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M ..., ~ N o 1 2 PENSTOCK ANALYSIS AND STIFFENER DESIGN EQUIVALENT FLANGE SECTIO~ PAGE 23 Definition _. ..._...h................._n...............__..__............................... ............... ..................... 43 24 Stress-strain relations n......_.....__......................................__..... ....__................u.....mn.... 45 25 Equations of equilibrium..................__.......................... .............mmn...__..........n 45 26 Solution of equations ...............................__ ...nno...__................................... 46 27 Restraints offered by pipe shelL.......... nmn__............_______.... ..mmmm__ ....m.............. 47 28 Length of equivalent flange..__......___ .......m._......... ..........__....._____.... 49 29 Outer stiffener ring flange._.......______ ...n..__............. ...........__________....__......._..______............ 49 CHAPTER III-STIFFENER RING ANALYSIS COMBINED SECTION RING LOADS 30 Pipe under pressure......m.......h................u..........n..........nu m.m..._......__ 31 Pipe partly fuIl....un......n..........m.m.........u........nuu.............. ____m_..m.... 55 56 STIFFENE:Il RING STRESSES 32 Pipe under pressure...........m..m...................,......................._.mummu. .....nm....___________ 56 33 Observed and computed stresses......uu....n.....ohhm.___........n...........m................m........ 61 34 Changes in curvature......................n... ........mnn....._____.......__...............____ un_..n,....___... 61 35 Pipe partly full..mm....h....',....................".....,..,.......h............,........h....m''''''''''''''',h.., 71 36 Original analysis....................nu..mn...m ..mn............mm___........mnm.mn...........nmm.n 72 37 Special solutions....__uhn...........................n..............n.................m.......__......................__ 75 38 Typical examples......._........_n.........n......nn..___..__.......___.....__n.nn_................................n.. 84 CHAPTER IV-FILLET INSERT THEORY 39 Secondary stresses without fillet............ .....m'u'...hh... 87 EQUILIBRIUM AND STRESS, STRAIN RELATIONS 40 Equations of equilibrium...._.....__...___ .................................. 89 41 Stress-strain relations...___...._._______.............n......____............................ .......................____ 90 PARABOLIC FILLET INSERT 42 Differential equation..____....._____................____........__............___....._. ....__.........._.._............... 91 43 Complementary function.........._____..______........___... m._.h.....______.......____.......... ........_....___. 91 44 Particular integral____.......___......___..____..______....... ........_................ m................___. 93 45 Complete solution ..______._....__ ___.._______._...... ..............___.___________. .........................__ 93 46 Boundary conditions ______...__..__..... ...........m___.........__ .........m..._ 96 STRAIGHT FILLET INSERT 47 Differential equation .............. ...................--...--..........._____....................._____h. 96 48 Complementary function...............................__.......m..m...___...................___m........_........ 97 49 Particular integral....................._________................___...................____._.........__.___..............___ 98 50 Complete solution ..__.___ ....__..............................__m.....___........___...__................... 98 51 'Typical reslll ts.... ..... ........ .. ...____.__.... ......__..h__...... ................ ._.......h_....._....__. ........_. ........ 100