Deflection of a flexible pipe culvert when stressed beyond the elastic limit

90 Pages·09.176 MB·English

by  Rahman, Abu Sharf Masih-ur

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nw m cnm tt a flexible mm m w m t
WWM $ m W § M SB8 I&SitlO LlHIf
JUbit §h&t Masiti-w IslBto.
A Dissertation Submitted ta tite
#r«i«®te faculty fa Partial M.filljs«at ®#
ft* x*N|«ijNHMaitt tat the tappet «f
DOCTOR or PHILOSOPHY
Major Subject* MiglMip iafiaearfiii
A pproved*
Zmm State GoUes*
1950
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UMI Number: DP13439
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• H i *
tmtM w eonsmss
HOfAXIOMS m * * * ♦ % * • * • * * > * * * * * * # # # . v
X* X!f1f?R£©BCXXCIll » *. #, * *; * * * * * * * * *, * * * * # « *, X
A. Usefulness of Flexible Pipes in Construction % *-* I
1* ®#ed for Investigation * * . # * * . * » • * * * « * 2
C« teview of Literature * * * * * , * • * . * * * * * 3
D* The Problem * * * * , * • * * * . # ; * * * # * * : # , * § 7
ix*. xhbqbbizcjl solbtxoh §p tm m m m f * ♦ * , * * * • 9
4* FUfe#«i«tal fhaory fi * . ■« * * * * * * * * * * * f
a* Idealised stress-strain diagram « « » • • * » 9
b* Curvature of a circular ring under load m*
presse# approximately as a function of its
Initial! radius and radial deflection * • .* * 11
«# Geometrical relationship between the change
! in curvature and circumferential strain la
the ring due to the load * ♦ « * . # * . * * 'ih
d. Equation for internal moment of a rectangular
section when it is partially In plastic
. . rang® of stress*. t * * . * . ■ * * * * * * * * Id
B* 4asmapti©as • « » * • * * » • ' * « * < * * * * * * IS
C* limitations to Hi® Assumptions # * # • * # # * * * , 19
B# Development of the Basic Differential Equations * 19
a* Elastic portion of the ring * * * • • * • . * * 19
b* Partially plastic portion of the ring * * * 21
!« General Solutions of the Differential Equations . 0
a*. Elastic portion * • * * * . . * * » * * » • * 2%**
b* Partially plastic portion * * * * * * * * * * 2%
The
f* Unknown Constants * « * • * # # * * » . . * * *. 31
0* She Boundary Conditions * * * * . * * . • * * . * ■ * $1
H. Determination of the Constants |2
X. Modifications of Moment Equations for the ling Due ■ ■
to the Change in its Geometry Under Load *, * *, *, %f
XXX* CQMPAftlSf* OP fffl THEORETICAL' SO LB f!« WITH 1®ULB
m m s m m corrugated metal pipes » * * . - * * * * * * **a
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If* discussion of mt m m m m m « * , » * * .» . * • * * ft
?* . possibilities m the masamt&i m this tsmcmt o
Vt.BXIBLI PIPB Of'OTif®'flilt JttKUMtAiO 03HDXTTQIB * $1
f I* CONCLU5IONS » # * * * * *>, * * ■# * * * * * * ♦. *■.<** *' Ijijtn
ni. m f m w a r n *■*. * '♦ * * « . * * . * > * * * . 62t>
tin. m m m m m w m m s , * . - * *. * * * * * * * * * * . « * 63fe
mam
ix, iPFiroa it mnxcnos m 4 flexible ciacinaa
due m its own weight, mwfmfm m t® smm * * * * &*
4* Deflection of a Fl«xlM© Cor rotated Pipe of
Structural Steel doe to its Own Weight,
Supported at the Base % * , 68
X* 4PP1»IX lit IMS TIC DHn.ECTION OF A F1K 1SL1
PIFF MDER 4 MODIFIED FIELD LOAD
wmzn&tt * % *• * * * » * * ■« * * * , * * * * , * * . 73
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w*fm
NOTATIONS
■ a S i, "
of tli© deflected ring BiMMA perpeisiiemlarly
to tli® line of action of the lead* senl-a&Jor m M of
■« elliptic rtsf*
% Badims of the defleeted ring measured along the line
of action of the loadf Ml-ainor axis of m elliptic
ring I width of the. ring:*
Id Thickness of the ring section,
id,© TMekness of the elastic- portion of the ring section
whets -Hi# ring section has partially gone Into plastic
tang© of stress*
1 ffedultts of elasticity of ring m-twial for tension
.and ooapresslont elliptic integral of the second kind*
« Ciretatforential strain in the ring section in# to
the concentrated loadf acdnXwi #f passive pressure
#f the sidefill*
’ f: Elliptic Integral of the first Mai* ■
1 lorisoatal thrust at m y section of the loaded-ring*
'fe.. 'IkImk Mrl»»t«l wn&t pressure on the ■flexible
pip# do# to the sidefill*
1- mmmrnt of inertia of the ring section
k -iohstsmts from 8o?fc»*» graphs-* ■
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•91:-
M Bending jsoiMfit*
$*oft&g' mmmt at the ring section directly master the
concentrated lead#
Mg Bending vonant at the section 90® tvm the section of
the ring directly tmder the eone«ntvatad ■ load*
H* Htatibnte resisting mmmt the ring section can develop
rfttamt going into the- plastic range of stress*
f Concentrated load.
Pg BbxImos concentrate# load the ring- «a& fee. subjected
to without putting it into plastic range of stress*
-t.,. -*r*R l r l/2
B, gES.
f Initial mean radius of the ring*
S unit stress*
$9 froportional limit*
t flilctaess of corrugated pipe
%
"
w Uniformly distributed land*
at Rectangular coordinate*.
y Radial deflection:©f the loaded ring at any section*
rectangular coordinate*
.fg ■ Radial deflection of the loaded ring in. the elastic
portion*
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y Radial deflection of the loaded ring in ' the partially
plastic portion*
lailal deflection of the leaded Hug directly aider
the leal*
?#*#!% *«4isl deflection ©f the tawdftd ring 90® fro* the
point of application of the load*
#® Differential length of ring*
* ingle iesoting the plastic bmmdey*
£ Dhtt noriaal strain*
% Unit strain at the proportional limit*
in Unit strain is the extras* igns? «ai outer fiber of
the loaded ring*
t* # » t In g le s ,
w 3 .1 ^2 ,
^ Curvature of the loaded ring at «gy section..'
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