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Dire Dawa Institute of Technology Department of Mechanical And Industrial Engineering

Dire Dawa Institute of Technology Department of Mechanical And Industrial Engineering. Chapter Five &Six DEFLECTION OF BEAMS By: Mesfin Dejene. 5.1 Introduction

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Dire Dawa Institute of Technology Department of Mechanical And Industrial Engineering

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  1. Dire DawaInstitute of TechnologyDepartment of Mechanical And Industrial Engineering Chapter Five &Six DEFLECTION OF BEAMS By: MesfinDejene

  2. 5.1 Introduction • The axis of a beam deflects from its initial position under action of applied forces. Accurate values for these beam deflections are sought in many practical cases: elements of machines must be sufficiently rigid to prevent misalignment and to maintain dimensional accuracy under load; in buildings, floor beams cannot deflect excessively to avoid the undesirable psychological effect of flexible floors on occupants and to minimize or prevent distress in brittle-finish materials; likewise, information on deformation characteristics of members is essential in the study of vibrations of machines as well as of stationary and flight structures. • The calculation of deflections is an important part of structural analysis and design. • Deflections are essential for example in the analysis of statically indeterminate structures and in dynamic analysis, as when investigating the vibration of aircraft or response of buildings to earthquakes. • Deflections are sometimes calculated in order to verify that they are within tolerable limits.

  3. P A v y A x B Fig.5.1

  4. 5.2. Relationship Between Loading, Shear Force, Bending Moment, Slope And Deflection.(Solution Method by Direct Integration)

  5. From Beam Theory, Boundary Conditions Refer figure 5.2.7(a) – (d) (a) Clamped Support: y(x1) = 0; y'(x1)=0; (b) Roller or Pinned Support: y(x1) = 0; M (x1)=0; (c) Free end: M (x1) = 0; V(x1) = 0; (d) Guided Support: y'(x1) = 0; V (x1)=0;

  6. Example: • Question: A Cantilever beam is subjected to a bending moment M at the force end. Take flexural rigidity to be constant and equal to EI. Find the equation of the elastic curve.

  7. In Some cases, it is not convenient to commence the integration procedure with the bending moment equation since this may be difficult to obtain. In such cases, it is often more convenient to commence with the equation for the loading at the general point XX of the beam. A typical example follows:

  8. Macaulay’s Method • The Macaulay’s method involves the general method of obtaining slopes and deflections (i.e. integrating the equation for M) will still apply provided that the term, W (x – a) is integrated with respect to (x – a) and not x.

  9. Case 2: Uniform Load Not reaching End of Beam

  10. Mohr’s Area-Moment Method • This method is used generally to obtain displacement and rotation at a single point on a beam. • This method makes use of the Moment - Area theorems given below. • Moment - Area Theorems • Referring the fig given,

  11. Referring the fig below • This is the first moment area theorem, Where P and Q are any two sections on the beam. i.echange in angle measured in radians between any two point P and Q on the elastic curve is equal to the M/EI area bounded by the ordinates through P and Q. • Referring to Figure, considering an element of the Elastic Curve,

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