Finite elements method for ship structure

2
Courtesy:
Dr. H. “Jerry” Qi;
University of Colorado

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Ship Collision
Impact Analysis

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Different Types of Structure

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 Created as numerical techniques for finding
approximate solutions to boundary value
problems for partial differential equations;
 FEM is based on a problem domain’s subdivision
into simpler parts—called finite elements—, and
on the calculus of variational methods to minimize
an associated error function.

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Your final grade depends on the overall
performance of the class.
For the evaluation, we will follow the MIST
standard instructions
But don’t worry about those;
Just lets try to understand the things

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Basic ideas of the finite element method
originated from advances in aircraft structural
analysis.
1941: Hrenikoff presented a solution of
elasticity problems using the “frame work
method”;
1943: Courant’s paper, which used piecewise
polynomial interpolation over triangular sub
regions to model torsion problems;
1955: A book by Argyris on energy theorems
Brief History [Chandrupatla and Belegundu]

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1956: Turner, et al. derived stiffness matrices
for truss, beam and other elements and
presented their findings;
1960: The term finite element was first coined
and used by Clough;
Early 1960s: Engineers used the method for
approximate solution of problems in stress
analysis, fluid flow, heat transfer and other
areas.
1967: The first book on finite elements by
Brief History

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Early 1970s: FEA was applied to nonlinear
problems and large deformations;
1972: Oden’s book on nonlinear cotinua
appeared;
1970s: Mathematical foundations were laid.
New element bdevelopment, convergence
studies and other related areas fall in this
category.
Brief History

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For Analyzing the Structures,
You need to know behavior of Structure
Lets review some basics of
Mechanics of Structures

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Fundamental of Stress Analysis
 Stress,
A
P


Unit is force per unit area and is denoted by the Greek letter sigma.
 Tensile Stress: When the bar is stretched by the
forces;
 Compressive Stress: If the forces are reversed in
direction, causing the bar to be Compressed;
 Normal Stress: Stresses act in a direction
perpendicular to the surface;
Normal stresses may be either tensile or compressive.
 Shear Stress: Stress acts parallel to the surface;
 Sign Convention: Tensile positive (+),
Compressive negative (-)

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Units of Stress and Some Basic Definitions
 Prismatic Bar: Straight structural member having the same cross
section throughout its length;
 Axial Force: Load directed along the axis of the member;
 Cross section: Section perpendicular to the longitudinal axis of
Structure;
 Units
USCS: psi (pounds per square inch), ksi (kilopounds per square inch)
SI: N/m2
(Newton per meter square - Pascal)
1 MPa = ?? N/m2
; 1 Mpa = ?? N/mm2
;

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Average Stress
The equation is valid only if the stress is
uniformly distributed over the cross section A i.e., if
force P acts through the centroid of the cross-
sectional area.
A
P


If the stress is not uniformly distributed, the stated
Equation is useful to calculate the average normal
stress on the cross section.

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Strain, 
 Strain, (elongation per unit length)
Where, : total elongation = (final – initial) length
L = initial length , Strain has no dimension
L

 
 If the bar is in tension, the strain is called a
tensile strain, representing an elongation or
stretching of the material.
 If the bar is in compression, the strain is a
compressive strain and the bar shortens.
 Tensile strain is usually taken as positive and
compressive strain as negative.
 A normal strain is associated with normal
stresses.

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Uniaxial Stress and Strain
 If the deformation is uniform throughout the volume, which requires
the bar be prismatic, the loads act through the centroids of the cross
sections, and the material be homogeneous (that is, the same
throughout all parts of the bar). The resulting state of stress and strain is
called uniaxial stress and strain.

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Mechanical Properties of Material
 The slope of the straight line from O to A is called the
modulus of elasticity;

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 Considerable elongation occurs with no noticeable increase in the tensile force
(from B to C). This phenomenon is known as yielding of the material, and point B
is called the yield point. The corresponding stress is known as the yield stress of
the steel.
 In the region from B to C the material becomes perfectly plastic, which means
that it deforms without an increase in the applied load.

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 The load eventually reaches its maximum value, and the corresponding stress (at
point D) is called the ultimate stress.
 Further stretching of the bar is actually accompanied by a reduction in the load,
and fracture finally occurs at a point such as E in Fig.
 The yield stress and ultimate stress of a material are also called the yield
strength and ultimate strength, respectively.

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 If the actual cross-sectional area at the narrow part of the neck
is used to calculate the stress, the true stress-strain curve (the
dashed line CE in Fig) is obtained. The total load the bar can carry
does indeed diminish after the ultimate stress is reached (as
shown by curve DE), but this reduction is due to the decrease in
area of the bar and not to a loss in strength of the material itself.
In reality, the material withstands an increase in true stress up to
failure (point E);
 Because most structures are expected to function at stresses
below the proportional limit, the conventional stress-strain curve
OABCDE, which is based upon the original cross-sectional area of
the specimen and is easy to determine, provides satisfactory
information for use in engineering design;

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Hooke’s Law
E is a constant of proportionality known as the modulus of elasticity for the
material. The modulus of elasticity is the slope of the stress-strain diagram in
the linearly elastic region. The units of E are the same as the units of stress.
Hooke’s law express the linear relationship between
stress and strain in simple tension or compression:

 E

More flexible materials have a lower modulus—plastics - from 0.7 to 14 Gpa
More stiff materials have a higher modulus—steel - 210 Gpa (approx.)
M o d u l u s o f e l a s ti c i t y i s o ft e n c a l l e d Yo u n g ’s m o d u l u s
Poisson’s Ratio
 Why “-” sign??
 Applicable only in uniaxial loading;
 Applicable for linearly elastic material;

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A
V


Stress
Shear
Average
1. Shear stresses on opposite (and
parallel) faces of an element are
equal in magnitude and opposite
in direction.
2. Shear stresses on adjacent (and perpendicular) faces of an element are
equal in magnitude and have directions such that both stresses point
toward, or both point away from, the line of intersection of the faces.
Acts tangential to the surface of the material;
Shear Stress
2
1 
 
requires
Condition
m
Equilibriu

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Shear Strain
Shear stresses have no tendency to
elongate or shorten the element in the x,
y, and z directions—Instead, the shear
stresses deform the element.
In picture, The angle  is a measure of
the distortion, or change in shape, of the
element and is called the shear strain.
Because shear strain is an angle, it is
usually measured in degrees or radians.
Positive and Negative Faces
A positive face has its outward normal
directed in the positive direction of a
coordinate axis. The opposite faces are
negative faces.

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Axially Loaded Members
Springs
K (Stiffness Constant): The force required to produce a unit
elongation;
f (flexibility constant): Elongation produced by a load of unit value.
Equations are also applicable to springs in compression.

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Prismatic Bars
Uniform Normal Stress,  = P/A
Axial Strain,  = /L
Longitudinal Stress,  = E
EA
PL

 
ns
Combinatio
Equations
Product EA  Axial Rigidity of the bar.
EA
L
f
L
EA
k




Bar
Prismatic
a
of
y
Flexibilit
Bar
Prismatic
a
of
Stiffness

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S T R E S S E L E M E N T S
 The most useful way
of representing the
stresses is to isolate
a small element of
material;
 An element of this
kind is called a stress
element.
 The dimensions of a
stress element are
assumed to be
infinitesimally small,
but for clarity we
draw the element to
a large scale;

Analysis of Stress and Strain
When the material is in plane stress in the xy plane, only
the x and y faces of the element are subjected to
stresses, and all stresses act parallel to the x and y axes.
A normal stress  has a subscript that identifies the
face on which the stress acts. The sign convention for
normal stresses is the familiar one, namely, tension is
positive and compression is negative.
A shear stress  has two subscripts—the first subscript denotes the face on
which the stress acts, and the second gives the direction on that face.
A shear stress is positive when the directions associated with its subscripts are plus-plus
or minus-minus; the stress is negative when the directions are plus-minus or minus-plus.
from equilibrium of the element:

Special Case – I (Uniaxial Stress)
(All stresses acting on the element are zero except for the normal stress x)
Special Case – II (Pure Shear)
(x = 0 and y = 0)
Special Case – III (Biaxial Stress)
(Element is subjected to normal stresses in both the x and y directions but
without any shear stresses)

Equilibrium of Element Volume
[Condition of Equilibrium]

Equilibrium of Element Volume
[Equations of Equilibrium]
Assignment #
01.01
Hint: 2.3 of S.S.
BHAVIKATTI

Physical problem
Mathematical model
Governed by differential equations of assumed
discrete system (discretized by appropriate finite
elements) with assumptions on loading, boundary
conditions, etc
Finite element solution
Improvement
Design improvements
Structural optimization
Assessment of results
The process of
FEM

On the front side of the
base plate, a uniform
normal pressure 100
MPa is applied and the
opposite side is
constrained.
Physical problem Finite element model
Results

Introduction to Finite Element Analysis
(FEA) or Finite Element Method (FEM)
 Finite element method (FEM) is one of the numerical
methods of solving differential equations that describe
many engineering problems. The FEM, originated in
the area of structural mechanics, has been extended to
other areas of solid mechanics and later to other fields
such as heat transfer, fluid dynamics, and electro-
magnetism

 Useful for problems with complicated geometries,
loadings, and material properties where analytical
solutions cannot be obtained.
 When a structural problem is given, it is important to
understand the following steps:
 Creation of the FE model of the given problem;
 Applying the boundary conditions and the loads;
 Solution of the finite element matrix equations;
and
 Interpretation and verification of the FE results.

FINITE ELEMENT ANALYSIS PROCEDURES
 Finite element analysis involves dividing the
structure into a set of contiguous elements. This
process is called discretization.
 Each element has a simple shape such as a line, a
triangle, or a rectangle, and is connected to other
elements by sharing "nodes." The unknowns for
each element are the displacements at the nodes.
These are also called degrees of freedom.
Displacement boundary conditions and applied
loads are then specified.

 The element level matrix equations are assembled to
form global level equations. The global matrix
equations are solved for the unknown
displacements, given the forces and boundary
conditions. From the displacements at the nodes,
strains and then stress in each element are
calculated. However, in practice there are many
difficulties in solving the real-life problems using
finite elements.

Advantages of the Finite Element
Method
The finite element method has been applied to
numerous problems, both structural and
nonstructural. This method has a number of
advantages that have made it very popular. They
include the ability to:
• Model irregularly shaped bodies quite easily;
• Handle general load conditions without
difficulty;
• Model bodies composed of several different
materials because the element equations are

• Handle unlimited numbers and kinds of
boundary conditions;
• Vary the size of the elements to make it
possible to use small elements where
necessary;
• Alter the finite element model relatively
easily and cheaply;
• Include dynamic effects;
• Handle nonlinear behavior existing with
large deformations and nonlinear materials.

The finite element method of structural analysis
enables the designer to detect stress, vibration, and
thermal problems during the design process and to
evaluate design changes before the construction of a
possible prototype. Thus confidence in the
acceptability of the prototype is enhanced. Moreover,
if used properly, the method can reduce the number
of prototypes that need to be built.

Even though the finite element method was initially
used for structural analysis, it has since been adapted
to many other disciplines in engineering and
mathematical physics, such as fluid flow, heat transfer,
electromagnetic potentials, soil mechanics, and
acoustics.

Assignment # 01.02
Hint: Appendix C.3 [Daryl L. Logan]
Derive equations for total strain for an isotropic
body subjected to triaxial stress

Assumption: The principle of superposition is hold; that is,
we assume that the resultant strain in a system due to
several forces is the algebraic sum of their individual
effects.

Assignment # 01.03
Hint: Appendix C.3 [Daryl L. Logan]
Stress - Strain relationship or constitutive matrix
for linear elastic and isotropic material

Plane Stress
from equilibrium of the element:

Methods of FEM and
General Steps

The Direct Approaches of FEM:
In the structural stress-analysis, determination of displacements and
stresses throughout the structure is key problem, which is in
equilibrium and is subjected to applied loads. For many structures, it is
difficult to determine the distribution of deformation using
conventional methods, and thus the finite element method is
necessarily used.
Traditionally there are two general direct approaches:
Force, or flexibility method
Unknown: Internal forces;
Governing Equations: First the equilibrium equations are used. Then
necessary additional equations are found by introducing compatibility
equations.
Result: A set of algebraic equations for determining the redundant or
unknown forces.

Displacement or stiffness method
Unknown: Displacements of the nodes;
Compatibility conditions requiring that elements connected at a
common node, along a common edge, or on a common surface before
loading remain connected at that node, edge, or surface after
deformation takes place are initially satisﬁed.
Governing Equations: Expressed in terms of nodal displacements
using the equations of equilibrium and an applicable law relating
forces to displacements.

Advantages of Displacement or stiffness method over Force, or
flexibility method:
 The two direct approaches result in different unknowns (forces or
displacements) in the analysis and different matrices associated
with their formulations (flexibilities or stiffnesses);
 For computational purposes, the displacement (or stiffness)
method is more desirable because its formulation is simpler for
most structural analysis problems. Furthermore, a vast majority of
general-purpose finite element programs have incorporated the
displacement formulation for solving structural problems.

Variational Method:
 The variational method includes a number of principles:
 The principle of minimum potential energy that applies to
materials behaving in a linear-elastic manner;
 The principle of virtual work. This principle applies more
generally to materials that behave in a linear-elastic fashion, as
well as those that behave in a nonlinear fashion.

• The primary characteristics of a finite element are
embodied in the element stiffness matrix;
• For a structural finite element, the stiffness matrix
contains the geometric and material behavior
information that indicates the resistance of the
element to deformation when subjected to loading;
• Such deformation may include axial, bending, shear,
and torsional effects;
• For finite elements used in nonstructural analyses,
such as fluid flow and heat transfer, the term
stiffness matrix is also used, since the matrix
represents the resistance of the element to change
when subjected to external influences.
Stiffness Matrix

 Spring Element:
 Introduction;
 Derivation of the Stiffness Matrix for a
Spring Element;
 Step 1 Select the Element Type;
 Step 2 Select a Displacement
Function;
 Step 3 Define the
Strain/Displacement and
Stress/Strain Relationships;
 Step 4 Derive the Element
Stiffness Matrix and Equations;
 Step 5 Assemble the Element
Equations to Obtain the Global
Equations and Introduce
Boundary Conditions;

 Spring Element:
 Example of a Spring Assemblage;
 Assembling the Total Stiffness Matrix by
Superposition (Direct Stiffness Method).
 Boundary Conditions:
 Homogeneous boundary conditions;
 Nonhomogeneous boundary conditions.

Example: 3.1 – 3.7
-- Daryl L. Logan

Different methods for deriving the
Element Stiffness Matrix and
Equations
Direct
Equilibrium
Method
Force or
ﬂexibility
method
Displacement
or stiffness
method
Methods of
Weighted
Residual/
Galerkin’s
Method
Principle of
virtual work
(elasticity
problem)
Variational
Method
Principle of
minimum potential
energy
Rayleigh – Ritz
Method
Castigliano’s
Theorem

 Galerkin’s Method:
 This is a method which can be applied to any
problem involving solution of a set of equations
subject to specified boundary values;
 In elasticity problems Galerkin’s method turns out
to be the principle of virtual work which may be
stated as, “a deformable body is in equilibrium
when the total work done by external forces is equal
to the total work done by internal forces”;
 The work done considered in the above derivation is
called virtual, since the forces and deformations
considered are not related.

 In calculus we know a function has extreme value
when its first derivative with respect to variables is
zero. The function is maximum, if the second
derivative is negative and is minimum, if its second
derivative is positive;
 The first derivative of function of a function is called
first variance. The function of a function is termed a
functional and the statement that the first variance of
functional is zero is termed as first variance attains a
stationary value;
 In many engineering problems there are such
functional, the first variance of which attain stationary
values. In elasticity problems potential energy of the
body of the structure is such functional.

 In solid mechanics it has been identified that total
potential energy is suitable functional, the first
variance of which yields equation of equilibrium
satisfying the boundary conditions;
 A deformable body is in equilibrium when the total
potential energy is having stationary value. By taking
second variance of potential energy, it has been
proved by researchers that the value is positive
definite;
 And hence it is concluded that the condition that value
of total potential energy is stationary correspond to
minimum value;

 Hence we have principle of minimum potential energy
in solid mechanics, which. may be stated as “of all the
possible displacement configurations a body can
assume which satisfy compatibility and boundary
conditions, the configuration satisfying equilibrium
makes the potential energy assume a minimum value”.
This is the variation principle in solid mechanics.

Finite elements method for ship structure

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