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Biaxial bending stresses and stress factors in trapezoidal and elliptical cross-sectional curved beams

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Summary

This paper deals with the general case of biaxial bending of a curved beam in space whose centerline is circularly curved. Stress equations at any point in the cross-section of the beam have been derived by using Winkler's method. Neutral planes were established and stress factors were found and tabulated for variable thicknesses, widths, and ratios of (R 2/R 1). Numerical examples for several cases are solved and are demonstrated in graphical manner.

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Abbreviations

a :

semi-major axis of an ellipse

a 1 :

distance from the centroidal point, along they axis to the interior fiber

a 2 :

distance from the centroidal point, along they axis to the exterior fiber

A :

cross sectional area

A 1 :

=a 2+a 1

A 3 :

=a 2 3+a 1 3

A 4 :

=a 2 4−a 1 4

b :

semi-minor axis of an ellipse

b 2 :

exterior fiber width

b 1 :

interior fiber width

b :

=b 2−b 1

F :

stress correction factor

h :

thickness of the beam

H :

\(H = \left( {\frac{{ba_2 }}{h} + b_1 } \right)\)

I y :

moment of inertia abouty-axis

I z :

moment of inertia aboutz-axis

I yz :

product of inertia

J y :

polar moment of inertia abouty-axis

J z :

polar moment of inertia aboutz-axis

J yz :

polar product of inertia

M y :

bending moment abouty-axis

M z :

bending moment aboutz-axis

R :

radius of curvature of the centroidal axis in the unloaded condition

R 1 :

=R 0−a 1

R 2 :

=R 0+a 2

x z :

fiber width parallel toz-axis

y :

fiber distance from centroidal point, along they-axis

z :

distance of point from Ccentroidal point, along thez-axis

σ E :

bending stress, elementary theory

σ W :

bending stress, Winkler's theory

References

  1. Winkler, E.: Formänderung und Festigkeit gekrümmter Körper, insbesondere der Ringe. Zivilingenieur, N.S.4, 232–246 (1858).

    Google Scholar 

  2. Golovin, H.: одна изь задчь статики упругаро тбла практическаго технолгическаго института санктпетербургскаго, pp. 373–410, 1881.

  3. Ribiere, M. C.: Sur l'équilibre d'élasticité des voutes en arc de cercle. Compt. Rend.108, 561–563 (1889); and: Sur les voûtes en arc de cercler encastre es aux naissances. Compt. Rend.132, 315–317 (1901).

    Google Scholar 

  4. Billewicz, W.: Analysis of stresses in circular elastic rings under the action of body forces and external loading. Doctorate thesis, Univ. of Michigan, 1932.

  5. Holland, M.: Pure bending of beams having initial curvature. The Aeronautical Journal78 (768), 570–573 (1974).

    Google Scholar 

  6. Anatanaras, D. G., Tzivanidis, G. I.: Pure bending of anI-section beam having initial curviture. The Aeronautical Journal82 (443), 523–526 (1978).

    Google Scholar 

  7. Qassem, W., Raftopoulos, D. D.: Stresses and stress factors in curved beams having initial curvature. ESP 22/85017, pp. 1–10, Oct. 7–9, 1985.

  8. Bresler, B.: Design criteria for reinforced columns under axial load and biaxial bending. ACI Proc.57, 483–490 (1960).

    Google Scholar 

  9. Chen, W. F., Shoraka, M. T.: Tangent stiffness method for biaxial bending of reinforced concrete columns. International Association for Bridge and Structural Engineering35-I, 1975.

  10. Meek, J. L.: Ultimate strength of columns with biaxially eccentric loads. ACI J Proc.60, (8), 1053–1064 (1963).

    Google Scholar 

  11. Pannel, F. N.: Failure surface for members in compression and biaxial bending. ACI J. Proc.61 (1), 129–140 (1963).

    Google Scholar 

  12. Rammurthy, L. N.: Investigation of the ultimate strength of square and rectangular columns under biaxial eccentric loads. ACI Publication SP-13, pp. 263–298, 1966.

    Google Scholar 

  13. Mattock, A. H., Kriz, L. B., Hognestad, E.: Rectangular concrete stress distribution on ultimate strength design. ACI J. Proc.57, 875–928 (1961).

    Google Scholar 

  14. Au, T.: Ultimate strength of rectangular concrete member subjected to unsymmetrical bending. ACI Proc.54, 657–674 (1958).

    Google Scholar 

  15. Brettle, H. J., Warner, F. R.: Ultimate strength design of rectangular reinforced concrete sections in compression and biaxial bending. Civil Engineering Transactions, pp. 101–110. Australia: Institute of Engineers April 1968.

    Google Scholar 

  16. Farah, A., Huggins, M. W.: Analysis of reinforced concrete columns subjected to longitudinal load and biaxial bending. ACI J. Proc.66 (7), 569–575 (1969).

    Google Scholar 

  17. Furlong, R. W.: Ultimate strength of square columns under biaxially eccentric loads. ACI J. Proc.32 (9), 1129–1140 (1961).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mechanical Engineering, The University of Toledo, 43606, Toledo, OH, USA

    D. D. Raftopoulos & W. Qassem

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  1. D. D. Raftopoulos
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  2. W. Qassem
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Raftopoulos, D.D., Qassem, W. Biaxial bending stresses and stress factors in trapezoidal and elliptical cross-sectional curved beams. Acta Mechanica 68, 71–94 (1987). https://doi.org/10.1007/BF01182017

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  • DOI: https://doi.org/10.1007/BF01182017

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