Abstract

Theaim of this paper is to examine the state of equilibrium of a flat textilestructure loaded by compression forces and deadweight. Only those forms ofdeformed structure were considered where its two ends were supported by pivotbearings, and lie on the immovable supporting plane. Such structure may be aflat textile structure (e.g. fabric) and above-mentioned boundary conditionsresult directly from behavior under specified loading. In the analysis, shapeof the deflection curve was determined for a given axial force and deadweight,and it was examined whether a given position is stable or unstable. Two shapeparameters occurring in the analysis are used for simulation of differentshapes of bending curve in the middle of compression. The analysis was made onthe basis of the energetic method, by examining potential energy of the system.Results may be used for different algorithms and programs for simulation offabric buckling, folding and for another application of textile mechanics.

1. Assumptionsof model and initial equations

Letus consider a flat textile structure of length l as its longitudinalsection loaded by compression force P and deadweight q as inFigure 1. The structure lies on immovable supporting plane and issupported on both ends by pivot bearings in points A and B. However, it issubject to Hooke's law while being bent, and the known relation for the bendingmoment isapplicable to it, where is the radius of curvature, and EI means thebending rigidity. In this case, the existence of the rigid base causes thelimitation of the y coordinate. It must be greater or equal to zero foreach value of the arc coordinate s, which is measured along thedeflection curve ().Theboundary conditions for this load scheme are the following:

Letus consider the infinitesimal section of the structure presented inFigure 2. The structure is inextensible, thus. Therefore we obtain the followinggeometrical condition.Writing the elementary equations of equilibrium for section from Figure 3,next multiplying by the appropriate virtual displacements x, y and, adding the sides and integrating within the limits from0 to l, we obtain

After integrating by parts and taking the boundary conditions, we have

The functional J[y] is the total potential energy of the system (potential of external and internal forces).

The Equation (3) is the necessary condition for the existence of extremum of the functional J[y] (see work [3]). If the equilibrium is stable (stability), then the potential energy reaches a minimum in the balance point. In the case of maximum potential energy, however, we are dealing with an unstable state of equilibrium (labile equilibrium).

2. Deflection curve and potential energy of the system

The deflection curve of the structure in the state of equilibrium should present the extremum of the functional (4). In this point the Ritz method has been applied. The equation of bending curve was assumed as follows

(5)

where A and B are the shape parameters, which are describing the different shape of the bending curve. The Equation (5) fulfils the boundary conditions for presented load scheme. Using the Ritz method the functional (4) becomes a function of two variables A and B,. The coefficients A and B may be evaluated from the necessary conditions of extremum of the function V

(6)

Using in Equation (4) the bending curve (5), and taking into consideration certain simplifications and boundary conditions we obtain the relationship for the total potential energy of the system

(7)

The function V(A,B) is the function of two variables A and B and of two constants, namely P and q.

3. The necessary conditions of state of equilibrium

At present the values of A and B will be found. For these parameters the above-mentioned structure will be in state of equilibrium under the external load. For that purpose function V (7) should be applied in Equations (6). To make further discussion more general, let us represent the parameters A and B, force P and deadweight q in the dimensionless form, relating them to Eulers critical force.

Let us make the following transformations: , where and . Thus, we obtain

The Equations (8) must be fulfilled simultaneously. We eliminate from Equations (8) first p, then w, obtaining

Equation (9) show the deadweight function w (a, b) of two variables a and b in the implicit form. The graph of function w (a,b) is presented in Figure 3. Similarly, the implicit function p (a, b) combines a and b related to the value of p. The graph of function p (a, b) is presented in Figure 4.

The structure is simultaneously loaded with continuous load w and force p, thus after superimposing both graphs w(a,b) and p(a,b) over each other the state of equilibrium occurs in crossing points of contour lines w = const. and p = const. In these points extremum of the functional (4) occurs. We expect also in his points the state of equilibrium because the conditions (6) are fulfilled. If it is true, it will be shown below. In Figure 5 it has been shown example point of state of equilibrium for p = 10 and w = 12.

4. Analysis of states of equilibrium

The equilibrium is stable (stability) if the potential energy V reaches a minimum in the balance point and unstable (labile equilibrium) if maximum (see works [1], [2] and [4]). The function V is the function of two variables V(a,b), thus the examination of equilibrium is simply the examination of extremum of function of two variables. Now, we must examine sufficient conditions of extremum of function V. The kind of extremum depends upon the sign of second derivative and the value of discriminant W, specified by formula:

. Therefore, if the necessary conditions (6) are fulfilled, and , then if the function V reaches in this point a minimum (stable equilibrium). Whereas , the function V reaches in this point a maximum (labile equilibrium). Using the relation (7) for the total potential energy of the system, we can find appropriate derivatives and discriminant W. In order to have dimensionless form of variables we multiply the W by , and the derivative by . Such multiplying cannot have influence on the sign of their values. Therefore, we have

It can be noted that only axial force p and parameters a and b have influence on the value of second derivative and of discriminant W. Thus changing the value p we can change the kind of equilibrium of the system. The sign of the second derivative and discriminant W has been found numerically. It has been obtained certain areas in which for some values a and b is stable equilibrium and for another values a and b is unstable equilibrium. It has been obtained also areas where the equilibrium dont exist (it comes from W0). For example, for p = 1,5 is only the area of stable equilibrium (it has been shown in Figure 6 dark area). The light area of the graph represents the points where the equilibrium doesnt exist.

Whereas in Figure 7 dark area represents the labile equilibrium area for p = 12.

5. Admissible values for the shape parameters

Values of the shape parameters A and B in Equation (5) cannot be arbitrary. In other words, these parameters cannot take the full range of values. Below there is a precise definition of the interval of admissible values of A and B. The existence of immovable supporting plane, as in Figure 1, leads to the geometrical condition for . Besides, from the inextensibility we have that for . Take into consideration dimensionless form of the variables and , we have

From hole area of a and b we choose only such values so that the inequalities (13) are satisfied. These inequalities have been calculated numerically. For example the extreme values of a and b are as follows:

. Admissible area of a and b have been show in Figure 8.

After analyzing the states of equilibrium it should be remembered that values of a and b should be from admissible area which is presented in Figure 8. Admissible are of a and b should be superimposed on the graphs from Figures 6-7.

6. Conclusions

Above-mentioned method of analyzing of a flat textile structure loaded by compression force and deadweight has been presented for the bending curve given by formula (5). Thanks to different parameters A and B we can simulate different shapes of bending curve (such as in the reality). Similarly, it can be carried out for different shapes of bending curve. For solving the problem the Ritz energetic method has been applied. Similar way to solving of the problem we can find in work [5] for the one-dimension problem. Due to nonlinearity numerical methods have been applied. These methods can be used without any problem for multi-dimension tasks with more than two shape parameters.

References

1. Dym C.L., Stability Theory and Its Applications to Structural Mechanics, Leyden Noordhoff Int. Publishing, 1974.
2. Naleszkiewicz J., Problems of Elastic Stability, PWN, Warsaw, 1958.
3. Forray M.J., Variational Calculus in Science and Engineering, McGraw-Hill, New York, 1968.
4. Timoshenko S.P., Gere J.M., Theory of Elastic Stability, McGraw-Hill, New York, 1961.
5. Szablewski P., Analysis of the Stability of a Flat Textile Structure, AUTEX Research Journal, Vol.6, No.4, 2006, p. 204-215.

About the Author:

The author is associated with Technical University of Łdź, Department of Technical Mechanics and Informatics, Lodz, Poland.

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