What are the torsion formula?
(T/J)=(τ/R)=(Gθ/L) T=torque J=polar moment of area for shafts τ=shear force R=radius G=shear modulus of material of shaft θ=angle of rotation L=length of shaft
How is J found for a shaft?
J=π (D^4 - d^4) *0.5 D is the external diameter. d is internal diameter for a cylinder
If you have a beam crossing a gap, and a force acts in the middle downwards, how do you find deflection?
FL^3 / 48EI
How do you find the perpendicular stresses with a transformed element?
Given in the exam
How do you find the shear stress of a transformed element?
What is the direction notation for stress and shear of a transformed element?
Stress move out from the centre of the element.
Shear along the top and right will point to the top right corner.
Shear along the bottom and left, will point to the bottom left corner.
A positve angle goes anit-clockwise.
What is Uniaxial stress?
Where there is no shear and stress only acts in one direction e.g. in a stress tester
What is Biaxial stress?
The element is subjected to normal stresses in both the x and y directions but without any shear stresses.
What are principle stresses and the principle planes?
The largest and smallest stress on an element.
The planes are what the principal stresses act.
The angle between the two planes is 90
How do you find the angle between the principle stresses?
How do you find the principle stresses for a transformed element?
When the principle stress is acting, what is the shear force acting on the transformed element?
How do you find the angle of a transformed element for maximum shear?
How would you find the maximum shear force on a transformed element?
How do you find the stress acting on a transformed element when there is maximum in plane shear stress?
Where is the centre of a mohr's circle?
The average stress
What is the radius of a mohr's circle?
Maximum in plane shear stress.
When can you assume a thin wall?
r is the radius of the pipe and t is the pipe thickness.
In a pipe, how is the horizontal stress found?
In a pipe, how is the longitudinal stress found?
How is the stress found in a sphereical vessel?
What is the safety factor?
n=tensile force / max allowible working stress
σy / σ1-σ3
What is castigliano's theorem for angle of roatation?
Θ=∫M/EI σM/σMi dx
Mi can equal 0 if needed.
What is castigliano's theorem in terms of bending moment?
y=∫(M/EI) (σM/σP) dx
P is the force acting at the point where we want to find deflection.
What is castigliano's theorem in terms of strain engery?
u=M2L / 2EI , where u is strain engergy
How is stress found in a spherical vessel?
longditudinal stress=Pr / 2t
In a mohr circle, what do the axises represent?
The X axis is the the normal stress δx
The y axis is the shear stress, positive at the bottom, negative at the top.
Practically speking, what are δx and δy?
δx are tensile forces
δy are compressive forces.
What are the principle triaxle stresses?
The three normal stresses that will act on a 3D element and result in no shear forces.
These principal stresses are assumed to have the magnitudes of maximum (δ1 ), intermediate (δ2 ) and minimum (δ3 ) intensity
What do the tri-axial stresses and forces look like on a 3D element?
What can you learn from the signs of the tri-axial stresses.
Tri-axial stress has an important implication for plane stress
Where the stress components have the same sign the maximum in-plane shear stress value is not the absolute maximum shear stress to which the material is subjected too
When two stress components have opposite signs (i.e. δmax and δmin are opposite signs), these stresses will be the maximum and minimum and the maximum in-plane shear stress equals the absolute maximum shear stress
What is tresca failure theory?
Remember to convert δy and δx to δ1 and δ2 with the equations used in the second week.
A hot rolled steel has a yield strength, σy of 100MPa. Estimate the factor of safety for the following principal stresses: A) 70, 70, 0 MPa
Assuming tresca failure.
If there is an 2D element understress and you find the principle normal stresses and they are both positive, what are the principle tri axial stresses?
σ1=large one, σ2=small one, σ3=0
When does von mises failure occour?
Definition of the Von Mises stress
What is the graph that shows the safe region of Tresca and Von mises
When you are calculating the deflection of a beam supported at two ends, and the force is symbetrical about the centre, how might you simplify the process?
Do the intergratatoin for the first half (between 0 and L/2) and multiply by 2
What is the equation for finding the slope of a deflected beam?
You will introduce a moment Mi which turns the beam in the direction you think it will go and it should cancel out by the end.
How do you derive the modified form of the castigiliano's equation for bending beams?
Strain energy equation given
State Castigliano’s theorem for beam deflection. Include a definition of all terms.