Chapter 1. Introduction

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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



楳i瑨攠b牡湣rf
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



楳i瑯捨散欠瑨攠獵ff楣楥i捹f
瑨攠
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.




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†
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




†††
瑨敯特⁥污獴






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




††
却牥獳敳†⡩湴敲(慬⁦牣攩




††
d敦牭慴楯n猠
ⴭ-

獴牡楮猠慮d⁤楳p污捥l敮t


††
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牮慬⁦牣敳⁴桡琠慣琠n⁴桥

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†††††

bd楥i









g牡癩瑡瑩rn慬⁦牣攠††

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⠱⤠(⁂dy⁦牣敳†††⁩湥牴楡⁦牣敳

楮瑩tn)







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

獴牥獳
n牭慬⁣潭pn敮琩








獨敡爠獴牥獳†⡳(敡爠捯mpn敮琠t














(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:

†
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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





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摩d灬慣p浥湴n捯c灯p敮瑳e潦慬a灯楮瑳t潦瑨攠扯by
摵物湧n摥景d浡瑩潮m慲a癥vy獭慬氠捯
浰慲m搠w楴栠
楴猠潲杩湡氠

摩浥湳楯d献



Problems (Exercise):



1.1.4, 1.1.2, new:




Chapter 2 Theory of Plane Problems


2.1 Plane Stress and Plane Strain






獰慴楡氠灲ab汥ls






慮攠灲eb汥l
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