Issuu on Google+

ÐÁÑÁÑÔÇÌÁ Â

ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ


ÐÁÑÁÑÔÇÌÁ Â

464

ÅÊÙÓ 2000 - Ó×ÏËÉÁ


ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

Â.1

ÁÓÕÌÌÅÔÑÏ ÓÕÓÔÇÌÁ

 (0.2 + 0.1 ⋅ η)⋅ ν y , η ≤ 3 ................. Óôñåðôïêáìðôéêüò ëõãéóìüò êáôÜ É-É Η ⋅ N / KI ≤  0.6 ⋅ ν y , η > 3   (0.2 + 0.1 ⋅ η)⋅ ν x , η ≤ 3 ...... Óôñåðôïêáìðôéêüò ëõãéóìüò êáôÜ ÉÉ-ÉÉ Η ⋅ N / K II ≤  0.6 ⋅ ν x , η > 3  üðïõ: I, II åßíáé ïé êýñéïé Üîïíåò åëáóôéêüôçôáò ôïé÷ùìÜôùí

Â.2

ÓÕÌÌÅÔÑÉÊÏ ÙÓ ÐÑÏÓ ÄÕÏ ÁÎÏÍÅÓ ÓÕÓÔÇÌÁ

I – I:

Ôßèåôáé

ν y =1 ...............................................Ìåôáöïñéêüò ëõãéóìüò

II – II:

Ôßèåôáé

ν x =1 ...............................................Ìåôáöïñéêüò ëõãéóìüò

III – III:

Â.3

 0,2 + 0,1 ⋅ η ......................Óôñåðôéêüò ëõãéóìüò rB H N / K III ≤  0,6 ⋅ η 

ÕÐÏËÏÃÉÓÌÏÓ ÔÙÍ ÓÕÍÔÅËÅÓÔÙÍ v x ÊÁÉ v y ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

Óôç ãåíéêÞ ðåñßðôùóç ôõ÷üíôïò óõóôÞìáôïò ôïé÷ùìÜôùí ôá äéáäï÷éêÜ âÞìáôá õðïëïãéóìïý åßíáé ôá áêüëïõèá:

Â.3.1 Ìçôñþï äõóêáìøßáò ÊÜèå êáôáêüñõöï óôïé÷åßï (i) ÷áñáêôçñßæåôáé áðü ôï êÝíôñï âÜñïõò G i , áðü ôï åëáóôéêü êÝíôñï K i êáé áðü ôïõò êýñéïõò Üîïíåò áäñÜíåéáò ( ξ i , ηi ) ôçò äéáôïìÞò ôïõ. Ïé ñïðÝò áäñÜíåéáò ùò ðñïò ôïõò Üîïíåò áõôïýò êáé ç óôñåâëùôéêÞ áäñÜíåéá ùò ðñïò ôï óçìåßï K i ãñÜöïíôáé áíôßóôïé÷á I ξi , I ηi êáé I κi . Óôï ôõ÷üí ãåíéêü óýóôçìá áíáöïñÜò O xyz ôï ìçôñþï äõóêáìøßáò ôïõ óõóôÞìáôïò ãñÜöåôáé (Ó÷Þìá. Â.1):

K xx  K =  K yx  K zx 

K xy K yy K zy

K xz   K yz  K zz 

ÅËËÇÍÉÊÏÓ ÊÁÍÏÍÉÓÌÏÓ ÙÐËÉÓÌÅÍÏÕ ÓÊÕÑÏÄÅÌÁÔÏÓ 2000

465

Â


ÐÁÑÁÑÔÇÌÁ Â

466

ÅÊÙÓ 2000 - Ó×ÏËÉÁ


ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

üðïõ:

(

)

(

)

Â

K xx = E ⋅ ∑ I ηi ⋅ συν 2 α i + I ξi ⋅ ηµ 2 α i = E ⋅ ∑ I yi i

i

K yy = E ⋅ ∑ I ηi ⋅ ηµ 2 α i + I ξi ⋅ συν 2 α i = E ⋅ ∑ I xi i

(

i

K zz = E ⋅ ∑ I κi + y i2 ⋅ I yi + x i2 ⋅ I xi − 2 ⋅ x i ⋅ y i ⋅ I xyi i

)

K xy = K yx = E ⋅ ∑ (I ηi − I ξi )⋅ ηµα i ⋅ συνα i = E ⋅ ∑ I xyi i

i

(

)

(

)

K xz = K zx = E ⋅ ∑ − y i ⋅ I yi + x i ⋅ I xyi i

K yz = K zy = E ⋅ ∑ − y i ⋅ I xyi + x i ⋅ I xi i

E

= ìÝôñï åëáóôéêüôçôáò óêõñïäÝìáôïò.

Â.3.2 Åëáóôéêü êÝíôñï - Êýñéïé Üîïíåò Ïé óõíôåôáãìÝíåò ôïõ åëáóôéêïý êÝíôñïõ Ê ôïõ óõóôÞìáôïò äßäïíôáé áðü ôéò ó÷Ýóåéò:

xκ =

K xx ⋅ K zy − K xy ⋅ K zx K xx ⋅ K yy − K 2xy

, yκ

=

K yx ⋅ K zy − K yy ⋅ K zx K xx ⋅ K yy − K 2xy

êáé ï ðñïóáíáôïëéóìüò ôùí êýñéùí áîüíùí åëáóôéêüôçôáò (É, ÉÉ) êáèïñßæåôáé áðü ôçí ãùíßá ωκ ôçò ó÷Ýóçò (Ó÷Þìá Â.1):

εϕ2ω κ =

2 ⋅ K xy K xx − K yy

Ç ïîåßá ãùíßá ωκ ðïõ ðñïêýðôåé áðü ôçí ðáñáðÜíù ó÷Ýóç (èåôéêÞ Þ áñíçôéêÞ) êáèïñßæåé ôçí èÝóç ôïõ Üîïíá É áí K xx > K yy Þ ôïõ Üîïíá ÉÉ áí K xx < K yy . Óôï óýóôçìá áíáöïñÜò ôùí êýñéùí áîüíùí åëáóôéêüôçôáò (É,ÉÉ,III) èá Ý÷ïõìå ôéò ìåôáöïñéêÝò äõóêáìøßåò ôïõ óõóôÞìáôïò:

ÅËËÇÍÉÊÏÓ ÊÁÍÏÍÉÓÌÏÓ ÙÐËÉÓÌÅÍÏÕ ÓÊÕÑÏÄÅÌÁÔÏÓ 2000

467


ÐÁÑÁÑÔÇÌÁ Â

468

ÅÊÙÓ 2000 - Ó×ÏËÉÁ


ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

K I = E ⋅ J II =

K II = E ⋅ J I =

K xx + K yy 2

K xx + K yy 2

2

 K xx − K yy   + K 2xy +   2  

 K xx − K yy −  2 

Â

2

  + K 2xy  

êáé ôçí óôñåâëùôéêÞ äõóêáìøßá:

K III = E ⋅ J κ = K zz − y 2κ ⋅ K xx − x 2κ ⋅ K yy + 2 ⋅ x κ ⋅ y κ ⋅ K xy

ηk

y

ξi ξk

II

ηi I

yi ηi

ξi

ωk

yk yi

αi

K

xi Ki ηn ξn

0

xk

x

xi

Ó÷Þìá Â.1: Óýóôçìá ôïé÷ùìÜôùí

Â.3.3 ÐáñÜëëçëç äéÜôáîç óôïé÷åßùí Óôçí åéäéêÞ ðåñßðôùóç êáôáêüñõöùí óôïé÷åßùí ìå ðáñÜëëçëç äéÜôáîç ôùí êýñéùí áîüíùí áäñÜíåéáò ( ξ i , ηi ) èá Ý÷ïõìå K xy = K yx = 0 êáé:

K yz = K zy = + E ⋅ ∑ (x i ⋅ I ξi )

K xx = E ⋅ ∑ I ηi , i

K yy = E ⋅ ∑ I ξi

(

i

K zz = E ⋅ ∑ I κi + y i2 ⋅ I ηi + x i2 ⋅ I ξi

,

i

K xz = K zx = −E ⋅ ∑ (y i ⋅ I ηi )

i

)

i

ÅËËÇÍÉÊÏÓ ÊÁÍÏÍÉÓÌÏÓ ÙÐËÉÓÌÅÍÏÕ ÓÊÕÑÏÄÅÌÁÔÏÓ 2000

469


ÐÁÑÁÑÔÇÌÁ Â

470

ÅÊÙÓ 2000 - Ó×ÏËÉÁ


ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

Ïé êýñéïé Üîïíåò (É, ÉÉ) èá Ý÷ïõí ôïí ßäéï ðñïóáíáôïëéóìü ìå ôïõò Üîïíåò ( ξ i , ηi ) êáé ïé óõíôåôáãìÝíåò ôïõ åëáóôéêïý êÝíôñïõ Ê èá åßíáé:

K yz

xκ =

Â

K , y κ = − zx K xx

K yy

Ïé êýñéåò äõóêáìøßåò ôïõ óõóôÞìáôïò äßäïíôáé áðü ôéò ó÷Ýóåéò:

K I = E ⋅ J II = K xx K II = E ⋅ J I = K yy K III = E ⋅ J κ = K zz − y 2κ ⋅ K xx − x 2κ ⋅ K yy Â.3.4 ÔéìÝò ôùí óõíôåëåóôþí v x êáé v y Èåùñïýìå ôï óýóôçìá áíáöïñÜò B xyz ìå áñ÷Þ Â ôï êÝíôñï ôùí áîïíéêþí äõíÜìåùí N i üëùí ôùí êáôáêüñõöùí óôïé÷åßùí óôç âÜóç ôïõò êáé Üîïíåò (x, y) ðáñÜëëçëïõò ðñïò ôïõò êýñéïõò Üîïíåò åëáóôéêüôçôáò (É, ÉÉ) (Ó÷Þìá Â.2). Áí åßíáé e x êáé e y ïé åêêåíôñüôçôåò ôïõ åëáóôéêïý êÝíôñïõ Ê ùò ðñïò ôï ðáñáðÜíù óýóôçìá áíáöïñÜò, ïé áäéÜóôáôïé óõíôåëåóôÝò v x êáé v y õðïëïãßæïíôáé áðü ôéò ó÷Ýóåéò:

v 2x

v 2y

2

 1 − λ2x 1 + λ2x = −  2  2

  + ε 2x  

 1 − λ2y −   2 

  + ε2 y  

=

1 + λ2y 2

2

üðïõ:

(

)

(

)

λ2x =

1 ⋅ K III K II + e 2x 2 rb

λ2y =

1 ⋅ K III K I + e 2y 2 rb

ε x = e x rb

ÅËËÇÍÉÊÏÓ ÊÁÍÏÍÉÓÌÏÓ ÙÐËÉÓÌÅÍÏÕ ÓÊÕÑÏÄÅÌÁÔÏÓ 2000

471


ÐÁÑÁÑÔÇÌÁ Â

472

ÅÊÙÓ 2000 - Ó×ÏËÉÁ


ÓÕÍÈÇÊÇ ÁÌÅÔÁÈÅÔÏÔÇÔÁÓ ÓÕÓÔÇÌÁÔÏÓ ÔÏÉ×ÙÌÁÔÙÍ

ε y = e y rb rb =

1 ⋅ ∑ ri2 ⋅ N i N i

áêôßíá åêôñïðÞò êáé riïé áðïóôÜóåéò ôùí áîïíéêþí äõíÜìåùí

N i áðü ôï êÝíôñï Â, ( N = ∑ N i ). i

Ãéá

λ x ≤ 1 Þ λ y ≤ 1 ç èåìåëéþäçò éäéïìïñöÞ ëõãéóìïý ôïõ óõóôÞìáôïò èá Ý÷åé

äåóðüæïíôá óôñåðôéêü ÷áñáêôÞñá, åíþ ãéá λ x > 1 êáé λ y > 1 èá Ý÷åé äåóðüæïíôá ìåôáöïñéêü ÷áñáêôÞñá (óôñåðôïêáìðôéêüò ëõãéóìüò). ÔÝëïò, ãéá e = 0 èá Ý÷ïõìå

v 2 = λ2 áí λ2 < 1 Þ v 2 = 1 áí λ2 > 1 .

z

III y ey

II I

Ni K ri

B

cx

x

Ó÷Þìá Â.2: Êåíôñïâáñéêü óýóôçìá áíáöïñÜò

ÅËËÇÍÉÊÏÓ ÊÁÍÏÍÉÓÌÏÓ ÙÐËÉÓÌÅÍÏÕ ÓÊÕÑÏÄÅÌÁÔÏÓ 2000

473

Â



26EKOS-463,474-supplementB