Stress Analysis Flashcards
What is the symbol for shear stress?
Shear strain?
Shear stress = tau
Shear strain = phi
What is an isotropic material?
One which exhibits uniform mechanical properties in all directions
This is generally true for metals, alloys and plastics, but not for biological materials like bone, tendon or ligament
Formula for shear stress?
SI Unit of shear stress?
Shear stress = shearing force/sheared area
tau = V/A
tau - shear stress
V - shearing force
A - sheared area
The SI unit is Pascal (N m^-2)
What will happen to a material undergoing shear stress?
It will be subject to angular deformation, which is quantified using shear strain (phi)
(i.e. tilting an object)
What is shear strain?
Shear strain = angle sheared = phi
It is the angle which the object has changed from its original position. Note the unit is RADIANS.
Formula for sheared strain?
tan(phi) = x/d
x = distance sheared d = distance between the 2 shearing forces
This is because a triangle is formed where phi is the angle between the origin and the top of the object, x is how far the top has travelled along the x axis, and d is the y axis.
However, in most cases the angle is so small it is less than 0.1 radians, so tan(phi) is pretty much equal to phi. Therefore:
phi = x/d
What is shear strength?
Formula?
The maximum shear stress the material can withstand before fracturing.
Shear strength = shear force required to fracture material/sheared area
What is the modulus of rigidity (G)?
Formula?
SI units?
The shear modulus - it is equal to the gradient of the shear-stress/shear-strain curve, up to a limiting stress
Modulus of rigidity = shear stress/shear strain
G = tau/phi
SI units = pascals
Do axial forces (tensile and compressive loads) also give rise to shear stress?
yes - and the shear stress causes the planes on the material to slip relative to one another, ultimately resulting in failure.
In axial loads, where does the largest shear stress occur?
How is it calculated?
At 45 degrees to the axial load
Since it is at an angle of 45 degrees, the shear stress is equal to HALF the axial stress:
tau(max) = sigma/2
Where tau - shear stress
and sigma - axial stress
How does shear stress in axial loadings relate to real life?
Although shear stress is half the stress acting axially, the shear stress may actually be limiting the material if the material is less than half as strong in shear as it is axially
e.g. cortical bone is less than half as strong in shear than it is in compression, and therefore tends to break at 45 degrees if a compressive load is applied
What is bending stress?
2 different types?
When a material is acted on by forces and moments that tend to bend or curve it. This will cause the material to be elongated in one side, and compressed in the other
2 types are Cantilever and 3-point
Pattern of stress and strain in an object subject to a bending load?
The stress and strain vary along the length of the bar. The strain (and therefore the stress) is greatest at the surfaces. Between the elongated and compressed side of the bar there will be an axis (or plane in 3D) where there is neither compressive or tensile stresses - the neutral axis
The largest stress will occur at the furthest point from the applied load
Equation to find the variation in stress for a segment subject to a bending load?
sigma = epsilonE = (y/r)E
sigma - stress in the segment under consideration
epsilon - strain in the segment
E - Young’s Modulus of the material
y - displacement of the segment from the neutral axis
r - radius of the circle containing the neutral axis
What is the stress in any layer of a material being bent dependent on?
Its displacement from the neutral axis
The further away, the greater the stress - the maximum stress will therefore occur at the surfaces if the material, which is where failure will occur
What is the bending moment?
An internal moment within a material which must balance the externally applied bending load in order to maintain static equilibrium. It can be calculated for any cross-section of a loaded bar by applying the principle of static equilibrium.
Point C is on a metal beam, which is being subject to cantilever bending by downward force F. C is x distance away from F. Write out an expression for the Moment centred around C
Mc - Fx = 0
Mc = Fx
What is the bending moment dependent on?
The bending force being applied and its displacement from the point of application of the bending force
What will a bending moment diagram show?
When is the diagram positive and negative?
The magnitude of the bending moment as you travel along the length of the bar. The greatest magnitude is at the furthest point away from the force, F, thus it is greatest when x (distance) = length of bar
The magnitude of the bending moment is positive when the moment causes sagging
The bending moment is negative when the moment causes hogging/arching
What is the bending strength of a beam dependent on?
The material, the cross-sectional area and the cross-sectional shape
In the equation for finding stress in a layer of material in a bent object, what does the symbol y indicate?
sigma = epsilonE = (y/r)E
y = displacement of the layer from the neutral axis. It can be deduced that a material with its mass distributed may from its neutral axis will be able to better resist any bending moments applied to it.
If 2 beams from the same material have identical cross-sectional areas but different shapes, the beam arranged in such a way that the majority of its material is distributed as far as possible from the neutral axis will be able to best resist a bending moment
General equation for the maximum bending moment a beam can resist?
Mmax = (sigmaI)/y(max)
Mmax - maximum bending moment
sigma - maximum bending stress
I(cap i) - second moment of area
y(max) - maximum displacement of the extreme layer of the beam from the neutral axis
(this can be altered depending on the material shape etc)
Relationship between stress in a particular layer of a beam, and the maximum bending moment that the beam can resist?
M/I = sigma/y = E/R
(MISYER)
Moment/Second moment of area
stress in layer/displacement from neutral axis
Young’s Modulus/Rupture point?
what is the second moment of area?
What is it dependent on?
Resistance to bending?
it is dependent on the cross-sectional shape of the beam - the further the material of a beam is concentrated away from its neutral axis, the greater its second moment of area