See Attachment: Use variant #7 Document Preview: Stability of Engineering Structures Spring 2013 Project 4 Subjects: Stability of cylindrical shells; design of beam-columns. In this project we...

Problem 1
At a depth h hydrostatis pressure given by
p gh
 is the density of water = 1 gram/cc, g is the acceleration due to gravity = 9.81 m/s2.
Let us consider the stresses due to a hydostatic pressure p.
The cylinder will experience a compressive force transfe
ed through the hemispherical ends along the X
direction and that force will be taken by the wall of the cylinder.
Force transmitted through the hemispherical ends =
2xF rp
where r is the outer radius.
The area of the cylinder wall (assuming wall thickness t to be small)
 
22 2A r r t rt     
Stress along x =
2943x
xx
F h
Pa
A t
  

when t is expressed in meter.
The circumferancial stress (Hoop stress) is given by
5886h
c
p
Pa
t t
  
when t is expressed in meter.
These two normal stresses work along two mutually perpendicular directions and there is no
other stress.
So, the von Misses criteria can be written as:
2 2 2
xx c Y  
where Y is the yield strength of the material.
This can be rewritten as:
2 2
22943 5886h h Y
t t
   
    
   

Solving this, we get
0.5
2 2 2 2
2
2943 5886h h
t
Y
 
  
 

This is the thickness obtained from the von Misses criteria.
For the buckling criteria we need to find out the critical value of the hydrostatic pressure pcr.
There are different formulations given by several researchers to find out the critical hydrostatic
pressure.
The critical hydrostatic pressure is given by
 
2.5
0.75
2 2.5
0.855
1
c
E t
p
L





This is taken from the following reference:
http:
www.dept.aoe.vt.edu/~cdhall/courses/aoe4065/NASADesignSPs/sp8007.pdf
Here, 
is a co
ection factor. The prescribed value for this case is
0.75 

Equating the critical hydrostatic pressure to the actual hydrostatic pressure, we get the
equired minimum thickness to meet the buckling criteria as:
 
 
2.5
0.75
2 2.5
0.4
0.75
2 2.5
0.855
1
1
0.855
E t
gh
L
L gh
t
E



 



 
  
 
 

For each case, we calculate the required thickness using the von Misses criteria and the
uckling criteria. The higher value among these two is the design thickness for that case.
The following table shows the values of required thicknesses at different depths for different
materials.
http:
www.dept.aoe.vt.edu/~cdhall/courses/aoe4065/NASADesignSPs/sp8007.pdf
Steel
We chose steel because of its high elastic modulus and yield strength.
The material properties are taken from the following page.
http:
www.roymech.co.uk/Useful_Tables/Matte
prop_metals.htm
The modulus of elasticity, yield strength and Poisson’s ratio of steel are 210 GPa,300 MPa and 0.3
espectively.
It can be noticed from the table that up to a depth of 1100 m , the buckling criteria governs the
design. After that the von Misses critera governs the design.

Depth (m)
Thickness (mm) (strength
criteria)
Thickness (mm) (cklingb
criteria) Design depth(mm)

100 2.1936 8.5641 8.5641

200 4.3872 11.3004 11.3004

300 6.5807 13.2901 13.2901

400 8.7743 14.911 14.911

500 10.9679 16.3031 16.3031

600 13.1615 17.5364 17.5364

700 15.3551 18.6518 18.6518

800 17.5487 19.6751 19.6751

900 19.7422 20.6243 20.6243

1000 21.9358 21.5121 21.9358

1100 24.1294 22.348 24.1294

1200 26.323 23.1395 26.323

1300 28.5166 23.8923 28.5166

1400 30.7102 24.6112 30.7102

1500 32.9037 25.2999 32.9037

1600 35.0973 25.9615 35.0973

1700 37.2909 26.5987 37.2909

1800 39.4845 27.2139 39.4845

1900 41.6781 27.8088 41.6781

2000 43.8717 28.3853 43.8717

2100 46.0652 28.9447 46.0652

2200 48.2588 29.4883 48.2588

2300 50.4524 30.0174 50.4524
http:
www.roymech.co.uk/Useful_Tables/Matte
prop_metals.htm

2400 52.646 30.5328 52.646

2500 54.8396 31.0354 54.8396


Titanium
The material properties are taken from the following page....