statics

Question 1

What are the torsion formula?

Answer 1

(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

Question 2

How is J found for a shaft?

Answer 2

J=π (D^4 - d^4) *0.5 D is the external diameter. d is internal diameter for a cylinder

Question 3

If you have a beam crossing a gap, and a force acts in the middle downwards, how do you find deflection?

Answer 3

FL^3 / 48EI

Question 4

How do you find the perpendicular stresses with a transformed element?

Answer 4

Given in the exam

Question 5

How do you find the shear stress of a transformed element?

Answer 5

Question 6

What is the direction notation for stress and shear of a transformed element?

Answer 6

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.

Question 7

What is Uniaxial stress?

Answer 7

Where there is no shear and stress only acts in one direction e.g. in a stress tester

Question 8

What is Biaxial stress?

Answer 8

The element is subjected to normal stresses in both the x and y directions but without any shear stresses.

Question 9

What are principle stresses and the principle planes?

Answer 9

The largest and smallest stress on an element.

The planes are what the principal stresses act.

The angle between the two planes is 90

Question 10

How do you find the angle between the principle stresses?

Answer 10

Question 11

How do you find the principle stresses for a transformed element?

Answer 11

Question 12

When the principle stress is acting, what is the shear force acting on the transformed element?

Answer 12

0

Question 13

How do you find the angle of a transformed element for maximum shear?

Answer 13

Question 14

How would you find the maximum shear force on a transformed element?

Answer 14

Question 15

How do you find the stress acting on a transformed element when there is maximum in plane shear stress?

Answer 15

Question 16

Where is the centre of a mohr’s circle?

Answer 16

The average stress

Question 17

What is the radius of a mohr’s circle?

Answer 17

Maximum in plane shear stress.

Question 18

When can you assume a thin wall?

Answer 18

r/t>10

r is the radius of the pipe and t is the pipe thickness.

Question 19

In a pipe, how is the horizontal stress found?

Answer 19

σ_y=σ_h=pr/tη

Question 20

In a pipe, how is the longitudinal stress found?

Answer 20

σ_x=σ_l=pr/2tη

Question 21

How is the stress found in a sphereical vessel?

Answer 21

σ_l=pr/2tη

Question 22

What is the safety factor?

Answer 22

n=tensile force / max allowible working stress

σ_y / σ₁-σ₃

Question 23

What is castigliano’s theorem for angle of roatation?

Answer 23

Θ=∫M/EI σM/σM_i dx

M_i can equal 0 if needed.

Question 24

What is castigliano’s theorem in terms of bending moment?

Answer 24

y=∫(M/EI) (σM/σP) dx

P is the force acting at the point where we want to find deflection.

Question 25

What is castigliano’s theorem in terms of strain engery?

Answer 25

u=M²L / 2EI , where u is strain engergy

σ_i=y_i=σu/σP_i

Question 26

How is stress found in a spherical vessel?

Answer 26

longditudinal stress=Pr / 2t

Question 27

In a mohr circle, what do the axises represent?

Answer 27

The X axis is the the normal stress δ_x

The y axis is the shear stress, positive at the bottom, negative at the top.

Question 28

Practically speking, what are δ_x and δ_y?

Answer 28

δ_x are tensile forces

δ_y are compressive forces.

Question 29

What are the principle triaxle stresses?

Answer 29

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

Question 30

What do the tri-axial stresses and forces look like on a 3D element?

Answer 30

Question 31

What can you learn from the signs of the tri-axial stresses.

Answer 31

 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. δ_max and δ_min are opposite signs), these stresses will be the maximum and minimum and the maximum in-plane shear stress equals the absolute maximum shear stress

Question 32

What is tresca failure theory?

Answer 32

Remember to convert δ_y and δ_x to δ₁ and δ₂ with the equations used in the second week.

Question 33

A hot rolled steel has a yield strength, σ_y of 100MPa. Estimate the factor of safety for the following principal stresses: A) 70, 70, 0 MPa

Assuming tresca failure.

Answer 33

Question 34

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?

Answer 34

σ1=large one, σ2=small one, σ3=0

Question 35

When does von mises failure occour?

Answer 35

Question 36

Definition of the Von Mises stress

Answer 36

Question 37

What is the graph that shows the safe region of Tresca and Von mises

Answer 37

Question 38

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?

Answer 38

Do the intergratatoin for the first half (between 0 and L/2) and multiply by 2

Question 39

What is the equation for finding the slope of a deflected beam?

Answer 39

You will introduce a moment M_i which turns the beam in the direction you think it will go and it should cancel out by the end.

Question 40

How do you derive the modified form of the castigiliano’s equation for bending beams?

Answer 40

Strain energy equation given

Question 41

State Castigliano’s theorem for beam deflection. Include a definition of all terms.

Answer 41