FINITE ELEMENT ANALYSIS OF CONVENTIONAL

WING AND FUSELAGE

BY

Flt. Lt. Muhammad Ismaeel

M.Sc. (Physics) (Punjab)

B.E. (Aerospace) Kar) P.E.

INTRODUCTION

Recent advances in structural technology have required greater accuracy and speed in the

analysis of structural system. This is particularly true in aerospace applications, where great

technological advances have been made in the development of efficient lightweight structures for

reliable and safe operation in severe environments. It is therefore not surprising that new

methods have been developed for the analysis of the complex structural configurations and

design used in aerospace engineering. In other fields of structural engineering, too, more refined

methods of analysis have been developed.

The methods of analysis which meet the requirements mentioned above use matrix algebra,

which is ideally suited for automatic computation on high-speed digital computers. In these

methods the digital computer is used not only for the solution of simultaneous equations but also

for the whole process of structural analysis from the initial input data to the final output.

Matrix methods are based on the concept of replacing the actual continous structure by a

mathematical model made up from structural elements of finite size (also referred to as Finite

elements) having known elastic and inertial properties that can be expressed in matrix form. The

properties of each element are calculated, using the theory of continous elastic media, while the

analysis of the entire structure is carried out for the assembly of the individual structural

elements.

METHODS OF STRUCTURAL ANALYSIS

Methods of structural analysis can be divided into two groups, analytical methods and numerical

methods. The limitations imposed by the analytical methods are well know only in special cases

are closed form solutions possible. Approximate solutions can be found for some simple

structural configurations, but in general for complex structures, analytical method cannot be used

and numerical methods must invariably be employed. Numerical methods of structural analysis

can be subdivided into two types, (1) numerical solutions of differential equations for

displacements or stress and (2) matrix methods based on discrete-element idealization i.e.

FINITE ELEMENT METHOD.

FINITE ELEMENT METHOD

In the F.E. Method the complete structural theory is developed ab initio in matrix algebra,

through all stages of analysis. The structure is first idealized into an assembly of discrete

structural elements with assumed form of displacement or stress distribution, and the complete

solution is then obtained by combining this individual approximate displacement or stress

distributions in a manner which satisfies the force-equilibrium and displacement compatibility at

the junction of these elements. Methods based on this approach appear to be suitable for the

analysis of complex structures. Also, the formulation of the analysis in the matrix algebra is

ideally suited for subsequent solution on digital computer and allows an easy and systematic

compilation of the required data.

Two complementary matrix methods of formulation of any structural problem are possible: (1)

The displacement method (stiffness method), where displacements are chosen as unknowns, and

(2) The force method (Flexibility method), where forces are unknowns.

THE EARLY USE OF FINITE ELEMENTS

Since the finite element concept is to a large extent physical rather than abstract in nature, it has

been is use in variety of forms for centuries. The basic idea has always been to replace an actual

problem by a simpler one. In formulating the simplified or idealized problem, use in made of the

So called FINITE ELEMENT.

For example, the circle was approximated by a regular polygon and the straight line then became

the finite element and in terms of its length both circumference and area were approximated in

terms of .

The determination of the value of was based on replacing the area of circle with slender

inscribed and subscribed, rectangles. Archimedes used finite elements for determining the

volume of solids.

STRUCTURAL ANALYSIS B FINITE ELEMENT METHOD

Modern finite element theory had its recognizable beginnings in the displacement (or stiffness)

method of structural analysis. The initial steps were based on a combination of elementary plus

intuitive reasoning. It was later shown that this early work could be fully developed from the

variational principles of elasticity which provides the more general and powerful formulation of

the subject. At the same time there is much to be gained by first considering the subject along the

intuitive lines upon which it was originally established.

Engineering structures generally consists of discrete parts fastened together to form a complete

system. Whether the final structure is a building, bridge, ship or airplane makes little difference.

Combination of sheets, plates beams, stiffeners etc, are welded, bolted or riveted together to form

the final unit. It is the task of the structural analyst to calculate the stresses and deformations of

such structures as they are subjected to all possible conditions of loading, temperature changes

and so on. An other type of structural unit to be analysed is continuum. This is generally

considered to be a single mass of material as found in a forging, solid-propellant rocket motor,

concrete dam and so on.

a. Discrete Attachment Structure; b. Continuous Attachment

Pin Jointed Truss structure: Two Flange spar.

Fig.1. Discrete and continuous structures

BASIC CONCEPTS OF FINITE ELEMENT METHOD

An eleastic continuum is an elastic system which has one or more surface continous through

some region in space. Examples of continua are shown in Fig. 2. Exact analysis of the plates and

shells having regular layouts may be accomplished using classical theory. An example of a

continuum which can not be solved using classical theory is the plate containing an irregularly

shaped hole. An approximate analysis of this plate may be made by dividing the plate into

discrete segments or finite elements.

After the continuum is divided into finite elements, the remaining steps of the analysis are

similar to the matrix displacement procedure.

DIVISION OF STRUCTURE INTO FINITE ELEMENTS.

Division of a framed structure into joints and members is for the most part automatic and usually

requires little engineering adjustment. The frame of figure 3a is automatically divided into six

members and six joints. If the beam deflection at mid span are desired, two joints may be

inserted as shown in Fig. 3b.

Replacement of the curved arch of fig. 3c bystraight members requires some adjustment.

Sufficient members must be selected so that the piecewise straight structure closely resembles

the curved structure. Thus the approximation of Fig. 3d is obviously poor whereas the

approximation of Fig 3(e) appears satisfactory.

A continuum can be subdivided into triangular, rectangular or trapezoidal elements of which

triangular is the simplest to solve. From the classical theory of elasticity, it is known that

discontinuities in the continuum cause abrupt stress increases around the discontinuity. Further

St. venant theorem states that these peak stresses decrease rapidly away from the discontinuity

thus both stress and strain gradients are high around a discontinuity and decrease as one moves

away from the stress raiser. These observations lead to the conclusion that small elements should

be used around the discontinuity or notch with the smallest elements at the corner of the notch

where stresses will occur parallel to the loading. Thus elements are selected as shown, in figure

4.

FIG. 2 : Examples of Continua

FIG. 3 : DIVISION of framed Structures into members and joints.

FIG. 4 : DIVISION of Continuum into elements

MATRIX DISPLACEMENT METHOD

Consider a structure in which node i (i = 1,2,3,--------n) is subjected to the forces Fxi , Fyyi , Fzi ,

in x, y, z directions respectively with displacements Ui , Vi , Wi in the corresponding directions.

The forces and the nodal displacements can be written as column matrices.

OR (F) (δ)

The generalized force system (F) can contain moment M and Torque T in addition to direct

forces in which case (§) will include rotation Ө.

For a complete structure, the nodal forces and displacement are related through a stiffness matrix

(K).

Fx , 1

Fy , 1

Fz , 1

Fx , 2

Fy , 2

Fz , 2

.

.

.

.

.

Fx , n

F , n

Fz , n

U1

V1

W1

Ux2

V2

W2

.

.

.

.

.

Un

Vn

Wn

,

(F) = (K) (δ) ---- (1)

Where (K) is a symmetric matrix of the form

K =

The element Kij (that is the element located in row i and column j) is known as the stiffness

influence coefficient (and Kij = Kji ).

For the analysis of trusses and frames, different stiffness matrices for the bar and beam elements

are required. But a generalized stiffness matrix for beam element (in space) has been developed.

All stiffness matrices for members of lower orders may be obtained by reducing this matrix to

the proper conditions

The generalized stiffness matrix is limited in its application to the beam (or bar) elements in local

coordinates. To accommodate for elements which are arbitrarily oriented in space, a coordinate

(local to global) transformation matrix is developed.

GENERAL SOLUTION PROCEDURE

The fundamental equation of the matrix displacement method is:-

(F) = [K] (δ) ---- (3)

1. The stiffness matrix for each element is computed in local coordinates and then

transformed into coordinates using equation.

[K] = [T]T

[K] [T] ------------ (4)

2. Adjustments for the boundary conditions are made and semi-Band system

stiffness matrix is formed in the rectangular form.

K11 K12 K13 …….. K1n

K21 K22 K23 …….. K2n

K31 K32 K33 …….. K3n

Kn1 Kn2 Kn3 …….. Knn

---- (2)

3. Loads are computed and arranged in column matrix according to the degree of

Freedom at the nodes.

4. Nodal displacement are determined from eqn. (3)

5. The nodal displacement for each element are extracted and the element reactions

are computed by using the element stiffness matrices.

The computer program PAFSTRAN has been developed on the lines of this solution procedure,

and the flow chart of the same has been attached to the paper as appendix ‘A’.

A few of the various problems solved by PAFSTRAN are being attached to this paper as

Appendix ‘B’.

CONCLUSION

PAFSTRAN is the most generalized computer program for the analysis of trusses and frames. Its

validity and reliability has been confirmed by solving a number of examples of various possible

cases. This program can handle as big a problem as could be conceived in practical life (such as

high voltage tower, bridge or frame work of a building) depending upon the computer memory

storage available. Another encouraging aspect of this program is its applicability to statically

indeterminate structures.

The complete structural analysis of wing and fuselage requires the analytical considerations of

bar, beam and plate elements along with the shear flow panels. The same program may be used

for the analysis of a plate element by including its stiffness matrix and carrying out minor

modifications in the main program.

REFERENCES

1. Azar J.J., Matrix Structural Analysis Permagon U.E. Series, OXFOR, 1972.

2. Fenner R.T., Computing for Engineers, Macmillan, London, 1974.

3. Jenkins W.M., Matrix and Digital Computer Methods is Structural Analysis, McGraw-

Hill, London, 1969.

4. Laursen H.I., Structural Analysis McGraw-Hill London, 1978.

5. Megson T.H.G., Aircraft Structures for Engineering Student, Edward Arnold, London,

1972.

6. Martin H.C., Introduction to Matrix Mehods of Structural Analysis, McGraw Hill-1966.

7. Prsemieniecki J.S., Theory of Structural analysis, McGraw-Hill, New York, 1968.

8. Rubinstein M.F., Matrix Computer Analysis of Structures, Prentice-Hall, 1966.

9. Vanderbiet M.D., Matrix Structural Analysis OPI, USA, 1974.

10. Willems and Lucas, Matrix Analysis for Structural Engineers, Prentice-Hall, London,

1968.

11. Martin H.C. and Carey G.F., Introduction to Finite element Analysis, Tata McGraw Hill,

New Delhi, 1975.