Lecture 11 - C

Subject: Mechanics of Solids-I CE-104:

Subject: Mechanics of Solids-I CE-104 Instructor: Prof. Dr. Akhtar Naeem Lecturer: Engr. Muhammad Nissar Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 1 N-W.F.P University of Engineering & Technology Peshawar Sec A: Lecture #11 : (Stresses in Beams) Chapter#5

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 11 Stresses in Beams Stresses in Beams: Forces and couples acting on the beam cause bending (flexural stresses) and shearing stresses on any cross section of the beam and deflection perpendicular to the longitudinal axis of the beam. If couples are applied to the ends of the beam and no forces act on it, the bending is said to be pure bending. If vertical forces are applied and it produces the bending, the bending is called ordinary bending.

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 12 Stresses in Beams ASSUMPTIONS: In using the following formulas for flexural and shearing stresses, it is assumed that a plane section of the beam normal to its longitudinal axis prior to loading remains plane after the forces and couples have been applied, and that the beam is initially straight and of uniform cross section and that the moduli of elasticity in tension and compression are equal.

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 14 Stresses in Beams Flexure Formula: Stresses caused by the bending moment are known as flexural or bending stresses. Consider a beam to be loaded as shown.

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 15 Stresses in Beams Consider a fiber at a distance y from the neutral axis, because of the beam’s curvature, as the effect of bending moment, the fiber is stretched by an amount of cd. Since the curvature of the beam is very small, bcd and Oba are considered as similar triangles. The strain on this fiber is By Hooke’s law, ε = σ / E, then

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 16 Stresses in Beams which means that the stress is proportional to the distance y from the neutral axis.

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 17 Stresses in Beams Considering a differential area dA at a distance y from N.A., the force acting over the area is The resultant of all the elemental moment about N.A. must be equal to the bending moment on the section.

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 18 Stresses in Beams but then substituting ρ = Ey / f b then

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 19 Stresses in Beams and The bending stress due to beams curvature is The beam curvature is:

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Instructor: Prof: Dr.Akhtar Naeem Lecturer: Engr Muhammad Nissar 20 Stresses in Beams where ρ is the radius of curvature of the beam in mm (in), M is the bending moment in N·mm (lb·in), fb is the flexural stress in MPa (psi), I is the centroidal moment of inertia in mm^4 (in^4), and c is the distance from the neutral axis to the outermost fiber in mm (in).