The objective function can be changed following the purpose of the design. The main interest of this work is in weight minimisation. The following expression is the weight evaluation function of tophat cross stiffened plate.
Weight of base plate is a product of the volume of composite material with the density of the material. Weight of the individual ply of base plate is,
(LBtbp)ρbp (5.49)
where L and B is length and width of base plate respectively. tbp is the thickness of the
individual ply and ρbp is the density of composite material of the individual ply.
As the base plate consists of nbp layers, the total weight of base plate (Wbp) can be
expressed as, Wbp =LB nbp X k=1 tbp(k)ρbp(k) (5.50)
Similarly, the weight of the crown element of a girder (Wcg) which consists of ncg layers
can be written as,
Wcg=B ncg
X
k=1
tcg(k)agρcg(k) (5.51)
wheretcg is the ply thickness of the crown element, ag is the width of the crown element
and ρcg is the density of composite material of the individual ply of the crown element.
For the web element of a girder which is comprised ofnwg layers, its weight (Wwg) can be
presented as, Wwg =B nwg X k=1 twg(k)hgρwg(k) (5.52)
where twg is the ply thickness of web element, hg is the height of web element and ρcg is
Wg =Wcg+ 2Wwg (5.53)
The weight of the beam is defined by the same method as the weight of girder Wg. The
total weight of the plate is,
WT =Wbp+gWg +bWb (5.54)
where g and b are the number of girders and beams respectively. Moreover, the weight function in Eq. 5.54 can be used for other types of plates. For a unidirectional stiffened plate,b = 0. For an unstiffened plate, g = 0 andb = 0.
Validation and testing
6.1
Introduction
The accuracy of the optimisation framework is primarily related to the structural analysis module and the genetic algorithm module. The analysis method of a stiffened composite plate is validated with a displacement method on the steel grillage and the results of equivalent elastic properties of symmetric laminate. For the unstiffened case, HSDT is implemented and validated with the results of Reddy (1997). The GA based optimisation procedure is tested for its convergence with different starting points and different oper- ators. Finally, the ability of the framework is demonstrated by comparing with ANSYS optimisation.
6.2
Grillage analysis
As the adapted grillage analysis is a combination of an analysis of steel grillage with equivalent elastic properties, each part is individually validated.
(a) Steel grillage:
The following steel grillage examples are used for validation and study purposes.
pressure of 137.900 kP a.
• 4×5 grillage: the grillage measures 6096 mm × 2540 mm and is acted on by an uniform pressure of 34.475 kP a.
Each example from Clarkson (1965) has been tested by two cases (I-beams and box- beams). The dimension of the longitudinal I-beam is 254 mm deep by 127 mm wide with 18.288 mm thick flange and 9.144 mm thick web. This gives a second moment of area I = 72465891.2 mm4. The dimension of the transverse I-beam is 69.85 mm deep
by 44.45 mm wide with 9.525 mm thick flanges, 5.08 mm thick web and this gives a second moment of areaI = 832462.85mm4. The dimension of the longitudinal box-beam
is 254.0 mm deep 127.0 mm wide with 18.288 mm thick flanges and 9.144 mm thick webs. This gives a second moment of area I = 80332665.14mm4. Transverse box-beams
69.85mmdeep by 44.45mmwide with 9.525 mmthick flanges and 5.08mmthick webs. This gives a second moment of area I = 886572.94mm4. We use a Young’s modulus for
steel of 206.87GP a.
To analyse the steel grillage, the following methods are implemented: the Force Method (FM) shown in Eq.3.13 presented by Jang et al. (1996), the Othotropic Plate Method
(OPM) shown in Eq.3.18 presented by Timosheko (1959) and the Energy Method (EM), shown in Chapter 5 which is developed by the author for composite materials has been adjusted in this example to account for steel.
From Table 6.1, the following points can be made:
• The authors Force Method program provides exactly the same solution as Clarkson. This confirms that all the program coding is correct.
• The Energy Method (EM) and Orthotropic Plate Method (OPM) solutions provide higher values than the exact solutions of the Force method (FM) but they have a
Table 6.1: Comparison between the results of the developed programs from energy method based Navier solution (EM), Orthrotropic Plate Method (OPM), Force Method (FM) and the results of Clarkson (1965) for the maximum deflectionδmax(mm) and maximum stress
of girderσg
max (MP a) and beam σbmax (MP a).
Grillage Beam Solution Clarkson Present
type (1965) FM OPM EM I δmax(mm) 10.95 10.95 11.01 11.01 4×4 σmax(MP a) 183.27 183.27 - 189.76 box δmax(mm) 9.63 9.63 9.93 9.93 σmax(MP a) 165.52 165.52 - 171.19 δmax(mm) 20.41 20.41 21.05 21.05 I σg max(MP a) 137.88 137.88 - 142.79 4×5 σb max(MP a) 205.35 205.35 - 206.82 δmax(mm) 18.34 18.34 19.10 19.10 box σg max(MP a) 125.37 125.37 - 129.59 σb max(MP a) 184.66 184.66 - 186.87
much lower computational time. It is not easy to obtain the stress solution from the Othotropic Plate Method (OPM).
(b) Equivalent elastic properties:
In the case of the unidirectional stiffened plate, the base plate element is under mem- brane mode in the x-direction and under bending mode in y-direction if stiffeners lay along x-direction. Therefore, this section shows the validation of the developed program for the membrane equivalent Young’s modulus in x-direction Em
x which can be evaluated
from Eq.5.1 and the bending equivalent Young’s modulus in y-direction (Eb
y) which can
be evaluated by the following equation.
Eyb =
12(D11D22−D122 )
t3D 11
(6.1)
The bending stiffness [D] is expressed as,
Dij = n X k (tkz¯2k+ t3 k 12)( ¯Qij)k (6.2)
in laminate and Qij is the transformed stiffness which is presented in Eq.5.4, Eq.5.5 and
Eq.5.5.
The equivalent Young’s modulus is evaluated from five laminates: [0/0/0/0], [0/90/90/0], [90/0/0/90], [45/-45/-45/45] and [0/45/-45/90]s. The lamina properties areE1 = 140GP a,
E2 = 10 GP a, G12 = 5 GP a and v12 = 0.3. Ply thickness (tk)=0.125 mm, all equal
through laminate.
Table 6.2: Comparison between the results of the developed program of equivalent elastic properties and those of Datoo (1991)
Laminate Equivalent Datoo (1991) Present elastic constants GP a GP a [0/0/0/0] Em x (membrane mode) 140 140 Eb y (bending mode) 10 10 [0/90/90/0] Em x (membrane mode) 75.5 75.36 Eb y (bending mode) 26.4 26.35 [90/0/0/90] Em x (membrane mode) 75.5 75.36 Eyb (bending mode) 124.7 124.21 [45/-45/-45/45] Em x (membrane mode) 17.7 17.74 [0/45/-45/90]s Em x (membrane mode) 54.1 54.07
From Table 6.2, it can be concluded that
• The author’s results are almost identical to Datoo’s results to the one decimal place precision presented by Datoo. The author is therefore confident with the validation. • The highest Em
x is obtained by laying all the fibres along the x-axis (zero fibre an-
gle). The lower the number of zero angle lamina, the lower the magnitude ofEm x .
(c) Shear stress calculation:
height is 50 mm. Crown and base plate width is 200 mm. The Young’s modulus of the crown and base plate elements is 54.1 GP a. The Young’s modulus of web (Ew) is 17.7
GP a. The section is subjected to shear forceQ= 10kN. The thickness of the crown and base plate elements is 1.0 mm. The thickness of the web is 0.5mm. SF1 and τ1 are shear
flow and shear stress at the corner of the crown element respectively. SF2 and τ2 are the
shear flow and the shear stress at the neutral axis (N.A.) of the cross section respectively.
Table 6.3: Comparison of the developed program shear stress calculation with Datoo (1991) Datoo (1991) Present SF1 (N/mm) 99 98.7226 τ1 (N/mm2) - 98.7226 SF2 (N/mm) 101 100.7413 τ2 (N/mm2) - 102.7600
From Table 6.3, it can be noticed that the developed program agrees well with the results by Datoo who ignored the accuracy after the decimal.