(1.9.1)--6.10 Chapter Sumary and Review材料力学材料力学.ppt

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1、 Mechanics of MaterialsChapter Sumary and ReviewChapter 6 Stresses in Beams(Advanced Topics)Welcome to Mechanics of Materials.This Chapter presented a number of specialized topics related to the bending of beams,including the analysis of composite beams,beams with inclined loads,unsymmetric beams,sh

2、ear stresses in thin-walled beams,and shear centers.This section gives some of the major concepts and findings presented in this chapter.1.Composite beams Moment-curvaturerelationship:Flexureformulasforcompositebeamsoftwomaterials：In the introduction to composite beams,specialized moment-curvature r

3、elationship and flexure formulas for composite beams of two materials were developed.Both materials must follow Hookes law,and the two parts of the beam must be adequately bonded so that they act as a single unit.The formulas here do not apply to reinforced concrete beams,which are not designed on t

4、he basis of linearly elastic behavior.Themodularratio:Themomentofinertiaofthetransformedsection:Thesimplifiedflexureformula:2.Transformed-section method A1A2The transformed-section method offers a convenient way of transforming the cross section of a composite beam into an equivalent cross section o

5、f an imaginary beam that is composed of only one material.With the introduction of the modular ratio n and the moment of inertia of the transformed section,a simplified flexure formula is given.3.Inclined load yzF Fz zF Fy yF F Analytical ProcedureSuperposition:Twoplanebendingareanalysedseparatelyan

6、dthecorrespondingresultsaresuperposed.Decomposition:DecomposetheloadalongthetwoCentroidalprincipalaxesofthecrosssectiontoobtaintwoorthogonalplanebending.If inclined loads act through the centroid of the cross section of beams with two axes of symmetry in the cross section,there will be no twisting o

7、f the beam about the longitudinal axis.For these beams,compute the bending stresses by resolving the inclined load into two componentsone acting in each plane of symmetry.The bending stresses were obtained from the flexure formula for each load component acting separately,and the final stresses obta

8、ined by superposing the separate stresses.*The stresses in the beam are very sensitive to slight changes in the direction of the load and to irregularities in the alignment of the beam itself.A general procedure for analyzing an unsymmetric beam:4.Bending of unsymmetric beams LocatethecentroidCandco

9、nstructasetofprincipalaxes;ResolvethebendingmomentMintocomponents;Thesuperpositionofthebendingstresses.Alternate procedure for analyzing an unsymmetric beam:A general procedure for computing normal stresses in unsymmetric beams acted on by any moment M is find the centroid,resolve the bending moment

10、 M into components,and superpose results of the flexure formula about the two principal centroidal axes to find normal stresses.An alternate approach is to use the generalized flexure formula with the original centroidal axes aligned with the beam cross section.5.Lateral load and shear centerC:thece

11、ntroidofthecrosssectiony,z:theprincipalcentroidalaxes*A lateral load acting on a beam produces bending without twisting only if it acts through the shear center.*The shear center(like the centroid)lies on any axis of symmetry。the shear center S and the centroid C coincide for a doubly symmetric cros

12、s section.6.Shear stresses in beams of thin-walled open cross sectionsThe shear stresses in beams of thin-walled open cross sections flow inward from the outermost edges,then down through the web,and finally outward to the edges of the bottom flange.Procedure for locating the shear center:7.Shear Ce

13、nters of Thin-Walled Open SectionsEvaluatetheshearstressesactingonthecrosssectionwhenbendingoccursaboutoneoftheprincipalaxes.Determinetheresultantofthosestresses.Theshearcenterislocatedonthelineofactionoftheresultant.Procedure for locating the shear center consists two main steps:first,evaluate the

14、shear stresses acting on the cross section when bending occurs about one of the principal axes;and second determine the resultant of those stresses.The shear center is located on the line of action of the resultant.8.Sections consisting of two intersecting narrow rectanglesAsimplelineofreasoning:The

15、 point of intersection of the two forces in the legs is the shear center.For all cross sections consisting of two thin,intersecting rectangles,the shear center S lies at the junction of the rectangles.9.Z SectionThe shear center of the Z-section coincides with the centroid.For any Z-section that is symmetric about the centroid,that is,any Z-section having identical flanges,same width and same thickness,the shear center of the Z-section coincides with the centroid.EndEnd of this section.