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11. Prismatic Beam Design

Design a beam to resist

both bending and

shear loads.

Develop methods used

for designing prismaticbeams.

Determine shape of fully

stressed beams.

Design of shafts based on

both bending and torsional moments.

CHAPTER OBJECTIVES

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CHAPTER OUTLINE

1. Basis for Beam Design


3. *Fully tressed Beams

!. *haft Design

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11.1 BASIS FOR BEA DESI!N

Beams are structural members designed to supportloadings perpendicular to their longitudinal a"es.

#nder load$ internal shear force and bendingmoment that vary from pt to pt along a"is of beam.

%"ial stress is ignored in design$ as it&s muchsmaller than the shear force and bending moment.

% beam designed to resist shear and bendingmoment is designed on the basis of strength.

'e use the shear and fle"ure formulae fromchapters ( and ) to design a beam$ only if the beamis homogeneous and displays linearelasticbehavior.

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11.1 BASIS FOR BEA DESI!N

%s sho+n$ e"ternal distributed and pt loads appliedto a beam is neglected +hen +e do stress analysis.

%dvanced analysis sho+s that the ma"imum value

of such stresses are of a small percentage

compared to bending stresses. %lso$ engineers prefer

to design for bearing

loads rather than pt

loads to spread the

load more evenly.

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-he actual bending and shear stresses must note"ceed the allo+able values specified in structural

and mechanical codes of practice.

'e need to determine the beam&s section modulus.

#sing fle"ure formula$ Mc/I$ +e have

0 is determined from the beam&s moment diagram$

and allo+able bending stress$ allo+ is specified in a

design code.

11." PRISATIC BEA DESI!N

( )1-11allow

dreq'

MS =

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nce Sreqdis no+n$ +e can determine thedimensions of the "section of the beam.

o+ever$ for beams +ith "section consisting of

various elements 4e.g. +ideflange section5$ then

an infinite no. of +eb and flange dimensions can becomputed.

6n practice$ the engineer +ill choose a commonly

manufactured standard shape from a handboo

that satisfies S7 Sre8&d.


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#se symmetric "section if allo+able bending stressis the same for tension and compression.

ther+ise$ +e use an unsymmetrical "section to

resist both the largest positive and negative

moment in the span. nce beam selected$ use shear formula

allo+ VQ/Itto chec that the allo+able shear stresshas not been e"ceeded.

9"ceptional cases are +hen the material used is

+ood.


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Fabricated Beams

1. Steel sections

0ost manufactured steel beams produced by rollinga hot ingot of steel till the desired shape is

produced. -he rolled shapes have properties that aretabulated in the %merican6nstitute of teel ;onstruction

4%6;5 manual. 4%ppendi" B5 'ide flange shapes defined by

their depth and +eight per unitlength.

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sections +hile the actual dimensions are smallerdue to sa+ing do+n the rough surfaces.

%ctual dimensions are to be used +hen performing

stress calculations.

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Fabricated Beams

3. Built-up Sections

% builtup section is constructed from t+o or more

parts ?oined together to form a single unit.

Based on 98n 111$ the momentresisting capacityof such a section +ill be greatest for the greatest

moment of inertia.

-hus$ most of the material should be built furthestfrom the neutral a"is.

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Fabricated Beams


For very large loads$ +e use a deep Ishapedsection to resist the moments.

-he sections are usually+elded or bolted to

form the builtup section.

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Fabricated Beams


'ooden bo" beams are made from

ply+ood +ebs and larger boards for the

flanges. For very large spans$ glulam beams are

used. uch members are made from

several boards gluelaminated together.

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60P@-%>-

Beams support loadings that are applied

perpendicular to their a"es.

6f they are designed on the basis of strength$ they

must resist allo+able shear and bending stresses. -he ma"imum bending stress in the beam is

assumed to be much greater than the localiAed

stresses caused by the application of loadings on

the surface of the beam.

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Procedure for analysis

hear and moment diagrams

Determine the ma"imum shear and moment in

the beam. -his is often done by constructing the

beam&s shear and moment diagrams. For builtup beams$ shear and moment diagrams

are useful for identifying regions +here the shear

and moment are e"cessively large and may

re8uire additional structural reinforcement or

fasteners.

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%verage normal stress

6f beam is relatively long$ it is designed by finding its

section modulus using the fle"ure formula$

Sre8&d Mma"/allo+. nce Sre8&dis determined$ the "sectional

dimensions for simple shapes can then be

computed$ since Sre8&d I/c. 6f rolledsteel sections are to be used$ several

possible values of Smay be selected from the

tables in %ppendi" B.

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%verage normal stress

;hoose the one +ith the smallest "sectional area$

since it has the least +eight and is therefore the

most economical.hear stress

>ormally$ beams that are short and carry large

loads$ especially those made of +ood$ are firstdesigned to resist shear and later checed against

the allo+ablebendingstress re8uirements.

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hear stress

#se the shear formula to chec to see that theallo+able shear stress is not e"ceeded

allo+CVma"Q/It.

6f the beam has a solid rectangular "section$ theshear formula becomes allo+C 1.,4Vma"/A)$98n ),$ and if the "section is a +ide flange$ it is

to assume that shear stress is constant overthe "sectional area of the beam&s +eb so that allo+C Vma"/A+eb$ +here A+ebis determined from theproduct of the beam&s depth and the +eb&sthicness.

11 P i ti B D i

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+ill only slightly increase Sre8&d. -hus$

3333' mm)10(1120mm)10(706

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E#APLE 11.1 $SOLN%

hear stress

ince beam is a +ideflange section$ the average

shear stress +ithin the +eb +ill be considered.

ere the +eb is assumed to e"tend from the very top

to the very bottom of the beam.From %ppendi" B$ for a '1:!=$ d !,, mm$tw : mm$ thus

#se a '!(=(=.

( ) ( ) !a100!a7.24mm8mm455 N1090

3

maxa"#

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E#APLE 11.&

aminated +ooden beam supports a uniform

distributed loading of 12 >/m. 6f the beam is to havea heightto+idth ratio of 1.,$ determine its smallest

+idth. -he allo+able bending stress is allo+ < 0Pa

and the allo+able shear stress is allo+ =.( 0Pa.>eglect the +eight of the beam.

11 Prismatic Beam Design

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E#APLE 11.& $SOLN%

hear and moment diagrams

upport reactions at Aand B

have been calculated and the

shear and moment diagrams

are sho+n.ere$ Vma" 2= >$

Mma" 1=.() >m.


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E#APLE 11.& $SOLN%

Bending stress

%pplying the fle"ure formula yields$

%ssume that +idth is a$ then height is h 1.,a. -hus$( )

323

allow

maxdreq' m00119.0

kN/m109

mkN67.10=

==

MS

( ) ( )

( )

m147.0

m003160.0

m00119.075.0

35.1121

33

3

dreq'

==

===

a

a

a

aa

c

I

S


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E#APLE 11.& $SOLN%

hear stress

%pplying shear formula for rectangular sections

4special case of ma" VQ/It5$ +e have

( )( ) ( ) ( )

!a60!a929.0

m147.05.1m147.0kN205.15.1 maxmax

.

A

V

>=

==


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2ote that although this result indicates that h = atx =$ it&s necessary that beam resist shear stress atthe supports$ that is$ h7 = at the supports.

b

PL

h allow

2

0 2

3

=

xL

hh

=202 2


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hafts +ith circular "section are used in many

types of mechanical e8uipment and machinery.

-hus$ they are sub?ected to cyclic or fatigue stress$

caused by the combined bending and torsional

loads they must transmit. tress concentrations +ill

also occur due to eys$

couplings$ and sudden

transitions in its"sectional area.

'11.) SHAFT DESI!N


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For e"ample$ +e can resolve

and replace the loads +iththeir e8uivalent components.

Bendingmoment diagrams for

the loads in each plane can bedra+n and resultant internal

moment at any section along

shaft is determined by vector

addition$ M (Mx2G Mz25.

'11.) SHAFT DESI!N


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6n addition$ the tor8ue diagram

can also be dra+n.

Based on the diagrams$ +e

investigate certain critical

sections +here thecombination of resultant moment and

tor8ue Tcreates the +orst stress situation.

-hen $+e apply fle"ure formula using the resultant

moment on the principal a"is of inertia.

'11.) SHAFT DESI!N


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'11.) SHAFT DESI!N

6n general$ critical element D4or C5

on the shaft is sub?ected to plane

stress as sho+n$ +here

J

Tc

I

Mc == and


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3ormally$ the "section of a beam is first designed

to resist the allo+able bending stress$ then the

allo+able shear stress is checed.

For rectangular sections$ allo+C 1., 4Vma"/A5.

For +ideflange sections$ +e use allo+C Vma"/A+eb.


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