NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 1 av 7
Page 1 of 7
Contact during the exam:
Professor Jørgen Amdahl
(735) 95544
EXAM IN SUBJECT TMR4205 BUCLING AND COLLAPSE OF MARINE
STRUCTURES
Thursday 07. June 2007
Time: kl 15.00 - 1900
Approved help (D):
Neither printed nor handwritten notes are permitted.
Approved, simple calculator is permitted.
Results available:
28. June 2007
The problem text is on 7 pages and includes 2 problems.
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 2 av 7
Page 2 of 7
PROBLEM 1
Average stress
End shortening
FIgure 1 Average stress versus end shortening for stiffened plate
a) Figure 1 shows the average stress versus end shortening for a stiffened rectangular
plate obtained by means of nonlinear finite element analysis or with the PULSE code.
Make an approximate copy of the curve and indicate the pre-buckling deformation
range, post buckling deformation range and the post collapse deformation range.
Indicate also in the diagram the elastic buckling stress, the ultimate strength and the
stress level at first yield; both in the plate middle plane and on plate top or bottom.
Sketch also the average stress – end shortening curve according to linear buckling
theory.
(a)
(b)
Figure 2 Rectangular plates with two different imperfections
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 3 av 7
Page 3 of 7
b) Figure 2 shows two identical rectangular plates with straight and simply supported
edges and with different imperfection shapes, but with the same amplitude. The plates
may be subjected to biaxial compression. The ultimate strength in uniaxial
compression is denoted xa and xb for plate a and plate b in x1-direction
(longitudinal direction) and ya and yb in x2-direction (transverse direction). Sketch
the capacity boundary/surface in the stress space (x ,y) for the two plates. No
calculations shall be performed, but the relative capacities shall be qualitatively
correct. Indicate the capacity envelope you will use in design, when you don’t know
the actual imperfections in the plate
c) Sketch qualitatively the longitudinal - and transverse normal stress distribution in
plate a for uniaxial compression in x1-direction and for plate b for uniaxial
compression in x2-direction. Indicate where you will find the hot spot (von Mises)
stresses. Indicate in the sketches the level of Euler buckling stress and ultimate
strength. Define the effective width of the plate.
d) It is desired to strengthen plates with an aspect ratio (length/width) of 3 by means of
secondary, transverse stiffeners. You have the option of using either one or two
secondary stiffeners for each plate field. Would you use one or two stiffeners, and
where would you apply it/them?
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 4 av 7
Page 4 of 7
PROBLEM 2
(see also given information at end of problem text)
a) Sketch the typical force (or stress) – end shortening relationship for a perfect and
imperfect cylindrical shell. What is the major reason for the difference in the behavior
for a perfect and imperfect shell? What is the major difference between the behavior
in the post-critical range of a cylindrical shell and a plate?
p
N
N
Axial compression
p
Lateral pressure
p
T
T
p
Torsion
Hydrostatic pressure
Figure 3 Load cases for a cylindrical shell
b) Figure 3 shows a cylindrical shell with radius, r, and thickness, t, subjected to various
load cases. Calculate the relevant membrane stress components for the four cases.
c) For the case with hydrostatic pressure sketch qualitatively how the circumferential
stress of the shell plate varies over one stiffener spacing when ring stiffeners are
introduced. The influence of both a large stiffener spacing and narrow stiffener
spacing shall be indicated. Compare with the stress for an unstiffened cylinder.
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 5 av 7
Page 5 of 7
r = 2.5 m , t = 20 mm, yield stress Y = 250 MPa
MPa
Q M
M Q
N
N
= 10 m
Q = 12 MN, N = 30 MN, M = 40 MNm
Figure 4 Unstiffened circular cylinder
d) An unstiffened cylindrical shell is subjected to combined bending, axial compression
and shear force, as shown in Figure 4. Find the maximum utilization of the cylinder
with respect to shell buckling. It is presupposed that ordinary beam theory is valid.
e) It is considered to improve the buckling capacity of the shell for axial compression by
introducing longitudinal stiffeners. What is the minimum number of stiffeners needed
for this purpose and what is the utilization with respect to shell buckling at this stage?
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 6 av 7
Page 6 of 7
Given information:
Thin-walled cylinder:
Moment of inertia : I   r 3t
Maximum shear stress:  
Q
 rt
For curved panel
2
2E  t  C
 
E 
121   2   s 
ZS 
s2
1  2
rt
For unstiffened cylindrical shell
2
2
2
ö
E æ
t
p

÷
çç ÷ C

1  2
sE=
Z
÷
12 (1- n 2)çè ø
rt
  
C   1  
 
2
Elasto-plastic buckling
Y
 Y   x  b    x 
1
2






 eq ,cr 
 Y eq 
4
 eqE
 eq   xE  bE  E  xE 
1 
Equivalent stress
 eq 
 x   b 2   x   b     2  3 2x
Utilisation:

  eq
 eq ,cr
NORGES TEKNISK-NATURVITENSKAPELIGE UNIVERSITET
Institutt for marin teknikk
Side 7 av 7
Page 7 of 7
Buckling coefficients for curved panels

Axial stress
Shear
Compressive ring
stress


0.702Z
4
2
æs ö
5.34 + 4 çç ÷
÷
çè ÷
ø
0.856
2
é æs ö2 ù
ê1 + ç ÷
ú
ê ççè ø÷
÷ú
ëê
ûú
1.04
r 

0.51 

 150 t 
S
s
Z
l
s
Z
l
S
34
S
0.5
0.6
0.6
Buckling coefficients for unstiffened cylinders

Axial Stress
Bending
Torsion and Shear
force
Lateral Pressure
Hydrostatic
Pressure


0.5
0.702 Z
r 

0.51 

 150t 
1
0.702 Z
r 

0.51 

 150t 
5.34
4
0.856Z
1.04 Z
0.6
0.6
2
1.04 Z
0.6
1
34
0.5