3 Deformations and Stress Patterns in Machine Components
3.2 DEFORMATIONS IN THE ELASTIC RANGE
There is a linear dependence between stress and deformation called Hooke’s law. It means that if we know the stress magnitude and the modulus of elasticity of the material, we can calculate the elastic deformation of the stressed piece. On the other hand, when the deformation is known, the stresses can be easily determined. This dependence is the basis for most stress-measuring instru-ments; they measure deformations and translate them into stresses.
But there is one more field to which Hooke’s law can be applied: if we know (or are able to make a true assumption about) the distribution of the deformations over the cross section of a part, we can calculate the maximal stress. To illustrate this idea, let’s consider the bending of a straight beam. The so-called ‘‘engineering approach” to strength calculation of beams is based on the hypothesis of plane cross sections (PCS). The idea of this hypothesis is that the cross sections of the beam, which are plane before bending (Figure 3.5a), remain plane after bending (Figure 3.5b).
With bending, the neighboring cross sections turn relative to each other by a certain angle j. The 1
Deformations and Stress Patterns in Machine Components 25
magnitude of this angle depends on the magnitude of the bending moment, the rigidity of the beam and, obviously, on the distance between the sections. Because the cross sections remain plane, the deformation of the beam between them is a linear function of the height. Across the width of the beam, the deformation doesn’t change (Figure 3.5c).
Let’s examine the word hypothesis. A hypothesis is a kind of assumption usually developed as follows: If we desire to work out the relation between appearances to explain or predict phenomena or processes and don’t have sufficient data to establish an exact dependence, we may begin by making some simplifying assumptions. Then, to check the validity of the hypothesis, these assump-tions and the dependence based on them are checked experimentally. If the correspondence between the experimental results and this invented dependence is not satisfactory, more appropriate assumptions should be made.
In our case, the hypothesis of PCS has been proved to be very exact and productive as applied to beams of a uniform cross section. But there are some exceptions to this rule (as with all rules).
For example, in the thin-walled beams shown in Figure 3.6, the stresses across the width are not uniform; hence, the plane cross sections become distorted after application of a load. Use of formulas based on the hypothesis of plane cross sections for such beams (with wide and thin flanges) may lead to a considerable error (the real stresses will be bigger than the calculated ones).
But let’s get back to the usual case when this hypothesis works with high accuracy.
Figure 3.7a shows a rod with a rectangular cross section loaded by bending moment M. The cross section of the rod is of variable height: h on the right and H on the left. A fillet of radius r makes the transition from the smaller height to the bigger one.
First, let’s define the stresses in the right part of the rod, where the cross section is of constant height h. For this part, the hypothesis of plane cross sections is valid. We consider a small element FIGURE 3.5 Bending deformation of a straight beam.
FIGURE 3.6 Deformation of thin-walled beams with wide thin shelves or flanges.
M M ϕ
(a)
(b)
(c)
26 Machine Elements: Life and Design
of the rod bounded by plane sections a-a-a and b-b-b (Figure 3.7a). Dashed lines show the shape of this element after bending. The boundary sections remain plane but they turn relative to each other by angle ϕ. With bending, the layers of the rod above the zero line (the neutral axis) become longer and the layers below the zero line become shorter, both by a variable amount δ (see Figure 3.7b).
Because the deformations are antisymmetric about the zero line, it is enough to consider only half of the element as shown in Figure 3.7b. (In this picture, one side, a-a, has been left unturned for convenience, and the other side, b-b, is rotated by the angle ϕ, which is assumed to be very small.)
The stress in the deformed layer according to Hooke’s law
(3.1) where
E= modulus of elasticity of the material (MPa)
ε= relative deformation (in this case, it is elongation) determined by the equation FIGURE 3.7A Plane sections and broken sections.
FIGURE 3.7B Strain distribution in a straight bar.
r
α H
ϕ α − ϕ
a ∆ a
b b
c cd
d
c d
a b M
0 h 0
M
σ=Eε
ε δ=
∆
b
∆
ϕ
0 a
δ
h/2 dy
y b
a
0
Deformations and Stress Patterns in Machine Components 27 As is clear from the picture, the magnitude of δ depends on the ordinate y:
Therefore, the stress in each layer is proportional to the distance y of the layer from the zero line:
As the values E, ϕ, and ∆ are constant in this problem, all of them may be substituted by one coefficient, say, K=Eϕ/∆:
(3.2) Now we are ready to calculate the stresses in this part of the rod. Elementary moment produced by a slice dy about the zero line
Here b= thickness (width) of the rod (not shown in the picture).
The sum of all the elementary moments, taken above and beneath the zero line, should be equal to the loading moment M:
From here,
The maximal stress (at y=h/2) from Equation 3.2:
(3.3) Fortunately, we didn’t make any mistakes and have got successfully the well-known equation for a rod of a constant rectangular cross section. Now, let’s try to deal with the fillet area. In this area, the hypothesis of PCS doesn’t work. For this case, the hypothesis of broken-line cross sections (BCS) suggested by Verhovsky1 works better. (The BCS hypothesis is based on experiments with rubber parts exposed to bending. The “cross sections” are just drawn on the side surface of the rubber bar, so that their shape under bending can be easily observed.) According to this hypothesis, the cross sections, which continue to be plane, are directed at a perpendicular to the curvilinear surface (section c-c-c, Figure 3.7a).
We consider a small element of the rod bounded by cross sections c-c-c and d-d-d (Figure 3.7a).
This element is adjoined to the beginning of the fillet and characterized by an angle α that is supposed to be very small. Also, here the dashed lines show the shape of this element after bending.
δ=yϕ
σ=Eϕ y
∆
σ =Ky
dM=σby dy⋅ =Kby dy2
M h dM Kbh
=2
∫
= 1230 2 /
K M
=12bh
3
σmax=Ky= M = ; bh
h M
bh 12
2 6
3 2
28 Machine Elements: Life and Design
In the same way as the previous case, we consider only half of the small element (c-c-d-d, Figure 3.7c).
One side of the element (c-c) is left unturned, and the other side, d-d, is rotated because of bending by an angle ϕ. Notice that in this case, the elongation δ is also proportional to the ordinate y:
In the previous case, the length of the deformed layers ∆ was constant, whereas in this case, the length is variable and determined as follows:
This result is easily obtainable, if we take into consideration that the length of the elementary layer depends linearly on its distance from point e. The layer with vertical coordinate y is at a distance of (r+h/2 – y) from point e. (Don’t forget: angle α is very small.)
That is, in the element c-c-d-d not only do the deformations grow as the layers get farther from the zero line, but also the length of the layers decreases, so the increase of the relative deformation (ε=δ/∆) with the increasing ordinate y is not linear but more intensive.
The stress in the layer at a distance y from the zero line is
Because in this problem, values E, α, and ϕ are constant, they can be substituted by one coefficient, say L=Eϕ/α. Then the maximal stress (at y=h/2) is given by
(3.4) FIGURE 3.7C Strain distribution in a filleted bar.
e
c
c d
0
d
dy
y
0 h/2 α−ϕ
ϕ
δ α
∆
δ=yϕ
∆ = +(r h/2−y)α
σ ε δ ϕ
* α
( / )
= = =
+ −
E E Ey
r h y
∆ 2
σ*max=L h r 2
Deformations and Stress Patterns in Machine Components 29 The elementary moment produced by a slice dy about the zero line is given by
The sum of all the elementary moments should be equal to the loading moment M:
After the L value is derived from this equation and inserted in Equation 3.4, the following formula for the maximal stress is obtained:
(3.5)
where
(3.6)
Let’s do comparative stress calculations for the straight part of the rod by Equation 3.3 and for the beginning of the fillet by Equation 3.5 and Equation 3.6. For this calculation, we can take b= h= 1, because, for the comparison, it is not important if it is 1 mm or in. (or whichever unit of length you want).
For the straight section, the maximal stress obtained from Equation 3.3 is
At the beginning of the fillet, the stress depends on the fillet radius. For example, if r= 0.2h, Φ= 0.1389, and the stress
The stress concentration factor Kt = 9/6 = 1.5. If the fillet radius r = 0.1h, Φ = 0.22, the maximal stress
and the stress concentration factor Kt= 11.36/6 = 1.89.
Here, shown clearly (and even visually), is the mechanism of the local stress raise called stress concentration. It is caused by a change in the shape of the machine component that modifies the deformation pattern. The place where the shape changes is called a stress raiser (or stress riser).
The sharper the change of the shape (in our case, the less the fillet radius), the higher the local stress.
dM by dy L y b
The BCS hypothesis is good for a visual presentation of the stress raiser mechanism. But it gives a correct result only when the bigger section is sufficiently larger than the smaller one, and the stress raiser is not too sharp. Specifically, in the case considered previously, according to R. E.
Peterson,3 Kt = 1.34–1.53 for r = 0.2h, and Kt = 1.48–1.93 for r = 0.1h depending on the ratio H/h.
The BCS hypothesis doesn’t take into account this factor, and in our calculation we got Kt = 1.5 and Kt = 1.89, respectively, i.e., close to the biggest values given by Peterson.
In most cases, the magnitude of the local stress can’t be determined by an analytical procedure.
Experimental methods and FEM are usually used for this purpose.
Another application of the BCS hypothesis is with a flat strip of rectangular cross section, in tension. If the strip is of constant cross section (Figure 3.8a), then obviously the hypothesis of PCS is valid. Under load, any small element of the strip bounded by sections a-a and b-b elongates evenly across the width of the strip (see the dashed line).
Figure 3.8b shows a strip with semicircular notches that has the same width w (in the narrow section). The cross sections, which keep their shape when loaded, look different (sections c-c and d-d). In the area of the notches, the length of the layers is significantly shorter than in the middle of the strip, and when section d-d is displaced under load (see dashed line), the relative elongation in this area will be greater. The stress distribution across the width of the strip will change correspondingly. Because in both cases the sum of all the stresses must be equal to the external load, the maximal stress in the notched specimen will be greater than that in the straight strip.
Let’s discuss more fully the meaning of the stress concentration factor (SCF), because it is not always clear. The maximal stress magnitude in the area of the stress raiser is determined using theory of elasticity techniques, by FEM, or experimentally. To calculate the SCF value, the maximal stress is divided by some “nominal” stress determined by methods of the strength of materials (so-called engineering methods). For the rod shown in Figure 3.7a, the nominal stress is determined from Equation 3.3. For the strip presented in Figure 3.8b, the nominal stress is obviously the ratio of the stretching force to the minimal cross-sectional area (through the bottoms of the notches). But if, for example, you are going to use SCF diagrams for a splined shaft, or a threaded shaft, or a shaft with a keyway, you have to figure out which cross section had been taken by the author when calculating the ‘‘nominal” stress: was it a circular cross section corresponding to the inner diameter of the teeth or thread, the real cross section, or something else? To calculate strength of the machine part, you should know the maximal stress. For that you calculate the nominal stress and then multiply it by the SCF. So until you know which cross section should be taken for the
‘‘nominal” calculation, you can’t use these diagrams.
FIGURE 3.8 Stress concentration in a notched strip in tension.
Tensile load Stress b
a
W W
d d
(a) (b)
b
a c c
The SCF obtained from such calculations is called theoretical SCF (Kt), and it enables (in principle) the designer to determine the maximal stress. At this point, the mathematics comes to an end, and some uncertainty begins. The fact is that in the trials for strength, the load capacity of the machine members reduces not by Kt times but less, by Ke times. The Ke value is called effective SCF, and that is what we really need to know but can’t. The interrelation between Kt and Ke is complicated and multifactored. Much depends on the loading condition. For example, under a static load, ductile materials (such are all the structural steels) are not sensitive to stress raisers.
When the local stress comes up to the yield point, this brings about a local plastic deformation that doesn’t harm the strength of the material but redistributes the stresses in such a way that the less-stressed volumes became more stressed in the elastic range. As the static load increases, the plastically deformed volume increases until it spreads across the whole section. This is the limiting state under static load whether there is a stress raiser or not, and thus here, Ke = 1.
Under a cyclic load, the stress raisers reduce the load capacity of the machine parts, but less than the Kt value predicts, depending on the feature called sensitivity to stress concentration. In practice, the Ke value is often derived from the following equation:
where q is a coefficient of sensitivity to stress concentration.
That coefficient depends on a number of factors:
The kind of material and existence in its structure of internal stress raisers (such as discon-tinuities in the form of foreign inclusions, defects on the grain boundaries, and disloca-tions). For instance, gray cast iron is hardly sensitive to sharp stress raisers under cyclic load, because its structure is pierced with flakes of graphite, which are sharp stress raisers incorporated into the material. And yet, the gray cast iron is sensitive to the stress concentration under the static load, because it is brittle, and stresses can’t be effectively redistributed by local plastic deformations.
The ductility of the material: The less the ratio of the yield stress to the ultimate stress of a steel, the less sensitive it is to the stress concentration. That means that high-strength steels are more sensitive to SCF.
The sharpness of the stress raiser: The sharper it is, the smaller the high-stressed volume and the stronger the effect of the stress release owing to local plastic deformations (for example, the sensitivity to the stress raiser is less).
The size of a part: Larger parts have greater (at the same sharpness of the stress raiser) high-stressed volume, yielding a less than helpful effect from local plastic deformation.
A host of other factors, such as quantity of load cycles and ratio of the mean stress to its amplitude.
Because most of these factors are difficult to take into account, the recommendations for choosing the q value are very approximate. But it is known that high-strength structural steels are very sensitive to stress concentration under cyclic load, and the recommended q value for them is about 0.90–0.95.
Some authors recommend (on the basis of experiments) coefficients Ke for a certain type of stress raiser (for example, threaded or splined shafts) related to certain steels. But in very important applications, in which totally reliable information about the permissible load is needed, the actual parts should be tested for strength and durability under real loading conditions.
Ke= +1 q K( t−1)