Engineering Transactions,

**1**, -, pp. 3-57, 1953### Kierunki Możliwych Usprawnień w Obliczeniach Łuków

This paper discusses the possible improvements in the statical computation of three main types of arches from the point of view of engineering practice. These are parabolic, elliptic (with vertical major axis) and circular arches.

The statical computation of arches representing parabolas of the second or third order leads to indefinite integrals, which are easy to calculate, if one assumes that the differential of the arch length, ds, is equal to its projection, dx, on the X-axis, or depends on this projection in a simple manner. The dependence between ds and dx is determined here by means of the method of least squares, and is expressed for a second order parabolic arch by Eqs. (22), (38) and (41), and for a third order parabolic arch by Eqs. (72), (73) and (74). The values of parameters g in function of rise-to-span ratio Ç=f :l are gathered in Tables 3 and 6. In addition, the case of a parabolic arch with a linear variability of cross-section is discussed. Eq. (54) represents the ration 1 : J expressed in the form of an algebraic function. Elliptic arches with vertical major axis are used in the design of arched culverts. Upper portions of these arches can be considered as curved beams of great curvature, i.e. of high r : e ratio. This is illustrated in Fig. 18. The question arises, whether elliptic arches should be treated as beams of great or of small curvature. The paper contains comparative computations, the results of which are represented graphically in Figs. 22-24. It can be seen from these graphs, that for practical calculations an elliptic arch of a culvert can be considered as a beam of small curvature. The dependence between the stresses in a lliptic culvert and the angle of internal friction of the loading soil is considered. The chapter, in which circular arches are discussed, contains tables and graphs facilitating the calculation of these arches. The basis for the determination of coefficients of the canonical equations of circular arches, (155)-(157) is the deformation of a bar representing a circular arch, built in at one end and free at the other, under the action of forces parellel to the axes of co-ordinates and located at the end and at an intermediate point of the bar (Figs. 34 and 36). These deformations are expressed as products of certain constants, (125), and coefficients depending on the angles O defined in Fig. 35. They are represented in Tables 7 and 8. By geometric addition of these deformations one obtains the dis- placements (150) determining the boundary conditions for an arch without articulations, of the auxiliary system represented in Fig. 39b. Equations (150) lead to Eqs. (155)-(157), the coefficients of which are tabulated in Tables 11 and 12. The problems of the influence of longitudinal forces on statically indeterminate quantities of an arch are discussed for each arch from in the corresponding chapters.

The statical computation of arches representing parabolas of the second or third order leads to indefinite integrals, which are easy to calculate, if one assumes that the differential of the arch length, ds, is equal to its projection, dx, on the X-axis, or depends on this projection in a simple manner. The dependence between ds and dx is determined here by means of the method of least squares, and is expressed for a second order parabolic arch by Eqs. (22), (38) and (41), and for a third order parabolic arch by Eqs. (72), (73) and (74). The values of parameters g in function of rise-to-span ratio Ç=f :l are gathered in Tables 3 and 6. In addition, the case of a parabolic arch with a linear variability of cross-section is discussed. Eq. (54) represents the ration 1 : J expressed in the form of an algebraic function. Elliptic arches with vertical major axis are used in the design of arched culverts. Upper portions of these arches can be considered as curved beams of great curvature, i.e. of high r : e ratio. This is illustrated in Fig. 18. The question arises, whether elliptic arches should be treated as beams of great or of small curvature. The paper contains comparative computations, the results of which are represented graphically in Figs. 22-24. It can be seen from these graphs, that for practical calculations an elliptic arch of a culvert can be considered as a beam of small curvature. The dependence between the stresses in a lliptic culvert and the angle of internal friction of the loading soil is considered. The chapter, in which circular arches are discussed, contains tables and graphs facilitating the calculation of these arches. The basis for the determination of coefficients of the canonical equations of circular arches, (155)-(157) is the deformation of a bar representing a circular arch, built in at one end and free at the other, under the action of forces parellel to the axes of co-ordinates and located at the end and at an intermediate point of the bar (Figs. 34 and 36). These deformations are expressed as products of certain constants, (125), and coefficients depending on the angles O defined in Fig. 35. They are represented in Tables 7 and 8. By geometric addition of these deformations one obtains the dis- placements (150) determining the boundary conditions for an arch without articulations, of the auxiliary system represented in Fig. 39b. Equations (150) lead to Eqs. (155)-(157), the coefficients of which are tabulated in Tables 11 and 12. The problems of the influence of longitudinal forces on statically indeterminate quantities of an arch are discussed for each arch from in the corresponding chapters.

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