Deformable body Mechanics Unit 2: stress analysis (class) Flashcards
What is shear stress?
slippage of surfaces or planes within a material
What is shear stress equal to?
shearing force/ sheared area
What are the units of shear stress?
pascal
What does shear strain quantify?
the angular deformation of a material undergoing shear stress
What unit do we use for shear strain?
radians
What is shear strength?
the maximum shearing force a material can withstand before it fractures
How do we calculate shear strength?
shear force at failure / sheared area
What is the modulus of rigidity equal to
a) as equation?
b) in shear stress-shear strain curve?
a) shear stress/ shear strain
b) the gradient up to a limiting stress
What symbol represents modulus of rigidity?
G
Where does the maximum shear stress occur?
45 degrees to axial loading
How do we calculate max shear stress?
axial stress/ 2
How can we relate modulus of rigidity to Young’s modullus?
G = E / 2 (1+v)
What is 1+v?
Poisson ratio
What is the modulus of rigidity equal to?
the gradient of a shear-stress/shear-strain curve up to a limiting stress
What does application of a load that tends to cause bending result in?
tension and compression so one side of the material will be elongated and the other compressed
give the two types of bending
cantilever
3 point fixation
If a bar is subjected to a bending load where will the stress be greatest?
on the outer surfaces
What do we call the area of a bar where there is neither tensile nor compressive stresses?
the neutral axis
- it stays the same length
What is the bending moment?
the internal moment within a bar that balances the external moments when a bending load is applied
What is the bending moment dependent on?
the applied bending force and its displacement from the point of application of the bending force
When does the maximum magnitude for bending moment occur?
at the maximum displacement
What is the bending resistance of a beam dependent on?
resistance of material (its E)
cross-sectional area
cross-sectional shape
How do we quantify the cross-sectional shape?
use the second moment of area (I)
Will a beam resist bending moments better if its mass is distributed
a) further from the neutral axis?
b) closer to the neutral axis?
A
What is the general bending equation for a beam?
MISYER
M = Sigma = E
I y R
bending moment = stress
2nd mom of area displacement of layer from neutral
axis
= youngs modulus
radius of circle containing neutral axis
what symbol represents shear stress?
τ tau
what is tan ϕ and therefore sheer strain equal to?
distance sheared / distance between two shearing forces
what is the unit for the modulus of rigidity?
pa
what causes torsional stress?
twisting due to the application of a moment
What is a bar under the action of a twisting moment in?
torsion
How does deformation and therefore shear strain of a bar under torsion vary?
zero at central axis and maximum at outer surface
Is it harder to twist a shorter or longer bar at a given moment?
shorter bar
What does
a) φ
b) θ
represent in torsion?
a) shear strain
b) angle of twist
What quantifies the resistance to shearing force in torsion?
modulus of rigidity (G): the higher it is the more resistance
What are the units for modulus of rigidity?
N/m2 or Pa
how can we calculate shear strain of a bar in torsion?
φ = θ x radius / length
how do we converts degrees to radians ?
multiply by π/180
How can re-distribution of material in a bar help to resist torsion?
hollow bar is more resistant than a solid bar
What is the polar second moment of area?
a measure of distribution of the material about a central axis
What symbol represents polar second moment of area?
J
What are the units for polar second moment of area?
metres to the power of 4
How do we calculate polar second moment of area for a solid circle?
J = π d4 / 2
What is the general equation for torsion?
M/J = T/R = Gθ/L
in the general equation for torsion what is
a) M?
b) T?
c) R?
d) G?
e) θ?
f) L?
g) J?
a) twisting moment
b) shear stress
c) radius of the cross section
d) modulus of rigidity
e) angle of twist
f) length of the bar
g) polar second moment of area
What is the angle of twist measured in?
radians
What is shear stress
a) proportional to?
b) inversely proportional to?
a) modulus of rigidity, angle of twist and distance of point of interest
b) length
How are long bones designed to resist torsional loads?
hollow with stronger outer cortex to maximise strength:weight ratio
Where do tibia fractures due to torsional loads tend to occur? why?
distally where there is a smaller J so it is less able to resist torsional loads and therefore is more likely to fracture
explain how a strain gauge works
gauge is stuck on to a surface and deforms with it to produce electrical resistance that is proportional to the strain and can be measured using an electrical circuit
give a positive and a negative for a strain gauge
+ve apply straight to structure so no need for a model
-ve work on a few points measuring a few directions so
no overall idea of a complete structure
explain photoelasticity
model of the structure is made from a polymer and polarised light is shone through while it is loaded in different ways to analyse patterns of stress
explain the finite element method
uses computer to make a mathematical model of something where a complex structure is broken down into small parts and loads can be analysed using equations
In the general bending equation what does y stand for?
displacement from neutral axis
How do we calculate polar second moment of area for a hollow circle?
J = π (d4outer - d4inner)/ 32
how do we calculate second moment of area for a circle?
I = πd4/64
ymax =1/2d
how do we calculate second moment of area for a rectangle?
I = bd3/12 ymax =1/2d
What does I stand for in the bending equation?
second moment of area
What does the polar second moment of area of a structure indicate?
how resistant the material is to axial twisting
What beam has the largest bending strength?
the I shaped beam
What does the moment of area indicate about a structure?
how resistant the structure is to bending
Which is more resistant to bending and twisting
a) hollow tube
b) solid bar
of the same diameter. Why?
hollow tube almost as resistant as solid bar because its mass is distributed further from the neutral axis so it has a greater second moment of area and polar second moment of area
Which characteristic of a twisted bar indicates its strength:weight ratio? Why?
polar second moment of area
hollow bar stronger per weight
material in the centre of a bar is effectively redundant
Why are hollow bars not stronger than solid bars in axial tension and compression?
changing to a hollow bar reduces the cross sectional area the force is applied on so axial strength would be reduced
What quantifies a materials ability to resist elongation and compression ?
ie its stiffness
young’s modulus
What is the symbol for shearing force?
V
