4. Local strength calculation
 Response patterns of typical ship structures
Primary, secondary and tertiary structure
Secondary response comprises the stress and
deflection of a single panel of stiffened plating,
e.g., the panel of bottom structure contained
between two adjacent transverse bulkheads.
The loading of the panel is normal to its plane
and the boundaries of the secondary panel are
usually formed by other secondary panels (side
shell and bulkheads) .
Tertiary response describes the out-ofplane deflection and associated stress of
an individual panel of plating. The
loading is normal to the panel, and its
boundaries are formed by the stiffeners of
the secondary panel of which it is a part .
4.1 Secondary structural response
 Orthotropic plate theory
Orthotropic plate theory refers to the theory of
bending of plates having different flexural
rigidities in the two orthogonal directions.
In applying this theory to panels having discrete
stiffeners we idealize the structure by assuming
that the structural properties of the stiffeners
may be approximated by their average values,
which are assumed to be distributed uniformly
over the width or length of the plate.
 Beam-on-elastic-foundation theory
The beam on elastic foundation solution is
suitable for a panel in which the stiffeners
are uniform and closely spaced in one
direction and more sparse in the other.
One of the latter members may be thought of
as an individual beam having an elastic
support at its point of intersection with each
of the closely-spaced orthogonal beams.
 Grillage theory
In the grillage method, each stiffener in the two
orthogonal sets of members is represented as
a simple beam.
 The finite element method
 Computational models for typical ship
grillages
Side
Girder
Bulkhead
Bulkhead
Floor
Side
Strength computational model of a bottom grillage
4.2 Tertiary structural response
 A longitudinal at bottom
Strength computational model
 A longitudinal at deck
Strength computational model:
Buckling (stability) computational model:
 A bottom shell in longitudinal framing
system
Strength computational model:
b
4
 4 4
a
Stress at mid-length of short side along
longitudinal direction (a/b>1.5-2.0)
b 2
 4  0.309q( ) MPa, q: pressure in MPa
t
 A bottom shell in longitudinal framing
system
Strength computational model:
b
4
 4 4
a
Stress at center along longitudinal direction
(a/b>1.5-2.0)
b 2
 4  0.075q( ) MPa, q: pressure in MPa
t
 A bottom shell in longitudinal framing
system
Strength computational model:
b
4
 4 4
a
Stress at mid-length of long side along
transverse direction (a/b>1.5-2.0)
b 2
 4  0.5q( ) MPa, q: pressure in MPa
t
 A bottom shell in
transverse framing system
Strength computational model:
b
4 4
a
Stress at mid-length of long side along
longitudinal direction (b/a>2.0)
a 2
 4  0.5q( ) MPa, q : pressure in MPa
t
 A bottom shell in
transverse framing
system
b
4 4
Strength computational model:
a
Stress at center along longitudinal direction
(b/a>2.0)
a 2
 4  0.25q( ) MPa, q : pressure in MPa
t
Strength computational load
1、Bottom Structure load
Including: Bottom shell, longitudinal at
bottom, grillage at bottom
the computational load
2、Broadside Structure load
3、Deck Structure load
4、 Bulkhead Structure load
4.3 Shear lag and effective breadth
Loading and resulting strain in flanges of simple beam
An important effect of this edge shear
loading of a plate member is a resulting
nonlinear variation of the longitudinal
stress distribution. This is in contrast to
the uniform stress distribution predicted
in the beam flanges by the elementary
beam equation.
In many practical cases, the departure
from the value predicted by the
elementary beam equation will be small.
But in certain combinations of loading
and structural geometry, the effect
referred to by the term shear lag must be
taken into consideration if an accurate
estimate of the maximum stress in the
member is to be made.
The effect of shear lag in a ship is to cause the stress
distribution in the deck, for example, to depart from
the constant value predicted by the elementary beam
equation. A typical distribution of the longitudinal
deck stress in a ship subject to a vertical sagging
load is sketched in the following figure.
Deck longitudinal stress, illustrating the effect of shear lag
The effective breadth,  b , is defined as the
breadth of plate that, if stressed uniformly
at the level across its width, would sustain
the same total load in the x-direction as the
non-uniformly stressed plate. Hence,
b
b  (1/  B )  x ( y)dy
0
The quantity  is called the plate
effectiveness.
Design analysis of the bottom shell in
transverse framing system
 Strength requirement for a panel
b 2
0.5q ( )    0.4 s
t
 Stability requirement for a panel
100t 2
)   s
b
 E  k1 20(
 Pressure satisfying both the requirements
of strength and buckling
 s2

H  100 p 
, p,  s in MPa
3
k1 2.5 10
 s , MPa
   1
k1  1.5
  0.5
  1,
220
240
300
350
400
H, m 12.9
15.4
24.0
32.6
42.6
b/t
37
35
32
29
27
H, m 6.4
7.7
12.0
16.3
21.3
49
45
41
38
k1  1.5 b/t
52
H satisfying the requirements of both strength
and buckling
b/t satisfying the requirement of buckling only
Conclusions:
1) If the buckling load is required to reach
yielding strength, the computed H is much
higher than real value of H for most real
ships and the value of b/t is quite small as
well. For this case, the structure design is
NOT reasonable.
2) If the buckling load is designed to reach
the half yielding strength, the use of low
strength steel (  s  300 MPa ) is Okay only.
Conclusions:
3) Reduction computation has to be considered
to determine the plating thickness due to the
poor stability performance of the bottom
shell in transverse frame system.
Design analysis of the bottom shell and
the longitudinal at bottom in
longitudinal framing system
 Strength requirement for a panel
b 2
0.309q ( )    0.3 s
t
 Strength requirement for a longitudinal
2
qba
  s
12W
f  Cw
3
2
 qba 

  f2
 12 s 
2
Cw 
f
W 2/3
 Bucking requirement for a panel
100t 2
 E  80(
)   s
b
 Bucking requirement for a longitudinal
E 
 2 Ei
a ( f  bet )
2
2
f
2
i  2 5f
Ci
 2.5 s
f  bt  b
s
a
f  0.49
b  f1
100 100
Design analysis of the plating
b 2
0.309q( )    0.3 s strength
t
100t 2
 E  80(
)   s buckling
b
H  100 p  

2
s
8.25 10
3
, p,  s in MPa
 s , MPa
   1
  0.5
 1
220
240
300
350
400
H, m 5.9
7.0
10.9
14.9
19.4
b/t
60
58
52
48
45
H, m 2.9
3.5
5.4
7.4
9.7
b/t
82
74
68
64
85
H satisfying the requirements of both
strength and buckling
b/t satisfying the requirement of buckling
only
Conclusions:
1) For high strength steel (  s  300 MPa ),
the size of plating is controlled by the
requirement of buckling.
2) For low strength steel (  s  240 MPa ),
the size of plating is controlled by the
requirement of strength.
Design analysis of the longitudinal
s
a
f  0.49
b  f1 , buckling
100 100
2
 qba 
f  Cw 3 
  f 2 , strength
 12 s 
2
2
 qba 
Cw 

12

f2
s 


f1
s a
0.49
b
100 100
2
3
s
a
f  0.49
b  f1 , buckling
100 100
2
 qba 
f  Cw 3 
  f 2 , strength
 12 s 
2
b
If  0.3,  s  294 MPa,   0.3, b / t =50,
a
f2
f

 0.15  0.20, q  0.04 MPa then
1
b
f1
The size of the longitudinal at deck is
controlled by the stability requirement.
The size of the longitudinal at bottom is
controlled by the strength requirement.