1

Finite Element Formulation for Analysis of Curved Composite

Panels at Elevated Temperature

Satyajit Sudhir Ghoman 1

Virginia Polytechnic Institute and State University, Blacksburg, VA, 24061

This paper outlines the finite element formulation of curved panels at elevated temperature,

a sub-part of the work performed in light of panel flutter analysis. A finite element method is

developed and presented to predict the behavior of curved panels subjected thermal loading.

The nonlinear FE formulation presented herein is based on following building blocks: 1)

Von Karman large deflection theory, 2) Marguerre shallow shell theory, and 3) Quasi-static

thermoelasticity. Constitutive relations for curved panel are derived and absorbed in the

formulation. In-plane forces and moments due to thermal loading are calculated. The

principle of virtual work is applied to develop the equations of motion of the system in

structural degrees of freedom. Mathematical expressions for element mass matrices, linear

and nonlinear stiffness matrices, thermal stiffness matrices, external mechanical loads and

applied thermal in-plane and moment loads are given. The Newton-Raphson method is used

to determine the panel deflection under the Static Thermal Loading (STL). Solution

procedure for determining critical buckling temperature is also outlined. Deflection of the

curved system is thoroughly investigated for 3D curved panels under increasing uniform or

linearly varying temperature gradient loading. Critical buckling temperatures are found out

for 3D flat plates of same geometry except the curvature, and used to define non-

dimensionalized thermal loading. The results showed that the system behavior alters

significantly when temperature effects come into the picture.

Nomenclature

a = Panel length

A, [A] = Element area, extension laminate stiffness matrix

[As] = Shear laminate stiffness matrix

b = Panel width

[B] = Coupling laminate stiffness matrix

[C] = Strain-Displacement relation matrix

[D] = Bending laminate stiffness matrix

E = Young‟s modulus of elasticity

G = Shear modulus

{G} = Slope vector

h = Panel thickness

H = Panel maximum height-rise

[H] = Interpolation functions matrix

[k] , [K] = Element and system linear stiffness matrices

L,M = Interpolation functions

[m], [M] = Element and system mass matrices

{M} = Bending moment vector

n, N = Number of layers, Interpolation function

{N}, [N] = In-plane force vector and in-plane force matrix

[n1] , [N1] = Element and system 1st order nonlinear system stiffness matrices

[n2], [N2] = Element and system 2nd order nonlinear system stiffness matrices

1 Graduate Research Assistant, Dept. of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State

University, Blacksburg, VA.

2

{p} = External forces vector

{R}, R = Shear force vector, Radius of curvature of panel

t, T = Time, Temperature

T0 = Uniform temperature across the panel

T1 = Linear thermal gradient across the thickness of the panel

[T] = Coordinate transformation matrix

u, v = In-plane displacements

W = Work done

{w} , {W}= Element nodal displacements, System nodal displacements

wo = Initial geometry of curved panel

x, y, z = Cartesian coordinates, z coordinate of internal coordinate system

α = Coefficient of thermal expansion

{ε} = Total strain vector

{κ} = Curvature vector

ν = Poisson‟s ratio

ρ = Panel mass density

Subscripts

b = Bending and rotational component

B = Coupling bending/rotational and in-plane components

cr = Critical

ext = External

int = Internal

k = Lamina layer

L = Linear

m = In-plane component

max = Maximum

min = Minimum

o = Curvature, initial conditions

s = shear, static component

stl = Static thermal load

t = Time dependent, time derivative, Temperature dependent component

u,v = In-plane components

uo, vo = In-plane components due to curved panel geometry

w = Out of plane component

wo = Out of plane component due to curvature

x, y = x and y direction, derivative with respect to x and y

Superscript

b = Bending, rotational component

m = In-plane component

o = Curved geometry

s = Shear, static component

stl = Static thermal load

T = Transpose

= Slope matrix

o = Curved geometry

I. Introduction

1.1 Nature of the problem

The investigation of the structural response of composite curved (skin) panels subjected to a combination of

various loads in the elevated temperature environment is a necessity. This paper outlines the finite element

formulation of composite curved panels at elevated temperature, which is developed as a sub-part of the work

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performed in light of panel flutter analysis for a supersonic air vehicle. The most important external loads are: (1)

the self-excited unsteady aerodynamic load, (2) the aerodynamic pre-loading or Static Aerodynamic Load (SAL)

dependent on panel curvature and dynamic pressure λ and (3) Uniform or varying temperature gradient across the

thickness of skin panel. Out of all these various loadings, this paper only addresses the loads due to temperature

effects of a typical supersonic flight. Temperature plays an important role as when a flight vehicle flies at supersonic

speed, friction within the viscous boundary layer and that between the viscous boundary layer and the vehicle

surface produces a heat exchange between the two fluid-structure mediums. At high speed, this exchange results in a

dissipation of the heat (energy) within the structure itself, raising subsequently the temperature of the vehicle air-

structure contact surface. This temperature induces in-plane forces and bending moments within the structure. These

in-plane forces and bending moments may cause instability and complex behavior of the curved panel itself. One of

the most interesting and important consequence of these induced in-plane forces also called thermal stresses, is when,

they are sufficiently large in compression can cause

panel buckling. The finite element formulation

presented in this paper is characterized by the feature

that it handles simultaneous application of

aerodynamic and thermo-elastic loads to the panel.

(To limit the focus of this paper, all the terms

involving corresponding aerodynamic loading are

neglected here, only the thermal loading terms and

their subsequent effects are presented).

1.2 Objective, Approach and Methodology

Even though a thorough literature review was conducted, it is not presented herein for the brevity of this paper.

It was seen that there is a limited number of papers in the open literature which tackle the problem of non-linear

analysis of curved composite panels at elevated temperatures thoroughly. Hence the main objective of this paper is

to build a better understanding of static behavior of curved panels at elevated temperatures. To achieve this objective,

a finite element formulation is developed to determine the panel response. Various temperature environments, such

as uniform temperature across the panel, or linearly varying temperature across the thickness of the panel are

considered. Curved panel static deflection and stiffness due to STL is determined accurately with the use of Newton-

Raphson iterative method.

The present approach uses the von Karman non-linear strain-displacement relation, the Marguerre curved plate

theory, the first order shear deformation theory, and quasi-static thermoelasticity. System equations of motion in

structural node degrees-of-freedom (DOF) are first obtained for 3-D curved panels. Critical buckling temperatures

are found out for 3D flat plates of same geometry but no curvature, and used to define non-dimensionalized thermal

loading. It is to be noted here that though curved panels don‟t have a definite Tcr, corresponding flat plate Tcr is used

here for comparison and non-dimensionalization purposes only. Curved panels undergo uniform and continuous

buckling as opposed to flat panels, which undergo a sudden buckling past Tcr loading. The response of the system is

then thoroughly investigated for 3D curved panels under increasing uniform or linearly varying temperature gradient

loading (Fig. 2).

II. Finite Element Formulation

In this section, the governing equation of motion (EOM) for a three dimensional curved panel subjected to a

static thermal load will be developed. The system level EOM will be expressed in terms of structural degrees of

freedom (DOF). The assumptions on which the following formulation is based are listed below.

1) The panel is composed of either isotropic material or a composite material. The theory incorporates Hook‟s

law and composite laminate theory for these materials respectively.

2) The curved panel could be either thin or thick. To include the effect of thick panels, first order shear

deformation theory is incorporated in the formulation.

3) High temperatures result in highly non-linear behavior of the panel. Hence, to include the effect of large

deflection, von Karman large deflection theory [1] is employed.

4) Out of numerous curved plate theories, Marguerre curved plate theory [2] is used to model the curved panel

element.

Fig. 1 - Curved panel geometry characterized by the

height-rise H/h

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Fig. 2 – Details of Linearly Varying Thermal

Loading.

5) Thermal loading on the panel is assumed to be quasi-static, and any thermal sink effects of stiffeners and

frames is neglected. Thus, temperature distribution over the surface of the panel is assumed to be uniform.

2.1 The Description of the Problem

Fig. 1 shows a 3-D curved panel. Geometry of the panel is defined by length a, width b, and thickness h. As

stated earlier, a panel could be isotropic, orthotropic or laminated. The steady-state temperature field subjected to the

panel is actually a function of all the three Cartesian coordinates, i.e., ΔT(x,y,z). Though, here in this paper, for the

sake of simplicity, a linear temperature variation across the thickness of the panel is assumed, i.e;

h

TzTT 1

0 (2.1)

where, T0 = average temperature i.e., temperature at the mid-plane of panel

T1 = linear temperature variation through the panel thickness.

Two cases of thermal loading are considered,

1) Uniform Temperature across the thickness ( T1 = 0)

2) Linearly varying temperature distribution across the

thickness given by Eq. (2.1).

Temperature induced thermal in-plane stresses and

transverse bending moments cause weakening, stiffening,

or even buckling of the panel. Interactions between

external loads and thermal effects result in complex

motions of the panel. Hence, this formulation aims at

developing a computationally cost effective solution

procedure and applying it to investigate the behavior of

curved panels under said loads.

2.2 Governing EOM: Finite Elements

Since the isotropic and orthotropic and symmetrical laminates can be treated as a special case of general

laminates, the EOM for general laminates is formulated herein.

2.2.1 MIN3 Element Displacement Vectors

In order to develop the curved panel governing EOM, a distinctive three-nodes finite element (MIN3) is

employed to discretize the panel system into many finite triangular elements. The element nodal displacement

vector w is comprised of components for each of the three triangular nodes. Each node is characterized by three

displacements and two rotations (5 DOF). The DOF are composed of the bending displacements iw , the normal

rotations xi with respect to x and yi with respect to y axes, and the in-plane displacements ii vu , respectively.

The element displacement vector is defined as

Tw

= b mw w w (2.2)

wherein, the nodal displacement vectors are defined as

T

bw= 321 www

(2.3)

T

w=

1 2 3 1 2 3 x x x y y y (2.4)

T

mw = 1 2 3 1 2 3u u u v v v (2.5)

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2.2.2 MIN3 Element Displacement Vectors

The initial geometry of any arbitrary curved panel is described by the function ),(0 yxw Here, the curved

geometry is characterized by the ratio H/h, where )],(max[ 0 yxwH and h is the thickness of the curved panel.

The element displacement functions used while deriving the EOM are defined as

),,(),,( tyxztyxuu yx (2.6)

),,(),,( tyxztyxvu xy (2.7)

),,( tyxwuz (2.8)

where ux , uy and uz are the displacement components along the Cartesian axes at any point within the element,

u, v and w are the displacements of the panel mid-plane, and x , y are the rotations of the panel mid-plane

normals about the x and y axes. The local coordinate z is defined as ),(0 yxwzz . In a brief form, the

displacements at any arbitrary point in the element can be expressed as

wHwHtyxw wbw ),,( (2.9)

wHtyx xx ),,( (2.10)

wHtyx yy ),,( (2.11)

mu wHtyxu ),,( (2.12)

mv wHtyxv ),,( (2.13)

2.2.3 Strain-Displacement Relationships

Non-linear Total Strain Deformation Vector

With the assumption of small in-plane strains and moderately large transverse displacement, the total strain

deformation vector can be written as

kzo

xy

y

x

(2.14)

where, o is the in-plane strain vector and k is the curvature vector. The detailed expression for the total

strain can be given as

xxyy

yx

xy

xoyyox

yoy

xox

yx

y

x

xy

y

x

xy

y

x

z

wwww

ww

ww

ww

w

w

vu

v

u

,,

,

,

,,,,

,,

,,

,,

2

,

2

,

,,

,

,

22

1

= o

m + o

b + o

wo + z (2.15)

where

o

m = Linear membrane strain vector

o

b = von Karman non-linear membrane stretching strain vector

o

wo =Curvature strain vector.

Total Transverse Shear Strain Deformation Vector

The total transverse shear strain deformation vector is given by

6

y

x

x

y

xz

yz

w

w

,

, (2.16)

By substituting u, v, w, and x , y with their respective interpolation functions as given in Eqs. (2.9) to

(2.13), the strain components can be expressed in the nodal displacement vectors, as follows,

The in-plane membrane strain vector is

mmm

xvyu

yv

xu

xy

y

x

o

m wCw

HH

H

H

vu

v

u

,,

,

,

,,

,

,

(2.17)

The von Karman non-linear stretching strain vector is

Gw

w

ww

w

w

ww

w

w

y

x

xy

y

x

yx

y

x

o

b 2

10

0

2

1

22

1

,

,

,,

,

,

,,

,2

,2

(2.18)

where

= Slope Matrix or Derivative matrix of transverse displacements, and

G = Slope Vector or Derivative vector of transverse displacements, which is defined as

w

H

Hw

H

H

w

wG

yw

xw

b

yw

xw

y

x

,

,

,

,

,

,

= bb wC +

wC (2.19)

Thus, wCwC bb

o

b 2

1 (2.20)

The in-plane strain vector due to curved geometry is expressed as

wCwCGw

w

ww

w

w

wwww

ww

ww

bboo

y

x

xoyo

yo

xo

xoyyox

yoy

xox

o

wo

,

,

,,

,

,

,,,,

,,

,,

0

0

(2.21)

Similarly, the bending curvature k can be expressed as

wCw

HH

H

H

b

xxyy

yx

xy

xxyy

yx

xy

,,

,

,

,,

,

,

(2.22)

The total strain vectors can thus be expressed in reduced form as functions of nodal displacements as follows

wCwCwCwCwC bbobbmm

o 2

1 (2.23)

wCb (2.24)

The total shear strain vector can be expressed in functions of the nodal displacements as

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w

HH

HHw

H

H

w

w

yxw

xyw

b

xw

yw

y

x

x

y

xz

yz

,

,

,

,

,

, (2.25)

In reduced form, the result is

wCwC bb (2.26)

2.2.4 Constitutive Relations

For a general composite lamina, the stress-strain relations for the kth

layer are given by:

TQT

QQQ

QQQ

QQQ

kk

kxy

y

x

xy

y

x

kkxy

y

x

k

662616

262212

161211

(2.27)

And

k

= kxz

yz

=

5545

4544

QQ

QQ

xz

yz

=

ksQ (2.28)

2.2.5 Resultant Laminate Forces and Moments

By integrating the stresses through each layer along the thickness of the laminate, we obtain the resultant

forces, moments and shear forces per unit length, which act on the composite laminated curved panel element.

The in-plane forces, bending moments and the shear forces can be expressed as

T

T

M

N

DB

BA

M

N

0

(2.29)

sAR (2.30)

TN = in-plane thermal loads, defined as

n

k

kkkkkk

h

h

kkT zzh

TzzTQzdzTQN

1

22

11

10

2

2

2

1)( (2.31)

TM = thermal bending moment, defined as

n

k

kkkkkk

h

h

kkT zzh

TzzTQzdzzTQM

1

33

1122

10

2

2

3

1

2

1)( (2.32)

2.2.6 System Element Matrices for Curved Panel

The governing EOM for laminated composite curved panel under consideration in the present work are derived

by using the Principle of Virtual Work. The principle states that for a system in equilibrium, the total work done by

internal and external forces for an infinitesimal virtual displacement is zero. i.e.

int 0extW W W (2.33)

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The virtual work of the internal forces over a curved panel element is given by

dARMkNWA

T

s

TT

0

int (2.34)

where s is the shear correction factor for the laminated composite element, defined as

1

2

11

ktr

ktrs

s (2.35)

2.2.7 Element EOM of Curved Panel System

Simplifying the expressions given by the virtual work principle, and regrouping the element mass, linear and

non-linear stiffness and loading terms, in a matrix form, the Element EOM emerges as

mmm

m

kk

kk

0

0

000

Element Linear Stiffness Matrix

000

0

0ss

b

s

b

s

bb

s kk

kk

Element Linear Shear Stiffness

000

000

000

mmb

mb

bmbbb

kk

kkk

kkk

Element Linear Stiffness Matrix due to

Curved Geometry

000

0

0TT

b

T

b

T

bb

kk

kk

Element Linear Stiffness Matrix due to

Thermal Effects

011

111

110

2

1

mmb

mb

bmb

nn

nnn

nn

Element Non-linear First Order Stiffness

Matrix

9

000

011

011

2

100

00

nn

nn

b

bbb

Element Non-linear First Order Stiffness

Matrix due to Curved Geometry

000

011

011

2

1bb

bb

NN

b

N

b

N

bb

nn

nn

Element Non-linear First Order Stiffness

Matrix including term bN

000

011

011

2

1mm

mm

NN

b

N

b

N

bb

nn

nn

Element Non-linear First Order Stiffness

Matrix including term mN

000

011

011

2

100

00

NN

b

N

b

N

bb

nn

nn

Element Non-linear First Order Stiffness

Matrix including terms bN0

and

0

N

m

b

b

bbb

w

w

w

nn

nn

000

022

022

3

1

Element Non-linear Second Order

Stiffness Matrix

T

m

T

p

p

0

Element Thermal Load Vector

0

o

o

T

Tb

p

p

Element Thermal Load Vector due to

Initial Slope Matrix o

We obtain global system EOM, expressed as follows, after assembling the element matrices over the complete

system domain, and applying the appropriate boundary conditions.

10

o

mb

TT

NNN

Ts

s

PP

WN

WNNNNN

WKKKK

23

1

111112

100

0

(2.36)

In a brief form, the above equation can be written as

stlL PWNNK

2

3

11

2

1 (2.37)

III. Solution Procedures

3.1 Solution Procedure for Deflection due to Static Thermal Loading

The system expressed in Eq. (2.37) can be written as a function of the static thermal deflection sW as

023

11

2

1

stlsssLs PWNNKW (3.1)

This function sW can be developed as a Taylor series expansion, which gives

i.e.

ss

s

s

s

s

s

sss

WWWd

Wd

WWd

WdWWW

0

(3.2)

The derivative of sW can be written as

stl

s

sssL

ss

s PWd

dWNNK

Wd

d

Wd

Wd

2

3

11

2

1 (3.3)

Noting that the linear stiffness matrix LK is constant, the static thermal load (STL) vector is independent of

sW , and after performing complex algebraic operations on non-linear stiffness matrices, we can simplify Eq. (3.3)

as

ssLas

s

s NNKAKWd

Wd21tan

(3.4)

This new stiffness matrix which results from the Taylor series expansion is termed as the tangent stiffness matrix

s

K tan which signifies the slope (tangent) of the deflected shape of panel under STL.

3.1.1 Newton Raphson Iterative Procedure for STL Deflection

To determine the static thermo-aerodynamic large deflection sW , Newton- Raphson iterative method is employed.

Putting

s

s

s KWd

Wdtan

in Eq. (3.2), we get

sssWWK tan (3.5)

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Substituting the expression for sW from Eq. (3.1) into Eq. (3.5),

sssLstlssWNNKPWK

2

3

11

2

1tan (3.6)

in abbreviated form, we can write

stlssPWK tan (3.7)

where, the unbalanced force vector is given by

sssLstlstl WNNKPP

2

3

11

2

1 (3.8)

The system equations as proposed by Eq. (3.8) can be solved for the incremental static thermal deflection sW

by employing Newton-Raphson method. For ith

iteration, Eq. (3.7) can be written in incremental form as

i

stl

i

s

i

sPWK

1

tan (3.9)

where i

s

i

s

i

sLstl

i

stl WNNKPP

2

3

11

2

1 (3.10)

The matrices isN1 and isN2 are evaluated with i

sW determined at ith

iteration step.

The updated static thermo-aerodynamic deflection is given by

11

i

s

i

s

i

s WWW (3.11)

This Newton-Raphson iterative scheme aims at decreasing the unbalanced load vector i

stlP gradually, so as to

decrease the incremental static deflection to a value that accounts for the tangent stiffness matrix s

K tan . At the

converged value of the deflection sW , the slope is defined by

s

stl

sWd

PdWK tan

(3.12)

3.1.2 Critical Buckling Temperature for Flat Plates

In case of purely thermal loading of the panels, high stresses are developed within the panel owing to the high

temperature. In a special case of uniformly heated isotropic or symmetrically laminated flat panels, the phenomenon

of thermal buckling (sudden „popping out‟ of the panel, beyond a certain temperature) is noticed. For all other panel

characteristics (such as curved panels, non-symmetrically laminated panels, panels experiencing thermal gradient

across the thickness etc.), thermal moments are experienced and panel undergoes a constant deformation instead of

sudden buckling. The uniform temperature at which the buckling phenomenon is observed is called critical -

buckling temperature, crT . crT is used normally as a reference temperature to non-dimensionalize the

temperature loading. For isotropic or symmetrically laminated flat panels subjected to uniform temperature across

the thickness, the critical buckling temperature crT is calculated by the procedure outlined below. For such a panel

configuration, the governing EOM can be obtained as shown below, by neglecting terms related to bending-

extension coupling, i.e. terms involving the laminate coupling stiffness matrix B , because in such a case, B =0.

m

Tm

bb

NT

m

b

PW

WNNK

K

K m 0

00

02

3

1

00

01

2

1

00

0

0

0 (3.13)

Furthermore, before buckling, the panel is flat, hence all the nonlinear matrices due to large deflection are null.

Hence, Eq.(3.13) can further be reduced to following two equations:

012

1

b

o

N

Tb WNKK m (3.14)

where the matrix oNmN1 is evaluated with o

mW as defined by

12

m

Tmo

m PKW

1 (3.15)

And m

T

m

m PWK (3.16)

Truncated Taylor series expansion of Eq.(3.14) can be written as

01

b

o

N

Tb WNKK m (3.17)

In view of Taylor series expansion in Eq.(3.17), Eq.(3.14) can be formulated as an Eigen-value problem as follows

o

N

TbmNKK 1

(3.18)

The critical buckling temperature crT is then given by

initialcr TT 1 (3.19)

Where, initialT is arbitrary uniform temperature loading used as initial guess. 1 is the lowest Eigen-

value of thermal buckling Eigen-problem in Eq.(3.18). is the corresponding mode shape of the buckling.

IV. Results and Discussion

4.1 Validation by Comparison

The design space for composite curved panels is vast, and it incorporates many governing factors such as

number of composite layers, laminate orientation, thickness of the composite panel, and directional material

properties. Design space for thermal loading is vast as well, incorporating factors such as uniform temperature on

both the sides of the panel, or linearly varying thermal gradient across the thickness of the panel, which in turn

decide the effects such as buckling of the panel. Though validation of the governing theory and corresponding

computer code was done at every step of development by comparison with the respective already-existing results, it

is beyond the scope of this paper to present all of them here owing to brevity. However, since isotropic materials can

be treated as a special case for composite materials, flat plates can be regarded as curved panels with zero curvature,

and 2D panels can be formulated as a special case of 3D panels, only the comparison made with 3D Graphite-Epoxy

composite panel subjected to uniform thermal loading is presented here.

4.1.1 Thermal Buckling Response of Composite Cylindrical Shallow Shell Panel For validation purposes, the comparison is made with 3D Graphite-Epoxy composite panel subjected to

uniform thermal loading, as analyzed by Przekop [5] is presented here. The physical and geometric parameters of

the 3-D composite fully clamped cylindrical panel (H/h = 3.337) were taken as:

Material Graphite-Epoxy

Geometry (inch) 10 x 15 x 0.05

Support Fully Clamped

Lamination (0/90)

E1 (psi) 26.24 x 106

E2 (psi) 1.49 x 106

ν 0.28

G12 = G13 = G23 (psi) 1.04 x 106

Density ρ (kg/m3) 1550

Coefficient of Thermal Expansion α1 (1/oF) -0.04 x 10

-6

Coefficient of Thermal Expansion α2 (1/oF) 16.7 x 10

-6

The panel was discretized with a mesh size of 17x17 encompassing 578 MIN3 elements. Uniform

temperature T0= 180 oF was applied to the panel, with no gradient across the thickness. (T1 = 0). Fig. 3-a shows the

comparison of deflection shapes along Y-axis for above-mentioned panel with those presented by Przekop. Fig. 3-b

shows the comparison of deflection shapes along X-axis. The agreement in the results is evident and as noticed by

Przekop, mode (1,1) is not dominant for the thermally deflected panel and therefore the maximum deflection does

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not occur at the center of the panel. Fig. 3-c shows the deflection shape of the panel in 3D, further explaining the

location of maximum deflection.

Ghoman Przekop

Fig. 3-a Comparison of Deflection shape along Y axis for x = 5.00” at T0 = 180 oF

Ghoman Przekop

Fig. 3-b Comparison of Deflection Shape along X axis for y = 3.75” and 7.5” at T0 = 180 oF

Fig. 3-c Panel Deflection Shape in 3D

Once the proposed theory and corresponding computer program were verified with above-mentioned as well as

numerous other benchmarks, cases were run subsequently and the results obtained thereby are presented in

forthcoming sections.

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4.2 Effect of Thermal Loading on Curved Composite (Graphite-Epoxy) Panel

Since composite materials find a lot of aerospace applications recently, it is of prime importance to study

composites in view of this paper. The design space for composite curved panels is vast, and it incorporates many

governing factors such as number of composite layers, laminate orientation, thickness of the composite panel, and

directional material properties. This paper employs a study of curved panel made of Graphite – Epoxy composite

material, with geometric, physical and material properties as given in the following table. A finer mesh size of (17 x

17) was used, thus employing 578 MIN3 elements.

Material Graphite-Epoxy

Geometry (inch) 12 x 12 x 0.05

Support Simply supported

Lamination (0/90)

E1 (psi) 22.5 x 106

E2 (psi) 1.17 x 106

ν 0.22

G12 = G13 (psi) 0.66 x 106

G23 0.4 x 106

Density ρ (kg/m3) 1550

Coefficient of Thermal Expansion α1 (1/oF) -0.04 x 10

-6

Coefficient of Thermal Expansion α2 (1/oF) 16.7 x 10

-6

Damping Parameter Ca 0.01

4.2.1. Effects of Increasing Uniform Temperature (T0) Across the Panel Thickness

The effects of increasing uniform temperature across the panel thickness, i.e. T0 from 0 to 250 oF, keeping T1 =

0, are presented here. A critical milestone toward the computation of the curved panel stability characteristics is the

determination of the thermostatic deflection shape sW . Prior to the determination of the thermostatic deflection

shape it is essential to evaluate the tangent matrix tanK using the static deflection sW from the previous step in

the Newton-Raphson iterative process, till convergence is attained. The tangent matrix, which represents the

stiffness of the deflected curved panel under the STL, is obtained by feeding the updated sW into the first-order

)(1 sWN and the second-order )(2 2

sWN stiffness matrices. The thermo-aerostatic deflection is determined

using the Newton-Raphson scheme outlined in the static analysis paragraph. The thermostatic deflection shape was

investigated for the panel‟s height-rise of H/h = 3. Since the thermostatic deflection shape is not uniform across the

y-axis in the 3-D case, the y-axis midline located after the eighth elements in the y-direction was chosen as a

representative of the deflection shape.

Deflections of panel at Y- midplane for various T0 loadings are plotted and presented in Fig. 4. These plots

throw light on panel behavior under the thermal loading. A lot many interesting phenomena are observed in these

plots. As seen in the plots, dotted line shows nominal shape of panel, while solid black line shows the buckled shape

of panel. As T0 increases, the buckling (deflection) increases as expected, and panel pops out more and more, due to

increasing T0.

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Figure 4- Mid-plane Deflections of Curved Gr-Ep Panel for Increasing Uniform Thermal Loading (T0)

Figure 6- Thermostatic Deflections of Curved Gr-Ep Panel for T0 = 0 - 50 oF, T1 = 0

oF

Fig. 5 shows thermostatic deflection of panel in 3D, at T0 = 0 oF and T0 = 50

oF. The effect of thermal buckling is

clearly seen in the deflection plot for T0 = 50 oF.

2. Effects of Increasing Linear Temperature Gradient (T1) Across the Panel Thickness

Fig. 7 shows the effect of increasing the linear temperature gradient T1 across the panel, while maintaining the

uniform temperature T0 = 50oF . We don‟t see any explicit and drastic changes in deflection shapes for increasing T1.

Thus increasing the linear temperature gradient across the panel thickness does not severely affect the deflection

shapes of the panel.

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Figure 7- Mid-plane Deflections of Curved Gr-Ep Panel for Increasing Linear Temperature Gradient (T1)

V. Conclusion

There are many papers dedicated toward the investigation of pre/post buckling analysis of flat plates in

elevated temperature regime, but a limited number of papers address the problem of thermal deflection of curved

composite panels. Hence, this paper is a step towards understanding the panoply of static and dynamic behavioral

nuances that constitute the buckling phenomenon of curved isotropic as well as thermal deflections of composite

panels, subjected to thermal loading. Such a better understanding will ensue the development of new design and

analysis tools for the new generation of civil/military supersonic aircrafts, spacecrafts and missiles in the offing, the

skins of which are typically exposed to higher temperatures accompanying supersonic flights. The development of

such tools is crucial, and of strategic necessity in order to avoid dramatic sudden failure of the aircraft skin.

Furthermore, the utilization of the FE analysis/design tools will also reduce the high costs associated with supersonic

wind tunnel experiments of surface panels. For future, beneficial will be the addition of effects of failure/ fatigue life

17

analysis under STL and also, combined random/aerodynamic/acoustic loads. Incorporating the deflection control

techniques by sensors and actuators, embedded shape memory alloys, etc. would be another area of focus.

References

1. Karman, von, T., "Festigkeitsprobleme in Maschinenbau," Encyclopadie der Mathematishen

Wissenschaften, IV, Chapter 27, 1910.

2. Marguerre, K., "Zur Theorie Der Gekrummten Platte Grosser Formanderung," Proceeding of the

5th International Congress for Applied Mechanics, Harvard University and the Massachusetts

Institute of Technology, Cambridge, MA, September 1938, edited by J. P. den Hartog and H.

Peters (one vol.), John Wiley and Sons, Inc., New York (USA), pp. 93-101, and Chapman and Hall Ltd.,

London (UK), 1939.

3. Houboult, J. C., “ A Study of Several Aerothermoelastic Problems of Aircraft Structures in High-Speed

Flight”, Ph.D. Thesis, Eidenossischen Technischen Hochschule, The Swiss Federal Institute of Technology,

Zurich, Switzerland, 1958.

4. Bolotin, V. V., "Non-conservative Problems of the Theory of Elastic Stability," Macmillan Co., New York,

1963, pp. 274-312.

5. Przekop, A., “Nonlinear Response and Fatigue Estimation of Aerospace Curved Surface Panels to Acoustic

and Thermal Loads,” Ph.D. Dissertation, Old Dominion University, Norfolk, VA, 2003.

6. Azzouz, M. S., “Nonlinear Flutter of Curved Panels Under Yawed Supersonic Flow Using Finite Elements,”

Ph.D. Dissertation, Old Dominion University, Norfolk, VA, 2005.

7. Ghoman, S. S., "Nonlinear Flutter Analysis of Curved Panels under Yawed Supersonic Flow and at

Elevated Temperatures Using Finite Elements", M.S. Thesis, Old Dominion University, Norfolk, VA, 2008.

Acknowledgement

Author would like to thank Dr. C. Mei, professor emeritus at Aerospace Engineering Department, Old

Dominion University, Virginia, USA and Dr. M.S. Azzouz, assistant professor at Midwestern State University,

Texas, USA, for their constant guidance and support. Author would also like to express gratitude towards Dr. R.K.

Kapania, Mitchell Professor at the department of Aerospace and Ocean Engineering, Virginia Tech, Virginia, USA.