PACIFIC ENGR 121 - Shear and Bending Moment Diagrams

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Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32BendingShear and Bending Moment Diagrams – covered in Statics and self-reviewed in Mechanics of MaterialsKnow terminology: simply-supported versus fixed; cantilevered, overhanging.Know sign conventions for positive versus negative shear and moments.If you can’t remember, memorize the shear and bending moment diagram for a simply-supported beam with a point load.A graphical method can be used to quickly visualize shear & bending moment diagramsBending Deformation of a Straight MemberAssumptions:straight, prismatic beam made of homogeneous material; cross-sectional area of beam is symmetrical with respect to an axis, and bending moment is applied perpendicular to axis of symmetry.Illustration of Bending for a BarDistortion of a Rubber Bar due to BendingAssumptionsLongitudinal axis does not charge length;Cross-sections remain plane;Any deformation of cross-section within its own plane can be neglected.Undeformed ElementDeformed ElementLongitudinal normal strain depends on y-position and radius of curvatureNormal Strain DistributionPoisson effects that cause εy & εz can be neglected. Recall assumption: any deformation of cross-section within its own plane can be neglected.The Flexure Formula From Hooke’s law, a linear variation in strain corresponds with a linear variation in stress.Bending Stress Variation6.32. Known: The ski supports the 180 lb weight of the man. The snow loading on its bottom surface is trapezoidal.Find: Determine the intensity, w, and draw the shear and bending moment diagrams for the ski.V, lbM, ft∙lbxx6.49. Known: A steel beam has the cross section shown. Find: Determine the largest internal moment the beam can resist if the moment is applied as follows:(a) About the z axis.(b) About the y axis.Data: The allowable stress for the steel is σallow = 24 ksi.6.54. Known: An aluminum strut has a cross-sectional area in the form of a cross.Find: Determine the bending stress acting at (a) Point A, and (b) Point B. Show the stresses acting on volume elements at these points. Data: The moment M = 8 kN∙m.6.59. Determine the largest bending stress developed in the member if it is subjected to an internal bending moment of M = 40 kN∙m.Unsymmetric BendingMoment Applied Along Principal AxisThe flexure formula applies for sections that are unsymmetric if the moment is applied about one the principal axes. Note that one of the principal axes is along an axis of symmetry, and the other principal axis is perpendicular to it.zIMyUnsymmetric Bending – (continued)Moment Arbitrarily AppliedA moment M applied in the yz plane at an angle θ has components Mz = M cosθ and My = M sinθ as shown below.The flexure formula can be applied to each moment component to obtain the resultant normal stress at any point on the cross-section.The stresses combine to form the resultant stress distribution below:Orientation of Neutral Axis (N.A.)6.102. The box beam is subjected to a bending moment of M = 15 kip∙ft directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. yzB CDAyzB CDAyzB CDA6.108. The 30 mm diameter shaft is subjected to the vertical and horizontal loadings of two pulleys as shown. It is supported on two journal bearings at A and B which offer no resistance to axial loading. Furthermore, the coupling to the motor at C can be assumed not to offer any support to the shaft. Determine the maximum bending stress developed in the shaft.Composite and Reinforced Concrete BeamsComposite beams consisting of layers with fibers, or rods strategically placed to increase stiffness and strength can be “designed” to resist bending.The transformed-section method can be used to analyze composite beams.First step – determine transformation factor, n, by calculating the ratio of the modulus of the stiff material relative to that of the less stiff material:Second step – use transformation factor, n, to develop an equivalent beam consisting of the same material. Use the flexure formula to determine the stress at each point on the transformed beam. Then, use the transformation factor, n, as a multiplier to determine the stress on the actual beam.Alternative Approach to transformed-section method.First step – determine alternative transformation factor, n′, by calculating the ratio of the modulus of the less stiff material relative to that of the stiffer material.Second step – use alternative transformation factor, n′, to develop an equivalent beam consisting of the same material. Use the flexure formula to determine the stress at each point on the transformed beam. Then, use alternative transformation factor, n′, as a multiplier to determine stress on the actual beam.The transformed-section method can also be used for reinforced concrete beams6.127. Known: A reinforced concrete beam is made using 2 steel reinforcing rods. Find: Determine the maximum moment M that can be applied to the section. Data: Allowable tensile strengths: (σst)allow = 40 ksi and (σconc)allow = 3 ksiElastic moduli: Est = 29(103) ksi and Econc = 3.8(103) ksi Assumption: the concrete cannot support a tensile stress.Stress ConcentrationsNotches, holes, or abrupt changes in the outer dimensions of a member’s cross-section can lead to stress concentrations at the discontinuity when a bending moment is applied to the member.Inelastic BendingThe flexure equation is valid up to the yield point, and before this point is reached the material behaves in a linear-elastic manner.After yield, elasto-plastic behavior can be assumed to determine normal stresses acting in a member.For both elastic and plastic analyses, three conditions must be met:Linear Normal Strain DistributionResultant Force Equals ZeroResultant Moment – must be equivalent to moment caused by stress distribution about the neutral axis.6-160. Determine the plastic section modulus & shape factor of the beam’s cross-section.6.167. Determine the plastic moment, MP, that can be supported by a beam having the cross section shown. σY = 30 ksi6.106. Known: The resultant moment acting on the aluminum strut has a magnitude of M = 520 N∙m and is directed as shown.Find: Determine the bending stress at points A and B. The


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