Chapter 6 Section 3,4 Bending Deformation, Strain and Stress in Beams Key Points: 1. Bending moment causes beam to deform. 2. X = longitudinal axis 3. Y = axis of symmetry 4. Neutral surface does not undergo a change in length 6.2 Bending Deformation and Strain Key Points: 1. Internal bending moment causes beam to deform. 2. For this case, top fibers in compression, bottom in tension. Key Points: 1. Neutral surface no change in length. 2. All cross-sections remain plane and perpendicular to longitudinal axis. 3. Neglect deformation of cros section within its own plane. u u u cy yss ss s =AA A =A A A= A A) (lim'lim0 0Radius of curvature Length = constant Length decreases cy =Says normal strain is linear Maximum at outer surface (where y = c) maxc c|.|

\| =cymaxc c|.|

\| =cy6.2 Bending Stress The Flexure Formula What about Stress???? Recall from section 6.1: Therefore, it follows that maxo o|.|

\| =cydA ycMdAcyy dA y ydF MAA A A}} } }=|.|

\|= = =2maxmax) (oo oSum moments about cut: This is the moment of inertia, I The Flexure Formula: IMc=maxoIMy= o Or in general: Max bending stress, psi Internal bending moment, lb-in Distance from NA to outer fiber, in Moment of inertia, in4 Examples: Find maximum moment Find area properties, I and c Calculate stress WHERE IS BENDING STRESS MAXIMUM??? Answer:Outer surface (furthest away from Neutral Axis) Value of x along length where moment is maximum!! Example: The T-shape beam is subjected to the loading below. 1. Draw shear and moment diagram. Identify location and magnitude of Mmax. 2. Determine location and magnitude of maximum bending stress and draw stress profile.Is the beam safe if the material is aluminum w/ oy = 15 ksi? 3. What is the largest internal moment the beam can resist if oallow = 2 ksi? 45K Statics: Example 1 - Pliers Given: Typical household pliers as shown. Find: Force applied to wire and force in pin that connects the two parts of the pliers. Side: what is the shear stress in pin and bending stress in handle?SofM Do this for homework. See solution Link Statics: Example 2 Crane Structure Given: Crane structure as shown. Find: Forces and FBDs for cables A-B and A-E, boom DEF and post AFC. Side: what is the normal stress in cables (average normal only) and normal stress in boom and post (combined loading)?SofM Do this for homework. See solution Link Example 4:Determine the resultant internal loadings acting on the cross sections located through points D and E of the frame.(1-114)