3.1.2.2 Solution on the Quasi-Transverse Shear Wave

GII2qðG22 G12Þ

xð1kxÞ þ xð0kÞ cos u hð0kÞðK þ sÞ

qG12 þ rkk0

xð1kx 1Þ þ xð0k 1Þ cos u hð0k 1ÞðK þ sÞ

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38	3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams
3.1.2.2 Solution on the Quasi-Transverse Shear Wave
Now we put G ¼ GII ¼				q
Now we put G ¼ GII ¼				G22 þ q 1rkk0	in all Eqs. 3.61–3.63 and 3.72–3.75, i.e.
write them on the quasi-transverse wave. As a result we obtain
	d
2GII1ðqG22 þ rkk0 Þ			hð0kÞ þ ayxð1kkÞ þ ðK þ sÞ gð0kÞ axxð1kkÞ
2GII1ðqG22 þ rkk0 Þ		ds	hð0kÞ þ ayxð1kkÞ þ ðK þ sÞ gð0kÞ axxð1kkÞ

¼GII1 qG21 þ qG22 þ 2r0kk x0ðkÞ sin u

(			)
þ GII1qG22 xð1kyÞ xð0kÞ sin u þ ðK þ sÞgð0kÞ			þ F2ðk 1ÞjG¼GII ; ð3:95Þ
d
2GII1ðqG22 þ rkk0 Þ		gð0kÞ axxð1kkÞ ðK þ sÞ hð0kÞ þ ayxð1kkÞ
2GII1ðqG22 þ rkk0 Þ	ds	gð0kÞ axxð1kkÞ ðK þ sÞ hð0kÞ þ ayxð1kkÞ

¼GII1 qG21 þ qG22 þ 2r0kk x0ðkÞ cos u

(

)

þ GII1qG22

xð1kxÞ þ xð0kÞ cos u ðK þ sÞhð0kÞ þ F3ðk 1ÞjG¼GII ;

ð3:96Þ

( d

)

2G 1

IA 1k

þ sÞ

ðq

yhðkÞ

ygðkÞ þ

2 þ rkkÞ

ds p xðkÞ

xgðkÞ þ

xhðkÞ

(

¼ GII1 qG12 þ qG22 þ 2rkk0

xð0kÞF ax cos u þ ay sin u þ xð1kxÞIx sin u

þ xð1kyÞIy cos u þ

)

xð0kÞ

ðIx

IyÞsin 2u ðK þ sÞ hð0kÞIx sin u gð0kÞIy cos u

(

þ GII1qG22F

ayxð1kyÞ axxð1kxÞ xð0kÞ ax cos u þ ay sin u

)

þ F7ðk 1ÞjG¼GII ;

ð3:97Þ

þ ðK þ sÞ aygð0kÞ þ axhð0kÞ

dx0

G 2

2G 1

ðk 1Þ

G 1

2 0

qð

þ rkk

1ÞxðkÞ

1 þ

rkk

ax cos u þ ay sin u xð1kk 1Þ gð0k 1Þ cos u þ hð0k 1Þ sin u

þ F1ðk 2ÞjG¼GII ;

ð3:98Þ

3.1 Theory of Thin-Walled Beams Based on 3D Equations

þqG21 þ 2qG22 þ 2r0kk GII1x1ðkk 1Þ sin u þ F4ðk 2ÞjG¼GII ;

n	o
GII2qðG22 G12Þ	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 ;

G 2

qð

1x sin

1y cos

wðkÞ xðkÞ

u þ xðkÞ

sin 2u þ

cos u ðK þ sÞ

xðkÞ

hðnkÞ

sin u þ gðkÞ

2G 1

sin

cos

þ rkk ds wðk 1Þ xðk 1Þ

u þ xðk 1Þ

x0ðk 1Þ 2 sin 2u þ h0ðk 1Þ sin u þ g0ðk 1Þ cos u ðK þ sÞ

þ F6ðk 2ÞjG¼GII :

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

	xð1kxÞ ¼ hð0kÞðK þ sÞ þ B1ðk 1Þ;					ð3:102Þ
	xð1kyÞ ¼ gð0kÞðK þ sÞ þ B2ðk 1Þ;					ð3:103Þ
dhð0kÞ	0	1k		dxð1kkÞ
	þ ðK þ sÞgðkÞ	¼ ðK þ sÞaxxðkÞ	ay		þ B3ðkÞ;	ð3:104Þ
ds	þ ðK þ sÞgðkÞ	¼ ðK þ sÞaxxðkÞ	ay	ds	þ B3ðkÞ;	ð3:104Þ
dg0	0	1k		dx1k
ðkÞ	0	1k		ðkÞ
ds	ðK þ sÞhðkÞ	¼ ðK þ sÞayxðkÞ	þ ax	ds	þ B4ðkÞ;	ð3:105Þ

40 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams

C	dx1k
C	ðkÞ	¼ B5ðkÞ þ F axB4ðkÞ ayB3ðkÞ
Ip	ds	¼ B5ðkÞ þ F axB4ðkÞ ayB3ðkÞ		;	ð3:106Þ
where IpC ¼ Ix þ Iy; and functions			BjðkÞðsÞ ði ¼ 1; 2; 3; 4; 5Þ are		presented in
Appendix 3.
The general solution of (3.104)–(3.106) has the form

hð0kÞ		Zs	B3ðkÞ sin v
hð0kÞ	¼ gð0kÞ sin v þ hð0kÞ cos v ayxð1kkÞ þ sin v		B3ðkÞ sin v
		s0
		Zs
	þ B4ðkÞ cos v ds þ cos v	B3ðkÞ cos v B4ðkÞ sin v ds;
		s0
gð0kÞ		Zs
gð0kÞ	¼ gð0kÞ cos v hð0kÞ sin v þ axxð1kkÞ þ cos v		B3ðkÞ sin v
		s0
		Zs
	þ B4ðkÞ cos v ds sin v	B3ðkÞ cos v B4ðkÞ sin v ds;
		s0

x1ðkkÞ ¼ ðIpCÞ 1 B5ðkÞ þ F axB4ðkÞ ayB3ðkÞ ds þ kð0kÞ; s0

where h0ðkÞ; g0ðkÞ; and kð0kÞ are arbitrary constants, and

v ¼ ðK þ sÞds:

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

At k = 0, for example, the values x0ð0Þ and wð0Þ vanish to zero, while the values

; g0

;

x1k

;

x1x

; and x1y take the form

ð0Þ

xð10kÞ ¼ const;

ð3:110Þ

hð00Þ

¼ gð00Þ sin v þ hð00Þ cos v ayxð10kÞ;

ð3:111Þ

gð00Þ

¼ gð00Þ cos v hð00Þ sin v þ axxð10kÞ;

ð3:112Þ

xð10xÞ ¼ hð00ÞðK þ sÞ;

ð3:113Þ

xð10yÞ ¼ gð00ÞðK þ sÞ;

ð3:114Þ

where g0

and h0

are arbitrary constants.

ð0Þ

Let

introduce

into

consideration

two

mutually

orthogonal

vectors:

and construct the vector d

and a a

nð0Þ

yxð0Þ;

xxð0Þ

o x; y

;

oð0Þ ¼

ð0Þ

hð00Þ þ ayxð10kÞ; gð00Þ axxð10kÞ

; where the vector Vð0Þ hð00Þ; gð00Þ

is the vector of the

transverse wave polarization. The vector dð0Þ has the constant modulus

gð00Þ

¼ const;

dð0Þ ¼

þ hð00Þ

and its angle of inclination to the x-axis is deﬁned by the formula

tgað0Þ

gð00Þ axxð10kÞ

gð00Þ cos v hð00Þ sin v

¼ tg

Uð0Þ v ;

sin

cos

hð0Þ

þ ayxð0Þ

ð0Þ

that is

að0Þ ¼ Uð0Þ v;

and

að0Þ ¼ v ¼ ðK þ sÞs ¼ ðK þ sÞGII ;

where tgUð0Þ ¼ gð00Þ hð00Þ

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

dx0

d2x0

2G 1

ðkÞ

ðk 1Þ

;

3:115

dz2

dx1x

d2x1x

2G 1

ðkÞ

ðk 1Þ

;

3:116

dz2

dx1y

d2x1y

2G 1

ðkÞ

ðk 1Þ

;

3:117

dz2

d2w

2G 1

ðkÞ

ðk 1Þ

;

3:118

dz2

GI 2q G12 G22

hð0kÞ þ ayxð1kkÞ ¼ 2GI qG22 þ rkk0

hð0k 1Þ þ ayxð1kk 1Þ

0 d2

2dxð1ky 2Þ

þ GI

qG2xðk 1Þ þ

qG2 þ rkk

hðk 2Þ þ ayxðk 2Þ qG2

;

dz2

ð3:119Þ

GI 2q G12 G22

gð0kÞ axxð1kkÞ ¼ 2GI 1

qG22 þ rkk0

gð0k 1Þ axxð1kk 1Þ

2 1x

0 d2

2dxð1kx 2Þ

þ GI

qG2xðk 1Þ

þ qG2 þ rkk

gðk 2Þ axxðk 2Þ qG2

;

dz2

ð3:120Þ

G22

GI 2q G12

IPAxð1kkÞ þ ayFhð0kÞ axFgð0kÞ

2G 1

IA 1k

Fa 0

þ rkk dz

P xðk 1Þ

xgðk 1Þ þ

yhðk 1Þ

þGI 1qG22F ayx1ðky 1Þ axx1ðkx 1Þ

þ qG22 þ rkk0

IPAxð1kk 2Þ Faxgð0k 2Þ þ Fayhð0k 2Þ

dz2

qG22F

ayxð1ky 2Þ axxð1kx 2Þ

ð3:121Þ

For the quasi-transverse wave, from Eqs. 3.95–3.101 we have

2GII1

qG22

þ rkk0

hð0kÞ þ ayxð1kkÞ ¼ GII1qG22xð1kyÞ

0 d2hð0k 1Þ

(

dxð1kk 1Þ

)

2 d

ð3:122Þ

þ qG2 þ rkk

qG2

xðk 1Þ

;

dz2

44 3 Transient Dynamics of Pre-Stressed Spatially Curved Thin-Walled Beams

2GII1

qG22 þ rkk0

gð0kÞ axxð1kkÞ ¼ GII1qG22xð1kxÞ

0 d2gð0k 1Þ

(

dxð1kk 1Þ

)

2 d

ð3:123Þ

qG2

þ rkk

qG2 dz

xðk 1Þ

;

dz2

2G 1

IA 1k

yhðkÞ

2 þ rkk dz

p xðkÞ

xgðkÞ

¼ GII1qG22F ayxð1kyÞ axxð1kxÞ

d2 n

þ qG22 þ rkk0

IpAxð1kk 1Þ Faxgð0k 1Þ þ Fayhð0k 1Þ

dz2

þ qG22F

axxð1kx 1Þ ayxð1ky 1Þ ;

G 2

2G 1

dx0

ðk 1Þ

xðkÞ

þ rkk

d2x0

ðk 2Þ

qG1

þ rkk

;

dz2

G 2

2G 1

dx1x

ðk 1Þ

xðkÞ

þ rkk

d2x1x

ðk 2Þ

qG1

þ rkk

;

dz2

G 2

2G 1

dx1y

ðk 1Þ

xðkÞ

þ rkk

d2x1y

ðk 2Þ

qG1

þ rkk

dz2

;

0 dwðk 1Þ

GII

wðkÞ ¼ 2GII

qG1

þ rkk

d2w

ðk 2Þ

qG1

þ rkk

dz2

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

GI G22

hðkÞ þ ayxðkÞ ¼

cðk 1Þ;

gðkÞ axxðkÞ

cðk 1Þ

;

G12 G22

A 1k

GIG22

IP xðkÞ þ ayFhðkÞ

axFgðkÞ

aycðk 1Þ

axcðk 1Þ

ð3:130Þ

G12 G22

Solving the set of Eqs. 3.130 yields

G G2

I 2

xðkÞ ¼ 0;

hðkÞ

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

	xð0kÞ ¼ xð1kxÞ ¼ xð1kyÞ ¼ wðkÞ ¼ 0;							ð3:132Þ
	hð0kÞ þ ayxð1kkÞ ¼ sð1kÞ ¼ const;
	gð0kÞ axxð1kkÞ ¼ sð2kÞ	¼ const;						ð3:133Þ
IPAxð1kkÞ þ ayFhð0kÞ axFgð0kÞ ¼ sð3kÞ ¼ const:
Solving (3.133) we have
			1	¼ cð1kkÞ			¼ const;
xð1kkÞ ¼ sð3kÞ þ sð2kÞax sð1kÞay		IpC	1	¼ cð1kkÞ			¼ const;
			1	¼ cðhkÞ			¼ const;
hð0kÞ ¼ sð1kÞ ay	sð3kÞ þ sð2kÞax sð1kÞay	IpC	1	¼ cðhkÞ			¼ const;	ð3:134Þ
					1	¼ cðgkÞ ¼ const:
gð0kÞ ¼ sð2kÞ þ ax sð3kÞ þ sð2kÞax sð1kÞay			IpC		1	¼ cðgkÞ ¼ const:

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

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