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LECTURE 1.1

Concept of Stress

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210/12/2011

Concept of Stress

Review of Statics Structure Free-Body Diagram Component Free-Body Diagram Method of Joints Stress Analysis Design Axial Loading: Normal Stress Centric & Eccentric Loading Shearing Stress

Shearing Stress Examples Problem Examples 1-4

Contents

Reference: Ferdinand P. Beer, E. Russell Johnston, Jr. & John T. DeWolf,

"Mechanics of materials" 5thedition, McGraw-Hill, 2005. (Textbook).

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Concept of Stress

The main objective of the study of the mechanics ofmaterials is to provide the future engineer with the means

of analyzing and designing various machines and load

bearing structures.

Both the analysis and design of a given structure involve

the determination of stressesand deformations. This

chapter is devoted to the concept of stress.

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Review of Statics

The structure is designed to support a 30 kN load

(Fig. 1.1)

Perform a static analysis to determine the internal

force in each structural member and the reactionforces at the supports

The structure consists of a boom and rod joined by

pins (zero moment connections) at the junctions andsupports

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Structure Free-Body Diagram

Structure is detached from supports

and the loads and reaction forces areindicated

Ayand Cycan not be determined from

these equations

kN30

0kN300

kN40

0

kN40

8.0*306.0*0

yy

yy

y

xx

xxx

x

xC

CA

CAF

AC

CAF

A

mkNAM

Conditions for static equilibrium:

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Component Free-Body Diagram

In addition to the complete structure, each

component must satisfy the conditions forstatic equilibrium

0

8.0*0

y

yB

A

mAM

Consider a free-body diagram for the boom:

kN30yC

substitute into the structure equilibrium

equation

Results:

kN30kN40kN40 yx CCA

Reaction forces are directed along

boom and rod

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Method of Joints

The boom and rod are

2-force members, i.e.,

the members are

subjected to only two

forces which are

applied at memberends

For equilibrium, the

forces must be parallel toan axis between the force

application points, equal

in magnitude, and in

opposite directions

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Stress Analysis

Conclusion: the strength of member BCisadequate

MPa165all

From the material properties for steel, theallowable stress is

From a statics analysis

FAB= 40 kN (compression)

FBC= 50 kN (tension)

Can the structure safely support the

30 kN load?

dBC= 20 mmMPa159

m10314

N1050

26-

3

A

PBC

At any section through member BC, theinternal force is 50 kN with a force

intensity or stressof

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Design Design of new structures requires selection of

appropriate materials and component

dimensions to meet performance requirements

For reasons based on cost, weight, availability,

etc., the choice is made to construct the rod

from aluminum (sall= 100 MPa). What is an

appropriate choice for the rod diameter?

An aluminum rod 26 mm or more in diameter is adequate

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Axial Loading: Normal Stress

The resultant of the internalforces for an axially loaded

member is normalto a section

cut perpendicular to the

member axis.

The force intensity on that

section is defined as the

normal stress.

A

P

A

Fave

A 0lim

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The normal stress at a

particular point may not be

equal to the average stress but

the resultant of the stress

distribution must satisfy

Axial Loading: Normal Stress (con.)

The detailed distribution ofstress is statically

indeterminate, i.e., can not

be found from statics alone.

Aave dAdFAP

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Centric & Eccentric Loading

A uniform distribution of stress in a

section infers that the line of actionfor the resultant of the internal

forces passes through the centroid of

the section.

A uniform distribution of stress is

only possible if the concentrated

loads on the end sections of two-

force members are applied at thesection centroids. This is referred

to ascentr ic loading.

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Centric & Eccentric Loading (con.)

If a two-force member is

eccentr ical ly loaded, thenthe resultant of the stress

distribution in a section

must yield an axial force

and a moment.

The stress distributions in

eccentrically loaded

members cannot be

uniform or symmetric.

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Shearing Stress

Forces Pand Pare applied transversely to the member

AB (F igs. 1.15& 16).

The corresponding average shear stress is,

The resultant of the internal shear force distribution is

defined as the shearof the section and is equal to the load

P.

Corresponding internal forces act in the plane of section C

and are called shearingforces.

A

P

ave

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Shearing Stress (con.)

Shear stress distribution variesfrom zero at the member

surfaces to maximum values

that may be much larger than

the average value.

The shear stress

distribution cannot be

assumed to be uniform.

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Shearing Stress Examples

Single Shear:

A

F

A

Pave

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A

F

A

P

2ave

Shearing Stress Examples (con.)

Double Shear:

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Problem Example 1

Three solid cylindrical barsAB,BC, and CEof cross-sectional

areasAAB,ABC,ACE, respectively, are welded together atBandC,and subjected to axial forcesP, 6P, 4P,P, 2Pas shown in Fig. 1.

Determine the normal stress in each bar. Given:LoadP= 20 kN.

Fig. 1

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Solution of Problem Example 1

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Problem Example 2

A stepped cylindrical bar of the cross-sectional areasAAB,ABC,

andACE, carries five axial forces as illustrated in Fig. 2. What isthe maximum value ofPfor the stresses not to exceed 100 MPa

in tension and 140 MPa in compression?

Fig. 2

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Solution of Problem Example 2

From solution of problem 1

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Problem Example 3

A cylindrical rodABCof diameters dAB and dBCfor partsABand

BC, respectively, is fixed at the upper and supports an axial loadP

at the free end (Fig. 3). Calculate the normal stress in portionAB.Requirement:that the normal stress in partBCis not to exceed 40

MPa.

Fig. 3

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Solution of Problem Example 3

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Problem Example 4

The wing of a monoplane is approximated as depicted in Fig. 4.

Determine the normal stress in rodACof the wing, if it has auniform cross section of 20 * 10-5m2.Assumptions: the air load

is uniformly distributed along the span of the wing as shown in

the figure.

Fig. 4

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Solution of Problem Example 4