Calculating Stress & Strain in Compressive Force on Component

[stackemup]

Homework Statement

The component shown in Fig 1 is made from a material with the following properties and is subjected to a compressive force of 5kN.

Material Properties :
Young’s Modulus of Elasticity – 200 GNm-2
Modulus of Rigidity – 90 GNm-2
Poisons ratio – 0.32

Calculate :
(a) The stress in :
(i) the circular section
(ii) the square section

(b) The strain in :
(i) The circular section
(ii) The square section

(c) The change in length of the component

(d) The change in diameter of the circular section

(e) The change in the 40mm dimension on the square section

(f) If the same component were subjected to a shear force of 7 kN as shown in FIG 2, calculate the shear strain in :
(i) The circular section
(ii) The square section

Homework Equations

unsure

The Attempt at a Solution

I have not made any attempt as of yet as I have not done this in a long time and need to be pointed in the right direction. I nwould greatly appreciate anyone who has the time, to take me through each question and guide me in the right direction. I am not asking for answers but for guidance on formulas to use and a bit of an explanation why each formula needs to be used. Thanks in advance.

 

[SteamKing]

Sorry, but PF is not a tutoring service. The rules state that you should at least make an attempt at a solution in order to receive help. You must have gotten this problem from some text or other source. Check that source for a discussion of basic concepts like stress and strain.

 

[stackemup]

Thanks for the info ... I will attempt and answer and post it.

 

[stackemup]

I have made an attempt on (a) but I am assuming it needs revising and quite a bit of work putting into it, but I would appreciate it if someone could help.
(a) The Stress in:
Stress = F / A
(i) The circular section
Area of cylinder = 2TTr^2 + h(2TTr)
= 7068.59 x 10^-3 m2
Force on cylinder
= 5 x 10^3
Stress
= 5 x 10^3 / 7068.59
= 707.3546
(ii) The square section
Area of square section = 1600 + 1600 + 2400 + 2400 + 2400 + 2400
= 12800 x 10^-3 m2
Force in square section
= 5 x 10^3
Stress
= 5 x 10^3 / 12800
= 0.390625

 

[SteamKing]

I have made an attempt on (a) but I am assuming it needs revising and quite a bit of work putting into it, but I would appreciate it if someone could help.
(a) The Stress in:
Stress = F / A
(i) The circular section
Area of cylinder = 2TTr^2 + h(2TTr)
= 7068.59 x 10^-3 m2

The area used for calculating stress is not the surface area of the cylinder but the cross-sectional area.

Force on cylinder
= 5 x 10^3
Stress
= 5 x 10^3 / 7068.59
= 707.3546
(ii) The square section
Area of square section = 1600 + 1600 + 2400 + 2400 + 2400 + 2400
= 12800 x 10^-3 m2
Force in square section
= 5 x 10^3
Stress
= 5 x 10^3 / 12800
= 0.390625

The area used for calculating stress is not the surface area of the block but the cross-sectional area.

The force is not being applied over the entire surface of the items but only on one particular portion of the part.

 

[BvU]

Are the dimensions in millimeters ? Check your area calculations !

 

[stackemup]

Heres my revised attempt -

(i) The circular section
CSA = TTr^2
= 706.8583 x 10^-3 m2
Force on cylinder
= 5 x 10^3
Stress
= 5 x 10^3 / 706.8583
= 7.0736
(ii) The square section
CSA = 1600 x 10^-3 m2
Force on square section
= 5 x 10^3
Stress
= 5 x 10^3 / 1600
= 3.125

 

[SteamKing]

Heres my revised attempt -

(i) The circular section
CSA = TTr^2
= 706.8583 x 10^-3 m2
Force on cylinder
= 5 x 10^3
Stress
= 5 x 10^3 / 706.8583
= 7.0736
(ii) The square section
CSA = 1600 x 10^-3 m2
Force on square section
= 5 x 10^3
Stress
= 5 x 10^3 / 1600
= 3.125

You're getting closer. You should always indicate the units on all calculation results.

The unit of stress in SI is the pascal. 1 pascal = 1 Newton / m²

You must be careful with the units for your area calculations. Although 1 meter = 1000 millimeters, 1 sq. meter ≠ 1000 sq. mm. Do you see why?

 

[stackemup]

How does this look?

(a) The Stress in:
Stress = F / A
(i) The circular section
CSA = TTr^2
Radius = 15 x 10^-3 m
CSA = 0.000707 m2
Force on cylinder
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.000707
= - 7.073553 Pa
(ii) The square section
CSA = 1600 x 10^-3 m2
Force on square section
= 5 x 10^3 N
Stress
= 5 x 10^3 / 1.6 m2
= - 3.125 Pa

Am I correct in thinking that compressive stresses are defined as negative in sign?

 

[SteamKing]

How does this look?

(a) The Stress in:
Stress = F / A
(i) The circular section
CSA = TTr^2
Radius = 15 x 10^-3 m
CSA = 0.000707 m2
Force on cylinder
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.000707
= - 7.073553 Pa

Check your arithmetic here.

(ii) The square section
CSA = 1600 x 10^-3 m2
Force on square section
= 5 x 10^3 N
Stress
= 5 x 10^3 / 1.6 m2
= - 3.125 Pa

Check your arithmetic here. You are also using the wrong value for the CSA.

Am I correct in thinking that compressive stresses are defined as negative in sign?

Yes, but that's a minor detail. It's more important to check your work to eliminate careless arithmetic mistakes or mistakes in copying numbers.

 

[stackemup]

(a) The Stress in:
Stress = F / A
(i) The circular section
CSA = TTr^2
Radius = 15 x 10^-3 m
CSA = 0.000707 m2
Force on cylinder
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.000707
= - 7.073553 MPa
(ii) The square section
CSA = 1600 x 10^-6 m2
Force on square section
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.0016 m2
= - 3.125 MPa

I have also attempted part B below.

(b) The strain in:

ε = dl / lo
= σ / E
= Stress / Young’s Modulus
(i) The circular section
Strain = 7.073553 / 200GNm
Strain = 3.53678 x 10^-8
(ii) The square section
Strain = 3.125 / 200GNm
Strain = 1.56250 x 10^-8

 

[SteamKing]

(a) The Stress in:
Stress = F / A
(i) The circular section
CSA = TTr^2
Radius = 15 x 10^-3 m
CSA = 0.000707 m2
Force on cylinder
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.000707
= 7.073553 MPa
(ii) The square section
CSA = 1600 x 10^-6 m2
Force on square section
= 5 x 10^3 N
Stress
= 5 x 10^3 / 0.0016 m2
= 3.125 MPa

I have also attempted part B below.

(b) The strain in:

ε = dl / lo
= σ / E
= Stress / Young’s Modulus
(i) The circular section
Strain = 7.073553 / 200GNm
Strain = 3.53678 x 10^-8
(ii) The square section
Strain = 3.125 / 200GNm
Strain = 1.56250 x 10^-8

For Part a), the stresses look good.

For Part b), you've backslid. The stresses you calculated are in units of MPa, not Pa. Also, Young's modulus is in units of GPa or GN/m², not GNm. You've got to use the correct values of the stress to calculate correct values of the strain. You've got to be vigilant about making these silly mistakes.

 

[stackemup]

This should now be correct -

(b) The strain in:

ε = dl / lo
= σ / E
= Stress / Young’s Modulus
(i) The circular section
Strain = 7.073553 / 200GN/m2
Strain = 0.0353678
(ii) The square section
Strain = 3.125 / 200GN/m2
Strain = 0.0156250

Thanks for the advice.

 

[BvU]

Not there yet! 7.07 MPa / 200 GPa is not 0.035. Same for (ii)

 

[stackemup]

Thanks for the support.

I am struggling to calculate part (d) The change in diameter of the circular section.

Is the below formula used to calculate this?

Change in diameter = - original diameter x Poisons ratio x (change in length / original length)

 

[BvU]

Think of something you expect to remain constant. Diameter x length would remain constant with your formula. Would that be logical ?

 

[stackemup]

My equation now doesn't seem correct.

I'm struggling to make any progress with this question.

 

[stackemup]

I believe the lateral strain multiplied by the diameter gives the increase in diameter.

The lateral strain is obtained from poissons ration, and the positive sign determines an increase in lateral dimensions, which is expected when an object is under compression.

 

[BvU]

Sorry, I completely misread your post #15. Says transverse strain = - Poisson's ratio times axial stress/Young modulus , which is completely correct. My post #16 is wrong and distracting.

 

[stackemup]

I am a litte confused now.

Is my post 18 correct?

 

[BvU]

Yes ! Apart from the typo: 0.1131
And the diameter was 30 mm, not 60.

 

[stackemup]

Thanks.