{'fraction_non_alphanumeric': 0.02514348182563542, 'fraction_numerical': 0.007242415960644985, 'mean_word_length': 4.61703760552571, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Deducing the form factors for shear used in the calculus of the displacements based on strain energy methods. Mathematical approach for currently used shapes

The paper presents an initial study concerning the form factors for shear, for a rectangular and for a circular cross section, being used an analytical method and a numerical study. The numerical study considers a division of the cross section in small areas and uses the power of the definitions in order to compute the according integrals. The accurate values of the form factors are increasing the accuracy of the displacements computed by the use of the strain energy methods. The knowledge resulted from this study will be used for several directions of development: calculus of the form factors for a ring-type cross section of variable ratio of the inner and outer diameters, calculus of the geometrical characteristics of an inclined circular segment and, using a Bool algebra that operates with geometrical shapes, for an inclined circular ring segment. These shapes may be used to analytically define the geometrical model of a complex composite section, i.e. a ship hull cross section. The according calculus relations are also useful for the development of customized design commands in CAD commercial applications. The paper is a result of the long run development of original computer based instruments in engineering of the authors.


Introduction
Several types of models may be developed when a phenomenon is being studied. Analytical models are using the classic knowledge in the field which is based on simplifying hypotheses. These assumptions were useful in order to minimise the amount of calculi by accepting a certain inaccuracy. Nowadays calculus instrument is the computer and it offers speed and accuracy when a certain volume of calculi must be performed. Moreover, it may be used for the integration of the various studies offering an overview regarding the phenomenon under investigation [1]. Regarding the analytical models, the computer may be used to disregard the classic hypotheses, for instance the small displacements assumption, by the use of the numerical integration [2]. In order to use the opportunities offered by the nowadays information technology development, the classic problems must be reformulated starting from the idea that the computer and the computer programming are the main calculus instruments. This approach includes the mathematical background of the problems which must be considered from a new perspective [3].

Analytical context and problem formulation
Calculus of the displacements of a given structure may be done using several approaches, for instance the method of initial parameters or using strain energy methods. In what follows, we consider the strain energy methods. An initial way to express the strain energy takes into account the stresses: where E is Young's modulus and G is the shear modulus or the modulus of rigidity.
For simple axial loads the normal stress may be computed using the relation the area of the cross section and the according strain energy is (7) For a structure consisting of ' N ' intervals for which the geometrical characteristics and the material constants are identical for each interval while the internal forces and moments may have a certain variation along the interval, the strain energy is [4]: (8) If we place a unit dummy load along the K  unknown displacement, and we remove the real loads, it results the functions of the internal forces and moments   x m Z on the same interval, ' i ', as the real internal forces and moments.
By applying the Reciprocal Work Theorem or Betti's Theorem, it results the expression of the unknown displacement In most of the 'classic' approaches the students are disregarding the effects of the internal forces. However, using the actual computing instruments and considering the accuracy of the results an important goal of the computer based analytical models, we must take into account the terms where the shear force appears in the previous relation, i.e. the Y k and Z k form factors for shear.

Discussion
The form factors may be computed analytically for simple shape cross sections. So far we haven't found proofs of the form factors for shear, the according values for simple shapes being taken from one book to another [5].
For complex composite shapes there may be conceived numerical approaches, in this case being necessary a new formulation of the problem.

Analytical approach for a rectangular section
Let us consider the calculus scheme presented in the following figure. As it may be noticed, we use the symmetry of the domain with respect to the vertical axis. Let us consider a point located at distance z in respect to the horizontal axis. Between the current point and the bottom most boundary of the section is the subdomain for which we consider the first moment of area. The first moment of area is: It results the law of variation: The Z k form factor for shear presented in (5) becomes It results (11)

Analytical approach for a circular section
Let us consider the circular cross section in the figure below. We consider a horizontal line at distance 0  z and, from this line, an infinite small distance, dz , which defines an infinite small area, dA , of a curvilinear rectangular region.
The area of the OBD triangle is The OC and BC segments may be expressed with respect to the R radius and the  angle in the BOC right angled triangle: The next stage is to evaluate the first moment of area.
The dA infinite small area may be expressed as the area of a curvilinear rectangle, i.e. dz BD dA   . (16) z coordinate may be also expressed with respect to radius R and angle  in the right angled triangle BOC : (17) The according derivative is: The expression of the infinite small area, dA , becomes The first moment of area is calculated as an integral for the infinite small area, dA : It results the expression: (20) The position of the centroid with respect to the centre of the circle is: (22).

The width of the section is
(23) In this case, the form factor is   (24) We consider the following product-to-sum identities: We evaluate In the (26) identity we replace By replacing (28) in (27), it results We evaluate    horizontal axis or the second moment of area with respect to the horizontal axis for a circular cross section, using the (35), (36) and (37) relations, figure 5. The results of the calculus based on the (35), (36) and (37) relations are presented in table 1. As it can be noticed, the errors are very small. Regarding the area, the error is zero because the i A  small areas are computed using the exact calculus relation and the area of the cross section is a summation of these values. Regarding the first moment of area, for the entire section its value is zero. If this value is used to compute the relative error, the denominator would be zero, therefore we consider the absolute error. As it can be noticed, the errors are very small, i.e. 1.36396·10 -10 . Moreover, the computer code offers the variation of the

Conclusions
The analytical approaches and the numerical studies presented in the paper offer the same results. The numerical approach is based on the power of the definitions and on the speed and accuracy of the computer based studies.