2E4: SOLIDS & STRUCTURES

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30 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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2E4: SOLIDS & STRUCTURES


Lecture 2

Dr.
Bidisha

Ghosh

Notes:
http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/So
lids & Structures



Statics









Statics
or the
bahaviour

of the rigid bodies under external
load
is studied using the three Newton’s Laws

(friction is
considered as well)



Mechanics of solids
deals with the internal changes/effects
of a body(deformable) to the application of external load.



Mechanics of Solids/Strength of materials/ Mechanics of
Materials/ Mechanics of deformable bodies






2

http://www.beam
-
wiki.org/wiki/Degree_of_freedom

http://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)

Degrees of Freedom (DOF)



3

There are 6 DOF for a rigid body;

When multiple rigid bodies are connected to develop a system, then the
number of DOF increases according to the number of bodies in the
system.


At the point of connection of two rigid bodies some DOF are suppressed
: ‘The concept of CONSTRAINTS’




Support Reactions


If a support prevents translation of a body in a given direction, a
reaction
force
is developed on the body in that direction.


If a support prevents
rotation
of a body in a given direction, a reaction
moment
is developed on the body in that direction
.


3 common types: roller, pinned/hinged, fixed


4

Equilibrium of Rigid Bodies


When a body is rigid, it does not deform.


When
the system of forces acting on a body has a zero resultant, then the
body is in equilibrium
.


Equilibrium of forces and moments
:


6 DOF: There can be 3 unknown displacements along axes and 3 unknown
rotations about axes. (Consider Cartesian Axis,
3D)


a)
Number of unknown displacements and rotations = 6


b)
Sums of forces along each axis = 0 give rise to 3 equations
.


c)
Sums of moments about each axis = 0 give rise to 3 equations.


d)
Number
of available equations
= no of unknowns: Unique
solution.


For (2D) planar problems,


Force
: Sum of forces in each direction is 0


Moment: Sum of moments about a point
is 0


3 equations and 3 unknowns




5

Determinacy

If a structure/machine can be solved using the equations of
equilibrium, then the structures are statically determinate.


http://wwwtw.vub.ac.be/werk/Mechanicasite/Statica/equilib/iv
-
vi.htm

http://www.iowadot.gov/subcommittee/bridgeterms.aspx


6

Process of solving STATICS problem

(summary of last 5 slides)

1.
What external loads are given?

2.
What are the support reactions?

Typically, the
reactions are
the unknowns of the problem.


3.
Draw Free Body
Diagram

The rigid bodies should be in force balance. Check if there is any unbalanced
force and sort it!


4.
Write
down the equations of force
equilibrium

a)
Define positive direction




b)
Sum up all forces in x
-
direction,

c)
Sum up all forces in y
-
direction,

d)
Take a point and sum up the moments about it,

e)
Solve the equations to find the unknowns.




7

Example
(As shown in class)

8

What is Mechanics of Solids??

What happens when bodies are not rigid but deformable??


Three main concepts:

1.
Stress

2.
Strain

3.
Hooke’s Law


What happens to deformable bodies when load is applied
?


Next two slides on what are the different types of loadings
possible!




9

External Loading

There are four basic types of loading (in order of
complexity).

Tension

Compression

Torsion

Bending

Sometimes, two or more basic types of loading can act
simultaneously on a member of a structure or machine.









Example of External Loading

This is a compression testing machine.

The different members are under different types of loading.

1.
The specimen tested is under compression.

2.
The two side bars (N) are under tension.

3.
The screw is subjected to twist or torsion.

4.
The crosshead is under bending.



Strain

A body responds to the application of
external forces by deforming and by
developing an internal force system, the
body keeps changing shape or
deforming until equilibrium is reached
between the external and internal
forces.


The intensity of deformation is called
STRAIN.

Deformation per unit length is called
strain.







25th May, 2010

Notes on Strain


Strain is the most significant factor in deformation
analysis.



Strain is involved in the experimental technique of
measuring stress. Stress is not measurable, but strain is.



Strain is a dimensionless and often expressed in 10
-
6
(
microstrain
)



Deformation should be compatible. Each deformed
portion of the member must fit together with adjacent
portion.

25th May, 2010

Internal Force: Stress

A body responds to the application of external
forces by deforming and by developing an
internal force system to hold together the
particles which forms the body.


The intensity of internal force is called STRESS.


The bar subjected to force P

From equilibrium, along the section
aa
, the particles/fibres are
subjected to force P in aggregate.

Assuming uniform cross
-
section, force per unit area

(total area A) ,





Diagram : Mechanics of materials (T.A. Philpot)





Hooke’s Law

A material which regains its shape when the external load
is removed is considered as ‘perfectly elastic’.



From tensile tests, it can be seen within the range of elastic behaviour of a
material the elongation is proportional to both the external load and the
length of the bar.




For linearly elastic materials, this Stress is proportional to
strain.




The factor of proportionality between stress and strain is
called, ‘Modulus of Elasticity’ or Young’s modulus.


E has the dimension of stress

25th May, 2010

How do we understand the behaviour of
deformable bodies?

How do we analyse the behaviour of deformable bodies when
external loads are applied on them??


THEORY OF ELASTICITY


This course is an introduction to the ‘Theory of Elasticity’


The principals of analysis of deformable bodies depend on:


Equilibrium Conditions:
Things should be in equilibrium

Material Behaviour:
Things should follow a force
-
deformation relation

Compatibility:
Deformations from all sides must match


16