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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsA beam is a structural member or machine component that is designed to support primarily forces acting perpendicular to the axis of the member.
Types of BeamsSimply
Supported(One pin, one roller)
Overhanging(One pin, one roller)
Cantilever(One fixed end)
Propped(One fixed end and one roller)
Continuous(Several pins and rollers)
Built-in(Both ends fixed)
Statically Determinate Statically Indeterminate
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in BeamsTypes of Supports Types of Loads
Roller:one unknown
Pin:two unknowns
Fixed:three unknowns
Concentrated loads
Distributed loads
Concentrated moments
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MEM202 Engineering Mechanics - Statics MEM
Shear Forces and Bending Moments in Beams
1. Determine the reactions, if necessary, using the free-body diagram of the overall beam.
2. Cut the beam at cross section where the shear force and bending moment are to be determined. Draw the free-body diagram.
3. Set up equilibrium equations for the free-body diagram and use them to determine the shear force and bending moment at the cross section.
4. Repeat steps 2 through 4 for as many cross sections as needed.
Procedure for determining shear forces and bending moments
( )axPwxRxMM
PwxRVF
raa
ry
−−−=⇒=
−−=⇒=
∑
∑
− 1
2
1
20
0
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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 P
LaAPBAF yyyy ⎟⎠⎞
⎜⎝⎛ −=⇒=−+=∑ 10
0==∑ xx AF
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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 2
0
( )xLqxMMqxxAM xxy −=⇒=++⋅−=∑ 20
2
2
yAx
AxV
xM
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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 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 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 Forces and Bending Moments in Beamsq
LA B
q
LA B
2qL
2qL
02 00 ==−= MqLVqw
2qL
xA
xVxM
⎟⎠⎞
⎜⎝⎛ −= xLqVx 2
( ) 000VVqxdxqwdx x
xx−=−=−= ∫∫
0
2
00 222MMxLxqdxxLqVdx x
xx−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−=⎟
⎠⎞
⎜⎝⎛ −= ∫∫ ( )xLqxM x −=
2
