LNB 30503 Ship Structures
TOPIC 2
SECTION MODULUS AND STRESS CALCULATION
4.1
SIMPLE BEAM THEORY
The total bending moment or the stress is calculated by the simple beam theory using the relation σ
y
=
M I
or
σ
=
M I y
=
M Z
where
σ M I
stress at distance y from the neutral axis bending moment second moment of area
I/y will have its smallest value when y is greatest; that is when y is measured to the extreme fibres, at the deck and bottom (keel! This value of I/y I/y or Z is is called the secti! "d#l#s and is the criterion of the strength of the girder in bending!
LNB 30503 Ship Structures
1tress at keel, σ keel
= M / Z keel 2 .0+!$ tonnes/m / 0!.'m * 2 &!.$tonnes/m '
1tress at deck, σ deck
= M / Z deck 2 .0+!$ tonnes/m / 0!++m * 2 %&!%0tonnes/m '
Ma%i"#" st&ess cc#&s at 'eel(
4.2
'eel
) *+.,-t!!es"4.
THE NEUTRAL A$IS
Theoretical analysis shows that the neutral axis occurs exactly at the centre of gravity of the cross section of the ob3ect or beam! 4ut with the plate and angle acting together as a beam the neutral axis is close to the plate! 564! 7aterial close to 6) contributes little to the strength of a beam8
4.2.1
Ne#t&al A%is / the H#ll
To locate the 6) of the hull, designers once again look for the centre of gravity of the hull cross section! These include only on those parts or members of the hull structure that
LNB 30503 Ship Structures
Solution 4.2
Ite"
Sca!tli!s
$0mm x 0$mm 0+*mm x *mm 0+*mm x 0$mm
=eb #ower lange
A&ea( a 3""2
Heiht h 3""
M"e!t / a&ea ah 3""0
2!d M"e!t / a&ea 2 ah 3""4
Lcal 2!d M"e!t / a&ea I 3""4
*++
0'+!$
.*.++
0.%.0.$+
.%.&!&&%
0&*.
0&!$
*****!$
'000..*
.+&0$'$!.%
$+%$
0!$
&*'*%!$
%.0.&.
0&'*00!.%
>ah 2 *%
>ah0 2 0&0$
%$>i 2 .$0'%$!$
>a 2 $'
Ste5 1 entre of gravity or height of neutral axis above baseline?
y 2
∑ ah = (*(((%( = ((*!//mm ≈ 114mm (($(' ∑a
Ste5 2
LNB 30503 Ship Structures
E$AMPLE 4.0
alculate second moment of area at neutral axis and hence section modulus for deck and keel! 4ending moment acting on ship at neutral axis is $+++ tm! ind stress at keel and main deck!
m
00 mm
* m
& mm
( * m
' mm
( ! $ m
mm 0 mm
0+ mm
LNB 30503 Ship Structures
Ste5 1
Ste5 2
Ste5 0
LNB 30503 Ship Structures
E$AMPLE 4.4
The mass distribution and buoyancy between sections of a ship, *++m in length, balanced on a hogging wave, are given below! The second moment of area of the midship section is %$0m ' and the neutral axis is .!*+m from the keel and .!%+m from the deck! alculate the maximum direct stresses (keel and deck given by the comparative calculation and the maximum sheer force and bending moment!
Stati!
Mass " 3MN
B#6a!c6 " 3MN
+!
+!++$
!&+.
+!+0$
0!+.*
+!%'
*!+0
*!+*
0!$&
$!&'
* +&'
& *&
0 * ' $ &
LNB 30503 Ship Structures
Solution 4.4
7aximum shear force 2 76 7aximum bending moment 2 76m σ 2
7×y I
Ceel stress 2 Aeck stress 2
2 12+.17MN" 2 2 101.-0MN" 2
LNB 30503 Ship Structures
4.4
SECTIONS 8ITH T8O MATERIALS
1ome ships9 strength cross section is composed of two different materials! Typically the hull may be steel and the superstructure aluminum! Dther materials used may be wood or reinforced plastic! In such a case it is convenient to think in terms of an effective modulus in one of the materials!
2+
and
Σ
that is? Σ
' # y + '
a
s
s
s
#a y a = +
The corresponding bending moment is? )M
2 Σ (σ s#sys σa #aya
' s # s y s + ' a #a y a = + ( (
LNB 30503 Ship Structures
arched in the opposite shear forces due to the stretch or compression and normal forces trying to keep the two in contact! The ability of the superstructure to accept these forces, and contribute to the section modulus for longitudinal bending, is regarded as an efficiency! It is expressed as? 1uperstructure efficiency 2
− σ a σ − σ
σ +
+
=here σo, σa and σ are the upper deck stresses if no superstructure were present, the stress calculated and that for a fully effective superstructure!
E$AMPLE 4.-
The midship section of a steel ship has the following particulars? ross:sectional area of longitudinal material 2 0!*m0 Aistance from neutral axis to upper deck 2 %!&m 1econd moment of area about the neutral axis 2 $m ' ) superstructure deck is to be added 0!&m above the upper deck! This deck is *m wide, 0mm thick and is constructed of aluminum alloy! If the ship must withstand a sagging bending moment of '$+76m, calculate the superstructure efficiency if with the
LNB 30503 Ship Structures
'$+ × .!.
1tress in the new deck (as effective steel 2
&*!0
= %!$ 76 / m
0
1tress in the deck as aluminium 2 %!$ / +!*00 2 00+!.& 76 / m 0 The superstructure efficiency relates to the effect of the superstructure on the stress in the upper deck of the main hull! The new stress in that deck, with the superstructure in the place, is given as $$ 76 / m 0! If the superstructure had been fully effective it would have been? '$+(%!& − +!0 &*!0
= $0!&* 76 / m
with no superstructure the stress was
Hence the superstructure efficiency 2
0
'$+ × %!& $ $/!.% $/!.%
0 = $!.% 76 / m
− $$
− $0!&*
= &0!&M
LNB 30503 Ship Structures
4.:
ACTUAL SECTION MODULUS CALCULATION
E%a"5le / Midshi5 Sca!tli! � Yahya Bin Samian, Department of arine !e"hnolo#y, $% , &! ' $e( 2))5
m !$ m
!%$ m
A-C #D6B# ) 2 0 cm0
!$ m
!$ m
!$ m
A-C F#)Tt 2 mm
1IA- F#)Tt 2 % mm #D6B 4CHA F#)Tt 2 & mm
0!+ m
LNB 30503 Ship Structures
ORMULA USED OR CALCULATIN; SECTION MODULUS
!Y*+ 1 : erti"al *late Data Re<#i&ed a!d Calc#lati! # 2 #ength (m
t 2 Thickness (mm
Dia&a" t
N 2 Aistance of entroid from keel (m L
) 2 )rea 2 # x t (m!mm st! 7oment 2 ) x N (m 0!mm 0nd! 7oment 2 ) x N 0 or st 7oment x N (m *!mm Io 2 Inertia 7oment 2 #* x t (m*!mm
= rom Ceel
!Y*+ 2 : -oriontal *late Data Re<#i&ed a!d Calc#lati! # 2 #ength (m t 2 Thickness (mm N 2 Aistance of entroid from keel (m ) 2 )rea 2 # x t (m!mm
Dia&a" t
LNB 30503 Ship Structures
E$AMPLE O MIDSHIP SECTION MODULUS CALCULATION
Act#al Se cti! Md#l #s Re<#i&ed Secti! Md#l#s 3E%a"5le 9ased ! La&e Ta!'e& #2 + m 42 & m 4 2 +!% 2 %!0% 0 2 +!+ 1)422.20 17 re@uired 2 m!cmO0
1afety actor 2
LNB 30503
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