chapter 3 stress-analysis1

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Mechanical Design I (MCE 321) L. Romdhane, SS 2016, 7:39 AM -- 1--

Summer 2016

Chapter 3Load and Stress Analysis

Mechanical Design 1(MCE 321)

Dr. Lotfi

Romdhane

[email protected]


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Chapter 3

Load and Stress Analysis


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3 Load and Stress Analysis

ChapterOutline

3-1 Equilibrium and Free-Body Diagrams

3-2 Shear Force and Bending Moments in Beams

3-3 Singularity Functions

3-4 Stress

3-5 Cartesian Stress Components

3-6 Mohrs Circle for Plane Stress

3-7

General Three-Dimensional Stress

3-8 Elastic Strain

3-9

Uniformly Distributed Stresses

3-10

Normal Stresses for Beams in Bending

3-11 Shear Stresses for Beams in Bending

3-12 Torsion

3-13 Stress Concentration

3-14 Stresses in Pressurized Cylinders

3-15 Stresses in Rotating Rings

3-16

Press and Shrink Fits

3-17 Temperature Effects

3-18

Curved Beams in Bending

3-19 Contact Stresses

3-20 Summary


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Shear Force and Bending Moments in Beams

If a beam with supports is cut at some section located at = 1and the left-

hand portion is removed as a free body, an internal shear force and

bending moment must act on the cut surface to ensure equilibrium.

Shear force and bending moment are related by the equation


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Sign Conventions for Bending and Shear

Fig. 33


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Distributed Load on Beam

Distributed load q(x) called load intensity

Units of force per unit length

When the bending is caused by a distributed load (),

the load intensitywith units of force per unit length, andis positive direction. It can be shown thatdifferentiating above equation results in


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Shear-Moment Diagrams


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Moment Diagrams

Two Planes

Add moments from

two planes as

perpendicular vectors


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Example 33


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Logic in Solid Mechanics


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Uniformly Distributed Stresses

The assumption of a uniform distribution of stress is frequently

made in design. The result is then often called pure tension, purecompression, or pure shear.

The stress is said to be uniformly distributed with

This assumption of uniform stress distribution requires that :

The bar be straight and of a homogeneous material

The line of action of the force contains the centroid of the section

The section be taken remote from the ends and from any discontinuityor abrupt change in cross section

Direct shear is usually assumed to be uniform across the crosssection, and is given by

The assumption of uniform stress is not accurate, particularly in thevicinity where the force is applied, but the assumption generallygives acceptable results.


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Shear Stress for Beams


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Normal Stress for Beams in Bending:

Assumptions

The equations for the normal bending stresses in straight beams are based onthe following assumptions

The beam is subjected to pure bending. This means that the shear force iszero, and that no torsion or axial loads are present ( for most engineeringapplications it is assumed that these loads affect the bending stressesminimally ).

The material is isotropicand homogeneous. The material obeys Hookes law. (linear)

The beam is initially straightwith a cross section that is constantthroughout the beam length.

The beam has an axis of symmetry in the plane of bending.

The proportions of the beam are such that it would fail by bendingratherthan by crushing, wrinkling, or sideway buckling.

Plane cross sections of the beam remain plane during bending.


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Normal Stress for Beams in Bending:

Analysis Elements of the beam coincident with the neutral plane have

zero stress.

The bending stress varies linearly withthe distance from the neutral axis, ,and is given bywhere is the second-area moment about the

axis.

Designating maxas the maximum magnitudeof the bending

stress, and as the maximum magnitudeof oftenwritten as

where = /is called the section modulus


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Torsion

Shear stresses develop throughout the cross section are given by

with

The assumptions used in the analysis are

The bar is acted upon by a pure torque, and the sections under consideration are remote

from the point of application of the load and from a change in diameter.

Adjacent cross sections originally plane and parallel remain plane and parallel after twisting,

and any radial line remains straight.

The material obeys Hookes law.

Any moment vector that is collinear with anaxis of a mechanical element is called atorque vector, because the moment causesthe element to be twisted about that axis.


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Types of loading (recap)

Bending y & z-axes

max

M

Z

IZ

c

Torsion x-axis

maxMcI

Tr

J

max

T

J

c

Axial force x-axisP

A

VQ

It

Trans. Shear

y & z-axes


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Combined Loading (recap)


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Example: Normal Stress

P Pey

A I


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Example: Normal Stress

(Simple Design Problem)

An offset link subjected to a force of 25kN is shown in the figure. The

Ultimate Strength of the material is 300 MPa. Determine the dimensionsof the cross-section. The desired factor of safety is 3.

= =

= (10 )= +

= +

=

2

3 10

2 =

3 = 100

0.2 100 750 = 0 = 25


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Stress and its Cartesian Components

The force distribution acting at a point on the surface have components in the

normal and tangential directions called normal stress and tangential shear stress,labeled by the Greek symbols and , respectively.

The units of stress in U.S Customary units are pounds per square inch (psi). For SI

units, stress is in Newtons per square meter (N/m2);1 N/m2 = 1 Pascal (Pa).

is labeled normal stress where indicates a normal stress and the subscript

indicates the direction of the surface normal. and are the shear stress components in the and directions, where the

first subscript indicates the direction of the surface normal whereas the second

subscript is the direction of the shear stress.


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Mohrs Circle for Plane Stress Formulas for the two principal stresses can

be obtained by substituting the angle as2

In a similar manner the two extreme-value

shear stresses are found to be

A graphical method for expressing the

relations developed in this section, called

ohrs circle diagram

, is a very effective

means of visualizing the stress state at a

point and keeping track of the various

components associated with plane stress. http://www.ijee.ie/OnlinePapers/Interactive/Philpot/mohr_learning_tool.htm
http://www.ijee.ie/OnlinePapers/Interactive/Philpot/mohr_learning_tool.htmhttp://www.ijee.ie/OnlinePapers/Interactive/Philpot/mohr_learning_tool.htmhttp://www.ijee.ie/OnlinePapers/Interactive/Philpot/mohr_learning_tool.htmhttp://www.ijee.ie/OnlinePapers/Interactive/Philpot/mohr_learning_tool.htm


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Example: Mohrs Circle

(Plane Stress)


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cm

cm

cm

cm

Example 1: Combined Loading(Simple Design Problem)

A single horizontal force P of 150 N is applied to the end Dof leverABD. Knowing that

portionABof the lever has a diameter of 1.2 cm, determine (a) the normal and shear

stresses on the element located at point Hand having sides parallel toxand yaxes; (b) the

principal planes and principal stresses at point H.

163.2 MPa

-74.7 MPa

N

27000 N.mm

15000 N.mm

88.4 MPa

79.6 MPa


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Example: Combined Loading

(Simple Design Problem)

The Shaft of an overhanging crank is subjected to a force P of 1kN. The shaft is

made up of a material with a shear yield strength of 190 N/mm2

. Determine thediameter of the shaft. The desired factor of safety is 2.


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Stress Concentration Any discontinuity in a machine part alters the stress distribution in

the neighborhood of the discontinuity so that the elementary stressequations no longer describe the state of stress in the part at theselocations.

Stress concentrations can arise from some irregularity not inherent inthe member, such as tool marks, holes, notches, grooves, or threads.

A theoretical, or geometric, stress concentration factor Kt or Kts

An example is shown in the Figure,that of a thin plate loaded in tensionwhere the plate contains a centrallylocated hole.


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Example: Stress Concentration


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Example: Stress Concentration


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Stress in Pressurized Cylinders

Cylindrical pressure vessels, hydraulic

cylinders, gun barrels, and pipes carryingfluids at high pressures develop both radial and tangential stresses

The tangential and radial stresses can be calculated by

The longitudinal stresses exist when the end reactions to the internalpressure are taken by the pressure vessel itself.

Thick walled


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Stress in Pressurized Cylinders

22

2

2

2

22

2

2

2

22

2

1

1

io

iil

o

io

iir

o

io

iit

rr

pr

endsclosedFor

r

r

rr

pr

r

r

rr

pr

Thin walled

p0=0

max

2

2

4

i i

t av

i i

t

i il

p d

tp d t

t

p dt

05.0ir

t


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Example: Pressurized CylindersAn aluminium alloy pressure vessel is made of tubing having an outer

diameter of 200 mm and a wall thickness of 6 mm.a) What pressure can the cylinder carry if the permissible tangential

stress is 82 MPa assuming thin walled theory?

b) On the basis of pressure found in a), compute the stress

components using the theory of thick walled cylinders


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Curved Beams in Bending The distribution of stress in a curved flexural member is determined by

assuming The cross section has an axis of symmetry in a plane along the length ofthe beam.

Plane cross sections remain plane after bending.

The modulus of elasticity is the same in tension as in compression.

The location of the neutral axis with respect to the center

of curvature O is given by the equation

The distribution is given by

where M is positive in the direction shown

The critical stresses occur at the inner andouter surfaces where =

and =


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Approximation:


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Example: Curved beamA utility hook was formed from a 25 diameter round rod into a geometry

shown in Figure. What are the stresses at the inner and outer surfaces at

section A-A if the load applied, = 4.

2

2 22

c i

n

c c

r r R

Rr

r r R

= 52.5

= 51.7

= 0.8

=

= 164.5

=

= 117.5

=

16= 0.744


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M h i l D i I (MCE 321) L R dh SS 2016 7 39 AM 39

Learning Outcomes:

Upon completion of the course, students will be able to: Identify types of stresses generated by external loads

Determine principal and maximum shear stresses

Define problems based on type of loading whether static or dynamic

Determine stress amplitudes and means

Select appropriate failure criteria Identify figures, tables, charts to calculate factors of safety and load limits

on parts or structures

Develop solutions for critical loads on compression members, pressurevessels and press fits based on their type

Select materials and size fasteners, welds, based on safety calculations

Design mechanical springs (material and size) to perform certain task forsafe applications.

chapter 3 stress-analysis1

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