Formula Sheets

# Formula Sheets - TABLE G-1 DEFLEGTIOI‘FS AND SLOPES 0F GANTILEVEB BEAMS y I 58 v = deﬂection in the y direction(positive upward A 3 it v =

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Unformatted text preview: TABLE G-1 DEFLEGTIOI‘FS AND SLOPES 0F GANTILEVEB BEAMS y I 58 v = deﬂection in the y direction (positive upward) % A 3 it v' = Liv/ix = slope of the deﬂection curve 55 z —v(L) : deﬂection at end B of the beam (positive downward) 05 = —v'(L) = angle of rotation at end B of the beam (positive clockwise) T L W‘ 63 E1 = constant 2 air We 1 q =— 6L2—4Lx+ 2 ’:———3L2-3Lx+ 2 m “ 24EI‘ J” V 6131‘ “ qL4 (1L3 a = — a = — B 8E1 3 6E] W q 4162 ﬂ 2 M = “WW - MHZ) mm“) x ﬁ—a'Jfb" v' — —q—(3a2 — 3ax + x2) (0 S x s a) GE] 3 3 qa qa = — 4 — ’ - —— s s 24EI( I a) V 6E1 (‘1 x L) 4 ‘5 qa qa A1 = .- = —— ’= —— I a V BE] v GE] 3 3 qa qa = 4L — a = — B 24EI( “) B 6E1 (Continued) 905 906 APPENDIX G Deflections and Slopes of Beams 3 9 v=—q—bx-2-(3L+3a—2x) (OSxSa) W 12EI i I qu |._a b v=—E(L+a—x) (DSxSa) q 4 3 2 2 3 4 = — — + — 4 + s s v MEI (x 4L): 6L x a x a ) (a x L) v' = —%(x3 3Lx3 + 3le — a3) (a s x s L) 2 - , qa b - qabL Ar = : = — 3L + ’ = — x a v 1251( a) v 251 53 = 41—01} — 4a3L + (.14), 9,3 = im 7 a3) 24E] 6131 P sz , Px _- l 12 6E” L x) v 2EI( L x) g PL3 PL2 5 = — 6 = — 3 BE] 3 2151 _2 5 11’ v = —-%(3a — x) v' = +%(2a — x) (0 s x s a) Ha‘ng‘J v : -£q:(3x — ) V' = ‘10—‘12 (a E I 5 L) 6E1 2E1 Atx=a 'v=-P—a3 '=—P—az 351 231 P512 P02 6 = — 3L — a = — B 6EI( ‘0 B 251 A/IQX2 I Max 6 v = — = —f g— ) 251 E1 ’ M 0 5 _ ML2 6 _ MOL 8 _ 251 3 E1 APPENDIX G Deﬂections and Slopes of Beams 907 M Mox2 Max 7 , 0 = ~ — —— s s 3—; 2131 El (0 x a) M a M a (1—qu = __0 _ F_ __L < <L ' 2E1 (2x a) v El ((1 x _ ) M 2 M Atx=a: v=— 0a v'=— 0a 2E1 El 5 7 M0” 2L 6 —ﬂ 3 2EI( ‘0 B _ El 2 40x 8 ‘10 =— 10L3—10L2+5L2—3 I . 40x 3 2 2 3 = a 4 — L +41: — 24LEI( L 6 x x x ) 5 _ 401:4 '9 _ (10143 E 30131 E 2451 2 ﬂ"0 =—q°x L37 0L2+3 9 v IZOLEIQO l x x) ‘ y “3035 1 2 3 = ~ * — + v 24LEIGH 61. x x ) llqoL‘ qoL‘ 40L 3 17x 1 3 2 3) - — I. — — 4 ~ + i v 37TWA“; cos 2L 8L 37r Lx 773x 12’ (“L (2 772Lx “21:2 SL2 sin 77x) #3151 2L 2110114 (10123 5 3 — 24 = 2 — 8 B 37451” ) 9” W351” ) ( Continued) 908 APPENDIX G Deflecﬁons and Slopes of Beams TABLE G-2 DEFLECTIONS AND StUPES 0F SIMPLE BEAMS v = deﬂection in the y direction (positive upward) v’ = dv/dx = slope of the deﬂection curve 5C = —v(L/2) = deﬂection at midpoint C of the beam (positive downward) x1 = distance from support A to point of maximum deﬂection Emu : —vm‘;x = maximum deﬂection (positive downward) 0,1 = —v'(0) = angle of rotation at left-hand end of the beam (positive clockwise) 33 = 12' (L) = angle of rotation at right—hand end of the beam (positive counterclockwise) 9'" 3 2 3 1 = — — 2 + v 2451“ I“ x) v' = ——q (L3 — 62x7- — 4x3) 24E! sqL“ 4L3 5 = gmax: —— 0 2 g = C _ 384121 A 3 2451 ‘1" 3 2 3 L 2 = — L — 24L + 1 s s — v 384EI(9 x 6x ) (0 x 2) v' = — q (91.3 — 72m2 + 6429) (0 Ex 5 5) 384E] 2 4L 3 2 2 3 (L ) =— 8 —24Lx +171. ~ —s 5L v 384E]( x x L ) 2 x v' = iii—(24x2 — 481.): + 171.2) (5 s x s L) 384151 2 511124 3.3L3 7211.3 5C = A = 9.9 = 768131 128E] 384E] 43‘ 4 3 2 2 2 2 2 3 = — — L + 4 L + 2a — 42:15: + s s 3 v 24LEI(a 4a a x Lx ) (0 x a) . q 4 3 2 2 2 2 2 1 z — — L+ L + 6 —1 + - S S v 24“Hm 4a 4a a x 2aLx 4Lx ) (0 x a) v = — qaz (-azL + 4L2): + alx — 6sz + 2:?) (a 5x5 L) 24LEI flag 2 2 2 v = _ 4L + e 12L + s S v 24LEI( a 1: L6): ) (a x L) \ 2 2 CI“ 2 ‘10 2 2 = 2L w 0 = 2L — 6" 24LEI( ‘0 B 24LE1( a ) APPENDIX 6 Deﬂections and Slopes of Beams 909 P1 2 2 . P 2 2 ( L) =— 31.74 =——L 74 s s— v 48EI( x) v 16EI( x) 0 x 2 PL3 PL2 5 — max = __ 3 = 3 : C 48E! A 3 165! , Pbx 2 2 2 . Pb 2 2 2 — — b i = —— — b — 3 s s v ELEIU. r x) v 6LEIU. x ) (O x a) _ Pab(L + b) Pab(L + a) A 6LE] B 6L5! Pb(3l.2 — 4b?) Pa(3L2 — 4a2) 2b 5 =— If Sb, 5 =— If“ ’ C ‘ 4851 a C 48E] L2 w [)2 Pb(L2 — 1:2)3’2 Ifazb, x = f and 5m: W ' 3 9V5 LEI PX 2 . P 2 2 : —— — — = —— — — 0 s s v 6E1(3al_ 3a x2) v ZEImL a x ) ( x a) = _ﬂ * 2_ 2 :3 _£ _ < < _ v 6EI(3LX 3x a ) v 2EAL 2x) (a 7 xﬁ L «1) Pa Pa(L — a) 5 = . - 3L2 —4 1 6 = =—— C 5am 24EI( a ) A 93 251 Mo M M : -i(2L2 — 3Lx + x2) v' = — 0 (2L2 A 6Lx + 3x2) ( ‘ 6LE1 sum M L2 g 8" Mug/M L mm 5 = 0 = 0 g z 0 0L \)3 015i 0‘ 0‘ C 1651 6" 3E1 3 6E] _" Q” L" \/37 MoL2 = 1 — *w d ax : x1 ( 3 ) a” 6‘" 9\/§EI MM 2 2 M0 2 2 ( L) =~ L—4 '=— L—12 OS 57 24LE1( x ) ” 24LEI( x ) x 2 6 _0 7 M01. 6 _ MOL C _ A _ 24E] ‘5' ’ 24E] (Continued) 910 APPENDIX E Deflections and Slopes of Beams M x r 9 v = — 6531mm — 3a2 — 2L2 — x2) (0 s x s a) M v’ = -6L;I(6aL — 3a1 7 2L2 — 3x2) (0 s x s a) M ab M Atx = a: v = — 3230;: — L) v' = — 3“glam — 3‘22 7 L2) M 0 2 2 0 2 2 = L — 3 — 2 = 7 9A 6LEIUSuc a L ) 03 6LE1(3a L ) 10 M°x( ) ' M”(L 21) = _ _ v = 2— _ v 2E1 2E1 MOL2 MOL # — : g = 6‘: 5”" SE] A 8 21:1 11 v = — 36%":E10L4 — 10L2x2 + 3x4) . ‘10 4 2 2 4 v = —36OLEI(7L ~ 30Lx +15x) 6 : SM“ 9 : 7Q0L3 0 : qoL3 C 768E] A 360E! B 4551 (10144 x1 = 0.5193L am = 0.00652 E1 40x 2 2 2 L 2 = 7 — S S _ l v 960LEI(5L 4x ) (0 x 2) * = — ‘10 (5L2 — 4x2)(L2 — 4x2) (0 s x s 5) 192LEI 2 qoL4 540131 5 — 0 = 0 = C "m" 120151 A B 192E] v='qoL4 sinﬂ v'=—q0L cosﬂ 774E] L 773E! L 4 3 40L QOL 58—Smax_ 4E1 l24:65: 7T3EI ...
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## This test prep was uploaded on 04/17/2008 for the course GENENG 234 taught by Professor Abdel-aal during the Spring '08 term at Wisc Platteville.

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Formula Sheets - TABLE G-1 DEFLEGTIOI‘FS AND SLOPES 0F GANTILEVEB BEAMS y I 58 v = deﬂection in the y direction(positive upward A 3 it v =

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