3-11 Shear Stresses Shear on a Beam Elementpkwon/me471/Lect 3.2.pdf · * The product of shear...
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1 3-11 Shear Stresses • How are the shear stresses due to a shear force distributed? V Not possible!!!! Shear on a Beam Element V+dV M V dx F 1 F 3 t dx ( ) Ib VQ Ib Q dx dM Q I dM dA y I dM F F bdx F dA I y dM M dA F dA I My dA F A A A A A = = = = − = = + = = = = ∫ ∫ ∫ ∫ ∫ τ τ σ σ 1 1 2 3 1 1 2 2 1 1 1 1 1 1 1 1 1 F 2 M+dM Q is the first moment w.r.t. the neutral axis. σ τ Rectangular & Circular Sections π 3 4r b h/2 h/2 N.P. A V h t Vbh bh h h b Q y h bh V Ib VQ y A y h y h b y h b ybdy ydA Q h y Area Yellow 2 3 8 12 ; 8 4 2 4 6 2 2 1 2 4 2 3 2 2 max 2 max 2 2 3 2 2 2 / = = = = − = = = + − = − = = = ∫ ∫ τ τ N.P. ( ) ( ) A V r V r r r V Ib VQ r r r y A Q 3 4 3 4 ) 2 ( 4 3 2 3 2 3 4 2 2 4 3 max 3 2 max = = = = = = = π π τ π π At the N.P. y y
Transcript of 3-11 Shear Stresses Shear on a Beam Elementpkwon/me471/Lect 3.2.pdf · * The product of shear...
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3-11 Shear Stresses
• How are the shear stresses due to a shear force distributed?
V
Not possible!!!!
Shear on a Beam Element
V+dV
M V
dx
F1
F3
t dx
( )
IbVQ
IbQ
dxdM
QIdMdAy
IdM
FFbdxF
dAI
ydMMdAF
dAIMydAF
A
AA
AA
==
==
−==
+==
==
∫
∫∫
∫∫
τ
τ
σ
σ
1
123
1122
1111
1
11
11
F2 M+dM
Q is the first moment w.r.t. the neutral axis.
σ τ
Rectangular & Circular Sections
π34r
b
h/2
h/2
N.P.
AV
htVbhbhhhbQ
yhbhV
IbVQ
yAyhyhb
yhbybdyydAQh
yAreaYellow
23
812;
842
46
221
2
42
32
2
max
2
max
22
3
222/
===⎟⎠
⎞⎜⎝
⎛=
⎟⎟⎠
⎞⎜⎜⎝
⎛−==
=⎟⎠
⎞⎜⎝
⎛ +⎟⎠
⎞⎜⎝
⎛ −=
⎟⎟⎠
⎞⎜⎜⎝
⎛−=== ∫∫
τ
τ
N.P.
( ) ( ) AV
rV
rr
rV
IbVQ
rrryAQ
34
34
)2(4
32
32
34
2
24
3
max
32
max
====
=⎟⎠
⎞⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛==
ππτ
ππ
At the N.P.
y y
2
Torsion • Torsion - the twisting of a straight bar loaded by twisting moment
called torque. • Examples: screwdriver, drive shaft and axles • Angle of Twist, θ(x) • Assuming the angle of twist is small, the length and radius do not
change.
x
y
z
θ(x)
x
θ(L)
c
ρ
Torsion
( ) ( ) const.21 == xx γγ
maxγρ
γ
θργ
γθρ
⎟⎠
⎞⎜⎝
⎛=
=
=
c
dxddxd
γ(x1)
γ(x2)
x1
x2
θ δ
γ
dx
c
θρτ
ρτ
τρ
τ
ρτ
drdc
dAc
T
cwhere
dAdMT
dMTM
AA
A A
Az
∫∫
∫ ∫∫∑
==
⎟⎠
⎞⎜⎝
⎛=
==
=−=
3max2max
max
0;0Torsion
JTr
JTc
== ττ ormax
6302533000TnFVH ==
Solid Shaft:
Tubular Shaft: ( ) ( )4444
44
322
322
ioio ddccJ
dcJ
−=−=
==
ππ
ππT Z
ρ c
τ
τ
ρ
τ
ρ
J: Polar Moment of Inertia
(hp) in English Units
ωTH = (W) in SI Units
T=torque, N·m ω=angular velocity, rad/s
F=force, lbf T=torque, lbf·in V=velocity, ft/min n=rpm,rev/m
Torsion of Noncircular Members
GbcTL
cbbcT
bcT
3
22max /8.13
βθ
ατ
=
⎟⎠
⎞⎜⎝
⎛ +≈=
yzτyxτ
xzτ
b/c α β
1.0 0.208 0.1406 1.2 0.219 0.1661 1.5 0.231 0.1958 2.0 0.246 0.229 2.5 0.258 0.249 3.0 0.267 0.263 4.0 0.282 0.281 5.0 0.291 0.291 10.0 0.312 0.312 ∞ 0.333 0.333
c
b
3
Closed Thin-walled Tubes
( )( ) ( ) mdAqpdsqqdsppdFdM
qdstdsdAdFtq
2
constant
====
===
==
ττ
τ
( ) ( )
tAT
tAAtrdstrdstT
m
mm
2
22
=
==== ∫∫τ
ττττ
tGATL
m
m21 4
=θ
* The product of shear stress times thickness of the wall is constant.
linemediansectionthebyenclosedarea=mA
linemediansectiontheofperimeter=mL
BBAA
BA
ttFFττ =
=
mA
Open thin-walled sections
Gcll τ
θθ == 1
213LcTcG == θτ
The Angle of twist
l
Stress Concentration
See Peterson’s Stress Concentration Factors
ots
ot
K
K
ττ
σσ
max
max
=
=
