Unit 2 - Stress analysis Flashcards
Name the 2 types of axial loading
Tensile
Compressive
Describe shear stress
When forces act in opposite directions to shear/slip surfaces within a material
What is the formula for shear stress?
Shear stress (τ - tau) = shearing force (V)/shearing area (A)
Pa
What is shear strain equivalent to?
Φ = Angle sheared (in rads)
What is shear strength & how is it calculated?
Maximum shear stress a material can withstand before failing
Shear strength = shear force at failure/sheared area
What is the modulus of rigidity?
G = shear stress/shear strain
Pa
Higher G = more resistant to shearing
Shear stress can also occur during axial loading
Where does the greatest shear stress occur in this instant & how is it calculated?
Largest shear stress occurs at 45 degrees to axial loading
Max shear stress = axial stress/2
Cortical bone <1/2 as strong in shear than compression & tends to break at 45 degrees to axial load
How are modulus of rigidity & Young’s modulus related?
G = E / 2(1+v)
v = Poisson ratio
Name 2 types of bending
Cantilever
3-point
What happens to the surfaces of a material when it is bent?
One side is compressed & the other elongated (tensile load)
What is the neutral plane of a material that is being bent?
An axis where there is no compressive or tensile stresses
What equation would you use to find variation in stress of a bent beam (bending stress)?
σ = εE = yE/r
E = Young’s modulus
y = displacement of segment under examination from neutral axis
r - radius of circle containing neutral axis
Where is a bent material most likely to fail?
At the surfaces
What is a bending moment?
A measure of the bending effect of an applied load at any point in a structure
M = Fd
When does the maximum bending moment of a bar occur?
When F is applied at largest distance possible (i.e. length of bar)
When is a bedning moment:
- positive?
- negative?
Positive = sagging bar )
Negative = hogging bar (
What 3 things is the bending strength of a beam dependent on?
Strength of material
Cross-sectional area
Cross-sectional shape
How would a beam with its mass distributed away from its neutral axis resist bending moments compared to once with mass closer to axis?
Better able to resist when mass distributed away from neutral axis
What is the second moment of area (I)?
A geometrical property of an area which reflects how its mass is distributed from neutral axis
Larger I = further material of the beam is concentrated away from neutral axis
How do you calculate the maximum bending moment that a beam can resist?
M(max) = σI / y(max)
σ = bending stress I = second moment of area y(max) = max displacement of extreme layer of beam neutral axis
What is the general equation for bending?
M/I = σ/y = E/r
MISYER
What is the equation for the second moment of area for:
- rectangular cross section
- circular cross section
I = bd^3 / 12 y(max) = d/2
I = πd^4 / 64 y(max) = d/2
(learn these 2 and can work out the rest)
What is torsional stress?
Stress caused by twisting due to application of a moment
Where is maximal torsional stress experienced?
At outer surface
How do you calculate shear stress in a bar being twisted?
t = Gθr / L
What is the equation for shear strain in a bar that is being twisted?
Φ = θr / L
Equal to angle of shear (constant along bar)
What is the polar second moment of area (J)?
A measure of the distribution of the material about the central axis
Units = m^4
What is the equation for the twisting moment of a bar?
M = JGθ / L
J = polar second moment of area G = modulus of rigidity θ = angle of twist (rads) L = length of bar
What is the polar second moment of area for a circular cross-section?
J = πd^4 / 32
Take away d^4 of inner tube from outer for hollow
What is the general equation for torsion?
M/J = t/r = Gθ/L
M = twisting moment J = polar second moment of area t = shear stress R = radius of cross section G = modulus of rigidity θ= angle of twist (in rads) L= length
Why is it better to have hollow bars if they are being twisted?
Outer surface place of greatest stress & at centre there is zero deformation therefore in a solid bar most of the material is being stressed below max allowable
When weight reduction and material saving are important, the material near the centre is effectively redundant.
To hollow the bar increases the strength to weight ratio
How are bones designed to resist torsional loads?
Hollow with strong cortical bone forming outer layer (maximising strength-weight ratio)
Why is distal tibia more prone to fractures from torsional loads?
Distal polar second moment of area of the tibia is smaller than the proximal
Although amount of bone tissue is the same the distal part is less able to resist torsion
Why may muscles contract if not causing a movement?
To alter the distribution of stress within bone
Muscles may contact to produce compressive loading on a bone and eliminate tensile loading (since bone is stronger in compression)
Why are tired athletes more likely to fracture?
Fatigued muscles less able to control distribution of stress within their bones
Name 3 methods of stress analysis that can be used for more complicated structures
Strain gauge
Photoelasticity
Finite element method
How does a strain gauge work?
A thin metal foil between 2 insulating films is cemented to surface of material required to measure strain
As material changes dimension with changes in stress, the length of the gauge will also change, altering resistance of foil (proportional to strain)
Describe photoelasticity
Useful for structures with complicated geometry/complicated loading
When certain plastics are subjected to force, some of their optical properties change to the stresses developed. Material becomes birefringent. Polarised light passing through material splits into 2 beams creating a fringe pattern
Stress can be quantified by counting number of fringes
Describe finite element modeling
Creates mathematical model & is reliant on power of modern computing to perform vast number of operations quickly
Splits structure up into various small sub-regions (elements)
Elements associated with a number of nodes for which equations are created and solved to calculate stress
