Ejemplo 2
L=5 m, b=200 mm, h=300 mm y E=200 GPa
Un elemento de tres nodos
interpolación de desplazamientos verticales N=[1−2325x2+66125x3−68625x4+243125x5x−65x2+1325x3−12125x4+4625x51625x2−32125x3+16625x4−85x2+3225x3−825x4+16625x5725x2−34125x3+52625x4−243125x5−15x2+15x3−8125x4+4625x5]T
interpolación de curvatura B=d2Ndx2=[−4625+396125x−816625x2+96625x3−125+7825x−144125x2+16125x33225−192125x+192625x2−165+19225x−9625x2+64125x31425−204125x+624625x2−96625x3−25+65x−96125x2+16125x3]T
matriz constitutiva D=E I=9×107 [N⋅m2]
reemplazando ∫50[−4625+396125x−816625x2+96625x3−125+7825x−144125x2+16125x33225−192125x+192625x2−165+19225x−9625x2+64125x31425−204125x+624625x2−96625x3−25+65x−96125x2+16125x3][9×107][−4625+396125x−816625x2+96625x3−125+7825x−144125x2+16125x33225−192125x+192625x2−165+19225x−9625x2+64125x31425−204125x+624625x2−96625x3−25+65x−96125x2+16125x3]Tdx[v1θ1v2θ2v3θ3]=∫50−12000[1−2325x2+66125x3−68625x4+243125x5x−65x2+1325x3−12125x4+4625x51625x2−32125x3+16625x4−85x2+3225x3−825x4+16625x5725x2−34125x3+52625x4−243125x5−15x2+15x3−8125x4+4625x5]dx+[F1M1F2M2F3M3]
integrando [1.05×1081.17×108−7.37×1071.97×108−3.1×1072.49×1071.17×1081.71×108−9.22×1071.65×108−2.49×1071.95×107−7.37×107−9.22×1071.47×1080−7.37×1079.22×1071.97×1081.65×10806.58×108−1.97×1081.65×108−3.1×107−2.49×107−7.37×107−1.97×1081.05×108−1.17×1082.49×1071.95×1079.22×1071.65×108−1.17×1081.71×108][v1θ1v2θ2v3θ3]=[−14000−5000−320000−140005000]+[F1M1F2M2F3M3]
reemplazando las condiciones de contorno [1.05×1081.17×108−7.37×1071.97×108−3.1×1072.49×1071.17×1081.71×108−9.22×1071.65×108−2.49×1071.95×107−7.37×107−9.22×1071.47×1080−7.37×1079.22×1071.97×1081.65×10806.58×108−1.97×1081.65×108−3.1×107−2.49×107−7.37×107−1.97×1081.05×108−1.17×1082.49×1071.95×1079.22×1071.65×108−1.17×1081.71×108][00v2θ2v3θ3]=[−14000−5000−320000−140005000]+[F1M10000]
sumando [1.05×1081.17×108−7.37×1071.97×108−3.1×1072.49×1071.17×1081.71×108−9.22×1071.65×108−2.49×1071.95×107−7.37×107−9.22×1071.47×1080−7.37×1079.22×1071.97×1081.65×10806.58×108−1.97×1081.65×108−3.1×107−2.49×107−7.37×107−1.97×1081.05×108−1.17×1082.49×1071.95×1079.22×1071.65×108−1.17×1081.71×108][00v2θ2v3θ3]=[F1−14000M1−5000−320000−140005000]
resolviendo F1=60000 [N]M1=150000 [N⋅m]v2=−3.69×10−3 [m]θ2=−2.43×10−3 [rad]v3=−1.04×10−2 [m]θ3=−2.78×10−3 [rad]
Desplazamientos, deformaciones, esfuerzos, etc.
Las funciones de forma en coordenadas locales se transforman a coordenadas globales usando: x=X−h
reemplazando h=0 x=X
desplazamientos verticales v=N v=[1−2325x2+66125x3−68625x4+243125x5x−65x2+1325x3−12125x4+4625x51625x2−32125x3+16625x4−85x2+3225x3−825x4+16625x5725x2−34125x3+52625x4−243125x5−15x2+15x3−8125x4+4625x5]T[00−3.69×10−3−2.43×10−3−1.04×10−2−2.78×10−3]=−11200x2+19000x3−118000x4=−11200X2+19000X3−118000X4 [m]
desplazamientos horizontales u=ydvdx=0.15[−1600X+13000X2−145000X3]=−14000X+120000X2−1300000X3 [m]
deformación normal εx=dudx=−14000+110000X−1100000X2
momentos flectores M=EId2vdx2=−150000+60000X−6000X2 [N⋅m]
cortante V=dMdx=60000−12000X [N]
esfuerzo normal σx=E εx=−50+20X−2X2 [MPa]
esfuerzo cortante τyz=VQI b=32−310X [MPa]
Dos elementos de tres nodos
Elemento 1
interpolación de desplazamientos verticales N=[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]T
interpolación de curvatura B=d2Ndx2=[−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3]T
matriz constitutiva D=E I=9×107 [N⋅m2]
reemplazando ∫2.50[−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3][9×107][−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3]Tdx[v1θ1v2θ2v3θ3]=∫2.50−12000[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]dx+[F1M1F2M2F3M3]
integrando [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×1074.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×107−5.9×108−3.69×1081.18×1090−5.9×1083.69×1087.9×1083.29×10801.32×109−7.9×1083.29×108−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×1089.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][v1θ1v2θ2v3θ3]=[−7000−1250−160000−70001250]+[F1M1F2M2F3M3]
Elemento 2
interpolación de desplazamientos verticales N=[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]T
interpolación de curvatura B=d2Ndx2=[−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3]T
matriz constitutiva D=E I=9×107 [N⋅m2]
reemplazando ∫2.50[−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3][9×107][−18425+3168125x−13056625x2+3072625x3−245+31225x−1152125x2+256125x312825−1536125x+3072625x2−325+76825x−76825x2+1024125x35625−1632125x+9984625x2−3072625x3−45+245x−768125x2+256125x3]Tdx[v1θ1v2θ2v3θ3]=∫2.50−12000[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]dx+[F1M1F2M2F3M3]
integrando [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×1074.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×107−5.9×108−3.69×1081.18×1090−5.9×1083.69×1087.9×1083.29×10801.32×109−7.9×1083.29×108−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×1089.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][v1θ1v2θ2v3θ3]=[−7000−1250−160000−70001250]+[F1M1F2M2F3M3]
Ensamblaje y solución
ensamblando matriz global [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×10700004.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×1070000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1088.38×108+8.38×108−4.68×108+4.68×10800009.96×1073.91×1073.69×1083.29×108−4.68×108+4.68×1083.41×108+3.41×10800000000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×10800009.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][v1θ1v2θ2v3+v1θ3+θ1v2θ2v3θ3]=[−7000−1250−160000−7000−70001250−1250−160000−70001250]+[F1M1F2M2F3+F1M3+M1F2M2F3M3]
sumando [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×10700004.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×1070000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1081.68×109000009.96×1073.91×1073.69×1083.29×10806.83×10800000000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×10800009.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][v1θ1v2θ2v3θ3v4θ4v5θ5]=[−7000−1250−160000−140000−160000−70001250]+[F1M1F2M2F3M3F4M4F5M5]
reemplazando las condiciones de contorno [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×10700004.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×1070000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1081.68×109000009.96×1073.91×1073.69×1083.29×10806.83×10800000000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×10800009.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][00v2θ2v3θ3v4θ4v5θ5]=[−7000−1250−160000−140000−160000−70001250]+[F1M100000000]
sumando [8.38×1084.68×108−5.9×1087.9×108−2.48×1089.96×10700004.68×1083.41×108−3.69×1083.29×108−9.96×1073.91×1070000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1081.68×109000009.96×1073.91×1073.69×1083.29×10806.83×10800000000−5.9×108−3.69×1081.18×1090−5.9×1083.69×10800007.9×1083.29×10801.32×109−7.9×1083.29×1080000−2.48×108−9.96×107−5.9×108−7.9×1088.38×108−4.68×10800009.96×1073.91×1073.69×1083.29×108−4.68×1083.41×108][00v2θ2v3θ3v4θ4v5θ5]=[F1−7000M1−1250−160000−140000−160000−70001250]
resolviendo F1=60000 [N]M1=150000 [N⋅m]v2=−1.1×10−3 [m]θ2=−1.61×10−3 [rad]v3=−3.69×10−3 [m]θ3=−2.43×10−3 [rad]v4=−6.96×10−3 [m]θ4=−2.73×10−3 [rad]v5=−1.04×10−2 [m]θ5=−2.78×10−3 [rad]
Desplazamientos, deformaciones, esfuerzos, etc.
Las funciones de forma en coordenadas locales se transforman a coordenadas globales usando x=X−h
Elemento 1
reemplazando h=0 x=X
desplazamientos verticales v=N v=[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]T[00−1.1×10−3−1.61×10−3−3.69×10−3−2.43×10−3]=−11200x2+19000x3−118000x4=−11200X2+19000X3−118000X4 [m]
desplazamientos horizontales u=ydvdx=0.15[−1600X+13000X2−145000X3]=−14000X+120000X2−1300000X3 [m]
deformación normal εx=dudx=−14000+110000X−1100000X2
momentos flectores M=EId2vdx2=−150000+60000X−6000X2 [N⋅m]
cortante V=dMdx=60000−12000X [N]
esfuerzo normal σx=E εx=−50+20X−2X2 [MPa]
esfuerzo cortante τyz=VQI b=32−310X [MPa]
Elemento 2
reemplazando h=2.5 x=X−2.5
desplazamientos verticales v=N v=[1−9225x2+528125x3−1088625x4+7683125x5x−125x2+5225x3−96125x4+64625x56425x2−256125x3+256625x4−165x2+12825x3−6425x4+256625x52825x2−272125x3+832625x4−7683125x5−25x2+45x3−64125x4+64625x5]T[−3.69×10−3−2.43×10−3−6.96×10−3−2.73×10−3−1.04×10−2−2.78×10−3]=−132304−1180x−1480x2−12250x3−118000x4=−11200X2+19000X3−118000X4 [m]
desplazamientos horizontales u=ydvdx=0.15[−1600X+13000X2−145000X3]=−14000X+120000X2−1300000X3 [m]
deformación normal εx=dudx=−14000+110000X−1100000X2
momentos flectores M=EId2vdx2=−150000+60000X−6000X2 [N⋅m]
cortante V=dMdx=60000−12000X [N]
esfuerzo normal σx=E εx=−50+20X−2X2 [MPa]
esfuerzo cortante τyz=VQI b=32−310X [MPa]