The hull structure consists of stiffened panels; bottom construction, side shell construction, upper deck construction, bulkhead, etc. Usually stiffened panels consist of plates, beams (small member, secondary member) and girders (big member, primary member). The plate receives loads such as water pressure, the beam supports the loads from the plate and the girder supports the loads from the beam.

A rational grillage structure which has
girders crossing each other (mutually-supported structure) is studied here from a strength viewpoint.

**What is a grillage?**

A grillage consists of two layers of “I” beams as shown in Figure. The load from the column is transferred to the base plate and the base plate transfers the load to the concrete. The concrete transfers the load to the top layer of “I” beams and then to the bottom layer of beams. The bottom layer of “I” beams would transfer the load to the concrete below and then to the rock underneath.

When assigning a cargo to a ship or barge, aside
from the physical fit of it into the vessel, the cargo weight and centre of
gravity must be taken into consideration. When dealing with heavy lift cargo
especially, there is a need to ensure that the vessels’ underdeck structure can
withstand the loads imposed by the cargo during seagoing conditions.

When the vessels’ internal structure is unable to withstand such loads, bespoke grillages need to be installed to ensure loads are distributed from the cargo support points into the vessels deck structure to avoid damage to either the cargo, the vessel or both.

## Grillage Structure

Figure 6.1.1 shows a stiffened panel with
length of longer edge *a*, shorter
edge *b *and uniform
load *p*. It is
common sense for the girders to be arranged in the direction of the shorter
span as the girders in the longer span are not so effective. Here common sense
is proved quantitatively.

As the results for fixed-boundary conditions
and for simply supported conditions are similar hereafter, the simply-supported
boundary condition is to be applied [1].

The maximum stress s_{y}* *is generated at the midpoint *O *in Fig. 6.1.1 and the stress s_{y}* *and the deflection d are as follows:

where d : deflection at point *O*.

Applying the following notations, maximum
stress s_{y}* *is described below.

*I _{x} *and

*I*are sectional moment of inertia with effective breadths

_{y}*m *and *n *are numbers of girders

*e _{x} *and

*e*are distances from center of gravity of section to face plates

_{y}spaces of girders *l _{x} *=

*b*

*/*(

*m*+1) and

*l*=

_{y}*a*

*/*(

*n*+1)

rigidity ratio per unit breadth *ix *= *I _{x}*

*/*

*l*and

_{x}*iy*=

*I*

_{y}*/*

*l*

_{y}rigidity ratio in longer and shorter edge
directions (mutually-supporting ratio)

a = *i _{x}*

*/*

*i*

_{y}_{}

Putting weight per unit area of grillage
structure as *W*_{1}, and weights per unit length of girders in longer and shorter
directions as *W _{x} *and

*W*respectively, the following relation is obtained.

_{y}Usually the same scantling girders are to be
applied for *X *and *Y *directions in a mutually-supported grillage
structure. Applying this principle the following results are obtained.

*I _{x}
*=

*I*=

_{y }Z_{x}*Z*=

_{y}W_{x}*W*

_{y}In Sect. 1.8, Optimum Design of Beam Section,
the relation between the section modulus of a beam and its weight per unit
length is explained. This principle can be applied to girders assuming the web
thickness is 12 mm. The result is shown in Eq. (6.1.7).

*W _{y} *= 1

*.*5

Where,

*W _{y} *: weight per unit length of girder in kgf/m

*Z _{y} *: section modulus of girder in cm

^{3}

Putting Eqs. (6.1.7) and (6.1.3) into (6.1.4)
and with the conditions described by (6.1.5) and (6.1.6), the weight per unit
area*W*1 of girders
is given by the following equation.

Where a grillage (mutually-supported)
arrangement is not applied, a girder arrangement in one direction is usually
applied. In this case the weight per unit area *W*_{0} is obtained
by putting a= 0 in Eq. (6.1.8). The weight ratio *W*_{1}*/**W*_{0} of the
mutually-supported and the one-direction girder arrangement is given as
follows:

The relation of Eq. (6.1.9) is shown in Fig.
6.1.2 with a along the horizontal axis and *b**/**a *as a parameter. It can be seen that the smaller *b**/**a*, which means a slender rectangular, and bigger a, mutually-supporting ratio, will bring a bigger weight
difference. It is important to note that the ratio*W*_{1}*/**W*_{0} is always
bigger than 1.0 which means the mutually-supported arrangement is always
heavier than the one-direction girder arrangement. This result is in agreement
with the designer’s common sense.

The stress s*y *at the point *O *in Fig. 6.1.1 in the shorter edge direction
is expressed by Eq. (6.1.1) and the stress s*x *at the same point *O *in the longer edge direction is expressed by
the following equations:

s_{x}*/*s_{y}* *is proportional to the square of *b**/**a *which means for a slender panel, s_{x}* *is very much smaller than s*y *and the mutually-supported condition will
disappear.

Even
in the case of a square panel the one-direction arrangement is better than the mutually-supported
arrangement because the weight ratio between two girders with section modulus *Zy *and one girder with section modulus 2*Zy *is:

And in
Fig. 6.1.2, *W*_{1}*/**W*_{0} = 1*.*41 for *b**/**a *= 1*.*0 and a= 1*.*0 is the same
story. Regarding the minimum weight of grillage structure, Yagi and Yasukawa’s
study is famous, and Kitamura’s study as a non-linear programming method is
also useful.