{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "symbol" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 264 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 35 "Ingenieria Geologica. C urso 2006-07" }}{PARA 257 "" 0 "" {TEXT 257 43 "LABORATORIO DE MECANIC A DE MEDIOS CONTINUOS" }}{PARA 258 "" 0 "" {TEXT 258 26 "Cuarta Sesion . Plasticidad" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Problema 3" }}{PARA 0 "" 0 "" {TEXT 260 123 "Suponie ndo un modelo de plasticidad de Von Mises, obtener la condicion de pla sticidad para los siguientes estados de carga:" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "1. tension uniaxial de traccion y de compresion " } {TEXT 259 1 "s" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(Linea rAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "contract := \+ proc(A,B)\nlocal c, i, j;\nc := 0;\nfor i from 1 to 3 do\nfor j from 1 to 3 do\nc := c + A[i,j]*B[i,j];\nend do;\nend do;\nend proc;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "mises := proc(sigma)\nlocal S;\nS := sigma-1/3*Trace(sigma)*IdentityMatrix(3);\nsqrt(3/2*contract (S,S));\nsimplify(%,assume=positive)\nend proc;" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 40 "sigma1 := <|<0,0,0>|<0,0,0>>;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "condicion := mises(sigma1)=s igma[f0];" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 14 "2. corte puro " } {TEXT 261 1 "t" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "sigma2 := \+ <<0,tau,0>||<0,0,0>>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "condicion := mises(sigma2)=sigma[f0];" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "3. tension uniaxial " }{TEXT 262 1 "s" }{TEXT -1 16 " mas corte puro " }{TEXT 263 1 "t" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "condicion := mises(sigma1+sigma2)=sigma[f0]; " }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Problema 4" }}{PARA 0 "" 0 "" {TEXT 264 111 "Para el criterio de Mohr-Coulomb,obtener el corte \+ de la superficie de fluencia con el plano de tension biaxial." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 80 "mohr_coulomb := (sigma[1]-sigma[3])+(sigma[1]+ sigma[3])*sin(phi)-2*c*cos(phi)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "st := solve(subs(sigma[3]=0,mohr_coulomb), sigma[1]); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "sc := solve(subs(sigma[ 1]=0,mohr_coulomb), sigma[3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "hexagono := [[st,st],[st,0],[0,sc],[sc,sc],[sc,0],[0,st]];" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "hexagono := subs(c=100,phi= Pi/6,hexagono);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plots[po lygonplot](hexagono,thickness=2);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 10 "Problema 5" }}{PARA 0 "" 0 "" {TEXT 265 24 "Ver enunciado entre gado." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "Modelo de Mohr-Coulomb" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT 266 21 "Meridiano de traccion" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 55 "sigma := + <2*s/3,-s/3,-s /3>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "u_oct := Normalize( <1,1,1>,Euclidean);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "ec1 \+ := xi=simplify(u_oct.sigma);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "vr := simplify(sigma-(u_oct.sigma)*u_oct);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "ec2 := rho=simplify(sqrt(vr.vr),assume=positi ve);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "solve(\{ec1,ec2\}, \{sigma_m,s\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "assign(% );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "mohr_coulomb := (sigm a[1]-sigma[3])+(sigma[1]+sigma[3])*sin(phi)-2*c*cos(phi)=0;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "rho := solve(mohr_coulomb,rh o);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "rho_enunciado := sqr t(2/3)*(2*c*cos(phi)-(2/sqrt(3))*xi*sin(phi))/(1+(1/3)*sin(phi));" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalb(simplify(rho-rho_enunc iado)=0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 43 "*Resolverlo para el meridiano de compresion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "*Modelo de Rankine" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 56 "Interpretaci\363n geometrica de las superficies de flue ncia" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots): with(plottools):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Direcci\363n de la trisectriz del \+ triedro de tensiones principales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "uh := Normalize(<1,1,1>,Euclidean); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Direcci\363n de la proyecci\363n del eje \+ <1,0,0> en el plano octa\351drico" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "up1 := simplify(Normalize(<1,0,0>-DotProduct(<1,0,0>, uh)*uh,Euclidean));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Direcci \363n normal a la anterior contenida en el plano octa\351drico" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "up1a:=simplify(uh &x up1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "Vector a partir de las coordena das locales del plano octa\351drico" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "vs := xi*uh+rho*cos(theta)*up1+rho*sin(theta)*up1a;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Componentes globales de un cil indro de radio 1 de eje la trisectric del triedro de tensiones princip ales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "vm1 := subs(rho=1,v s[1]);\nvm2 := subs(rho=1,vs[2]);\nvm3 := subs(rho=1,vs[3]);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Centro de la base del cilindro a d istancia 3 del plano octa\351drico" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "basea := evalm(subs(\{rho=0,theta=0,xi=3\},vs));" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "Cilindro de von Mises" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "cilindro_mises := \n plot3d([ vm1,vm2,vm3],xi=-3..3,theta=0..2*Pi,axes=normal,scaling=constrained,co lor=gold),\n spacecurve([xi,xi,xi],xi=0..3,color=black,linestyl e=DASH,thickness=3),\n plots[arrow](basea,uh,color=red,width=[0 .15,relative]),\n plots[arrow](basea,up1,color=blue,width=[0.15 ,relative]),\n textplot3d(\{[3,0,0,`sigma1`],[0,3,0,`sigma2`],[ 0,0,3,`sigma3`]\},color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "display([cilindro_mises]);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Distintas trayect orias de tensiones (uniaxial, biaxial, tracci\363n compuesta y compres i\363n compuesta)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 270 "uniax := spacecurve([s,0,0],s=0..3,color=green,thickness=3):\nbiax := space curve([s,s,0],s=0..3,color=blue,thickness=3):\nstrac := x -> spacecur ve([s/4,s/2,2*s/3],s=0..x,color=red,thickness=3):\nscomp := x -> spac ecurve([-s/3,-2*s/3,-s/2],s=0..x,color=brown,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "display(strac(5),scomp(3),axes=norm al);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "sols := fsolve(\{s/4=vm1,s/2=vm2,2*s/3=vm3\}, \{s=100, theta=Pi, xi=3\}); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "inter_strac := sphere(s ubs(sols, [s/4,s/2,2*s/3]),0.2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sols := fsolve(\{-s/3=vm1,-2*s/3=vm2,-s/3=vm3\}, \{s= 100, theta=0, xi=-30\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "inter_scomp := sphere(subs(sols, [-s/3,-2*s/3,-s/2]),0.2):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "display(cilindro_mises,uniax ,biax,strac(5),scomp(5),inter_strac,inter_scomp);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "Otra forma: construcci\363n del ci lindro de von Mises en componentes de las tensiones principales" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "sigmam:=1/3*(sigma1+sigma2+ sigma3);\nec_xi := xi=sqrt(3)*sigmam;\nec_rho := rho=sqrt((sigma1-sigm am)^2+(sigma2-sigmam)^2+(sigma3-sigmam)^2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "implicitplot3d(rhs(ec_rho)^2=1,sigma1=0..3,sigma2 =0..3,sigma3=0..3,grid=[20,20,20],scaling=constrained,axes=normal);" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Superficie de fluencia de Mohr-C oulomb" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "mohr_coulomb := ( sigma1-sigma3)+(sigma1+sigma3)*sin(phi)-2*c*cos(phi)=0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "mc1 := subs(c=10,phi=Pi/6,mohr_coul omb);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "sigmam_eh:=solve(s ubs(sigma1=x,sigma3=x,mc1),x); # valor de x en la superficie de fluenc ia" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "sigma1_u := solve(sub s(sigma3=0,mc1)); # valor de sigma1 en la superficie de fluencia" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "vertice := [sigmam_eh,sigmam _eh,sigmam_eh]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 361 "gmc1 := polygonplot3d([\n[vertice,[sigma1_u,0,0],[sigma1_u,sigma1_u,0]],\n[ve rtice,[sigma1_u,sigma1_u,0],[0,sigma1_u,0]],\n[vertice,[0,sigma1_u,0], [0,sigma1_u,sigma1_u]],\n[vertice,[0,sigma1_u,sigma1_u],[0,0,sigma1_u] ],\n[vertice,[0,0,sigma1_u],[sigma1_u,0,sigma1_u]],\n[vertice,[sigma1_ u,0,sigma1_u],[sigma1_u,0,0]]\n],axes=normal):\ndisplay(gmc1,strac(30) ,scomp(10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "Representaci\363 n de la pir\341mide de Mohr-Coulomb con base normal a la trisectriz de l triedro de tensiones principales" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "st := solve(subs(sigma1=sm+2*s/3,sigma3=sm-s/3,sm=-1 0,mc1));\nsc := solve(subs(sigma1=sm+s/3,sigma3=sm-2*s/3,sm=-10,mc1)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "sm := -10;\nsigmat1 := [sm+2*st/3,sm-st/3,sm-st/3];\nsigmac1 := [sm+sc/3,sm+sc/3,sm-2*sc/3]; \nsigmat2 := [sm-st/3,sm+2*st/3,sm-st/3];\nsigmac2 := [sm-2*sc/3,sm+sc /3,sm+sc/3];\nsigmat3 := [sm-st/3,sm-st/3,sm+2*st/3];\nsigmac3 := [sm+ sc/3,sm-2*sc/3,sm+sc/3];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "baseb := [-10,-10,-10];\nuhb := [-10,-10,-10];\nupp1 := convert(si mplify(10*sqrt(3)*up1),list);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 404 "spacecurve([x,x,x],x=-10..10*sqrt(3),color=black,thickness=3), \nplots[arrow](baseb,uhb,color=red,width=[0.15,relative]),\nplots[arro w](baseb,upp1,color=blue,width=[0.15,relative]),\npolygonplot3d([\n[ve rtice,sigmat1,sigmac1],\n[vertice,sigmac1,sigmat2],\n[vertice,sigmat2, sigmac2],\n[vertice,sigmac2,sigmat3],\n[vertice,sigmat3,sigmac3],\n[ve rtice,sigmac3,sigmat1]\n],axes=normal),\nstrac(30),scomp(30):\ndisplay (%);" }}}}}{MARK "4" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }