## J

20 mm from the ANA. The second moments of area give the moment of inertia about the ANA and this can be corrected for the position of the true NA. This is illustrated for the diagrammatic cross section shown in Figure 7.8, noting that the second moments of thin horizontal members about their own centroids will be negligible.

It will be noted that material in the centre of the top two decks is not included. This is to compensate for the large hatch openings in these decks. Because of the ship's symmetry about its middle line, it is adequate to carry out the calculation for one side of the ship and then double the resulting answer. In this example the ANA has been taken at the keel.

From Table 7.1, the height of the NA 4.135

0.799

Second moment of area of half section about keel = 40.008 + 2.565 = 42.573 m4

and about the actual

For the whole section the Z values are: 42.326

7.82

42.326

5.18

Table 7.1 Calculation of properties of simplified section in Figure 7.8

Item Scantlings Area Lever Moment Second Second moment

(m2) about about moment about own keel keel (in4) cent,raid

Item Scantlings Area Lever Moment Second Second moment

(m2) about about moment about own keel keel (in4) cent,raid

Upper deck |
6 |
X |
0.022 |
0.132 |
13 |
1.716 |
22.308 |
0 |

Second deck |
6 |
X |
0.016 |
0.096 |
10 |
0.960 |
9.600 |
0 |

Side shell |
13 |
X |
0.014 |
0.182 |
6.5 |
1.183 |
7.690 |
2.563 |

Tank top |
10 |
X |
0.018 |
0.180 |
1.5 |
0.270 |
0.405 |
0 |

Bottom shell |
10 |
X |
0.020 |
0.200 |
0 |
0 |
0 |
0 |

1.5 |
X |
0.006 |
0.009 |
0.75 |
0.007 |
0.005 |
0,002 | |

Summations |
0.799 |
4.136 |
40.008 |
2.565 |

If the bending moments for this structure are those calculated in Example 7.1, the stresses can be found. The still water stresses are:

5.41

8.17

The wave bending stresses are: 718

5.41

8.17

This gives total stresses in the deck of 280.0 or 14.6 MN/m2 compression, and in the keel stresses of 185.5 or 9.7 MN/m2 tension, depending upon whether the ship is sagging or hogging.

The sagging stresses would be too high for a mild steel ship and some action would be needed. One way would be to spread the central 5000 tonne load uniformly over the whole ship length. This would reduce the stresses to 132.7 MN/m2 in the deck and 87.9 MN/m2 in the keel. If it was desired to increase the section modulus to reduce the stresses, the best place to add material would be in the keel or upper deck, whichever had the higher stress. That is, to add material as far from the

NA as possible. However, the change in the lower of the two original stresses must be watched. The general problem of adding material to a cross section is discussed later.

### Sections with two materials

Some ships' strength cross secdon is composed of two different materials. Typically the hull may be steel and the superstructure aluminium. Other 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. Usually this would be in terms of steel.

The stress, o, in a beam at a point z from the NA is Ez/R, where R is the radius of curvature. Provided transverse sections of the beam or ship remain plane, this relationship will hold as the extension or strain at any given z will be the same. For equilibrium of the section, the net force across it must be zero. Hence using subscripts s and a for steel and aluminium:

that is:

The corresponding bending moment is:

Ve R

where /E is the effective second moment of area.

The composite cross-section can therefore be considered made up of material s, usually steel, if an effective area of material a is used in place of the actual area. The effective area is the actual multiplied by the ratio E^/E^. For different steels the ratio is effectively unity, for aluminium alloy/steel it is about | and for grp/steel it is about i^.

### Changes to section modulus

It is often desirable to change the section modulus in the early design stages. The effect of changes is not always immediately obvious. Consider the addition of an element of structure above the neutral axis, but below the upper deck, as in Figure 7.9. Assume the element is of area a and that the original section had area A, and radius of gyration k. With the addition of a the NA is raised by az/(A + a) = <3z.

Figure 7.9 Changing section modulus

The new second moment of inertia is:

For a given bending moment, the stesses will not increase if the section modulus is not reduced. This condition is that:

I + ôI I ôl ôz -- - is greater than zero, that is — is greater than —

As depicted ÔI is positive and ôz/z2 is negative so ôl/1 is always greater than ôz/z at the deck provided the material is added within the section. At the keel the condition becomes:

Aaz2

—--must be greater than az/zx (A + a) or z must be greater

than A2/2]

Thus to achieve a reduction in keel stress the material must be added at a height greater than $/zx from the neutral axis.

Corresponding reladonships can be worked out for material added below the neutral axis. If the new material is added above the upper deck then the maximum stress will occur in it rather than in the main deck. It can be shown that in this case there is a minimum area thai must be added at any given height in order to reduce the stress in the deck.

### STRESS CONCENTRATIONS

So far only the general stresses in a structure have been considered. There are several reasons why local stresses may exceed considerably those in the vicinity. The design may introduce points at which the loads in a large structural element are led into a relatively small member. It is useful in looking at a structure to consider where the load in a member can go next. If there is no natural, and even, 'flow' then a concentration of stress can occur. Some such details are bound to arise at times, in way of large deck openings for instance, or where the superstructure ends. In such cases the designer must take care to minimize the stress concentration. Well rounded corners to hatch openings are essential and added thickness of plating abreast the hatches reduces the stress for a given load. The magnitude of this effect can be illustrated by the case of an elliptical hole in an infinitely wide plate subject to uniform tensile stress across the width. If the long axis of the ellipse is 2a and the minor axis is 2b, then with the long axis across the plate the stresses at the ends of the long axis will be augmented by a factor [1 + (2a/b)]. If the hole is circular this concentration factor becomes 3. There will be a compressive stress at the ends of the minor axis equal in magnitude to the tensile stress in the plate. In practice there is litde advantage in giving a hatch corner a radius of more than about 15 per cent of the the hatch width. The side of the hatch should be aligned with the direction of stress otherwise there could be a further stress penalty of about 25 per cent,

Apart from design features built into the ship, stress concentrations can be introduced as the ship is built. Structural members may not be accurately aligned either side of a bulkhead or floor. This is why important members are made continuous and less important members are made intercostal, that is they are cut and secured either side of the continuous member. Other concentrations are occasioned by defects in the welding and other forming processes. Provided the size of these defects is not large, local redistribution of stresses can occur due to yielding of the material. However large defects, found perhaps as a result of radiographic inspection, should be repaired.

### Built-in stresses

Taking mild steel as the usual material from which ships are built, the plates and sections used will already have been subject to strain before construction starts. They may have been rolled and unevenly cooled. Then in the shipyard they will be shaped and then welded. As a result they will already have residual stresses and strains before the ship itself is subject to any load. These built-in stresses can be quite large and even exceed the yield stress locally. Built-in stresses are difficult to estimate but in frigates8 it was found that welding the longitudinals introduced a compressive stress of 50MPa in the hull plating, balanced by regions local to the weld where the tensile stresses reached yield.

### Cracking and brittle fracture

In any practical structure cracks are bound to occur. Indeed the build process makes it almost inevitable that there will be a range of cracklike defects present before the ship goes to sea for the first time. This is not in itself serious but cracks must be looked for and corrected before they can cause a failure. They can extend due to fatigue or brittle fracture mechanisms. Even in rough weather fatigue cracks grow only slowly, at a rate measured in mm/s. On the other hand, under certain conditions, a brittle fracture cam propagate at about 500 m/s. The MVKurdistan broke in two in 19799 due to brittle fracture. The MV Tyne Bridge suffered a four metre crack10. At one time it was thought that thin plating did not suffer brittle fracture but this was disproved by the experience of RN frigates off Iceland in the 1970s. It is therefore vital to avoid the possiblity of brittle fracture. The only way of ensuring this is to use steels which are not subject to this type of failure under service conditions encountered11.

The factors governing brittle fracture are the stress level, crack length and material toughness. Toughness depends upon the material composition, temperature and strain rate. In structural steels failure at low temperature is by cleavage. Once a crack is initiated the energy required to cause it to propagate is so low that it can be supplied from the release of elastic energy stored in the structure. Failure is then very rapid. At higher temperatures fracture initiation is by growth and coalescence of voids and subsequent extension occurs only by increased load or displacement12. The temperature for transition from one fracture mode to the other is called the transition temperature. It is a function of loading rate, structural thickness, notch acuity and material microstructure.

Unfortunately there is no simple physical test to which a material can be subjected that will determine whether it is likely to be satisfactory in terms of britde fracture. This is because the behaviour of the structure depends upon its geometry and method of loading. The choice is between a simple test like the Charpy test and a more elaborate and expensive test under more representative conditions such as the Robertson crack arrest test. The Charpy test is still widely used for quality control.

Since cracks will occur, it is necessary to use steels which have good crack arrest properties. It is recommended11 that one with a crack arrest toughness of 150 to 200MPa(m)°5 is used. To provide a high level of assurance that britde fracture will not occur, a Charpy crystallinity of less than 70 per cent at 0°C should be chosen. For good crack arrest capability and virtually guaranteed fracture initiation avoidance, the Charpy crystallinity at 0°C should be less than 50 per cent. Special crack arrest strakes are provided in some designs. The steel for these should show a completely fibrous Charpy fracture at 0°C.

### Fatigue

Fatigue is by far and away the most common cause of failure13 in general engineering structures. It is of considerable importance in ships which are usually expected to remain in service for 20 years or more. Even when there is no initial defect present, repeated stressing of a member causes a crack to form on the surface after a certain number of cycles. This crack will propagate with continued stress repetitions. Any initial crack-like defect will propagate with stress cycling. Crack initiation and crack propagation are different in nature and need to be considered separately.

Characteristically a fatigue failure, which can occur at stress levels lower than yield, is smooth and usually stepped. If the applied stressing is of constant amplitude the fracture can be expected to occur after a defined number of cycles. Plotting the stress amplitude against the number of reversals to failure gives the traditional S-N curve for the material under test. The number of reversals is larger the lower the applied stress until, for some materials including carbon steels, failure does not occur no matter how many reversals are applied. There is some evidence, however, that for steels under corrosive conditions there is no lower limit. The lower level of stress is known as the fatigue limit.

For steel it is found that a log-log plot of the S-N data yields two straight lines as in Figure 7.10. Further, laboratory tests14 of a range of

Life cycles

Figure 7.10 S-N curve typical welded joints have yielded a series of log-log S-N lines of equal slope.

The standard data refers to constant amplitude of stressing. Under these conditions the results are not too sensitive to the mean stress level provided it is less than the elastic limit. At sea, however, a ship is subject to varying conditions. This can be treated as a spectrum for loading in the same way as motions are treated. A transfer function can be used to relate the stress range under spectrum loading to that under constant amplitude loading. Based on the welded joint tests referred to above14, it has been suggested that the permissible stress levels, assuming twenty million cycles as typical for a merchant ship's life, can be taken as four times that from the constant amplitude tests. This should be associated with a safety factor of four thirds.

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