# Optimal Fuzzy and Dynamics Design of Ecological Sandwich Panel Vessel Roofs

Authors: Heikki Martikka; Erkki Taitokari
Journal: Advances in Fuzzy Systems
 (2011)

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## Abstract

In this study the basic engineering principles, goals, and constraints are all combined with fuzzy methodology and applied to optimally design sandwich panel circular plate roofs for large vessels loaded statically and dynamically. These panels are made up of two stiff, strong veneer skins separated by vertical and peripheral stiffener plates. Advantages are high strength, lightweight, and sustainability. In the present approach, first the goals and constraints of the end user are identified and expressed as decision variables which are formulated using the engineering variables for materials, geometry, and function. Then same consistent fuzzy satisfaction functions are formed over the desired ranges to suit the customer's desires. The risk of extreme dynamic loadings exciting resonance is studied by natural frequency and mode analysis by FEM and analytical models. The results show the most critical locations and give guidelines for innovative remedies of the concept before detailed FEM analyses to finalize the design.


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## Body

## 1. Introduction

Sandwich panels with stiff light cores offer many advantages compared to solid products such as high bending and buckling load-bearing capacities and weight saving without sacrificing reliability. Especially in large industrial roofs, these properties promise benefits provided the structures are optimally designed also.This case study is based on an industrial design task of a very large circular plate veneer sandwich roof.The conventional analytical optimisation methods are cumbersome and not suitable for engineering design work as discussed by Gibson and Ashby [1] and Ashby [2] from the point of view of optimal material selection. The present approach is based on the fuzzy goal aggregation ideas in design optimisation presented by Diaz [3]. In this study the same fuzzy formulation is used consistently to describe the goals and constraints of the end user.In engineering optimisation at concept stage, most tasks are highly nonlinear, and fuzzy and design variables are few and discrete.  This approach is used by Martikka and Pöllänen [4], who have applied multiobjective optimisation combined with technical laws and heuristics.  Martikka [5]  has applied fuzzy optimum methodology to the design of industrial composite structures with the goal of maximising the customer satisfaction. Taitokari and Martikka [6]  have studied the dynamics design of vessels of fibre-reinforced plastic with steel shafts for fluid mixing using microstructural design.  The design of wood structures is based on the micro- and macrostructural modelling of wood which is discussed   in [1] and on the basic theory of analysis and performance of fibre composites by Agarwal and Broutman [7]. The global megatrends of our technological civilisation studied by Marchetti [8] are important guidelines for doing ecodesign engineering work and choosing sustainable materials like wood. Hassan [9] discusses value-focused thinking on the environmental selection of wall structures.  Ecological wood surface protection methods developed recently [10] promote   optimal utilisation of wood in very large industrial roofs and also large buildings and bridges. In this task basic engineering as presented by Blake [11] is needed. For solving complex structural engineering tasks, the FEM is an indispensable tool [12]. Using it may overcome the know-how barriers of purely analytical structural dynamics theory as discussed by Soedel [13].One future goal is to utilise biomimicry to copy the ingenious optimal designs in nature into technological products.


## 2. Basic Mechanics of Sandwich Panel Roofs

Basic mechanics is needed to understand and to utilise the sandwich design. Basic concepts for sandwich roofs are   illustrated in Figures1–3.Figure 1
Stress definitions. Material removal from the web can be activated at a defined areaV.Figure 2
Roof models for cylindrical vessels. (a) Model 1 with no centre support. (b) Model 2 with support. (c) Cone and sphere forms. (d) Stresses on face.Figure 3
Geometry of a roof sector. Number of stiffeners isnw=3.Large circular roofs are commonly made as flat circular plates, cones, spherical, or some curved form as shown in Figure2.Geometry definitions of a roof sector are shown in Figure3.


## 3. Design Variables

Design variables are classified into three main types.


### 3.1. Functional Design Variables

Among these are loads, supports, reliability, and factors of safety. Load is external static pressurep=0.01 MPa on the top of the roof. It can be changed to design variable.  Self-weight pressure due to veneer is about pself=3·ρveneer·tg=0.001MPa=0.1p. Now it is considered as included in the total load caused by snow loads.


### 3.2. Geometric Variables

Outer radius of roof isR0=a=10 m. Radius from centre is r, and the relative radius ρ=r/R0. Design variable options are shown in Table 1.Table 1
Design variable options.
Variable12345678910d (id), height  (m).2000.220.240.260.30.340.360.400.450.50t (itt), face  (m)0.010.0150.02.0250.03nw (iw), radial webs1234—ρ(irho)=r/R00.10.50.9—


### 3.3. Material Variables

Material variables for wood and for veneer are shown in Table2  to illustrate its variability of strength properties. Now only veneer is used in the roof as main load-bearing material. Allowable design value strengths are used as parts of the factors of safety of their decision variables.Table 2
Material selection data for optimisation.
E,  MPaR, MPaIm=2,EIm=2,RIm=5,EIm=5,RIm=3,EwoodIm=3,RwoodVeneer,EVeneer,R, ifa=1, faceELσLU350008003500801000090 t, 43 c500050ETσTU80001003500805004500060EzσzU80080 c,12 t3500808006500020GLTτLTU2500251300306507200020GTzτTzU250025130030507200020GzLτzLU2500251300306507200020HereE is elastic modulus and R is strength property. Material codes are Im=2  for FRP  beam, Im=5 for FRP glue, and Im=3 for wood. Main material axes:  L: Longitudinal, T: Transverse, z: normal to plate. c: compression value, t: tensile value.


## 3.1. Functional Design Variables

Among these are loads, supports, reliability, and factors of safety. Load is external static pressurep=0.01 MPa on the top of the roof. It can be changed to design variable.  Self-weight pressure due to veneer is about pself=3·ρveneer·tg=0.001MPa=0.1p. Now it is considered as included in the total load caused by snow loads.


## 3.2. Geometric Variables

Outer radius of roof isR0=a=10 m. Radius from centre is r, and the relative radius ρ=r/R0. Design variable options are shown in Table 1.Table 1
Design variable options.
Variable12345678910d (id), height  (m).2000.220.240.260.30.340.360.400.450.50t (itt), face  (m)0.010.0150.02.0250.03nw (iw), radial webs1234—ρ(irho)=r/R00.10.50.9—


## 3.3. Material Variables

Material variables for wood and for veneer are shown in Table2  to illustrate its variability of strength properties. Now only veneer is used in the roof as main load-bearing material. Allowable design value strengths are used as parts of the factors of safety of their decision variables.Table 2
Material selection data for optimisation.
E,  MPaR, MPaIm=2,EIm=2,RIm=5,EIm=5,RIm=3,EwoodIm=3,RwoodVeneer,EVeneer,R, ifa=1, faceELσLU350008003500801000090 t, 43 c500050ETσTU80001003500805004500060EzσzU80080 c,12 t3500808006500020GLTτLTU2500251300306507200020GTzτTzU250025130030507200020GzLτzLU2500251300306507200020HereE is elastic modulus and R is strength property. Material codes are Im=2  for FRP  beam, Im=5 for FRP glue, and Im=3 for wood. Main material axes:  L: Longitudinal, T: Transverse, z: normal to plate. c: compression value, t: tensile value.


## 4. Stresses and Deflections

### 4.1. Roof with No Centre Support

The dominant stress state in the roof sandwich is the face normal stress. Bending stresses at two heights from neutral plane below and on top of the flange are(1)σ1=MDz1E1,σ2=MDz2E2.
Here  M=Mxb (Nm/m) (m) = (Nm)  is true moment  and  Mx  (Nm/m) is line moment,  b is width of the plate, E1 and E2 are Young’s moduli,  and D is bending stiffness. For a material layer k, one obtains   (2)σk=Ekε(z)=Ek(ε0+βz).
Thus            (3)σk=NΣEA/Ek+MzΣEI/Ek.
Here ε is strain at distance z from neutral plane, N is normal force, and A is cross-sectional area. Bending stress and bending stiffness are(4)σb=EkMzΣEI=EkMzD,D=ΣEkIk=D′bα,Ik=∫z2dAk.
Deflection of plate at radius r from centre is(5)w(r)=pr0464D′(1+ν)[2(3+ν)C1-(1+ν)C0],C0=1-ρ4,C1=1-ρ2,ρ=rR0.
The stiffness is due to face sheets and the grid of web plates in radial and peripheral directions. It is assumed that the plate is equally stiffened(6)D=EI=D′bα,bα=rα=ρR0α,EI=(EI)faces+(EI)webs.
This may be expanded as (7)EI=(Ef1+Ef2)bαt(12d)2+nwtwd312.
Line moments and shear are (8)mr=-D′(w,rr+νrw,r)=pr0216D′(3+ν)C1,mt=-D′(νw,rr+1rw,r)=pr0216D′[2(1-ν)+(1+3ν)C1],qr=-pr02ρ,D′=Dbα.
Some abbreviations are advantageous in moments(9)mr=pr0216D′(3+ν)C1=GR,R=(3+ν)C1,G=pr0264D′,mt=pr0216D′T=GT,T=[2(1-ν)+(1+3ν)C1].
The stresses are (10)σr=mrtd=GRtd,σθ=mθtd=GTtd.
Shear force is (11)qr=-pr02ρ⟹Qrtot=qrbα.
Shear stress is maximal at midweb with stress concentration factor Kτ(12)τ=KτQrtottwd⟹Q[N]=Kτq[Nm]bα[m]⟹τ=Kτqbαbαd=Kτqrd,τ=qrdKτ=-12pr0dρKτ,Kτ=1.5.
For orthotropic plates the failure criterion   of Tsai Hill is applicable [9](13)(σLσLU)2-(σLσLU)(σTσTU)+(σTσTU)2+(τLTτLTU)2≤1,
Glue joint is used between face sheet and webs and its failure criterion is (14)(τLTτLTUglue)2≤1,((1/2)p(r0/d)KττLTUglue)2≤1.


### 4.2. Roof with Centre Support

Now the boundary conditions for the roof are(15)w(r=0)=0,w(r=R0)=0,w(r)=Gr2(ρ2-1).
The moments are calculated with (8) and stresses with (10).


## 4.1. Roof with No Centre Support

The dominant stress state in the roof sandwich is the face normal stress. Bending stresses at two heights from neutral plane below and on top of the flange are(1)σ1=MDz1E1,σ2=MDz2E2.
Here  M=Mxb (Nm/m) (m) = (Nm)  is true moment  and  Mx  (Nm/m) is line moment,  b is width of the plate, E1 and E2 are Young’s moduli,  and D is bending stiffness. For a material layer k, one obtains   (2)σk=Ekε(z)=Ek(ε0+βz).
Thus            (3)σk=NΣEA/Ek+MzΣEI/Ek.
Here ε is strain at distance z from neutral plane, N is normal force, and A is cross-sectional area. Bending stress and bending stiffness are(4)σb=EkMzΣEI=EkMzD,D=ΣEkIk=D′bα,Ik=∫z2dAk.
Deflection of plate at radius r from centre is(5)w(r)=pr0464D′(1+ν)[2(3+ν)C1-(1+ν)C0],C0=1-ρ4,C1=1-ρ2,ρ=rR0.
The stiffness is due to face sheets and the grid of web plates in radial and peripheral directions. It is assumed that the plate is equally stiffened(6)D=EI=D′bα,bα=rα=ρR0α,EI=(EI)faces+(EI)webs.
This may be expanded as (7)EI=(Ef1+Ef2)bαt(12d)2+nwtwd312.
Line moments and shear are (8)mr=-D′(w,rr+νrw,r)=pr0216D′(3+ν)C1,mt=-D′(νw,rr+1rw,r)=pr0216D′[2(1-ν)+(1+3ν)C1],qr=-pr02ρ,D′=Dbα.
Some abbreviations are advantageous in moments(9)mr=pr0216D′(3+ν)C1=GR,R=(3+ν)C1,G=pr0264D′,mt=pr0216D′T=GT,T=[2(1-ν)+(1+3ν)C1].
The stresses are (10)σr=mrtd=GRtd,σθ=mθtd=GTtd.
Shear force is (11)qr=-pr02ρ⟹Qrtot=qrbα.
Shear stress is maximal at midweb with stress concentration factor Kτ(12)τ=KτQrtottwd⟹Q[N]=Kτq[Nm]bα[m]⟹τ=Kτqbαbαd=Kτqrd,τ=qrdKτ=-12pr0dρKτ,Kτ=1.5.
For orthotropic plates the failure criterion   of Tsai Hill is applicable [9](13)(σLσLU)2-(σLσLU)(σTσTU)+(σTσTU)2+(τLTτLTU)2≤1,
Glue joint is used between face sheet and webs and its failure criterion is (14)(τLTτLTUglue)2≤1,((1/2)p(r0/d)KττLTUglue)2≤1.


## 4.2. Roof with Centre Support

Now the boundary conditions for the roof are(15)w(r=0)=0,w(r=R0)=0,w(r)=Gr2(ρ2-1).
The moments are calculated with (8) and stresses with (10).


## 5. Dynamical Modelling of Roof with Centre Support

### 5.1. Basic Theory

Calculation of the eigenfrequencies is a difficult mathematical task. Thus some approximate modelling is useful to get fast rough answers. A sketch of a mode shape is illustrated in Figure4.Figure 4
Sketch of an assumed lowest mode shape.According to Soedel [13],  the equation of motion of a circular plate can be solved by using the method of separation of variables (16)w=W⋅R(r)Θ(θ)T(t).
Here t  is time, Θ is angle, r is radius from centre, and  W is the amplitude factor. The radial function is given by the Bessel functions. A trial solution is (17)R(r)=AJ1(λr)+BY1(λr)⟹Θ(θ)=C1cos⁡αθ+C2sin⁡αθ,T(t)=sin⁡ωt,w=W⋅RΘT.
The equation for the radial function is(18)R,rr+1rR,r+(±λ2-α2r2)R=0,
where(19)λ2=ωρhD.


### 5.2. Simple Model for Lowest Mode

First a very simple function with two parameters is used to get reasonable estimates(20)R(r)=rpsin⁡br.
One parameter is solved by substituting this to boundary conditions  (21)w(0)=W⋅R(0)ΘT=0,a=R0,w(a)=W⋅R(a)ΘT=0⟹sin⁡ba=0,ba=0,π,2π,nπ.
The angle function is made to fit to the assumed shape(22)Θ(0)=C1cos⁡α0+C2sin⁡α0=1⟶C1=1,Θ(12π)=C1cos⁡α12π+C2sin⁡α12π=0,C2=-cos⁡α(1/2)πsin⁡α(1/2)π,b=nπa.
Thus a simple angle model is (23)Θ(θ)=cos⁡αθ-sin⁡αθtan(π/2)α,α=32.
The parameters of the simple trial function for describing the first mode can be determined by obtaining first the necessary derivatives   (24)R(r)=rpsin⁡br,R,r=prp-1sin⁡br+rpbcos⁡br,R,rr=p(p-1)rp-2sin⁡br+2rp-1pbcos⁡br-rpb2cos⁡br.
Substitution of these into (18) gives  (25)p(p-1)rp-2sin⁡br+2rp-1pbcos⁡br-rpb2sin⁡br+prp-2sin⁡br+rp-1bcos⁡br+(±λ2-α2r2)rpsin⁡br=0.
From this, one obtains(26)rp-2(p(p-1)-α2)sin⁡br+rp-1(2p+1)bcos⁡br-rp(b2±λ2)sin⁡br=0⟹p=-12,α2=p(p-1)=34,b2=±λ2.
The present simple model gives at lowest mode n=1(27)(λa)simple≈3.14,λ2a2=ωa2ρhD′=±b2a2=n2π2.
The bending stiffness per unit length of the plate is(28)D′=1bαD=1bα12Ebαth2=12Eth2.
Density of sandwich plate is(29)ρ=mV=ρw2bαLtbαLh=ρw2th.


### 5.3. More Accurate Model

Now the models by Soedel [13] on the inplane vibrations of circular plates are applied. The solutions are given by the Bessel functions(30)R=WJ1(λr).
This function satisfies the boundary conditions(31)J1(λa)=0,d2dr2J1(λr)|r=a=0,J1(λ0)=0,d2dr2J1(λr)|r=0=0.
The eigenfrequencies corresponding to the roots of the Bessel function are numbered m=0,1,2,3.The physical meaning is thatm represents the number of interior node circles. Now the centre column support constraints to the modes in such a way that the second root probably gives the lowest mode. The lowest roots are (λa)m=2.403,3.832,5.135,6.379 [13]. The Eigen frequency corresponding to m is(32)ωm=((λa)ma)2D′ρh.
Substitution gives (33)ωm=((λa)ma)2(1/2)Eth2ρw(2t/h)h.
Thus(34)ωm=((λa)ma)2h2Eρw.
The frequency can be calculated by this using the data for the roof(35)h=0.3,E=5000MPa,ρw=500kg/m3,a=10m,(λa)2=3.832,⟹f=11Hz.
The simple model gave (λa)simple=3.14  and fsimple=7 Hz. This is less than the theoretical frequency:  ftheor=11. It is close to the accurate FEM values which are within the range:    fFEM=10–13 Hz.


## 5.1. Basic Theory

Calculation of the eigenfrequencies is a difficult mathematical task. Thus some approximate modelling is useful to get fast rough answers. A sketch of a mode shape is illustrated in Figure4.Figure 4
Sketch of an assumed lowest mode shape.According to Soedel [13],  the equation of motion of a circular plate can be solved by using the method of separation of variables (16)w=W⋅R(r)Θ(θ)T(t).
Here t  is time, Θ is angle, r is radius from centre, and  W is the amplitude factor. The radial function is given by the Bessel functions. A trial solution is (17)R(r)=AJ1(λr)+BY1(λr)⟹Θ(θ)=C1cos⁡αθ+C2sin⁡αθ,T(t)=sin⁡ωt,w=W⋅RΘT.
The equation for the radial function is(18)R,rr+1rR,r+(±λ2-α2r2)R=0,
where(19)λ2=ωρhD.


## 5.2. Simple Model for Lowest Mode

First a very simple function with two parameters is used to get reasonable estimates(20)R(r)=rpsin⁡br.
One parameter is solved by substituting this to boundary conditions  (21)w(0)=W⋅R(0)ΘT=0,a=R0,w(a)=W⋅R(a)ΘT=0⟹sin⁡ba=0,ba=0,π,2π,nπ.
The angle function is made to fit to the assumed shape(22)Θ(0)=C1cos⁡α0+C2sin⁡α0=1⟶C1=1,Θ(12π)=C1cos⁡α12π+C2sin⁡α12π=0,C2=-cos⁡α(1/2)πsin⁡α(1/2)π,b=nπa.
Thus a simple angle model is (23)Θ(θ)=cos⁡αθ-sin⁡αθtan(π/2)α,α=32.
The parameters of the simple trial function for describing the first mode can be determined by obtaining first the necessary derivatives   (24)R(r)=rpsin⁡br,R,r=prp-1sin⁡br+rpbcos⁡br,R,rr=p(p-1)rp-2sin⁡br+2rp-1pbcos⁡br-rpb2cos⁡br.
Substitution of these into (18) gives  (25)p(p-1)rp-2sin⁡br+2rp-1pbcos⁡br-rpb2sin⁡br+prp-2sin⁡br+rp-1bcos⁡br+(±λ2-α2r2)rpsin⁡br=0.
From this, one obtains(26)rp-2(p(p-1)-α2)sin⁡br+rp-1(2p+1)bcos⁡br-rp(b2±λ2)sin⁡br=0⟹p=-12,α2=p(p-1)=34,b2=±λ2.
The present simple model gives at lowest mode n=1(27)(λa)simple≈3.14,λ2a2=ωa2ρhD′=±b2a2=n2π2.
The bending stiffness per unit length of the plate is(28)D′=1bαD=1bα12Ebαth2=12Eth2.
Density of sandwich plate is(29)ρ=mV=ρw2bαLtbαLh=ρw2th.


## 5.3. More Accurate Model

Now the models by Soedel [13] on the inplane vibrations of circular plates are applied. The solutions are given by the Bessel functions(30)R=WJ1(λr).
This function satisfies the boundary conditions(31)J1(λa)=0,d2dr2J1(λr)|r=a=0,J1(λ0)=0,d2dr2J1(λr)|r=0=0.
The eigenfrequencies corresponding to the roots of the Bessel function are numbered m=0,1,2,3.The physical meaning is thatm represents the number of interior node circles. Now the centre column support constraints to the modes in such a way that the second root probably gives the lowest mode. The lowest roots are (λa)m=2.403,3.832,5.135,6.379 [13]. The Eigen frequency corresponding to m is(32)ωm=((λa)ma)2D′ρh.
Substitution gives (33)ωm=((λa)ma)2(1/2)Eth2ρw(2t/h)h.
Thus(34)ωm=((λa)ma)2h2Eρw.
The frequency can be calculated by this using the data for the roof(35)h=0.3,E=5000MPa,ρw=500kg/m3,a=10m,(λa)2=3.832,⟹f=11Hz.
The simple model gave (λa)simple=3.14  and fsimple=7 Hz. This is less than the theoretical frequency:  ftheor=11. It is close to the accurate FEM values which are within the range:    fFEM=10–13 Hz.


## 6. Goal and Constraint Formulation

In the concept stage the essential design variables are few, discrete, and their relationships are highly nonlinear. A fast enough search method is exhaustive learning search. Now all goals and constraints are formulated consistently by one flexible fuzzy function. This is illustrated in Table3 and Figure 5.Table 3
Skewness   parameter  values.
abcdep10.10.1155p250.1150.1x1max0.020.50.50.50.98Figure 5
Principle of modelling of the general satisfaction functions.In the design algorithm the satisfaction function is defined for each decision variables by inputting the left and right limits and two bias parameters p. The left skewed option a is useful to get low-cost designs. Flat shape allows indifferent choice of s. The location of maximum can be shifted. The call is CALL pzz(smin, smax, p1, p2, s, P(s)). The output is the satisfaction function P(s). The total event s is intersection of separate events(36)s=s1∩s2∩s3∩s4∩s5∩s6.
Satisfaction on this event s is(37)P(s)⟹P(s)=P(s1)⋅P(s2)⋅⋯⋅P(sn).
The design goal is to maximise this product.All design variables are now discrete. A concept is formulated by making trial selections from the lists. Then decision variable and their satisfaction are calculated. Total satisfaction is calculated for each concept alternative. The one with maximal satisfaction is optimal.


## 7. Decision Variables

The desired range for decision variablesk is R(sk)=skmin<sk<skmax  and satisfaction on it is  P(sk).Decision variables1 is face sheet failure risk.  Face sheet failure by biaxial stressing is predicted by the Tsai Hill failure criterion in:  (38)FCf=(σLσLU)2-(σLσLU)(σTσTU)+(σTσTU)2+(τLTτLTU)2≤1.
The desired range for the decision variable is from small to 0.7. The bias favours small failure risk(39)R(s1)=s1,min⁡=10-5<s1<s1,max⁡=0.7,s1=FCf,Ps(1)=P(s1).Decision variables2  is maximum deflection  wmax  at midplate. This was obtained for first setting r=0 and using the optimal dimensions for d and t(40)ρ=0,C0=1-ρ4=1,C1=1-ρ2=1,wmax⁡=w(ρ=0)=pr0464D′(1+ν)[2(3+ν)-(1+ν)].
The desired range for the decision variable is from small to high, 0.7. This high-limit option is used to explore extreme effects due to high loads. But the biased satisfaction function favours desired moderate deflections and excluds unacceptable large values(41)s2=wmax⁡,Ps(2)=P(s2).Decision variables3 is web shear factor of safety. Web shear occurs on nw plates of thickness tw and height d  of the total plate. The line shear qr acts on each sector at radius ρ. Sector angle is α=12 deg. Shear stress concentration factor is 1.5(42)α=2π30,bα=rα=ρR0α⟹Qrtot=qrbα.
The desired range for the decision variable is from small to 0.2. The bias favours small value of s3(43)s3=τwebτLTuweb,τLTuweb=1.5Qrtottwd1nw,tw=t,Ps(3)=P(s3).
Here τLTuweb is the shear strength of the web material, and its thickness is set same as the face sheet thickness t.Decision variables4 is related to the risk of local sheet buckling of the compressed plate in radial direction.A square plate surrogate model is used to model the buckling risk for the roof plates. A conservative model for the inelastic buckling strength is given by Blak [11]. Now it is assumed that it is freely supported at the stiffened edges. The support factor depends on the edge support stiffness condition. The smallest value is chosen.Kp=3.62 for simple-simple edges,Kp=4.9 for simple-fixed edges,Kp=6.3 for fixed-fixed edges.The inelastic reduction factor for these edge support types is aboutη=0.7. Width is b and thickness  is  t(44)b=bα1nw,Kp=3.62,η=(EtE)1/2≈0.7,Scr=KpηEf(tb)2.
Elastic modulus definitions are shown in Figure 6.Figure 6
Material model and geometrical definitions.The desired range for the decision variable is from small to 0.3. The bias favours small value ofs4(45)s4=σrScr,Ps(4)=P(s4).Decision variables5 is related to the local sheet buckling of the compressed plate in tangential direction.The desired range for the decision variable is from small to 0.3. The bias favours small value of s3(46)s5=σtScr,Ps(5)=P(s5)Decision variables6 is the cost  (47)Cost=cf(mf+mw),mf=2ρf12R02αt,mw=nwρftR0d.
Satisfaction on it is (48)Ps(6)=P(s6).Decision variablesf is the lowest Eigen frequency   (49)s7=f1.
The desired range is tentatively f1=5–25 Hz. The frequency for the theoretical model with thickness h=0.3 m gave f1=11 Hz. The statically optimal roof with no centre support had larger thickness h=0.5 giving higher frequency  f1′=(0,5/0.3)·11=18. Thus full satisfaction is obtained readily giving (50)Ps(7)=P(s7)≈1.


## 8. Results of Optimisation

Results are shown in Figure8 and Tables 4 and 5.Table 4
Results of optimisation. The properties at each relative radius apply to the whole plate. Model 1 with no centre supports.
(a)Relative radiusρ=r/R0ρ=0.1d (mm), t (mm), nw500, 30, 1wmax,middle (mm), fFEM243Pg,total satisfaction0.13Ps(1) satisfaction on s1=FCf0.93, 0.096Ps(2) satisfaction on s2=wmax0.57, 0.243Ps(3) satisfaction on s3=τweb/τLTuweb0.76, 0.0052Ps(4) satisfaction on s4=σr/Scr, Scr.86, 0.052, 260Ps(5) satisfaction on s5=σt/Scr, N5=1/s50.97, 0.072, 14Ps(6) satisfaction on s6=cost0.38,     14000σTuf,     σr,   N50,   13.6,      3.7σTuf,      σt,  N60, 19,         3.15τweb,   τLTuweb0.1, 20(b)ρ=r/R0ρ=0.5ρ=0.9d (mm),  t (mm), nw500,        30,   3500,      30,    3wmax,middle (mm), fFEM274300Pg,  satisfaction0.0530.056Ps(1),    s1=FCf0.927,    0.06.97,      0.063Ps(2),     s2=wmax0.5,    0.2740.45,  0.3Ps(3),     s30.93,     0.0441,  0.14Ps(4),     s40.54, 0.110.65,   0.09Ps(5),    s5, N5=1/s50.99,    0.18, 5.5.91,  0.43, 2.3Ps(6),     s6=cost0.21,  194100.21,   19410σLuf, σr,  N50,    10,  550,     2.6,  19σTuf, σt, N60,    17,  3.560,     12,  5τweb, τLTuweb1.04, 202.8,     20Table 5
Results of optimisation. The properties at each relative radius apply to the whole plate. Model 2 with centre support.
Relative radiusρ=r/R0ρ=R0/1.4=rcritd (mm),  t (mm), nw500,         10,       3wmax,middle (mm), fFEM30Pg,    total satisfaction0.49Ps(1)  satisfaction on s1=FCf0.985,   0.054Ps(2)  satisfaction on s2=wmax0.9982,  0.03Ps(3)  satisfaction on s3=τweb/τLTuweb0.757,  0.0052Ps(4)  satisfaction on s4=σr/Scr, Scr0.838.   0.05, 260Ps(5) satisfaction on s5=σt/Scr, N5=1/s50.85,    0.015,    65Ps(6) satisfaction on s6=cost0.75,     6470σLuf,     σr,   N50,       13,     3.12σTuf,      σt,  N60,       4,         2.7τweb,   τLTuweb0.1, 20Two different models are considered in Figure7.Figure 7
(a) Roof with no centre support, model 1. (b) Roof with centre support, model 2.Figure 8
Total satisfactionPg, cost, and safety factor N5 versus relative radius. Model with no roof support.


### 8.1. Model with No Centre Supports

As shown in Figure8 the satisfaction is maximal at inner radius and decreases towards outer radius. Main reason is that density of web stiffening is larger at inner radii. This shows that more stiffeners at the outer radius would increase the total satisfaction on the roof.The cost is now defined as the cost of one sector with angleα=12 deg. The face thickness is variable. The stresses are obtained at each radius depending on the geometrical selections. The middeflection is constrained to a desired range.  The face and web wall were 0.03 m and the roof height was 0.5 m. It is assumed that cost is obtained with uniform thickness. From relative radius 0.5 to 1, the cost, satisfaction, and factor of safety are reasonably constant.


### 8.2. Model with a Centre Support

Results are shown in Table5.


### 8.3. Comparison of Models

Model 1 with no centre support required roof total heightd=0.5 m and face t=0.030 m resulting in maximum deflection of 0.3 m. The cost was 19 k units and satisfaction Pg=0.057. This deflection was unacceptably large. Therefore this concept was rejected, and only the roof with centre support is considered further in detail.Model 2 with centre support requiredd=0.5 and the face only t=0.010 m. Maximum deflection was zero at centre and about 0.007 m at half radius. This deflection divided by radius is 0.007/10=0.0007 and is acceptable. The satisfaction is Pg=0.49 and cost 6.5 k units. This shows that adding a center support is profitable if there are no harmful side effects.In the final design still some detailed changes were made and also a change to slightly conical form.


## 8.1. Model with No Centre Supports

As shown in Figure8 the satisfaction is maximal at inner radius and decreases towards outer radius. Main reason is that density of web stiffening is larger at inner radii. This shows that more stiffeners at the outer radius would increase the total satisfaction on the roof.The cost is now defined as the cost of one sector with angleα=12 deg. The face thickness is variable. The stresses are obtained at each radius depending on the geometrical selections. The middeflection is constrained to a desired range.  The face and web wall were 0.03 m and the roof height was 0.5 m. It is assumed that cost is obtained with uniform thickness. From relative radius 0.5 to 1, the cost, satisfaction, and factor of safety are reasonably constant.


## 8.2. Model with a Centre Support

Results are shown in Table5.


## 8.3. Comparison of Models

Model 1 with no centre support required roof total heightd=0.5 m and face t=0.030 m resulting in maximum deflection of 0.3 m. The cost was 19 k units and satisfaction Pg=0.057. This deflection was unacceptably large. Therefore this concept was rejected, and only the roof with centre support is considered further in detail.Model 2 with centre support requiredd=0.5 and the face only t=0.010 m. Maximum deflection was zero at centre and about 0.007 m at half radius. This deflection divided by radius is 0.007/10=0.0007 and is acceptable. The satisfaction is Pg=0.49 and cost 6.5 k units. This shows that adding a center support is profitable if there are no harmful side effects.In the final design still some detailed changes were made and also a change to slightly conical form.


## 9. Results of FEM Checking

### 9.1. Results of Static Calculations

Results are shown in Figures9, 10, and 11. Boundary conditions are shown in Figure 9. The degrees of freedom are maximally six (u1,u2,u3,θ1,θ2,θ3) along the axes shown. Here the notation of fixity is 1: fixed, 0: free;  A: (010000), B: (111000), C: (010101). The roof is freely supported without bending moments at circular opening, the centre with radius R0=0.5 m, and at the outer radius at the vessel cylinder. The radial sides are given symmetry plane boundary conditions.   Static pressure was p=0.01 MPa at upper plate. Total height d=0.33 m, t=0.03 m for face and webs, elastic modulus for veneer E=5000 MPa (Table 2), and Poisson’s ratio v=0.3. The lowest Eigenfrequencies were 10 by FEM and about 11 analytically. They are satisfactorily close to each other considering the large difference between the models.(a) Deformed geometry  and maximal principal stress range  +7.5–−13.9 MPa. Minimal principal stress  range 3.9–−33.4. Total deformation  is at  aboutR0/1.4 radius and is 72.4 mm with pressure p=0.01 MPa. (b) Internal web stiffeners and FEM boundary conditions.
(a)(b)Figure 10
The first buckling mode with factor of safety 8.24. The plates with largest unsupported areas with high compressive stresses buckle first.Figure 11
The first buckling mode for the whole plate model with factor of  safety 8.24.


### 9.2. Results of Dynamical Calculations

These are shown in Figures12, 13, 14, and 15.Figure 12
The lowest eigen mode corresponding to eigenfrequency of 9.74 Hz. Analytical  simple model gave roughlyf=11 Hz.Figure 13
The second  eigen mode corresponding to eigenfrequency of  10.48 Hz. More complex models are shown in Figures14 and 15.Figure 14
The second  eigen mode corresponding to eigenfrequency of  13.01 Hz.Figure 15
The third  eigen mode corresponding to eigenfrequency of  16.89 Hz.


## 9.1. Results of Static Calculations

Results are shown in Figures9, 10, and 11. Boundary conditions are shown in Figure 9. The degrees of freedom are maximally six (u1,u2,u3,θ1,θ2,θ3) along the axes shown. Here the notation of fixity is 1: fixed, 0: free;  A: (010000), B: (111000), C: (010101). The roof is freely supported without bending moments at circular opening, the centre with radius R0=0.5 m, and at the outer radius at the vessel cylinder. The radial sides are given symmetry plane boundary conditions.   Static pressure was p=0.01 MPa at upper plate. Total height d=0.33 m, t=0.03 m for face and webs, elastic modulus for veneer E=5000 MPa (Table 2), and Poisson’s ratio v=0.3. The lowest Eigenfrequencies were 10 by FEM and about 11 analytically. They are satisfactorily close to each other considering the large difference between the models.(a) Deformed geometry  and maximal principal stress range  +7.5–−13.9 MPa. Minimal principal stress  range 3.9–−33.4. Total deformation  is at  aboutR0/1.4 radius and is 72.4 mm with pressure p=0.01 MPa. (b) Internal web stiffeners and FEM boundary conditions.
(a)(b)Figure 10
The first buckling mode with factor of safety 8.24. The plates with largest unsupported areas with high compressive stresses buckle first.Figure 11
The first buckling mode for the whole plate model with factor of  safety 8.24.


## 9.2. Results of Dynamical Calculations

These are shown in Figures12, 13, 14, and 15.Figure 12
The lowest eigen mode corresponding to eigenfrequency of 9.74 Hz. Analytical  simple model gave roughlyf=11 Hz.Figure 13
The second  eigen mode corresponding to eigenfrequency of  10.48 Hz. More complex models are shown in Figures14 and 15.Figure 14
The second  eigen mode corresponding to eigenfrequency of  13.01 Hz.Figure 15
The third  eigen mode corresponding to eigenfrequency of  16.89 Hz.


## 10. Conclusion

We have combined in this approach basic mechanics with fuzzy goal formulations and heuristics to obtain optimised sandwich panel roof of veneer. In nature, analogous ingeniously designed and biomanufactured structures are already widely and successfully used by animals and plants.One evident advantage of the present method over the conventional ones is that more useful and reasonable optima can be found since all relevant desires like societal and ecological ones are considered simultaneously during the design. At this stage most of the total life cycle costs and sustainability are determined.Dynamic methods are important tools to design structures with desired static and dynamic stability under extreme wind or seismic loads.  The lowest frequencies by FEM were with rounding 10, 11, and 13 Hz. Analytical method gave the lowest as   11 Hz.  The utility of the simple models is that they can be used in exploratory optimum concept design. The results show that a reasonable trade-off type optimum between two different constructions is obtained.The future aim is to generalise this approach of   combining basic mechanics with new goal formulations and FEM fine-tuning.Using this design methodology designers of machines can better mimic the ingenious sustainable solutions found in nature and select the   right future megatrends of technology   in their design and development work.



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*Source: 879049-2011-10-05.xml*