J-~t m~J ~*_~L)J :~~:~m f~- .' .' '-!-~-,0-, fJ1-,;l-=j~fuJ~ eLCfL'~~

-

~~&4A-{ ~~PJC~{fJt~~~~tn*:t~1t~~), ' +'i/~~I1~- J1~J-t---- -t4} 1

, . ," .

s~" ~[I +~~f2~~1t~£

4~ + 2 G(A,el( J j

-to .• Ul: :',..' ~ .+2. ..

. -

-

-

:

-

-(tJ,-

I

-------->q" -= 64 +, ----.,--, _<2,-1.; c 6 z-, ~ee... 1f~J

c ,~~~J~-,(3-~t}

6'-?J ..... 0 3-1

°-1-4-= °4.'

tj I ~ -&, 6 '3-3-2-2-'iUL-ttM· -aN .8-2-B~ 03-Z.

dZ-4 'C c5~_20;-

2> 3"\- .~ o~~.

.

L~ r fflh o/~(7)-~(zzL? .JJ-t'/(-t.tAJ~.-f,. _ ~ -

.

. .

, .

-

.. - .. - ; .

~I(-

jp) do = d44 ~ I-k-[4- + sri?) + Ze'(F)]

~4) 5,~ ~ $21 " fk [ CP0 rn,) + 1f(f me) + Yrl)7f\ f nf} +Zerm) * ZerJ~3.~

~)) °/ 3 ~ °81 = ~ [ CPC(3IlJ') -> 1fI0 m,) + Y~).'If~ ~,) 4- L90 m,)" ZerfJe4"I>L)]

u) ~14 "0-4l = ~ [4J~) + 2.v(ff-) + 4e If-€j J

- t7) 622 "" ~k. [I + y'(~ rn,) '" 'if~0 rI.) + 2e. <Cf m,) 4. 26'~n,j J28)_ 0n" 632 = fr [~V) + rrrnJ'!{'(f m- + y~II\,)1(01J,) + ze(f"'.)rm,)

.... ZeqoO!JE)1n. f](z'f) 624 e 04t e IT [¢l f"')" 0/r1I1+)'!¥~o-{J + 1Yf0,) + Ze(3Jll.jS0tJ + Z~n ,)J

- -

ao) 633 C (k-[ I + '0/'f \'l1~) -I- <.( l (rYJ-..) .... Ze'/( m,) + '1 e'( t7.) ~1 .- tJ- b 04= 64-~' r-k.Ctp(f OZ) + 'tyrlll,) CY 0CJ + '1\"o/>n,) .... Z~ft>';1 tJ ;~rn~

-(2 -

l (~0 6 '0 c:,-$-Z-2~~-~-2-k

--

~L~~ s~ii~<ep~).

-Jf~ . . 624 =~4~Cq>-~ n,J + 1Kjp n0~ ;':~~'f. l'l,~)J ' '

~) Ol~~4-3~fk-C¢~-f-nS .-+ ~.J-:~:2-6-Cf [)~~ .

(2~4U /(JUlq M¢Cf~~f4-±~Y'"~) · G-0" (rCff6 +~ '+~ -~ 4- 2ifJJ~)(, . 7 /

= =t -~)( .e. ~f>< '~ 4-.~~~~, •-

qL~~CCJ!t) t'7 Jx ~ ..

- ,

-/3-

~.? --~,,' -= d4 LJ::: 1_ _ -'-- -,--- _

1:'--fS-~

-,-----,C19d2.2.~~7>-3-.-"<::~fr:...!.:k..:::....-------------~--

_--'~"='4~~---::-O!~-e-~l-·~-.1!: ~-~)'----------~-~--1-

;;_. ---z ~ - --I-

f) e 15 .¥:. @~-~ 1 I

~~ ~"\i- 4-3-k..:;-- -' ---!--',

-~ et/U/,U'~ ~/4 J&-j-CffW6bU~J'V~ (ie' - ,K' )-J-L",

.-

-

-

..'- -- ,

,.- .

/4-

JI ~O} 3z~ 6'3 0 ) J4cr"h j ) Z ( '3 , 4-

k) ) k~) k~ 4 u+ 4- J~',d, j fv 1- I ~

Uj~rl#WvL ~4t et 5 I

feu 1f eb-r-rnLS ~

~ .fa flu fr'-<"(JutaV ~)1J:,-<UJ ~;oa.ds .J flu

12/ ) R~ I R~ 4- R4- ftu

'fl i.U

$,,, -t- 1?, $IL + f<z6,; r 7?lOn t- f?+ 61 .~ _12!i

0'20 + 12 6qt -+ ~? J2.' + 7<i DZ ?; + R", 0t4 .:: - -Rz./k-z.

'0 ~l0.3L +- 'R~Jn ;- ~E3~ R;(.. 0.34- R~~

+ Co ~

'lo - ~

a40 -\- 'R\ a4t + 1<-2 J4~ + ~j64~ t- R'4 '-o~4 ::: - R4!k+

crv __R, (O,l-:Ie ~.) -4- «1. o(!. ~ .1(~ 013 -t- 1(4-0t4-R, () 12 -T Rz (~Z7.~t) T 1<.3 62.3 + R4 oZ4

t<-\ 0rs- - -\- R-z Ol.~ - '" -l<~ {15-33~ ~) + 4 0 ~4-R\b(4 + R~6z.'\- T -t<'j 51> + t(4(64-+-~)

1: _

'I'r:- -- "'0·

, I~' A + I·6. t-. - - 4=- ----'~-z-k~ -' ~k k~,.----------------,--------

/i1> = I, lq4113, C . _ Eh _ '--z.-cr-.CJOV-(7(Ti-O-.·-3-------'-------~

__...-.-:.- --=-"k.-__'- -r'Z- - \:'2::·~0~Z.-.-8-e - ~. r1: ....7~Z_------

.

.-L - .3 3 5_Ll o-~.. ~~dl_0--2f2e_·l~~__~~~K - 0~

. ~ \ ..e o 4l;</ 1'. to - 4 - .',

- 1-·--.-~ -

.'

~/A;k)~-

Q,~~ t::: ~ . ~;;-e-(l-v.\~l-

- l:)L4: 2. 0 . e.-C"l'lL. CD") p t') 't-

-~~~_.n~~ __.. Jd ~~_LL3 ~ :7. - 8. 35'1?' . ./' l ' \.

L~42.0~_~_-='<_'~),~4kcL ..:-:·.__~__._ .~ ___'\ F-. - I

3 4 . '~ 8 1 '35"Cf :- ~·e.47'~II, 3S"f:?

---02i~~O-4 -4 <1:• to

,0,000'3 tll.1I'-- - ,.

10'(7 3. 82* ''':' ..

-e •

- ~2-1--.i& leu·· ~~~Slj'_.j~~~'tZ~

..( edt 4- ~ '. ~g. - m.,-)-YYl~)-Y->-'-:.--i-f-)

.

-/~ - .

R .-;:::. O'z.o-~)

6~l +- ;-z.2-

- 5-30

_.{f~ ~e

~~t.

. 3

-f( =;j'4c

(53)-L1- .d4 4- + k:if-

--

~&d~ e5if,> {ff,ty£ ·~tL un~·n "

ja ~A'•. (~ ~ ,_.~ 01f.];m'7

-o . ~

,

SGttfH£- :

-/7 -

J V) My flo' 1 Y ~ ) 1'1 It ) n 6

@

•

- liB-

~.. u( f-1 1f

~ ~I • /I :

~~/ ~

~ =' (O!- -(" ( -~I"i)c -

~~ ~ t-~t·(-~C'>< -~I)C ')

~ - 13 e- -t,( (-~ -+ U t ,..)d.<. ~

­<-

- dQ.. ~ - f)L

..__ u,.£n) t~

-/q-

, 'C- , 4 _ (ls:\ ~

--:/:1:; - D ~_1c:"t- - - t 4 ~,*-J'~' ~

---- ------'----""'------------- ------

-(~~-,y -=- ".~ . ~'1-f < c £ -t-~J-(~ + ~}r'~r~rLt(-(-fbj-

--------- ----. ~ ~=--'~'-- = -.

,

&;~c.!ibJ_1--iuj 6W~ to .-lk -eAJ~dd ~

8- ~ 13 ...

~

:

-20-

(E6) H;- ~it ~ 1- 1fF) -Z(?>t j (rV-)1(O'i) M6T:-V:e[CP(~J-I] .

.(60 ) VFJ :. -:- 1t±[ JCt~L) - (>e if~ !.J.) ] ~

~) VB ~ ¥ i [j(( ~C) - (~ J

- ! t/.Uti _ ou<. /' /7; flu n a~ t-C/kdi a--~d

k - i(! ·,;{ If .t {o C' !J ~6Ue.n 6<

~ c/ /(~u1c ~7f(' <Me ) ~/f;-<X

A-UUt- ~ 4A 4tn-vU ui. /1'- .a4c )

ft1 HR· -= P~ D-4'{(.c) - Zt-C j P))

- {6~ - N6° ~ '~1!i LtjJe/lj -/1 ,_~LJj C(fr" +, ,i: [.}16C) / Cl( [4'C)I

- If£)- Q~-" - ¥, ±[J0=£) - c]

-z.( -

!1 ~ -' - -:!E-. :1-.,(f~J .. 13 - 6f~L__~ ---------,--------

C).

(fA) LX-/t) _. +"fr)'~'..

W-f~J 2(de. -r-<- -35:82<.. ~ 2X.J5"7S2~ e

I- 4 71.'2~ x g a;;tf-t.:l:-r- ~ .-

7/.~4~. 'e~{l 81:_=-~-e-I-3

,o·lo€i7 5"'4-85" . .. -

-

lU'ek -I-W··. tip (h. 0 ~. C6Pt) .. ~~. . . --' ~---- --.--~-- ~

~rH 7-~ ~u~ ~a~. ...

.(;J()) .11ft0 90

8r~ -

"-

&)(!) .~~

'fvL- 'C -t-J~/ -13

(/

(]zj GL~ - -;0 f.' .

i

\.-

03) (2a~ -t-.~

" /-~

~ ..... -

.

,-22-

-z~ -

::.. !

-~) -51-6 = /) - .---- ---,--:'---::---'------'---------

I - '\ "5': - 90 -. !!2!..I~"J '-°ir-f): - k ;€:------------'----

"I

(S,4' ~' ~ .~~/-7-----GL~_'t::._fc; ;e

,f _ ~ - J.i4aL~Q40 '- ~-----~----'---"---------------

1-----,--~~1u,-=----o/~) kCMJ_~k d'U£ io ., ...vu~....::=~~d~ -~tu- tr~~~ ~~L.)CJ)--bJff3J-t.J67-

tlL7 '

d

.-----'-------------,~---------

----<Cfo£t),_-l2,. = ~-~_Q·CJ~__~ ._6"t-t,

(IC? ' \.. 0 ~ .6 30 .. c ~_!Jk~=~l='=--= _---"t.-<.:.J2 j ~:l (j -L 1<-.e.(L -f- .:L

. ~ 3 + ~~-----:2:~-Qc-~-------------------,

4-.--

.ffI

(fob) ./£ - 0_.

. I

·--2.-k2~~) z.~ k. 2.

-:=. ,1& , r-k2+-g-k(gIJLL12z. =~

2-R;:~-e +

.$- l1'\i~ Z. k"k 3~-3

2 k 3{fl_b~ R -:: ,- -(/1<3 -tZr<." - k:se 2k, 8 -~"'-_~I to

c..K.4. r/GlG$-(5~-0 . 4' 1=-- "<:

~-4-:4tk~'t---ZKR4 ": .

Z~-k4f k~

-

.,

...

-2D-

rY~'orlj - f='--'-'--"--......:....,I,-ftt--~~?:-j_~=---_.....-:_A_~_-- _

/ez) /P ~ 0 I ,~. '_ 7-,-,~-__n-_,_--==-- ---..:..-~,-/'lP:,~---f--

------ -2.-k-z.---------~----W~-____f-

-,(§;#---ga- '" f2---tkr-hZ-kc=--c :_'-__ c:... , -!..-I~-______,___g-~-~-V,-.:\'r-I\---+)-

1,-------------:@!j-,---,-~ - _13 ;'1.:2~-----------------;--~--- --,-'1-I

~ \

I----t.'~_(."____I<_&---=~=j4=_,;;==vn.=------------~:-~'"m ,

-~-i !!j~' Gt.1J lZi~ "C Ar/ k. '~~

1--------1(-=-.__ ----111-4-[;) -1F~~~--------/I:::"";L ~ j} /J. ~

1-----------"jJ-~ -.f=--fl. ~~-.------Ut----------

)2 ((- ~ ~) -

'Ehb1 ----_-1<-i -:;. r""l. Y/Iu ~T'--,-"--

£h /, -k-......__,_-,C-z- --=)J;;=~'-I-:J/_ellU_''3._----n/,/-!'H_'_-/~_'_,~__(_---'-

I - 0

-2(, -

- 2- --15.. - z:; ;

,(3-k-;;--t--2.-k £tJ·b· E hi I-l--b =+-:-2-h3-r- .1.-1- (3~ ~v +2I' _

.;...r-Z-

~ 14r.YJk

@;) 121'- 0 'fi:>~.~

-2'A" I

(§~.f( I = .. ' 2- ... -"'-_. ffikr~z,

f31l\z. c.rt--~- - ~-- ,

-- --

;214.3(£~ '~ ,1hj/~

'~~TWr-2-t I

~"'" - --:a.a ~ -- ~4wP --- Z744

~!) ({' ~/1?'4-: 14- .Lf(-'A 4 ~-2-t

...... ---'--~ -=- -~ .............. -----~..e- /Uc.&'on, 4

L

l~~ 2)._.,

A, - en-e-Sj t5VY-ut 7 6F l-

-6- c. /J--W a'd~~s ~)-- --

jJ, 3O~ y 2.), ,

'7 f

P' C-o ~-'L\ G t"z ,'-t-I:;

AtiUti at> -- eo

(Ua~i~-6'-,=

r--'UJ'Ufl'f4.. 0 ~

,------- ------

-2.7-

-----

--,

-L-t--'-4

-----'f I

_ -= 12~ .. Ri.

-27ti- -

/Uu~_~ _

,--_~lZ, co

.. . 2 (li--;::-6--;tj.1ZI.~i ' tA~ +-Zf;

I

1_-@)~-~,-c..-fj1-~r=------------

• 3

, 'C.

-~~-----------'---R2---bz~~----~-----

-------_.--:..../1-,~_/;t Z ,;-;;f? A"Z"I7/ohhf:Jc'-7 _

-~--~~.~~---~-----------------

,----------~-b-'---,------=---,;b-----.--

,-----'----'-12....=.....c1-~-·1~-~ , =;: ~4';;;:; " ~Xd---~-

. 7.Q.",~~=-J-b L JIY b I_ T C . (), j q C? '~L Irr-& I------- . 1 =r~-=-.--------

Ikut-'7"-------------~-----'---

1 0)

~~--------------= ..A - b t

~ t ~ _@';?-UtU

U/t;fh Icrr;t;tM:-~ s &ffI' lbl)

"~ ,ffy<"'f~ f ftfl/4