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Совершенствование методики прочностного расчета двутавровой балки и ее изложения в курсах технических дисциплин..pdf

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378.14.015.62:378.147.227:539.3
,
. .
. .
,
. .
Yu.S. Holodnyak, A.V. Perig, I.A. Matveev
, .
Donbass State Engineering Academy, Kramatorsk, Ukraine
,
I-BAR STRENGTH COMPUTATION METHODOLOGY
PERFECTION FOR TEACHING
OF ENGINEERING DISCIPLINES
,
,
,
.
.
,
-
,
,
-
.
:
,
,
,
,
.
The equivalent stress distribution along the height of an I-bar cross-section has been studied. The
zones of maximum equivalent stress in conjunction with the I-bar geometric parameters and power load
at the section have been found. The conditions for maximum equivalent stresses have been specified to
not exceed the allowable values. An improved method for the comprehensive assessment of I-bar
strength against the normal, tangential and equivalent stresses has been proposed. This computational
approach may be recommended to lecturers and students of technical universities as well as engineering specialists in the field of strength computations.
Keywords: I-bar, equivalent stresses, strength, design procedure, quotient space.
,
.
,
77
,
,
,
.
,
-
,
,
.
.
,
.
.],
[1–4
(2)
kmax
,
-
,
(1),
,
(3)
[j]
:
σ max ≤ [σ] ,
(1)
τmax ≤ [τ] ,
(2)
σmax ≤ [σ] .
(3)
,
[k]
jmax
,
– σmax ,
,
,
-
,
,
,
,
.
.
.
[5, 6],
,
M
,
Q,
(1)
78
(2)
,
-
,
jmax
kmax.
[j], . .
, . .
-
(3)
-
.
.
–
,
-
,
,
.
.1
)
j
[7].
(
k
. 1.
,
-
-
:
σ=
–
M
y,
Jx
(4)
:
τ=
QS x( )
dJ x



0≤ y≤
τ=
QS x( )
bJ x



h
h
− t < y ≤ ,
2
2
y
y
h 
− t ,
2 
(5)
79
Q–
M
x
; Jx –
,
); Sx(y) –
y.
(
(6)
.
k
. 1),
= σ2 + ατ2 ,
g=3–
.
-
:
σ
g=4–
,
x(
,
,
.
(4)
(6)
σ
M
= 
 Jx
(5),
2
 QS ( y ) 

y  + α x  .
 dJ 
x 


2
(7)
Sx(y)
-
Sx:
1
S x( y ) = S x − ⋅ dy 2 .
2
(8)
(8)
j
(7),
-
:
2
2
M 
 Q 
1
2 
y +α
= 
 S x − ⋅ dy   .
2

 Jx 
 dJ x 
σ
(9)
GNU CAS
Maxima
j
y
(9)
.
:
(
)
2
2
2 ⋅ M 2 y 2 ⋅ αQ S x − 0,5 ⋅ dy y
−
dσ
I x2
dI x2
1
= ⋅
= 0.
dy
2
2
2 2
2 2
α
−
⋅
0,5
Q
S
dy
M y
x
+
2
Ix
d 2 I x2
(
(10),
:
80
(10)
)
y,
(9)
-
yextr = 0, ±
(
)
1
2 ⋅ αd αQ 2 S x − M 2 d .
αdQ
(11)
(11),
2
2
(10)
0,5
(M/Q) < ( · Sx/d) ,
–
(
–
2
x).
· Q · Sx ≤ M 2 · d,
(11)
,
: yextr = 0.
· Q · Sx > M · d,
.
(11)
0,5
(M/Q) ≥ ( · Sx/d) ,
;
,
(9)
M/Q,
.
-
:
M 
αS x
.
  =
Q
d
 
,
=3
30 – 351,7
,
10
60 – 610,5
.
,
123,8
(
,
-
),
y:
d 2σ
dy 2
(12)
M2· d =
(M
).
(12)
· Q2 · Sx,
(12)
(M/Q) < (M/Q) ,
(M/Q) > (M/Q) ,
.
(12)
-
2
d − αQ 2 S x
QJ x S x α
M2 · d >
,
–
M2· d <
( 0) =
,
· Q2 · Sx,
–
· Q2 · Sx,
.
(M/Q) = (M/Q) ,
(12)
,
.
,
,
,
M/Q
–
y,
-
:
d 4σ
dσ4
( 0) =
–
3 ⋅ αQd
> 0,
Sx J x
[8].
81
,
(M/Q) ≥ (M/Q)
(M/Q) < (M/Q) –
,
yextr = ±
(9)
-
.
(
)
1
2 ⋅ αd αQ 2 S x − M 2 d .
αdQ
(13)
-
y
(13),
d 2σ
dy 2
(14)
( yextr ) =
(
)
2 ⋅ αQ 2 S x − M 2 d Q αd
dJ x M
2
( 2 ⋅ αQ S
2
x
− M 2d
· Q2 · Sx > M 2 · d,
,
,
.
)
(14)
.
(M/Q) < (M/Q) ,
(9)
.
,
(M/Q) < (M/Q)
(9)
,
(13).
,
,
,
:
yextr = ±
(M/Q) = 0 ( . .
x:
M=0
2 
M2 
α
−
S
d .
 x
αd 
Q2 
Q ≠ 0)
max
yextr
=
2 ⋅ Sx
.
d
10
287,2
,
x
101,1
,
30 –
60 – 498,5
,
( h/2).
M/Q
(M/Q)
lim yextr
(M/Q)
(y = 0):
2

M 
αS 
2 
2 
 αS x −   d  =
=
αS x − x d  = 0.


d 
αd 
αd 
Q


,
(M/Q) < (M/Q) ,
82
,
M/Q
(13)
-
,
,
(y = 0),
K (yk = (h/2) – t),
(
.
. 1).
(
-
)
M/Q,
.
(M/Q) < (M/Q)
K.
(M/Q) > (M/Q) –
(M/Q)
σ
,
(0) = σK
(15)
.
(9):
(0) =
σ
α
QS x
;
dJ x
(16)
2
σK
(16)
 M  y2
1
α 

= Q   K2 + 2 2  S x − ⋅ dyK2  .
2

 Q  Jx d Jx 
(17)
2
(17)
(15)
-
M/Q,
M M 
α
=  =
Q Q
dyK
=3
2
1


S x2 −  S x − ⋅ dyK2  .
2


(M/Q)
30 – 330,2
,
(M/Q) < (M/Q) .
,
:
10 – 118,2
60 – 559,5
, . .
(M/Q) ≥ (M/Q) , . .
,
(9)
,
,
K.
M/Q
,
,
:
M/Q
.
,
–
,
K;
σ
max
(M/Q)
M/Q
.
,
.2
30
M = 30,703·106 ·
(
(
3), 35,170·106 ·
1), 33,020·106 ·
(
(
4), 37,000·106 ·
-
(9),
:
= 3; Q = 105 ;
2), 34,115·106 ·
(
5).
83
σ max
,
[j], . .
(3).
.
σ max = α
QS x
= ατmax .
dJ x
(18)
. 2.
M /Q
ヽ 30: (M/Q)1 = 307,03
; (M/Q)2 = 330,20
;
(M/Q)3 = 341,15
; (M/Q)4 = 351,70
; (M/Q)5 =
= 370,00
[2],
(18)
τmax =
[k],
,
σ max
≤ [ τ] .
α
[j]:
[ τ] =
(19)
84
(20)
(19)
,
[ σ] .
(20)
α
σ max ≤ [σ] , . .
(3).
j
K.
:
2
σK
 MyK 
 Q
= 
 +α
 Jx 
 dJ x
2
1

2 
 S x − ⋅ dyK   =
2


2
2
 y 
 α 

1
2 
M  K  + Q2 
 S x − ⋅ dyK   .
 J [σ] 
2
 
 x

 dJ x [ σ ] 
= [ σ]
2
,
,
(
(21)
M
:
)
Q
(22)
dJ x [ σ]
(23)
Sx α
.
,
-
[ M ] = Wx [σ] ,
[Q ] =
,
(21)
M
(3)
Q,
.
(21)
,
(3):
σ max = [σ],
2
[ σ]
2
 y 
 α 
1

M  K  + Q2 
S x − ⋅ dyK2   = [ σ ] .



2
 
 J x [ σ] 
 dJ x [ σ] 
2
–
:
,
2
(24)
(24),
-
2
 y 
 α 

1
2 
M  K  + Q 2 
 S x − ⋅ dyK   = 1.
2
 
 J x [σ] 
 dJ x [ σ] 
2
(25)
:
yK
1
= ,
J x [ σ] β
(
α S x − 0,5 ⋅ dyK2
dJ x [ σ]
(26)
) = 1.
γ
(27)
85
(26)
(27) (25),
:
M 2 Q2
+ 2 = 1.
β2
γ
,
M
-
Q,
:
(
(3), . .
J x [ σ]
,
yk
β=
γ=
dJ x [ σ ]
α S x − 0,5 ⋅ dyk2
)
.
M
Q
(22)
(23):
M ≤ [M ] ,
(28)
Q ≤ [Q ] .
(29)
(28) (29)
,
M
,
(1)
,
, d = 6,5
,
Q
(3),
,
-
(2).
.
4
ヽ 30 (Jx = 7080·104
, yK = 139,8
3
3
3
, Wx = 472·10
, [j] = 150 / 2),
g = 3, Sx = 268·10
. 3 [7].
,
[Q] = 148,715 · .
3
)
· ;
= 194,910
; [M] = 70,800
(
· ;
. 4,
(
M2 = 63
· ; Q3 = 73,5
Q5 = 73,5
; M5 = 44,1
M7 = 66,4 · .
86
= 75,966
. 3): Q1 = 81,5
; M1 = 0
; M3 = 63
· ; Q4 = 73,5
· ; Q6 = 73,5
; M6 = 0
· ; Q2 = 18,5 ;
; M4 = 4,1
· ;
· ; Q7 = 0
;
. 3.
Q
M
(
(1),
(2)
.
. 4),
(3)
.
.
. 4.
,
(1).
[M],
-
87
[Q],
(330,21
1, 4
,
),
6
(2).
M
Q
,
,
– 2, 3, 5
(3)
(2).
M/Q
7–
,
,
-
,
(3).
. 4,
(ヽ 30)
,
.
.
,
-
,
,
-
.
,
.
,
.
(
)
,
,
,
-
;
.
,
,
.
,
88
-
,
-
,
,
-
,
»
«
.
–
,
,
.
-
,
(
,
)
-
.
.
,
.
[
1.
.];
2.
.
3.
:
4.
5.
:
.
.
/ . .
., 1986. – 775 .
:
. . .
.–
:
. .,
. .
. – .:
, 1989. – 624 .
. .
:
., 1989. – 311 .
. .,
. .
:
.
. – .:
, 2001. – 88 .
. .,
. .,
. .
// :
ヽ 1 (6). – . 74–80.
6.
;
/
,
/
2011. – . 463–469.
7. :
( :
:
. .
, . .
2007. – 220 .
8.
. .
2008. – . 1. – 624 .
;
:
. .
,
.
.
. .
-
:
,
. .
:
:
[
.
[
: .];
.
. –
;
.
:
–
.
,
: : »
) /
,
.–
.:
.–
:;. – 2006. –
. –
«
:
.
;
.] //
.
;
-
,
1.11.2012
89
donetsk.ua).
–
, .
,
.
,
,
(84313,
,
, 72, -mail: texmex@dgma.
–
,
(84313,
,
, .
, .
alexander.perig@dgma.donetsk.ua, olexander.perig@gmail.com).
,
.
,
, 72, -mail:
–
,
(84313,
,
, .
, 72, -mail: matveev.ivan.dgma@gmail.com).
-
Holodnyak Yuri Sergeevich – Candidate of Technical Sciences, Associate
Professor, Donbass State Engineering Academy (84313, Ukraine, Kramatorsk,
Shkadinova st., 72, -mail: texmex@dgma.donetsk.ua).
Perig Aleksandr Viktorovich – Candidate of Technical Sciences, Senior
Lecturer, Donbass State Engineering Academy (84313, Ukraine, Kramatorsk,
Shkadinova st., 72, -mail: alexander.perig@dgma.donetsk.ua, olexander.perig@
gmail.com).
Matveev Ivan Anatolyevich – Student, Donbass State Engineering
Academy (84313, Ukraine, Kramatorsk, Shkadinova st., 72, -mail: matveev.ivan.
dgma@gmail.com).
90
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