This paper deals with a specific method of examine the state of equilibrium of a flat textile structure. This kind of structure is modelled as an inextensible elastica loaded with its dead weight and axial force. The elastica represents, as an example, a longitudinal section of a fabric. It is assumed that the elastica rests on a flat, immovable base. We considered only those forms of deformed elastica where it's the two ends were supported by pivot bearings, and the tangent at those points lay on the immovable supporting plane. In the analysis, the shape of the deflection curve was determined for a given axial force, and it was examined whether a given position is stable or unstable. The analysis was carried out on the basis of the energetic method, by examining potential energy of the system. The investigations can be used for simulation of fabric buckling, folding and another applications of textile mechanics.
Keywords
stability, fabrics, textile mechanics, elastica, deflection curve, states of equilibrium
1. Assumptions of model and initial equations
Let us consider a flat fabric resting on a fixed, immovable base (Figure 1). Under the action of compressive forces, folds arise on its surface, which remain there due to the occurrence of friction forces.
Depending on the friction force quantity, those folds will remain or disappear after the action of deforming forces. In order to enable a thorough examination of stability of the deformed fabric, a substitute model was assumed which was limited to its deformed shape, i.e. to the fold. To generalize our further considerations, the term "elastica" in place of "fabric" is introduced hereafter.
Let heavy elastica be loaded with the axial force P and continuous load q in the coordinate system as in Figure 2. The elastica rests on a flat, fixed base and is supported on both ends by pivot bearings. It is inextensible so it cannot change its length l under the influence of loads acting on it.
However, it is subject to Hooke's law while being bent, and the known relation for the bending moment M is applicable to it
6. Discussion of the range of values of axial force and continuous load
In point 3 it was stated that a cannot take any optional value due to the specific length of the elastica. Admissible values of a belong to the interval: 0< a < where .
On analyzing Figure 4 it is seen that if we increase the value w, then the stable curvilinear solutions as located right from the curve g, will be for the value w above a certain amount greater than .
Since we try to be always within the admissible limits of the value a, then let us consider what the maximum value w should be for stable curvilinear solutions as to belong still to the interval.
Let the value ak defined by the equation (38) be less than . By virtue of the above
- If p
- If p=pk, then for 0
- The situation of p>pk is illustrated as example in Figure 5 together with the diagrams of the force p for w=3 and of the potential energy v in case when w=3 and p=8>pk (for our example pk=7,3399 and ak=0,1103).
In point a=0 the derivative . Based on Figure 5 it can be clearly seen that while increasing the value a from point a=0 the function v is increasing up to the local maximum which is attained at a=ans. Then the function is decreasing till the local minimum occurring at a=as, and increases again.
To conclude, if p
Eventually, for p>pk there are three forms of equilibrium.
1. Rectilinear form for a=0 corresponds with the state of stable equilibrium.
2. Curvilinear form corresponding to the left part of the curve (for a=ans) is unstable (local maximum of energy v).
3. Curvilinear form corresponding to the right part of the curve (for a=as) is stable (local minimum of energy v).
Let us now calculate the points as and ans.
Considering that a>0 let multiply both sides of the equation (34) by a. We obtain then
The equation (39) is a cubic equation with respect to a.
Roots of a have to be calculated with the given p and w. According to the earlier analysis, for p>pk and under assumption that a>0 there should be two roots, respectively of as and ans, while ans
Approximately, the boundary value for w amounts to .
Similarly, when looking at Figure 5 it can be seen that with fixed w increasing of the force p above a certain value results in the value as greater than the admissible one.
Since the value as depends not only on the value p, but also on the value λ (dependent in turn on w), thus for various w the maximal values of the axial force pmax will be different, and above them there are no more stable curvilinear solutions in the discussed interval of admissible values of a.
In case when w=0 it is sufficient for the value am as defined by the formula (36) was less than .
The approximate value of the maximal axial force in case of w=0 amounts to .
For the second case of w>0, the values of pmax for subsequent are calculated numerically.
To this end, with a fixed w, the value as was calculated from the formula (44) for subsequent forces p increasing by even steps, beginning from the value pk, till the moment of exceeding the value .
Then, the calculation was repeated for the next value of w. From the obtained values, a diagram of maximal axial force pmax as function of continuous load w was drawn up.
Based on the formula (37), also a diagram of the critical force pk as function of continuous load w was made. Both the diagrams are presented in Figure 6.
7. Conclusions
In conclusion let us sum up the problem of stability of the discussed elastica in case when w>0.
As follows from the above discussion, in this case the rectilinear form of equilibrium will always be stable, while we consider infinitesimal deviations from the point of balance. It can be seen clearly that with the increasing value p the local maximum of energy occurs at a value approaching closer and closer to a=0, but not reaching it. This only proves the fact that for large values of p it is easier to unbalance the system when it is in a stable, rectilinear state of equilibrium, causing some finite displacement to it. In order, however, for the system to assume a new, curvilinear form of stable equilibrium, it is necessary to pass the maximum of potential energy corresponding to the unstable form of equilibrium (Figure 5). The greater the force p is, the less displacement is needed for the system to assume a new form of equilibrium.
The force pk as defined by the formula (37) can be called the critical force, above which (except for the rectilinear form of stability) there is also a curvilinear form of stable equilibrium of the system.
Due to the assumption of inextensibility of the elastica there arose the limitation of the value of the shape parameter a, which has to be less than . Thus, in turn, some limitations for the value w and axial force p followed in the determination of the curvilinear stable solutions as in the limits of admissible values a, which are illustrated in Figure 6.
References
1. Dym C L 1974 Stability Theory and Its Applications to Structural Mechanics (Leyden: Noordhoff Int. Publishing)
2. Naleszkiewicz J 1958 Problems of Elastic Stability (Warsaw: PWN)
3. Forray M J 1968 Variational Calculus in Science and Engineering (New York: McGraw-Hill)
4. Timoshenko S P and Gere J M 1961 Theory of Elastic Stability (New York: McGraw-Hill)
About the author:
PhD Piotr Szablewski is a lecturer at Technical University of Lodz, Department of Technical Mechanics and Informatics, Lodz, Poland. He has over 18 years experience in the teaching of mechanics and numerical methods in engineering. Particularly, he is interested in theoretical methods and numerical simulation.
He is interested in theoretical mechanics and its applications in woven and non-woven textiles (particularly - textile composites). He tries to develop textile mechanics on Department of Technical Mechanics and Informatics - Technical University of Lodz. During the last three years he has worked on applications of numerical method for building geometrical models for textile composites and predicting mechanical parameters. Currently he�s working on simulation and optimization geometrical parameters of composites, to obtain higher quality for structural strength of textile composites.
He has also experience in working with international student groups in the field of mechanics.
PhD Piotr Szablewski
Department of Technical Mechanics and Informatics
Technical University of Lodz
Zeromskiego 116
90-924 Lodz
Poland
e-mail: piotr.szablewski@p.lodz.pl
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No simplifications are applied to the curvature as is done with the theory of bending beams, because big deformations are involved here.
In this case, existence of the rigid base causes limitation of the y coordinate. It must be greater or equal to zero for each value of the arc coordinate s, which is measured along the deflection curve.
The boundary conditions for this load scheme are the following
Zero M moment at the points of support A and B results from the fact that apart from the fold, the elastica rests flat on the base and its curvature amounts to zero. From that fact it also results that the tangent at the points of support must be horizontal. Thus, we get additional boundary conditions, namely
Let us consider the infinitesimal elastica section presented in Figure 3.
As it has been already mentioned, the elastica is inextensible, thus
Therefore we have the following geometrical condition:
Basing on the above equations of equilibrium we get the principle of virtual work on the virtual displacements . To do this, we multiply the equations (5) by appropriate virtual displacements. Then, by adding the sides and integrating within the limits from 0 to l we obtain
The obtained equation (9) represents the principle of virtual work, which provides that in the state of equilibrium the sum of work of all actual forces (both external and internal) acting on the system, for any virtual displacements, is equal to zero.
The functional occurring in the square brackets in formula (10) is total potential energy of the system (potential of external and internal forces).
If equilibrium is stable (stability), then the potential energy reaches minimum in the balance point. In the case of maximum potential energy, we deal however with the unstable state of equilibrium (labile equilibrium) [1], [2].
2. Deflection curve
As it is already known the deflection curve of the elastica in the state of equilibrium should present the functional extremum (11), respectively minimum for stable equilibrium, and maximum for labile one. In order to determine the functional extremum, let us take for granted the equation of the deflection curve, which fulfils given boundary conditions. Let the deflection curve be described by the formula
where A and B are coefficients unknown for the time being.
It can be easily noticed that the function (13) fulfils the boundary conditions (2).
As regards the additional boundary conditions (3) concerning the derivative dy / ds, we obtain from them relationship between the A and B coefficients in the form of
The range of admissible values of A parameter will be presented in the next point.
3. Admissible values for the shape parameter
We have the deflection curve defined by the equation (14). The shape parameter A occurring in the equation will be hereafter presented in the dimensionless form , related to the length l. This parameter cannot take full range of values. Below there is a precise definition of the interval of admissible values of a.
In the model assumptions it was stated that the existence of a fixed base in the system imposes the condition
Apart from that, in the model assumptions the condition (4) was given due to inextensibility of the elastica, which concerns the derivative . Applying it now, and substituting we obtain that for
because to satisfy the inequality (17) it is sufficient to substitute the function (18) with its maximal value.
On examining the function (18) it can be proved that in the given interval it has only one maximum amounting to
4. Potential energy of the system
Let us consider for the functional (11) the deflection function described by the relation (14). After substituting this function in the equation (11) J[y] becomes a function of a single variable A.
J [y] = V(A) .
To find the value of A coefficient, let Ritz method be applied [3]. The method uses the necessary condition of existence of V(A) function extremum, that is the equation
dV / dA = 0. (21)
The formula (11) must be first transformed and individual integrals calculated.
For the first addend it follows that
Before calculation of the third addend, it is necessary to represent the curvature in a somewhat different form. It is known that
This type integrals are discussed among others in publications [4] and [2]. As it is impossible to represent the result of the above integration in the form of elementary functions, an approximate solution is to be applied. To do this, the numerator and denominator of the integral are multiplied by
A better approximation can be obtained by subsequent multiplication of the numerator and denominator of the formula (26) by and so on. Applying in (27) the formulae for the first and second derivatives of the y function, it is obtained after integration.
The potential energy is so the function of the single variable A which can be called a variable parameter of shape, and two constants connected with the external load, namely P and q.
5. Analysis of states of equilibrium
To begin analysis of the states of equilibrium, the above-mentioned condition (21) is to be applied, on basis of which the value of the shape parameter A can be determined. It is the parameter on that the kind of equilibrium depends with a given load defined by P and q. Thus, we have
Since the relation (34) was obtained by use of the formula (21) expressing the necessary condition for existence of the function extremum, thus points lying on the p curves correspond to extremum of the function of potential energy. Location of the minimum and maximum of energy must still be defined.
Here, the second derivative of potential energy is used as equated to zero
The curve g described by the equation (35) is a diagram of the compressing force p represented as function of the parameter a but corresponding only to the points for which the second derivative (or in another way ) is equal to zero. Figure 4 presents dependence of the force p on the dimensionless parameter a for several values w, and the drawn curve g. This is a boundary curve. Right from it, on each of the p curves, with w>0 there are points for which , which corresponds to the minimum of potential energy v, that is to the state of stable equilibrium. It should be noted moreover that the curve g crosses the functions p in their minimal points. The boundary value of the shape parameter img35.gif is also marked in the diagram.
Now let us discuss in more detail the states of equilibrium for two possible cases of continuous load w (w=0 and w>0). It should be remembered that everything is considered with the condition A >0 or, which follows a >0.
Case I (w=0).
Here are considered Figure 4 and the formulae of potential energy v and its derivatives. Substituting w=0 in the formula (33) we have
It can be noticed that minimal value of the force p amounts to pk0=5 and it occurs with a=0.
- If p
- If p=pk0, then the function v also has its minimum for a=0, but it is more flat in this point (for a=0 all differential coefficients of the function v with respect to a up to the third degree inclusive are equal to zero, while ).
- In case when p>pk0, then the derivative when a >0 has already two zero points:
one for a=0, the other for a=am defined by the formula
It can be checked that for a=0 the derivative , so in this point there is the maximum of potential energy v. On the other hand, for a=am the derivative , so there is the minimum of potential energy v.
To conclude, for p< pk0 there exists only the rectilinear form of equilibrium, that is the stable position is only for a=0.
However if p>pk0, there are two positions of equilibrium. First one, for a=0 is unstable, whereas the other, for am defined by the formula (36) is the position of stable equilibrium.
Case II (w>0)
Like above, here are considered the formulae of potential energy v and its derivatives.
Basing on Figure 4 it can be seen that the minimal value of the force p, which for further consideration will be marked as pk, is greater than it was in the previous situation for w=0 (pk>pk0=5).
It can be also seen that the minimum occurs with a>0. Let this point be designated as ak like in Figure 5. To determine the values pk and ak the minimum of the function p given by the formula (34) must be found. After appropriate transformation, we obtain
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