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belek pryzmatycznych na spr\352\277ystym podlo\277u Prof. Marcin Kaminski, D.Sc., Ph.D.Chair of Mechanics of Materials, Technical University of L\363dz, Al. Politechniki 6, 93-590 L\363dz, POLAND, email: Marcin.Kaminski@p.lodz.pl L\363dz, January 2008 Abstract: This script is entirely devoted to the static problem of a deflection of the linear elastic istotropic prismatic beams resting on the homogeneous linear elastic and isotropic foundation. It is demonstrated below how one can symbolically solve the problems of such beams consisting of one, two and three different intervals with spatially varying distributed load applied along the beam, different concentrated forces and moments. It is also possible to define some variability of the bending stiffness as well as the elastic foundation parameter as piecewise constant functions along the beam. This program is able to solve symbolically fourth order differential equations of the beam equilibrium, then - to determine the integration constants from the additional boundary and continuity conditions for the beam intervals. Finally, it computes maxima and minima of deflections, bending moments and shear forces and prepares the diagrams of all those functions. This script may be easily modified to extend the beam analysis towards the structural sensitivity studies and/or reliability analysis for the beams resting on the elastic subsoils. Keywords: elastic beams, elastic foundation, ordinary differential equations, beam diagrams
<Text-field layout="Heading 1" style="Heading 1"><Font encoding="ISO8859-1"> Belka jednorodna na jednorodnym podlo\277u poddana obci\271\277eniu stalemu rownomiernie rozlo\277onemu </Font></Text-field>restart: with(plots): with(Optimization): Warning, the name changecoords has been redefined Analizujemy tutaj lini\352 ugi\352cia dla belki pryzmatycznej o stalym przekroju le\277\271cej na winklerowskim spr\352\277ystym podlo\277u. Material, z kt\363rego wykonana jest belka jest r\363wnie\277 liniowo-spr\352\277ysty i izotropowy. R\363wnanie r\363\277niczkowe opisuj\271ce poszukiwane ugi\352cie ma posta\346 nast\352puj\271c\271: Diff(y(x),x$4)-4*beta^4*y(x)-q/EJ=0;NiMvLCgtJSVEaWZmRzYkLSUieUc2IyUieEctJSIkRzYkRisiIiUiIiIqKEYvRjApJSViZXRhR0YvRjBGKEYwISIiKiYlInFHRjAlI0VKR0Y0RjQiIiE=gdzie beta=sqrt(sqrt(k/4/EJ));NiMvJSViZXRhRywkKigiIiMhIiJGJyMiIiJGJyomJSJrR0YqJSNFSkdGKCNGKiIiJUYqWarunki brzegowe niezb\352dne do obliczenia stalych calkowania to bc1,bc2 odpowiadaj\271ce lewemu ko\361cowi belki oraz bc3, bc4 dla prawego ko\361ca belki. Inne warunki brzegowe, jak np. zerowanie si\352 k\271ta ugi\352cia na osi symetrii belki nie s\271 tutaj konieczne. de:=diff(y(x),x$4)-4*beta^4*y(x)-q/EJ=0;NiM+JSNkZUcvLCgtJSVkaWZmRzYkLSUieUc2IyUieEctJSIkRzYkRi0iIiUiIiIqKEYxRjIpJSViZXRhR0YxRjJGKkYyISIiKiYlInFHRjIlI0VKR0Y2RjYiIiE=Znajdujemy najpierw rozwi\271zanie og\363lne dla r\363wnania r\363\277niczkowego linii ugi\352cia dsolve(de,y(x)); assign(%);NiMvLSUieUc2IyUieEcsLCoqIiIlISIiJSViZXRhRyEiJSUjRUpHRislInFHIiIiRisqJiUkX0MxR0YwLSUkY29zRzYjKigiIiMjRjBGN0YsRjBGJ0YwRjBGMComJSRfQzJHRjAtJSRleHBHRjVGMEYwKiYlJF9DM0dGMC0lJHNpbkdGNUYwRjAqJiUkX0M0R0YwLUY8NiMsJEY2RitGMEYwKolejno stosujemy warunki brzegowe dla lewego brzegu wla\305\223ciwe dla swobodnego ko\303\261ca bc1:=simplify(subs(x=0,y(x))=0);NiM+JSRiYzFHLywkKiYjIiIiIiIlRikqKCwsJSJxRyEiIiosRipGKSUkX0MxR0YpLSUkY29zRzYjIiIhRikpJSViZXRhR0YqRiklI0VKR0YpRikqKkYqRiklJF9DMkdGKUY1RilGN0YpRikqLEYqRiklJF9DM0dGKS0lJHNpbkdGM0YpRjVGKUY3RilGKSoqRipGKSUkX0M0R0YpRjVGKUY3RilGKUYpRjYhIiVGN0YuRilGKUY0bc2:=simplify(subs(x=L,y(x))=0);NiM+JSRiYzJHLywkKiYjIiIiIiIlRikqKCwsJSJxRyEiIiosRipGKSUkX0MxR0YpLSUkY29zRzYjKigiIiMjRilGNSUlYmV0YUdGKSUiTEdGKUYpKUY3RipGKSUjRUpHRilGKSosRipGKSUkX0MyR0YpLSUkZXhwR0YzRilGOUYpRjpGKUYpKixGKkYpJSRfQzNHRiktJSRzaW5HRjNGKUY5RilGOkYpRikqLEYqRiklJF9DNEdGKS1GPjYjLCRGNEYuRilGOUYpRjpGKUYpRilGNyEiJUY6Ri5GKUYpIiIha tak\277e warunki brzegowe dla ko\361ca prawego bc3:=simplify(subs(x=0,diff(y(x),x$2))=0);NiM+JSRiYzNHLywkKigiIiMiIiIpJSViZXRhR0YoRiksKiomJSRfQzFHRiktJSRjb3NHNiMiIiFGKUYpJSRfQzJHISIiKiYlJF9DM0dGKS0lJHNpbkdGMUYpRiklJF9DNEdGNEYpRjRGMg==bc4:=simplify(subs(x=L,diff(y(x),x$2))=0);NiM+JSRiYzRHLywkKigiIiMiIiIpJSViZXRhR0YoRiksKiomJSRfQzFHRiktJSRjb3NHNiMqKEYoI0YpRihGK0YpJSJMR0YpRilGKSomJSRfQzJHRiktJSRleHBHRjFGKSEiIiomJSRfQzNHRiktJSRzaW5HRjFGKUYpKiYlJF9DNEdGKS1GODYjLCRGMkY5RilGOUYpRjkiIiE=Potem wyznaczamy warto\305\223ci tych stalych poprzez rozwiazanie nastepuj\302\271cego ukladu r\303\263wna\303\261: solve({bc1,bc2,bc3,bc4},{_C1,_C2,_C3,_C4}): assign(%):Uzyskujemy w ten spos\363b ostateczn\271 lini\352 ugi\352cia dla rozpatrywanej belki. simplify(y(x)): W celu sporz\302\271dzenia wykres\303\263w wprowadzamy okre\305\223lone warto\305\223ci parametr\303\263w gruntu oraz belki - k, m, EJ oraz L. Znajdujemy jednocze\305\223nie warto\305\223ci liczbowe odpowiadaj\302\271ce ekstremom poszczeg\303\263lnych funkcji. k:=1: q:=1: EJ:=1: L:=1: beta:=sqrt(sqrt(k/4/EJ)): Wykres ugi\352cia belki jest nast\352puj\271cy: y:=y(x): fmax:=Maximize(y,x=0..L): evalf(fmax); fmin:=Minimize(y,x=0..L): evalf(fmin); p1:=plot(-y,x=0..L, title=`Ugi\352cie belki`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},font=[TIMES, BOLD, 16]);Kolejno sporz\271dzamy wykres k\271t\363w ugi\352cia dla tej belki s:=diff(y,x): plot(s,x=0..L, title=`K\271t ugi\352cia belki`,axes=boxed,font=[TIMES, BOLD, 16]);Nast\352pnie mamy wykres momentu zginaj\271cego m:=-EJ*diff(y,x$2): Mmax:=Maximize(m,x=0..L): evalf(Mmax); Mmin:=Minimize(m,x=0..L): evalf(Mmin); plot(m,x=0..L, title=`Moment zginaj\271cy`,axes=boxed,font=[TIMES, BOLD, 16]);oraz wykres sily tn\302\271cej na dlugo\305\223ci belki t:=-EJ*diff(y,x$3): Tmax:=Maximize(t,x=0..L): evalf(Tmax); Tmin:=Minimize(t,x=0..L): evalf(Tmin); p1:=plot(t,x=0..L, title=`Sila tn\271ca`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},font=[TIMES, BOLD, 16]);NiM3JCQiK2wwazo4ISM2NyMvJSJ4RyQiKzAsKytdISM1NiM3JCQhK2l2YnZGISNFNyMvJSJ4RyQiIiIiIiE=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Jak wynika z tych wykres\363w, funkcja ugi\352cia i momentu zginaj\271cego s\271 symetryczne wzgl\352dem osi x=L/2, podczas gdy wykres k\302\271ta ugi\303\252cia oraz sily tn\302\271cej jako pochodne rz\303\252du nieparzystego s\302\271 antysymetryczne wzgl\303\252dem tej samej prostej. Warto\305\223\303\246 maksymaln\302\271 ugi\303\252cia otrzymujemy oczywi\305\223cie dla warto\305\223ci x, dla kt\303\263rej k\302\271t jest r\303\263wny 0 i zmienia znak w otoczeniu tego punktu (moment zginaj\302\271cy i sia tn\302\271ca wykazuj\302\271 podobn\302\271 zale\302\277no\305\223\303\246).
<Text-field layout="Heading 1" style="Heading 1"><Font encoding="ISO8859-1"> Jednorodna belka obci\271\277ona sil\271 skupion\271 oraz skupionym momentem oraz dwoma rodzajami obcia\277enia r\363wnomiernie rozlo\277onego </Font></Text-field>restart: with(plots): with(Optimization): Warning, the name changecoords has been redefined Ze wzgl\303\252du na zmienno\305\223\303\246 obci\302\271\302\277enia na dlugo\305\223ci belki dzielimy j\302\271 na dwa rozlaczne przedzialy (gdzie obowi\302\271zuj\302\271 r\303\263\302\277ne r\303\263wnania r\303\263\302\277niczkowe ugi\303\252cia), co odzwierciedla zmiany w sztywno\305\223ci gi\303\252tnej, parametrach podlo\302\277a a tak\302\277e r\303\263\302\277ne warunki brzegowe. Zasadnicz\302\271 r\303\263\302\277nic\302\271 w stosunku do poprzednio rozwi\302\271zanego zadania jest obecno\305\223\303\246 obci\302\271\302\277e\303\261 r\303\263wnomiernie rozlo\302\277onych na dlugo\305\223ci przedzial\303\263w - trapezowego q1(x) (staego + trojkatnego) na pierwszym odcinku oraz stalego q2(x) na odcinku drugim. R\303\263wnania r\303\263\302\277niczkowe ugi\303\252cia oznaczamy jako de1 oraz de2, w kt\303\263rych wyst\303\252puj\302\271 stale _C1,..., _C8 (po 4 stale dla ka\302\277dego odcinka). Warunki brzegowe bc1,bc2 odzwierciedlaj\302\271 sytuacj\303\252 na lewym ko\303\261cu schematu statycznego, natomiast bc3, bc4 - na prawym ko\303\261cu ukladu. W warunkach bc5, bc6, bc7, bc8 rozpoznajemy warunki zszycia dla punktu wsp\303\263lnego obydwu linii ugi\303\252cia, czyli dla x=L1. de1:=diff(y[1](x),x$4)-4*beta1^4*y[1](x)-q1(x)/EJ1=0;NiM+JSRkZTFHLywoLSUlZGlmZkc2JC0mJSJ5RzYjIiIiNiMlInhHLSUiJEc2JEYwIiIlRi4qKEY0Ri4pJSZiZXRhMUdGNEYuRipGLiEiIiomLSUjcTFHRi9GLiUkRUoxR0Y4RjgiIiE=de2:=diff(y[2](x),x$4)-4*beta2^4*y[2](x)-q2(x)/EJ2=0;NiM+JSRkZTJHLywoLSUlZGlmZkc2JC0mJSJ5RzYjIiIjNiMlInhHLSUiJEc2JEYwIiIlIiIiKihGNEY1KSUmYmV0YTJHRjRGNUYqRjUhIiIqJi0lI3EyR0YvRjUlJEVKMkdGOUY5IiIhDefiniujemy obci\271\277enia q1(x) oraz q2(x) zgodnie z wprowadzonym schematem statycznym. Mamy q0:=5: q1(x):=q0*(1+2*x/L1): q2(x):=q0: Kolejno otrzymujemy rozwi\271zania og\363lne dla tych rowna\361 ro\277niczkowych jako dsolve(de1,y[1](x)); assign(%);dsolve(de2,y[2](x)); assign(%);NiMvLSYlInlHNiMiIiI2IyUieEcsLCouIiImRigiIiUhIiIsJiUjTDFHRigqJiIiI0YoRipGKEYoRiglJmJldGExRyEiJUYxRi8lJEVKMUdGL0YvKiYlJF9DMUdGKC0lJGV4cEc2IyooRjMjRihGM0Y0RihGKkYoRihGKComJSRfQzJHRigtJSRzaW5HRjtGKEYoKiYlJF9DM0dGKC0lJGNvc0dGO0YoRigqJiUkX0M0R0YoLUY6NiMsJEY8Ri9GKEYoNiMvLSYlInlHNiMiIiM2IyUieEcsLCoqIiImIiIiIiIlISIiJSZiZXRhMkchIiUlJEVKMkdGMEYwKiYlJF9DMUdGLi0lJGV4cEc2IyooRigjRi5GKEYxRi5GKkYuRi5GLiomJSRfQzJHRi4tJSRzaW5HRjhGLkYuKiYlJF9DM0dGLi0lJGNvc0dGOEYuRi4qJiUkX0M0R0YuLUY3NiMsJEY5RjBGLkYuWprowadzamy nowe stale calkowania dla drugiej linii ugi\352cia: _C5 .. _C8 zamiast _C1 .. _C4y[2](x):=subs(_C1=_C5,_C2=_C6,_C3=_C7,_C4=_C8,y[2](x)); NiM+LSYlInlHNiMiIiM2IyUieEcsLCoqIiImIiIiIiIlISIiJSZiZXRhMkchIiUlJEVKMkdGMEYwKiYlJF9DNUdGLi0lJGV4cEc2IyooRigjRi5GKEYxRi5GKkYuRi5GLiomJSRfQzZHRi4tJSRzaW5HRjhGLkYuKiYlJF9DN0dGLi0lJGNvc0dGOEYuRi4qJiUkX0M4R0YuLUY3NiMsJEY5RjBGLkYuStosujemy kolejno warunki brzegowe typowe dla swobodnego ko\361ca na lewym brzegu bc1:=simplify(subs(x=0,diff(y[1](x),x$2))=0);NiM+JSRiYzFHLywkKigiIiMiIiIpJSZiZXRhMUdGKEYpLColJF9DMUchIiIqJiUkX0MyR0YpLSUkc2luRzYjIiIhRilGKSomJSRfQzNHRiktJSRjb3NHRjNGKUYpJSRfQzRHRi5GKUYuRjQ=bc2:=simplify(subs(x=0,diff(y[1](x),x$3))=0);NiM+JSRiYzJHLywkKioiIiMiIiIpJSZiZXRhMUciIiRGKUYoI0YpRigsKCUkX0MxRyEiIiUkX0MyR0YpJSRfQzRHRilGKUYwIiIhi kolejno takie same warunki dla prawego brzegu bc3:=simplify(subs(x=L1+L2,diff(y[2](x),x$2))=0);NiM+JSRiYzNHLywkKigiIiMiIiIpJSZiZXRhMkdGKEYpLCoqJiUkX0M1R0YpLSUkZXhwRzYjKihGKCNGKUYoRitGKSwmJSNMMUdGKSUjTDJHRilGKUYpRikqJiUkX0M2R0YpLSUkc2luR0YxRikhIiIqJiUkX0M3R0YpLSUkY29zR0YxRilGOyomJSRfQzhHRiktRjA2IywkRjJGO0YpRilGKUYpIiIhbc4:=simplify(subs(x=L1+L2,diff(y[2](x),x$3))=0);NiM+JSRiYzRHLywkKioiIiMiIiIpJSZiZXRhMkciIiRGKUYoI0YpRigsKiomJSRfQzVHRiktJSRleHBHNiMqKEYoRi1GK0YpLCYlI0wxR0YpJSNMMkdGKUYpRilGKSomJSRfQzZHRiktJSRjb3NHRjNGKSEiIiomJSRfQzdHRiktJSRzaW5HRjNGKUYpKiYlJF9DOEdGKS1GMjYjLCRGNEY8RilGPEYpRikiIiE=Nast\303\252pnie wprowadzamy warunki ci\302\271glo\305\223ci poszczeg\303\263lnych funkcji dla x=L, w kt\303\263rych uwzgl\303\252niamy skokow\302\271 zmian\303\252 momentu oraz sily tn\302\271cej w dw\303\263ch ostatnich warunkach bc5:=simplify(subs(x=L1,y[1](x))=subs(x=L1,y[2](x)));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bc6:=simplify(subs(x=L1,diff(y[1](x),x))=subs(x=L1,diff(y[2](x),x)));NiM+JSRiYzZHLywkKiYjIiIiIiIjRikqKiwsIiImISIiKjBGKkYpJSRfQzFHRilGKkYoKSUmYmV0YTFHRi1GKS0lJGV4cEc2IyooRipGKEYyRiklI0wxR0YpRilGN0YpJSRFSjFHRilGKSowRipGKSUkX0MyR0YpLSUkY29zR0Y1RilGKkYoRjFGKUY3RilGOEYpRikqMEYqRiklJF9DM0dGKS0lJHNpbkdGNUYpRipGKEYxRilGN0YpRjhGKUYuKjBGKkYpJSRfQzRHRilGKkYoRjFGKS1GNDYjLCRGNkYuRilGN0YpRjhGKUYuRilGMiEiJUY3Ri5GOEYuRilGKSooRipGKCUmYmV0YTJHRiksKiomJSRfQzVHRiktRjQ2IyooRipGKEZIRilGN0YpRilGKSomJSRfQzZHRiktRjxGTUYpRikqJiUkX0M3R0YpLUZARk1GKUYuKiYlJF9DOEdGKS1GNDYjLCRGTkYuRilGLkYpbc7:=EJ1*simplify(subs(x=L1,diff(y[1](x),x$2))=M+EJ2*subs(x=L1,diff(y[2](x),x$2)));NiM+JSRiYzdHLywkKioiIiMiIiIlJEVKMUdGKSklJmJldGExR0YoRiksKiomJSRfQzFHRiktJSRleHBHNiMqKEYoI0YpRihGLEYpJSNMMUdGKUYpISIiKiYlJF9DMkdGKS0lJHNpbkdGMkYpRikqJiUkX0MzR0YpLSUkY29zR0YyRilGKSomJSRfQzRHRiktRjE2IywkRjNGNkYpRjZGKUY2KiZGKkYpLCwlIk1HRikqLEYoRiklJEVKMkdGKSUkX0M1R0YpKSUmYmV0YTJHRihGKS1GMTYjKihGKEY0RktGKUY1RilGKUYpKixGKEYpRkhGKSUkX0M2R0YpLUY6Rk1GKUZKRilGNiosRihGKUZIRiklJF9DN0dGKS1GPkZNRilGSkYpRjYqLEYoRilGSEYpJSRfQzhHRilGSkYpLUYxNiMsJEZORjZGKUYpRik=bc8:=EJ1*simplify(subs(x=L1,diff(y[1](x),x$3))=P+EJ2*subs(x=L1,diff(y[2](x),x$3)));NiM+JSRiYzhHLywkKiwiIiMiIiIlJEVKMUdGKSklJmJldGExRyIiJEYpRigjRilGKCwqKiYlJF9DMUdGKS0lJGV4cEc2IyooRihGLkYsRiklI0wxR0YpRikhIiIqJiUkX0MyR0YpLSUkY29zR0Y0RilGKSomJSRfQzNHRiktJSRzaW5HRjRGKUY3KiYlJF9DNEdGKS1GMzYjLCRGNUY3RilGKUYpRjcqJkYqRiksLCUiUEdGKSouRihGKSUkRUoyR0YpJSRfQzVHRikpJSZiZXRhMkdGLUYpRihGLi1GMzYjKihGKEYuRkxGKUY2RilGKUYpKi5GKEYpRklGKSUkX0M2R0YpLUY7Rk5GKUYoRi5GS0YpRjcqLkYoRilGSUYpJSRfQzdHRiktRj9GTkYpRihGLkZLRilGKSouRihGKUZJRiklJF9DOEdGKUZLRilGKEYuLUYzNiMsJEZPRjdGKUY3Rik=Ostatecznie wyznaczamy warto\305\223ci wszystkich stalych calkowania jako rozwi\302\271zanie dla nast\303\252puj\302\271cego ukladu r\303\263wna\303\261 solve({bc1,bc2,bc3,bc4,bc5,bc6,bc7,bc8},{_C1,_C2,_C3,_C4,_C5,_C6,_C7,_C8}): assign(%):Upraszczamy posta\346 funkcji ugi\352cia simplify(y[1](x)): simplify(y[2](x)): oraz definiujemy warto\305\223ci parametr\303\263w k, m, EJ, L w celu wykonania odpowiednich rysunk\303\263w k1:=1: q1:=1: EJ1:=1: L1:=1: k2:=1: q2:=1: EJ2:=1: L2:=1: beta1:=sqrt(sqrt(k1/4/EJ1)): beta2:=sqrt(sqrt(k2/4/EJ2)): P:=1: M:=1: L:=L1+L2: Otrzymujemy wykres linii ugi\352cia y1:=y[1](x): y2:=y[2](x): deflection:=piecewise(x<L1,y1,x<L,y2): fmax:=Maximize(deflection,x=0..L): evalf(fmax); fmin:=Minimize(deflection,x=0..L): evalf(fmin); plot(deflection,x=0..L, title=`Linia ugi\352cia belki`,axes=boxed,axes=boxed,font=[TIMES, BOLD, 16]);kolejno znajdujemy wykres k\271t\363w ugi\352cia dla rozpatrywanej belki y1:=diff(y[1](x),x): y2:=diff(y[2](x),x): Slope:=piecewise(x<L1,y1,x<L,y2): plot(Slope,x=0..L, title=`K\271t ugi\352cia belki`,axes=boxed,axes=boxed,font=[TIMES, BOLD, 16]);Wykres momentu zginaj\302\271cego musi mie\303\246 nieci\302\271glo\305\223\303\246 w miejscu polaczenia y1(x) oraz y2(x) o wielko\305\223ci M y1:=-EJ1*diff(y[1](x),x$2): y2:=-EJ2*diff(y[2](x),x$2): moment:=piecewise(x<L1,y1,x<L,y2): Mmax:=Maximize(moment,x=0..L): evalf(Mmax); Mmin:=Minimize(moment,x=0..L): evalf(Mmin); p1:=plot(moment,x=0..L, title=`Moment zginajacy`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},axes=boxed,font=[TIMES, BOLD, 16]); Wykres sily tn\302\271cej musi wykazywa\303\246 podobn\302\271 wla\305\223ciwo\305\223\303\246 - skok o wielko\305\223\303\246 P dla wsp\303\263lrz\303\252dnej x=L1, gdzie przylo\302\277ono obci\302\271\302\277enie zewn\303\252trzney1:=-EJ1*diff(y[1](x),x$3): y2:=-EJ2*diff(y[2](x),x$3): shear:=piecewise(x<L1,y1,x<L,y2): Tmax:=Maximize(shear,x=0..L): evalf(Tmax); Tmin:=Minimize(shear,x=0..L): evalf(Tmin); p1:=plot(shear,x=0..L, title=`Sila tn\271ca`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},axes=boxed,font=[TIMES, BOLD, 16]); 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<Text-field layout="Heading 1" style="Heading 1"><Font encoding="ISO8859-1"> Belka wieloprz\352slowa na spr\352\277ystym podlo\277u </Font></Text-field>restart: with(plots): with(Optimization): with(LinearAlgebra): Warning, the name changecoords has been redefined Na zako\361czenie rozpatrzymy lini\352 ugi\352cia belki tr\363jprz\352slowej pokazanej na powy\277szym rysunku, gdzie dla ka\277dego odcinka obowi\271zuje analogiczne r\363wnanie linii ugi\352cia: Diff(y(x),x$4)-4*beta^4*y(x)-q/EJ=0;NiMvLCgtJSVEaWZmRzYkLSUieUc2IyUieEctJSIkRzYkRisiIiUiIiIqKEYvRjApJSViZXRhR0YvRjBGKEYwISIiKiYlInFHRjAlI0VKR0Y0RjQiIiE=i gdzie oczywi\305\223cie jest beta=sqrt(sqrt(k/4/EJ));NiMvJSViZXRhRywkKigiIiMhIiJGJyMiIiJGJyomJSJrR0YqJSNFSkdGKCNGKiIiJUYqW tym przypadku dokonujemy podzialu calej dlugo\305\223ci belki na trzy rozl\302\271czne podprzedzialy w celu uwzgl\303\252dnienia zmian sztywno\305\223ci, obci\302\271\302\277en i warunkow gruntowych. Zauwazmy, ze jak poprzednio, r\303\263wnanie to redukuje si\303\252 do klasycznego r\303\263wnania linii ugi\303\252cia belki zgodnie z teori\302\271 rz\303\252du pierwszego poprzez podstawienie k=0. R\303\263wnania r\303\263\302\277niczkowe odpowiadaj\302\271ce temu podzialowi to odpowiednio de1, de2 oraz de3. Ich czterokrotne calkowanie powoduje pojawienie sie stalych calkowania C1,..., _C12, kt\303\263rych warto\305\223ci znajdujemy z warunkow brzegowych bc1,bc2 dla lewego ko\303\261ca struktury, warunk\303\263w bc3 oraz bc4 - dla prawego ko\303\261ca belki oraz warunk\303\263w zszycia odpowiadaj\302\271cych wsp\303\263lrz\303\252dnym x=L1 oraz x=L2. de1:=diff(y[1](x),x$4)-4*beta1^4*y[1](x)-q1/EJ1=0;NiM+JSRkZTFHLywoLSUlZGlmZkc2JC0mJSJ5RzYjIiIiNiMlInhHLSUiJEc2JEYwIiIlRi4qKEY0Ri4pJSZiZXRhMUdGNEYuRipGLiEiIiomJSNxMUdGLiUkRUoxR0Y4RjgiIiE=de2:=diff(y[2](x),x$4)-4*beta2^4*y[2](x)-q2/EJ2=0;NiM+JSRkZTJHLywoLSUlZGlmZkc2JC0mJSJ5RzYjIiIjNiMlInhHLSUiJEc2JEYwIiIlIiIiKihGNEY1KSUmYmV0YTJHRjRGNUYqRjUhIiIqJiUjcTJHRjUlJEVKMkdGOUY5IiIhde3:=diff(y[3](x),x$4)-4*beta3^4*y[3](x)-q3/EJ3=0;NiM+JSRkZTNHLywoLSUlZGlmZkc2JC0mJSJ5RzYjIiIkNiMlInhHLSUiJEc2JEYwIiIlIiIiKihGNEY1KSUmYmV0YTNHRjRGNUYqRjUhIiIqJiUjcTNHRjUlJEVKM0dGOUY5IiIhKolejno wyznaczamy rozwi\271zania og\363lne dla tych r\363wna\361 jako dsolve(de1,y[1](x)); assign(%);dsolve(de2,y[2](x)); assign(%);NiMvLSYlInlHNiMiIiI2IyUieEcsLCoqIiIlISIiJSZiZXRhMUchIiUlJEVKMUdGLiUjcTFHRihGLiomJSRfQzFHRigtJSRzaW5HNiMqKCIiIyNGKEY5Ri9GKEYqRihGKEYoKiYlJF9DMkdGKC0lJGV4cEdGN0YoRigqJiUkX0MzR0YoLSUkY29zR0Y3RihGKComJSRfQzRHRigtRj42IywkRjhGLkYoRig=NiMvLSYlInlHNiMiIiM2IyUieEcsLCoqIiIlISIiJSZiZXRhMkchIiUlJEVKMkdGLiUjcTJHIiIiRi4qJiUkX0MxR0YzLSUkZXhwRzYjKihGKCNGM0YoRi9GM0YqRjNGM0YzKiYlJF9DMkdGMy0lJHNpbkdGOEYzRjMqJiUkX0MzR0YzLSUkY29zR0Y4RjNGMyomJSRfQzRHRjMtRjc2IywkRjlGLkYzRjM=dsolve(de3,y[3](x)); assign(%);NiMvLSYlInlHNiMiIiQ2IyUieEcsLCoqIiIlISIiJSZiZXRhM0chIiUlJEVKM0dGLiUjcTNHIiIiRi4qJiUkX0MxR0YzLSUkZXhwRzYjKigiIiMjRjNGOkYvRjNGKkYzRjNGMyomJSRfQzJHRjMtJSRzaW5HRjhGM0YzKiYlJF9DM0dGMy0lJGNvc0dGOEYzRjMqJiUkX0M0R0YzLUY3NiMsJEY5Ri5GM0YzWprowadzamy stale _C5 .. _C8 zamiast _C1 .. _C4 dla okre\305\223lenia funkcji y2(x) oraz stale _C9 .. _C12 zamiast _C1 .. _C4 - w celu zdefiniowania linii ugi\303\252cia y3(x) y[2](x):=subs(_C1=_C5,_C2=_C6,_C3=_C7,_C4=_C8,y[2](x)); NiM+LSYlInlHNiMiIiM2IyUieEcsLCoqIiIlISIiJSZiZXRhMkchIiUlJEVKMkdGLiUjcTJHIiIiRi4qJiUkX0M1R0YzLSUkZXhwRzYjKihGKCNGM0YoRi9GM0YqRjNGM0YzKiYlJF9DNkdGMy0lJHNpbkdGOEYzRjMqJiUkX0M3R0YzLSUkY29zR0Y4RjNGMyomJSRfQzhHRjMtRjc2IywkRjlGLkYzRjM=y[3](x):=subs(_C1=_C9,_C2=_C10,_C3=_C11,_C4=_C12,y[3](x));NiM+LSYlInlHNiMiIiQ2IyUieEcsLCoqIiIlISIiJSZiZXRhM0chIiUlJEVKM0dGLiUjcTNHIiIiRi4qJiUkX0M5R0YzLSUkZXhwRzYjKigiIiMjRjNGOkYvRjNGKkYzRjNGMyomJSVfQzEwR0YzLSUkc2luR0Y4RjNGMyomJSVfQzExR0YzLSUkY29zR0Y4RjNGMyomJSVfQzEyR0YzLUY3NiMsJEY5Ri5GM0YzWprowadzamy warunki brzegowe typowe dla swobodnego ko\361ca na lewym ko\361cu belki bc1:=simplify(subs(x=0,diff(y[1](x),x$2))=0); NiM+JSRiYzFHLywkKigiIiMiIiIpJSZiZXRhMUdGKEYpLCoqJiUkX0MxR0YpLSUkc2luRzYjIiIhRilGKSUkX0MyRyEiIiomJSRfQzNHRiktJSRjb3NHRjFGKUYpJSRfQzRHRjRGKUY0RjI=bc2:=simplify(subs(x=0,diff(y[1](x),x$3))=0);NiM+JSRiYzJHLywkKioiIiMiIiJGKCNGKUYoKSUmYmV0YTFHIiIkRiksKCUkX0MxR0YpJSRfQzJHISIiJSRfQzRHRilGKUYxIiIhoraz warunki na prawym brzegu typowe dla swobodnego ko\361ca bc3:=simplify(subs(x=L,diff(y[3](x),x$2))=0);NiM+JSRiYzNHLywkKigiIiMiIiIpJSZiZXRhM0dGKEYpLCoqJiUkX0M5R0YpLSUkZXhwRzYjKihGKCNGKUYoRitGKSUiTEdGKUYpRikqJiUlX0MxMEdGKS0lJHNpbkdGMUYpISIiKiYlJV9DMTFHRiktJSRjb3NHRjFGKUY5KiYlJV9DMTJHRiktRjA2IywkRjJGOUYpRilGKUYpIiIhbc4:=simplify(subs(x=L,diff(y[3](x),x$3))=0);NiM+JSRiYzRHLywkKioiIiMiIiIpJSZiZXRhM0ciIiRGKUYoI0YpRigsKiomJSRfQzlHRiktJSRleHBHNiMqKEYoRi1GK0YpJSJMR0YpRikhIiIqJiUlX0MxMEdGKS0lJGNvc0dGM0YpRikqJiUlX0MxMUdGKS0lJHNpbkdGM0YpRjYqJiUlX0MxMkdGKS1GMjYjLCRGNEY2RilGKUYpRjYiIiE=Kolejno wprowadzamy warunki ci\302\271glo\305\223ci dla poszczeg\303\263lnych pochodnych funkcji ugi\303\252\303\246 y1(x) i y2(x) dla wsp\303\263lrz\303\252dnych x=L oraz zszycia funkcji y2 oraz y3 w x=2L bc5:=simplify(subs(x=L1,y[1](x))-subs(x=L1,y[2](x))=0);NiM+JSRiYzVHLywkKiYjIiIiIiIlRikqLCw2KiglI3ExR0YpKSUmYmV0YTJHRipGKSUkRUoyR0YpISIiKjBGKkYpJSRfQzFHRiktJSRzaW5HNiMqKCIiIyNGKUY5JSZiZXRhMUdGKSUjTDFHRilGKSlGO0YqRiklJEVKMUdGKUYvRilGMUYpRikqMEYqRiklJF9DMkdGKS0lJGV4cEdGN0YpRj1GKUY+RilGL0YpRjFGKUYpKjBGKkYpJSRfQzNHRiktJSRjb3NHRjdGKUY9RilGPkYpRi9GKUYxRilGKSowRipGKSUkX0M0R0YpLUZCNiMsJEY4RjJGKUY9RilGPkYpRi9GKUYxRilGKSooJSNxMkdGKUY9RilGPkYpRikqMEYqRiklJF9DNUdGKS1GQjYjKihGOUY6RjBGKUY8RilGKUY9RilGPkYpRi9GKUYxRilGMiowRipGKSUkX0M2R0YpLUY2RlFGKUY9RilGPkYpRi9GKUYxRilGMiowRipGKSUkX0M3R0YpLUZGRlFGKUY9RilGPkYpRi9GKUYxRilGMiowRipGKSUkX0M4R0YpLUZCNiMsJEZSRjJGKUY9RilGPkYpRi9GKUYxRilGMkYpRjshIiVGPkYyRjBGaG5GMUYyRilGKSIiIQ==bc6:=simplify(subs(x=L1,diff(y[1](x),x))-subs(x=L1,diff(y[2](x),x))=0);NiM+JSRiYzZHLywkKiYiIiMjIiIiRigsMiooJSRfQzFHRiotJSRjb3NHNiMqKEYoRiklJmJldGExR0YqJSNMMUdGKkYqRjJGKiEiIiooJSRfQzJHRipGMkYqLSUkZXhwR0YwRipGNCooJSRfQzNHRiotJSRzaW5HRjBGKkYyRipGKiooJSRfQzRHRipGMkYqLUY4NiMsJEYxRjRGKkYqKiglJF9DNUdGKiUmYmV0YTJHRiotRjg2IyooRihGKUZERipGM0YqRipGKiooJSRfQzZHRiotRi9GRkYqRkRGKkYqKiglJF9DN0dGKi1GPEZGRipGREYqRjQqKCUkX0M4R0YqRkRGKi1GODYjLCRGR0Y0RipGNEYqRjQiIiE=bc7:=simplify(subs(x=L1,EJ1*diff(y[1](x),x$2))-subs(x=L1,EJ2*diff(y[2](x),x$2))=0);NiM+JSRiYzdHLywyKiwiIiMiIiIlJEVKMUdGKSUkX0MxR0YpLSUkc2luRzYjKihGKCNGKUYoJSZiZXRhMUdGKSUjTDFHRilGKSlGMUYoRikhIiIqLEYoRilGKkYpJSRfQzJHRilGM0YpLSUkZXhwR0YuRilGKSosRihGKUYqRiklJF9DM0dGKS0lJGNvc0dGLkYpRjNGKUY0KixGKEYpRipGKSUkX0M0R0YpRjNGKS1GODYjLCRGL0Y0RilGKSosRihGKSUkRUoyR0YpJSRfQzVHRikpJSZiZXRhMkdGKEYpLUY4NiMqKEYoRjBGRkYpRjJGKUYpRjQqLEYoRilGQ0YpJSRfQzZHRiktRi1GSEYpRkVGKUYpKixGKEYpRkNGKSUkX0M3R0YpLUY8RkhGKUZFRilGKSosRihGKUZDRiklJF9DOEdGKUZFRiktRjg2IywkRklGNEYpRjQiIiE=bc8:=simplify(subs(x=L1,EJ1*diff(y[1](x),x$3))-subs(x=L1,EJ2*diff(y[2](x),x$3))=P);NiM+JSRiYzhHLywkKigiIiMiIiJGKCNGKUYoLDIqKiUkRUoxR0YpJSRfQzFHRiktJSRjb3NHNiMqKEYoRiolJmJldGExR0YpJSNMMUdGKUYpKUYzIiIkRikhIiIqKkYtRiklJF9DMkdGKUY1RiktJSRleHBHRjFGKUYpKipGLUYpJSRfQzNHRiktJSRzaW5HRjFGKUY1RilGKSoqRi1GKSUkX0M0R0YpRjVGKS1GOzYjLCRGMkY3RilGNyoqJSRFSjJHRiklJF9DNUdGKSklJmJldGEyR0Y2RiktRjs2IyooRihGKkZJRilGNEYpRilGNyoqRkZGKSUkX0M2R0YpLUYwRktGKUZIRilGKSoqRkZGKSUkX0M3R0YpLUY/RktGKUZIRilGNyoqRkZGKSUkX0M4R0YpRkhGKS1GOzYjLCRGTEY3RilGKUYpRiklIlBHbc9:=simplify(subs(x=L21,y[2](x))-subs(x=L21,y[3](x))=0);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bc10:=simplify(subs(x=L21,diff(y[2](x),x))-subs(x=L21,diff(y[3](x),x))=0);NiM+JSViYzEwRy8qJiIiIyMiIiJGJywyKiglJF9DNUdGKSUmYmV0YTJHRiktJSRleHBHNiMqKEYnRihGLUYpJSRMMjFHRilGKUYpKiglJF9DNkdGKS0lJGNvc0dGMEYpRi1GKUYpKiglJF9DN0dGKS0lJHNpbkdGMEYpRi1GKSEiIiooJSRfQzhHRilGLUYpLUYvNiMsJEYxRjtGKUY7KiglJF9DOUdGKSUmYmV0YTNHRiktRi82IyooRidGKEZDRilGMkYpRilGOyooJSVfQzEwR0YpLUY2RkVGKUZDRilGOyooJSVfQzExR0YpLUY6RkVGKUZDRilGKSooJSVfQzEyR0YpRkNGKS1GLzYjLCRGRkY7RilGKUYpIiIhbc11:=simplify(subs(x=L21,EJ2*diff(y[2](x),x$2))-subs(x=L21,EJ3*diff(y[3](x),x$2))=M);NiM+JSViYzExRy8sMiosIiIjIiIiJSRFSjJHRiklJF9DNUdGKSklJmJldGEyR0YoRiktJSRleHBHNiMqKEYoI0YpRihGLUYpJSRMMjFHRilGKUYpKixGKEYpRipGKSUkX0M2R0YpLSUkc2luR0YwRilGLEYpISIiKixGKEYpRipGKSUkX0M3R0YpLSUkY29zR0YwRilGLEYpRjgqLEYoRilGKkYpJSRfQzhHRilGLEYpLUYvNiMsJEYxRjhGKUYpKixGKEYpJSRFSjNHRiklJF9DOUdGKSklJmJldGEzR0YoRiktRi82IyooRihGMkZGRilGM0YpRilGOCosRihGKUZDRiklJV9DMTBHRiktRjdGSEYpRkVGKUYpKixGKEYpRkNGKSUlX0MxMUdGKS1GPEZIRilGRUYpRikqLEYoRilGQ0YpJSVfQzEyR0YpRkVGKS1GLzYjLCRGSUY4RilGOCUiTUc=bc12:=simplify(subs(x=L21,EJ2*diff(y[2](x),x$3))-subs(x=L21,EJ3*diff(y[3](x),x$3))=0);NiM+JSViYzEyRy8sJCooIiIjIiIiRigjRilGKCwyKiolJEVKMkdGKSUkX0M1R0YpKSUmYmV0YTJHIiIkRiktJSRleHBHNiMqKEYoRipGMEYpJSRMMjFHRilGKSEiIioqRi1GKSUkX0M2R0YpLSUkY29zR0Y0RilGL0YpRikqKkYtRiklJF9DN0dGKS0lJHNpbkdGNEYpRi9GKUY3KipGLUYpJSRfQzhHRilGL0YpLUYzNiMsJEY1RjdGKUYpKiolJEVKM0dGKSUkX0M5R0YpKSUmYmV0YTNHRjFGKS1GMzYjKihGKEYqRklGKUY2RilGKUYpKipGRkYpJSVfQzEwR0YpLUY7RktGKUZIRilGNyoqRkZGKSUlX0MxMUdGKS1GP0ZLRilGSEYpRikqKkZGRiklJV9DMTJHRilGSEYpLUYzNiMsJEZMRjdGKUY3RilGNyIiIQ==Definiujemy warto\305\223ci parametr\303\263w k, m, EJ, L oddzielnie w ka\302\277dym z wprowadzonych przedzial\303\263w w celu wykonania odpowiednich wykres\303\263w k1:=10: q1:=10: EJ1:=2000: L1:=5: beta1:=sqrt(sqrt(k1/4/EJ1)): k2:=10: q2:=0: EJ2:=210: L2:=6: beta2:=sqrt(sqrt(k2/4/EJ2)): k3:=10: q3:=0: EJ3:=300: L3:=4: beta3:=sqrt(sqrt(k3/4/EJ3)): P:=100.0: M:=10.0: L:=L1+L2+L3: L21:=L1+L2: evalf(bc1): evalf(bc2): evalf(bc3): evalf(bc4): evalf(bc5): evalf(bc6): evalf(bc7): evalf(bc8): evalf(bc9): evalf(bc10): evalf(bc11): evalf(bc12):Rozwi\271zujemy ukad r\363wna\361 wynikaj\271cy z warunk\363w brzegowych w celu wyznaczenia stalych calkowania i mamy fsolve({bc1,bc2,bc3,bc4,bc5,bc6,bc7,bc8,bc9,bc10,bc11,bc12},{_C1,_C2,_C3,_C4,_C5,_C6,_C7,_C8,_C9,_C10,_C11,_C12}): assign(%):Upraszczamy jednocze\305\223nie r\303\263wnania linii ugi\303\252cia w poszczeg\303\263lnych przedzialach simplify(y[1](x)): simplify(y[2](x)): simplify(y[3](x)): Kolejno otrzymujemy wykres linii ugi\352cia y1:=y[1](x): y2:=y[2](x): y3:=y[3](x): deflection:=piecewise(x<L,y1,x<L1+L2,y2,x<L,y3): fmax:=Maximize(deflection,x=0..L): evalf(fmax); fmin:=Minimize(deflection,x=0..L): evalf(fmin); plot(deflection,x=0..L, title=`Linia ugiecia belki`,axes=boxed,font=[TIMES, BOLD, 16]); oraz wykres funkcji k\271ta ugi\352cia y1:=diff(y[1](x),x): y2:=diff(y[2](x),x): y3:=diff(y[3](x),x): Slope:=piecewise(x<L1,y1,x<L1+L2,y2,x<L,y3):p1:=plot(Slope,x=0..L, title=`Kat ugiecia belki`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},font=[TIMES, BOLD, 16]); wykres momentu zginaj\271cego y1:=-EJ1*diff(y[1](x),x$2): y2:=-EJ2*diff(y[2](x),x$2): y3:=-EJ3*diff(y[3](x),x$2): moment:=piecewise(x<L1,y1,x<L1+L2,y2,x<L,y3): Mmax:=Maximize(moment,x=0..L): evalf(Mmax); Mmin:=Minimize(moment,x=0..L): evalf(Mmin); p1:=plot(moment,x=0..L, title=`Wykres momentu zginajacego`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},font=[TIMES, BOLD, 16]); oraz sily tn\271cej y1:=-EJ1*diff(y[1](x),x$3): y2:=-EJ2*diff(y[2](x),x$3): y3:=-EJ3*diff(y[3](x),x$3): shear:=piecewise(x<L1,y1,x<L1+L2,y2,x<L,y3): Tmax:=Maximize(shear,x=0..L): evalf(Tmax); Tmin:=Minimize(shear,x=0..L): evalf(Tmin); p1:=plot(shear,x=0..L, title=`Wykres sily tnacej`,axes=boxed): p2:=plot(0,x=0..L,axes=boxed,colour=black): display({p1,p2},font=[TIMES, BOLD, 16]);NiM3JCQiKyd5KSplNSIhIik3Iy8lInhHJCIrKioqKioqKlwiRiY=NiM3JCQhK1YiKnorRyEjNTcjLyUieEckIitmJTM5ayIhIzw=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NiM3JCQiK0AsXylIKSEiKTcjLyUieEckIistVDcpKioqISIqNiM3JCQhK1pdKXAsIiEiKDcjLyUieEckIismKioqKioqKlwhIio=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Zauwa\302\277my, \302\277e tak jak w poprzednim przypadku, zastosowanie obci\302\271\302\277e\303\261 skupionych musi wplywa\303\246 na skokow\302\271 zmienno\305\223\303\246 odpowiedniego wykresu. Dodajmy, \302\277e mo\302\277emy zastosowa\303\246 tutaj r\303\263wnie\302\277 inne warunki podparcia wla\305\223ciwe dla zwyklych belek. I tak np. mo\302\277emy zastosowa\303\246 tutaj teleskop albo zamocowanie (utwierdzenie) - je\305\223li jakakolwiek inna podpora pojawi si\303\252 dla 0<x<L, w\303\263wczas dwa z r\303\263wna\303\261 bc5-bc12 musi zawiera\346 dodatkowo warunki zerowania si\352 przemieszcze\361 i moment\363w (jak dla przegubowego podparcia). Gdy stosujemy podpor\352 dla x=0 lub x=L, w\363wczas warunki brzegowe bc1-bc4 musz\271 by\346 odpowiednio zmienione poprzez np. wprowadzenie y(x)=0 oraz diff(y(x),x)=0 w przypadku utwierdzenia jednego z ko\361c\363w belki.
ReferenceS. Timoshenko, J.N. Goodier, Theory of Elasticity, second edition, McGraw-Hill Book Company Ltd., New York-Toronto-London, 1951. Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit 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