NORMAL AND SHEAR STRESSES IN OPEN SECTIONS

8/9/2019 NORMAL AND SHEAR STRESSES IN OPEN SECTIONS

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NANYANG TECHNOLOGICAL UNIVERSITY

SCHOOL OF MECHANICAL AND PRODUCTION ENGINEERING

MP2071 LABORATORY 2A

P2.3 NORMAL AND SHEAR STRESSES IN OPEN

SECTIONS

1


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CONTENTS

1) Introduction

2) Objectives of experiment

3) Procedure

4) Calculations and theoretical data

5) xperimental results

!) "raphical representation of data

#) $iscussion

%) Conclusion

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I. INTRODUCTION

&ppl'in( a load perpendicular to the axis of an open section beam results in bendin(

t*istin( shearin( and deformation+ $ifferent structural ph'sics of the beam *ill 'ield

distinct characteristicall' behaviour+

,o*ever man' other components come into pla' *hen the load is applied and upon

observation of its ph'sical behaviour+ In this experiment *e focus on stud'in( the effect

of appl'in( offset loadin(s to open section beams that are -./ shaped and -C/ shaped

orientated+

0i(ure 1 sho*s the application of the load P s'mmetricall' throu(h the vertical axis of an

.shaped open section beam+ he beam *ill experience deformation due to bendin(

shearin( but no t*istin(+ his corresponds to its ph'sical shear stress concentration and

normal stress distribution as sho*n in fi(ure 3)+ his is illustrated in the bendin(

moment and shear stress euations as follo*s

It

VQ

I

My

ave

x

=

−=

τ

σ

3


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0i( 1 ffect of loadin( P directl' on the vertical axis passin( throu(h the centroid of the . shaped orientated beam+

,o*ever *hen the beams orientation is turned anticloc6*ise to form a -C/ shaped beam

the same load P applied throu(h its centroid it *ill experience effects of torsion

(twisting) bendin( and shearin( deformations as sho*n in fi(ure 2+ his is illustrated in

the bendin( moment and shear stress euations as follo*s

due to pure bendin( onl')

due to other additional effects of torue)

0i( 2 ffect of loadin( P directl' on the vertical axis passin( throu(h the centroid of the

C shaped orientated beam+

4

It

VQ

I

My

ave

x

≠

−=

τ

σ


5/19

he difference in ph'sical behaviour bet*een the 2 orientated beams could be due to the

different shear stress and normal stress distribution alon( side the thin *alls 0i(ure 3 and

4)+ his is because the centroid of each beam orientation is differentl located as *ell

as the shear center.

Note that the direct loading of ! as described abo"e has a similar effect of doing a

do#ble loading test which will be e$%lained in the %roced#re.

& single loading test (loading of ! at onl ' side of the beam) is carried o#t to f#rther

enhance the effect of twisting and torsion on both orientation t%e beams.

0i(+ 3 7 8hear stress

distributions for .

shaped orientated open

section beam

0i(+ 4 7 8hear stress

distributions for C

shaped orientated open

section beam

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II. OECTI*ES

1+) o stud' the characteristics of bendin( shearin( and t*istin(

deformation9deflection9stress components in the thin*alled open section beam

under the load applied to

i) a .shaped orientated thin *alled section beam : lon( and short

ii) a Cshaped orientated thin *alled section beam : lon( and short

2) o extricate the difference bet*een the effects of bendin( shearin( and t*istin(

due to double loadin( and sin(lesided loadin( applied to long and short beams

of both -shape/ orientations+

III. !ROCEDURE

Do#ble+loading test

1+) he fixed load P of ! 6() is applied sim#ltaneo#sl e,#idistant from the

centroid of the subject open section beam+

2+) he process starts b' loadin( from the furthest distance on both sides of the beam

and comin( in*ards to*ards the centroid and neutral axis of the beam+

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3+) he left and ri(ht deflections -& and - on the (au(es is noted respectivel'+ he

si(n convention considered is positive for cloc6*ise deflection and ne(ative for

anticloc6*ise deflection on the (au(e meter+

4+) ;eadin(s are ta6en at loadin( points 1


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2+) he process starts b' loadin( from the furthest distance on right side of the beam

and comin( in*ards to*ards the centroid and neutral axis of the beam and

throu(h to the left side+

3+) he left and ri(ht deflections -& and - on the (au(es is noted respectivel'+ he

si(n convention considered is positive for cloc6*ise deflection and ne(ative for

anticloc6*ise deflection on the (au(e meter+

4+) ;eadin(s are ta6en at loadin( points ?1


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his section is to obtain individual data based on the dimensions of the specimens

involved+ ,ence a theoretical result can be obtained and then compared to the

experimental results+

Specification of dimensions

$eflection due to bendin(

@ bm > PA3

3Inn

*here P is shear force is 'oun( modulus of steel !B"pa)

nn > ht2 92?htbt92)?tbb2t)92

2ht?tb2t)

b

h

A

t

Inn

nn

F

Icc

cc

B


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Goment of inertia for rectan(le

I>1912 bh3 1)

Parallel axis theorem

IH>I?&d2 2)

sin( euation 1) and 2)

Inn > 2Dth3912?hth92 )2E?b2t)t3912?tb2t) t92)2

Icc > ht3912?2htb92t92)2?tb2t)3912

8hear centre

e > th2 b2

4Icc

T1EORETIC&0 D&T& for OT1 N and C sha%ed orientated beams.

Short

beam

0ong beam

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2nn 9mm 1#+34 1#+2B1

2cc 9mm 25+4% 2532

Inn 9J1


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Aon( Keam. 8hape in mm)

Distance -& - (-&+-)56 (-7-&)561


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!<


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5< 0.085 -0.025 0.055 0.030

!< 0.120 -0.100 0.110 0.010

#< 0.265 -0.300 0.283 -0.018

%< 0.320 -0.420 0.370 -0.050

B< 0.365 -0.500 0.433 -0.068

1


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%<


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T%e(0ong)

E$%erimental 9.:;mm 9.96mm

Theoretical 9.'


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?. hrou(h 8in(le Aoadin( test for . 8hape for both len(th it is found has an

increased value+ his could be due to more si(nificant effect of torsion or

t*istin( as the load is onl' applied on one side+

8or C Sha%e Do#ble 0oading E$%erimental and Theoretical "al#e of

deflection d#e to !#re ending 4oment

T%e++0ong

E$%erimental 9.'@V

Fh

I

Vth

4

2

V

h

I

thb

4

22

Kecause *e consider the 0 > = but there is an an(le formed due to t*istin(+ &s a result

0 =+

1#


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0 > = sin + herefore *e onl' need to consider of eQ >θ sin4

22

I

thb.

ThereforeA (E$%erimental) eB e (Theoretical)

2+ he ratio of lon( beam shear center is (reater than the ratio of short beam shear

center because of the of the lon( beam is (reater than the of the short beam+

• Further discussion on single loading

Why the deflection due to bending moment is not constant?

'. 0rom the formula > the torsion *ill chan(e and var' proportionall' *ith the

an(le + Compare the deflection bet*een the t*o beam the is a bi((er than the

short beam due to deflection for lon( beam+

6. 8ince the an(le for both beam have the different an(le+ he an(le for lon( beam

is lar(er than short beam+ Kased on the (raph *e compared both beamQs *e

found out lon( beam have the bi((er slope as compare to the short ones+

:. $ue to torsion *ill cause the chan(e in an(le the plane shear stress varies

accordin( the an(le + Lhen has chan(ed the R *ill chan(e+ ,ence the

deflection due to bendin( moment is different

Other possible explanations/discussions

• he excessive and irre(ular beam deflection or deformation resulted from

continuous loadin( could result in a small de(ree of plastic deformation in

1%


19/19

the beamQs *alls+ his could contribute to var'in( readin(s and then

apparent errors+

• Lhen the beam is orientated in a C shape orientation as compared to an .

shape orientation the forces actin( on the thin *alls of the beam are

distributed alon( a different spatial orientation (ivin( much more

allo*ance for bendin( at joints bet*een the thin *alls+ hus this could

possibl' explain the distinct differences *hen the beam is rotated B<

de(rees to a C shape orientated beam+

*II. CONC0USION

hrou(h this experiment it is apparent that it is critical to stud' the behavior of

thin *all beams and ho* the' react and deform correspondin( to the ph'sical structural

orientation and the *a' that the load is applied+ It is important for en(ineers to bear in

mind all the factors that come into pla' that causes deformation li6e normal and shear

stresses torsion and torues and bendin( moments and (ive it a better understandin( to

aid in desi(nin( a structure that do not fail readil' under harsh circumstances+

1B

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NORMAL AND SHEAR STRESSES IN OPEN SECTIONS