a 1,000-word essay. In the essay, you must summarize what you consider to be the key concepts from the chapters covered, and you must include at least three photographs that you've taken to help explain the concepts.
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6186695_67669617_engr213e_week4_chapter_0628129.pdf
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TORSION
Chapter 6
Perspec-ve The analysis of circular sha9s under torsional loading is very similar to the analysis of bars under axial loading. There are equa-ons for stress on the cross-‐sec-on, stresses on oblique planes, and overall deforma-on. The overall problem approach is very similar. But for some reason torsional loading is more confusing, probably because the loading mode is more difficult to visualize. And there are more details to track: an unfamiliar geometric term (polar moment of iner-a, J), gear factors to calculate, and unfamiliar units of mechanical power. Be careful with details and work methodically as you begin.
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INTRODUCTION 6.1
Machinery Components • This class will focus exclusively on sha9s of circular (possibly
hollow) cross-‐sec-on • These are used in machinery that transmits power • The circular cross-‐sec-on greatly simplifies the overall
mechanics • The mechanics of structural elements of non-‐circular cross-‐
sec-on, for example an I-‐beam that is subjected to torsional loading, is much more complex and le9 to subsequent classes
• We will also skip the sec-ons on sta-cally indeterminate torsion and stress concentra-ons, as these are very similar to axial loading
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TORSIONAL SHEAR STRAIN
6.2
Plane Sec-ons Remain Plane • For a circular sha9, twis-ng does not warp a cross-‐sec-on • This allows the defini-on of shear strain (review sec-on 2.3)
to be applied directly to the sha9 surface:
This is the defining illustra-on for circular sha9s in torsion. The red gamma is shear strain (angle change in radians) on the surface when comparing the before-‐twist line (C’-‐D”) to the a9er-‐twist line (C’-‐D’). The shear strain on the surface is linked with deforma-on of radial lines on the cross-‐ sec-on.
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Non-‐Uniform Strain • The simple geometry of deforma-on, and linking surface and
cross-‐sec-on deforma-ons, leads to the following very important conclusion:
Shear strain is zero at the center of a circular sha9 under torsional loading, and varies
linearly moving outward toward the surface.
It is maximum at the outer surface.
TORSIONAL SHEAR STRESS
6.3
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Stress and Torque • Two steps are now taken in the deriva-on of equa-ons we
will use in the solu-on of torsional mechanics problems
1. Linear elas-c behavior (Hooke’s Law) is assumed, meaning that shear stress has the same overall spa-al pa`ern as shear strain
2. Resultant torque on a cross-‐sec-on is equated with the integrated shear stress
Axial Loading Torsion
• Equa-on for stress:
• Geometric Factor: – Cross-‐sec-onal area, A
• Equa-on for stress:
• Geometric Factor: – Polar moment, J
τ = T ρ J
σ = F A
J = πr4
2 = πd4
32
J = π 2
R4 − r4⎡⎣ ⎤⎦ = π 32
D4 − d4⎡⎣ ⎤⎦
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Interpreta-on
τ = T ρ J
This single term, distance from the center of the sha9, indicates that the shear stress is zero at that loca-on. Since there is no stress, that bit of material is not doing anything. Most power transmission sha9s are hollow for this reason.
Treat J just like the A of the axial stress equa-on: the relevant geometric factor for torsional loading. As soon as you iden-fy a torsion problem, think “polar moment”. It is there because of the non-‐uniform nature of stress over the cross-‐sec-on, and reflects the greater role of material toward the outside of the sha9.
Applied vs. Internal Resultant • And please remember this essen-al fact:
• The torque parameter T is not an applied torque, but the
internal resultant torque that acts at a par-cular cross-‐ sec-on, as revealed by virtual cuts of the structure
τ = T ρ J
cut cut cut cut
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STRESSES ON OBLIQUE PLANES
6.4
More than Just Shear Stress • When you think “torsion”, you naturally think “shear stress” • But more than just shear stresses are present • Compare the following with Figure 1.8, sec-on 1.5:
And realize that torsion can lead to normal stress failure, such as the tensile stress failure shown here.
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TORSIONAL DEFORMATIONS
6.5
Compare with Axial Loading
• Things to remember about this formula:
– The result is in Radians – The term T is the internal reac-on torque – The terms T, J, and G must all be constant over the length L – There are summed and integral versions if the terms are not constant
• Torsional deforma-ons are completely analogous to axial deforma-ons – Compare with sec-on 5.3 of the text
φ = TL JG
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TORSIONAL SIGN CONVENTIONS
6.6
Shear and Normal Stresses • The sign of a normal stress is very important
– Tension and compression have very different effects on material
• The sign of shear stress is not as significant – It indicates direc-on of poten-al slip along a plane – But for most materials, magnitude and not direc-on is important
• For combined loading cases (later in the course) – We need to track whether shear stresses add or subtract – We need a sign for stress transforma-ons
• Prac-ce visualizing the direc-on of shear stress ac-on
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GEARS IN TORSION ASSEMBLIES
6.7
Gear Ra-o • Gear ra-os are easy to calculate, but confusing to apply
– Ra-o can be calculated from radius, diameter, or number of teeth
• Ra-o affects torque and angular deflec-on differently – The situa-on is analogous to pulleys, and reflects energy conserva-on – If you decrease li9ing force using a pulley assemble, you must increase
the distance you pull through to li9 a load – If you decrease torque with a gear assembly, you must increase the
angular deflec-on of the sha9
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Focus on the Interac-on • Remember this figure:
– Gears develop equal force where they contact each other
– Since moment arm lengths are different, and torque is force -mes moment arm, the torques must be different
– The larger gear/sha9 carries the larger torque
– The smaller gear/sha9 carries the smaller torque
– Angular deflec-on of the sha9s follows the opposite pa`ern
POWER TRANSMISSION
6.8
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It’s All About Units • Nothing complicated here:
– Power is torque -mes angular velocity
• But the units are confusing – Radians per second is not a common angular velocity measure – Horsepower is an old but s-ll used power unit
• Note the different forms of the power equa-on for angular velocity expressed in either frequency or rpm
• Note the conversion between lb•9/sec and hp
That’s it for Torsion • We do not cover the remaining torsion sec-ons in the book,
6.9 – 6.12
