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1. ICBT CAMPUS-MT HND in Mechanical Engineering Module : Strengths of Materials Lecturer : Mr. Mihiran Galagedara Batch : Batch 02 Copies : 17 Date : 27/01/2012 Session : 12

2. Columns By: Mihiran Galagedara B.Sc. Eng (Hons)UOM, AMIESL Faculty of Engineering and Construction, ICBT Campus-MT, Sri Lanka.

3. Analysis of pure bending has been limited to members subjected to bending couples acting in a plane of symmetry. Members remain symmetric and bend in the plane of symmetry. The neutral axis of the cross section coincides with the axis of the couple

4. Columns are the vertical prismatic members supporting axial loads . In practical applications, the loads applied on the columns are not always axial or act symmetrically to the cross section. Therefore theories need to be modified to describe such situations i.e. for skew loading (Bending of symmetrical sections about axes other than the axes of symmetry ) Introduction

5. Consider the simple rectangular-section beam shown. It is subjected to a load inclined to the axes of symmetry. 1. Skew Loading In such cases bending will take place about an inclined axis i.e. the N.A. will be inclined at some angle to the XX axis and deflections will take place perpendicular to the N.A.  

6. In this kind of cases it is convenient to resolve the load P, and hence the applied moment , into its components parallel with the axes of symmetry and to apply the simple bending theory to the resulting bending about both axes . Skew Loading contd…

7. It is thus assumed that simple bending takes place simultaneously about both axes of symmetry The total stress at any point (x, y) being given by combining the results of the separate bending actions algebraically The normal conventions for the signs of the stress is used, i.e . tension-positive, compression-negative. Skew Loading contd…

8. Skew Loading contd… The equation of the N.A. is obtained by setting equation to zero,

9. 2. Combined bending and direct stress- eccentric loading

10. (a) Eccentric loading on one axis There are numerous examples in engineering practice where tensile or compressive loads on sections are not applied through the centroid of the section Which thus will introduce not only tension or compression as the case may be but also considerable bending effects .

11. Combined bending and direct stress- eccentric loading contd… In concrete applications , for example, where the material is considerably weaker in tension than in compression, any bending and hence tensile stresses which are introduced can often cause severe problems .

12. Combined bending and direct stress- eccentric loading contd… Consider the beam shown in the figure where the load has been applied at an eccentricity e from one axis of symmetry .

13. Combined bending and direct stress- eccentric loading contd… The stress at any point is determined by calculating the bending stress at the point on the basis of the simple bending theory and combining this with the direct stress (load/area), taking due account of sign,

14. Combined bending and direct stress- eccentric loading contd… The positive sign between the two terms of the expression is used when both parts have the same effect and the negative sign when one produces tension and the other compression . It should now be clear that any eccentric load can be treated as precisely equivalent to a direct load acting through the centroid plus an applied moment about an axis through the centroid.

15. Combined bending and direct stress- eccentric loading contd… The distribution of stress across the section can be given by

16. Combined bending and direct stress- eccentric loading contd… The equation of the N.A. can be obtained by setting equal to zero  

17. In some cases the applied load will not be applied on either of the axes of symmetry so that there will now be a direct stress effect plus simultaneous bending about both axes. Thus, for the section shown with the load applied at P with eccentricities of h and k, the total stress at any point (x, y) is given by

18. Eccentric loading on two axes contd…

19. Again the equation of the N.A. is obtained by equating eqn. (4.26) to zero, when This equation is a linear equation in x and y so that the N.A. is a straight line such as SS which may or may not cut the section.

20. 3. “Middle-quarter ” and “middle-third” rules

21. Introduction As cast iron and concrete are not strong in tension, considerable problems may arise when they are subjected to eccentric loads. It will be convenient provided that the load is applied within certain defined areas, no tension will be produced whatever the magnitude of the applied compressive load .

22. “Middle-third ” Middle Third Rule states that no tension is developed in a wall or foundation if the resultant force lies within the middle third of the structure. Consider , the rectangular cross-section of The stress at any point (x, y) is given by eqn

23. “Middle-third ” Thus, with a compressive load applied, the most severe tension stresses are introduced when the last two terms have their maximum value and are tensile in effect,

24. “Middle-third ” Thus, with a compressive load applied, the most severe tension stresses are introduced when the last two terms have their maximum value and are tensile in effect, For no tension to result in the section, must be equated to zero,

25. “Middle-third ” This is a linear expression in h and k producing the line SS in the figure below.

26. “Middle-third ” If we follow the same procedure for other three quadrants, we’ll get the shaded area with diagonals of b/3 and d/3 Hence termed the middle third rule .

27. “Middle-quarter” rule for circular columns whatever the position of application of P, an axis of symmetry will pass through this position So that the problem reduces to one of eccentricity about a single axis of symmetry.

28. “Middle-quarter” contd… Therefore for zero tensile stress in the presence of an eccentric compressive load

29. “Middle-quarter” contd… Thus the limiting region for application of the load is the shaded circular area of diameter d/4 which is termed the middle quarter.

30. References 1. Hearn E.J, (2000), Mechanics of Materials, ISBN 0 7506 3265 8