STRESS DUE TO BENDING STRESS DUE TO AXIAL

STRESS DUE TO BENDING

STRESS DUE TO AXIAL FORCES Distance Stress due to purely axial forces gives direct stress. –either tensile or compressive. i. e. The stress is the same value across the section stress = F A Stress variation across member

Top fibre in. . . . compression Bottom fibre in. . Tension Point where tension changes to compression is called. . . . NEUTRAL AXIS HERE THE STRESS IS. . . ZERO Distance Consider beam under simple u. d. l. + Stress variation across member

Internal Moment of Resistance • Imagine beam cut along section • Imagine forces needed to maintain stability push pull • These two forces constitute a COUPLE which resists the external moment trying to bend the beam.

THE EXTERNAL MOMENT AT A SECTION WILL INDICATE THE INTERNAL COUPLE GENERATED INSIDE THE BEAM SCTION F d F • Knowing depth of beam will give us the “lever arm” applicable to this couple • Hence we can deduce the tensile and compressive stress on the beam section External BENDING MOMENT, or applied moment, due to the loading on the beam = Internal moment of resistance or couple on the beam section • The forces constituting the couple vary along the section due to the variation from compression to tension • The resistance the section offers is dependant on the beam cross sectional shape. • The material of the beam also affects the resistance (or strength) • from the above and with a bit of dfferential calculus we can formulate a general solution to beams in bending.

General theory of bending Where M = Moment of resistance = Applied external Bending Moment I = 2 nd Moment of Area about the Neutral Axis (also known as Moment of Inertia) f = permissible stress, or stress in beam material y = distance from neutral axis to (usually) outer fibre of beam section E = Young’s Modulus of beam material R = Radius of curvature of the beam (the deflected shape)

I = 2 nd Moment of Area • This is an indication of the distribution of, and hence the resistance to bending, of material in the beam cross section. • Also note that the two area properties, I & y can be combined to give the Section Modulus Z. Units: cm 3