MECHANICS OF MATERIALS: NORMAL & SHEAR STRESS Normal Stress Caused by Bending: Recall shear and moment calculation and graphing techniques: Load (w) Shear (V) Moment (M) 푀 = ∫ 푉 푑푥 = ∬ 푤 푑푥 0 0 0 −푤 (constant) 푤푥 (linear) 푤푥2 2 푑 푑 (parabolic) 푤 = 푤 = 2 푀 푑푥 푑푥 −푤푥 (linear) 푤푥2(parabolic) 푤푥3 −푤푥2 (parabolic) 푤푥3 푤푥4 The Neutral Axis is the axis at which a member in bending experiences no normal stress: When bending, we observe how stress varies as a point moves away from the Neutral Axis. 푦 푦 푦 휀 = − ⋅ 휀 such that 휎 = 퐸 (− ) ⋅ 휀 and 휎 = − ⋅ 휎 푐 푚푎푥 푥 푐 푚푎푥 푥 푐 푚푎푥 푀푐 푀푦 The negative sign is necessary 2 because for a positive 푦 point, the 퐼 = ∫ 푦 푑퐴 = or 휎푥 = − for any 푦 휎푚푎푥 퐼 beam experiences compression The Parallel Axis Theorem is often 1 used for objects that are not strictly Rectangle 퐼 = 푏ℎ3 푟푒푐푡푎푛푔푙푒 12 rectangular or circular, but rather are 1 Circle 퐼 = 휋푅4 comprised of several shapes: 푐푖푟푐푙푒 4 1 3 Triangle 퐼푡푟푖푎푛푔푙푒 = 푏ℎ ′ 2 36 퐼푖 = 퐼 + 퐴푖푑푖 1 4 Semi-circle 퐼푠푒푚푖푐푖푟푐푙푒 = 휋푅 8 푏 is in the direction parallel (∥) to the NA ℎ is in the direction perpendicular (⊥) to the NA Unsymmetrical bending due to moments at angles requires the moment vector be decomposed: The location of the neutral axis is measured by 휙 offset from the positive z-axis. By knowing the neutral axis experiences no normal stress, 푀푧푦 푀푦푧 휎푥 = 0 = − + , 퐼푧 퐼푦 푀 푦 푀푦푧 푧 = 퐼푧 퐼푦 푦 푀푠푛휃 퐼 퐼 푦 = ⋅ 푧 = 푡푎푛휃 ⋅ 푧 = 푧 푀푐표푠휃 퐼푦 퐼푦 푧 Such that 퐼 푡푎푛휙 = 푧 ⋅ 푡푎푛휃 퐼푦 The ? quadrants depend on the angle of the moment Shear Stress Caused By Shear Force: There are two directions in which shear force acts: longitudinal (along the length of the beam) and transverse (on cut plane) Distance from NA to the 푉푄 푦̅ 1. Horizontal shear (longitudinal): ∆퐻 = ∆푥 퐼 centroid of A 푉푄 푉푚푎푥 Area above the point or above 2. Transverse shear: 휏푎푣푔 = and 휏푚푎푥 = 1.5 ⋅ 퐼푡 퐴 퐴 the Shear Plane (opposite side ∆퐻 푉푄 Shear flow (shear force per unit length): 푞 = = of NA) ∆푥 퐼 Moment of inertia of entire How to determine Q: 퐼 object (independent of 퐴 or 푦̅) 푉 Transverse force applied Thickness of the object at the 푡 point observed or Shear Plane For more information, visit a tutor. All appointments are available in-person at the Student Success Center, located in the Library, or online. Adapted from Hibbeler, R.C. (2014). Mechanics of Materials (9th Edition). Boston, MA: Prentice Hall. .