# Chapter 1. Introduction

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30 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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Chapter 1. Introduction

1.1

Content of Theory of Elasticity

1.2 Important C
oncept in Theory of

Elasticity

1.3 Basic Assumptions

1.4

Problems

1.1 Contents of
Theory of Elasticity

Theory of Elasticity

Solid Mechanics

which deals with the
stress and displacements

in
elastic
solids

produced by
external forces

or
changes in temperature
.

The

purpose of stu
dy

strength, stiffness and stability

of structural and machine
elements.

Solid Mechanics I
----

bar

(
Mechanics of Materials
)

Solid Mechanics II
----

b
ar system

(
Structure Mechanics
)

bars

plates

†
Solid Mechanics

Solid Mechanics III
----

blocks

(
Theory of Elasticity
) dams

shells

Solid Mechanics VIII

(Theory of Plasticity)

beam

mech. of mater.

䙯爠數慭p汥†††

†
b敡e†††††††††††††††††⁴桥特⁯⁥污獴楣楴

†
p污瑥

†††††††
m散栠ef⁭慴敲a

䙯爠數慭p汥†††

†
p污瑥

†††

䩯楮琠 慰p汩捡瑩ln f 瑨攠 慢v攠
three branches

of solid
mechanics

-------------

Finite Element Method (FEM)

1.2 Some important concepts in

theory of elasticity

䕸瑥牮慬⁦牣敳

††

††
d敦牭慴楯n猠
ⴭ-

††

††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††††
†††††

bd楥i

g牡癩瑡瑩rn慬⁦牣攠††

†††††
⠱⤠(⁂dy⁦牣敳†††⁩湥牴楡⁦牣敳

d敦楮楴楯n⁯⁢dy⁦牣攺

F

=

(vector quantity)

Component of F
---

X, Y, Z, the projections of F

on x, y, z axis

Dimension is [force][length]
, e.x., N/m
.

Fig. 1.2.1

pressure (in water, atmosphere)

(2) Surface force

contact force

definition:

[force] [length]

Components of F along x, y, z axes denote
d by

Fig. 1.2.2

(3) The
internal forces

produced by
external forces

Stress
at a

point: definition

S =

n牭慬

(4) The
stress state

at
a point

Definition of the
stress component

and
its sign

( Note : differences with the def
inition in solid mechanics II)

Relations between shear stresses

We will show that the stress state on any section through the
point can be calculated if we know the 6 stress components,
i.e.,
the 6 stress components completely d
efine the stress
state at a point.

(5) Deformation:
By
deformation

we mean the change of
shape

of a body

6 strain components

completely define the
deformation condition

(or strain
condition) at that
point

(6) Displacement
:
By

displacement
(unit: length)

we
mean the change of position, the displacement components in
the x, y, z axes are denoted by u, v, w respectively.

All the above

at a point
vary
with the
position of the point considered, so th
ey are
functions of
coordinates

in space.

1.3 Basic assumptions in theory of elasticity

(1) The body is continuous, so

can be

expressed by continuous functions in space

(2) The body is perfectly elastic
----

wholly obeys
Hook's law of elasticity
----

linear relations between
stress components and strain components.

(3) The body is homogeneous , i.e., the elastic
properties are

the same throughout the

body
--
elastic constants will be

independent of the
loc
ation in the body.

(4) The body is isotropic so that the elastic properties
are the

same in all directions, thus the elastic
constants will be

independent of the orientation
of coordinate axes.

example:

†

y獴慬汩湥n捥牡r楣猠慮s獴敥汳

†
w潯搠慮a晩扥f牥楮景牣敤e捯c灯p楴i

⠵⤠周攠摩d灬慣p浥湴n慮a獴牡楮猠慲a獭慬氬椮攮,瑨攠

Problems (Exercise):

1.1.4, 1.1.2, new:

Chapter 2 Theory of Plane Problems

2.1 Plane Stress and Plane Strain

ⴭⴭ

plane stress

and
plane strain problems

(1) plane stress problem (2) Plane strain problem

and plane stress condition and plane strain condition

Example: thin plate Example: dam

2.2 Equation of Equilibrium
in
Plane Problems