MEM202 Engineering Mechanics - Statics Shear Forces …cac542/L20.pdf · MEM202 Engineering...
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Transcript of MEM202 Engineering Mechanics - Statics Shear Forces …cac542/L20.pdf · MEM202 Engineering...
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsSign Convention
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsP
aL
A B
( ) ( ) PLaBaPLBM yyA
=== 0
yAx
AxV
xM PLaAVVAF yxxyy
==== 10
PxLaxAMMxAM yxxy
===+= 10
P
yAa
xA
xVxM P
LaVVPAF xxyy
=== 0
PaLxMMaxPxAM xxy
==++= 10)(
ax 0For
Lxa For
ReactionsP
yA yB
aL
A BxA PL
aAPBAF yyyy
==+= 10
0== xx AF
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsLxa For ax 0For
PLaVVPAF xxyV
=== 0
PaLxMMPaxAM xxy
==+= 10
PLaAVVAF yxxyV
==== 10
PxLaxAMMxAM yxxy
===+= 10
P
PLa
1
BR
a LA B
PLa
PxLa
1 PaL
x
1Pa
LaM
= 1max
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in Beamsq
LA B
Reactions ( ) ( )( )2
02 qLBLLqLBM yyA ===q
LA B
yA yB
xA 20 qLAqLBAF yyyy ==+=
0== xx AF
=== xLqVVqxAF xxyy 20
( )xLqxMMqxxAM xxy ==++= 2022
yAx
AxV
xM
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MEM202 Engineering Mechanics - Statics MEM
Example
Free-body diagram #1
Free-body diagram #2
( ) ( ) ( )( )k 11 ;k 9
023030924Aabout ==
==AB
B
RRqPRM
ft 15
ft 15ft 9
( ) k 6014152.011 === VVFV( ) ( )( )
k-ft 5.58
0215152.0)9(1415
=
=+=M
MVM x
( ) k 6015302.09 ==+= VVFV( ) ( )( ) k-ft 5.580215152.099 ==+= MMM
Reactions
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsRelations among w, V, and M
( )xw
L
A Bx dx
( )wdxdV
dVVwdxVFy=
=++=or
0
wdxdV
=
12
12
2
1
2
1
2
1
VwdxV
VVdVwdx
x
x
V
V
x
x
+=
==
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsRelations among w, V, and M
( )
( )( )2
or
0 2
2
2
dxwVdxdM
dMM
dxwVdxMMO
+=
=++
=
VdxdM
=
( )xw
L
A Bx dx
12
12
2
1
2
1
2
1
MVdxM
MMdMVdx
x
x
M
M
x
x
+=
==
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MEM202 Engineering Mechanics - Statics MEM
Shear-Force and Bending Moment Diagrams
dxdMV
dxdVw ==
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MEM202 Engineering Mechanics - Statics MEM
Shear-Force and Bending Moment DiagramsOne Concentrated Load
Several Concentrated Loads
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MEM202 Engineering Mechanics - Statics MEM
Shear-Force and Bending Moment DiagramsUniform load over the entire span
Uniform load over part of the span.
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MEM202 Engineering Mechanics - Statics MEM
Shear-Force and Bending Moment Diagrams
Cantilever beam with two concentrated loads.
Cantilever beam with a uniform load.
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MEM202 Engineering Mechanics - Statics MEM
Shear-Force and Bending Moment DiagramsBeam with an overhang and a concentrated moment
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in Beams
AR BR
( )( )( ) 0
221000
600032000
02100060002000
2
=+
+
+=
=+=
Mx
xxM
VxF
aa
y( ) ( )( )( ) ( ) 022000561000
13200010=++
+= AB RM
lb 6000=AR
lb-ft 80006000500lb 60001000
2 +=
+=
xxMxV
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in Beamsq
LA B
q
LA B
2qL
2qL
02 00
=== MqLVqw
2qL
xA
xVxM
= xLqVx 2
( ) 000 VVqxdxqwdx xxx
===
0
2
00 222MMxLxqdxxLqVdx x
xx=
=
= ( )xL
qxM x = 2
