OPINIONS ABOUT ALLOTTING STRAINS ON THE LAYERS OF A LAMINATED MONOLITH COMPOSITE

A previous paper stated the method of determination of the neutral surface of a composite monolithic, laminate composite, organizing some of the component layers, go to the method of distribution of tasks developed in one given case. This regard illustrates several possible options of which results can enable a proper assessment of the maximum strains and alignment layers, ensuring a maximum lifting capacity. The study can be developed to optimize the organization of layers, depending on the practical purpose of using the composite structure (mechanical, thermal and / or sound).


INTRODUCTION
The minimum consumption of building materials in mechanical structures, usually, but even more in the equipment of great burden, such as those in the process industries, represent a special interest both for the researchers in the field, and for manufacturers and users.Substances are present here, in general, with high working parameters (pressures, temperatures) and high chemical and / or mechanical aggressiveness.These aspects are to be taken into consideration since the projection, in different stages of their work, as when in operation, for metallic materials as well as for composite materials [1][2][3][4][5][6][7][8][9].In addition to fulfilling the safety of technical regulations, the authors respond responsibly with regard to the general or specific costs as well [10][11].It was a key reason for which in a previous work of the authors [12] it was studied the method of determining the neutral position of the surface of a laminated composite, generally with non-symmetrical alignment or a symmetrical (mirrored) one.In this case the sharing variants of loads allotted to component layers are exposed, for static or dynamic applications [13][14][15][16].Such an analysis allows finding, finally, the effective strains developed in layers, and the possibility of comparing the maximum tensions / resistances, to adopt a reasoned decision for the studied case.
Developing the study of the presence of the outside strains on a monolithic laminated board, in this paper, we consider the following aspects: -it is accepted as known the position of a neutral surface of a board with a given structure, geometric and material; -accepting the conditions of existence of the same specific linear deformation for all layers, these will be subjected to proportional loads with axial rigidities in longitudinal and transverse direction; the analysis takes into account a constant thickness of each layer, and thus, the same cross-sectional area; the longitudinal elasticity modules and the rigidities of the layers are accepted constant throughout the thickness of the layer concerned; -considering the same spins of the layers under the action of uniform bending moments, accepting the position of the neutral surface or not, based on the values attributed to each layer, the expressions of the corresponding tensions are deducted: it proposes three assessment alternatives of the strains developed; -the same spins are under consideration when allotting the unitary moments of twisting on the layers of a composite, namely to establish formulas for determining the torsional tensions which overlap with the other values of the strains, of a different nature, under the request of an elastic global strain; -establishing the methodology for allotting the uniform shearing forces in the plans layers and the normal one to their neutral surfaces.

THE DISTRIBUTION OF AN AXIAL FORCE ON A LAMINAE/LAYERS OF A LAYERED COMPOSITE
It is considered a longitudinal or transverse total force that can develop strain of stretching or compression, irrespective of the position of the neutral surface of the composite in the following (Figure 1).The force in question may be allotted in proportion, considering the same values of the specific linear strains of all laminae.If a board is loaded biaxially it is considered the following expressions: in which case the distributed forces on the component layers have the relationships (equation ( 2)): ; ; Therefore, the normal strains created in each layer along the axis is determined with the relationships (equation which may be of stretching or compressing, as appropriate. -Along the axis O y (equation ( 4)): The force acting on each layer, along the axis O y, produces the specific linear deformation: deducting, further, equation ( 5): In the previous relationships were also noted the (longitudinal) rigidities when streching / compressing the elements noted 'j' , respectively the equivalent (longitudinal) rigidities characteristic of the same type of strain: -Along the axis O x (equation ( 8)): ; -Along the axis O y (equation ( 9)): ;

DISTRIBUTION OF UNIFORM BENDING AND TWISTING MOMENTS ON THE LAMINAE OF A LAYERED COMPOSITE MONOLIT A. The distribution of the unitary bending moments Variant I
In terms of the same rotation (curvature) of the surfaces of composite and laminae, knowing the position of the neutral surface [12,13] and the arrow 'w' of the board, set for different board types and modes of support, strained statically or dynamically [17][18][19][20][21], it derives the following expressions for the uniform bending moments: -Along the axis O x (equation ( 10)): where (Figure 2, equations ( 11)-( 13)): The normal tensions, stretching and compressing on the two surfaces of the layer have the form (equation ( 14)): 2 6.
x j x j j M     -Along the axis O y (Equations ( 15)-( 18)): , ; 1 ; y j y j s j i j x y j y x j y x j x j s j s j x y j x y j y The normal tension, stretching and compressing the two surfaces of the layer have the form (equation ( 19)): 2 6.
Note: In the above it was found the average temperature mj T on the layer thickness.

Variant II
Considering the position of the neutral surface of the composite board, established based on the above methodology, under the same curvature conditions, uniform bending moments can be distributed as follows: -Along the axis Ox: Note: In the following it is considered neutral surface position of the composite [12,13], the distribution of the total bending moment being made based on the rigidity of the assembly and of each layer for the same rotation amount of the la Assuming a layered composite monolith (with layers glued intimately with each other, without slip under the action of external load / loads) and of unique spins , it is expressed the following distribution, for the axis Ox (taking into account the flexural rigidity of the layers, with unitary wide and a height j  ) [16]: so that (equations ( 20)-( 22)): 12 0,5 0,5 6 ; x j x j j x j j j x j j j x j The above equalities can be written by considering the flexural stiffness of the layers (equations ( 23)-( 25)): respectively the bending stiffness equivalent to that of the laminated board: In the previous relationships with xj d were noted the distances from the neutral surface of the composite to the layer j , in case of the appropriate section.Typically, the layers' thicknesses are identical ( x j y j j in the cross sections of the composite.If the longitudinal elasticity modules are different, the neutral positions of the surfaces in the two cross sections are different, so that the analysis can be done distinctly. Normal stresses can be evaluated with expressions of the form (14). -Along the axis O y (equations ( 26)-( 28)): The tensions developed along the axis Oycan be calculated with relations of the form (19).

Note:
For each layer of the composite stretching tensions occur on the lower surface, and compression ones on the the upper one, regarding the purpose of the uniform bending time (Figure 3).In accordance with [16] the composite layers develop uniform forces and normal proper tensions, according with the intensity of the uniform bending moment, with expressions like: -Along the axis O x -Along the axis O y: Note: the normal tensions calculated with the equalities (29) and (30) constant on the thickness of the respective layer, add to those developed by uniform bending moments they produce.

Variant III:
Note: In the following we will consider the position of the neutral composite laminated surface -Figure 4   3 or with regard to the correspondence: getting to: 3, Note: in the equality (33) it was not taken into account the sign of the tensions involved in the equality.
Based on the corresponding triangles congruence the following expressions are deducted for the tensions developed at the interfaces among the layers of the composite, written as: The tensions on the inferior side of the element   , namely the upper one of the element , j equation (39) Note: It is noted from the above equality the possibility of assessing the normal tension developed at the top of the element / layer (the compression one, as referred to above).
Taking into account Figure 4 the following relationships are deducted (equation ( 40)): From the contents of Figures 4 and 5 it is easily set the relationship for the calculation of the tension developed on the lower surface of the element j (equation ( 41)): and further ((equations ( 42) -( 45)): ; Note: The bearing capacity of the composite is accepted when the maximum tension is below the / allowable tension/resistance of the material from which the layer in question is carried out.

B. The distribution of uniform twisting moments
For a moment (total) twist / torque T M (acting in a plane x O z -Figure 6) on a layered composite it is considered the same rotation of all layers so that (equation ( 46)): so that the uniform twisting moments allotted to a layer is denoted by the phrase where it was used the coefficient dependent on the geometric simplex j b  in this case accepts the value 0,333 , and 3 jj bG     representing the torsional rigidity of the layer, to which belongs the torsional modulus of elasticity j G [22].As a result there develops the maximum shear tensions (equation ( 48)) [22]:

DISTRIBUTION OF CROSS-CUTTING FORCES (SHEAR) ON THE LAMINAE/ LAYERS OF A COMPOSITE -Along the axis Ox
(along the layers of the composite)

Variant I
Accepting an identical rotation of the layers of the monolithic composite, the unitary force  49)) [16]: where j G is the modulus of transverse (shear) elasticity of the element / layer j (Figure 6) On the basis of the above shear strains can be evaluated from each layer of the composite, with a uniform distribution over the entire cross-sectional area (situated in the plane x O z -Figure 6, equation ( 52)): Variant II The paper [16] considering the distribution of a total cutting / (shear) force in a plane x O z (Figure 6) presents the following calculation relation [D.I. Juravski's formula (1821-1891)] of shear tensions along the dimension / width b with a maximum value in the middle of it (equation ( 53)): where it was appealed to the previous notations.It easily appears that 1,5 .
For a well argued study it is recommended comparing the results offered by earlier formulas to take an appropriate decision to the case and ensure the normal functioning of the structure analyzed.
-Along the axis Oy (in a transverse direction to the layers of the composite) For a composite of a sandwich type ( with three layers) paper [16] gives the following methodology for evaluating the maximum shear strains at the interfaces between the outer layers 12  and 23  respectively the one from the middle layer, under the action of a total force oriented along the axis Oz (Figure 6, equation (54)).
  in which these measures were used (equation ( 55)-( 57)): Note: The values of shear strains, determined by the equalities (52) are inscribed on a parabolic graph [16].
, the shear strain is constant over the entire thickness of the middle layer of the composite, and equal to the values in the areas of separation of the outer layers, so that (equation ( 58)) [16]: Other notations: , x j y j AA  cross-sectional area along the axis Oxor Oy, corresponding to the layer   ,, D D D  flexural rigidity; , x j y j EE  modules of longitudinal elasticity of the layers j , along two reference axes , x j y j FF  ; the forces spread across the layers, along the axes of reference , x j y j II  ; the moments of inertia of the rectangular sections with width and height equal to the current layer unit j ; , x j y j MM  uniform bending moments relative to and axes Ox

CONCLUSIONS
Expanding the study determining the neutral position of the surface of a monolithic layered composite, in the contents of the present work it is going to be shown the methods of allotting the uniform shear forces or the normal ones, respectively of the uniform bending or twisting moments.As noted in the analysis presented above, it is taken into account some working hypothesis, as the same specific linear strains, respectively spins, of each layer, in accordance with the composite itself.Depending on the types of plane boards, of various external loads and modes of support to their specific contours, it is determined the normal uniform forces or the uniform bending and twisting moments.Further, the form stability of the board itself can be analyzed, or of the single layers, as well as the influences manifested.The carrying capacity of the layers, respectively of the intrinsic board is determined by the maximum allowable strain values compared with the allowable values, or the structural stability.It is noted that based on the results found, one may move to a proper organization of the layers or to an optimization imposed by the technical and economic nature.

Fig. 1 .
Fig. 1.The distribution of unidirectional force on the strata of the composite.-Along the axis O x (equation (1)):

Fig. 2 .
Fig. 2. The neutral position of the surface of the lamina.
The moments of inertia xj I from the previous relationships are calculated relative to its own axis of symmetry   0 xj I , also taking into account J. Steiner's relationship [22]:

Fig. 3 .
Fig. 3. Distribution of bending moment on the composite layers.


located within the pale of the element j , characterized by the quotas sc  and ic  of the board with a total thickness .c The following inequalities are noted (equation (31)): the meaning of the uniform bending moment M in Figure5, it is accepted the hypothesis of equivalence of this with the distribution of some normal streching tension (width ic  ) and of compression (width sc  ), resulting in the equality (equations (32)-(38)):

Fig. 5 .
Fig. 5.The distribution of bending moment on the composite layer.

Figure 6 ,
Figure 6, the uniform force

Fig. 6 .
Fig. 6.Distribution of shear forces on the composite layers.

,
thickness of the composite; wthe arrow of the board     the thickness of a laminated composite respectively the distances measured from the outer surfaces to the characteristic neutral surface; stiffness equivalent of the monolithic laminated board; of the board along the axes Ox and .Oy Given that, in general, the thicknesses of the layer are equal in the directions Oxand Oy, it is inferred 1.