# 2D Nonlinear Finite Element Analysis of Masonry Bedjoint

FINITE ELEMENT METHOD 11

1.2.2 Solution

Finite element method uses algebraic equations in matrix form. We will use a

simple example for explaining the solution procedure.

Fig. 1.2: Finite element of rectangular shape Fig. 1.3: Common nodes

In Fig. 1.2 we see a finite element of rectangular shape. Writing an equation for

each of the nodes describing a displacement of a node, as a function of its co-nodes,

gives us 8 equations from the whole element. It forms a matrix of 8 equations.

Continuing according fundamental laws of mechanics we relate displacements to

stresses. From stresses we obtain strain energy and from this we derive potential

energy and finally from minimum potential energy we obtain a pair of system

equations for the complete element. Every element can be described by the stiffness

matrix:

� �

F �

�_{K}�^{�}_{d}^{�}

(1.1)

The matrix operates with a vector displacement d, which is a displacement d of the

whole element. K is called a stiffness matrix. Displacement and stiffness are related

to load F. Stiffness matrix of the element with 8 nodes can be written:

�

K

�

�^{K}

�

�^{K}

�

� _{�}

�

�^{K}

11

21

81

K

K

K

�

12

22

82

�

�

�

�

K

K

K

�

18

28

88

�

�

�

�

�

�

(1.2)

Carrying out this process for every element in mesh we get a stiffness matrix for each

element. Next step is to combine all individual matrices together to obtain a stiffness

of a whole system. Any two neighboured elements have got nodes in common (Fig.

1.3). Individual matrices can be combined by a merging technique – this process is

called assembly (Fig. 1.4). Using standard procedure we eliminate parts of the matrix.

Rows of the matrix represent a set of simultaneous equations. We solve a first

equation and substitute to remaining ones. At the end, when last matrix is added, we

are left with a solution of a single node – its displacement. We are getting backwards,

applying the acquired information, in to the rest of equations, as a key of a system,

FINITE ELEMENT METHOD 12

until every displacement of node is obtained. From these results corresponding

stresses can be calculated.

Fig. 1.4: Merged area

1.2.1 Post-processing

Postprocessor software uses methods for sorting, printing and plotting

solutions of finite element method. There are many possible ways to present the

results: sort element stresses according intensity, check equilibrium, plot deformed

structure, animate dynamic model behaviour. As usual, every computer output

needs to be checked by an engineering judgment, if the solution is physically

possible.

1.3 Linearity

Linear analysis is based on linear stress-strain relationship, where Hook’s law

is applied. This method can be used only in elastic part of stress-strain diagram.

Results from different loads are compatible, because of linear superposition validity.

We use linear analysis method mainly because the solutions are easier to find,

computational requirements are small and solutions can be superposed on each

other.

In many cases, linear analysis is not suitable and nonlinear analysis is

necessary when:

� designing high performance components

� trying to establish the cause of failure

� simulating true material behaviour

� better understanding of physical phenomena is needed

FINITE ELEMENT METHOD 13

1.3.1 Non-Linear Analysis

The first type is a geometric nonlinear analysis. It is used in cases with large

strains, displacements, and rotations and other. A good example is an airplane wing,

on which large load affects. After every load step a new geometry of a component is

redefined by adding nodal displacements. This guarantees that the component

geometry will have true adapted (deformed) shape for next load step.

The second type is a material nonlinear analysis. Material nonlinearities occur

when the stress-strain or force-displacement law is not linear, or when material

characteristics change with the applied loads. In this case, total load on component is

applied in small steps. In each step, nonlinear stress-strain relationship is

approximated by a linear one. Because our model is controled by a displacement it is

advantageous that the displacement values can be modified after each load step

depending on demands.

1.4 Applications

The Finite element method is very universal tool has got a world-wide usage

in many mechanical engineering disciplines (for example in biomechanical,

aeronautical and automotive industries). The main aim is the design and

development of products. An optimized design means the cost reduction by

minimizing a weight and material consumption. The FEM results allow detailed

visualization of stresses and displacements distribution (also shows weak parts of

model, places with inappropriate bend or twist...). This method is also very practical,

there is no need to physically test the product every time a part or material

characteristic is changed (experimental analysis is expensive and we do not create a

vain models).

Perfect examples of FEM usage are crash-tests (Fig. 1.5). FEM allows designing

a prototype, testing it virtually, redesigning and manufacturing it. The

implementation of FEM has significantly decreased the time to take products from

concept to the production line. In conclusion, benefits of FEM involve increased

accuracy, improved design and better understanding into critical design parameters,

virtual modelling, fewer physical prototypes, faster and less expensive design cycle

and increased productivity.

Fig. 1.5: The FEM application in crash-tests

FINITE ELEMENT METHOD 14

1.5 Limitations of FEM

Even trough, FEM is very universal and effective, and can enable designers to

obtain information about the behaviour of various structures with almost arbitrary

loads, the achieved results must be carefully examined before they can be used.

The most considerable limitation of the finite element method is that the

accuracy of the obtained solution is mostly a function of the mesh resolution. A

problem around point loads and supports is obvious. Large forces on a very small

area mean almost infinite stresses. These places with highly concentrated stress must

be carefully analysed with the use of an adequately refined mesh.

Reaching a solution with finite element method often requires a significant

amount of computer’s and user’s time and it is important not to underestimate user’s

knowledge and experiences. Present packages can solve many sophisticated

problems; there is a strong will to get results without doing the hard work of

thinking. Modern finite element packages are powerful tools that are expandable to

mechanical design and analyses. They become so approachable that a common user

has to avoid making easy mistakes.

1.6 Convergence criteria

The finite element method provides a numerical solution to an overall

problem. It may be expected that the solution converges to the exact solution under

certain circumstances. It can be shown that the displacement formulation of the

method leads to the actual stiffness of the structure. Hence the sequence of

progressively finer meshes is expected to convergence to the exact solution if

presumed element displacement fields satisfy certain criteria.

� The displacement field within an element must be continuous. It does not

yield a discontinuous value of function but more like a smooth variation of

function, which do not involve openings, overlap or jumps. This condition can

be satisfied by using a polynomials for the displacement model, for example:

2 3

W � C_{1 }� C_{2}x � C_{3}x � C_{4}x �... _{(1.3)}

� The displacement polynomial should include a constant term, representing a

rigid body displacement, which should occur at any unrestrained component

when subjected to external load. It also should contain linear terms, which on

differentiation give constant strain terms. Constant strain is the logical

condition as the element size reduces to a point in the limit.

� Compatibility of the displacement and its derivatives, up to required order,

must be satisfied across inter-element boundaries. Otherwise, the

displacement solution may result in separated or overlapped inter-element

boundaries.

FINITE ELEMENT METHOD 15

� Beside the convergence and compatibility requirements, one of the important

considerations in choosing proper terms is that the element should have no

preferred direction. It means that the displacement shapes will not change

with a change in local coordinate system. This property is known as geometric

isotropy.

1.7 Nonlinear solution methods

It is not possible to solve the nonlinear equation system in one single step as it

could be done for linear equation system. From this reason the solution must be

applying iterative algorithms.

When we use the loading as a number of increments it is called an

incremental-iterative solution method. These methods are also known as path

following algorithms, because the structural behaviour is examined for every step of

the load history and then for the complete load path.

In can be shown in equation 1.4, that total load vector F depends on n scalar

increase factors, the load factors λ_{i}. The single load vectors F_{i }represent the nodal

force values caused by random load collectives of point, line and surface.

F � �_{1}F_{1 }� �_{2}F_{2 }�... � �_{n}F

n

(1.4)

To simplify following considerations we can restrict to only one parameter

load system. It means, all load factors will be constant and only one will be

incremented. Simplification the load factor only one is remaining – λ and F_{0 }is a

reference load.

F

� �F

0

(1.5)

Algorithms that allow us to follow an arbitrary nonlinear path in the loaddisplacement-space

are called following algorithms. Two of them are introduced

next.

1.7.1 Newton-Raphson method

The Newton-Raphson method is one of the most frequently used iteration

scheme. It is based on the incremental solution concept. For a constant current load

level the iterative corrections of displacement are obtained from the out of balance

forces. From this reason the tangential stiffness matrix is used. The two following

attributes describe this method:

� It is pure force controlled method. The load level is constant and algorithm is

trying to iteratively find an equilibrium situation for that considered load

level.

FINITE ELEMENT METHOD 16

� While we use the exact tangential stiffness matrix in the iteration the algorithm

converges in a quadratic form. It is obvious that the error approximation

decreases with quadratic order from one to the next iteration.

Suppose that initial displacements d_{0 }are known, the first guess of nodal

displacement for load F is obtained by solving a set of linear algebraic equations

where

K_{T }_{(0)}

d

1

� F ^{(1.6)}

K

(

� K d )

T

0) T

(

0

(1.7)

is a tangent stiffness matrix calculated for initial displacements.

As the displacements d_{1 }are most probably not accurate, the equilibrium equation 1.8

(matrix representation of a set of nonlinear algebraic equations for unknown nodal

displacements d) is not satisfied

K d ) � F

(

1

(1.8)

and that means there are unbalanced (or residual) nodal forces

k

1

� K d ) � F

(

1

(1.9)

By calculating new tangential stiffness matrix

K

(

� K d )

T

1) T

(

1

(1.10)

and solving new set of algebraic linear equations

K

T (

�d � r

1) 1 1

(1.11)

we will get an improved solution

d

2

d_{1 }� �d_{1}

� (1.12)

k_{2 }� K d ) � F � 0

(

2

If the procedure is iterated until the sufficiently accurate solution is

obtained. The iteration process is shown in Fig. 1.6.

FINITE ELEMENT METHOD 17

Fig. 1.6: The Newton-Raphson method

1.7.2 Modified Newton-Raphson method

The modified Newton-Raphson method is kind of simpler way how to reach

equation equilibrium. The method is based on using the same tangent stiffness

matrix for all iterations. The stiffness is determined once at the beginning of the

iterative process and is used unchanged until the equilibrium is reached. The two

following attributes describe this method:

� It is pure force controlled method. The load level is constant and algorithm is

trying to iteratively find equilibrium for that considered load level.

� Unlike the standard Newton-Raphson method, the algorithm demonstrates

slower convergence, but there is no need to re-establish the stiffness matrix

after every iteration. On the other hand, number of iterations is usually larger.

There are some occasions, when the tangent stiffness must be modified,

especially when calculation is not able to converge due to zero stiffness of some

elements.

FINITE ELEMENT METHOD 18

Fig. 1.7: The Modified Newton-Raphson method

THEORY OF ELASTICITY 19

2 Theory of elasticity

2.1.1 Elasticity

Almost all structural materials exhibit an elastic response at a certain range of

deformation. For example, if some external forces produce deformation of a structure

and they do not reach the elastic limit, the deformation disappears and no residual

deformation remains.

2.1.2 Stress

In the most simplest case of a prismatic bar loaded by a tensile force uniformly

distributed over the ends (Fig. 2.1), we can assume the internal forces will be also

uniformly distributed over any cross section n-n’.

Fig. 2.1: Prismatic bar

Then the intensity of this distribution (the stress) can be obtained by dividing the

total tensile force F by the area A of the cross section n-n’.

In the general case, when stress is not distributed uniformly over cross section

(Fig. 2.2), to obtain the magnitude of stress appears on small area δA, cut out of the

section area n-n’ at any point 0, we assume that the forces appear across this

elemental area can be reduced to a resultant δF.

Fig. 2.2: General object

THEORY OF ELASTICITY 20

The limiting value δF/δA ratio presents the magnitude of the stress acting on the

cross section n-n’ at the point 0. Direction of the stress is the limiting direction of the

resultant δF.

In the general case when the stress direction is related to the area δA on which

it acts, the stress is composed of normal and shearing stresses acting in the plane of

the area δA.

2.1.1 Notation for forces and stresses

There are two kinds of external forces which may act on bodies. The first are

forces distributed over the surface of the body (for example the pressure of one body

to another or hydrostatic pressure). The second are forces distributed through the

volume of a body (for example gravitational forces, magnetic forces or if the body is

subjected to acceleration, inertial forces).

We use to mark normal stress with letter σ and shearing stress with letter τ. To

differ on which plane the stresses are acting it is used to add subscripts to these

letters. If we imagine a small cubic element at a point 0 (Fig. 2.3) with sides parallel to

the coordinate axes, the stress acting on the side of this element takes as positive

direction with the axis orientation.

Fig. 2.2: Infinitesimal cube subjected to stresses

The shearing stress has got similar notation except it consists of a two

subscripts. The first one indicates the direction of the normal to the plane and the

second indicates the direction of the component of the stress.