strength of materials - basic concepts
TRANSCRIPT
-
8/10/2019 Strength of Materials - Basic Concepts
1/17
1/25/2008
Basic
Concepts
4
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
BASIC CONCEPTSBASIC CONCEPTS
PRINCIPLES: Governing principles
St. Venant's principle
INTERNAL FORCES AND STRESSES: Internal forces Direct stress
Shear stress
DISPLACEMENTS AND STRAINS: Displacements
Linear strain
Shear strain
MATERIAL BEHAVIOUR: Stress-Strain relationships
Superposition principle
Material properties
Strain energy
Material failure
Time effects
-
8/10/2019 Strength of Materials - Basic Concepts
2/17
1/25/2008
Basic
Concepts
5
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
GOVERNING PRINCIPLESGOVERNING PRINCIPLES
The way in which materials transmit loads is
governed by two basic principles:
Equilibrium: the sum of forces and moments on abody or any part of the body must be equal to zero.
Certain problems can be solved using only
equilibrium considerations. These are known as
statically determinate.
Compatibility: the movements resulting from the
external loads must be internally compatible (i.e. the
material must not break) and compatible with the
external support conditions.
F F Mx y= = = 0 0 0; ;
-
8/10/2019 Strength of Materials - Basic Concepts
3/17
1/25/2008
Basic
Concepts
6
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
ST.ST. VENANT'SVENANT'S PRINCIPLEPRINCIPLE
A useful further principle is St. Venants Principle: no
matter how complex the distribution of external
forces at a small region on the surface of a body is,
the resulting effect at a small distance away will onlydepend on the statically equivalent force.
-
8/10/2019 Strength of Materials - Basic Concepts
4/17
1/25/2008
Basic
Concepts
7
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
INTERNAL FORCESINTERNAL FORCES
Consider a bar with an external load and its Free
Body Diagram:
Taking a cut through a section of a bar, equilibrium
and Newtons third law of action and reaction implythe existence of an equal internal force acting on
each section of the bar:
On a given slice we have:
F F
F
F F F F
F F
-
8/10/2019 Strength of Materials - Basic Concepts
5/17
1/25/2008
Basic
Concepts
8
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
DIRECT STRESSDIRECT STRESS
Stress is the amount of internal force per unit area:
Units: Newtons/metre2 or N/m2 or Pascal. Typically
engineers use MN/m2, i.e. 106 N/m2 or N/mm2.
Stress can be tensile (+) or compressive (-):
FF
A=
F
A
Tension (+) Compression (-)
-
8/10/2019 Strength of Materials - Basic Concepts
6/17
1/25/2008
Basic
Concepts
9
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
SHEAR STRESSSHEAR STRESS
The force acting on an area may be normal or
tangential to the area. The direct stress is then the
normal force per unit area and the shear stress is
the tangential force per unit area:
Signs:
= =
F
A
F
An t
and Fn
F
FFt
A
x
y
+
-
8/10/2019 Strength of Materials - Basic Concepts
7/17
1/25/2008
Basic
Concepts
10
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
DISPLACEMENTSDISPLACEMENTS
As a result of the external actions materials will
deform. This deformation manifests itself in small
movements or displacements of material points. It
has units of length (m or mm):
In the case of a beam where the displacement is
perpendicular to the structure, it is known as
deflection:
u F
F
d
-
8/10/2019 Strength of Materials - Basic Concepts
8/17
1/25/2008
Basic
Concepts
11
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
LINEAR STRAINLINEAR STRAIN
All materials deform when subject to external actions
such as loads or temperature changes. The
deformation, i.e. change in shape is measured by the
strain at a point:
Linear Strain is defined as the change in length over
the initial length:
Strain is dimensionless. It is often given as a %.
=l
l
ll
l l+
-
8/10/2019 Strength of Materials - Basic Concepts
9/17
1/25/2008
Basic
Concepts
12
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
SHEAR STRAINSHEAR STRAIN
Deformation can also imply distortion which is
measured by the shear strain as the change inangle:
The shear strain is dimensionless and often given asa percentage %.
-
8/10/2019 Strength of Materials - Basic Concepts
10/17
1/25/2008
Basic
Concepts
13
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
STRESSSTRESS -- STRAIN RELATIONSHIPSSTRAIN RELATIONSHIPS
Derived from tensile tests:
Strain is related to stress via the stress-strain curve :
Proportionality
LimitLinear
Elastic
Range
Breaking
Point
Strain Gauge
F F
-
8/10/2019 Strength of Materials - Basic Concepts
11/17
1/25/2008
Basic
Concepts
14
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
SUPERPOSITION PRINCIPLESUPERPOSITION PRINCIPLE
In the linear elastic range the effect of more than one
load can be obtained by adding the effect of each
individual load acting alone:
F1
F2
F1
F2
=
+
-
8/10/2019 Strength of Materials - Basic Concepts
12/17
1/25/2008
Basic
Concepts
15
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
MATERIAL PROPERTIESMATERIAL PROPERTIES
In the elastic range, direct stress is proportional to
linear strain. The proportionality coefficient is
Youngs Modulus E of the material:
Shear stress is proportional to shear strain. The
proportionality coefficient is the Shear Modulus G:
Thermal effects. Changes in temperature lead to alinear strain which is proportional to the temperature
change. The proportionality coefficient is the
Coefficient of Thermal Expansion :
=E
= G
T T=
-
8/10/2019 Strength of Materials - Basic Concepts
13/17
1/25/2008
Basic
Concepts
16
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
POISSON'S RATIOPOISSON'S RATIO
The result of a direct stress in one direction is a
direct strain in the same direction plus a lateral
strain:
The ratio between direct and lateral strain is given byPoissons coefficient (typically 0.3):
l
dd
l
1 2= = l
l and
d
d
=
2
1
-
8/10/2019 Strength of Materials - Basic Concepts
14/17
1/25/2008
Basic
Concepts
17
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
MATERIAL PARAMETERS: Typical ValuesMATERIAL PARAMETERS: Typical Values
Material E(GN/m2) (
oC
-1) u(MN/m
2) l(MN/m
2)
Mild Steel 200 1.2 10-5
370 280
High Steel 200 1.3 10-5 1550 770
Concrete T 14 1.2 10-5
3 -
Concrete C 14 1.2 10-5
30 -
Carbon Fibre 170 - 1400 -
Glass Fibre60 - 1600 -
Aluminium 70 2.3 10-5
430 280
Titanium 120 0.9 10-5
690 385
Magnesium 45 2.7 10-5
280 155
-
8/10/2019 Strength of Materials - Basic Concepts
15/17
1/25/2008
Basic
Concepts
18
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
STRAIN ENERGYSTRAIN ENERGY
When a material is deformed, the work done by the
external forces is accumulated as elastic strainenergy in the material.
The strain energy per unit volume w is the areaunder the stress-strain relationship:
For linear elastic materials w is:
F
W F A V = = =z zzd d dl l
F
w E E= = =12
12
2 12
2
w= z ( ) d
-
8/10/2019 Strength of Materials - Basic Concepts
16/17
1/25/2008
Basic
Concepts
19
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
MATERIAL FAILUREMATERIAL FAILURE
All materials fail at different values of stress.
Depending on the amount of strain (or strain energy)
before failure, the material is said to be brittle or
ductile:
Breaking Point Breaking Point
DUCTILE MATERIAL BRITTLE MATERIAL
-
8/10/2019 Strength of Materials - Basic Concepts
17/17
1/25/2008
Basic
Concepts
20
School of
Engineering
Prof. J. BonetProf. J. Bonet
EGEG--120120
Strength ofStrength of
MaterialsMaterials
TIME EFFECTSTIME EFFECTS
Creep: the deformation of materials under load
increases with time:
Fatigue: materials subject to cyclic loads eventuallyfail at a lower than the short term failure stress:
Mild Steel
Aluminium
Endurancelimit
tertiary creep
secondary creep
primary creep
t
u
No. Cycles104 105 107106