an sys notes

7/27/2019 An Sys Notes

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Principle stressesPrinciple stresses are the results of combinations of tensile and shear stresses at the same point

on the cross-section of a piece of loaded structure. Although these two stresses are from different

causes, the plane on which they act can be imagined to be rotated and when this is done the sizeof each stress varies. Some of the shear stress can then become a tensile stress and the opposite

also is applicable.

The rules for this are defined by a circle diagram with the tension or compression stress plottedhorizontally and the shear stress vertically. A point on the circle defines the angle at which the

particular combination of stress applies and this is the combination of stress first found and

described above.

On the horizontal axis the circle cuts it in two places. These are the values of maximum and

minimum principle stresses, when all of the stress is expressed in terms of a tension (or

compression) form.

Von Mises stress

In an elastic body that is subject to a system of loads in 3 dimensions, a complex 3 dimensional system

of stresses is developed (as you might imagine). That is, at any point within the body there are stressesacting in different directions, and the direction and magnitude of stresses changes from point to point.

The Von Mises criterion is a formula for calculating whether the stress combination at a given point will

cause failure.

There are three "Principal Stresses" that can be calculated at any point, acting in the x, y, and z

directions. (The x,y, and z directions are the "principal axes" for the point and their orientation changes

from point to point, but that is a technical issue.)

Von Mises found that, even though none of the principal stresses exceeds the yield stress of the

material, it is possible for yielding to result from the combination of stresses. The Von Mises criteria is a

formula for combining these 3 stresses into an equivalent stress, which is then compared to the yieldstress of the material. (The yield stress is a known property of the material, and is usually considered to

be the failure stress.)

The equivalent stress is often called the "Von Mises Stress" as a shorthand description. It is not really a

stress, but a number that is used as an index. If the "Von Mises Stress" exceeds the yield stress, then the

material is considered to be at the failure condition.


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The formula is actually pretty simple, if you want to know it:

(S1-S2)^2 + (S2-S3)^2 + (S3-S1)^2 = 2Se^2

Where S1, S2 and S3 are the principal stresses and Se is the equivalent stress, or "Von Mises Stress".

Finding the principal stresses at any point in the body is the tricky part.

Shear strength

Subjected to forces which cause it to twist, or one face to slide relative to an opposite face, a

material is said to be in shear (Figure 5). Compared to tensile and compressive stress and strain,the shear forces act over an area which is in line with the forces.

Figure 5: Shear stress applied to an object

The force per unit area is referred to as the shear stress, denoted by the symbol (Greek letter

tau), where

Its unit is the pascal (Pa), where force is measured in newtons (N) and area in square metres.


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When shear stress is applied, there will be an angular change in dimension, just as there is a

change in length when materials are under tension or compression. Shear strain, denoted by the

symbol (Greek letter gamma), is defined by

where the angular deformation, symbol (Greek letter phi) is expressed in radians. The last

approximate equality results from the fact that the tangent of a small angle is almost the same asthe angle expressed in radians. This is the reason why some texts give the radian as the unit of

strain. Both shear strain and angular deformation are ratios, so have no units. However, it is not

unusual for shear strain to be quoted in %, as with tensile strain.

Shear stresses are most evident where lap joints are fastened together and forces applied to pull

them apart, but are also seen when rods are twisted, or laminated boards bent.

The shear strength of a material is the maximum stress that it can withstand in shear beforefailure occurs. For example, punching, cropping and guillotining all apply shear stresses of more

than the maximum shear stress for that material.

As with Hookes Law for tensile stress, most metals have a shear stress which is proportional to

the shear strain. And in a similar way to Youngs modulus, the gradient of the graph is referredto as the shear modulus ormodulus of rigidity. Again the SI unit

3for shear modulus is the

pascal (Pa).

3 You are very likely to find Youngs modulus and shear modulus quoted in psi (pounds force

per square inch) or kpsi (thousands of psi). To convert to MPa, multiply the figure in kpsi by

6.89. Watch the units! You should also expect there to be very wide variations in the figuresquoted, as these depend critically on alloy composition and work hardening (for metals), on

purity (for ceramics) and on formulation (for polymers).

Table 2: Shear strength and shear modulus for selected materials

material shear strength MPa modulus of rigidity GPa

96% alumina 330

304 stainless steel 186 73

copper 42 44

aluminium 30 26

Sn63 solder 28 6

epoxy resin 1040

Stiffness


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The stiffness of a material is an important aspect of PCB design, being the ability of the material

to resist bending. When a board is bent, one surface stretches and the inside of the radius is

compressed. The more a material bends, the more the outer surface stretches and the internalsurface contracts. A stiff material is one that gives a relatively small change in length when

subject to tension or compression, in other words, a small value of strain/stress.

However, on the basis that stiff = good, a natural feeling that this should be a larger figure means

that we actually quote the ratio of stress/strain. So a stiff material has a high value of Youngs

modulus. From Table 1 you will be aware of the very wide range of properties in electronicmaterials. Note that the metals in this list are much stiffer than polymers, but well below the

stiffness of a typical ceramic. However, this stiffness is accompanied by extreme brittleness. One

of the features of a metal is that it is unlikely to shatter, as would a piece of glass or ceramic, but

it will show permanent deformation when forces are appliedask any car body shop!

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