Lecture 17 Bending Stress：讲座17弯曲应力.doc

EGR 236 Properties and Mechanics of Materials Spring 2013Lecture 17: Bending StressToday: - Homework questions: - New Topics: - Some more Shear-Moment Diagrams - Stress in Beams subjected to Bending - Moment of Inertia - Homework: Read Section 6:3-4 Work Problems from Chap 6: 37, 73, 75, 80Following todays class you should be able to:- State where the maximum stresses will be found in a beam subjected to bending loads.- Be able to calculate the maximum stress in a beam caused by a bending load- Understand the difference between compressive and tensile bending stress.Beams:Whenever there is an internal bending moment, M, acting on the cross section of a beam, this will set up normal stresses in the beam.FBL-x xMVVFAThe stress set up by such a bending moment tends to be maximum as you reach the outer fibers of the beam. There is also a location in the cross section where the stress drops to zero along what is called the neutral pNeutralAxisMVVMtenThe neutral axis runs through the centroid of the area of the cross section.The stress increases linearly with the distance from the neutral axis.So the farther away a portion of the cross section is from the centroid, the larger will be the normal stress due to pNeutralAxisMVVMtenOn one side of the neutral axis the stresses will be tensile. On the other side the stresses will be compressive.The magnitude of these normal stresses are given by a formula known as the flexure formula where M is the resultant internal bending moment, c is the distance from the neutral axis I is the moment inertia of the cross section about the neutral axisNote that the negative sign is an indication of whether the stress is tensile or compressive. Negative means a compressive stress. Procedure for analysis of beams subjected to bending.1) Determine the internal reactions. Find the distribution of V and M along the length of beam. This is usually done by drawing the Shear and Moment diagrams.2) Determine the Section Properties. This may include needing to a) locate the neutral axis through the centroid of the area b) determine the moment of inertia about the centroid3) Determine the maximum normal compressive and tensile stresses by use of the flexure formulaExample 1: A T-shaped beam built from two wooden boards has the loading shown below. Determine the maximum compressive and tensile stress in the beam.4181w = 5 lb/ftAB4 ftSolution:Load Diagramw = 5 lb/ftAB4 ftw = 5 lb/ft10 lb10 lbShearDiagramMomentDiagram-10 lb +10 lb10 lb-ftStep 1: Find the internal bending moment by drawing the shear and moment diagrams:Maximum internal bending moment is 10 lb-ft at the middle of the span.4181ycentStep 2: Determine the section properties:a) find the centroidal axis: ycentoid b) find the moment of inertia about the centroid.4181d1d2Recall that moment of inertia of a body can be found by adding the centroidal inertia to the parallel axis theorem component. For this cross section (which has two rectangular shapes) and compNeutralAxisten4181 Total inertia 3) Find the maximum bending stresses: Use Mmax = 10 lb-ftcompression of 4.33 psitension of 6.8 psi Example 2: A beam is built from a hollow circular tube (outside diameter 4 and inside diameter 3) that supports the loading shown below.Determine the stress along at points A, B and C at the location of the span with the maximum bending stress.Load Diagram1 kipABShearDiagramMomentDiagram2 kip3152Solution:a) Find the support loads, then draw the Shear and Moment Diagram.b) Find Section Properties 4 3BAC cross sectionc) Find Bending StressesLoad Diagram1 kipABShearDiagramMomentDiagram2 kip31521.95kip1.05 kip1.05 kip0.05 kip-1.95 kip3.15 kip-in3.9 kip-inSolution:a) Find the support loads, then draw the Shear and Moment Diagram. Note: Maximum stress occurs at the point of the maximum moment: The point where the 2 kip load is applied. Mmax = 3.9 kip inb) Find Section Propertie