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INSTRUCTIONAL MANUAL
OF
UNSYMMETRICAL
BENDDING APPARATUS


ENGINEERING MODELS & EQUIPMENT
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By:
UNSYMMETRICAL BENDING APPARATUS
CONTENTS: Page No.
1.0 Theory 03
2.0 Objective 06
3.0 Apparatus 06
4.0 Suggested experimeta! "or# 06
$.0 %esu!ts & 'iscussios 06
6.0 Samp!e 'ata Sheet 06
(.0 )recautios 0(
UNSYMMETRICAL BENDING APPARATUS
1.0 THEORY:
A member may be subjected to a bedig momet* "hich acts o a p!ae ic!ied to
the pricipa! axis +say,. This type o- bedig does ot occur i a p!ae o- symmetry o-
the cross sectio* it is ca!!ed usymmetrica! bedig. Sice the prob!em re!ated to
-!exure i geera! di--ers -rom symmetrica! bedig* it may be termed as s#e"
bedig.

Oe o- the basic assumptios i derivig the -!exura! -ormu!a
I
MY
f =
is that the
p!ae o- the !oad is perpedicu!ar to the eutra! axis. .very cross/sectio has got t"o
mutua!!y perpedicu!ar pricipa! axis o- iertia* about oe o- "hich the momet o-
iertia is the maximum ad about the other a miimum. 0t ca be sho" that a
symmetric axis o- cross/sectio is oe o- the pricipa! axis ad oe at right ag!es to
the same "i!! be the other pricipa! axis.
1or beams havig usymmetrica! cross/sectio such as ag!e +2, or chae! +3,
sectios* i- the p!ae o- !oadig is ot coicidet "ith or para!!e! to oe o- the
pricipa! axis* the bedig is ot simp!e. 0 that case it is said to be usymmetrica! or
o/uip!aar bedig.
0 the preset experimet -or a cati!ever beam o- a ag!e sectio* the p!ae o-
!oadig is a!"ays #ept vertica! ad the ag!e iro cati!ever beam itse!- is rotated
through ag!es i steps o- 4$
0
.
4osiderig the positio o- ag!e iro "herei the p!ae o- !oadig ma#es a ag!e
"ith 5/5 axis o- the sectio* "hich is oe o- the pricipa! axes o- the sectio. The
compoets o- the vertica! !oad ) a!og 5/5 ad 6/6 axis are )cos ad )si
respective!y.
The de-!ectio 6 ad 5 a!og 6/6 ad 5/5 axis respective!y are give by
VV
EI
L P
U
3
. si
3

=
+1,
UU
EI
L P
V
3
. cos
3

=
+2,
ad the magitude o- resu!tat de-!ectio oo* is give by
( ) ( )
2 2
V U + = +3,
7
5ertica! axis
ad its directio is give by
U
V

=
1
ta
+4,
"here* is the ic!iatio o- the resu!tat de-!ectio "ith the 6/6 axes. This resu!tat
disp!acemet is perpedicu!ar to the eutra! axis / but ot i the p!ae o- the !oad ).
OO8 9
O8) 9 5
O8: 9 ;
O) 9 6
O: 9 7
7
7< 6
;
5
A
;
x
4
5
O
4$
0
=
x
6
>ori?ota! axis
A<
;<
=< 7<
;<
O
1
4<
Deflecte !o"#t#o$
y
y
6

= =

=
OP
P O
U
V 8
ta

VV
UU
EI
L P
EI
L P
3
. si
3
. cos
3
3

9
cot
UU
VV
I
I
+$,
1or the ag!e sectio used i the preset experimet 0uu ad 0vv ca be #o" -rom the
tab!es o- =ureau o- 0dia Stadards had boo#* -or properties o- stadard sectios.
There-ore -or a give ag!e * the magitude o- ag!e ca be -oud out.
The hori?ota! ad vertica! compoets o- the de-!ectio ca be ca!cu!ated o the basis
o- the geometry avai!ab!e as sho" i -ig. 0t ca be see@
, + si
, + cos


+ =
+ =
Y
X
+6,
Simi!ar!y*


cos si
si cos
V U Y
V U X
+ =
=
+(,
There-ore* the procedure o- ca!cu!atig the de-!ectios "ou!d be

;
5
7
u
;

5
O

u
y

A0/ +B,
+B,

O<
u
7
5 ;
+A0/,
)
4a!cu!ate 6 ad 5 usig eCuatios +1, ad +2,.
4ompute usig eCuatios +3,.
4ompute usig eCuatios +4, ad to chec# the va!ues by usig the eCuatio +$,
4a!cu!ate the reCuired va!ues o- ; ad 7 usig eCuatios +$, ad +6, separate!y.
%.0. OB&ECTI'E:
To study the behavior o- a cati!ever beam uder symmetrica! ad usymmetrica!
bedig.
(.0 APPARATUS:
Apparatus cosist o- a ag!e o- si?e 1D x 1D x 1EFD or i eCuiva!et metric uits o-
!egth F0cm is tied as a cati!ever beam. The beam is -ixed at oe ed such that the
rotatio o- 4$
0
iterva!s ca be give ad c!amped such that the pricipa! axis o- its
cross/sectio may be ic!ied at ay ag!e "ith the hori?ota! ad vertica! p!aes.
A!so arragemet is provided to app!y vertica! !oad at the -ree ed o- the cati!ever
ad to measure hori?ota! ad vertica! de-!ectio o- the -ree ed. A dia! gauge "ith
magetic base is supp!ied "ith the apparatus.
).0 SUGGESTED E*PERIMENTAL +OR,:
Step1@ 4!amp the beam at ?ero positio ad put a "eight o- $00gms +$G, o the hager ad
ta#e the ?ero !oadig o the beam to activate the member.
Step2@ Set the dia! gauges to ?ero readig to measure vertica! ad hori?ota! disp!acemet at
the -ree ed o- the beam.
Step3@ 2oad the beam i steps o- 1#g +10G, up to 4#g ad ote the vertica! ad hori?ota!
de-!ectios each time.
Step4@ %epeat the steps +1, to +3, turig the beam through 4$
0
iterva!s. )rob!em o-
usymmetrica! bedig "i!! arise o!y i those cases "here the !egs o- the ag!e
sectio are i hori?ota! ad vertica! positios. 0 those cases both vertica! ad
hori?ota! de-!ectios eed to measure.
-.0 RESULTS AND DISCUSSIONS:
4ompute the theoretica! de-!ectios ad compare "ith those measured experimeta!!y.
..0 SAMPLE DATA SHEET:
Hateria! o- beam 9
7oug<s modu!us o- the materia! +., 9
Spa o- cati!ever beam +2, 9
Sectioa! properties
Si?e 9
0xx 9 cm
4
0yy 9 cm
4
0uu 9 cm
4
0vv 9 cm
4
Area 9 cm
4
S!.
Go.
Ag!e
+degree,
2oad
+#g,+G,
Observed de-!ectio
+mm,
Heasured de-!ectios
+mm,
; 7 ; 7
1.
2.
3.
4.
$.
6.
(.
F.
0
4$
0
A0
0
13$
0
1F0
0
22$
0
2(0
0
31$
0
/.0PRECAUTIONS:
Ta#e care to see that you do ot exert -orce o the -ree ed o- the cati!ever beam.
)ut the !oad o the hager gradua!!y "ithout ay jer#.
)er-orm the test at a !ocatio* "hich is -ree -rom vibratio.