1/1 SOE 1032 SOLID MECHANICS

Website www.ex.ac.uk~TWDavies/solid_mechanics

course organisation,lecture notes, tutorial problems,deadlines

Course book:J M Gere, Mechanics of Materials, Nelson Thornes, 2003, £29.

1/2 TUTORIALS Three groups, A, B, C Three sessions Monday, Tuesday and Friday

weeks 8, 9, 10, 11

1/3 LABORATORY SESSIONS

ONE DEMONSTRATION Week 10 Write up deadline Friday week 11 Hand in to 307 and get it date

stamped.

1/4 LEARNING TRIANGLE

ME YOU

BOOK

1/5 DEFORMATIONS to be studied

Static objects Extension or compression of a rod

under an axial load Twisting of a rod by applied torque Bending of a beam subjected to

point loads, uniformly distributed loads and bending moments

1/6 BASIC SCIENCE USED Newton’s 3rd Law, equilibrium

(STATICS)

Auxiliary relationships based on material properties – e.g. Hooke’s Law

1/7 POSSIBLE MATERIALS

NATURAL

MANUFACTURED

1/8 NATURAL MATERIALS

WOOD

STONE

1/9 MANUFACTURED MATERIALS METALS (examples used in this

course)

PLASTICS

CONCRETE AND BRICK

CERAMICS AND GLASS

1/10 SCOPE OF COURSE STRESS AND STRAIN IN SIMPLE

SYSTEMS DEFORMATIONS IN TENSION,

COMPRESSION & TORSION STRESS & BENDING IN BEAMS MOHR’S CIRCLE i.e. essential parts of Chapters 1 to

5 and 7 (see reading list).

1/11 NORMAL STRESS Axial force per unit X-sectional area P/A = N/m2 or Pa (like pressure) Tensile stress (positive) Compressive stress (negative) eg m=100 kg held by rod of A = 1

cm2

g = 10ms-2

=P/A = mg/A = 1000/10-4 = 10 MPa

1/12 SHEAR STRESS

TANGENTIAL FORCE PER UNIT AREA

P/A = N/m2 or Pa

1/13 NORMAL STRAIN Change in length caused by

normal stress = /L (dimensionless) Tensile strain Compressive strain eg = 2 mm, L = 2 m, then = 1

mm/m or = 0.1%

1/14 UNIAXIAL STRESS Conditions are that: Deformation is uniform throughout

the volume (prismatic bar) which requires that:

Loads act through the centroid Material is homogeneous See Section 1.2 in book

1/15 LINE OF ACTION OF AXIAL FORCES FOR UNIFORM STRESS DISTRIBUTION

Prismatic bar of arbitrary cross-section A

Axial forces P producing uniformly distributed stresses = P/A

1/16 BALANCE THE MOMENTS

1/17 Mechanical Properties Strength

compression tension shear

Elasticity, plasticity, ductility,creep Stiffness, flexibilty Used to relate deformation to

applied force

1/18 Mechanical Testing Tensile test machine

1/19 Tensile test for mild steel

1/20 Nominal and true SS Nominal stress based on initial area True stress based on necked area Nominal strain based on initial

length True strain based on current length Use nominal values when operating

within elastic limit

1/21 Test data – linear scale

1/22 TENSILE TEST SPECIMEN

1/23 BRITTLE MATERIAL

1/24 CAST IRON

-1000

-800

-600

-400

-200

0

200

400

-0.03 -0.02 -0.01 0 0.01

Compression

Tension

1/24 STONE

Compression

Tension

Strain

Stress

5-200MPa

1/26 WOOD Not an isotropic or homogeneous

material Stronger and stiffer along the grain Stronger in tension than compression Fibres buckle in compression Very high strength/weight ratio Very high stiffness/weight ratio

1/27 COMPRESSION TEST

1/28 Compression test - concrete

1/29 ELASTICITY

1/30 PLASTICITY

1/31 DUCTILITY

1/32 CREEP

1/33 LINEAR ELASTICITY STRAIGHT LINE PORTION OF

STRESS-STRAIN CURVE = E. (Hooke’s Law) E is the modulus of elasticity or

Young’s Modulus and is the slope of the curve

For stiff materials E is high (steel 200GPa)

For plastics E is low (1 to 10 GPa)

1/34 POISSON’S RATIO Tensile stretching of a bar results in

lateral contraction or strain (and v v) For homogeneous materials axial

strain is proportional to lateral strain Poisson’s Ratio = - (lateral/axial

strain) = - (’ / ) For a bar in tension is positive and ’

is negative, and v.v. for compression.

1/35 Poisson (1781-1840)

Normal values 0.25-0.35 for metals Concrete about 0.2 Cork about 0 (makes it a good

stopper) Auxetic materials have NEGATIVE

1/36 Axial and lateral deformation

L

BB..

= /L

1/37 Volume change V1=L*B2 (original volume)

V2=L(1+ )*(B(1- * ))2 (final volume)

V2=L B 2(1+ -2 -22+ 22 +32)

V2L B 2(1+ -2)

1/38 DILATION V=L B 2 (1 -2) (change in volume) V/V1=(1 -2) or (1 -2)/E = DILATION

Max value of is 0.5 Since is 1/4 to 1/3 then dilation is /3 to /2

1/39 SUMMARY Stress - Force/Area

Strain – (Extension or compression)/Length

For some materials and subcritical loads strain is proportional to stress (Hooke)

Change in length proportional to change in width (Poisson). Shrinkage/expansion.

Characteristics determined by experiment