PANDEO EN COLUMNAS
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ÍNDICE pp. INTRODUCCI N………………………………………………..……….
1
Pandeo en Columnas………………….……………………………..….
3
Determinar el Pandeo en Columnas………………..……………..….
4
Carga Crítica………………………………….……...……………..……
9
Pandeo flexional………………….…………………………………..….
9
Pandeo local……………………………………………………………...
10
Pandeo global…………………………………..………………………..
11
Plano de pandeo………………………………..………………………..
13
Teoría de la bifurcación………………………..………………………..
13
Pandeo torsional………………………………..………………………..
14
Curva Elástica…………………………………..………………………..
15
Métodos Energéticos…………………………..………………………..
17
Dimensionado de elementos sometidos a pandeo…………………..
18
Esbeltez y coeficientes de pandeo……………………………………..
20
Columnas Esbeltas……………………………………………………..
22
Consideración de los efectos de la esbeltez…………………………..
24
CONCLUSIONES……....………………………………………………..
26
REFERENCIAS BIBLIOGRÁFICAS…………………………………..
28
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INTRODUCCIÓN En efecto, se estableció que la selección de elementos estructurales y de máquinas se basa en tres características: resistencia, rigidez y estabilidad. Los procedimientos de análisis de esfuerzos y deformaciones se estudiaron en detalle en los capítulos anteriores. En este segmento, se tratará la cuestión de la posible inestabilidad de sistemas estructurales. En tales problemas, se deben hallar parámetros críticos adicionales que determinen si es posible una configuración o patrón de deformación dado para un sistema particular. Este problema es diferente de cualquier otro de los vistos anteriormente.
Como un ejemplo intuitivo sencillo considérese una barra de diámetro D sometida a una fuerza axial de compresión. Si tal barra actuando como “columna”, fuera de longitud D, no surgiría ninguna cuestión acerca de la
inestabilidad y este miembro corto podría soportar una fuerza considerable. Por otra parte, la misma barra tuviera una longitud de varios diámetros, al ser sometida a una fuerza axial aún menor que la que puede soportar la pieza corta podría llegar a ser lateralmente inestable, presentándose en ella pandeo lateral y podría fallar o sufrir colapso. Una regla delgada ordinaria, si se somete a compresión axial, fallará de esta manera. La consideración de la sola resistencia del material no es suficiente para predecir el comportamiento de tal miembro.
El mismo fenómeno se presenta en numerosas otras situaciones en que existen esfuerzos de compresión. Placas delgadas, completamente capaces de resistir cargas de tracción, resultan muy ineficaces para transmitir compresión. Vigas angostas, sin arriostramiento lateral, pueden doblarse
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diseñados, pueden deformarse gravemente por la presión externa y asumir formas que difieren en forma notable de su configuración geométrica original.
Un tubo de pared delgada puede arrugarse o plegarse como papel de seda cuando se somete a torsión. Durante algunas etapas de su encendido, las delgadas cubiertas de los cohetes o proyectiles autopropulsados se cargan críticamente a compresión. Estos son problemas de primordial importancia en el diseño de ingeniería. Además, por lo general los fenómenos de pandeo o arrugamiento que se observan en miembros cargados ocurren más bien repentinamente. Por esta razón, muchas de las fallas estructurales por pandeo son espectaculares y muy peligrosas.
El enorme número de problemas de inestabilidad o pandeo de estructuras sugerido por la lista anterior está fuera del alcance de este trabajo. Aquí solo se considerará el problema de la columna. Utilizándolo como ejemplo, sin embargo se ponen de relieve las características esenciales del fenómeno de pandeo y algunos procedimientos básicos para su análisis.
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PANDEO DE COLUMNA Se denomina “columna” a los elementos que trabajan bajo compresión.
Generalmente las columnas fallan a cargas menores a las que producirían falla por fluencia o fractura del material, la falla de las columnas es denominada “pandeo” y es una falla por pérdida de función de la columna.
Por lo tanto, en el diseño de elementos que se encuentren a compresión es necesario hacer un análisis de pandeo. El pandeo es un fenómeno de inestabilidad elástica que puede darse en elementos comprimidos esbeltos, y que se manifiesta por la aparición de desplazamientos importantes transversales a la dirección principal de compresión.
En ingeniería estructural el fenómeno aparece principalmente en pilares y columnas, y se traduce en la aparición de una flexión adicional en el pilar cuando se halla sometido a la acción de esfuerzos axiales de cierta importancia. La aparición de deflexión por pandeo limita severamente la resistencia en compresión de un pilar o cualquier tipo de pieza esbelta. Eventualmente, a partir de cierto valor de la carga axial de compresión, denominada carga crítica de pandeo, puede producirse una situación de inestabilidad elástica y entonces fácilmente la deformación aumentará produciendo tensiones adicionales que superarán la tensión de rotura, provocando la ruina del elemento estructural.
Desde la perspectiva más general, el diseño de las columnas consiste básicamente en seleccionar una sección transversal adecuada para la misma, con armadura para soportar las combinaciones requeridas de cargas axiales mayoradas Pu y momentos (de primer orden) mayorados Mu, incluyendo la consideración de los efectos de la esbeltez de la columna
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DETERMINAR EL PANDEO EN COLUMNAS El pandeo es un fenómeno de inestabilidad elástica que puede darse en elementos comprimidos esbeltos, y que se manifiesta por la aparición de desplazamientos importantes transversales a la dirección principal de compresión. En ingeniería estructural, el fenómeno aparece principalmente en pilares y columnas, y se traduce en la aparición de una flexión adicional en el pilar cuando se halla sometido a la acción de esfuerzos axiales de cierta importancia.
La aparición de deflexión por pandeo limita severamente la resistencia en compresión de un pilar o cualquier tipo de pieza esbelta. Eventualmente, a partir de cierto valor de la carga axial de compresión, denominada carga crítica de pandeo, puede producirse una situación de inestabilidad elástica y entonces fácilmente la deformación aumentará produciendo tensiones adicionales que superarán la tensión de rotura, provocando la ruina del elemento estructural.
Además del pandeo flexional ordinario, ordinario, existe el pandeo torsional o inestabilidad elástica provocado por un momento torsor excesivo. Existen diferentes maneras o modos de fallo por pandeo. Para un elemento estructural frecuentemente hay que verificar varios de ellos y garantizar que las cargas están lejos de las cargas críticas asociadas a cada modo o manera de pandear. Los modos típicos son:
Pandeo flexional. Modo de pandeo en el cual un elemento en compresión se flecta lateralmente sin giro ni cambios en su sección transversal.
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Pandeo flexo-torsional. Modo de pandeo en el cual un elemento en compresión se flecta y gira simultáneamente sin cambios en su sección transversal.
Pandeo lateral-torsional. Modo de pandeo de un elemento a flexión que involucra deflexión normal al plano de flexión y, de manera simultánea, giro alrededor del centro de corte Los pilares y barras comprimidas de celosías pueden presentar diversos modos de fallo en función de su esbeltez mecánica:
Los pilares muy esbeltos suelen fallar por pandeo elástico y son sensibles tanto al pandeo local el propio pilar como al pandeo global de la estructura completa.
En los pilares de esbeltez media las imperfecciones constructivas como
las
heterogeneidades
son
particularmente
importantes
pudiéndose presentar pandeo anelástico.
Los pilares de muy baja esbeltez fallan por exceso de compresión, antes de que los efectos del pandeo resulten importantes. El pandeo local es el que aparece en piezas o elementos aislados o que estructuralmente pueden considerarse aislados. En este caso la magnitud de la carga crítica viene dada según el caso por la fórmula
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carga crítica global de cierto tipo de estructuras (por ejemplo en entramados de cúpulas monocapa) es mucho menor que la carga crítica (local) de cada uno de sus elementos.
El estudio de la estabilidad estructural trata por lo tanto de determinar los valores de las cargas críticas que provocan el colapso por grandes deformaciones.
Para
este
estudio
es
necesario
suponer
que
las
deformaciones no son pequeñas, y en consecuencia la posición deformada de la estructura no puede confundirse con la posición sin deformar. Por lo tanto las ecuaciones de equilibrio se deben plantear ahora en la posición deformada, y no en la inicial.
Los conceptos de carga crítica y estabilidad del equilibrio pueden ponerse de manifiesto con gran facilidad mediante el siguiente ejemplo sencillo. Considérese el sistema mostrado en la figura siguiente, en el que la barra se supone infinitamente rígida, y por lo tanto sólo el muelle de torsión acumula energía. Un análisis de primer orden, planteando el equilibrio en la posición indeformada, indica que la barra está sometida a una compresión de valor P y que el resorte está descargado. Sin embargo, si se plantea el equilibrio en la posición deformada se obtiene que el resorte tenga un par de valor
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Es posible obtener más información sobre la estabilidad del sistema efectuando un análisis de segundo orden, considerando la expresión exacta del potencial total del sistema:
Para que haya equilibrio este potencial debe ser estacionario:
Esta ecuación se satisface de dos formas. Si θ=0 cualquier valor de P la satisface, lo cual corresponde con la solución del análisis de primer orden, que permite cualquier valor de P . También se cumple la ecuación de equilibrio anterior si el valor de P es: P k L = θ sinθ Esta Esta es la relación entre la carga axial P y el giro θ en cualquier posición de equilibrio, y en ella se observa que para θ=0 la carga vale P=k/L. Esto indica que k/L es un valor crítico de la carga, que hace pasar a la barra de la situación inicial θ=0 a a una situación en la que la barra comienza a girar y el muelle empieza a tener esfuerzo.
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Para estudiar la estabilidad del sistema se calcula la derivada segunda del potencial:
Para la solución de primer orden, con θ=0 , es decir con P
se sustituye el valor de la carga crítica en la derivada del potencial:
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CARGA CRÍTICA Tomando como ejemplo el caso de una viga colocada verticalmente con los extremos articulados de manera que pueda flexarse en cualquier sentido, si le aplicamos una carga H , se flexará. Si después le aplicamos gradualmente una fuerza P , no habrá ningún cambio de esfuerzo si al mismo tiempo que aumenta P vamos vamos disminuyendo H para para que la deflexión o flecha δ permanezca igual (el esfuerzo es directamente proporcional a la flecha o
deformación).
El Momento Flector en el centro del tramo L será: M=(H/2)*(L/2) + P* δ . Cuando H = 0, M cr = P cr * δ , es decir que P cr es la “carga critica” necesaria para mantener la columna en su posición deflectada sin ningún empuje lateral. Cualquier aumento de P por encima de dicho valor P cr hará hará aumentar la flecha, lo que aumentara el momento, lo que a su vez incrementara δ , entre otros. Hasta que la columna falla por pandeo. La carga crítica es, pues, la máxima carga axial bajo la cual una columna permanece recta pero en una condición tan inestable que un pequeño empuje lateral la hará flexar.
Pandeo flexional
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En los pilares de esbeltez media las imperfecciones constructivas como las heterogeneidades son particularmente importantes pudiéndose presentar pandeo anelástico.
Los pilares de muy baja esbeltez fallan por exceso de compresión, antes de que los efectos del pandeo resulten importantes.
Pandeo local El pandeo local es el que aparece en piezas o elementos aislados o que estructuralmente pueden considerarse aislados. En este caso la magnitud de la carga crítica viene dada según el caso por la fórmula de Leonhard Euler o la de Engesser. La carga crítica de Euler depende de la longitud de la pieza, del material, de su sección transversal y de las condiciones de unión, vinculación o sujeción en los extremos. Para una pieza que puede
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Figura 1. Modelo de los distintos tipos de pandeo de Euler. Como se puede ver, según las coacciones externas de la viga, la deformación debida al pandeo será distinta. Fuente: Ortiz, L. (2007)
Al producto producto
se le llama llama longitud de pandeo. pandeo.
Pandeo global En una estructura compleja formada por barras y otros elementos
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Las ecuaciones que gobiernan el comportamiento de la estructura son:
Ecuación de equilibrio:
Relación elástica entre acortamiento y esfuerzo axial:
Relación geométrica de las configuraciones no-deformada y deformadas:
Donde: N , esfuerzo axial de cada una de las barras; ΔL Δ L, acortamiento
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Cada una estas cargas presenta modos de fallo diferentes en la estructura. De entre los dos posibles modos de fallo por pandeo ocurrirá el que presente un ángulo de aparición mayor donde estos ángulos vienen dados por:
Plano de pandeo El plano de pandeo se refiere ref iere al plano que contendrá contend rá el inicio de la la deformada de una pieza sometida a compresión dominante. El plano de pandeo contiene el eje baricéntrico de la viga y sobre él la deflexión por pandeo es máxima. Para una pieza sometida sólo a compresión sin momentos apreciables adicionales, el plano de pandeo coincidirá con el
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Donde:
P c c , carga crítica corregida por las imperfecciones. C(d), es una constante que depende del patrón de imperfección dado por
. , es un parámetro escalar que cuantifica el grado de imperfección para un
patrón de imperfección
dado.
Si las imperfecciones tienen naturaleza estadística y vienen dadas por una distribución normal multivariante entonces la carga crítica tiene una distribución dada por:
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Donde las nuevas magnitudes son: Im, es el momento de inercia mínimo en flexión. I w w , J , son respectivamente el módulo de alabeo y el módulo de torsión. G, el módulo de elasticidad transversal.
Y el resto de magnitudes tienen el mismo significado que para el pandeo flexional puro. En piezas donde el momento de alabeo es despreciable puede usarse la expresión aproximada:
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El valor de la carga crítica es precisamente el que hace que dichas relaciones se cumplan. El método de Euler para barras aisladas es un ejemplo de uso de este método. Por ejemplo para determinar la carga crítica de un pilar empotrado en su base y libre en el extremo tratamos de resolver la ecuación de la curva elástica bajo las siguientes condiciones:
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Métodos Energéticos Para estructuras de una cierta complejidad el método anterior resulta de muy difícil aplicación, ya que requiere integrar un número elevado de ecuaciones diferenciales para cada elemento lineal de la estructura. Un método
aproximado
consiste
en
presuponer
aproximadamente
las
deformaciones asociadas al pandeo, que satisfaga las condiciones de contorno en los extremos de las piezas, y en igualar la energía de deformación W int con el trabajo exterior realizado por la fuerza que produce el int con
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En ingeniería estructural existe una necesidad práctica de dimensionar los elementos lineales sometidos a compresión con la suficiente sección transversal como para que no fallen por pandeo. La sección transversal necesaria para que eso no ocurra es muchas veces mayor que la que sería necesaria para soportar un esfuerzo de tracción de la misma magnitud (entre 1,5 y 6 veces en la mayoría de casos). La mayoría de normas usan un coeficiente de reducción de la resistencia cuando el esfuerzo sobre el elemento lineal es de compresión y no de tracción. El Eurocódigo por ejemplo da para la resistencia de un pilar sometido a compresión y tracción
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Donde: es el esfuerzo axial a que está sometido el elemento. N x es M y y , M z z, son los momentos flectores medidos según las dos direcciones
principales de inercia. W y y, W z z, son los momentos resistentes asociados a los momentos
principales de inercia de la sección transversal. En situaciones donde las tensiones tangentes y el alabeo seccional de la sección sean importantes debe substituirse el miembro antes del signo de menor que, por la tensión de Von Mises y en la expresión de Navier debe
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En otros casos más complejos con otras condiciones en los extremos, con sección variable, entre otros, la carga crítica anterior debe ser corregida por un factor constante. En piezas de sección constante puede definirse además
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resistencia debida al probable efecto de pandeo sobre el mismo. Existen varias maneras, todas ellas esencialmente equivalentes, de tratar esta reducción
de
la
resistencia
por
efecto
del
pandeo,
por
ejemplo
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COLUMNAS ESBELTAS La esbeltez de una columna se expresa en términos de su relación de
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El factor de esbeltez depende de la longitud libre de pandeo del elemento. Esta longitud está regida por el tipo de unión de los extremos del elemento a analizar. Elementos sin restricción a rotación se pandean en toda
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