C-16-C-302.1.1 & 1.2

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 : Introduction , Important terms Duration : 100 min Revised By : Dr.K.LakshmiPathiD.Haseena N.SrujanaM.YerriSwamy H.O.D : Dr.K.LakshmiPathi Venue : G.P.W, Hindupur 1 C-16-C-302.1.1 & 1.2

2. Objectives On completion of this period, you would be able to understand Beam, Effect of loading on beams Neutral layer, Neutral axis, Modulus of section and Moment of resistance C-16-C-302.1.1 & 1.2

3. What is a Beam ? Beam is a structural member sufficiently long compared to lateral dimensions Supported along its length Subjected to transverse loads C-16-C-302.1.1 & 1.2

4. Beam LOADING BEAM SUPPORTS C-16-C-302.1.1 & 1.2

5. Beam Under Loading C-16-C-302.1.1 & 1.2

6. What happens when beam is subjected to external loads? Beam bends and undergoes deformation Every section is subjected to Bending moment and Shear force BM & SF vary across the length of beam depending upon the loading C-16-C-302.1.1 & 1.2

7. What happens when beam is subjected to external loads? • Internal stresses develop in the beam to resist Bending moment i.e, Bending stresses • Internal stresses develop to resist Shear force i.e, Shear stresses C-16-C-302.1.1 & 1.2

8. ShearForce • Shear force at a section of a beam is the algebraic sum of the forces on either side of that section C-16-C-302.1.1 & 1.2

9. Bending Moment • Bending Moment at a section of the beam is the algebraic sum of the moments of all the forces either to the left or right of the section C-16-C-302.1.1 & 1.2

10. + ve SF Sign Convention for SF Left up and Right down force at a section may be considered as + ve SF. C-16-C-302.1.1 & 1.2

11. - ve SF Left down and Right up force at a section may be considered as – ve SF Note: The above sign convention is not mandatory, suitable sign convention may be adopted as per the convenience C-16-C-302.1.1 & 1.2 11

12. + ve BM Sign Convention for BM The moment due to upward forces on the left or right of a section is considered as + ve BM or Sagging Bending Moment C-16-C-302.1.1 & 1.2 12

13. (- v e) Hogging - Ve BM The moment due to downward forces on the left or right of a section is considered as - ve BM or Hogging Bending Moment C-16-C-302.1.1 & 1.2 13 13 13

14. Relation between Loading, Shear Force and Bending Moment Simply supported beam subjected to u dl over entire span C-16-C-302.1.1 & 1.2

15. Consider the equilibrium of the portion of the beam between sections 1-1 and 2-2. This small portion of the beam of the length dx is at a distance of x from the left hand support as shown in the above figure. Let F = Shear force at the section 1-1, F + dF = Shear force at the section 2-2, M = Bending moment at section 1-1, M + dM = Bending moment at section 2 -2. C-16-C-302.1.1 & 1.2

16. The forces and moments acting on the length ‘dx’ are • The force F acting vertically upwards at the section 1-1, (ii) The force F +dF acting vertically downwards at the section 2-2, (iii) The load w .dx acting downwards, • The moments M and (M +dM) acting at sections 1-1 and 2-2- respectively. C-16-C-302.1.1 & 1.2

17. The portion of the length dx is in equilibrium Hence, resolving the forces acting on this part vertically, F – w. dx – (F + dF) = 0 or - dF = w .dx or dF /dx = -w.  The above equation shows that the rate of change of shear force is equal to rate of loading. ‘-’sign indicates that with increase of (x) the value of SF decreases C-16-C-302.1.1 & 1.2

18. Taking the moments about the Section 2-2, we get M – w.dx.dx/2 + F .dx = M + dM or - w(dx)2/2 + F .dx = dM Neglecting the higher powers of the small quantities, F .dx = dM or F = dM/dx or dM /dx = F C-16-C-302.1.1 & 1.2

19. The above equation shows that the rate of change of bending moment is equal to shear force at the section  For the maximum bending moment, the shear force is zero But in practice, the bending moment may be maximum where shear forces changes its sign C-16-C-302.1.1 & 1.2

20. Pure bending / Simple bending Pure bending / Simple bending occurs when the beam bends with Constant Bending Moment Completely free from Shear Force In the above Fig, the pure bending exists in between the two applied loads S.F. Diagram B.M. Diagram Fig.1 C-16-C-302.1.1 & 1.2

21. 9C303.1 & 2

22. What happens during bending of beams? When a beam is subjected to bending due to transverse loading, the material of the C/S of the beam is subjected to bending stress (or) flexural stress Bending stresses are of two types: Bending stress in compression (fc or c) Bending stress in tension (ft or t) C-16-C-302.1.1 & 1.2

23. All the upper layers get contracted –compressive strains & compressive stresses develop All lower layers get extended – tensile strains & tensile stresses develop A layer in between the two extreme layers, which neither contracted nor extended – Neutral layer – no strain & stresses Beam subjected to sagging /positive Bending moment C-16-C-302.1.1 & 1.2

24. Beams subjected to hogging (Negative) Bending moment All the upper layers get extended –tensile strains & tensile stresses develop All lower layers get contracted – compressive strains & compressive stresses develop A layer in between the two extreme layers which neither contracts nor extended -Neutral layer – no strain & stresses C-16-C-302.1.1 & 1.2

25. W A B C L W/2 + W/2 - SFD +WL/4 BMD Bending moment & Shear force diagram for Simply Supported Beam A + wL/2 SFD - wL/2 + wL2/8 BMD C-16-C-302.1.1 & 1.2

26. 50kN kN 50 50 50 50 + 0 Shear Force Diagram (SFD) -50 25 0 kN-m -25 - -50 -75 -100 -125 Bending Moment Diagram (BMD) Bending moment & Shear force diagram for Cantilever Beam 2.5m C-16-C-302.1.1 & 1.2 26

27. Neutral Layer Layer which neither contracts nor extends due to BM (no strains and stresses in it) It always passes through the centroidal axis of the beam section 9C303.1 & 2

28. Neutral Axis • Intersection of neutral layer with the cross section of the beam 9C303.1 & 2

29. C-16-C-302.1.1 & 1.2

30. Neutral Layer Layer which neither contracts nor extends due to BM (no strains and stresses in it) It always passes through the centroidal axis of the beam section Neutral Axis Intersection of neutral layer with the cross section of the beam C-16-C-302.1.1 & 1.2

31. Neutral Axis Click Here C-16-C-302.1.1 & 1.2

32. C-16-C-302.1.1 & 1.2

33. 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-302.1.1 & 1.2 33

34. C-16-C-302.1.1 & 1.2

35. Moment of resistance (M.R) It is defined as the product of bending stress and modulus of section (or) It is defined as the resistance offered by the cross section of the beam for the external bending moment M = σmax x Z C-16-C-302.1.1 & 1.2 35

36. Summary • In this lesson we have discussed about • Beam subjected to pure bending • Neutral layer & Neutral axis • Moment of Resistance • Modulus of Section C-16-C-302.1.1 & 1.2 36

37. Quiz Pure bending of a beam means SF in the beam is uniform throughout b) BM is uniform throughout c) BM in the beam is zero d) None C-16-C-302.1.1 & 1.2 37

38. At neutral axis of the beam a) SF is zero b) BM is zero c) Bending stress is zero d) Bending stress is max C-16-C-302.1.1 & 1.2 38

39. 3. Section modulus is IY b) I/Y c) Y/I d) IY/2 C-16-C-302.1.1 & 1.2

40. Frequently asked question • Define the following terms • Neutral Axis • Modulus of Section • Moment of Resistance C-16-C-302.1.1 & 1.2

41. Frequently asked question • Define the following terms • Neutral Axis • Modulus of Section • Moment of Resistance C-16-C-302.1.1 & 1.2

42. Write the equations to calculate Modulus of Section and • Moment of Resistance of the following sections: • a) Rectangular • b) Square • c) Circular • d) Hollow Circular C-16-C-302.1.1 & 1.2