Mechanics unit 4 - stress analysis deck 1 Flashcards
List all of the 6 different types of loading that a structure may be subjected to
- Unloaded
- Tension
- Compression
- Bending
- Shear
- Torsion
How does the mechanical properties of biological materials like bone, tendon or ligament differ in general from metals, alloys and plastics ?
- The materials like metals, alloys and plastics are isotopic - that is they exhibit uniform mechanical properties in all directions (this is what we have considered so far in the previous units)
- But this is not true for biological materials
State the symbols used to represent stress, strain, shear stress and shear strain
- σ = stress
- ε = strain
- τ = shear stress
- Φ = shear strain
So far in the previous units what type of loading has been considered?
Axial loading - loads that are applied along a geometric axis of the structure under load. It may be compressive or tensile.
What type of loading do structures normally undergo and therefore will be covered in this unit?
Structures normally undergo much more complex loading than just axial loading that results in shearing stresses, bending stresses, and torsional stresses
When do axial stresses occur and give an example of both types of axial stresses
Axial stresses occur when a material is subjected to tensile or compressive forces acting along a geometric axis.
- An example of tensile stresses is those which arise in the radius and ulna when an object is lifted
- An example of compressive stresses is those which arise in the vertebrae that are supporting the upper body mass
Define what axial stress is
This is the force per unit area
State the equation for axial stress
Define what axial strain is
This is the ratio of the change in length to the original length
State the equation for axial strain
For an elongation and a compression which is termed as +ve and which is -ve ?
- +ve for elongation
- -ve for compression
What does the ability of a material to resist elongation and compression depend on ?
Its stiffness - given by its youngs modulus
Define what youngs modulus is
This is the ratio of the stress to strain
State the equation for youngs modulus
Do SAQ 1 pg. 3 stress analysis unit 4
Ans in workbook
Define what shear stress is
It is defined as being equal to the magnitude of the shearing force divided by the sheared area
What causes shear stress to occur and give some examples of situations in which they would occur in
Shear stresses are caused by forces acting in oppposite directions to shear ot slip surfaces or planes within a material
State the equation for calculating shear stress
SI units = Pa
What will a material undergoing shear stress also undergo?
An angular deformation known as shear strain
Define shear strain
This is the angle sheared
What units are used when calculating shear strain ?
Radians NOT degrees
Think strain still essentially has no units if you consider the units for modulus of rigidity (G)
What is the equation used to calculate the shear strain ?
Note it is the very bottom equation to remember:
shear strain = x / d
Do worked example on pg. 5 stress analysis unit 4
Ans in workbook
Define what shear strength is
- This is the max shear stress that the material can withstand before fracturing
- e.g. this is how you cut paper with scissors as you apply a shear stress equal to the shear strength of the paper in order to cut it
State the equation for calculating the shear strength of a material
What is the shear modulus known as and what is it equal to?
The modulus of rigidity - this is equal to the gradient of the shear-stress / shear-strain curve up to a limiting stress
State the equation for calculating the modulus of rigidity
Or G = τ / Φ
What are the SI units of modulus of rigidity?
Pa (pascal)
What does all tensile and compressive forces (axial loading) give rise to ?
They also give rise to shear stresses as well as axial ones
At what angle does the largest shear stress occur to the axial loading placed on a material and what is it equal to?
- In the planes at 45 degrees to the axial loading
- The largest shear stress is equal to half the axial loading here
State the equation used to calculate the maximum shear stress and state its units
SI untis = Pa or N m-2
Although the max shear stress arising in a material is half that acting axially on the material, why might the shear stress still be the limiting factor in which results in fracturing/ breaking of the material?
Because the material may be lss than half as strong in shear as it is axially
Why does cortical bone tend to break at 45 degrees when an axial applied compressive load is placed on it ?
Because cortical bone is < half as strong in shear than in compression
Do worked example, SAQ 2 & 3 pg. 7 stress analysis unit 4
Ans in workbook
When do bending stresses arise in a material ?
They arise when a material is acted upon by forces and moments that tend to bend or curve it. Causing one side to be elongated and the other to be compressed
What are the 2 main forms of bending which occur ?
Cantilever and 3-point bending
Describe how the stresses and strain vary long the length of a material e.g. a bar when it is subjected to a bending load
The strain (and therefore the stress) is greatest at the surfaces of the bar/ material since the elongation and compression is greater at the surfaces. Then as you move centrally from either the elongated or compressed side to the centre of the bar / material the strain and stresses will decrease until you reach the neutral axis or the neutral layer / plane where they will be 0
Define what the neutral plane is
This is the plane in a material where there is neither tensile or compressive stresses
Where does the neutral axis lie in a material?
- For structures with a symmetrical cross-section (e.g. circular or rectangular) it lies at the geometric centre.
- For more complex structures the position of the neutral axis will depend on its geometry e.g. femur
Do SAQ 4 pg. 8 stress analysis unit 4
Ans in workbook
Consider how the stress in a beam varies e.g. a straight beam bent to form the arc of a circle
What can be done to find the variation in stress in the beam?
State what the stress of any layer of a material is dependent on ?
- It is dependent on its displacement from the neutral axis. The further the layer is from the axis, the greater the stress.
- The max stress with therefore occur at the surfaces of the beam meaning the material will fail at the surfaces rather than within the material
Do SAQ 5 pg. 9 stress analysis unit 4
Ans in workbook
When a bar/ material is subjected to a bending load what must happen to ensure static equilibrium is maintained?
An internal moment known as the bending moment must balance the externally applied moments if static equilibrium is to be maintained
Define what the bending moment is
It is a measure of the bending effect of an applied load at any point in a structure
What rule must be applied to allow the bending moment to be calculated for any cross-section of a loaded bar?
By applying the principle of static equilibrium (specifically rotational equilibrium)
Consider the example shown and write down how to calculate the bending moment
What is the bending moment dependent on ?
The applied bending force and its displacement from the point of application of the bending force
What can be drawn to illustrate the bending moment on a bar?
A bending moment diagram which is a graph of the bending moment against the distance along the bar.
The max magnitude for the bending moment occurs when the displacement (x), from the applied bending force (F) is equal to the length of the bar (L), that is M = FL
Describe the sign convention of bending moments
Bending moments follow the sagging and hogging sign conventions
- Bending moments that cause sagging of a beam are +ve
- Bending moments that cause hogging of a beam are -ve
