Marek

AUTEX Research Journal, Vol. 5, No1, March 2005 © AUTEX

ANALYSIS OF PRODUCTS

TRANSVERSE

DEFORMABILITY

OF

SPACER

Marek Musioł Former junior lecturer of the Department of Technical Mechanics Faculty of Textile Engineering and Marketing Technical University of Lodz ul. Zeromskiego 116, 90-543 Lodz, Poland

Abstract The paper presents a model of transverse deformation in a textile product consisting of two external layers combined with deformed elements in the middle layer. Transverse deformability of textiles is particularly significant for several-layer products, and is decisive for the utility properties of such products.

Key words: textiles, mechanic, transverse deformation, modelling The analysis is aimed at investigating the transverse deformability of laminates. The transverse deformability of textiles is particularly significant for several-layer products. Such deformability is decisive for the utility properties of these products. It may thus be crucial for such products as carpets, floor covering, many kinds of furniture covering and upholstery products, etc. The paper demonstrates changes in the connecting element’s height under a transverse force. Transverse deformability was modelled with the use of the system of differential equations of equilibrium. An equilibrium state of the connector in its curvilinear deformation is sought. The results were obtained in the form of diagrams in which the shortening of the connector is shown depending on the force action and mechanical properties of the connector. external upper layer connectors external lower layer Figure 1. Diagram of load applied to a textile

∆x B

ϕB

l y

ϕA

x

A http://www.autexrj.org/No1-2005/0107.pdf

xk

There are many methods of interleaving the connectors with external layers. The method of fixing connectors may be defined as elastic, i.e. intermediate between total and articulated fixation . This paper discusses the simplest case of connecting the connector with external layers. Such a model is shown in Figure 2 below. Figure 2. Connector model, l - total connector height before loading, B – articulated fixation with external upper layer, A – articulated fixation with external lower layer, ∆x – change of connector’s height under loading ∆y force P, ϕb, ϕA – angles beetwen the line of the connector and perpendicular lines at point B and A

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AUTEX Research Journal, Vol. 5, No1, March 2005 © AUTEX

A change in the connector height under force load P is designated as ∆x (Figure 2).

P H

ϕB

y ϕA

Figure 3. Load diagram

x

s H P

A connector element of ds length is separated from the object and released from constraints by introducing a component of internal forces H, V and a bending moment M, which are functions of the position s of cross-section H=H(s); V=V(s); M=M(s). Figure 4. A fragment of ds length connector

V+dV

Such a system should satisfy the conditions of equilibrium:

dy

H+dH M+dM

ϕ

dx

ds x

H M V

EI

∑ Px = −

dV =0 ds

(1)

∑ Py = −

dH =0 ds

(2)

∑ M =Vdy − Hdx − dM = 0

(3)

The above system of equations is complemented by a physical relation that complies with Hooke’s law,

y

dϕ =M ds

(4)

and the geometrical relations

dx = cos ϕ ds

(5)

dy = sin ϕ ds

(6)

The following boundary conditions accompany Equations (1) through (6): for s=0

for s=l

x=0

y=0

y=0

M =0

M =0

V =P

(7)

Following Equations (1) and (2), we have H=0 and V=P, and after inserting (3) to the equation we receive:

dM + QPkr sin ϕ = 0 ds

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(8)

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AUTEX Research Journal, Vol. 5, No1, March 2005 © AUTEX

Q stands here for a ratio of transverse force applied to the product in relation to a critical buckling force

Q=

P ; Pkr

Pkr =

π 2 EI l2

Finally, the system of equations is reduced to four equations: (4), (5), (6) and (8). The system of equations is solved with use of a numerical shot method; the problem with boundary conditions is reduced to one of substitution, with initial conditions for s=0. When arbitrarily assuming the value of angle ϕ (0 ) = ϕ A , we have all the initial conditions given:

ϕ (0) = ϕ A M (0) = 0 x(0) = 0

(9)

y (0) = 0 As a result of numerically solving theequations with initial conditions (9), the values of moments for s=l depending on the load applied with use of Q force and at ϕ A assumed angle areobtained.

M B = M B (Q,ϕ A ) Based on the moment values obtained at the upper end, we looked for such moments that reach a zero value (Figure 5).

M (ϕ A , Q ) = 0

(10)

Following the solution of this equation, a ratio between load and initial angle is obtained:

Q = Q(ϕ A ) Figure 5 illustrates the solution obtained in the form of three diagrams corresponding to various forms of deformation. Figure 6 shows some deformation forms of the connector corresponding to various angles. The range of initial angle was assumed as

0 < ϕ A < π , load range 1 < Q < 10 .

P Pkr Figure 5. Values of function M (ϕ A , Q ) = 0

Some forms of the connector deflection are shown in Figure 6 below.

ϕ

When solving the system of equations, and knowing the initial angle ϕ, the dependence (11) is obtained, based on which the diagram shown in Figure 7 is generated.

Using some geometrical relations (5), the shortening of the connector may be described with the use of the following dependence: l

∆x = l − ∫ cos(ϕ )ds

(11)

0

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AUTEX Research Journal, Vol. 5, No1, March 2005 © AUTEX

Figure 6. Some shapes of the connector

where l- means the initial length of the connector. The formula is true for Q>1. When deformations are small, the connector length may be approximately calculated with use of the following dependence:

 ∆x  Pl 2 =  2 − 11,76 l  π EI 

(12)

∆x l

Q

Figure 7. Diagram of displacements

For small deformations such as zero to approximately 0.3, a linear diagram was obtained, despite the nonlinear theory. When the connector deflections are larger, the diagram is nonlinear in nature, and described by the dependence (11).

Conclusions The changes in the connector height in small deformations have a linear nature, despite a nonlinear theory being applied. In deformations larger than 30% of the initial connector length, the diagram patterns are curvilinear.

References 1. Kobza W., Modelowanie zginania tekstylnych wyrobów kompozytowych (in Polish), PAN, 2000. 2. Musioł M., Wyboczenie prętów zanurzonych w ośrodku sprężystym i zamocowanych sprężyście, Modelowanie i symulacja komputerowa w technice (in Polish), WSI, Łódź, 2002.

∇∆

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