Sourav Term Paper

FINITE ELEMENT ANALYSIS OF FOLDED PLATES

TERM PAPER LEADING TO THESIS

By

SOURAV CHANDA

Class Roll No. – MCE 001410402009

Under the Guidance

Prof. (Dr.) Arup Guha Niyogi

DEPARTMENT OF CIVIL ENGINEERING

JADAVPUR UNIVERSITY

2015

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CERTIFICATE

This is to certify that this term paper leading to thesis entitled “Finite Element Analysis of

Folded Plates” has been prepared by Shri Sourav Chanda (Class Roll no. – MCE

001410402009) for fulfillment of the requirements for the award of MCE degree, is a record of

work carried out under my supervision and guidance. I hereby approve this term paper leading to

thesis for submission and presentation.

Dr. Arup Guha Niyogi

Professor of Civil Engineering

Jadavpur University

Kolkata

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ACKNOWLEDGEMENT

I am extremely grateful and deeply indebted to Prof. ARUP GUHA NIYOGI, Professor of Civil

Engineering, Jadavpur University, Kolkata for his excellent guidance and whole hearted

involvement during my term paper without whose invaluable suggestions, meticulous efforts,

versatility and untiring guidance, this project would not have been feasible. I am also indebted to

him for his encouragement and moral support throughout this programme.

It is my privilege and pleasure to express my deep sense of gratitude and sincere thanks to him

for his timely help, valuable support and suggestions which have helped to orient my Term

paper.

Place:

Date: SOURAV CHANDA

MCE 1st year – 2014-15

Class Roll No. 001410402009

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FINITE ELEMENT ANALYSIS OF FOLDED PLATES

Introduction

Structural shapes are usually complicated and it is difficult to solve the governing differential

equations directly for such end conditions and initial conditions. Thus one resort to numerical

methods to obtain approximate solutions of these complex problems that are reasonably accurate.

Finite element method is one such approximate method, where the structure is subdivided into

finite number of smaller pieces, where the governing equations are applied to obtain elemental

system equations, which are then assembled, end conditions and initial conditions are inserted

and solved for. It is a domain type solution method where the solution provided knowledge of

the domain at nodes that connect the elements.

Folded plate is an assemblage of flat plates, all of which do not lie in the same plane. They

provide better cladding coverage due to added stiffness compared to flat plates and,

simultaneously, provide better architectural ambience. North light folded plates are inevitable in

factory roofs. Aviation and maritime industries do make use of folded plates to build aircraft and

ships.

For two dimensional structures like plate and shells, we are faced with the difficulty regarding

the choice of discretization. In such cases, the continuum will be assumed to consist of imaginary

divisions. Thus the real continuum is divided into a finite number of discrete elements.

In a continuum that is divided into a mesh, two adjacent elements placed side by side will have a

common edge. Its true representation will result in a more complicated analysis. Therefore, in

order to make the analysis simpler, it is assumed that the elements are connected at the nodal

points only and the nodal continuity requirement is to be satisfied. Once the discretization is

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made, the analysis follows a rather set procedure. Finite element analysis can be done by two

approaches:

Force based Finite Element Method

Displacement based Finite Element Method

Among these, displacement based finite element method is more popular. Here, the stiffness

matrix of the individual elements needs to be formulated. The forces actually distributed in the

real structure are transformed to act as nodal points.

In the finite element analysis, the continuum is divided into a ‘finite’ number of elements, having

‘finite’ dimensions and reducing the continuum having ‘infinite’ degrees of freedom to ‘finite’

degrees of freedom.

Literature Review

In continuum mechanics, plate theories are mathematical descriptions of the mechanics of flat

plates that draws on the theory of beams. Plates are defined as plane structural elements with a

small thickness compared to the planar dimensions. The typical thickness to width ratio of a plate

structure is less than 0.1.

A plate theory takes advantage of this disparity in length scale to reduce the full three

dimensional solid mechanics problem to a two dimensional problem. The aim of plate theory is

to calculate the deformation and stresses in a plate subjected to loads.

Of the numerous plate theories that have been developed since the late 19th century, two are

widely accepted and used in engineering. These are

The Kirchhoff – Love theory of plates (classical plate theory)

The Mindlin – Reissner theory of plates (first order shear plate theory)

Kirchhoff – Love theory for thin plates

The Kirchhoff – Love theory is an extension of Euler–Bernoulli beam theory to thin plates. The

theory was developed in 1888 by Love using assumptions proposed by Kirchhoff. It is assumed

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that a mid surface plane can be used to represent the three dimensional plate in two dimensional

form.

The following kinematic assumptions that are made in this theory:

Straight lines normal to the mid surface remain straight after deformation.

Straight lines normal to the mid surface remain normal to the mid surface after

deformation.

The thickness of the plate does not change during a deformation.

Figure 1

Let us consider the coordinate axes x and y in the middle plane of the plate and the z axis

perpendicular to that plane. Let us further consider an element cut out of the plate by two

pairs of planes parallel to the xz and yz planes as shown in figure 1. In addition to the

bending moments Mx and My, the twisting moments Mxy and vertical shear forces are

considered on the sides of the element. The magnitude of these shear forces per unit

length parallel to the y and x axes we denote by Qx and Qy, respectively, so that

(a)

We must also consider the load distributed over the upper surface of the plate. The

intensity of this load we denote by q, so that the load acting on the element is qdxdy.

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Figure 2

Projecting all the forces acting on the element onto the z axis we obtain the following

equation of equilibrium:

From which

(b)

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Taking moments of all the forces acting on the element with respect to the x axis, we

obtain the equation of equilibrium

(c)

The moment of the load q and the moment due to change in the force Qy are neglected in

this equation, since they are small quantities of a higher order than those retained. After

simplification the equation becomes,

..........(d)

In the same manner, by taking moments with respect to the y axis we obtain,

............(e)

Since there are no forces in the x and y directions and no moments with respect to the z

axis, the three equations (b), (d) and (e) completely define the equilibrium of the element.

Let us eliminate the shearing forces Qx and Qy from these equations by determining them

from equations (d) and (e) and substituting into equation (b). In this manner, we obtain

...........(f)

Observing that Myx = - Mxy, by virtue of τxy = τyx, we finally represent the equation

equilibrium (f) in the following form:

.............(g)

Now,

...........(h)

............(i)

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Substituting these expressions in equation (g), we obtain

...............(j)

This is the governing differential equation of laterally loaded thin plates.

Mindlin–Reissner Plate Theory:

Although Kirchhoff hypothesis provides comparatively simple analytical solutions for most of

the cases, it also suffers from some limitations. For example, Kirchhoff plate element cannot

rotate independently of the position of the mid – surface. As a result, problems occur at

boundaries, where the undefined transverse shear stresses are necessary especially for thick

plates. Also, the Kirchhoff theory is only applicable for analysis of plates with smaller

deformations, as higher order terms of strain – displacement relationship cannot be neglected for

large deformations. Moreover, as plate deflects its transverse stiffness changes. Hence, only for

small deformations the transverse stiffness can be assumed to be constant.

Contrary, Mindlin–Reissner plate theory is applied for analysis of thick plates, where the shear

deformations are considered, rotations and lateral deflections are decoupled. It does not require

the cross – sections to be perpendicular to the axial forces after deformation. I t basically

depends on the following assumptions:

The deflections of the plate are small.

Normal to the plate mid – surface before deformation remains straight, but is not

necessarily normal to it after deformation.

Stresses normal to the mid – surface are negligible.

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Folded Plates:

Folded plate is an assemblage of flat plates, all of which do not lie in the same plane. They

provide better cladding coverage due to added stiffness compared to flat plates and,

simultaneously, provide better architectural ambience. North light folded plates are inevitable in

factory roofs. Aviation and maritime industries do make use of folded plates to build aircraft and

ships. Hipped plates are ideally suited for a variety of structures such as factory buildings,

assembly halls, go downs, auditoriums and gymnasia, requiring large column free area.

The different types of folded plates used for the various types of structure are shown below:

Figure: Different types of folded plates

Folded plates have certain advantages as:

Simple shuttering involving only straight planks are required.

Moveable form work can be used resulting in speedy construction.

Simpler diaphragms are used in place of complicated traverses required for shells.

The design computations are simpler.

Folded plates consume slightly higher quantities of concrete and steel, but the increased

cost on this account is more than offset by the lower shuttering costs.

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With their light reflecting geometry and pleasing outlines, folded plates are aesthetically

comparable with shells.

Structural Behaviour of Folded Plates

Folded plates resist the system of transverse loads by ‘plate’ and ‘slab’ action. The loads acting

normal to each plate causes transverse bending between the junctions of the plates, which can be

considered as imaginary supports of a continuous slab. This transverse bending termed as ‘Slab

action’. The transverse moments developed in the plate can be determined by a continuous beam

analysis assuming the supports to be at the junctions of the plates.

The plate being supported at their ends on transverses, bend under the action of loads in their

own plate as shown in figure . The longitudinal bending of the plates in their own plate is termed

as ‘Plate action’. The bending stresses resulting from plate action may be considered to have a

linear distribution across each plate, with maximum intensity at the centre of span section.

Formulation of Displacement based Finite Element Method

The displacement based finite element method can be regarded as an extension of the

displacement method of analysis of beam and truss structures, and it is therefore valuable to

review the analysis process. The basic steps in the analysis of a beam and truss structure using

the displacement method are the following:

a) Idealize the total structure as an assemblage of beam and truss elements that are

interconnected at structural joints.

b) Identify the unknown joint displacements that completely define the displacement

response of the structural idealisation.

c) Formulate force balance equations corresponding to the unknown joint displacements and

solve these equations.

d) With the beam and truss element end displacements known, calculate the internal element

stress distributions.

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e) Interpret, based on the assumptions used, the displacements and stresses predicted by the

solution of the structural idealisation.

Derivation of stiffness matrix of a flat rectangular shell element:

A simple rectangular flat shell element can be obtained by superimposing the plate bending

behaviour of rectangular plate element and the plane stress behaviour of that element. The

resulting element is as shown in figure below. The element can be employed to model

assemblages of flat plates (e.g. folded plate structures) and also curved shells. For actual

analysis, more effective shell elements are available.

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Before discussion of analysis of flat shell element, we will focus on the plate bending behaviour

of rectangular plate bending element and plane stress behaviour of that element.

Derivation of stiffness matrix for rectangular plate bending element

Fig. 2: Rectangular Plate Bending Element

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Let us consider a rectangular plate bending element of thickness ‘h’ as shown in fig. 2 figure

above. As shown in figure, this element considered has three degrees of freedom per node.

Therefore, it is necessary to have 12 unknown generalized coordinates, α1, α2, ………, α12, in the

displacement assumption for w.

The polynomial used is,

Hence,

We can now calculate δw/δx and δw/δy:

And

Using the conditions,

We can construct the matrix A, obtaining

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Where,

which can be shown to be always singular.

To evaluate the matrix E, we recall that in plate bending analysis curvatures and moments are

used as generalised strains and stresses. Calculating the required derivatives of (b) and (c), we

obtain,

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Hence we have,

For plane stress problem, the constitutive matrix C is given by,

With the matrices ϕ, A and E given in (a), (d) and (f) and the material constitutive matrix given

above, the element stiffness matrix can be calculated as,

KB = ∫ ETCE dV

Derivation of Stiffness Matrix for a plane stress problem:

Let us consider a cantilever plate as shown in figure below.

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For finite element analysis, the plate can be idealised as shown below.

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The plate is acting in plane stress conditions. For an isotropic linear elastic material the stress –

strain matrix is defined using Young’s modulus E and Poisson’s ratio ᶹ as

The displacement transformation matrix H(2) of element 2 relates the element internal

displacements to the nodal point displacements,

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Where U is a vector listing all nodal point displacements of the structure,

To derive the matrix H(2) in (a) we recognize that there are four nodal point displacements each

for expressing u(x,y) and v(x,y). Hence, we can assume that the local element displacements u

and v are given in the following form of polynomials in the local coordinate variables x and y:

The unknown coefficients α1, .............., β4, which are also called generalised coordinates, will be

expressed in terms of the unknown element nodal point displacements u1, .............., u4 and

v1, ............, v4. Defining

We can write (c) in matrix form:

Where

And

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Equation (e) must hold for all nodal points of the element; therefore, using (d), we have

In which

And

Solving from (f) for α and substituting into (e), we obtain

Where the fact that no superscript is used on H indicates that the displacement interpolation

matrix is defined corresponding to the element nodal point displacements in (d),

The displacement functions in H could also have been established by inspection. Let Hij be the

(i,j)th element of H; then H11 corresponds to a function that varies linearly in x and y, is unity at

x = 1, y = 1 and is zero at the other three element nodes.

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With H given in (h) we have

The strain – displacement matrix can now directly be obtained from (g). In plane stress

conditions the element strains are

Where

Using (g) and recognizing that the elements in A-1 are independent of x and y, we obtain

Where

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Hence, the strain – displacement matrix corresponding to the local element degrees of freedom

is,

The matrix B could also have been calculated directly by operating on the rows of the matrix H

in (h).

Let Bij be the (i,j)th element of B; then we now have

Where the element degrees of freedom and assemblage degrees of freedom are ordered as in (d)

and (b).

Similarly, E matrix can be found out for all the nodes and then assembled, and consequently

stiffness matrix for the plate element under plane stress condition can be obtained as,

KM = ∫ ETCE dV

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Derivation of stiffness matrix of flat rectangular shell element:

Let KB and KM be the stiffness matrices, in the local coordinate system, corresponding to the

bending and membrane behaviour of the element, respectively. Then the shell element stiffness

matrix KS is

The folded plate structures should be idealised as shown below.

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Since we deal in these analyses with six degrees of freedom per node, the element stiffness

matrices corresponding to the global degrees of freedom are calculated using the transformation

matrix as,

Where,

And T is the transformation matrix between the local and global element degrees of freedom.

References:

[1] Finite Element Procedures by Klaus – Jurgen Bathe

[2] Matrix and Finite Element Analyses of Structure by Madhujit Mukhopadhyay & Abdul

Hamid Sheikh

[3] Theory of Plates and Shells by Stephen P. Timoshenko & S. Woinowsky – Krieger

[4] Advanced Reinforced Concrete Design by Dr. N. Krishna Raju

[email protected]

There are a few suggestions”

1. The introduction started all of a sudden. I added a few lines

2. There should be a literature review: Talk about plates, classical plate theory, Mindlin

theory, then about folded plates, books and journal papers on folded plates. Search

sciencedirect from University for further info. For presentation, they may be presented

briefly in a single table. Discuss on this aspect

3. At the end you could add a table explaining your programme during conducting your

thesis.

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