# Lec4 Pure Bending

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\| = =}} }oo ooo oo o ng Substituti2Atta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Beam Section Properties The maximum normal stress due to bending, modulus section inertia of moment section = === =cISI SMIMcmoA beam section with a larger section modulus will have a lower maximum stress Consider a rectangular beam cross section, Ah bhhbhcIS6136131212= = = =Between two beams with the same cross sectional area, the beam with the greater depth will be more effective in resisting bending. Structural steel beams are designed to have a large section modulus. Atta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Properties of American Standard Shapes Atta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Deformations in a Transverse Cross Section Deformation due to bending moment M is quantified by the curvature of the neutral surface EIMIMcEc Ec cm m== = =1 1 o c Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero, vvc cvvc c y yx z x y = = = = Expansion above the neutral surface and contraction below it cause an in-plane curvature, curvature c anticlasti 1= =' vAtta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Sample Problem 4.2 A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine (a) the maximum tensile and compressive stresses, (b) the radius of curvature. SOLUTION: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. ( ) + =='2d A I IAA yYx Apply the elastic flexural formula to find the maximum tensile and compressive stresses. IMcm= o Calculate the curvature EIM=1Atta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Sample Problem 4.2 SOLUTION: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. mm 38300010 1143===AA yY = = = = 3333 210 114 300010 4 2 20 1200 30 40 210 90 50 1800 90 20 1mm , mm , mm Area,A y AA y y( ) ( )( ) ( )4 9 - 32 31212 31212 31212m 10 868 mm 10 86818 1200 40 30 12 1800 20 90 = = + + + = + = + ='Id A bh d A I IxAtta ur Rehman Shah, Lecturer HITEC University Taxila cantt Mechanics of Materials - I Chapter 4: Pure Bending Sample Problem 4.2 Apply the elastic flexural formula to find the maximum tensile and compressive stresses. 4 94 9mm 10 868m 038 . 0 m kN 3mm 10 868m 022 . 0 m kN 3 = = = ==Ic MIc MIMcBBAAmoooMPa 0 . 76 + =AoMPa 3 . 131 =Bo Calculate the curvature ( )( )4 9 -m 10 868 GPa 165m kN 31==EIMm 7 . 47m 10 95 . 2011 - 3= =

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