- •Preface
- •Contents
- •1 Introduction
- •References
- •2.1…Review of Dynamic Engineering Theories of Thin-Walled Beams of Open Section
- •References
- •3.1…Theory of Thin-Walled Beams Based on 3D Equations of the Theory of Elasticity
- •3.1.1 Problem Formulation and Governing Equations
- •3.1.2.1 Solution on the Quasi-Longitudinal Wave
- •3.1.2.2 Solution on the Quasi-Transverse Shear Wave
- •3.2…Construction of the Desired Wave Fields in Terms of the Ray Series
- •References
- •4.2.3 Numerical Example
- •Appendix
- •References
- •5 Conclusion
- •6.3…The Main Kinematic and Dynamic Characteristics of the Wave Surface
- •Reference
38 |
3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams |
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3.1.2.2 Solution on the Quasi-Transverse Shear Wave |
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Now we put G ¼ GII ¼ |
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in all Eqs. 3.61–3.63 and 3.72–3.75, i.e. |
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write them on the quasi-transverse wave. As a result we obtain |
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2GII1ðqG22 þ rkk0 Þ |
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hð0kÞ þ ayxð1kkÞ þ ðK þ sÞ gð0kÞ axxð1kkÞ |
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¼GII1 qG21 þ qG22 þ 2r0kk x0ðkÞ sin u
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þ GII1qG22 xð1kyÞ xð0kÞ sin u þ ðK þ sÞgð0kÞ |
þ F2ðk 1ÞjG¼GII ; ð3:95Þ |
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2GII1ðqG22 þ rkk0 Þ |
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gð0kÞ axxð1kkÞ ðK þ sÞ hð0kÞ þ ayxð1kkÞ |
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¼GII1 qG21 þ qG22 þ 2r0kk x0ðkÞ cos u
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þ GII1qG22 |
xð1kxÞ þ xð0kÞ cos u ðK þ sÞhð0kÞ þ F3ðk 1ÞjG¼GII ; |
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ð3:96Þ |
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2G 1 |
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IA 1k |
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yhðkÞ |
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¼ GII1 qG12 þ qG22 þ 2rkk0 |
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xð0kÞF ax cos u þ ay sin u þ xð1kxÞIx sin u |
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þ xð1kyÞIy cos u þ |
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IyÞsin 2u ðK þ sÞ hð0kÞIx sin u gð0kÞIy cos u |
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þ GII1qG22F |
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ayxð1kyÞ axxð1kxÞ xð0kÞ ax cos u þ ay sin u |
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þ F7ðk 1ÞjG¼GII ; |
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ð3:97Þ |
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þ ðK þ sÞ aygð0kÞ þ axhð0kÞ |
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G 2 |
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2G 1 |
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ax cos u þ ay sin u xð1kk 1Þ gð0k 1Þ cos u þ hð0k 1Þ sin u |
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þ F1ðk 2ÞjG¼GII ; |
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ð3:98Þ |
3.1 Theory of Thin-Walled Beams Based on 3D Equations
GII2qðG22 G12Þ |
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xð1kxÞ þ xð0kÞ cos u hð0kÞðK þ sÞ |
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d n |
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qG12 þ rkk0 |
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xð1kx 1Þ þ xð0k 1Þ cos u hð0k 1ÞðK þ sÞ |
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þqG21 þ 2qG22 þ 2r0kk GII1x1ðkk 1Þ sin u þ F4ðk 2ÞjG¼GII ;
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xð1kyÞ xð0kÞ sin u þ gð0kÞðK þ sÞ |
¼2GII1 qG21 þ r0kk dds nx1ðky 1Þ x0ðk 1Þ sin u þ g0ðk 1ÞðK þ sÞo
þqG21 þ 2qG22 þ 2r0kk GII1x1ðkk 1Þ cos u þ F5ðk 2ÞjG¼GII ;
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G 2 |
qð |
G2 |
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1x sin |
1y cos |
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sin 2u þ |
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cos u ðK þ sÞ |
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2G 1 |
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þ rkk ds wðk 1Þ xðk 1Þ |
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x0ðk 1Þ 2 sin 2u þ h0ðk 1Þ sin u þ g0ðk 1Þ cos u ðK þ sÞ
þ F6ðk 2ÞjG¼GII :
39
ð3:99Þ
ð3:100Þ
ð3:101Þ
From (3.95)–(3.101) it is seen that on the quasi-transverse wave, in contradistinction to the quasi-longitudinal wave, the discontinuities h0ðkÞ; g0ðkÞ; and x1ðkkÞ are determined by the differential Eqs. 3.95–3.97 within the accuracy of arbitrary constants, while the discontinuities x0ðkÞ; x1ðkxÞ; x1ðkyÞ; and wðkÞ are defined from the the algebraic Eqs. 3.98–3.101, in so doing the discontinuities x0ðkÞ and wðkÞ according to (3.98) and (3.101), respectively, have the higher order than the discontinuities h0ðkÞ; g0ðkÞ; and x1ðkkÞ; while the discontinuities x1ðkxÞ and x1ðkyÞ; as it follows from (3.99)
and (3.100), have the same order as the primarily components on this wave.
For arbitrary magnitudes of k, the set of Eqs. 3.95–3.97, 3.99 and 3.100 can be rewritten as
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xð1kxÞ ¼ hð0kÞðK þ sÞ þ B1ðk 1Þ; |
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ð3:102Þ |
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xð1kyÞ ¼ gð0kÞðK þ sÞ þ B2ðk 1Þ; |
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ð3:103Þ |
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þ ðK þ sÞgðkÞ |
¼ ðK þ sÞaxxðkÞ |
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þ B3ðkÞ; |
ð3:104Þ |
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þ B4ðkÞ; |
ð3:105Þ |
40 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams
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dx1k |
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ð3:106Þ |
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where IpC ¼ Ix þ Iy; and functions |
BjðkÞðsÞ ði ¼ 1; 2; 3; 4; 5Þ are |
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Appendix 3. |
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The general solution of (3.104)–(3.106) has the form |
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¼ gð0kÞ sin v þ hð0kÞ cos v ayxð1kkÞ þ sin v |
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þ B4ðkÞ cos v ds þ cos v |
B3ðkÞ cos v B4ðkÞ sin v ds; |
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¼ gð0kÞ cos v hð0kÞ sin v þ axxð1kkÞ þ cos v |
B3ðkÞ sin v |
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B3ðkÞ cos v B4ðkÞ sin v ds; |
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x1ðkkÞ ¼ ðIpCÞ 1 B5ðkÞ þ F axB4ðkÞ ayB3ðkÞ ds þ kð0kÞ; s0
where h0ðkÞ; g0ðkÞ; and kð0kÞ are arbitrary constants, and
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v ¼ ðK þ sÞds:
s0
ð3:107Þ
ð3:108Þ
ð3:109Þ
Reference to Eqs. 3.107–3.109 shows that the main values on the quasi-transverse wave, which define the type of this wave, i.e. h0ðkÞ; g0ðkÞ; and x1ðkkÞ; are interconnected with each other, since they are expressed in terms of x1ðkkÞ via the shear center coordinates which, in the general case of the beam’s cross-section, are different from those of the center of gravity. The values x1ðkxÞ and x1ðkyÞ; according to (3.102) and (3.103), ultimately depend on the value x1ðkkÞ as well, and this coupling is supported by the torsion s; the value K ¼ du=ds; and by the angle v: In other words, in the case of a spatially twisted thin-walled beam of open section, on the twisting-shear wave there occur flexural motions of the same order as the discontinuities in the velocities of the transverse translatory motions and in the angular velocity of rotation of the cross-section.
3.1 Theory of Thin-Walled Beams Based on 3D Equations |
41 |
At k = 0, for example, the values x0ð0Þ and wð0Þ vanish to zero, while the values
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xð10kÞ ¼ const; |
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ð3:110Þ |
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xð10xÞ ¼ hð00ÞðK þ sÞ; |
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xð10yÞ ¼ gð00ÞðK þ sÞ; |
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ð3:114Þ |
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o x; y |
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transverse wave polarization. The vector dð0Þ has the constant modulus |
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dð0Þ ¼ |
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gð00Þ cos v hð00Þ sin v |
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that is
að0Þ ¼ Uð0Þ v;
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að0Þ ¼ v ¼ ðK þ sÞs ¼ ðK þ sÞGII ;
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where tgUð0Þ ¼ gð00Þ hð00Þ |
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42 |
3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams |
Fig. 3.3 Scheme of the location of the polarization vector of the quasi-transverse wave
In other words, on the wave surface, the vector dð0Þ remaining constant in magnitude on the wave front rotates around the point Bð0Þ with the angular velocity ðK þ sÞGII ; in so doing the polarization vector during its rotation around the point C changes both in magnitude and in the direction (Fig. 3.3). We shall name the point Bð0Þ as the center of rotation.
Note that for curvilinear beams of solid section the center of rotation coincides with the center of gravity of the beam’s cross section.
If the cross section of the open section beam possesses two axes of symmetry, then ax ¼ ay ¼ 0; and the center of rotation coincides with the center of gravity.
From the comparison of Eqs. 2.14 and 3.110–3.114 it follows that three values h0ðkÞ; g0ðkÞ and x1ðkkÞ are related to each other both in the technical theory of thinwalled beams of open section [6] and in the theory developed here. However, in contrast to the technical theory, where these values are interrelated on three transverse waves which propagate with the velocities depending on the geometrical characteristics of the thin-walled beams of open section, in the present theory
this coupling takes place only on one wave propagating with the natural velocity q
GII ¼ qG22 þ q 1r0kk independent of the thin-walled beam geometry.
3.1.3Particular Case of a Straight Thin-Walled Beam of Open Section
In the case of a straight thin-walled beam of open section, the governing equations are considerably simplified since ¼ K ¼ s ¼ 0 and s = z. Thus, for the quasilongitudinal wave from (3.76)–(3.82) we have
3.1 Theory of Thin-Walled Beams Based on 3D Equations |
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2G 1 |
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¼ |
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ðk 1Þ |
; |
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ð |
3:115 |
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dz |
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dx1x |
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d2x1x |
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2G 1 |
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¼ |
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ðk 1Þ |
; |
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ð |
3:116 |
Þ |
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dz |
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dx1y |
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d2x1y |
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2G 1 |
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¼ |
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ðk 1Þ |
; |
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ð |
3:117 |
Þ |
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dz |
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dw |
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d2w |
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2G 1 |
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¼ |
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ðk 1Þ |
; |
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ð |
3:118 |
Þ |
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d |
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GI 2q G12 G22 |
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hð0kÞ þ ayxð1kkÞ ¼ 2GI qG22 þ rkk0 |
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hð0k 1Þ þ ayxð1kk 1Þ |
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1 |
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1y |
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0 d2 |
0 |
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1k |
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2dxð1ky 2Þ |
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þ GI |
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qG2xðk 1Þ þ |
qG2 þ rkk |
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hðk 2Þ þ ayxðk 2Þ qG2 |
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ð3:119Þ |
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GI 2q G12 G22 |
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gð0kÞ axxð1kkÞ ¼ 2GI 1 |
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qG22 þ rkk0 |
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gð0k 1Þ axxð1kk 1Þ |
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2 1x |
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0 d2 |
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1k |
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2dxð1kx 2Þ |
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þ GI |
qG2xðk 1Þ |
þ qG2 þ rkk |
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gðk 2Þ axxðk 2Þ qG2 |
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dz2 |
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ð3:120Þ |
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G22 |
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GI 2q G12 |
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IPAxð1kkÞ þ ayFhð0kÞ axFgð0kÞ |
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2G 1 |
G2 |
0 |
d |
IA 1k |
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Fa |
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Fa 0 |
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¼ |
I |
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q |
2 |
þ rkk dz |
P xðk 1Þ |
xgðk 1Þ þ |
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yhðk 1Þ |
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þGI 1qG22F ayx1ðky 1Þ axx1ðkx 1Þ
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d2 |
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þ qG22 þ rkk0 |
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IPAxð1kk 2Þ Faxgð0k 2Þ þ Fayhð0k 2Þ |
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qG22F |
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ayxð1ky 2Þ axxð1kx 2Þ |
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ð3:121Þ |
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dz |
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For the quasi-transverse wave, from Eqs. 3.95–3.101 we have |
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d |
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2GII1 |
qG22 |
þ rkk0 |
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hð0kÞ þ ayxð1kkÞ ¼ GII1qG22xð1kyÞ |
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0 d2hð0k 1Þ |
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dxð1kk 1Þ |
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2 |
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2 d |
1y |
ay |
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ð3:122Þ |
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þ qG2 þ rkk |
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qG2 |
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xðk 1Þ |
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; |
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dz2 |
dz |
dz |
44 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams
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d |
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2GII1 |
qG22 þ rkk0 |
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gð0kÞ axxð1kkÞ ¼ GII1qG22xð1kxÞ |
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dz |
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0 d2gð0k 1Þ |
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dxð1kk 1Þ |
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2 d |
1x |
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ð3:123Þ |
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þ |
qG2 |
þ rkk |
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qG2 dz |
xðk 1Þ |
þ |
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ax |
; |
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dz2 |
dz |
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d |
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2G 1 |
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G2 |
0 |
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IA 1k |
Fa |
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q |
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þ |
yhðkÞ |
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II |
2 þ rkk dz |
p xðkÞ |
xgðkÞ |
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¼ GII1qG22F ayxð1kyÞ axxð1kxÞ |
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d2 n |
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o |
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þ qG22 þ rkk0 |
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IpAxð1kk 1Þ Faxgð0k 1Þ þ Fayhð0k 1Þ |
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þ qG22F |
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axxð1kx 1Þ ayxð1ky 1Þ ; |
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dz |
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G 2 |
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2G 1 |
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dx0 |
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G2 |
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ðk 1Þ |
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II |
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xðkÞ |
¼ |
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þ rkk |
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dz |
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d2x0 |
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ðk 2Þ |
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þ |
qG1 |
þ rkk |
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; |
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dz2 |
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G 2 |
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1x |
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2G 1 |
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dx1x |
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G2 |
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ðk 1Þ |
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xðkÞ |
¼ |
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þ rkk |
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dz |
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ðk 2Þ |
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þ |
qG1 |
þ rkk |
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; |
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dz2 |
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G 2 |
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G2 |
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2G 1 |
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G2 |
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0 |
dx1y |
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G2 |
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ðk 1Þ |
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II |
q |
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xðkÞ |
¼ |
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þ rkk |
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dz |
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ðk 2Þ |
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þ |
qG1 |
þ rkk |
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dz2 |
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; |
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2 |
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2 |
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2 |
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0 dwðk 1Þ |
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GII |
q |
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G2 |
G1 |
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wðkÞ ¼ 2GII |
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qG1 |
þ rkk |
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dz |
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þ |
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d2w |
ðk 2Þ |
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qG1 |
þ rkk |
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dz2 |
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: |
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ð3:124Þ
ð3:125Þ
ð3:126Þ
ð3:127Þ
ð3:128Þ
From Eqs. 3.115–3.118 it follows that on the quasi-longitudinal wave for k 0
xð0kÞ |
¼ cð0kÞ |
¼ const; |
xð1kxÞ |
¼ cð1kxÞ |
¼ const; |
|
xð1kyÞ |
¼ cð1kyÞ |
¼ const; |
wðkÞ |
¼ cðwkÞ |
¼ const; |
ð3:129Þ |
3.1 Theory of Thin-Walled Beams Based on 3D Equations |
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45 |
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0 |
1k |
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GI G22 |
1y |
0 |
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1k |
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GI G22 |
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1x |
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||||
hðkÞ þ ayxðkÞ ¼ |
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cðk 1Þ; |
gðkÞ axxðkÞ |
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¼ |
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cðk 1Þ |
; |
|||||||||
G12 G22 |
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G12 G22 |
||||||||||||||||
A 1k |
0 |
0 |
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GIG22 |
1y |
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1x |
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IP xðkÞ þ ayFhðkÞ |
axFgðkÞ |
¼ |
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aycðk 1Þ |
axcðk 1Þ |
: |
ð3:130Þ |
|||||||||||
G12 G22 |
||||||||||||||||||
Solving the set of Eqs. 3.130 yields |
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1k |
0 |
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G G2 |
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1y |
0 |
G |
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G2 |
1x |
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I 2 |
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I |
2 |
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|||||||||
xðkÞ ¼ 0; |
hðkÞ |
¼ |
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cðk 1Þ; |
gðkÞ |
¼ G12 G22 |
cðk 1Þ: |
ð3:131Þ |
||||||||||
G12 G22 |
From Eqs. 3.122–3.128 it follows that on the quasi-transverse wave for k 0
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xð0kÞ ¼ xð1kxÞ ¼ xð1kyÞ ¼ wðkÞ ¼ 0; |
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ð3:132Þ |
||||||
|
hð0kÞ þ ayxð1kkÞ ¼ sð1kÞ ¼ const; |
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|||||
|
gð0kÞ axxð1kkÞ ¼ sð2kÞ |
¼ const; |
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ð3:133Þ |
||||
IPAxð1kkÞ þ ayFhð0kÞ axFgð0kÞ ¼ sð3kÞ ¼ const: |
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||||||||
Solving (3.133) we have |
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1 |
¼ cð1kkÞ |
¼ const; |
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xð1kkÞ ¼ sð3kÞ þ sð2kÞax sð1kÞay |
IpC |
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1 |
¼ cðhkÞ |
¼ const; |
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hð0kÞ ¼ sð1kÞ ay |
sð3kÞ þ sð2kÞax sð1kÞay |
IpC |
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ð3:134Þ |
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1 |
¼ cðgkÞ ¼ const: |
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gð0kÞ ¼ sð2kÞ þ ax sð3kÞ þ sð2kÞax sð1kÞay |
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IpC |
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Thus, for the particular case of a straight untwisted thin-walled beam of open profile, it has been found that the transient transverse wave is a pure sheartorsional mode as it follows from Eq. 3.132, while on the quasi-longitudinal wave the ‘admixed’ components of secondary shear occur of the higher order than the main values what is verified by Eq. 3.131.
If we put ax ¼ ay ¼ 0 in Eqs. 3.115–3.134 and neglect warping motions, then we could also obtain, as a particular case, the solution for a straight untwisted rod of a massive cross-section, which coincides with that resulting for a straight rod from Eqs. 3.43–3.48 in Rossikhin and Shitikova [2], when ¼ K ¼ s ¼ 0: This result verifies the validity of the solution obtained for a thin-walled beam of open section.
As for the technical theory by Korbut and Lazarev [7], then there is no transition from the solution for a thin-walled beam to that for a straight beam with a massive cross-section, i.e. to the Timoshenko beam. It is well known that the