Lecture 2 – Loads, traction, stress Stress

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Stress

Lecture 2 – Loads, traction, stress

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Introduction

• One important step in mechanical design is the determination of the internal stresses, once the external load are assigned, and to assess that they do not exceed material allowables.

• Internal stress - that differ from surface or contact stresses that are generated where the load are applied - are those associated with the internal forces that are created by external loads for a body in equilibrium.

• The same concept holds for complex geometry and loads. To evaluate these stresses is not a straightforward matter, suffice to say here that they will invariably be non-uniform over a

surface, that is, the stress at some particle will differ from the stress at a neighbouring particle

a slender block of material; (a) under the action of external forces F, (b) internal normal stress σ, (c) internal normal and

shear stress

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Tractions and the Physical Meaning of Internal Stress

• All materials have a complex molecular microstructure and each molecule exerts a force on each of its

neighbors. The complex interaction of countless

molecular forces maintains a body in equilibrium in its unstressed state.

• When the body is disturbed and deformed into a new equilibrium position, net forces act. An imaginary

plane can be drawn through the material

• The force exerted by the molecules above the plane on the material below the plane and can be attractive or repulsive. Different planes can be taken through the same portion of material and, in general, a different force will act on the plane.

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The traction at some particular point in a material is defined as follows:

• take a plane of surface area S through the point, on which acts a force F.

• shrink the plane – as it shrinks in size both S and F get smaller, and the direction in which the force acts may change, but eventually the ratio F / S will remain constant and the force will act in a

particular direction.

• limiting value of this ratio of force over surface area is defined as the traction vector (or stress vector)

An infinite number of traction vectors act at any single point, since an infinite number of different planes pass through a point

For this reason the plane on which the traction vector acts must be specified; this can be done by specifying the normal n to the surface on which the traction acts

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The traction vector can be decomposed into

components which act normal and parallel to the surface upon which it acts. These components are called the stress components, or simply stresses, and are denoted by the symbol s ; subscripts are added to signify the surface on which the stresses act and the directions in which the stresses act.

Sign Convention for Stress Components The following convention is used:

• the stress is positive when the direction of the normal and the direction of the stress

component are both positive or both negative

• the stress is negative when one of the

directions is positive and the other is negative

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Type of stress

Normal stress.

The resisting area is normal to the internal force.

• Tensile stress. Is the stress induced in a body when subjected to two equal and opposite pulls (tensile force) as a result of which there is the tendency to increase in length

• Compressive stress. Stress induced in a body when subjected to equal and opposite pushes as a a result of which there is a tendency of decrease in lenght

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Type of stress

Combined stress.

A condition of stress that cannot be represented by a single resultant stress

• Shear stress. Forces parallel to the area

resisting the force cause shear stress. It differ from tensile and compressive stresses. Known also as tangential stress

• Torsional stress. The stress and deformation induced in a circular shaft by a twisting

moment.

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Simple stress

In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction).

Three such simple stress situations, that are often encountered in engineering design, are:

• Uniaxial normal stress

• Simple shear stress

• Isotropic normal stress

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Simple stress

• Uniaxial normal stress

This stress state occurs in a straight rod of uniform section and homogeneous material subjected to tension by opposite forces of magnitude F along its axis.

If the system is in equilibrium and not changing with time, and the weight of the bar can be

neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity

through the full cross-sectional area, A.

Therefore, the stress, σ throughout the bar, across any horizontal surface, is expressed by:

𝜎 = 𝐹 𝐴

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Simple stress

• Simple shear stress

This type of stress occurs when a uniformly thick layer of elastic material is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer

Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F.

Assuming that the direction of the forces is known, the stress across M can be expressed simply by the single number τ , calculated simply with the magnitude of those forces, F and the cross sectional area, A.

For any plane S that is perpendicular to the layer, the net internal

force across S, and hence the stress, will be zero. 𝜏 = 𝐹

𝐴

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Simple stress

• Isotropic normal stress (hydrostatic stress)

Another simple type of stress occurs when the material body is under equal compression or tension in all directions.

This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces —

provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected.

The force across any section S of the cube must balance the forces applied below the section. In the three sections shown, the forces are 𝐹 (top right), 𝐹 2 (bottom left), and 𝐹 3/2 while the area of S is 𝐴, 𝐴 2 and 𝐴 3/2, respectively.

So the stress across S is F/A in all three cases.

𝜎 = 𝐹 𝐴

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Representation of a three-dimensional stress state

STRESS TENSOR. A tridimensional stress state is described by the Cauchy stress tensor σ, (or true stress tensor)

The tensor consists of nine components

that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration.

The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n.

The SI units of both stress tensor and stress vector are N/m², corresponding to the stress scalar. The unit vector is dimensionless.

𝜎_𝑖𝑗 =

𝜎₁₁ 𝜎₁₂ 𝜎₁₃ 𝜎₂₁ 𝜎₂₂ 𝜎₂₃ 𝜎₃₁ 𝜎₃₂ 𝜎₃₃

≡

𝜎_𝑥 𝜏_𝑥𝑦 𝜏_𝑥𝑧 𝜏_𝑦𝑥 𝜎_𝑦 𝜏_𝑦𝑧 𝜏_𝑧𝑥 𝜏_𝑧𝑦 𝜎_𝑧

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STRESS DEVIATOR TENSOR. The stress tensor can be always expressed as the sum of two terms:

1. a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor, which is responsible for the change the volume of the stressed body;

2. a deviatoric component called the stress deviator tensor, which is responsible for the shape change at constant volume

Where 𝜋 = 𝜎_𝑘𝑘 3 is the pressure.

The deviatoric stress tensor can be obtained by subtracting the hydrostatic stress tensor from the Cauchy stress tensor:

𝜎_𝑖𝑗 = 𝑠_𝑖𝑗 + 1

3𝜎_𝑘𝑘𝛿_𝑖𝑗

𝑠_𝑖𝑗 = 𝜎_𝑖𝑗 − 𝜋𝛿_𝑖𝑗

𝑠_𝑖𝑗 =

𝑠₁₁ 𝑠₁₂ 𝑠₁₃ 𝑠₂₁ 𝑠₂₂ 𝑠₂₃ 𝑠₃₁ 𝑠₃₂ 𝑠₃₃

≡

𝜎₁₁ − 𝜋 𝜎₁₂ 𝜎₁₃ 𝜎₂₁ 𝜎_𝑦 − 𝜋 𝜎₂₃ 𝜎₃₁ 𝜎₃₂ 𝜎_𝑧 − 𝜋

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Principal stresses and stress invariants. At every point in a stressed body there are at least three planes, called principal planes, with normal vectors, called principal directions,

where the corresponding stress vector is perpendicular to the plane and where there are no shear stresses.

The three stresses normal to these principal planes are called principal stresses.

The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates.

The components of the stress tensor σ_ijdepend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical quantity and as such, it is independent of the coordinate system chosen to represent it.

There are certain invariants associated with every tensor which are also independent of the coordinate system.

𝜎_𝑖𝑗 =

𝜎₁₁ 𝜎₁₂ 𝜎₁₃ 𝜎₂₁ 𝜎₂₂ 𝜎₂₃ 𝜎₃₁ 𝜎₃₂ 𝜎₃₃ ≡

𝜎₁ 0 0 0 𝜎₂ 0 0 0 𝜎₃

𝐼₁ = 𝜎₁₁ + 𝜎₂₂ + 𝜎₃₃

𝐼₂ = 𝜎₁₁𝜎₂₂ + 𝜎₂₂𝜎₃₃ + 𝜎₃₃𝜎₁₁ − 𝜎₁₂² − 𝜎₂₃² − 𝜎₃₁² 𝐼₃ = 𝑑𝑒𝑡 𝜎_𝑖𝑗

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The Cauchy stress tensor obeys the tensor

transformation law under a change in the system of coordinates.

A graphical representation of this transformation law is the Mohr's circle for stress.

The Mohr circle is 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, σ_n, and ordinate, τ_n, 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.

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Mohr circle for two-dimensional stress state

In two dimensions, the stress tensor at a given material point P with respect to any two

perpendicular directions is completely defined by only three stress components.

For the particular coordinate system ( x , y ) these stress components are: the normal stresses σ_x and σ_y, and the shear stress τ_{x y}

From the balance of angular momentum, τ_xy = τ_yx

Thus, the Cauchy stress tensor can be written as:

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Plane stress: s_z=0 Planes strain: e_z=0

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

These two equations are the parametric equations of the Mohr circle.

𝜎_𝑛 = 1

2 𝜎_𝑥 + 𝜎_𝑦 + 1

2 𝜎_𝑥 − 𝜎_𝑦 𝑐𝑜𝑠2𝜃 + 𝜏_𝑥𝑦𝑠𝑖𝑛2𝜃 𝜏_𝑛 = −1

2 𝜎_𝑥 − 𝜎_𝑦 𝑠𝑖𝑛2𝜃 + 𝜏_𝑥𝑦𝑐𝑜𝑠2𝜃

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Eliminating the parameter 2θ from these parametric equations will yield the non-parametric equation of the Mohr circle:

With center in C ¹

2 𝜎_𝑥 + 𝜎_𝑦 , 0 𝜎_𝑛 − 1

2 𝜎_𝑥 + 𝜎_𝑦

2

+ 𝜏_𝑛² = 𝑅²

𝑅 = 1

2 𝜎_𝑥 − 𝜎_𝑦

2

+ 𝜏_𝑛²

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Mohr circle: applications examples

Scope of the design is the determination of the maximum stress resulting from external load (operation and exceptional)

Step 1. Determine the point and locations where stress is the highest and more critical.

Step 2. From the general information on s and t, determined the principal stresses

Step 3. Select a failure criterion for the specific material

Step 4. Determined material allowables

Step 5. Compare calculated stress with allowable provide a safety factor.

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SIMPLE TENSION

All point of the section of an element are equally stressed.

There no preferential choices for the most critical point: select a reference cube element around the point and construct the stress state.

t

s x

y z

s_x

𝜎₂ = 𝜎₃ = 0 𝜎₁ = 𝜎_𝑥 𝜎_𝑖𝑗 =

𝜎_𝑥 0 0 0 0 0 0 0 0

=

𝜎₁ 0 0 0 0 0 0 0 0

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SIMPLE TORSION

The most stressed point are those at the maximum radius of the section of the element

Z is a principal axis because the shear stress is 0 t

s x

y z

t_xy

t_yx

𝜎₃

𝜏_𝑥𝑦

𝜎_𝑖𝑗 =

0 𝜏_𝑥𝑦 0 𝜏_𝑦𝑥 0 0

0 0 0

=

𝜎₁ 0 0 0 0 0 0 0 𝜎₃

𝜎₂ 𝜎₁

𝜎₁ = 𝜎₃ = 𝜏_𝑥𝑦

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BENDING AND TORSION

The most stressed point are those at the maximum radius of the section of the element

Z is a principal axis because the shear stress is 0 t

s x

y z

t_xy

t_yx

𝜎₃

𝜏_𝑚𝑎𝑥 𝜎_𝑖𝑗 =

𝜎_𝑥 𝜏_𝑥𝑦 0 𝜏_𝑦𝑥 0 0

0 0 0

=

𝜎₁ 0 0 0 0 0 0 0 𝜎₃

𝜎₂ 𝜎₁

𝜎₁ = 𝜎₃ = 𝜏_𝑥𝑦

s_x

𝜏_𝑥𝑦 𝜎_𝑥

𝜏_𝑥𝑦

𝜎₁ = 𝜎_𝑥

2 + 𝜎_𝑥 2

2

+ 𝜏_𝑥𝑦² 𝜎₂ =0 𝜎₃ = 𝜎_𝑥

2 − 𝜎_𝑥 2

2

+ 𝜏_𝑥𝑦²

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Suggested readings

• http://www.mech.utah.edu/~brannon/public/Mohrs_Circle.pdf

• Schaum's Outline of Strength of Materials, Fifth Edition (Schaum's Outline Series) Fifth (5th) Edition Paperback – September 12, 2010

• Strength of Materials (Dover Books on Physics) Reprinted Edition by J. P. Den Hartog, ISBN-10: 0486607550