Axisymmetric stress Flashcards
Give the two definitions for integrity
- An unimpaired condition
- The quality of being whole and complete
What are the dimensions of an axi-symmetric stress distribution?
3D stress distribution:
-Radial, Hoop, Axial
What are the key components for which axi-symmetric stress is considered?
-Thick disks subjected to rotation
-Thick cylinders subjected to pressure loading
What is represented by R(B), sigma(theta), sigma(r) and dr?
R(B): body force
sigma(theta): hoop stress
sigma(r): radial stress
dr: thickness of element
What is the first step in setting up axi-symmetrical calculations?
Set up equilibrium equation on element
What assumptions are made to simplify the axi-symmetric equilibrium equation?
-sin(dtheta/2) = dtheta/2
-Second order terms can be neglected
-Stress field is asymmetric
-Stresses vary only with radius
What must be insured due to the deformation of the material of the element?
Compatibility of displacements
How is compatibility of displacements ensured?
Considering the geometry of a typical displacement/deformation using ELASTIC HOOKE’S LAW relationships
What is represented by:
u, du, r, dr?
u: radial displacement at r
du: deformation
r: distance from centre to inner edge
dr: original thickness
Give the strains (not in terms of stresses) for E(r), E(A), E(theta)
(Where E is epsilon)
E(r) = du/dr
E(A) = dw/dA
E(theta) = u/r
Give the equation and units for R(B) in relation to centrifugal force
R(B) = rhoomega^2r
In rad/s
What assumption is made about E(A) in a THICK DISK and why?
The axial strain is constant with radius
The plane sections are assumed to remain plane
How many boundary conditions are required to solve for A and B?
Two, as there are two unknowns
What does the assumption of E(A) being constant lead to?
dE(A)/dr = 0
What assumption is made about axial stress in a THIN DISK?
Sigma(A) = 0
What does assuming sigma(A) = 0 lead to?
A different set of integration constant equations (Jim pls fix this wording I’ve forgotten maths)
What boundary conditions are used for a solid rotating shaft?
sigma(Ro) = 0
B = 0
Explain the boundary conditions for a solid rotating shafts
At Ro there is nothing to ‘push’ against, so there is no area for a stress to occur
In the centre, B/0 would be infinite if B=/= 0
What are the boundary conditions for a hollow rotating shaft?
Sigma(Ro) and sigma(R1) are 0
Explain the boundary conditions for a hollow rotating shaft
There is no stress as there is nothing to ‘push’ against- no area for force to be applied to
How do you solve the equations for sigma(R1) and sigma(R0)?
Sigma(R1) - sigma(R0)
What do the Lame equations apply to?
Thick cylinders subjected to internal and external pressure
What happens to the body force term in the Lame equations?
Body forces are 0 as omega = 0
What happens to the radial and hoop stresses as r decreases?
They both increase
What sign do hoop and radial stresses have for internal pressure conditions?
Hoop: positive
Radial: negative
What are the hoop and radial stresses also, due to the problems being axisymmetric?
They are principal stresses
sigma(1) = hoop stress
sigma(2) = radial stress
How can max shear stress be calculated using the principal stress equivalents?
tau(max) = (sigma(1)-sigma(2)
What does k represent in an internally pressurised cylinder?
r(o)/r(i)
Why would an initial compressive hoop stress be induced in a cylinder?
Compression is negative, so initial pressurisation (which is tensile, so positive) overcomes the initial negative, bringing hoop stress to 0
How can one contain a higher pressure in practice?
-Shrink fit
-Wind wire onto cylinder under tension
-Overstrain once with a pressure beyond yield, to leave compressive residual stress distribution near bore
How does the appearance of the hoop strress vs r graph change from a single to compound cylinder?
Initially lower, then spikes at the interface to above
What is needed to produce a shrink fit?
Outer diameter of inner cylinder must be slightly greater than inner diameter of outer cylinder
Called ‘diametral interference’
What equation can be used to calculate diameter change in a shrink fit?
deltaD = D epsilon(theta)
Where epsilon(theta) is hoop strain
What happens to the calculation for diametral interference when both cylinders are the same material?
Poisson’s ratio can be ignored and E can be factored out
For an interference fit of a sleeve on a solid shaft, what are the initial conditions?
sigma(r) = sigma(theta) = C = -P (uniform across shaft)
B = 0 to avoid infinite stress at r = 0
