Axial Loading

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MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

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Axial Loading

Materials for this chapter are taken from :
1. 2. Ferdinand P. Beer, E. Russell Johnston,Jr, John T. Dewolf, David F. Mazurek “ Mechanics of Materials” 5th Edition in SI units R.C.Hibbeler “ Mechanics of Materials “ Seventh Edition Ch 2 - 1

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Introduction
Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced under loading. Statics analyses alone are not sufficient. Considering structures as deformable allows determination of member forces and reactions which are statically indeterminate. Determination of the stress distribution within a member also requires consideration of deformations in the member. Chapter 2 is concerned with deformation of a structural member under axial loading. Later chapters will deal with torsional and pure bending loads.

Ch 2 - 2

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Saint-Venant’s Principle
Saint-Venant’s Principle

states that both

localized deformation and stress tend to “even out” at a distance sufficiently removed from these regions.

Ch 2 - 3

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Normal Strain under Axial Loading

P σ = = stress A

ε=

δ
L

2P P σ = = 2A A

= normal strain

ε =

δ

L

P σ= A 2δ δ = ε= 2L L
Ch 2 - 4

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Stress-Strain Test

rutlandplastics.co.uk

www.tensilkut.com

deeshaimpex.com

Tensile Test Machines & Specimens
Ch 2 - 5

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Stress-Strain Test: Ductile Materials

necking rupture

Test specimen with tensile load elongation occurs until “necking” and rupture. Ch 2 - 6

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Stress-Strain Test: Brittle Materials

Pottery, glass, and cast iron Material that breaks suddenly under stress at a point just beyond its elastic limit Brittle materials may also break suddenly when given a sharp knock

Ch 2 - 7

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Deformations of Members under Axial Loading
From Hooke’s Law:

σ = Eε

ε=

σ
E

=

P AE

From the definition of strain:

ε=

δ
L

Equating and solving for the deformation,

PL δ = AE
With variations in loading, cross-section or material properties,

PL δ =∑ i i

i Ai Ei
Ch 2 - 8

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

Example 2.1
75 kN 45 kN 30 kN

STEPS: Divide the rod into components at the load application points. Apply a free-body analysis on each component internal force Evaluate the total of the component deflections. to determine the

120 mm

120 mm

200 mm

E = 200 GPa D = 100mm d = 50mm

Determine the deformation of the steel rod shown under the given loads.

Ch 2 - 9

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of Studies

SOLUTION: Divide the rod into three components:
A B C D 30 kN

Apply free-body analysis to each component to determine internal forces, P = 30kN 1 P2 = −15kN P3 = 60kN
30 kN

75 kN

45 kN P1 45 kN

Evaluate total deflection,
30 kN

P2

δ =∑
i

PLi 1  PL1 P2 L2 P3 L3  i =  1 + +  Ai Ei E  A1 A2 A3 

75 kN P3

45 kN 30 kN

 ( 30 × 103 ) 0.2 ( −15 × 103 ) 0.12 ( 60 × 103 ) 0.12  1 = + +   200 × 109  1.96 × 10−3 7.854 × 10−3 7.854 × 10−3    = 1.874 × 10−5 m Ans.

L1 = 200mm A1 = 1.96 × 10−3 m 2

L2 = L3 = 120mm A2 = A3 = 7.854 × 10-3 m 2

Ch 2 - 10

MEC411

Faculty of Mechanical Engineering Engineering Mechanics Centre of...
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