OBJECTIVE 1) To examine examine how shear force force varies varies with with an increasing increasing point point load. ) To examine examine how shear force force varies at the the c!t position position of the "eam for vario!s vario!s loading loading conditions.
#E$%&I&' O(TCOE 1) ) ,) -)
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I&T%O/(CTIO& $&/ T0EO% Beams are one of the most common elements fo!nds in str!ct!res. 2hen a hori3ontal mem"er of a str!ct!re 4"eam) is loaded with loads in the vertical direction it will "end d!e to the ind!ced reactions of s!ch loads. The amo!nt of "ending of the "eam will depend on the amo!nt and t+pe of loads length of the "eam elasticit+ and t+pe of the "eam. If the ends of a "eam are restrained longit!dinall+ "+ its s!pport or if a "eam is a component of a contin!o!s frame axial force ma+ also develop. If the axial force is small the t+pical sit!ation for most "eams can "e neglected when the mem"er is designed. In the case of reinforced concrete "eams small val!es of axial compression act!all+ prod!ce a modest increase 4on the order of 5 to 16 percent) in the flex!ral strength of the mem"er. To design design a "eam the engineer m!st constr!ct the shear and moment c!rves c! rves to determine the location and magnit!de of the maxim!m val!es of these forces. Except Excep t for short heavil+ loaded "eams whose dimensions d imensions are controlled "+ shear re7!irements the proportion of the cross section are determined "+ the magnit!de of the maxim!m moment in the span. $fter a section is si3ed at the point of maxim!m moment the design is completed "+ verif+ing that the shear stresses at the point of maxim!m shear !s!all+ ad8acent to a s!pport are e7!al to or less than the allowa"le shear strength of the material. 9inall+ the the deflection prod!ced "+ " + service loads m!st "e chec*ed to ens!re that the mem"er has ade7!ate stiffness. #imits on deflection are set "+ str!ct!ral codes. To provide this information graphicall+ graphicall+ we constr!ct shear and moment c!rves. c! rves. These c!rves which prefera"l+ sho!ld "e drawn to scale consist of val!es of shear and moment plotted as ordinates against distance along the axis of the "eam. $ltho!gh we can constr!ct shear and moment c!rves "+ c!tting free "odies at intervals along the axis a xis of a "eam and writing e7!ation of e7!ili"ri!m to esta"lish the val!es of shear and moment at partic!lar section it is m!ch simpler to constr!ct constr!ct these c!rves from the "asic relationships that exist "etween load shear and moment. Bending moment at an+ section of a "eam is defined to "e the alge"raic s!m of the moment at the sectioning developed "+ vertical components of external forces applied on the "eam "+ considering the left or the right of ass!med section or !n"alanced
moment at the sectioning to the left or the right of the ass!med section. Variation of "ending moment along "eam can "e vis!ali3ed "+ Bending oment /iagram 4B/) which is defined as a diagram that shows variations of "ending moment along the "eam considered. The final step in the design of a "eam is to verif+ that it does not deflect excessivel+. Beams that are excessivel+ flexi"le !ndergo large deflections that can damage attached nonstr!ct!ral constr!ction: plaster ceiling masonr+ walls and rigid piping for example ma+ crac*. ;ince most "eams are span short distances sa+ !p to ,6 or -6 ft are man!fact!red with a constant cross sections to minimi3e cost the+ have excess flex!ral capacit+ at all sections except the one at which maxim!m moment occ!rs. Beams are t+picall+ classified "+ the manner in which the+ are s!pported. $ "eam s!pported "+ a pin at the one end and a roller at the other end is called a simpl+ s!pported "eam. If the end of the simpl+ s!pported "eam extends over a s!pport it is referred to as a "eam with an overhang. $ cantilever "eam is fixed at the one end against translation and rotation. Beams are s!pported "+ several intermediate s!pport are ca lled contin!o!s "eam. If "oth ends of a "eam are fixed "+ the s!pport the "eam is termed fixed ended. 9ixed ended "eams are not commonl+ constr!cted in practice "!t the val!es of end moments in them prod!ced "+ vario!s t+pes of load are !sed extensivel+ as the starting point in several methods of anal+sis for indeterminate str!ct!res.
9ig. 1 : ;hear 9orce and Bending oment
9ig. : Change of ;hape d!e to ;hear 9orce
There are a n!m"er of ass!mptions that were made in order to develop the Elastic Theory of Bending . These are: 1) The "eam has a constant prismatic cross C/ in the image "elow). 5) The ne!tral plane of a "eam is a plane whose length is !nchanged "+ the "eam?s deformation. This plane passes thro!gh the centroid of the cross
Part 1
2
a % $
@c!tA #
% B
9ig!re 1
;hear force at left of the section Sc W ( L-a ) ..e7!ation 1 L ;hear force at the right of the c!t section Sc -Wa e7!ation L
Part 2
(se this statement : “The shear force at the ‘cut’ is equal to the algebraic sum of the force acting to the left or right of the cut”
$DD$%$T(; 1) eas!ring 9orce achine ) #oad
D%OCE/(%E Part 1 1) Chec* the /igital 9orce /ispla+ meter reads 3ero with no load. ) Dlace a hanger with a 166g mass to the left of the @c!tA. ,) %ecord the /igital 9orce /ispla+ reading in Ta"le 1. %epeat !sing an+ masses "etween 66g and 566g. Convert the mass into a load in &ewton 4m!ltipl+ "+ .F1). Shear Force at the cut (N) = Displayed Force. -) Calc!late the theoretical ;hear 9orce at the c!t and complete the Ta"le 1. Part 2
1) Chec* the /igital 9orce /ispla+ meter 3ero with no load. ) Caref!ll+ load the "eam with the hangers in an+ positions and loads as example in 9ig!re 9ig!re , and 9ig!re - and complete Ta"le . ,) %ecord the /igital 9orce /ispla+ reading where : Shear Force at the cut (N) = Displayed Force. -) Calc!late the s!pport reaction 4% $ and % B) and calc!lated the theoretical ;hear 9orce at the c!t.
1-6mm
% $
@c!tA
% B
21 66g 41.G&)
9ig!re
% $
6mm
21
2
@c!tA
G6mm
2here = 21 > 2 an+ load "etween 166g to 566g 9ig!re ,
% B
% $
6mm
21
@c!tA
% B
2 -66mm 2here = 21 > 2 an+ load "etween 166g to 566g
9ig!re -
%E;(#T Mass *(g)
6 66 56 ,66 ,56 -66
Load (N)
6 1.G .-5, .-, ,.-,,.-
Force
Experimental Shear Force
heoretical Shear Force
(N)
(N)
(N)
6 1.66 1.-6 1.G6 1.F6 .16
6 1.66 1.-6 1.G6 1.F6 .16
6 6.F6, 1.661.61.-65 1.G65
* Use any mass between 200g to 00g a!le "
No
Mass"
Mass#
$"
$#
Force
(g)
(g)
(N)
(N)
(N)
heoretical
Experimental
Shear Force (N)
% & (N)
%' (N)
Shear Force (Nm)
, -
66 66 66
6 ,66 ,66
1.G 1.G 1.G
6 .-, .-,
< 6.56 .G6 6.H6
< 6.56 .G6 6.H6
.5FG .1F5 1.-F
a!le #
/$T$ $&$#;I; For a!le " (Part 1) !rom !ig"re #$ 2
a % $
@c!tA #
% B
For ; Mass g ! 2""
#oad & 66 x .F1 1666
1.G &
9orce & 1.66 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
1.66 &
Theoretical ;hear 9orce & ;c 2 4# a) # 1.G x 46.-- 6.G) 6.- 6.F6, &
For ; Mass g ! 2#"
#oad & 56 x .F1 1666
.-5, &
9orce & 1.-6 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
1.-6 &
< 6.G.H6 ,.G5H
< 6.G.H6 6.H1,
Theoretical ;hear 9orce & ;c 2 4# a) # .-5, x 46.-- 6.G) 6.- 1.66- &
For ; Mass g ! $""
#oad & ,66 x .F1 1666
.-, &
9orce & 1.G6 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
1.G6 &
Theoretical ;hear 9orce & ;c 2 4# a) # .-, x 46.-- 6.G) 6.- 1.6- &
For ; Mass g ! $#"
#oad & ,56 x .F1 1666
,.-,- &
9orce & 1.F6 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
1.F6 &
Theoretical ;hear 9orce & ;c 2 4# a) # ,.-,- x 46.-- 6.G) 6.- 1.-65 &
For ; Mass g ! %""
#oad & -66 x .F1 1666
,.- &
9orce & .16 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
.16 &
Theoretical ;hear 9orce & ;c 2 4# a) # ,.- x 46.-- 6.G) 6.- 1.G65 &
For a!le # (Part 2) !rom !ig"re 2$
1-6mm
% $
@c!tA
21 66g 41.G&) 9orce & < 6.56 & Experimental ;hear 9orce & /ispla+ed 9orce (%hear !orce at a c"t& ')
< 6.56 &
K 6 K9x 6 K9+ 6 KB 6 =
<1.G 46.5F) L % $ 46.--) 6 % $ 1.1,F 6.-% $ .5FG &
K9x 6 K9+ 6
= %B L .5FG 1.G 6 % B 1.G .5FG % B <6.G- &
Theoretical ;hear 9orce &
< 2a # < 41.G) x 46.1-) < 6.G- &
!rom !ig"re $
% $
6mm
21
G6mm
2
@c!tA
% B
% B
9orce & .G6 & Experimental ;hear 9orce & /ispla+ed 9orce .G6 &
(%hear !orce at a c"t& ')
K 6 K9x 6 K9+ 6 K$ 6 =
% B 46.--) .-,46.G) 1.G46.) 6 % B 1.1H 6.-% B .H6 &
K9x 6 K9+ 6
= %$ 1.G .-, L .H6 6 % $ 1.G .-, .H6 % $ .1F5 &
Theoretical ;hear 9orce &
=
=
4
−
−
4
−
−
W 1 a L
)
−
4
−
W a L
1.EG ) 6. ) 6.--
−
)
4
−
.E-, ) 6.G ) 6.--
6.F1 L 1.H, .H6 &
!rom !ig"re $
% $ -6mm
21
@c!tA 2 -66mm
%B
9orce & 6.H6 & Experimental ;hear 9orce & /ispla+ed 9orce 6.H6 &
(%hear !orce at a c"t& ')
K 6 K9x 6 K9+ 6 KB 6 =
<1.G 46.) .-,46.6-) L % $ 46.--) 6 % $ 6.5- 6.-% $ 1.-F &
K9x 6 K9+ 6
= %B L 1.-F 1.G .-, 6 % B 1.G L .-, 1.-F % B ,.G5H &
Theoretical ;hear 9orce &
=
=
4
[
−
4
W L L
−
−
a
]
)
−
.E-,[ 6.--
4
−
−
W 1 a L
6 .- ]
6.--
< 6.GF 4<6.F1) 6.H1, &
/I;C(;;IO& art " 1) &eri'e equation 1
!rom !ig"re #$ 2
a
@c!tA
)
)
−
4
−
1.EG ) 6. 6.--
)
#
% $
% B
#et = KB 6 4 % $ x # ) 2 4 # a ) 6 % $ 2 4 # a ) # %ince the force at the c"t is e+"al to the algebraic s"m of the force acting to the left or right of the c"t$ Therefore& ;C % $ ;c 2 4 # a ) # #et = K$ 6 4 <% B x # ) 4 2 x a ) 6 % B 4 < 2 x a ) # Therefore =
;C 4 < 2 x a ) #
2here
2 #oad a C!t section from % $ # #ength from % $ to % B
This e7!ation is !sed to determine the val!e of ;hear 9orce "+ theor+. 2 is a load place !pon the @c!tA section with the length of a. # is total length from % $ to % B. 2) Plot a grah hich comare *our e+erimental result to those *ou calculate, using theor*-
Dlease see graph 1 as attached.
$) .omment on the shae of the grah- /hat ,oes it tell *ou about ho 0hear Force 'aries ,ue to an increase, loa,
9rom the ;hear 9orce vers!s #oad graph we plotted in this experiment a linear
graph was o"tained for "oth Experimental ;hear 9orce and Theoretical ;hear 9orce val!es. Both graphs are linear and go thro!gh the origin 466) which tell !s that ;hear 9orce does not exist when no load was applied on the "eam. 9rom the graph we can notice that when the load applied on the "eam was increase the ;hear 9orce will also increase. This indicate that ;hear 9orce is linearl+ proportional (,ositie) to the load appl+ on the "eam :
;hear 9orce M #oad
%) &oes the equation *ou use, accuratel* re,ict the beha'ior of the beam
es the e7!ation %c
. W(L / a) L that we !sed in this experiment for
Theoretical ;hear 9orce calc!lation acc!ratel+ predict the "ehavior of the "eam. This is "eca!se from the 'raph 1 plotted we can notice that when the load we placed at the "eam was increased the val!e of ;hear 9orce also increased. This indicate that ;hear 9orce is linearl+ proportional (,ositie) to the load appl+ on the "eam. Eam,le $ 9rom the experiment when a .-5, & load was applied on the "eam at the @c!tA the Experimental ;hear 9orce o"tained was 1.-6 &. 9rom the calc!lation done for Theoretical ;hear 9orce "+ !sing the %c . W(L / a)L e7!ation the ;hear 9orce we o"tain was 1.-5 &. This indicates that this e7!ation can acc!ratel+ predict the "ehaviors of the "eam.
art # 1) .omment on ho the results of the e+eriments comare ith those calculate, using the theor*
9rom the experiments done "+ o!r gro!p we fo!nd that there is onl+ a small difference "etween the val!es of Experimental ;hear 9orce and the Theoretical ;hear 9orce. 9or fig!re and fig!re , the val!e of the Experimental ;hear 9orce is almost the same compare to the Theoretical ;hear 9orce. 2hile for the fig!re -
the val!e of the Theoretical ;hear 9orce is higher than the val!e of the Experimental Bending oment. %eferring to this res!lts we concl!de that the differences "etween the val!e of the experiment and theor+ was pro"a"l+ ca!se "+ the mista*e done "+ o!r gro!p mem"er when ta*ing the val!e for the force when it was hang on the "eam.
2) &oes the e+eriment roof that the shear force at the ‘cut’ is equal to the algebraic sum of the forces acting to the left or right of the cut- f not h*
es the experiment proof that the shear force at the @c!tA is e7!al to the alge"raic s!m of the forces acting to the left or right of the c!t. This is "eca!se from the val!e of 21 2 % $ and % B we can concl!de that 21 L 2 % $ L % B !ig"re 2 21 L 2 % $ L % B 1.G & L 6
.5FG & L 4<6.G- &) 1.G &
!ig"re 21 L 2 % $ L % B 1.G & L .-,
.1F5 & L .H6 & -.65 &
!ig"re 21 L 2 % $ L % B 1.G & L .-,
1.-F & L ,.G5H & -.65 &
$) Plot the shear force ,iagram for loa, cases in Figure 2$ an, %-
Dlease see graph and , as attached.
%) .omment on the shae of the grah- /hat ,oes it tell *ou about ho 0hear Force 'aries ,ue to 'arious loa,ing con,ition
9rom ;9/ 'raph for 9ig!re we o"tained in 'raph we can noticed that when a loading <1.G & is p!t at the end of the "eam 4left side of % $) the val!e of the
shear force ca!se "+ this load is negative. %eaction 9orce at $ is e7!al to .5FG & and therefore the total ;hear 9orce at this point is L 6.G- &. &egative force of <6.G- & at B "alances the ;hear 9orce at $ and th!s total ;hear 9orce at B is 3ero.
9rom ;9/ 'raph for 9ig!re , we o"tained in 'raph when a loading <1.G & and <.-, & are "oth place at the length of 6 mm and G6 mm from the right side of % $ calc!lation reveal that reaction force at $ is L .1F5 & and reaction force at B is L .H6 &. The graph also indicates that ;hear 9orce on the negative part is e7!ivalent to the positive part that is e7!al to 3ero.
9rom ;9/ 'raph for 9ig!re - we o"tained in 'raph , we can concl!de that when a loading of 1.G & and .-, & are "oth place -6 mm and -66 mm from the right side of % $ calc!lation reveal that reaction force at $ is L 1.-F & and reaction force at B is L ,.G5H &. The graph also tells !s that ;hear 9orce on the negative part is e7!ili"ri!m to the positive part that is 3ero.
9rom "oth ;9/ 'raph o"tained from the 'raph and 'raph , the shape of the graph is close at the "oth end of the origin. This indicate that ;hear 9orce will change according to the load appl+ to the "eam. This happens to ens!re that ;hear 9orce at left side is e7!al to the ;hear 9orce at the right side to create e7!ili"ri!m.
CO&C#(;IO& 9rom this experiment o!r gro!p managed to examine how shear force varies with an increasing point load. O!r gro!p also managed to examine how shear force varies at the c!t position of the "eam for vario!s loading conditions. 9or part one experiment we concl!de that when the load we place at "eam is increase the ;hear 9orce will also increase. Th!s we concl!de that ;hear 9orce is linearl+ proportional (,ositie) to the load appl+ on the "eam.
2hile for the part two experiment we concl!de that from the ;9/ graph draw "+ o!r gro!p in this experiment we noticed that ;hear 9orce normall+ will happen at an+ point on the "eam when a load is appl+ at the @c!tA. The res!lt from the experiment also indicate that ;hear 9orce at the @c!tA section is e7!al to the forces acting at "oth right and left side of the @c!tA section on the "eam.
%E9E%E&CE; !sof $hamad 4661). Ne*ani* Bahan /an ;tr!*t!r. ala+sia: (niversiti Te*nologi ala+sia ;*!dai Johor /ar!l TaA3im. %. C. 0i""eler 4666). Nechanic Of aterials. -th. ed. England: Drentice 0all International Inc.
