Section 2

Tension, Compression and ShearIn this section the concept of stress will be introduced, and this will be applied to components that are in a state of tension, compression, and shear. Strain measurement methods will also be briefly discussed.

© Loughborough University 2010. This work is licensed under a Creative Commons Attribution 2.0 Licence.

Contents• Introduction• Stress Analysis• Engineering Stress• Normal (Engineering) Stress• Saint Venant’s Principle (End Effect)• Normal Strain• Strain Measurement• Wheatstone Bridge• Further Strain Measurement Methods• Shear Stress• Failure of Bolted and Riveted Joints• Fastener Shear Failure• Bearing Failure of Plate• Pure Shear• Complementary Shear Stresses• Shear Strain• Stresses on an Inclined Plane• Credits & Notices

Introduction

• Previous section dealt with analysis of rigid body members– Internal forces regarded as

point internal forces– Cross-sectional dimension

and area of contact of no importance

• This section will deal with distributed internal forces– Cross-sectional dimension

and area of contact of great importance

y

z

x

P1

P2

P3

P4

A

P

P

P1

P2

Stress Analysis

• For stress analysis:– Magnitude, direction, location of external load– Geometry of structure– Deformation of structure

• Deformation in the form of:– Tension– Compression– Shear– Torsion– Flexure– Combination of above!

• In this module, we will concentrate on tension, compression and shear

Engineering Stress

• Consider a slender structure such as a bar or rod of length L subject to a pair of tensile forces at its ends– The structure is in equilibrium– The self-weight of the structure is negligible compared to the

magnitude of the applied forces– The structure is made of an isotropic and homogeneous material

• The ratio of the resultant force Px to the cross sectional area A at mn is called the Engineering Stress

L

Px

Px m

ny

x

zA

Px

Px

F.B.D:-

m

n

Normal (Engineering) Stress

• The subscript x denotes the direction of the force or stress (in cartesian coordinates)

• Engineering stress is an average intensity of the internal forces acting on the original cross-sectional area

• When the cross-sectional area on which stress acts is normal to the applied force – termed normal stress

• Tensile stress is positive, compressive stress is negative• Stress is a second order tensor• Units of stress:

– SI: N/m2 or Pascal (Pa) – practically expressed as Pa 106 or MPa (N/mm2)

– Imperial: lbf/in2 or psi – practically expressed as psi 103 or ksi

A

Pxx

Example 2.1

• An civil aircraft fuselage can be considered as a cylinder of radius r and skin thickness t. If the cabin is pressurised to a pressure p, calculate the stress in the circumferential (hoop, H) and longitudinal (axial, L) directions. Which is the greater?

Answer: H=pr/t, L=pr/2t

Saint Venant’s Principle (End Effect)

• At the point of application of the load on a structure there will be a stress concentration

• The above discussion of engineering stress does not consider this concentration of stress since we are dealing with an average intensity of internal forces

• The principle of St. Venant is therefore applied to overcome this problem:-– “that while statically equivalent systems of forces acting on a

body produce substantially different local effects the stresses at sections distant from the surface of the loading are essentially the same”

P

Pb

A

A

B

B >b Stress Distributions

Section A-ASection B-B

Normal Strain

• Consider a bar under a tensile load– If the load is increased gradually, the bar will deform

to a new length L′• The elongation of the bar is the difference

between the new length and the original length• The strain is defined as the ratio of the

elongation and the original length

L′

L

PL

LL

L

L-L

x

Deformation:

Strain:

Strain Measurement

• Strain is a dimensionless quantity, and is expressed as either a percentage or a microstrain:

• Electric resistance strain gauge used to measure surface strains– Insulating polymer foil base and

electric circuit printed (etched) onto surface

– Various gauge configurations available

10L

% 100L

6xx

Wheatstone Bridge• Very sensitive resistive circuit enabling strain to be measured• One active gauge in circuit – resistance change results in imbalance

in circuit – measured as a potential difference at Vo

Vi

R2

R4

R1

Vo

RgActive Gauge

(stressed)

Dummy Gauge (un-stressed)

Vo

Quarter Bridge without temperature compensation

Quarter Bridge with temperature compensation

R

dR

K

1

a

Further Strain Measurement Methods

• There are many non-contact and full-field methods developed to measure the surface strain of a structure:– Thermoelastic Stress Analysis– Photoelasticity– Móire Interferometry– Speckle Pattern Interferometry– Digital Image Correlation– Brittle Coatings

Shear Stress• Consider a riveted joint:

• The action of the external load creates a state of direct shear on the rivet shank

• The blue arrows represents internal resistance to the load

• The shear stress is given by:

Ps Ps

s

s

A

P

As

PsPs

Typical failure of rivet shank

Example 2.2

• A metal bolt of 10mm diameter connects a bar to a clevis. What is the shear stress developed in the pin if a load of 10kN is applied to the connection?

PP

Answer: s=63.7 MPa

Example 2.3

• Three steel sheets are joined by two rivets as shown below. If the rivets have diameters of 15mm and the maximum shear strength in the rivets is 210 MPa, what force P is required to cause the rivets to fail in shear?

P

P/2

P

PP/2

Answer: P=148.4 kN

Failure of Bolted and Riveted Joints

• Many aeronautical and automotive structures are comprised of thin plate fabrications

• There are many modes of failure for bolted and riveted connections used to join thin plates together

• In joint stressing, you must check every possible mode of failure before certifying the joint fit for service

• Assumption of analysis – fastener fills hole completely – perfect fit, no ‘play’

Fastener Shear Failure

• Load carried by a single rivet = Pb• Define all as the maximum allowable

shearing stress of the material of the rivet• Maximum force that rivet can sustain:

b4

dP

4

dPb

all2

max

2

all

P P

aa

t

P per unit

width

P per unit

width

d

b

bPbPb

Bearing Failure of Plate• The bearing pressure between

the rivets and the plates may become excessive, resulting in plate ‘bruising’

• Average bearing stress between a rivet and its surrounding hole is:

• If b is the stress at which either the rivet or the plate fails in bearing then:

td

Pbb

b

tdP b

max

d

b

Pb

Pb

PbPb

t

Plate Tearing

• The most heavily stressed regions of the plates are on sections such as e-e

• Average tensile stress on this reduced plate area is:

• If the plate material has an ultimate tensile strength of ult (full definition to follow later in module), then:

t)db(

Pbave

b

t)db(P ult

max

d

b

Pb

Pb

Pb

b

(b-d)

t

Plate Shear-Out

• Plate shear-out may occur on planes such as c-c, with the result that the whole block of material c-c-c-c is sheared (ejected) out of the plate

• If ult is the ultimate shearing stress of the plate material, then failure is on the point of occurring when:

b

at2P

at2Pb

ultmax

ult

a

b

Pb

Pb

Pb

b

Pb

Pb

Pb

Plate Bursting

• The plates may fail due to the development of large tensile stresses, leading to a bursting-type of failure

• This is difficult to estimate and will not be considered further

Pb

Pb Pb

Pb Pb

Pb

Pure Shear• Consider a square element of unit thickness with only shear

stresses acting:

• Shear stresses act on the surfaces of the element or parallel to its associated planes Double subscript required– First subscript denotes the normal of the plane, second is the direction

of the shear load• Shear stress is positive when normal to plane and direction are

either BOTH in the positive direction or BOTH negative

xy

yx

xyyx

dx

dy

A

CB

D

Complementary Shear Stresses

• Vertical and horizontal force equilibrium shows that xy on face AB is equal to xy on face CD

• Moment equilibrium shows that xy on AB is equal to yx on AD:

yxxy

yxxy 02

dy)dx1(2

2

dx)dy1(2

xy

yx

xyyx

dx

dy

A

CB

D

Shear Strain

• Shear stresses per se do not cause any volume change, but result in distortion of the element

• The angle is a measure of shear strain or change in shape of the element

• Units of shear strain is the radian or dimensionless. Sign convention follows shear stresses

• When a shear strain is small, it is approximated by:

y

xA

D

B

C

/2

/2

xy

yx

y

xA D

B C

xy

yx

a h

h

atan

Stresses on an Inclined Plane

• On plane normal to applied load:

• On inclined plane n′n′:

0 ,A

P

P P

n A′

n

n′

n′A

Ps

Pn

P

n′

n′

cossinA

P

A

P ,

A

cosP

A

P s2

n

Example 2.4

• A short steel bar with a square cross-section of 25mm by 25mm is subjected to an axial compressive load of 120kN. Calculate a complete set of complementary stresses on the cross-sectional plane oriented at 30 to the normal cross section.

Answer: =-144 Mpa, =-83.14 MPa

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Credits