Strength of Materials - Basic Concepts

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


2/17

1/25/2008

Basic

Concepts

5

School of

Engineering


EGEG--120120


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; ;


3/17

1/25/2008

Basic

Concepts

6

School of

Engineering


EGEG--120120


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.


4/17

1/25/2008

Basic

Concepts

7

School of

Engineering


EGEG--120120


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


5/17

1/25/2008

Basic

Concepts

8

School of

Engineering


EGEG--120120


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 (-)


6/17

1/25/2008

Basic

Concepts

9

School of

Engineering


EGEG--120120


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

+


7/17

1/25/2008

Basic

Concepts

10

School of

Engineering


EGEG--120120


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


8/17

1/25/2008

Basic

Concepts

11

School of

Engineering


EGEG--120120


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+


9/17

1/25/2008

Basic

Concepts

12

School of

Engineering


EGEG--120120


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 %.


10/17

1/25/2008

Basic

Concepts

13

School of

Engineering


EGEG--120120


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


11/17

1/25/2008

Basic

Concepts

14

School of

Engineering


EGEG--120120


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

=

+


12/17

1/25/2008

Basic

Concepts

15

School of

Engineering


EGEG--120120


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=


13/17

1/25/2008

Basic

Concepts

16

School of

Engineering


EGEG--120120


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


14/17

1/25/2008

Basic

Concepts

17

School of

Engineering


EGEG--120120


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


15/17

1/25/2008

Basic

Concepts

18

School of

Engineering


EGEG--120120


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


16/17

1/25/2008

Basic

Concepts

19

School of

Engineering


EGEG--120120


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


17/17

1/25/2008

Basic

Concepts

20

School of

Engineering


EGEG--120120


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

Strength of Materials - Basic Concepts

Documents

Transcript of Strength of Materials - Basic Concepts