El círculo de Mohr es una representación gráfica bidimensional de la ley de transformación del tensor de tensiones de Cauchy .
El círculo de Mohr se utiliza a menudo en cálculos relacionados con la ingeniería mecánica para la resistencia de los materiales , la ingeniería geotécnica para la resistencia de los suelos y la ingeniería estructural para la resistencia de las estructuras construidas. También se utiliza para calcular tensiones en muchos planos reduciéndolas a componentes verticales y horizontales. Estos se denominan planos principales en los que se calculan las tensiones principales ; El círculo de Mohr también se puede utilizar para encontrar los planos principales y las tensiones principales en una representación gráfica, y es una de las formas más fáciles de hacerlo. [1]
Después de realizar un análisis de tensión en un cuerpo material asumido como un continuo , los componentes del tensor de tensión de Cauchy en un punto particular del material se conocen con respecto a un sistema de coordenadas . El círculo de Mohr se utiliza luego para determinar gráficamente los componentes de tensión que actúan sobre un sistema de coordenadas girado, es decir, actúan sobre un plano de orientación diferente que pasa por ese punto.
La abscisa y ordenada (,) de cada punto del círculo son las magnitudes de los componentes del esfuerzo normal y del esfuerzo cortante , respectivamente, que actúan sobre el sistema de coordenadas girado. En otras palabras, el círculo es el lugar geométrico de los puntos que representan el estado de tensión en planos individuales en todas sus orientaciones, donde los ejes representan los ejes principales del elemento de tensión.
El ingeniero alemán del siglo XIX Karl Culmann fue el primero en concebir una representación gráfica de las tensiones considerando las tensiones longitudinales y verticales en vigas horizontales durante la flexión . Su trabajo inspiró a su compañero ingeniero alemán Christian Otto Mohr (el homónimo del círculo), quien lo extendió a tensiones bidimensionales y tridimensionales y desarrolló un criterio de falla basado en el círculo de tensiones. [2]
Métodos gráficos alternativos para la representación del estado de tensión en un punto incluyen el elipsoide estrés de Lamé y cuádrica estrés de Cauchy .
El círculo de Mohr se puede aplicar a cualquier matriz tensorial simétrica de 2x2 , incluidos los tensores de deformación y momento de inercia .
Motivación
Las fuerzas internas se producen entre las partículas de un objeto deformable, asumido como un continuo , como reacción a fuerzas externas aplicadas, es decir, fuerzas superficiales o fuerzas corporales . Esta reacción se deriva de las leyes del movimiento de Euler para un continuo, que son equivalentes a las leyes del movimiento de Newton para una partícula. Una medida de la intensidad de estas fuerzas internas se llama estrés . Debido a que el objeto se asume como un continuo, estas fuerzas internas se distribuyen continuamente dentro del volumen del objeto.
En ingeniería, por ejemplo, estructural , mecánica o geotécnica , la distribución de tensiones dentro de un objeto, por ejemplo tensiones en una masa rocosa alrededor de un túnel, alas de avión o columnas de edificios, se determina mediante un análisis de tensiones . El cálculo de la distribución de tensiones implica la determinación de tensiones en cada punto (partícula de material) del objeto. Según Cauchy , la tensión en cualquier punto de un objeto (Figura 2), asumida como un continuo, está completamente definida por los nueve componentes de la tensión.de un tensor de segundo orden de tipo (2,0) conocido como tensor de tensión de Cauchy ,:
Después de que se haya determinado la distribución de la tensión dentro del objeto con respecto a un sistema de coordenadas , puede ser necesario calcular los componentes del tensor de tensión en un punto particular del material con respecto a un sistema de coordenadas rotado , es decir, las tensiones que actúan sobre un plano con una orientación diferente que pasan por ese punto de interés, formando un ángulo con el sistema de coordenadas. (Figura 3). Por ejemplo, es de interés encontrar el esfuerzo normal máximo y el esfuerzo cortante máximo, así como la orientación de los planos sobre los que actúan. Para lograr esto, es necesario realizar una transformación tensorial bajo una rotación del sistema de coordenadas. De la definición de tensor , el tensor de tensión de Cauchy obedece a la ley de transformación del tensor . Una representación gráfica de esta ley de transformación para el tensor de tensión de Cauchy es el círculo de Mohr para la tensión.
Círculo de Mohr para el estado de tensión bidimensional
En dos dimensiones, el tensor de tensión en un punto material dado con respecto a cualesquiera dos direcciones perpendiculares está completamente definido por sólo tres componentes de tensión. Para el sistema de coordenadas particular estos componentes de estrés son: los estreses normales y , y el esfuerzo cortante . A partir del equilibrio del momento angular, se puede demostrar la simetría del tensor de tensión de Cauchy. Esta simetría implica que. Por lo tanto, el tensor de tensión de Cauchy se puede escribir como:
El objetivo es utilizar el círculo de Mohr para encontrar los componentes de la tensión. y en un sistema de coordenadas rotado , es decir, en un plano de orientación diferente que pasa por y perpendicular a la -plano (Figura 4). El sistema de coordenadas rotado hace un ángulo con el sistema de coordenadas original .
Ecuación del círculo de Mohr
Para derivar la ecuación del círculo de Mohr para los casos bidimensionales de tensión plana y deformación plana , primero considere un elemento material infinitesimal bidimensional alrededor de un punto material (Figura 4), con una unidad de área en la dirección paralela a la - plano, es decir, perpendicular a la página o pantalla.
A partir del equilibrio de fuerzas sobre el elemento infinitesimal, las magnitudes de la tensión normal y el esfuerzo cortante están dados por:
Derivación de las ecuaciones paramétricas del círculo de Mohr - Equilibrio de fuerzas Del equilibrio de fuerzas en la dirección de (eje) (Figura 4), y sabiendo que el área del plano donde actos es , tenemos: Sin embargo, sabiendo que
obtenemos
Ahora, a partir del equilibrio de fuerzas en la dirección de (eje) (Figura 4), y sabiendo que el área del plano donde actos es , tenemos:
Sin embargo, sabiendo que
obtenemos
Ambas ecuaciones también se pueden obtener aplicando la ley de transformación del tensor sobre el tensor de tensiones de Cauchy conocido, que es equivalente a realizar el equilibrio estático de fuerzas en la dirección de y .
Derivación de ecuaciones paramétricas del círculo de Mohr - Transformación tensorial La ley de transformación del tensor de tensión se puede establecer como Expandiendo el lado derecho y sabiendo que y , tenemos:
Sin embargo, sabiendo que
obtenemos
Sin embargo, sabiendo que
we obtain
It is not necessary at this moment to calculate the stress component acting on the plane perpendicular to the plane of action of as it is not required for deriving the equation for the Mohr circle.
These two equations are the parametric equations of the Mohr circle. In these equations, is the parameter, and and are the coordinates. This means that by choosing a coordinate system with abscissa and ordinate , giving values to the parameter will place the points obtained lying on a circle.
Eliminating the parameter from these parametric equations will yield the non-parametric equation of the Mohr circle. This can be achieved by rearranging the equations for and , first transposing the first term in the first equation and squaring both sides of each of the equations then adding them. Thus we have
where
This is the equation of a circle (the Mohr circle) of the form
with radius centered at a point with coordinates in the coordinate system.
Sign conventions
There are two separate sets of sign conventions that need to be considered when using the Mohr Circle: One sign convention for stress components in the "physical space", and another for stress components in the "Mohr-Circle-space". In addition, within each of the two set of sign conventions, the engineering mechanics (structural engineering and mechanical engineering) literature follows a different sign convention from the geomechanics literature. There is no standard sign convention, and the choice of a particular sign convention is influenced by convenience for calculation and interpretation for the particular problem in hand. A more detailed explanation of these sign conventions is presented below.
The previous derivation for the equation of the Mohr Circle using Figure 4 follows the engineering mechanics sign convention. The engineering mechanics sign convention will be used for this article.
Physical-space sign convention
From the convention of the Cauchy stress tensor (Figure 3 and Figure 4), the first subscript in the stress components denotes the face on which the stress component acts, and the second subscript indicates the direction of the stress component. Thus is the shear stress acting on the face with normal vector in the positive direction of the -axis, and in the positive direction of the -axis.
In the physical-space sign convention, positive normal stresses are outward to the plane of action (tension), and negative normal stresses are inward to the plane of action (compression) (Figure 5).
In the physical-space sign convention, positive shear stresses act on positive faces of the material element in the positive direction of an axis. Also, positive shear stresses act on negative faces of the material element in the negative direction of an axis. A positive face has its normal vector in the positive direction of an axis, and a negative face has its normal vector in the negative direction of an axis. For example, the shear stresses and are positive because they act on positive faces, and they act as well in the positive direction of the -axis and the -axis, respectively (Figure 3). Similarly, the respective opposite shear stresses and acting in the negative faces have a negative sign because they act in the negative direction of the -axis and -axis, respectively.
Mohr-circle-space sign convention
In the Mohr-circle-space sign convention, normal stresses have the same sign as normal stresses in the physical-space sign convention: positive normal stresses act outward to the plane of action, and negative normal stresses act inward to the plane of action.
Shear stresses, however, have a different convention in the Mohr-circle space compared to the convention in the physical space. In the Mohr-circle-space sign convention, positive shear stresses rotate the material element in the counterclockwise direction, and negative shear stresses rotate the material in the clockwise direction. This way, the shear stress component is positive in the Mohr-circle space, and the shear stress component is negative in the Mohr-circle space.
Two options exist for drawing the Mohr-circle space, which produce a mathematically correct Mohr circle:
- Positive shear stresses are plotted upward (Figure 5, sign convention #1)
- Positive shear stresses are plotted downward, i.e., the -axis is inverted (Figure 5, sign convention #2).
Plotting positive shear stresses upward makes the angle on the Mohr circle have a positive rotation clockwise, which is opposite to the physical space convention. That is why some authors[3] prefer plotting positive shear stresses downward, which makes the angle on the Mohr circle have a positive rotation counterclockwise, similar to the physical space convention for shear stresses.
To overcome the "issue" of having the shear stress axis downward in the Mohr-circle space, there is an alternative sign convention where positive shear stresses are assumed to rotate the material element in the clockwise direction and negative shear stresses are assumed to rotate the material element in the counterclockwise direction (Figure 5, option 3). This way, positive shear stresses are plotted upward in the Mohr-circle space and the angle has a positive rotation counterclockwise in the Mohr-circle space. This alternative sign convention produces a circle that is identical to the sign convention #2 in Figure 5 because a positive shear stress is also a counterclockwise shear stress, and both are plotted downward. Also, a negative shear stress is a clockwise shear stress, and both are plotted upward.
This article follows the engineering mechanics sign convention for the physical space and the alternative sign convention for the Mohr-circle space (sign convention #3 in Figure 5)
Drawing Mohr's circle
Assuming we know the stress components , , and at a point in the object under study, as shown in Figure 4, the following are the steps to construct the Mohr circle for the state of stresses at :
- Draw the Cartesian coordinate system with a horizontal -axis and a vertical -axis.
- Plot two points and in the space corresponding to the known stress components on both perpendicular planes and , respectively (Figure 4 and 6), following the chosen sign convention.
- Draw the diameter of the circle by joining points and with a straight line .
- Draw the Mohr Circle. The centre of the circle is the midpoint of the diameter line , which corresponds to the intersection of this line with the axis.
Finding principal normal stresses
The magnitude of the principal stresses are the abscissas of the points and (Figure 6) where the circle intersects the -axis. The magnitude of the major principal stress is always the greatest absolute value of the abscissa of any of these two points. Likewise, the magnitude of the minor principal stress is always the lowest absolute value of the abscissa of these two points. As expected, the ordinates of these two points are zero, corresponding to the magnitude of the shear stress components on the principal planes. Alternatively, the values of the principal stresses can be found by
where the magnitude of the average normal stress is the abscissa of the centre , given by
and the length of the radius of the circle (based on the equation of a circle passing through two points), is given by
Finding maximum and minimum shear stresses
The maximum and minimum shear stresses correspond to the ordinates of the highest and lowest points on the circle, respectively. These points are located at the intersection of the circle with the vertical line passing through the center of the circle, . Thus, the magnitude of the maximum and minimum shear stresses are equal to the value of the circle's radius
Finding stress components on an arbitrary plane
As mentioned before, after the two-dimensional stress analysis has been performed we know the stress components , , and at a material point . These stress components act in two perpendicular planes and passing through as shown in Figure 5 and 6. The Mohr circle is used to find the stress components and , i.e., coordinates of any point on the circle, acting on any other plane passing through making an angle with the plane . For this, two approaches can be used: the double angle, and the Pole or origin of planes.
Double angle
As shown in Figure 6, to determine the stress components acting on a plane at an angle counterclockwise to the plane on which acts, we travel an angle in the same counterclockwise direction around the circle from the known stress point to point , i.e., an angle between lines and in the Mohr circle.
The double angle approach relies on the fact that the angle between the normal vectors to any two physical planes passing through (Figure 4) is half the angle between two lines joining their corresponding stress points on the Mohr circle and the centre of the circle.
This double angle relation comes from the fact that the parametric equations for the Mohr circle are a function of . It can also be seen that the planes and in the material element around of Figure 5 are separated by an angle , which in the Mohr circle is represented by a angle (double the angle).
Pole or origin of planes
The second approach involves the determination of a point on the Mohr circle called the pole or the origin of planes. Any straight line drawn from the pole will intersect the Mohr circle at a point that represents the state of stress on a plane inclined at the same orientation (parallel) in space as that line. Therefore, knowing the stress components and on any particular plane, one can draw a line parallel to that plane through the particular coordinates and on the Mohr circle and find the pole as the intersection of such line with the Mohr circle. As an example, let's assume we have a state of stress with stress components , , and , as shown on Figure 7. First, we can draw a line from point parallel to the plane of action of , or, if we choose otherwise, a line from point parallel to the plane of action of . The intersection of any of these two lines with the Mohr circle is the pole. Once the pole has been determined, to find the state of stress on a plane making an angle with the vertical, or in other words a plane having its normal vector forming an angle with the horizontal plane, then we can draw a line from the pole parallel to that plane (See Figure 7). The normal and shear stresses on that plane are then the coordinates of the point of intersection between the line and the Mohr circle.
Finding the orientation of the principal planes
The orientation of the planes where the maximum and minimum principal stresses act, also known as principal planes, can be determined by measuring in the Mohr circle the angles ∠BOC and ∠BOE, respectively, and taking half of each of those angles. Thus, the angle ∠BOC between and is double the angle which the major principal plane makes with plane .
Angles and can also be found from the following equation
This equation defines two values for which are apart (Figure). This equation can be derived directly from the geometry of the circle, or by making the parametric equation of the circle for equal to zero (the shear stress in the principal planes is always zero).
Example
Assume a material element under a state of stress as shown in Figure 8 and Figure 9, with the plane of one of its sides oriented 10° with respect to the horizontal plane. Using the Mohr circle, find:
- The orientation of their planes of action.
- The maximum shear stresses and orientation of their planes of action.
- The stress components on a horizontal plane.
Check the answers using the stress transformation formulas or the stress transformation law.
Solution: Following the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the material element in this example are:
- .
Following the steps for drawing the Mohr circle for this particular state of stress, we first draw a Cartesian coordinate system with the -axis upward.
We then plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as show in both Figure 8 and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on each plane rotating the material element clockwise. This way, the shear stress acting on plane B is negative and the shear stress acting on plane A is positive. The diameter of the circle is the line joining point A and B. The centre of the circle is the intersection of this line with the -axis. Knowing both the location of the centre and length of the diameter, we are able to plot the Mohr circle for this particular state of stress.
The abscissas of both points E and C (Figure 8 and Figure 9) intersecting the -axis are the magnitudes of the minimum and maximum normal stresses, respectively; the ordinates of both points E and C are the magnitudes of the shear stresses acting on both the minor and major principal planes, respectively, which is zero for principal planes.
Even though the idea for using the Mohr circle is to graphically find different stress components by actually measuring the coordinates for different points on the circle, it is more convenient to confirm the results analytically. Thus, the radius and the abscissa of the centre of the circle are
and the principal stresses are
The coordinates for both points H and G (Figure 8 and Figure 9) are the magnitudes of the minimum and maximum shear stresses, respectively; the abscissas for both points H and G are the magnitudes for the normal stresses acting on the same planes where the minimum and maximum shear stresses act, respectively. The magnitudes of the minimum and maximum shear stresses can be found analytically by
and the normal stresses acting on the same planes where the minimum and maximum shear stresses act are equal to
We can choose to either use the double angle approach (Figure 8) or the Pole approach (Figure 9) to find the orientation of the principal normal stresses and principal shear stresses.
Using the double angle approach we measure the angles ∠BOC and ∠BOE in the Mohr Circle (Figure 8) to find double the angle the major principal stress and the minor principal stress make with plane B in the physical space. To obtain a more accurate value for these angles, instead of manually measuring the angles, we can use the analytical expression
One solution is: . From inspection of Figure 8, this value corresponds to the angle ∠BOE. Thus, the minor principal angle is
Then, the major principal angle is
Remember that in this particular example and are angles with respect to the plane of action of (oriented in the -axis)and not angles with respect to the plane of action of (oriented in the -axis).
Using the Pole approach, we first localize the Pole or origin of planes. For this, we draw through point A on the Mohr circle a line inclined 10° with the horizontal, or, in other words, a line parallel to plane A where acts. The Pole is where this line intersects the Mohr circle (Figure 9). To confirm the location of the Pole, we could draw a line through point B on the Mohr circle parallel to the plane B where acts. This line would also intersect the Mohr circle at the Pole (Figure 9).
From the Pole, we draw lines to different points on the Mohr circle. The coordinates of the points where these lines intersect the Mohr circle indicate the stress components acting on a plane in the physical space having the same inclination as the line. For instance, the line from the Pole to point C in the circle has the same inclination as the plane in the physical space where acts. This plane makes an angle of 63.435° with plane B, both in the Mohr-circle space and in the physical space. In the same way, lines are traced from the Pole to points E, D, F, G and H to find the stress components on planes with the same orientation.
Círculo de Mohr para un estado tridimensional general de tensiones
To construct the Mohr circle for a general three-dimensional case of stresses at a point, the values of the principal stresses and their principal directions must be first evaluated.
Considering the principal axes as the coordinate system, instead of the general , , coordinate system, and assuming that , then the normal and shear components of the stress vector , for a given plane with unit vector , satisfy the following equations
Knowing that , we can solve for , , , using the Gauss elimination method which yields
Since , and is non-negative, the numerators from these equations satisfy
- as the denominator and
- as the denominator and
- as the denominator and
These expressions can be rewritten as
which are the equations of the three Mohr's circles for stress , , and , with radii , , and , and their centres with coordinates , , , respectively.
These equations for the Mohr circles show that all admissible stress points lie on these circles or within the shaded area enclosed by them (see Figure 10). Stress points satisfying the equation for circle lie on, or outside circle . Stress points satisfying the equation for circle lie on, or inside circle . And finally, stress points satisfying the equation for circle lie on, or outside circle .
Ver también
- Critical plane analysis
Referencias
- ^ "Principal stress and principal plane". www.engineeringapps.net. Retrieved 2019-12-25.
- ^ Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0-415-27297-1.
- ^ Gere, James M. (2013). Mechanics of Materials. Goodno, Barry J. (8th ed.). Stamford, CT: Cengage Learning. ISBN 9781111577735.
Bibliografía
- Beer, Ferdinand Pierre; Elwood Russell Johnston; John T. DeWolf (1992). Mechanics of Materials. McGraw-Hill Professional. ISBN 0-07-112939-1.
- Brady, B.H.G.; E.T. Brown (1993). Rock Mechanics For Underground Mining (Third ed.). Kluwer Academic Publisher. pp. 17–29. ISBN 0-412-47550-2.
- Davis, R. O.; Selvadurai. A. P. S. (1996). Elasticity and geomechanics. Cambridge University Press. pp. 16–26. ISBN 0-521-49827-9.
- Holtz, Robert D.; Kovacs, William D. (1981). An introduction to geotechnical engineering. Prentice-Hall civil engineering and engineering mechanics series. Prentice-Hall. ISBN 0-13-484394-0.
- Jaeger, John Conrad; Cook, N.G.W.; Zimmerman, R.W. (2007). Fundamentals of rock mechanics (Fourth ed.). Wiley-Blackwell. pp. 9–41. ISBN 978-0-632-05759-7.
- Jumikis, Alfreds R. (1969). Theoretical soil mechanics: with practical applications to soil mechanics and foundation engineering. Van Nostrand Reinhold Co. ISBN 0-442-04199-3.
- Parry, Richard Hawley Grey (2004). Mohr circles, stress paths and geotechnics (2 ed.). Taylor & Francis. pp. 1–30. ISBN 0-415-27297-1.
- Timoshenko, Stephen P.; James Norman Goodier (1970). Theory of Elasticity (Third ed.). McGraw-Hill International Editions. ISBN 0-07-085805-5.
- Timoshenko, Stephen P. (1983). History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. Dover Books on Physics. Dover Publications. ISBN 0-486-61187-6.
enlaces externos
- Mohr's Circle and more circles by Rebecca Brannon
- DoITPoMS Teaching and Learning Package- "Stress Analysis and Mohr's Circle"