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NED University of Engineering & Technology Department of Petroleum Engineering Lecture #05 CE-212 - Mechanics of Solids DEFLECTIONS IN BEAMS INTRODUCTION  Bending in beams is accompanied by deflections and rotations.  Deflection relates to the movement of the beams vertically downwards or upwards, while rotation relates to the angle of deformation. ( is also the slope of the deflection curve, (  The comprehensive term displacement is used to indicate both deflection and rotation.
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  NED University of Engineering & Technology Department of Petroleum Engineering 1 By Dr. Huma Khalid Lecture #05 CE-212 - Mechanics of Solids DEFLECTIONS IN BEAMS INTRODUCTION    Bending in beams is accompanied by deflections and rotations.    Deflection   relates to the movement of the beams vertically downwards or upwards, while rotation   relates to the angle of deformation. (    is also the slope of the deflection curve, (         The comprehensive term displacement is used to indicate both deflection and rotation.    In addition to safety checks (e.g. limiting stresses), designers must avoid large deflections which are associated with poor appearance and with too much flexibility.    Large deflections can render the structure unserviceable, or even endanger it or neighbouring structures. DIFFERENTIAL   EQUATION   OF   THE   ELASTIC   DEFLECTION   CUREVE    From previous knowledge      where    is the curvature of the beam.    Relates the curvature         to the moment   as   where    is the flexural rigidity of the beam. Combining both equations yields: which is the basic differential equation of the elastic deflection curve. Fig. D1 depicts each of these parameters for a simply supported beam loaded as shown.  NED University of Engineering & Technology Department of Petroleum Engineering 2 By Dr. Huma Khalid    Eq. (D1) can be integrated (for a given beam load case) once to obtain the angle of rotation or slope.    Then twice to obtain the deflection    (provided that the bending moment    is known).    Therefore, the method is sometimes called the method of successive integration. Often, both    and    are constant along the beam, which simplifies the integration of eq. (D1). Boundary Conditions    The integration of eq. (D1) yields constants, which can be determined by knowing the  boundary conditions (BC), i.e. prior knowledge of slope and deflection at given points on the beam.    The sketch below depicts deflection and slope values for different boundary conditions.  NED University of Engineering & Technology Department of Petroleum Engineering 3 By Dr. Huma Khalid    Question : Determine the status of M ,   and   at points A, B and C of the beam  below.    Answer  NED University of Engineering & Technology Department of Petroleum Engineering 4 By Dr. Huma Khalid Example D1 Determine the maximum deflection of the cantilever beam shown below. Solution Select the x-axis and y-axis as shown above. From eq. (D1),         . Therefore, the slope at any point on the cantilever can be achieved by integration, i.e.:
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