6
258 7 9 V M 6 2 6 3 6 1 z 6 1 6.1b 6 1c spn
259 1 2 w w 1 w 2 M () (b) (c) b c I T I T 7 12 W bem I S bem web flnge 6 1 1 5 R M 1 2 M w w 1 w 2 6 5 6 1 1 ww 1 w 2 2 M 1 2 3 6 1 stticll determinte stticll indeterminte 1 12 6 2 2 combined bem
26 w Hinge 3 2 6 3 2 e M e 6 m e 1 knm e 5 N. m w 14 kn R M 1. 2. w 6 3b 3.
261 w e e M e /4 M e /2 (b) () /3 e w /2 e R M R M e /4 M e /2 (c) b c 6 3 w 2 F = : R 1 M w = M e e e 2 2 3 2 3 1 2 : M M w 2M 4 + = = 4 6 + = e e 1 1 F = : R 2 + w = R = 2 w e e 2 2 R = M = 794 N in. = 79. 4 kn m R R = = 4 N 4kN 6 3c R M 15 kn w 1 kn m 6 4
262 = 15 kn w = 1 kn/m () = 2 m = 4 m 15 kn w = 4 kn (b) R 2 m 4 m R R 2 b w R R 6 4b F = : R M M c = : 15( 2) 4( 2)+ R ( 4)= R = 12. 5kN 15 6 R 4 4 2 = ( ) ( )+ ( )= R = = 42. 5 kn F = : +. +. = 15 42 5 4 12 5 R R 6 4b 6 5 M
263 6 5b R R R F = : R = M R M = : ( ) 4 = = 4 + M R 3 3 M = : R ( )+ 4 M = R = 4 M 3 M M F = : + + + = 4 4 6 5b M /4 /4 /2 R M /4 /4 /2 R R M b H 6 6H H H 1 kn H H w 1 2kN m w 2 4kN
264 = 1 kn w 1 = 2 kn/m w 2 = 4 kn/m 1 m 6 kn 3 kn 1 m Hinge 2 m 2 m H 3 m H H H R () (c) 1 kn 2 kn/m 4 kn/m 1 kn H R M H R M H H R R R (b) (d) b d H H 6 6b 4 6 6c d 6 6 6 6c2 2 2 32 3kN4 32 6kN F = : H M H = : R ( 3) 6( 2) 3()= 1 R = F = : H 3 6+ R = H = 4 kn = 5 kn H H H 6 6d F = : R F M = : R 1 4 = R = 14 kn = : M 1 2 4 4 M = 36 kn m = 36 kn m = ( ) ( )=
265 H R R M VM 1 6 6 7 2 6 7b 6 7c 2 w SF SM V = w M = 1 2 w 2 w w M M w V V () (b) w /2 M V (c) V M b c w V M
266 V M 6 5 V M 1 6 2 6 8 V M 6 8b +V -V +M -M () +V -V +M -M (b) b 6 9 w 1 kn m
267 5kN 1 6 m V M w = 1 kn/m = 5 kn () 1 m 4 m 1 m 1 kn/m 5 kn (b) R = 15 kn R = 3 kn 1 kn/m (c) 15 kn M V 6-1 kn/m V 5 kn (d) M 3 kn b d 6 9b R 15 kn R 3 kn 6 9c V M 6 9c 1,, 5 F = : 15 1( 1) V = 1 M = : 15 + 1 ( 1) + M = 2
268 2 V = 1 + 25 M = 5 + 25 5 V M 3 4 m V 9kN M 22 2 kn. m 6 9d V M SF SM 3 4 m V M = 3 + 1( 1. 6)+ 5 = 9 kn. = 3( 1. 6) 5( 6 3. 4) 1 2. 6 1 26 ( ) 1 = 22. 2 kn m 2 3 4 m 6 1 6 4 6.1 6 1 w 2 kip ft 4 kips 7 ft V M = 4 kips w = 2 kip/ft w = 4 kn/m = 1 kn 2 m 1 m 1 m 7 ft 1 ft 5 ft 5 ft 6 1 6 2 6.3 6 3 V M 6.2 6 2 w 2kN m 1 kn () (b) V M 6 3
269 6.4 6 4 FE V M V M 2 kn E F 2 N/m 6.7 6 7 w () V M (b) V M w 1 m 1 m.8 m 4 m 6 4 6 7 6.5 6 5 () V M (b) V M 6.8 6 8 1 kn m 3 kn m () V M (b) 4m V M 2 lb/ft 3 kn/m 3 ft 3 ft 1 kn/m 6 m 6 5 6 8 6.6 2 kips M 1 kip. ft 6 ft V M = 2 kips M = 1 kips. ft 6.9 H H H H w 6 lb ft 6 9 () R R R (b) H V M (c) 3 ft V M 4 ft 1 ft 6 ft 6 6
27 1.2 kips w = 6 lb/ft 3 kip.ft H Hinge 4 ft 2 ft 2 ft 4 ft 6 ft 6 9 6 12 6.1 H 6 1 () R R M (b) H V M w H Hinge /3 w /2 /2 6 13 6.14 6.15 6 14 6 15 VM V M 6 1 1 kn 5 kn 6.11 E 6 11 E w 5 N m 1 3kN 2 4kN () V M (b) V M 1 m 2 m 1 m 2 kn 6 14.8 m 1.5 m w = 5 N/m 1 = 3 kn 2 = 4 kn E 2 m 2 m 1.5 m 1.5 m 2 kn/m 4 kn 6 11 6.12 6.13 6 12 6 13 VM V M 2 m 4 m 2 m 6 15
271 6.16 6.17 6 16 6 17 V M V M 2 kn 6 kn /4 2 m E 2 m Hinge 1 m 2 m 2 m 6 19 F /3 /3 /3 2 ft 2 kips 4 6 16 3 25 lb/ft 6.2 6 2 w 4MN m w V M V M 2 ft 4 ft 4 ft w = 4 MN/m Uniform rection 6 17 6.18 6.19 6 18 6 19 V M V M 3 m 4 m 6 2 w r 3 m 3 kn/m 6 6 kn/m 2 m 4 m 2 m 2 m 6 18
272 6 6 6 8 6 1 z w M M w d w d/2 d V d V + dv M + dm () (b) b wm VM 6 1 d w 6 1b d V M dv dm 6 1b w d w d 6 1b F = : V V + dv wd ( )+ =
273 dv d = w 6 1 w w dv d = dv d = 6 1 = V V wd V V 6 2 6 2b 6 1 w 6 2 6 1b M = : M + dm M V d w d d ( ) = 2 dm d = V 6 3 V 6 3 6 3 6 3
274 = M M V d M M 6 4 6 4b 6 1 6 3 6 3 6 4 V M 6 4 6 8 6 1 6.3 6 4 V M V M V M
275 6 6 6 9 1. 2. 6 2 6 1 3. 6 4 6 3 6 1 6 9 6 11 dv d dm d
276 6 11 6 11b R b = R = b () (b) R = b/ R = / (c) b/ V 1 M 1 (e) V b/ -/ M b/ (d) V 2 M 2 (f) b/ b cd e f
277 V M 6 11c,, SF SM 6 11e f 6 11d,, SF SM 6 11e f V V 2 M b = M = 1 1 b = = 2 b b b = ( ) 6 11e f 1 6kN E 2 1 kn 6 12 1 kn 6 12b R 16 kn M 1 kn. m 6 12c,, 4 SF SM V 1 6 M 1 6 6 12d SF SM 4,, 1 V 2 16 M 2 16 1 1 b b 6 12e 6 12f 2 kn. m
278 2 = 1 kn kips 1 = 6 kips kn 2 ft m 6 kips kn 1 kn kips () 4 mft 6 ft m (b) 2 kip.ft kn. m M = 1 kip.ft kn. m R = 16 kips kn 6 kn kips V, kips kn (c) V 1 M 1 (e) -6-16 1 kn kip.ft. m M, kip.ft kn. m (d) V 2 M 2 1-16 kips kn -4 b cd e f (f) -24-1 6 13 15 m 1 3 m 1kN 6 14 6.14b R, 6 14c 6 14d V M SF SM,, r Weight Stnd V 1 M 1,,
279 () (c) (b) R = R = V M 1 (e) V 1 - (d) V 2 M 2 ( f) M - 6 13 b cd e f V 2 M 2 6 14e 6 14f V m 1 kn M m 15 N. m M 1 6 15 w () (b) R R w 2 6 15b () V M 6 15c SF SM
28 w w () R = w/2 R = w/2 (b) V w/2 w (c) V M w/2 m = /2 (d) w 2 /8 -w/2 M (e) bc d e 1 V = w w 2 1 1 M = w w 2 2 2 c d 2 6 15d 6 15e V dm d c m 2 d M m w 2 8 (b) 1 V = w w m = 2 6 15d V w 2 dv d w V V = w V = 1 w 2 w 2 V
281 m 2 1 6 15e M w 2 = w = w 1 2 m w M = 1 = 1 w m 2 2 2 8 2 w 2 8 2 6 16 2 kn w 4kN m 6 16b R 25 kn R 35 kn = 2 kn w = 4 kn/m 2 kn 4 kn/m (b) 4 m 4 m () 2 m R = 25 kn V, kn 4 m 4 m 2 m R = 35 kn 25 (c) 1 9 8 ñ11 2-27 3 68 M, kn. m (d) 7.7 m -8 b c d
282 6 16c V R 25 kn w6 2 V V 44 16 V 9 kn V 2 kn V 11 kn V V 44 16 V 27 kn 35 kn V 8kN V V 42 8 V 27 kn 6 16c 1 68 kn. m 2 76 kn. m 3 8kN. m 6 4 M M 68 M M 76 M M 8 M 68 kn. m M 8 kn. m M M m 68 kn. m 6 17 E E w 8kN m 1 2 kn E 2 6kN 6 17b R 12 8 kn R 19 2 kn 6 17c V 6 2 1 V V = V ( )( )= = 2 8 1. 5 6 6kN E
283 w = 8 kn/m 1 = 2 kn 2 = 6 kn 8 /1.5 8 kn/m 2 kn 6 kn 1.5 m 1.5 m 1 m 1 m E 1.5 m 1.5 m 1 m 1 m R = 12.8 kn R = 19.2 kn E () (b) V, kn 6.8 6 (c) -6 E -13.2 M, kn. m 7.2 (d) E -3-6 b c d 12 8 kn 6 8 kn 2 kn 13 2 kn 6kN E E 6kN V m 13 2 kn 6 17d M 6 4 M M = 15. 1 8 d M = 3 kn m 2 15. M M 68. 15. M 72. kn m = ( ) = M M = 13. 2() 1 M = 6 kn m M M = 61 E () M E = M E 3 E M m 7 2 kn
284 6 5 6 6 6.21 6.22 6 12 6 13 V M w w 6.23 6.24 6 14 6 15 V M 6 31 6.25 6.26 6 16 6 17 V M 6.27 6.28 6 18 6 19 V M 6.29 6 2 V M 6.3 6 3 V M 6.32 6 32 w V M /4 /2 /4 6 32 6.33 6 33 w o V M w w /3 /3 /3 6 3 6.31 w w 6 31 V M 6 33 6.34 6 34 E V M
285 /2 /4 /4 E 6.38 6 38 w V M w 6 34 6.35 6 35 2 V M 2 6 35 2/3 /3 6 38 6.39 6 39 w V M 6.36 6 36 2w w V M 2w w w /2 /2 6 39 w /2 /2 6 36 6.37 w 2 kip ft 2 kips 6 37 V M = 2 kips w = 2 kip/ft 4 ft 2 ft 6.4 6 4 V M 1.5 kn 3 kn/m Hinge 3 m 2 m 1 m 1 m 6 4 6.41 6.42 6 41 6 42 V M 6 37
286 4 kips 3 kips/ft w /4 /2 /4 6 ft 6 ft 6 46 6 41 6.47 6.48 6 47 6 48 V M 2/3 /3 12 kn 3 kn 6.43 6.44 6 43 6 44 V M 6 42 w 2 m 1 m.5 m 6 47 8 kips 6 kip. ft 3 ft 6 ft 3 ft 6 43 6 48 w 6.49 E 6 49 6kN E 6kN 2kN. m V M /2 /2 6 kn 6 kn 6 44 2 kn. m 6.45 6.46 6 45 6 46 V M 1.5 m 1.5 m 1 m 1 m E 6 49 k 2 6.5 6.51 6 41 6 42 6.52 6.53 6 43 6 44 6 45
287 6.54 6.55 6 45 6 46 6.56 6 56 () T (b) M z (c) z M z 6.576.58 6 57 6 58 M.4 m 3 kn 6 kn r =.2 m 5 kn 2 kn V, kn 55.6 m r 6 56-5.4 m 2 m 1 m 1 m 6 57-45 6.59 6.6 6 57 6 58 M 6.61 6.62 6 61 6 62 M 2 ft V, kips V, kips 5-3 1.4 6 ft 4 ft -2.6 V, kn 1 4 ft 2nd-degree curve 6 59 2 4 ft 6 ft 4 ft 6 6 4-5 V, kn 25 5 1 V, kips -2 3 m 1 m 6 61 6 2 m -25 6 m 2 m 6 58 4-1 4 ft -9 8 ft 4 ft 6 62
288 1 w V M singulrit function Mcul n n n 2 2 Mcul - Mcul n = ( ) n < for n ( n =, 12,,...) 6 5 1-1 n = () n 6 18-2 n = 1 (b) 6 5 n = 2 (c) n
289 < = 1 1 < = 6 6 Mcul n d = n + 1 n+1 for n 6 7 6 2 3~8 n n 1 prbolic spndrel V wd M Vd w() = M < - > -2 w() = < - > -1 w() = w< - > w() = b w < - > 1 w w() = b 2 < - > 2 w() = w< - 1 > w< - 2 >
29 w() = b w < - 1 > 1 b w < - 2 > 1 w < - 2 > w() = w < - 1 > b w < - 1 > 1 b w < - 2 > 1 w 6 19 ( ) = = w 1 = 6 8 w n 1 M ( ) = w M 2 = M = 6 9 w n 2 6 8 6 9 n n+1 d = for n < (n = 1, 2,...) 6 1 n 1 6 2 1 2 6 1 6 5 6 1 6 8 6 1 6 2
291 w () w () / - -1 () w () / = M / 2 w () M - -2 (b) b M 6 2 M V M () w () w () 6 2 M R R
292 2 M M = : R M R ( 3 ) ( 2 )= = + 3 3 2 M M F = : R + + = R = 3 3 3 3 6 8 6 9 6 2 1 2 1 ( ) = + + w R o M 2 R 3 2 1 1 b 1 b V M w R b M V ( )= M + 2 3 3 1 M 1 M ( )= M 2 3 3 + 1 c (b) 6 8 6 1 6 2 se 1 se 2 w( )= M 2 2 1 1 V ( )= M 2 + 1 M ( )= M 2 + 1 1 + 2 d e M M 3 e ( )= = ( )= + = M M 3 M 3 1 2 1 1 1 = M = M 1 3 3 3 3 d e c V M b
293 6 21 6 21 w M = R R w 2 4 + = = 1 : 8 w w 3 F = : R + = R = w 2 8 8 6 21b 6 2 2 6 3 1 w( )= w o w + w 2 w 8 8 1 1 f f 3 1 V ( )= w w + w 2 w 8 8 1 1 g w w () /2 /2 3/8 w w 1/8 w R R /2 /2 (b) V 3/8 w (c) m = 3/8 -w/8 M 9w 2 /128 w 2 /16 w b c
294 3 1 1 1 M ( )= w w + w 2 w 8 2 2 8 1 2 2 1 h M V g V 2 3 w w m 8 = 3 8 m = h g h M m = 3 3 w w w 1 3 8 8 2 8 = 9 128 2 6 1 6 4 6 22 1 4 m w 1 2 kn m w 2 3 kn m M 5 kn. m 6 22b M w 1 w 2 () M w 1 w 2 w 2 - w 1 (b) M R b
295 6 22b R = ( w + w )= 14. ( 2 2 2+ 3)= 3. 5 kn 1 2 M = M + 2 ( 2 w + w )= + ( + )= 1 2 6 7 8 5 14. 14 24 7. 41kN m 6 i 6 2 1267 1 2 2 w( )= R M M w 2 + w 3 w w 2 1 w w + 1 2 1 2 3 1 + w w 1 ( ) 2 1 1 3 6 12 3 V M w w w( )= R 1 M 2 2 2 1 M w 2 1 1 2 2 1 = 35. 74. 5 14. 2 28. 1. 4 28. 1 1 j w m kn m 6 7 6.63 6 63 M w V M M = 6.64 b w 6 64 w V M b w b 6 63 6 64
296 6.65 6 65 E () (b) 1m 1 kn E 3 6 65 6.66 w 6 66 w V M 6 68 () wv M (b) M = 4 kn w = 9 kn/m 3 m 1.5 m 6 68 6.69 E 6 6 () wv M (b) M w E /3 /3 /3 /2 /2 6 66 6 69 6.67 6 68 w V M 6.7 6 7 () (b) = 4 kips 5 ft 1 ft M = 8 kip.ft = 1 kn w = 15 kn/m = 1 kn 6 67 1 m 2 m 1 m 6.68 6 7
297 6.71 6 71 M wv M w M = 16 kn.m = 2 kn 3 6 74 2 m 2 m 2 m 6 71 6.75 2 6.72 6.73 w 6 72 6 73 w 1 w 2 () wv M (b) M 6 75 w 6.76 6 76 /2 /2 () wvm (b) 6 72 2ft 4 kipsw 1 5 kip ft w w w 2 /2 /2 6 76 6 73 6.74 w 6 74 () (b)
298 V M stticll determinte bem stticll indeterminte bem 1 12 6 5 dv d w dm d V V V M V M V M V M dv d = w = V V wd dm d = V = M M V d n = ( ) n < for n ( n =, 12,,...) = 1 < = 1 < d = n n + 1 n+1 for n
299 w( )= = 2 w( )= M = M 1 = = n n+1 d = for n ( n = 1, 2,...) 6.1 S. H. rndll, N.. hl, nd T. J. rdner, n Introduction to the Mechnics of Solids, 2nd with SI units, McGrw-Hill, New York, 1978. 6.2 E.. opov, Introduction to Mechnics of Solids, rentice Hll, Upper Sddle River, NJ, 1968. 6.3.. Ugurl, Mechnics of Mterils, McGrw-Hill, New York, 1991. 6.4 W.. ilke, lebsch s Method for em eflections, Journl of Engineering Eduction, Jnur 1964, pp. 17 174.