# otpornost materijala - treća, četvrta i peta nedelja predavanja

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27 , U 1. ( ) . N,N=const, (), N const. ................................................................................................................................... ,N = N (z), A = A (z). , , : N 0, TBxB = TByB = MBxB = MByB = MBtB = 0.(1) : 1. : BzB 0. BzxB = BzyB = 0. 2.( ): ., . : BzB (x, y, z) = BzB (z) BzB (x, y) = const,B zxB = B zy B= 0 28 : BzB = BzB, BzB (x, y, z) = Bz B(z) = BzB (z) BzB (x, y) = const . UU=constU,, . 3.( ) z N dAAz = , 4.0 = = Ax zM dA y , 5.0 = = Ay zM dA x , . : ( ) zz z = , : ( ) ( ) z N dA zAz= , (3) : ( )( )Az Nzz= . (4) . 0 0 = = = = xA Ax z z zS S dA y dA y 0 0 = = = = yA Ay z z zS S dA x dA x . (5) xy ,N . U = (z) : 29 dz (dz). ( )( )( ) z Az Nzz= ( )( ) ( )( ) z A Ez NEzzzz= = .(6) : ( ) ( ) dz z dzdzdzzz z = = ,z : ( )( ) = = = z zzzdzz A Ez Ndz dz z0 0 0 , (7) l,:

( )( ) = = = l lzldzz A Ez Ndz dz l0 0 0 (8) N = const, A = const, : AN= ,A EN= , zA ENz z = = ,lA ENl l = = . (9) ( ) / l () . l, = i ii iA El Nl .(10) ................................................................................................................................... 30 ( )( )d zz Az N |||

\|=maxmax,(11) ( )( )d zz A Ez N |||

\|=maxmax,(12) ( ) ( )dz A z N ,(13) ( )( )dz Nz A , ( = const) dNAmax . (14) SAINTVENANT BzB . . - : , . , . . 31 ( ) T zz = .(15) :1. 0, N = 0, 2.T 0 N 0. 0, N = 0 :T z dz T dz zz zz = = = 0 0 ( ), T l dz T ll = = 0( ). (16) [ [[ [P-1P ] ]] ] =1210P -6PKP-1P,=21000kN/cmP2P. ( ), (t). .. T 0 N 0 : UA (z), N (z), const UUN = const, A = const, =const ( )( ) z Az Nz= , AN= , ( )( ) ( )( )Tz EAz NTEzzzz + = + = , TEAN + = , ( )( )dz0 0 |||

\| + = = z zzTz EAz Ndz z ,T z zEANz + = , ( )( )dz0 0 |||

\| + = = l lzTz EAz Ndz l .(17)T l lEANl + = .(18) , : - ( ) - . . 32 ( ),, . , () . , : -n - ( , ) -s - , k = n s , -k-(),k . ,, ,k, , . , , n. . ( ). 33 : = = 0 cos: 0n z z n idA dA N , = = 0 sin: 0n z z n idA dA T .(1) : ( ) 2 cos 121cos2+ = =z z n, 2 sin21cos sinz z n= = . (2) : ( )max0z n n = = = ; 2 4 maxzn n =||

\|= = ,(3) 2 4 zn =||

\|; ( ) 0 0 =n .(4) ,zy, n , n ,4 3 2 1n n n1,n2,n3,n4 n1, n2, n3, n4 . : 1 = , 2 = + 2, 3 = + , 4 = + 3 2, : 3 1 n n = 4 2 n n = , = = = =4 3 2 1 n n n n.(5) = = = =4 3 2 1 n n n n .34 (), (). , . . x = Ex+ |||

\|Ey y =||

\|Ex +Ey

, : ( )( ). 0,1,1=+ = =xyy x yy x xEE (6) : 1.( ) ( ) , 0 cos s dA sin cos dA: 0n= + + = in dA Ni 2.( ) ( ) . 0 sin s dA cos cos dA: 0n= + = in dA Ti (7) 35 : 2 sin cos sin 2 = =n, ( ) 2 cos sin cos2 2 = =n,(8) : 2 0 : 0 = = =n; 4 :max m = =n; 4 : 0 = =n; 2 0 :max = = =n. =4, , . , ., . . (6), x , y , : ( ). 0, 1,=+ = == =xyy xy xE (9) . : ( ) =||

\| = 1 1 dddd d d , : ( )( ) +=+ = = =||

\|1111'' '2 '2 ' '2 4 ddddddtg . (10) , , 2 2 tg . : 36 ++= +=||

\|2221212 412 42 4tgtgtg tgtg tgtg ,(11) : +=+2211,(12) : = 2 . (13) (9) ( ) =+ = GE1 2 , = G (14) ( )[ ]2L /F 1 2 + =EG . . ( ) .,, , , . . , : , , , . 37 2. () . , , . : : M t = const[ [[ [F L] ]] ], M t const[ [[ [F L] ]] ] , : () : m = dM / dz [ [[ [F L / L] ]] ] mt = const. , m t = m t (z). , , .,,o , . 38 0 =t zM M ,0 = = =y xT T N ,0 = =y xM M . 1. - 0z , 0zx 0zy , 0z= , - z

2. - , , - . 3. - r rG =, R RG =.(1) A= A (z). ., , : 39 1.0 = =Ax zxT dA , 2.0 = =Ay zyT dA , 6.( ) = = A At z zx zyM dA dA y x .(2) ( ) dz d A A = = ( ) = =dzd,(3) dzd = - ( ). : ( ) ( ) ( ) z G G zz = = , .(4) ( (( (z) )) ) ( ) ( ) z M dA zAt z = ,,(5) (4)(5), : ( )( )( ) z I Gz Mzt0= ,(6)

: ( )( )max0max|||

\|= z I Gz Mt . M t (z) = M t = const, G I0 (z) = G I0 = const, ( ) constI GMzt== = 0 , 0maxmaxI GMt= , . 40 ( (( (z) )) ) ( )( )( )( )dzz dz I Gz Mzt == 0 : ( ) ( )( )( )dzz I Gz Mdz z zztz = =000 , : ( ) ( )( )( )dzz I Gz Mdz z l zltl = = =000 .(7) M t (z) = M t = constG I0 (z) = G I0 = const, : ( ) constI GMzt== = 0 ,( ) zI GMz zt= =0 ,( ) lI GMl l zt= = =0 [rad], (8) (G I0) / l () . z (3) ( )( )( ) =z Iz Mztz0, .(: 0 R( r))(9) . 5.8 ( )( )( )max0max =z Iz Mzt, (10) 41 : ( ) ( )( ) z Wrz I z I00max0= = [cm3] - .(11) . , ( )( )( ) z Wz Mzt0max= , (12) : ( )( )max0max|||

\|=z Wz Mt . : I0 (z) = I0 = const, W0 (z) = const, 0maxmaxWMt= . .................................................................................................................................. ( )( )( )dtz Wz Mz |||

\|=max0max, ( ) z A A = , (13) dtWM 0max, const A = . ( )( )( )dtz I Gz Mz |||

\|= max0max.(14) ( ) ( ) ( )d tz W z M 0- ,(15) ( ) ( )d tz I G z M 0 - . ( )( )dtz Mz W0, dtMWmax0- , (16) ( )( )dtGz Mz I 0, dtGMI max0 - . (17) ................................................................................................................................... 42 ( )( )( )dtz Wz Mz =0max. ) ( )( )( )( ) ( )dtz M z dz dz Iz W= =16 2300, : ( )( ) ( )3 35 16dtdtz M z Mz d [cm]. (18) ) ( )4300116 2 = =ddIW , d d0= , ( )( )( )( )( )343415116 dtdtz M z Mz d[cm].(19) ( )( )( )dtz I Gz Mz = 0max ) ( )( ) ( )dtGz M z dz I =3240, : ( )( )4] [32cm rad Gz Mz ddt [cm].(20) 43 / m . rad / cm, / m rad / cm : 100 180((

=((

m cmraddo ) ( )( )( )441 ] [32 cm rad Gz Mz ddt[cm].(21) . - , , . : - , - , .