2E4: SOLIDS & STRUCTURES

Lecture 3

Dr. Bidisha GhoshNotes: http://www.tcd.ie/civileng/Staff/Bidisha.Ghosh/Solids & Structures

Stress-Strain Curves

•Ductile materials are those which can yield and undergo significant deformation in normal temperature before breaking.•Brittle materials rupture with little deformation.

strainstrain

stress

Stress

Tips• We will be always working within elastic limit• The slope of the stress-strain curve is E,

modulus of elasticity• E indicates stiffness, i.e. how much force is

needed for unit deformation.• Key equations:

• We can measure deformation and so we can measure strain.

• Measuring stress is very difficult….WHY?

3

;Pl

EAE

Terminology/jargons

Prismatic bar: A straight rodCross-section: A section perpendicular to the longitudinal axisAxial: Something along the longitudinal axisSimply supported: one end hinged/pinned and the other on rollersAll axial loads are centric if not specified otherwise, i.e. load passes through the centre of the cross-section.

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Stress: more formally

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.Along the cross-section aa, the particles/fibres are subjected to force P in aggregate. Internal forces develop on each particle.The resultant of the internal forces on aa is called a stress resultant. The stress resultant F and P should be collinear to ensure equilibrium.Assuming uniform cross-section, force per unit area (total area A) ,

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

P

A

Notes on stress

The definition is for uniform stress over a cross-section.This holds true if:1. The bar is homogenous and prismatic2. Centric loading3. Section under consideration is distant from

point of application of load

In case non-uniform stress distribution we work with average stress.

(St. Venant’s principle deals with the issue of non-uniform stress near boundaries.)

(Von Mises Stress: A sophisticated way of defining stress for ductile material)

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Units (SI) 1• Unit of distance metre(m) or millimetre (mm)

or centimetre (mm)

• Unit of force, Newton (N). Force can be expressed as mass,

1kilogram (kg) force = 1x9.81N ≈10N 1tonne = 9806N

• Unit of time seconds (sec)

1 3

21 10 10 ;1 10 ;mm cm mcm m

Units 2Stress = ; Unit is N/m2

This is called PASCALGenerally, MPa (Mega Pascal or 106Pa) is used to express stress. 1Mpa = 106Pa = 106N/m2 = = 1N/mm2

E or Young’s modulus has same units as stress but is often expressed in Gpa.

1Gpa = 103MPa = 1kN/mm2

PA

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6 2

10

10

N

mm

Example 1(Taken from Ugural, pg 33, ex 2.2 )

Find largest stress in the bar for P = 20KN(assume positive sign is for tensile normal stress)

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Shear Stress

A shear stress is produced whenever the applied forces cause one section of the to slide past the adjacent section.

Take the example of the clevis and bracket. The pin connection in them is under a condition of double shear (along mn and pq plane). From equilibrium, v = p/2

Direct Shear Stress

The average shearing stress (assuming uniform distribution)

Since this shear stress is caused by the direct action of applied force it is called direct shear.

However, shear stress is never as simple as discussed and generally cannot be assumed uniform or averaged out.

Direct shear occurs to an area parallel to applied load.

2avgs s

V P

A A

Bearing StressIn the clevis bracket diagram, the pin creates stresses in members connecting them along the bearing area or surface of contact.

Considering equilibrium, the stress is,

Bearing stress is a special type of normal stress that occurs on the surface between two separate interacting members.

It is assumed that the area of contact is the ‘projected contact area’ rather than the actual area.

(area of contact of clevis or area of contact of bracket)

b

p

P

P

A

M1.1

M1.1

M1.1

M1.1

M1.1

M1.1

M1.1

M1.1