Page 224 - Marks Calculation for Machine Design

P1: Shibu/Sanjay
14:35
January 4, 2005
Brown˙C05
Brown.cls
STRENGTH OF MACHINES
206
These three transformation equations between the unrotated stresses and the rotated
stresses were presented in Sec. 5.1 as follows:
σ xx + σ yy σ xx − σ yy
σ x x = + cos 2θ + τ xy sin 2θ (5.1)

2 2
σ xx + σ yy σ xx − σ yy
σ y y = − cos 2θ − τ xy sin 2θ (5.2)

2 2
σ xx − σ yy
τ x y =− sin 2θ + τ xy cos 2θ (5.3)

2
Furthermore, it was shown that there is a special angle of rotation (φ p ), determined by
the following equation,
2τ xy
tan 2φ p = (5.9)
σ xx − σ yy
that if this special angle of rotation is substituted in Eqs. (5.1) to (5.3), then a maximum
principalstress(σ 1 )andaminimumprincipalstress(σ 2 )wouldresult,givenbythefollowing
equations,
σ 1 = σ avg + τ max (5.15)
σ 2 = σ avg − τ max (5.16)
and where the shear stress on the element showing (σ 1 ) and (σ 2 ) would be zero.
The first term on the right-hand side of Eqs. (5.15) and (5.16) is the average stress (σ avg )
determined by the following equation,
σ xx + σ yy
σ avg = (5.14)
2
and the second terms (τ max ) and (τ min ) are determined from the following equations:

2

σ xx − σ yy
2
τ max = + τ xy (5.12)
2
τ min =−τ max (5.13)
To provide a check on these calculations, the following relationship must always be
satisfied between the principal stresses and the unrotated stresses:
σ 1 + σ 2 = σ xx + σ yy (5.17)
It was also shown in Sec. 5.1 that the maximum and minimum shear stresses occur in
an element rotated 45 degrees from the angle (φ p ), denoted (φ s ), and determined from the
following equation:
σ xx − σ yy
tan 2φ s =− (5.10)
2τ xy
However, it was also shown that if the angle (φ p ) for the principal stresses is already
known, then the angle (φ s ) can be found from the relationship:
φ s = φ p ± 45 ◦ (5.11)
It was stated in Sec. 5.1 without proof that if the angle (φ p ) for the maximum principal
stress (σ 1 ) were known, then the angle (φ s ) for the maximum shear stress (τ max ) could be
found from the following equation:
φ s = φ p − 45 ◦ (5.18)