EAS209 L15_ PureBending- spring 2011(1)

EAS209 L15_ PureBending- spring 2011(1) - EAS 209-Spring...

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EAS 209-Spring 2011 Instructors: Christine Human 2/7/2011 1 Lecture 15– Beams: Pure Bending Lecture 15: Pure Bending To date we have studied: Axial Loading (bars and cables) CHAPTERS 2 Torsional Loading (shafts and tubes) CHAPTER 3 Now we will start analyzing bending in beams CHAPTERS 4,5,6,8,9 Today’s Objective Define pure bending Develop elastic flexure formula I My x We will start by analyzing only statically indeterminate beams Today’s Homework Relationship between normal stress and applied moment EAS 209-Spring 2011 Instructors: Christine Human 2/7/2011 2 Lecture 15– Beams: Pure Bending Classification of Beams Typical Beams Supports and reactions Roller - vertical reaction Pinned - vertical and horizontal reaction Fixed - vertical, horizontal and moment reaction
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EAS 209-Spring 2011 Instructors: Christine Human 2/7/2011 3 Lecture 15– Beams: Pure Bending Pure bending Consider the beam below (which represents a barbell held overhead by a weightlifter) Within the central section, M =constant, V =0 The center section of the beam is in Pure Bending Shear force diagram 80 lb 80x12=960 lb.in Bending moment diagram Note how the beam will deform – center portion in pure bending will form a circular arc EAS 209-Spring 2011 Instructors: Christine Human 2/7/2011 4 Lecture 15– Beams: Pure Bending Sign Convention M’ M’ Hogging Sagging +ve moment Compression in top Tension in bottom -ve moment Tension in top Compression in bottom
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EAS 209-Spring 2011 Instructors: Christine Human 2/7/2011 5 Lecture 15– Beams: Pure Bending Symmetric Member in Pure Bending Unfortunately, the stress distribution is statically indeterminate. We need to consider the beam deformations. Consider a prismatic member possessing a plane of symmetry and subjected to equal and opposite couples M & M’ acting in the plane of symmetry Look at equilibrium on a section perpendicular to the axis of the member and consider the three equilibrium equations containing x . M
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EAS209 L15_ PureBending- spring 2011(1) - EAS 209-Spring...

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