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Theory of Simple Bending: Assumptions and Neutral Axis - C-16-C-302

This lesson module covers the assumptions made in the theory of simple bending and the concept of the neutral axis. Learn how to find stresses in unsymmetrical sections and the calculation of section modulus.

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Theory of Simple Bending: Assumptions and Neutral Axis - C-16-C-302

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  1. e LESSON MODULE FOR C-16 CURRICULUM, SBTET& ANDHRA PRADESH Year/Semester : III semester Branch : Civil Engineering Subject : Strength of Materials& Theory of Structures Subject code : C-302 Topic : Theory of Simple Bending Sub topic : Assumptions of Theory of Simple Bending Duration : 100 min Revised By : Dr.K.Lakshmi Pathi D.Haseena N.Srujana M.Yerri Swamy H.O.D : Dr.K.Lakshmi Pathi Venue : G.P.W, Hindupur C-16-C-302.1.3&1.4

  2. Objectives • On completion of this period, you would be able to understand • The basic assumptions in theory of simple bending • The concept of Neutral Axis • Finding stresses in un – symmetrical sections C-16-C-302.1.3&1.4

  3. Recap • In the previous session we have discussed about • Definition of beam, Bending moment, Shear force • Sign convention ,Relation between S.F., B.M & intensity of loading (w) • Neutral layer, Neutral axis, Modulus of section, Moment of resistance • Concept of theory of simple bending/pure bending C-16-C-302.1.3&1.4

  4. R Assumptions in theory of pure bending/simple bending • The beam is initially straight before the application of loads on it • The material of the beam is homogeneous and isotropic – same elastic properties in all directions and at all points C-16-C-302.1.3&1.4

  5. R Assumptions in theory of pure bending/simple bending • Elastic limit is not exceeded • Transverse sections which are plane before bending remain plane after bending C-16-C-302.1.3&1.4

  6. R Assumptions in theory of pure bending/simple bending • Each layer of the beam is free to expand or contract independently of the layers above or below it • Young’s modulus of elasticity ‘E’ of the material is same in tension and compression C-16-C-302.1.3&1.4

  7. R Assumptions in theory of pure bending/simple bending • Beam section is symmetrical about the plane of bending • There is no resultant pull or push on the cross section of the beam C-16-C-302.1.3&1.4

  8. R Assumptions in theory of pure bending/simple bending • The loads are applied in the plane of bending • The transverse section of the beam is symmetrical about a line passing through the centre of gravity in the plane of bending C-16-C-302.1.3&1.4

  9. R Assumptions in theory of pure bending/simple bending • The radius of curvature of the beam before bending is very large in comparison to the transverse dimension • As a result of a bending moment or couple, length of beam will take up a curved shape C-16-C-302.1.3&1.4

  10. R Assumptions in theory of pure bending/simple bending • A very short length may be treated as a part of the arc of a circle • It follows that at the outer radii the material will be subjected to tension and at the inner radii is subjected to compression. C-16-C-302.1.3&1.4

  11. R Assumptions in theory of pure bending/simple bending • At some radius there will be no stress • This layer of the material is the neutral layer or neutral axis C-16-C-302.1.3&1.4

  12. ASSUMPTIONS OF THEORY OF PURE BENDING C-16-C-302.1.3&1.4

  13. ASSUMPTIONS OF THEORY OF PURE BENDING • Click Here C-16-C-302.1.3&1.4

  14. Bending stresses in a beam • Above the neutral axis bending stresses are compressive • Below the neutral axis bending stresses are tensile C-16-C-302.1.3&1.4

  15. Bending Stress in a Beam • Click Here C-16-C-302.1.3&1.4

  16. Neutral axis for symmetrical sections For symmetrical beams, NA lies exactly at the center of the depth yc = yt = D/2 Extreme fibre stresses are equal Fig.2 C-16-C-302.1.3&1.4 16

  17. C-16-C-302.1.3&1.4

  18. C-16-C-302.1.3&1.4

  19. Neutral axis for unsymmetrical sections If the NA does not lie exactly at the centre of the depth, the beam is known as unsymmetrical beam   yc≠ yt Extreme fibre stresses are not equal Fig.3 C-16-C-302.1.3&1.4 19

  20. Problems • An unsymmetrical simply supported beam of depth 480mm is subjected to compressive stress of 20N/mm2 and tension stress of 80N/mm2. Find the position of NA from the top? σc=20N/mm2 A N 480 - X σt=80N/mm2 C-16-C-302.1.3&1.4

  21. σc=40N/mm2 σt=? 2)A triangular beam of depth 300 mm is subjected to a max comp stress of 40N/mm2 at the top. Find the tensile stress at the bottom of the beam? N A C-16-C-302.1.3&1.4

  22. σc=? σt=30N/mm2 3)A triangular beam of depth 240 mm is subjected to a tensile stress 30 N/mm2 find the corresponding comp. stress? C-16-C-302.1.3&1.4

  23. 4)An unsymmetrical beam of depth 500mm is subjected to tensile stress of 120N/mm2 and comp. stress of 80N/mm2. Locate the position of NA from the top? σc=80N/mm2 x N A 500-x σt=120N/mm2 C-16-C-302.1.3&1.4

  24. 5) An unsymmetrical beam of depth 300mm is subjected to a comp stress of 40N/mm2 and tension stress of 160 N/mm2 at the bottom. Find the position of NA from the top? σc=40N/mm2 x N A 300-x σt=160 N/mm2 C-16-C-302.1.3&1.4

  25. Calculation of Section Modulusfor Standard Sections C-16-C-302.1.3&1.4 25

  26. Rectangular Section y N A C-16-C-302.1.3&1.4 26

  27. Section modulus (Z)(or) modulus of section It is the ratio of moment of inertia and distance of the most extreme fibre from the NA It is denoted by ‘Z’ Z = Moment of Inertia / Centroidal distance of extreme fibre Z= I / Ymax C-16-C-302.1.3&1.4 27

  28. Square Section C-16-C-302.1.3&1.4 28

  29. Circular Section C-16-C-302.1.3&1.4 29

  30. Hollow Circular Section C-16-C-302.1.3&1.4 30

  31. Hollow Rectangular Section C-16-C-302.1.3&1.4 31

  32. Section modulus (z) for Unsymmetrical section Y c N A Y t Z = I / Ymax C-16-C-302.1.3&1.4

  33. Triangular Section Yc N A Yt C-16-C-302.1.3&1.4 33

  34. C-16-C-302.1.3&1.4

  35. C-16-C-302.1.3&1.4 35

  36. C-16-C-302.1.3&1.4 36

  37. C-16-C-302.1.3&1.4 37

  38. C-16-C-302.1.3&1.4 38

  39. C-16-C-302.1.3&1.4 39

  40. C-16-C-302.1.3&1.4 40

  41. C-16-C-302.1.3&1.4 41

  42. C-16-C-302.1.3&1.4 42

  43. C-16-C-302.1.3&1.4 43

  44. C-16-C-302.1.3&1.4 44

  45. C-16-C-302.1.3&1.4 45

  46. C-16-C-302.1.3&1.4 46

  47. C-16-C-302.1.3&1.4 47

  48. C-16-C-302.1.3&1.4 48

  49. C-16-C-302.1.3&1.4 49

  50. C-16-C-302.1.3&1.4 50

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