Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

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Week 8 § 3-9 Strain Energy in Torsion and Pure Shear Fig. 3-34 Prismatic bar in pure torsion. P GI TL = φ Fig. 3-35 Torque-rotation diagram for a bar in pure torsion (linearly elastic material). 2 T U = (類似軸向受力元件之 2 δ T W U = = P GI TL = φ L GI GI L T U P P 2 2 2 2 φ = = (單位 JouleN.m ftlb1
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    27-May-2022
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Transcript of Week 8 ꆱ 3-9 Strain Energy in Torsion and Pure Shear

Chapter 1Fig. 3-34 Prismatic bar in pure torsion.
PGI
TL =φ
Fig. 3-35 Torque-rotation diagram for a bar in pure torsion (linearly elastic material).
PGI TL
1
Shear Force
Vτht
Strainenergy density
3-10
©
b150N.m 1.6mG80Gpa
P79.52×103 mm4
Uc4.56 J
a distributed torque.
a T(x)tx
work by the applied torqueStrain energy of the bar
WU
Fbτbtbdx Fcτctcdx FbFc
τbtbτctc Shear Flow fτtconst
7
Fig. 3-41 Cross section of thin-walled tube.
Total shear force on the element f ds The moment
dTrfds
Fig. 3-42 Thin-walled circular tube.
©2001 Brooks/Cole, a division of Thomson Learning, Inc. Thomson Learning™ is a trademark
Thin-walled rectangular tube
9
Fig. 3-40
10
tdsdx
dx t ds
J2πr3t Rectangular tube
Fig. 3-44 Angle of twist φ for a thin-walled tube.
GJ TL
GJ LTUWT
Week 8
3-13
©
Fig. 3-46 Example 3-14. Comparison of circular and square tubes.
Same material Same length Same wall thickness Same cross-sectional area Circular tube Am1πr2
J12πr3t A12πrt
Beams
15
Simply supported beam Cantilever beam Beam with a overhang
©
20
01
Br
oo
ks
/C
ol
e,
Types of Loads Concentrated load Distributed load Linearly varying load Simple beam Fig 4-2a
ΣFhoriz0 HAP1cosα0
HAP1cosα
ΣMB0 RAL(P1sinα)LaP2(LP)qc2 / 20
ΣMA0 RBL(P1sinα)aP2bqc(Lc / 2)0
( )( ) L
cLqc
ΣMB0 RALP4LaM10
ΣMA0 R4aPBLM10
( ) L
§ 4-3 Shear Forces Bending Moments
©
Sign Convention Fig. 4-5 Sign
conventions for shear force V and bending moment M.
in the positive direction
©
©
ΣFvert0RAPV0
VRAP L
L MPV 0
©
XL 2 0
XL L Lq
2
0
0
1
B
r
o
Fig. 4-9 Example 4-3. Shear force and bending moment in a beam with an overhang.
q20 lb/ft
V6K
M58.5 K-ft