Iv describ e the po int x a bout whi ch yo u take mom

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Unformatted text preview: +I--,---, - IV\ /. Describ e the po int x a bout whi ch yo u take mom ent s: Solution (in terms o f M f, L, q and x) M(x) =- - - --'-"- Y - - -- 1 - - - - - - - - - - - - -- pt) ( I pt) x - - - - ( I pt) =0 (I _ x= (I pt ) - (I pts) 2. (40 points). C h, 9 I ntegration M ethod : A simply supported beam A B s upporting a distributed load o f m aximum intensity q=PIL is shown in the figure . Assume t hat Y oung's m odulus E and the moment o f i nertia I are given. Determine the equation for q(x), Use the integration method as presented in class to derive the equation o f the deflection curve O(x) for this structure as well as for V(x), M(x), a nd {)(x). F urthermore, obtain formulas for the deflection 8c at point C and for the angle o f rotations (JA and (Jo at the ends A and B . D etermine the reactions R Ay and R By at the ends. T he forces R Ay and R By m ust point in the positive y direction. (a). (4 pts) Write down the four b oundary c onditions that apply for this structure. (b). (4 pts) Write down the four differential equations that are needed in solving this problem: (b) 4pts (a) 4 pts I») Diff. Eq. #1 : B .C.#I: Men)::: () ~) ;. ­ q B .C. #2: a.c #3: B.C. #4: rC l) .4~B 0' ( D) - 0 Q ~ M G· ) - v I' ,) J )( u:­ Diff. Eq. #2: q=PIL .~ A.(t- u.) _ t-A. 6 ) .­ &. 'I ­ t:. r --------'Diff. Eq. # 3: cL~:' z: Diff Eq. #4: x~ -----_----=:..:~- (14 pts) Method: [For full credit, do n ot t ake short cuts. Show all y our work.] U se n ext p age as n eeded ~(y )~ e r P = J \I C:~ ") ~ y = - U'/( l -;- (' M ( 'i ) M =' ) - ~ (y: );:: -- fjt.­ P I\..­ ;:'} ~ \ )! 1../(1- ) ;:.. : c.<... ' \ z.. L \ Co ) ~ () "9 '2. z: - PL 1. 2-1­ GL - ----- ­ -- - ...... .1 ( -~ + ~t f ~'E 'I: 'b(f) =­ 0 ( 1-­ - \VI LL') :;.. ( j ~ P ~ J 7. . + ~ ( 'f ) :: M -p/ + c , -\ ~C r ) k~ .: . 4­ b L. ~fJ( Y ) d 'f ~ be 0.) -::­ 0 ::=) { o (L.) :: D r.: Write y our final a nswers h ere: (18 pts) R Ay ~ ,h "" = V(x) = - . PVk. t j M (x) = 8(x) = - py.l/2 L -i-J.......... t:r: (ff (2pts) - ' --'--- + - --""""""""'--->--.....J:L= -_ _-=+---_ ~::t---<L_ W ..L (_ 0i­ l ')=-.1 . O(x) =_ =....c._------" ..40 I~_ ,- P 24 -c: T Lr be = (2 pts) _ _-'----+-'-'"-----'--_--"--"' ''''--- ...,-----"-- _ ' _ =_ _ (2 p ts) - - - - - - ~ r; P L-I---.~'-=--_"'____r"'__ ' / 'S 9 4 - - - - - - - - - - - - (2 p ts) (2 pts) 2 {g el< ) 3. (40 points) . Discontinuitv function Method: A cantilever beamACB supporting a distributed load as shown in the figure. Assume that Young's modulus E and the moment of inertia I are given. Determine the equation for qtx). Use the discontinuity function method as presented in class to derive the equation of the deflection curve 8(x) for this structure as well as for V(.\"), M (x), and ()(x). Furthermore, obtain formulas for the deflection ~ at point B and for the angle of rotations at the end B. Determine the reactions MA and RAJ' at the wall. The force R Ay must point in the positive y direction and the moment MA must point in the positive z direction. en (a). (4 pts) Write down the f our boundary conditions that appl y for this structure. (b). (4 p ts) Write down the f our differential equations that are need ed in s o lving thi s problem: ( a) 4 pts ( b) 4pt s Diff. Eg. # I : B.C. #1: L) - 0 )Ie q=PIL Diff. Eq. #2 : ; . M('II- ) \J(~) Diff. Eq. #3 : d-@{~) M ( ~) B .C. #2: " ,,(\..)-:;0 B.C. #3: 1 t(o) ::. O B .C. #4: j i ( 0 ) ,;.. Diff. Eq . #4 : C x =U2 g>< = --;r; J a(lt) '";T; x=L 0 PL/lf (2 pts) (2 pts) (2 pts) (2 pts) 4 ~ ~ ­:.r e 4 (7<)...
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This note was uploaded on 09/18/2013 for the course AME 3143 taught by Professor Staff during the Fall '11 term at The University of Oklahoma.

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