Structural Mechanics Unit 1 Flashcards
what is the difference between a structural material and a structure
structure - is an arrangement of one or more materials in a way that is designed to sustained loads
structural material - any material that may be used to construct a structure
what are the symbols for stress, strain and co-efficient of viscosity
stress - σ (sigma)
strain - ε (epsilon)
co-efficient of viscosity - η (eta)
what is stress denied as, the equation and units
force per cross-sectional area
stress = force/area
units = newton per metre squared or Pascal
what would handle more force, a bar with a bigger or small cross-sectional area
bar with a bigger cross-sectional area
in regards to strain, what would the difference be in elongation of 2 bars, 1 of which is longer than the other
the longest bar would elongate more
if bar 1, is twice the length of bar 2, it will elongate twice as much
what does the stress-strain curve show
how the material deforms/behaves as it is loaded
on a stress-strain curve, what does the letters P,E,Y,H,V and R represent
P = proportional limit E = elastic limit Y = yield strength H = strain hardening V = ultimate stress R = rupture
in regards to the stress-strain curve, was is the relationship between stress and strain at very small loads
there is a linear relationship between stress and strain
i.e. if stress doubles, strain doubles
what happens at P = proportional limit
relationship between stress and strain is not proportional anymore
what happens at E = Elastic limit
before here, if the load is removed from the material, the material will recover back to its original shape and size
this is the elastic region
after the elastic limit, the material will NOT return to its original shape and size after the load is removed
what happens at Y = Yield point
after this point, the material will undergo considerable elongation without an increase in stress
- highlighted by the flatness of the region on the graph
- material is displaying perfect plastic behaviour (no elastic recovery)
what happens after the E = Elastic limit
material is in the plastic region
material deforms instantly under applied load
material may partially recover to original size and shape when load is removed, but NOT completely like in the elastic region
what are the section of the stress-strain curve in order they happen
elastic region
plastic region
strain hardening
necking
what is happening in the strain hardening region
the material is undergoing changes in its atomic and crystalline structure
results in an increased resistance to further deformation
what is U = ultimate strength point on the stress-strain curve
occurs at highest point on the graph
after this point, strain increases with a reduction in stress and ‘necking’ happens
the stress the bar can withstand decreases, NOT sue to any loss of material but due to reduction in cross-sectional area of the bar
how can the true stress-strain curve be obtained
calculate the stress at the narrowest part of the neck
what is the rupture point, R
point at which the material breaks
stress at this point is called the rupture strength
what is a material that can only handle a small amount of strain before breaking described as
brittle
what is a material that deformed plastically before breaking described as
ductile
what is the difference between ductile and brittle materials
a brittle material ruptures after a small amount of strain whilst a ductile material can deform considerably before rupturing
what is Hooke’s Law
Up to a certain level of stress (the proportional limit), the strain is proportional to the applied stress
what is the equation for Young’s modulus and the other name for it
Youngs Modulus (E) = stress/strain
Unit = N m-2 or Pa
modulus of elasticity
what does a large young modulus mean
that the material requires a large amount of stress is required to produce a small strain i.e. material is stiff
vice versa, a small young’s modulus means only a small amount of stress is needed to produce a big strain i.e. material is flexible
what is the definition of rigidity and equation
ability to resist axial deformation
rigidity = E x cross sectional area
[rigidity = EA]
Unit = Newton (N)
what is stiffness (k) defined as and what is the equation
Force required to produce a unit deflection (i.e.. force required to elongate or shorten the bar by 1 metre)
stiffness = applied force/change in length
[k = F/trianglel]
unit = N m-1
how can the equation of stiffness be rearranged using young’s modulus
k = F/trianglel
To
k = EA/l
what is the definition of flexibility, equation and unit
Deflection under a unit load
Flexibility = length / stiffness
f = l / EA or l / k
Units = m N -1
in regards to flexibility and stiffness, what would in increase in the length of a bar mean
reduction in stiffness
increase in flexibility
what does it mean when a material displays viscous behaviour
material does not deform instantaneously when a load is applied
the strain [stretching] is prolonged
the material will not return to its original shape and size after the load is removed
can viscous behaviour be represented by Hooke’s Law or Young’s modulus
No
as viscous materials are dependant upon the strain rate not the stress
equation for viscous behaviour
η (Coefficient of Viscoscity) = stress/strain rate
Strain rate = change in strain/change in time
unit = N m-2 .s [Newton per metre squared second] or Pa .s [pascal per seconds]
what is viscoelastic behaviour
a combination of both elastic and viscous behaviour
e.g. cartilage and cortical bone
what is meant when a material displays ‘creep’
the material continues to deform over time when a CONSTANT LOAD is applied
what are examples of materials that creep
wood - will creep noticeably at only a few hours at room temp
[materials which are exposed to high temperatures are vulnerable to creep and may even fracture because of it]
what is meant by ‘stress relaxation’
if a material is kept under CONSTANT STRAIN then the STRESS in it will gradually diminish over time
[due to the change in the ordering of the atoms in the material]
what are the types of loadinf
Axial ( tension and compression)
Shear
Bending
Torsion
in axial strain, what does it mean if the strain is positive and negative
positive = elongation
negative = compression
what is shear stress
slippage of surfaces or planes within a material
caused by forces acting in OPPOSITE directions
what is 2 examples of shear stress in orthopaedics
a screw being sheared by a fracture fixation plate and bone
bone cement being sheared by the hip prosthesis and bone
what is the symbol, equation and unit for shear stress
tau = τ
V = shearing force A = shearing area
τ = V/A
unit = Pa
what is shear strain and how would you calculate it
angle sheared [in radians]
tan φ = x / d
where φ = angle
x = distance tilted forward
d = length
what is the definition of shear strength and the equation for it
the max shear stress a material can withstand before failing
shear strength = shear force at failure / sheared area
what is the value for the relationship between shear stress and shear strain called and what is the equation
Modulus of Rigidity (G) =
shear stress / shear strain
units = N m-2 or Pa
at what angle does the largess shear stress occur
at 45 degrees TO THE AXIAL LOADING
what is the equation to calculate max shear stress
max shear stress =
axial stress / 2
Or
τmax = σ / 2
what happens to cortical bone when it is applied with an AXIAL COMPRESSIVE LOAD
as cortical bone is less than half as strong in shear than in compression, it will tend to break at 45 degrees to an axial compressive load
what is bending stress and the 2 types we will looks at
Application of loading tending to cause bending results in both tension and compression
Cantilever (think of a diving board)
3 point bending
what is the neutral plane in regards to bending stress
plane where there is neither tensile or compressive stresses
i.e. no changes
[neutral axis maintains the same length when a beam is bent, neither compressed or elongated\
if you were to put a bending force on a bar, where would the strain and stress be greatest
at the surface
since elongation and compression is greater at the surfaces
what is the definition of a bending moment
a measure of the bending effect of an applied load at any point in a structure
what is the bending moment dependent on
the applied bending force and its displacement from the point of application of the bending force
[page 10 of the unit 2 notes to see bending moment diagrams]
what is the equation for calculation bending moment
M = FL
F = applied bending force L = length of the bar
what is a positive and negative bending moment called
positive = sagging
[happy face]
negative = hogging
[think of a diver when they jump on a diving board and it curves in a sad face]
what is bending strength of a beam dependant on
strength of a material
cross sectional area
cross sectional shape
what is the Second Moment of Area
quantifies resistance of a substance to bending
i.e. the further the material of a beam is concentrated away from its neural axis, the larger its second moment of area
unit = m ^4 symbol = I
what is the equation for calculating the second moment of area for a rectangle, a circle, and a circle with a hollow centre
Rectangle:
I =bd^3 / 12
[where b = breadth and d= depth]
Circle:
I = pie x D^4 / 64
[where D = diameter]
Circle with a hollow centre”
I = pie (D^4 - d^4) / 64
[where D = diameter and d = second diameter]
what structure is best at bending
the more hollow a structure, the better it is a bending
what is the general equation for bending
M/I = sigma/y = E/r
M = moment I = second moment of area sigma = stress y = displacement from the neutral axis E = young's modulus r = radius of the circle
what is the aid memoir to remember the general equation for bending
MISYER
what is the equation to calculate the maximum bending moment
Mmax = sigma x I / ymax
sigma = maximum bending stress I = second moment of area ymax = max displacement of the extreme layer of the beam from the neutral axis
unit = Nm [Newton Metre]
what is the ymax values for a rectangle, a circle and a circle with a hollow centre
rectangle:
ymax = 1/2d
circle:
ymax = 1/2d
circle w/ hollow centre
ymax = 1/2d outer
what is an example of a bending fracture in bones
Boot top fracture seen in skiers
what causes torsional stresses
twisting due to the application of a moment
one end is fixed, the other end is being twisted
when is a circular bar said to be in pure torsion
when its cross-section retains its shape
i.e. remains circular and its radius is unchanged
the angle of twist varies along the length of the bar, when is it at its maximum
at the outer surface
[further you get away from the neutral axis, the higher the stress/strain]
what is the angle of twist measured
radians
what is the shear strain equal to
equal to the angle of shear
[constant along the length go the bar]
what is the equation to calculate torsional strain
angle of twist x radius / length
what is the equation to calculate torsional stress
modulus of rigidity x angle of twist x radius / length
how do you calculate modulus of rigidity
shear stress / shear strain
units = N m^-2 or Pa
[higher the number, harder it is to generate shear/strain]
what is the Polar Second Moment of Area [J] a measure of
a measure of the distribution of the material about the central axis
unit = m^4
what is the equation for the applied twisting moment
M = J [G x angle of twist/lenght of the bar]
what is the general equation for torsion
M/J = τ/R = Gθ/L
where: M=Twisting moment J= polar second moment of area τ=shear stress R= radius of cross section G= modulus of rigidity θ= angle of twist L= length
what is the equation for polar second moment of area for circle and a hollow circle
Circle:
J = pie x d^4 / 32
Hollow Circle:
J = pie(d^4 outer - d^4 inner) /32
what is the shear stress inversely proportional to
the length
where is the maximum shear stress in a bar located
at the outer surface
where is the minimum shear strain in the bar located
at the centre
how are bones structured to resist torsional loads
hollow with strong cortical bone on the outer layer
maximises strength-to-weight ratio
the tibia is most likely to fracture due to torsional loads - most fractures are found distally, why is this?
the distal polar second moment of area of the tibia is SMALLER than the proximal polar second moment of area
amount of bone tissue is the SAME, the distal part is less able to resist torsional loads and therefore, most likely to fracture
how does muscle activity reduce the chance of fracture
muscles contract to alter stress distribution within the bone
muscle contracts > produces a compressive load on a bone and eliminates any tensile loading
[bones stronger in compression than tension]
what do strain gauges do and whats an advantage of them
offer a means of measuring strain, and thus the stress, at the surfaces of a structure
consist of a very thin metal foil located between 2 pieces of thin insulating film
they can be applied to the actual surface under study
what are the methods to perform stress analysis on a structure
strain gauges
photo-elasticity
Finite Element methods (FEA)
how does FEA modelling work
done on a computer
- blue = compressive stress
- red = tensile stress
what factors are important in material failure
magnitude of the applied load
rate of speed at which the load is applied
no. of times that the load is applied
what is the difference between ductile and brittle fractures
ductile
- fracture occurs AFTER plastic deformation
- material shows “necking”
brittle
- fracture occurs W/OUT plastic deformation
- i.e. no ‘necking’
what test allows the stress-strain curve to be drawn
tensile test
what is the name given to the point on a stress-strain curve where a material fractures
rupture strength of the material
how does the rupture strength and the ultimate strength of a material differ
rupture strength - stress when a material will fracture
ultimate - maximum stress calculated [calculate at the point of necking in a ductile material]
to summerise when will a material fracture
when it is subjected to a load greater than its ultimate strength
how do ductile fractures form
- application of tensile load
- formation of microscopic voids (small holes) at the centre of the bar
- stress increases, voids grow
- voids connect with each other to form cavities
- actual metal to metal contact is reduced
- unable to support applied load and complete fracture occurs
what causes the formation of the voids
high stress causes separation of the metal at grain boundaries or at the interfaces between the metal grains and inclusions
what is the characteristic look of a ductile fracture
“necking” and a small shear lip [gives a cup and cone appearance]
when will a ductile material act like a brittle one
if it has been exposed to fatigue loading
how do brittle fractures vary from ductile fractures
occurs suddenly
fracture surface is flat, perpendicular to the load and has a granular appearance
has CHEVRON pattern
summarise the difference in appearance between ductile and brittle fractures
ductile - cup and cone appearance with granulated central portion
brittle - flat, granular cross-section with a chevron fracture
why are sharp corners, i.e. on a boat, more prone to failing
the stress is concentrated on 1 area
[curved corner solve this problem]
what areas are likely to be points of high stress in a structure
places where there is a SHARP CHANGE in shape
[the sharper the change, the higher the stress conc]
what shapes are prone to fracture
at the tip of cracks or notches
[phenomenon of concentrated stress at the tip of a crack or notch is called NOTCH SENSITIVITY]
what is an impact loading
a sudden intense blow/load
Impacts by masses create loads that are greater than equivalent mass static loading conditions.
[i.e. a impact load = to a static load may result in fracture, but the gradually applied static load may not]
what is the name of the test used to test a structures resistance to an impact load and how does it work
Charpy Impact Test
- Heavy pendulum is released from a known height
- as pendulum reaches bottom of its trajectory it strikes and breaks the test specimen
- then continues until it reaches the peak of its swing
- the height reached by the pendulum at the end of the swing is lower than the height from which is was released
- thus the difference is the energy absorbed by the specimen as it if fractured
what is the equation for the charpy impact test
calculating potential energy
PE = W (ho - hf)
ho = original height hf = final height W = weight of pendulum
or PE = mg (ho - hf)
m = mass g = gravity
Unit = Joules [J]
what can influence a materials ability to absorb energy and how
the temperature
- a material will be able to absorb more energy as the temp increases
- due to an increasing temperature changing the material from BRITTLE TO DUCTILE
- yield strength decreases
- ultimate tensile strength decreases
- strain increases
[ductile material is tougher than a brittle one]
when testing a material, how could you take account for the difference in ability to absorb energy at difference temperatures
a series of impact tests at different temperatures
what is a fatigue fracture
a fracture caused by repeated loading
Load required is lower than for failure due to application of a steady load
[Failure due to combination of magnitude of load and repeated no. of loadings]
[orthopaedic implants are prone to fatigue fractures]
what is the appearance of a fatigue fracture
2 regions:
- a relatively smooth region marked by concentric markings that may allow the origin of the fracture to be found [sometimes called clam shell markings]
- then either a granular or fibrous region
- granular appearance indicates a brittle fracture
- fibrous appearance indicates a ductile fracture
what is fatigue life
Expressed in cycles
- is the number of cycles a structure can withstand before a fracture
Dependent on applied stress
Reduced by surface defects
Below Endurance Limit = no failure
what are factors influencing fatigue life
Stress Geometry Surface quality Material Type Residual stresses
Internal defects:
• Direction of loading
• Grain size
• Environment
how can corrosion be prevented
creating an alloy
- put a layer on that metal that is already corroded and cannot be corroded anymore
- called PASSIVATION LAYEr
what parts of metal are particularly prone to attack from corrosions
imperfections on the surface of the metal
- the imperfections develop to crevices after being corroded
what happens at the crevice that has developed from corrosion
gives rise to stress concentrations in the structure
eventually leads to fracture
how does corrosion affect the fatigue behaviour of metal
reduces the fatigue resistance of metals, results in a lower fatigue life and no endurance limit
[endurance limit = range of cyclic stress that can be applied to the material w/out causing fatigue failure]
what are the 2 groups of metals
ferrous metal
non-ferrous metal
what is the most common ferrous alloys
steel
- alloy of iron and carbon
[stainless steel
- allow of iron, chromium, nickel and carbon
- corrosion resistant]
what part of the stainless steel alloy gives it its corrosion resistant property
chromium
what are examples of non-ferrous metals and alloys used in biomechanics
titanium
titanium based alloys
- used in heart value components, joint replacement endoprostheses, # fixation plate
what are the main advantages of titanium based alloys
lower density compared to steel
higher strength-to-weight ratios compared to Aluminium
excellent corrosion resistance
what are the main disadvantages of titanium based alloys
high material cost compared to steel and aluminium
low young’s modulus (110 GPa) compared to steel (200 GPa)
- i.e. will deform more under a given load
what is 316L
stainless steel used in orthopaedics
[10.6 - 18 % chromium, 10 - 14 % nickel, 2 % manganese, 0.035% carbon]
what is the difference in properties between low carbon and high carbon steels
Low carbon steels- Low strength and hardness but good ductility
High carbon steels-High strength and hardness but brittle
what are properties of polymers
light weight corrosion resistant low tensile strength easily manufactured low density
based on long chains of monomers
what is the stress-strain behaviours of polymers
non-linear and time dependent
exhibit both elastic and plastic behaviour
what are the 2 main subtypes of polymers
plastics
elastomers
what are the 2 main categories of plastics
thermoplastic
thermoset
what are features of thermoplastic
display plastic behaviour at high temperatures
there structure is stable at high temps, so can be heated, cooled, re-heated or reformed w/out altering their behaviour
examples of thermoplastics and there uses in orthopaedics
Polyethylene - acetabular cups
Polypropylene - orthoses
Polymethyl methacrylate [PMMA] - bone cement
what are elastomers better known as and why
rubbers
can deform by enormous amounts w/out permanent shape change
e.g. a normal rubber can be stretched to 7 times its original length, 700% strain, and still return to its original shape/size
what is the structure of ceramics and an example of one
Crystalline in structure
e.g. Diamonds
what are properties of ceramics
very hard but brittle high melting point low electrical and thermal conductivity good chemical and thermal stability high compressive strength
[Diamond E=1200 GN m-2. Highest known young’s modulus]
what property does ceramics not exhibit
creep
what is the use for ceramics in orthopaedics
used for heads of hip prostheses
when are composite materials formed
when 2 or more materials are joined to give a combination of properties that can not be obtained from original materials
what are the 3 categories of composite materials
particulate e.g. concrete
fibre e.g. fibreglass
laminar e.g. plywood
what is features of particulate composite materials
hard brittle material dispersed within a softer more ductile material
e.g. concrete, mixture of gravel and cement
what is features of fibre composite materials
fibres of strong, stiff brittle material within a softer, more ductile material
improves strength, fatigue resistance, stiffness and strength-to-weight ratio
e.g. fibreglass, contains glass fibres embedded in a polymer
what is features of laminar composite materials
several different forms
very thin coating may cover a material to improve corrosion resistance
thick layers may be laminated together to improve strength i.e. plywood
compare properties of bone cement and cortical bone
- in compression, in tension, young’s modulus, which is more ductile, fatigue cycles, overall
Both are stronger in compression than in tension
- but cortical bone in 3-5 times strong than bone cement
Cortical bone has higher young’s modulus
- i.e. is more stiffer
Bone cement elongates further at fracture
- more ductile than cortical bone
Cortical bone is able to withstand more than twice as many fatigue cycles as bone cement
Overall
- cortical bone is stronger, stiffer and more brittle material than bone cement.
