1.0 TITLE TITLE
Deflection of Curved Bars 2.0 INTRODUCTION
The deflection of a beam or bars must be often be limited in order to provide integrity and stability of structure or machine. Plus, code restrictions often require these members not vibrate or deflect severely in order to safely support their intended loading. This experiment helps us to show some ind of deflection and how to calculate the deflection value by using Castigliano!s Theorem and mae a comparison between result of the experiment and the theory. 3.0 OBJECTIVE
To validate validate the formula of deflection of curved bars based on Castigliano!s Theorem. 4.0 APPAR APPARAT ATUS US
T"C#$%P&"'T curved bar apparatus, quarter circle and circle bars. 5.0 TH THEO EORY RY
(hen a force is exerted on an elastic ob)ect, wor is done on it will deform. The wor energy which is stored in the elastic ob)ect is called strain energy. The elastic ob)ect will return to its original shape when the force is released. *train energy can be caused by pressure, tension, bending, torsion, shear force, etc. Castigliano!s theorem states that the deflection caused by any external load is equal to the partial differential of the strain energy with respect to that external load. +t every point of action of these forces there are deflections. igure - shows an elastic ob)ect which is acted upon by forces (-, (, (/, 0., (n. +t every point of action of these forces fo rces there are deflections. de flections. +ccording to Castigliano!s Theorem, the deflections at these points are1
2$ 3 4- 5 2$ 3 4 5 2$ 3 4 /5 00..5 2$ 3 4 n 2(2( 2(/ 2( n (ith
$ (
3 3
total strain energy force at point
4
3
deflection at respective (
The deflection of point + in the direction of the force is equal to the partial differential of the total strain energy $ for the system with respect to (- 6force acting at point +7.
(/
( C
B
x/
x +
(@ x@
D
xx
(-
" (
A. Quartr C!r"# Bar
or conservation of energy, the total strain energy which is caused by bending moment & is, $ 3 8 &9"% ds 6-7 The bending moment for a quarter circle bar, at any section dθ is, & 3 Pv:sin; *o, the strain energy stored due to a load Pv is, $ 3
6@7
The horiAontal deflection can be calculated by applying an imaginary force P in the horiAontal direction. rom Castigliano!s Theorem, the horiAontal is given by,
> 3 d $9d P 3 P: /9"%
67
(ith : 3 radius of quarter circle bar 3 -mm " 3 elastic modulus of bar material 3 E'9m % 3 second moment of area of the cross section % 3 bh/ 9 - b 3 width of the cross section 3 mm d 3 thicness of the cross section 3 /mm
igure 6a7 #uarter Circle Bar
igure 6b7 Circular Bar
B. C!r"u#ar Bar
The deflection which is caused by force P? is, δ?
3
∂$ ∂Pv
P:F 3 "%
π ( G ) @ π P:F
3 .-@H "% with the values of :, " and % the as above. The cross section of the bar is x / mm.
$.0 PROCEDURE
A. Quartr C!r"# Bar
6i7 6ii7 6iii7 6iv7 6v7
+ hanger attached to the free end of the bar. Two dial gauges clamped at their allocated points. . g load applied and the vertical and horiAontal deflections are recorded. Ioads added in increment of . g until a maximum load of -.J g. The deflections recorded for each loading.
B. C!r"u#ar Bar
6i7
6ii7
The above method followed but with a load increment of . g until a maximum load of @ g. The vertical displacements for each loading recorded.
$.0 RESULTS
Ioad . .@ .J .= -. -. -.@ -.J
?ertical deflection, P- 6mm7 %ncreasing load Decreasing load +verage ./= .@.@ .=/ -. .H -./ -.@ -./= -.= -.H/ -.== ./@ .@ ./= .= .= .=/. /./ /./ /.KH /.KH /.KH
oriAontal deflection,P 6mm7 %ncreasing load Decreasing load +verage .-= ./ .K .@= .KJ .J .=H -. .H -./ -./ -.H -.= -.J -.J -.=J -.=H -.== .- . .-H .@H .@H .@H
Table for quarter circular bar
Ioad . -. -. . . /. /. @.
%ncreasing load . .@.J .=/ -./ -. -.@/ -.J@
?ertical deflection, P 6mm7 Decreasing load ..@ .J/ .=@ -. -.J -.@J -.J@
Table for circular bar
+verage ..@.J/ .=@ -.@ -. -.@ -.J@
Ioad 6g7
?ertical deflection
oriAontal deflection
. .@ .J .= -. -. -.@ -.J
6mm7 .@J .H -./=K -.=@H ./-.KK@ /./J /.JH=
6mm7 .H@ .=H .==/ -.-KK -.@K -.KJJ .J ./@
Table for deflections of quarter bar for each loading using theory analysis
Ioad 6g7
?ertical deflection
. -. -. . . /. /. @.
6mm7 .-H .@/= .JK .=KJ -.H -./-@ -./@ -.K
Table for deflections of circular bar for each loading using theory analysis
$.1 E%a&'# () Ca#"u#at!(*
a7 #uarter Circle Bar i.
*econd moment of area of the cross section
% 3 bd/ 9 - 3 x - G/ x 6/ x -G/7/ - 3 .J x -G-- m@ ii. ?ertical deflection 6theory7 .g, δv
3 π P:/ @ "% 3
6 . x H.=- 7 6.-7/ 6@7 6 x -H 7 6.J x -G--7 π
3 .@J mm
iii.
*econd moment of area of the cross section % 3 bd/ 9 - 3 x - G/ x 6/ x -G/7/ - 3 .J x -G-- m@
iv. oriAontal deflection 6theory7 .g, δ
3 P:/ "%
3
6. x H.=-7 6.-7/ 67 6 x -H 7 6.J x -G--7
3 .H@ mm
b7 Circular bar i.
*econd moment of area of the cross section % 3 bh/ 9 - 3 x - G/ x 6/ x -G/7/ - 3 .J x -G-- m@ ii. Deflection 6theory7 .g, δ
3 .-@HP:/ "% 3 .-@H 6 . x H.=- 7 6.-7/ 6 x -H 7 6.J x -G--7 3 .-H mm
+.0 DISCUSSION
Eraph is straight lines with a positive slope. *o, its can be said that load is linearly dependent with vertical deflection for quarter circle bar. the vertical deflection will
increase when the load increases. rom the graph plotted, the result from the experiment is match or same with the results that calculated using theorem castiglianoLs. Theoretically the straight lines obtained are through the origin point. *lope for both of the lines are almost the same value. This shows that the error occur in the experiment are very small and the results can be accepted. Eraph obtained is straight line with a positive gradient. "ven though the graph is straight line, but there is a difference between the experiment results and theoretical results. Besides that, theoretical result is bigger compare than experiment results. There are differences in the both results because there are some errors occurred during the experiment. The errors may be caused by the Castigliano!s Theorem formula which is not recommended to be used in the calculation for horiAontal deflection for quarter circle bar. This is because the forces applied are only assumptions values and not the real values. But, it still can use for compare with theory and experiment. Eraph in a straight line shape obtained for load 6'7 versus vertical deflection v 6mm7 for circular bar. The gradient for both of the lines are positive. The lines are through the origin point.
There differences between the experiment results and theoretical results are very small compare to graph of load 6'7 versus horiAontal deflection 6mm7 for quarter circle bar. This shows that the experiment results and calculation results are almost the same. "ven though the errors are very small, but it could not be neglected and have to be taen in consider.
ro the graphs plotted, we can conclude that deflection is linearly dependent on applied load. Deflection will be increase, when the applied load increases.
rom the data that has been recorded, there are a lot of errors occurred causes the differences between the experiment results and theoretical results. The factors which cause the errors are1 • • • • •
,-
+pplied load is not a standard value The bar is already in deformed condition before the experiment begin The mistaes done by the person who recording the readings :andom errors caused by the equipments Dial gauge is not placed at the exact place and always dislocated T/ aa*ta () u!* Cat!#!a*( T/(r&
Castigliano!s theorem can be used to measure deflection of curved beams, but other theorems such as &acaulay theorem can be used only to measure vertical or horiAontal bar. This are the advantages of using Castigliano!s theorem compared to other theories. i. This theorem can measured deflection of the bar that bend but &acaulay methods can only measured deflection of vertical and horiAontal. ii. The value of young &odulus " is fixed according to the material been used. iii. Calculation that is to get the value of beam deflection is more simple if the differences between load is solve first before we integrate or other solution. iv. This method is using the !dummy! where it function is to tae the point that is not mentioned the actual load. ,"
B! Cat!#!a*( T/(r& (t/r &t/( t/at "a* - u t( &aur t/ -a& )#"t!(* ar67
i.
Macaulay method M 'eed an
equation for all beams. This method is using all
three basics equations to find the value of deflection and beam slope. ii.
Double integration method M This method is having /
equations to find the
value of deflection that is overall beam moment equation, deflection equation and slope equation. %f the load situation is changed along the beam, the moment equation is also changed. Besides Castigliano!s method, other methods such as &acaulay method, *uperposition method, and &oment +rea method can be used to calculate beam deflection.
8.0 CONCLUSION
+s a conclusion, this experiment has achieved its main ob)ective to validate the formula of deflection of curved bars based on Castigliano!s Theorem. The formulas are1
?ertical deflection 6quarter circle bar7 3 πP: /9@"% Deflection 6circular bar7 3 .-@HP: /9"%
"ven though the formula for horiAontal deflection for quarter circle bar is given as 3 P:/9@"%, but the results obtained from experiment is much bigger compare to the calculation done using that formula. +ccording to reference boo &echanics of "ngineering &aterials, the formula for horiAontal deflection for quarter circle bar is 3 P:/9"%. This formula is almost matching with results obtained from the experiment. *o the more exact formula to calculate the horiAontal deflection for quarter circle bar is 3 P:/9"%.
The conclusion are1 i. with using the Castigliano!s Theorem ðod in calculate the bar deflection, it more easier if compared with other method. ii. the result of deflection value is not far between experimental and theoretical. iii. this experiment is succesful that is the deflection bar formula according to the Castigliano!s Theorem can be approved it punctuality for getting the beam deflection. There are some differences of results obtained from the experiment and the theorem. This is because of the errors occurred during the experiment. owever the overall conclusion is, deflection is linearly dependent on the applied weight.
9.0 RE:ERENCES
B((;6
a)
Benham, P.P. and Crawford, :. N., &echanics of "ngineering &aterials, -HH@, Iongman *cientific O Technical, "ngland.
b)
Beer O ".:ussell Nohnston, Nr.6-HH7. &echanics of &aterials, &etrics "ditions, &cErawGill Boo Company
c)
-. Clive I. Dyme 6-HHK7, Structural Modelling & Analysis Cambridge $niversity Press, $Q. Page1 J-
d)
. :.C ibbeler 6--7, Mechanics of Materials Eighth Edition in S.I Units Prentice all %nc, Page1 JH O J-H.
e)
/. (illiam +. 'ash 6-HH=7, Schaum’s of Theory & Problems of Strength of Materials &cEraw ill, Page 1 @K@
I*tr*t6
http199www.public.iastate.edu9Rfanous9ce//9virtualwor9homepage.html http199en.wiipedia.org9wii9&acaulaySKsmethod http199www.codecogs.com9reference9engineering9materials9curvedbeams.phpUsec=
