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Bending From Wikipedia, the free encyclopedia For other uses, see Bending (disambiguation) . "Flexure" redirects here. For joints that bend, see living hinge . For bearings that operate by bending, see flexure bearing . Bending of an I-beam Continuum mechanics Laws[show] Solid mechanics [show] Fluid mechanics [show] Rheology [show] Scientists[show] v t e In Applied mechanics , bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

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BendingFrom Wikipedia, the free encyclopediaFor other uses, see Bending (disambiguation)."Flexure" redirects here. For joints that bend, see living hinge. For bearings that operate by bending, see flexure bearing.

Bending of an I-beam

Continuum mechanics

Laws[show]

Solid mechanics [show]

Fluid mechanics [show]

Rheology [show]

Scientists[show]

v

t

e

In Applied mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element.

The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two.[1] When the length is considerably longer than the width and the thickness, the element is called a beam. For example, a closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending.

On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending.

In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. Therefore, to make the usage of the term more precise, engineers refer to a specific object such as; the bending of rods,[2] the bending of beams,[1] the bending of plates,[3] the bending of shells [2] and so on.

Contents

1 Quasistatic bending of beams o 1.1 Euler–Bernoulli bending theoryo 1.2 Extensions of Euler-Bernoulli beam bending theory

1.2.1 Plastic bending 1.2.2 Complex or asymmetrical bending 1.2.3 Large bending deformation

o 1.3 Timoshenko bending theory 2 Dynamic bending of beams

o 2.1 Euler-Bernoulli theory 2.1.1 Free vibrations

o 2.2 Timoshenko-Rayleigh theory 2.2.1 Free vibrations

3 Quasistatic bending of plates o 3.1 Kirchhoff-Love theory of plateso 3.2 Mindlin-Reissner theory of plates

4 Dynamic bending of plates o 4.1 Dynamics of thin Kirchhoff plates

5 See also 6 References 7 External links

Quasistatic bending of beams

A beam deforms and stresses develop inside it when a transverse load is applied on it. In the quasistatic case, the amount of bending deflection and the stresses that develop are assumed not to change over time. In a horizontal beam supported at the ends and loaded downwards in the middle, the material at the over-side of the beam is compressed while the material at the underside is stretched. There are two forms of internal stresses caused by lateral loads:

Shear stress parallel to the lateral loading plus complementary shear stress on planes perpendicular to the load direction;

Direct compressive stress in the upper region of the beam, and direct tensile stress in the lower region of the beam.

These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This bending moment resists the sagging deformation characteristic of a beam experiencing bending. The stress distribution in a beam can be predicted quite accurately when some simplifying assumptions are used.[1]

Euler–Bernoulli bending theory

Main article: Euler–Bernoulli beam equation

Element of a bent beam: the fibers form concentric arcs, the top fibers are compressed and bottom fibers stretched.

Bending moments in a beam

In the Euler–Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'. In other words, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the yield stress of the material. For stresses that exceed yield, refer to article plastic bending. At yield, the maximum stress experienced in the section (at the furthest points from the neutral axis of the beam) is defined as the flexural strength.

Consider beams where the following are true:

The beam is originally straight and slender, and any taper is slight

The material is isotropic (or orthotropic), linear elastic, and homogeneous across any cross section (but not necessarily along its length)

Only small deflections are considered

In this case, the equation describing beam deflection ( ) can be approximated as:

where the second derivative of its deflected shape with respect to is interpreted as its curvature, is the Young's modulus, is the area moment of inertia of the cross-section, and is the

internal bending moment in the beam.

If, in addition, the beam is homogeneous along its length as well, and not tapered (i.e. constant

cross section), and deflects under an applied transverse load , it can be shown that:[1]

This is the Euler-Bernoulli equation for beam bending.

After a solution for the displacement of the beam has been obtained, the bending moment ( )

and shear force ( ) in the beam can be calculated using the relations

Simple beam bending is often analyzed with the Euler-Bernoulli beam equation. The conditions for using simple bending theory are:[4]

1. The beam is subject to pure bending. This means that the shear force is zero, and that no torsional or axial loads are present.

2. The material is isotropic (or orthotropic) and homogeneous.3. The material obeys Hooke's law (it is linearly elastic and will not deform plastically).4. The beam is initially straight with a cross section that is constant throughout the beam

length.5. The beam has an axis of symmetry in the plane of bending.6. The proportions of the beam are such that it would fail by bending rather than by

crushing, wrinkling or sideways buckling.7. Cross-sections of the beam remain plane during bending.

Deflection of a beam deflected symmetrically and principle of superposition

Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I -beams ) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.

The classic formula for determining the bending stress in a beam under simple bending is:[5]

where

is the bending stress M - the moment about the neutral axis y - the perpendicular distance to the neutral axis Ix - the second moment of area about the neutral axis x.

Extensions of Euler-Bernoulli beam bending theory

Plastic bending

Main article: Plastic bending

The equation is valid only when the stress at the extreme fiber (i.e., the portion of the beam farthest from the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures.

Complex or asymmetrical bending

The equation above is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical sections, the axial stress in the beam is given by

[6]

where are the coordinates of a point on the cross section at which the stress is to be

determined as shown to the right, and are the bending moments about the y and z

centroid axes, and are the second moments of area (distinct from moments of inertia) about

the y and z axes, and is the product of moments of area. Using this equation it is possible to calculate the bending stress at any point on the beam cross section regardless of moment

orientation or cross-sectional shape. Note that do not change from one point to another on the cross section.

Large bending deformation

For large deformations of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made:

1. Assumption of flat sections - before and after deformation the considered section of body remains flat (i.e., is not swirled).

2. Shear and normal stresses in this section that are perpendicular to the normal vector of cross section have no influence on normal stresses that are parallel to this section.

Large bending considerations should be implemented when the bending radius is smaller than ten section heights h:

With those assumptions the stress in large bending is calculated as:

where

is the normal forceis the section areais the bending moment

is the local bending radius (the radius of bending at the current section)

is the area moment of inertia along the x-axis, at the place (see Steiner's theorem)is the position along y-axis on the section area in which the stress is calculated.

When bending radius approaches infinity and , the original formula is back:

.

Timoshenko bending theory

Main article: Timoshenko beam theory

Deformation of a Timoshenko beam. The normal rotates by an amount which is not equal to

.

In 1921, Timoshenko improved upon the Euler-Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are:

normals to the axis of the beam remain straight after deformation there is no change in beam thickness after deformation

However, normals to the axis are not required to remain perpendicular to the axis after deformation.

The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is[7]

where is the area moment of inertia of the cross-section, is the cross-sectional area, is the

shear modulus, is a shear correction factor, and is an applied transverse load. For materials with Poisson's ratios ( ) close to 0.3, the shear correction factor for a rectangular cross-section is approximately

The rotation ( ) of the normal is described by the equation

The bending moment ( ) and the shear force ( ) are given by

Dynamic bending of beams

The dynamic bending of beams,[8] also known as flexural vibrations of beams, was first investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane rotation. In 1921 Stephen Timoshenko improved the theory further by incorporating the effect of shear on the dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of vibration where the dynamic Euler-Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams continue to be used widely by engineers.

Euler-Bernoulli theory

Main article: Euler-Bernoulli beam equation

The Euler-Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams

of constant cross-section under an applied transverse load is[7]

where is the Young's modulus, is the area moment of inertia of the cross-section, is the deflection of the neutral axis of the beam, and is mass per unit length of the beam.

Free vibrations

For the situation where there is no transverse load on the beam, the bending equation takes the form

Free, harmonic vibrations of the beam can then be expressed as

and the bending equation can be written as

The general solution of the above equation is

where are constants and

The mode shapes of a cantilevered I-beam

1st lateral bending

1st torsional

1st vertical bending

2nd lateral bending

2nd torsional

2nd vertical bending

Timoshenko-Rayleigh theory

Main article: Timoshenko beam theory

In 1877, Rayleigh proposed an improvement to the dynamic Euler-Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko-Rayleigh theory.

The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is [7][9]

where is the polar moment of inertia of the cross-section, is the mass per unit length of the beam, is the density of the beam, is the cross-sectional area, is the shear modulus, and is a shear correction factor. For materials with Poisson's ratios ( ) close to 0.3, the shear correction factor are approximately

Free vibrations

For free, harmonic vibrations the Timoshenko-Rayleigh equations take the form

This equation can be solved by noting that all the derivatives of must have the same form to

cancel out and hence as solution of the form may be expected. This observation leads to the characteristic equation

The solutions of this quartic equation are

where

The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as

Quasistatic bending of plates

Main article: Plate theory

Deformation of a thin plate highlighting the displacement, the mid-surface (red) and the normal to the mid-surface (blue)

The defining feature of beams is that one of the dimensions is much larger than the other two. A structure is called a plate when it is flat and one of its dimensions is much smaller than the other two. There are several theories that attempt to describe the deformation and stress in a plate under applied loads two of which have been used widely. These are

the Kirchhoff-Love theory of plates (also called classical plate theory) the Mindlin-Reissner plate theory (also called the first-order shear theory of plates)

Kirchhoff-Love theory of plates

The assumptions of Kirchhoff-Love theory are

straight lines normal to the mid-surface remain straight after deformation straight lines normal to the mid-surface remain normal to the mid-surface after

deformation the thickness of the plate does not change during a deformation.

These assumptions imply that

where is the displacement of a point in the plate and is the displacement of the mid-surface.

The strain-displacement relations are

The equilibrium equations are

where is an applied load normal to the surface of the plate.

In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external load can be written as

In direct tensor notation,

Mindlin-Reissner theory of plates

The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not necessarily normal to the mid-surface after deformation. The displacements of the plate are given by

where are the rotations of the normal.

The strain-displacement relations that result from these assumptions are

where is a shear correction factor.

The equilibrium equations are

where

Dynamic bending of plates

Main article: Plate theory

Dynamics of thin Kirchhoff plates

The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are

where, for a plate with density ,

and

The figures below show some vibrational modes of a circular plate.

mode k = 0, p = 1

mode k = 0, p = 2

mode k = 1, p = 2

Mohr's circleFrom Wikipedia, the free encyclopedia

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (December 2014)

Figure 1. Mohr's circles for a three-dimensional state of stress

Mohr's circle, named after Christian Otto Mohr, is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor.

After performing a stress analysis on a material body assumed as a continuum, the components of the Cauchy stress tensor at a particular material point are known with respect to a coordinate system. The Mohr circle is then used to determine graphically the stress components acting on a rotated coordinate system, i.e., acting on a differently oriented plane passing through that point.

The abscissa, , and ordinate, , of each point on the circle, are the magnitudes of the normal stress and shear stress components, respectively, acting on the rotated coordinate system. In other words, the circle is the locus of points that represent the state of stress on individual planes at all their orientations, where the axes represent the principal axes of the stress element.

Karl Culmann was the first to conceive a graphical representation for stresses while considering longitudinal and vertical stresses in horizontal beams during bending. Mohr's contribution extended the use of this representation for both two- and three-dimensional stresses and developed a failure criterion based on the stress circle.[1]

Alternative graphical methods for the representation of the stress state at a point include the Lame's stress ellipsoid and Cauchy's stress quadric.

The Mohr circle can be applied to any symmetric 2x2 tensor matrix, including the strain and moment of inertia tensors.

Contents

1 Motivation for the Mohr Circle 2 Mohr's circle for two-dimensional state of stress

o 2.1 Equation of the Mohr circle o 2.2 Sign conventions

2.2.1 Physical-space sign convention 2.2.2 Mohr-circle-space sign convention

o 2.3 Drawing Mohr's circle o 2.4 Finding principal normal stresses o 2.5 Finding principal shear stresses o 2.6 Finding stress components on an arbitrary plane

2.6.1 Double angle 2.6.2 Pole or origin of planes

o 2.7 Finding the orientation of the principal planes o 2.8 Example

3 Mohr's circle for a general three-dimensional state of stresses 4 References 5 Bibliography 6 External links

Motivation for the Mohr Circle

Figure 2 Stress in a loaded deformable material body assumed as a continuum.

Internal forces are produced between the particles of a deformable object, assumed as a continuum, as a reaction to applied external forces, i.e., either surface forces or body forces. This reaction follows from Euler's laws of motion for a continuum, which are equivalent to Newton's laws of motion for a particle. A measure of the intensity of these internal forces is called stress.[2] Because the object is assumed as a continuum, these internal forces are distributed continuously within the volume of the object.

In engineering, e.g., structural, mechanical, or geotechnical, the stress distribution within an object, for instance stresses in a rock mass around a tunnel, airplane wings, or building columns, is determined through a stress analysis. Calculating the stress distribution implies the determination of stresses at every point (material particle) in the object. According to Cauchy, the stress at any point in an object (Figure 2), assumed as a continuum, is completely defined by the nine stress components of a second order tensor of type (2,0) known as the Cauchy stress tensor, :

Figure 3 - Stress transformation at a point in a continuum under plane stress conditions.

After the stress distribution within the object has been determined with respect to a coordinate

system , it may be necessary to calculate the components of the stress tensor at a particular

material point with respect to a rotated coordinate system , i.e., the stresses acting on a plane with a different orientation passing through that point of interest —forming an angle with

the coordinate system (Figure 3). For example, it is of interest to find the maximum normal stress and maximum shear stress, as well as the orientation of the planes where they act upon. To achieve this, it is necessary to perform a tensor transformation under a rotation of the coordinate system. From the definition of tensor, the Cauchy stress tensor obeys the tensor transformation law. A graphical representation of this transformation law for the Cauchy stress tensor is the Mohr circle for stress.

Mohr's circle for two-dimensional state of stress

Figure 4 - Stress components at a plane passing through a point in a continuum under plane stress conditions.

In two dimensions, the stress tensor at a given material point with respect to any two perpendicular directions is completely defined by only three stress components. For the

particular coordinate system these stress components are: the normal stresses and , and the shear stress . From the balance of angular momentum, the symmetry of the Cauchy stress tensor can be demonstrated. This symmetry implies that . Thus, the Cauchy stress tensor can be written as:

The objective is to use the Mohr circle to find the stress components and on a rotated

coordinate system , i.e., on a differently oriented plane passing through and

perpendicular to the - plane (Figure 4). The rotated coordinate system makes an angle

with the original coordinate system .

Equation of the Mohr circle

To derive the equation of the Mohr circle for the two-dimensional cases of plane stress and plane strain, first consider a two-dimensional infinitesimal material element around a material point (Figure 4), with a unit area in the direction parallel to the - plane, i.e., perpendicular to the page or screen.

From equilibrium of forces on the infinitesimal element, the magnitudes of the normal stress and the shear stress are given by:

[show]Derivation of Mohr's circle parametric equations - Equilibrium of forces

Both equations can also be obtained by applying the tensor transformation law on the known Cauchy stress tensor, which is equivalent to performing the static equilibrium of forces in the direction of and .

[show]Derivation of Mohr's circle parametric equations - Tensor transformation

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 positive sign because they act in the negative direction of the -axis and -axis, respectively.

Mohr-circle-space sign convention

Figure 5. Engineering mechanics sign convention for drawing the Mohr circle. This article follows sign-convention # 3, as shown.

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:

1. Positive shear stresses are plotted upward (Figure 5, sign convention #1)2. 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 (references needed) 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

Figure 6. Mohr's circle for plane stress and plane strain conditions (double angle approach). After a stress analysis, the stress components , , and at a material point are known. These stress components act on two perpendicular planes and passing through . The coordinates of point and on the Mohr circle are the stress components acting on the planes and of the material element, respectively. The Mohr circle is then used to find the stress components and , i.e., coordinates of any stress point on the circle, acting on any other plane passing through . The angle between the lines and is double the angle

between the normal vectors of planes and passing through .

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 :

1. Draw the Cartesian coordinate system with a horizontal -axis and a vertical -axis.

2. 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.

3. Draw the diameter of the circle by joining points and with a straight line .4. 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 principal shear stresses

The maximum and minimum shear stresses correspond to the abscissa 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

Figure 7. Mohr's circle for plane stress and plane strain conditions (Pole approach). 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.

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

Figure 8

Figure 9

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 10º with respect to the horizontal plane. Using the Mohr circle, find:

The principal stresses and 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 ordinates 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

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 10º 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 63.435º 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.

Mohr's circle for a general three-dimensional state of stresses

Figure 10. Mohr's circle for a three-dimensional state of stress

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 the 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 .