institute of applied mechanics8-0 viii.3-1 timoshenko beams (1) elementary beam theory...
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Institute of Applied Mechanics8- 1
VIII.3-1 Timoshenko Beams (1)
Elementary beam theory (Euler-Bernoulli beam theory)
Timoshenko beam theory
1. A plane normal to the beam axis in the undeformed state remains plane in the deformed state.
2. All the points on a normal cross-sectional plane have the same transverse displacement.
3. There is no stretch along the beam axis. no thickness stretch
4. A plane normal to the beam axis in the undeformed state remains normal in the deformed state.
13
131 xwu
xxu
13
131 '
xwu
xwxu neglect shear deformation!!
In both beam theory, only stress resultants (sum over cross section area) are considered.
3D problems 1D problems !!
??2 u
qx3
x1
Assume: b 0 symmetric axis
x3
x2
3213321321 ,,,, ,0 xxxtxxxttt
Institute of Applied Mechanics8- 2
A
Ad0 3,322,221,12
33221,3 Ex
VIII.3-2 Timoshenko Beams (2)
1,21
13
1,311
w
x
33221111 E
1,1313 2 w
x3
x2
x3 q
x1
strain field:
13
131 xwu
xxu
??other ij
stress field:
ijkkijijE 1
0
0
0
3,332,231,13
3,322,221,12
3,312,211,11
equations of equilibrium:
??2 u
033
AAd
3322221122 nnnt
B.C.
CA
snnAx
ddd
d33222212
1
CA
stAx
dd d
d212
1
geometry, loading: symmetric w.r.t. x3-axis
0d12 AA
3213321321 ,,,, ,0 xxxtxxxttt
symmetric axis
prismatic beam: n1 = 0: =
Ok!!
12 odd function of x2
Institute of Applied Mechanics8- 3
AA
AxxAxx
ddd
d313,3312,321311
1
A
Ad0 3,332,231,13
VIII.3-3 Timoshenko Beams (3)0
0
0
3,332,231,13
3,322,221,12
3,312,211,11
3332231133 nnnt
CA
snnAx
ddd
d33322313
1
CA
stAx
ddd
d313
1
3213321321 ,,,, ,0 xxxtxxxttt
prismatic beam: n1 = 0: =
11d
dxp
x
Q
A
Ax d0 33,312,211,11
Qstxx
MC
dd
d13
1
Qx
M
1d
d
3312211111 nnnt
A
CAsnnx
x
Mdd
d
d313312213
1
0d
d
0d
d
11
1
xpx
Q
Qx
MSummary:
AAQ d13
A
AxM d113
A
AExx d33221,33
A
AxEI d333221,
A
Aw d1,2
Cstxp d31
Institute of Applied Mechanics8- 4
VIII.3-4 Timoshenko Beams (4)
0d
d
d
d
1
2
1
p
x
wA
x0
d
d
d
d
d
d
1
2
11
x
wA
xEI
x
0d
d
0d
d
11
1
xpx
Q
Qx
M
A
AxEIM d333221,
A
AwQ d1,2
is used to adjust the approximate theory to agree with the 3D theory.
When = 0.3, = 0.850 for rectangular cross-section and 0.886 for circular cross-section.
A
Ax d33322 Approximations:
1. Neglect
2. Replace by 2 : shear factor, a correction factor
Timoshenko beam equation
Institute of Applied Mechanics8- 5
VIII.3-5 Remarks
1. Euler-Bernoulli beam theory neglects shear deformation
0d
d
d
d
1
2
1
p
x
wA
x
0d
d
d
d
d
d
1
2
11
x
wA
xEI
x
1dd xw
21
2
d
d
x
wEIM 12
1
2
21
2
d
d
d
dxp
x
wEI
x
0d
d
0d
d
11
1
xpx
Q
Qx
M
1,EIM
A
AwQ d1,2
21
2
1 d
d
d
d
x
wEI
xQ
2
2. The Timoshenko beam theory accounts for flexural as well as shear
deformation. While the Euler-Bernoulli beam theory accounts
only for flexural deformation.
3. Two B.C.s are required at both ends either w or Q either dw/dx1 or M
Institute of Applied Mechanics8- 6
VIII.3-6 Example (1)0
d
d
d
d
1
2
1
p
x
wA
x
0d
d
d
d
d
d
1
2
11
x
wA
xEI
x
0d
d
0d
d
11
1
xpx
Q
Qx
M
1,EIM
A
AwQ d1,2
q
L
cross-sectional area Amoment of inertia Icorrection factor 2
C
stxp d31
0d
d
d
d
1
2
1
p
x
wA
x
0d
d
d
d
d
d
1
2
11
x
wA
xEI
x
'2 wAEI
11 cxEI
p
211212
cxcxEI
p
31221
131 26
cxcxc
xEI
p
B.C.s:0)()0(
0)()0(
LMM
Lww
0)0(by M
EI
pLc
21 0)( from LM ??)( 1 xw
0 pEI
x3
x1
Institute of Applied Mechanics8- 7
VIII.3-7 Example (2)3
21
31 46
cxEI
pLx
EI
p
0d
d
d
d
d
d
1
2
11
x
wA
xEI
x
0462 3
21
31
21
cx
EI
pLx
EI
pwA
EI
pLx
EI
pEI
3
21
3112 46
22
1cx
EI
pLx
EI
ppLpx
Aw
EI
pLx
EI
p
21
41331
411
212 12242
1cxcx
EI
pLx
EI
ppLxpx
Aw
B.C.s:0)()0( Lww
0)0(by w
EI
pLc
24
3
3 0)( from Lw
1
331
411
212 24122422
1x
EI
pLx
EI
pLx
EI
px
pLx
p
Aw
Institute of Applied Mechanics8- 8
VIII.3-8 Example (3)EI
pLx
EI
pLx
EI
p
2446
321
31 1
331
411
212 24122422
1x
EI
pLx
EI
pLx
EI
px
pLx
p
Aw
1d
d
x
MQ
Lxpx
xpL
xp
M 11
121 222
1,EIM
A
AwQ d1,2
0d
d
0d
d
11
1
xpx
Q
Qx
M
2
2d
d
1
11
1
Lxp
Lxpx
x