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DOI: http://dx.doi.org/10.20534/AJT-16-9.10-28-30
Kuliev Sabir,
Azerbaijan University of Architecture and Construction
Kazymov Musa, Azerbaijan State University of Oil and Industry E-mail: rahimova_mahluqa@mail.ru
Study of cracks formation in curved bars and rocks
Abstract: As it is know, different load-gripping devices, so called hooks, are used in the performance of loading and unloading works of mobile installations, cranes.
Years-long use of load hooks showed that the wrong selection of the material, production technology as well as violations of the exploitation regime lead to the appearance ofhollows, cracks, slag inclusions, which can cause the breakdown of hooks and emergency situations at production site. Hence, during the design of hooks, it is required to calculate its exact durability and crack-resistance.
Keywords: load-gripping devices, high curve bar, stress-intensity factor, cracks, buckling loads, normal fracture.
The work [1] presents the study of the stress condition A principle moment consists in the condition
of load-gripping devices, hooks, and defines total stress in of the limit equilibrium. The simplest variant of this
critical points B and C of cross-section (Figure 1). condition, based on physical ideas of Griffith, was for-
Having defined the maximal total stress in a criti- mulated by Irwin in case of a normal fracture [2; 3].
cal point of the cross-section of the bar during the cal- Irwin showed that the appearance of cracks in a brittle
culation of hook durability, it is required to determine or quasi-brittle body takes place when stress intensity
critical values of external load (i. e. the weight of a lifted at the top of the crack (k) reaches some (constant, for
load) at which cracks formation and local or complete this material) value. destruction of the body begin.
Figure1. Curved bar (hook)
Study of cracks formation in curved bars and rocks
k = K^ (!)
Vn
The constant kc characterizes resistance of the material to destruction, which is determined experimentally. At uniaxial tension of the body, the value of the critical stress Pnp = &np is defined by the correlation
-IE (2)
Pkp is a mean value of technical durability of this material at uniaxial tension [<Jh ].
minPkh - 0,97P = [(7, ] (3)
According to calculations obtained on the basis of accepted Griffith-Irwin hypothesis, the data about the spread of the crack accords well with experimental data [4].
Since our equation simultaneously includes bending moment (Mbend) and tension force, P we took into account the following criterion of brittle fracture specified in the works of Knowles, Vang, Engie and Williams [4; 5]:
K
1 + v 3 + v
Klbend KC
(4)
Here, Rlcakulated is a coefficient of stress intensity at tension, which is defined on the basis of the following asymptotic formula (in polar coordinates r and 0):
r e 36
a = 5cos— + cos— r 2 2
K (calculated) e 36
ae =—1 .— 3cos— + cos—
6 4V27 2 2
e
36
< „ = sin—+ sin-2 2
ar
K
(calculated )
4>/2r
6 36 5 cos— + cos— 2 2
3 6+ 36 3co s— + co s— 22
6 36 sin—+ sin— 22
(5)
In numerical calculations, the value r is accepted as r = 0,011/, where 6 is the angle between the axis ox and their polar radius vector r.
Klbmd is a coefficient of stress intensity at pure bending of the moment Mbend, which is defined according to the following asymptotic formula:
0 N 36
a.
|(3 + 5v )■ cos—-(7 + v )cos—
K" g (5 + 3) 6
ae =—-\ i— -—(5 + 3v)-cos—
6 2(3 + v)Jlr hy J 2
a66=-( - v)sin66 + (7 + v)siny
a.
Kb
2(3+v)*j2r
r- N 6 . 36
(3 + 5v)cos--(7+v)cos—
22
6 36
(5 + 3v)cos— + (7+v)cos— 22
6 36
-(1 — v)sin— + (7 + v)sin— 22
(6)
where v is a Poisson ration for many makes of steel 0,2 " v" 0,3 and coefficient > is defined by the following formula:
6M„, [i -cos2a]-\fj
K(bend) _
2h2
M,.
is a moment defined by the formula:
( d , MuSr = P ■ x = pl 2 + h1
(7)
(8)
where, P is force applied to the hook (weight of the lifted load); d is the diameter of inner circumference of the hook (curved bar); h is the distance from inner circumference to the center of gravity of the cross-section of the bar (hook); l is the length of supposed crack (maximal length of the crack will be equal to the length of the base of trapezoid, i. e. lmax = a).
In numerical calculations, the value of technical durability (durability limit of the bar material) [cb] for different makes of steel is selected from respective reference books. Below is the values [cb] for some makes of steel: Material:
[cb] = CT -10 34 kG/mm 2; [cj = CT - 25 46 kG/mm 2; [cj = CT - 40 58 kG/mm 2; C4 (12-28) - grey cast iron 12 kG/mm 2 - ofstree 50 kG/mm 2 - in compression [cb]= CT - 20x - 80 kG/mm 2 Final stress in the critical point of the section of the bar (hook) is defined according to the following formula
GB = Gg calculated + G Bbend (9)
P
where cgcalailated = — is normal stress at the tension with the force P.
The stress of the bending is related to M, , and
O bend
defined by the formula
^ = ^T' k (10)
F
where Mbend is bending moment defined by the formula
Mbnd = P [ 2 + h ) (11)
K is the ratio depending on typical sizes of section, which is defined as:
K = -I. z o + z; z„ r + z'
At the section of tasks by the method of the theory of elasticity, the stress at bending abend is defined by the formula
at which the body starts destruction, according to the following formulas.
" 10
G bend = MS
a 2b2
i b ■ ln—-
b2 ■ ln- + a 2ln- + b2 - a2
5= 4
K
(12)
PyTl
= 4a
where a is a numerical value of stress 5g Critical values of load P
Pd =
■ cd 10
■h ]
(13)
(14)
= (b2 - a2 )2 - 4ab2 ln bj
Knowing the final stress oe, we define the coefficient of stress intensity (K1) and the value of critical load Pcd,
0,97 ■ 4a
Conclusions. Equations obtained by us for final stress allow defining the coefficient of stress intensity and the value of critical force at which the body start destruction.
References:
1. Кулиев С. А., Казимов М. И. Напряженное состояние грузоподъемных крюков. Восточно-Европейский журнал передовых технологий. - 1/1 (73) - 2015. - Украина, - С. 67-72.
2. The theory of rupture-Proc. First Intern. Congr. Appl. Mech. Delft - 1924. - 55-63.
3. Analysis of stresses and strains near the and of a crack traversing a plate. - t. Appl. Mech. - 1957, - 24, - № 3, - P. 361-364.
4. Эрдоган И. С. О развитии трещин в палстинках под действием продольной и поперечной нагрузок технической механики (труды Американское общество инженеров механиков) - 1963. - Т. 85. - д.4, - С. 49-51.
5. Math T.and Phys, - 1960, - V. 39, - No 3. - P. 223-236.
6. Appl T. Mech., - 1961, - V. 28, - No 3. - P. 372-378.
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