Summary-For the first time in the theory of solid cargo fastening there has been investigated a case when the cargo is in motion in relation to the wagon floor with acceleration r a , its speed being at this moment equal to r v . There have been set out the results of analytical investigation of cargo shift in dynamics and accordingly elongation and tension in flexible fastening elements under the action of plane force system. It has been established that the longitudinal force perceived by the flexible fastening elements in value is smaller than the force obtained when inertia in relative motion (at rest) is not taken into account. Hence, the cargo shift lengthwise the wagon in this case will be smaller. This, in its turn, will affect the decrease of elongation value and consequently the decrease of the effort of every flexible element, thus increasing their load-carrying capacity.

Keywords: cargo, thrust bars, flexible fastening elements, cargo shift in dynamics, efforts in flexible fastening elements.


# I.

Formulation of a Problem ormulas derived for determining efforts in flexible fastening elements of the cargo under the action of longitudinal and vertical forces presented in Appendix 8-Technical conditions [1,2], (as has been pointed out in [3 -22]) have been the result of incompletely solved problems when the longitudinal force value perceived by fastening means according to the gravity power of cargo G is understated (i. e. is always within the limits (0,97 ÷ 1,2)G) while during shunting collisions in a hump-yard or emergency braking this force may vary within -(1,2 ÷ 2)G . Moreover, they don't take into account the efforts of preliminary twisting of every fastening wire R0, without which the cargo is not liable to dispatching. Just because due to effort R0 the cargo is pressed against the wagon floor, friction force is increased. In [1,2] there is no mention of the notion «shift of the cargo lengthwise the wagon" and hence, no mention of "elongation of each fastening element" to the value of which the efforts in each fastening elements are according to Hooke's law directly proportional. As a result, the efforts of each fastening element have one and the same value, which disagrees with reality. It should be noted that in [3 -22] a technical problem of cargo fastening under the action of space force system and, as a special case, under the action of plane force system, is solved within the fundamental law of dynamics during relative motion at rest. Unfortunately, there has not been yet considered the case when the cargo is moving lengthwise the wagon floor with acceleration r a its speed at the moment being equal to r v [23,24].

On this basis it can be noted that determining of cargo shift lengthwise the wagon floor and correspondingly elongation and efforts in each fastening element during cargo motion with acceleration lengthwise the wagon floor at a given relative speed is an urgent technical problem for transport research.


# a) Problem Formulation In Dynamics (It is for the first time that the problem is set)

To derive an analytical formula of cargo shift lengthwise the wagon, elongation and efforts in flexible fastening elements in case of the cargo moving in relation to the wagon floor with acceleration r a at speed r v , as in case of motion of deformable thread on an imperfect curved surface [25].


# b) Problem Specification

As in [7], let us consider the case, when cargo with gravity force G , located on the wagon on down grade at angle 0 ? (rad. 0.006 ÷ 0.021 or 0.344 ÷ 1.2 degrees which agrees with grade within 6 ÷ 21°/ °°) in the mode of both brake release and service braking is kept from lengthwise shifting by flexible fastening elements. The contours of the cargo when it is placed on the wagon the effective area makes it possible to use thrust and/or spacer wooden bars (Fig. 1a, b). b -aerodynamic resistance force [24].


# c) Man-Made Assumption

In working out a computable model as in [17] we assume wagon frame to be the major constrain for the cargo (object) and flexible elastic fastening elements and thrust bar to be additional constraints [10,18,23,24].

We assume that effective longitudinal and vertical forces are perceived by flexible elastic fastening  


# F F x <

).

As it is known [7,10,18], external constraint reaction (non-ideal) R is resolved into normal and 
tangent ? F component, i.e. ? + = F N R . Coordinates

# d) Formation of Dynamic Model

We apply theoretically to the mass center of material system (cargo) C just as in Fig. 1a  II.


# Methods OF Solution

The formation of dynamic and constructing a mathematical model of cargo movement on a wagon is based on classical concepts and provisions of theoretical mechanics (for example, Constraint and their reactions, the Principle of ties release of the fundamental law of dynamics of the relative motions of records) [23,24]).


# a) Problem Analytical Solution

Unlike in [7,10,18], for deriving an engineering formula we will use the fundamental law of relative transferring cargo motion during rolling stock movement along tangent described by the equation in vector form
? e r I I R F a M + + + = , (1)
where r a is cargo relative acceleration ( or acceleration of cargo relative to the wagon floor).

As applied to the problem in question Let us assume that as in [25], cargo is in motion lengthwise the wagon floor with acceleration r a and let its speed be r v at the moment. Then equation (1) in projections upon coordinate axes Ox and Oz is presented in the form ( )
G F = is active force, ? + = F N R is reactive force,rx x x x ? r x ex Ma R F F F G I = ? ? ? ? + . bar b. ? ;(2)
( )
rz z r z ez z Ma N F F I G = + ? ? ? ? b. ,(3)

# Global Journal of Researches in Engineering

Volume XIII Issue X Version I 
) (i
F means that the force is dependent on the number of fastening elements but it doesn't mean that it is to be summed according to i.


# Elastic force
) (i
F has only one value.


# III.


# Results o f Solution a) Mathematical Solution of The Problem

In compliance with this and applied to (2) and (3) the following can be derived
) ? sin( 0 G G x = ; ? ? = = + = ? ? 1 1 ; 0 n i ix n i ix x R R F ) ? cos( 0 b b. r x r F F = ; (2?) ) ? cos( 0 G G z = ; ? ? = = + = ? ? 1 1 ; 0 n i iz n i iz z R R F ) ? sin( 0 b b. r x r F F = ; (3?)
According to the Coulomb law we'll write down
fN F ? ? , (4)
where f is sliding friction coefficient f = 0,7f coh with allowance made for f coh being the coefficient of cohesion friction between contact surfaces of cargo and wagon floor which are accepted according to reference data) [7,10,17,18]. Substituting ( 4 
? ? ? ? ? ? + + + ? = ? ? = = ? ? 1 0 b 1 0 ) ? sin( 0 ) ) ? cos( ( n i r iz n i iz ez F R R I G f F ? (5)
Just as in [10,17,18] let us present expression (5)  
in the form e F F F ? ? ? + = . elast ,(6) where. ) ? cos( ) ? sin( 0 ) ) ? cos( ( 0 ) ? sin( bar. 0 b 1 0 b 0 1 1 1 0 ? ? ? ? rx x r n i r iz ez n i iz n i ix n i ix ex Ma R F F R I G f R f R R G I = ? ? ? ? ? ? ? ? ? + + ? ? ? ? ? ? + ? ? ? ? = = = = (7) ( ) ( ) ( ) . ) ? sin( ) ? cos( 0 0 ) ? cos( ) ? sin( bar. 0 0 b 1 0 0 1 1 ? ? ? rx x r n i iz ix ez ex n i iz n i ix Ma R f F fR R fI f G I R f R ? ? + ? + ? ? + ? + = + ? ? ? = = =
We rewrite the derived expression in the form 
( ) ( ) ( ) . ) ? sin( ) ? cos( 0 0 ) ? cos( ) ? sin( bar. 0 0 b 1 0 0 ? rx x r n i iz ix ez ex x Ma R f F fR R fI f G I F ? ? ? + ? + ? ? + ? + = ? ? =
Here we are to take into account the fact that x R bar. is a design value of fastening thrust elements reactions (thrust bars) calculated either according to arbitrarily chosen or just as in [22] in accordance with scientifically grounded number of fastening elements (nail) in agreement with clearance outline. For example, After elementary manipulations with the above expression we determine projections of elastic forces (effort or tension) of i-flexible fastening elements onto axis Ox it is in this way that reactions of thrust bars are determined (12 [21]):


# Global Journal of Researches in Engineering

[R nail ] = 1, 08 is an assumed value of force per one fastening item (nail), (Table 32 Appendix 14 to International Rail Freight Transportation Agreement); n bar.? is accepted number of thrust bars according to cargo location and fastening scheme, item (Fig. 1); k 1 is a strength coefficient of fastening thrust bars, taking into account the state of wagon floor, items (usually accepted to be 0,5 ÷ 0,6); n nail.? is accepted value of needed number of nails per each thrust bar, item;
R bar.? = k 1 ?n nail.? ?n bar.? ?[R nail ]
is allowable load per one fastening item (nail) (46 Appendix 14 to). For example, for a nail with ?4and length 100 ÷ 120 mm By introducing notions of "shearing" and "retentive" forces [7,10,11,18] we rewrite the above expression
. ret . shear F F F x ? = ? ,(9)
where
) ? sin( 0 . shear G I F ex + = ( ) ( ) ( ) . ) ? sin( ) ? cos( 0 0 ) ? cos( bar. 0 0 b 1 0 . reet ? rx x r n i iz ix ez Ma R f F fR R I G f F + + + + + + + ? = ? =(11)
From now on, for simplicity of problem solving we'll study a case when cargo is retained lengthwise the wagon by i-flexible fastening elements (
? n i , 1 =
) and thrust bar (Fig. 1). Then according to the method of determining deformations at minor displacement we project a new point position first onto "original" or "old" direction of thrust element [17, 18 -21]. As a thrust element is arbitrarily located in space for calculating the projection it is necessary to make use of the method of double projection the way it is done in theoretical mechanics for arbitrarily located force [10,15,18]. Based on this we'll derive a formula for finding elongation of fastening flexible elements depending on cargo shift along wagon  It is obvious that elongation in the flexible elastic fastening element will occur only when there is cargo shift lengthwise the wagon at value x ? .

According to [7, 10, and 15] the movement of cargo lengthwise the wagon as one-mass oscillatory system can be presented in the following way (Fig. 3).  R ] = 0,047 ??, for a nail of ?6 and length 150 ÷ 200 mm = ?N. According to the standards of European countries for a fastening item of ?5mm and length 100 ÷ 150 [ = 1, 25 ?N, and for an item of ?6mm and length 150 ÷ 200mm [ = 1,5 ?N [26].

[ ] ]

1,08

nail.

[
R ] nail. R R (10a) nail. nail.
Designations in Fig. 3 and 4 are the same as in In expressions ( 8) and ( 11) projections of elastic forces (effort and tension) of i-flexible elastic fastening elements onto longitudinal and vertical axes Ox and Oz are determined according to the formulas
? ? = = ? ? = ? ? 1 1 n i i i i i n i xi l l a c R ; ? ? = = = ? ? 1 1 0 0 n i i i i n i xi l a R R ;(13)
;
? ? 1 1 ? ? = = ? ? = n i i i i i n i zi l l h c R , 0 0 ? ? 1 1 ? ? = = = n i i i i n i zi l h R R (14)
or taking into account (12), In formulas ( 13) and ( 14) rigidity of i-flexible fastening element (kN/m) with a number of threads n i (item), diameter d i (mm) and l i of length (m) of fastening wire: Putting the first expressions (13a) and (14a) in (8) we obtain , ? ? 1 1
i i n i i i i n i xi l a x l a c R ? ? ? = ? ? = = ? ? 1 1 ; ? ? = = = ? ? 1 1 0 0 n i i i i n i xi l a R R ; (13?) ; ? ? 1 1 ? ? = = ? ? ? = n i i i i i i n i zi l h x l a c R , 0 0 ? ? 1 1 ? ? = = = n i i i i n i zi l h R R (14?)x n i i i i i i i i n i i i i F l h x l a c f l a x l a c ? = ? ? ? + ? ? ? ? ? = = or . ? 1 x i i n i i i i i i F l a x l a l h f c ? = ? ? ? ? ? ? ? ? ? ? ? + ? =
The above expression with consideration for (15) has the form n . 854 , 7
? 1 2 x i i i i i i i i i i F l a x l a l h f l n d ? = ? ? ? ? ? ? ? ? ? ? ? + ? =
Hence we can find cargo shift lengthwise the wagon , 854 , 7
? 1 2 i i n i i i i i i i i x l a l a l h f l n d F x ? = ? ? ? ? ? ? ? ? ? + ? = ? (16)
where . long
F F x ? = ?
? is longitudinal force, determined by formula ( 9), (10) and (11) with consideration for second expressions ( 13), ( 14):
) ? sin( 0 . shear G I F ex + = ; (10) ( ) ( ) . ) ? sin( ) ? cos( 0 ) ? cos( bar. 0 0 b 1 0 . ret ? rx x r n i i i i i i ez Ma R f F l a l h f R I G f F + + + + + ? ? ? ? ? ? ? ? + + ? = ? = (11?)
Here, if aerodynamic resistance force b r F acts from the cargo rear back, this force should be put in the formula with a negative sign with consideration for coordinates of its application. Just as in [7,10,15) cargo shift lengthwise the wagon x ? is the distance from the cargo butt surface that is able to provide joint performance of flexible and thrust fastening means if a thrust bar is nailed to the wagon floor from the cargo butt at a distance less than x ? .

It can be observed from ( 16) that first, cargo shift lengthwise the wagon will occur only when schedule from (10) and 11a) there will be excluded descend angle ? (i. e. 0 0 = ? ). In these cases (10) and (11a) will be   That is why (11b) will have a simple form of: (11?)
rx x n i r i i i i i ez Ma R F l a l h f R I G f F + + + + ? ? ? ? ? ? ? ? + + ? = ? =(
While solving practical problems by using formula ( 9) or ( 16) just as in [10,15,17,18]  Using the derived value of cargo shift lengthwise the wagon x ? , just as in [10,18] in compliance with Hooke's law (as the product of (15) multiplied by ( 12)) we determine effort (tension) i R in i-  (17) or with consideration for (12) [ ], case will also be smaller. This, in its turn, will affect the decrease of elongation value and hence the decrease of the effort of each fastening element meanwhile increasing their loading capacity. The results obtained in analytical investigation are an important contribution to the theory of cargo fastening. 


# Global
a![Figure1 a : Diagram of allocating cargo and thrust bar on the wagonmoving down grade on the tangent 1 -wagon , 2 -cargo, 3 -thrust bar](image-2.png "Figure1 a :")
![elements](image-3.png "")
![.e. lose a constraint). An additional constraint (a thrust bar) is also a non-ideal and non-retentive (single-sided) one, it prevents the cargo from shifting from the contact plane to one side (to the right) and not keeping it from shifting to the other side (to the left).Let us assume that flexible fastening elements are pre-tensioned by efforts R0 i (for example, R01 = R0 11 = R0 2 = R0 22 = 20 kN), and they increase normal constituent of constraint reaction (platform floor), therefore, cargo and floor cohesion force coh F (and hence sliding friction x F slid. meaning that coh. slid.](image-4.png "")
![, ), points of application of external constraints reaction R are not known and are to be defined.](image-5.png "")
![to the object. We will take into account the fact that these forces exert influence on the external constraints (platform and fastening means). As to the object (cargo), we formally apply external forces(reactive forces) -normal component of wagon floor reaction and tangent component of this reaction as ultimate friction force (the force of cargo cohesion with wagon floor) coh or sliding friction force x F slid. ; reactions i R0 of preliminary twistings of fastening wire and bar reaction x R . bar (Fig.2 a, b) [17].](image-6.png "F")
2![Figure 2 a : Dynamic model of cargo on the wagon moving along the tangent on down grade](image-7.png "Figure 2 a:")
![vertical transferring inertia forces, ? I is Carioles inertia force (for the problem in question it is assumed to be equal to zero).](image-8.png "")
















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