山农双语材力6.ppt

Mechanics of Materials CHAPTER 6 DEFORMATION IN BENDING 61 Deformation in bending in engineering62 Approximate differential equation of the deflection curve 63 Integration method to determine the deformation in bending64 Superposition method to determine deformation in bending 65 Simple statically indeterminate beam66 How to decrease deformation in bending261 Deformation in bending in engineeringStudy range：Calculation of the displacement of the straight beam with equal sections in symmetric bending.Study object：checking rigidify of the beam；Solving problems about statically indeterminate beams（to provide complementary equations for the geometric-deformation conditions of the beam)341).Deflection：3、The relation between the angle of rotation and the deflection curve：1、Two basic displacement quantities of to measure deformation of the beamSmall deflection x2).Angle of rotation：2、deflection curve：w=f(x)62 Approximate differential equation of the deflection curvePvCq qC1y51、Approximate differential equation of the deflection curveFormula(2)is the approximate differential equation of the deflection curve.Small deformation(1)(2)6yyMM7For the straight beam with the same shape and equal section area,the approximate differential equation of the deflection curve may be written as the following form：81.Integration of the differential equation63 Integration method to determine the deformation in bending9Displacement conditions at the supports：Continuity conditions：Smooth conditions：2.Boundary conditions of the displacementPABCPD10ABlaCM11ALFCabBEAhDALFCabkBxy12Example 1 Determine the deflection curves、maximum deflections and maximum angles of rotation of the following equal-section straight beams.Set up the coordinates and write out the bending-moment equationWrite out the differential equation and integrate itDeterminate the integral constants by the boundary conditionsSolution：PLxf13Write out the equation of the deflection curve and plot its curveThe maximum deflection and the maximum angle of rotationxfPL14Solution：Set up the coordinates and write out the bending-moment equation xfPLa Write out the differential equation and integrate it Example 2 Determine the deflection curves、maximum deflections and maximum angles of rotation of the following equal-section straight beams.15Determine the integral constants by boundary conditionsPLaxf16Write out the equation of the elastic curve and plot its curveThe maximum deflection and the maximum angle of rotationPLaxfw-17ALFCabyxxBCxACBSample 3 Determine the deflection curves、maximum deflections and maximum angles of rotation of the following equal-section straight beams.Solution：18BCAC（1）（2）（3）（4）19BCAC20 ：21 x223 3、Rigidity conditions of the beamWhere is called the permissible angle of rotation；w is called the permissible deflection.In general we can do three kinds of calculations about the rigidity by theses conditions：、Check the rigidity：(In civil engineering，the strength often-plays an important role and the rigidity plays a secondary role except special cases.）、Design the dimension of the section；、Determine the permissible load。2364 Superposition method to determine deformation in bending1 1、Superposition of loads:The deformation of a structure due to the action of multi-loads is equal to the algebraic sum of deformation resulting from the separate action of each load.2、Superposition of structural forms（rigidization method of segment by segment）：）：24Solution:Decompose the loads as shown in the figureqqPP=+AAABBB CaaDetermine the deformations of the beam under the action of simple loads by looking up the table.Example 4 Determine the angle of rotation of point A and the deflection of point C by the principle of superposition.25qqPP=+AAABBB CaaSum up26www27www2829Example 5 5 Explanation of the superposition of structural forms(digitization method of segment by segment)：=+PL1L2ABCBCPL2w1w2EquivalencexfxfwPL1L2ABCRigidize the segment ACPL1L2ABCRigidize the segment BC PL1L2ABCMxfEquivalence30w?www31ww32w331、Treatment method：Combining the compatibility equation of deformation、physical equation with equilibrium equations to determine the whole unknown forces.Solution：Set up the primary beam Determine the degree of statically indeterminacy.The structure in which redundant constraints are substituted by reactionsprimary structure。=q0LABLq0MABAq0LRBABxf65 Simple statically indeterminate beam34Geometric equationcompatibility equation of deformation+q0LRBAB=RBABq0ABPhysical equationrelation between the deformation and forcesComplementary equationSolve other problems（reaction、stress、deformation etc.）35Geometric equationcompatibility equation of deformationSolution：Set up the primary beam=Example 10 Determine the reaction at end B in the structure as shown in the figure.LBCxfq0LRBABCq0LRBAB=RBAB+q0AB36=LBCxfq0LRBABCRBAB+q0ABPhysical equationrelation between the deformation and forcesComplementary equation Solve other problems（reaction、stress、deformation etc.）3766 How to decrease deformation in bendingRigidity：1 1.Decrease Decrease M Mmaxmax 2 2.Increase Increase I IZ Z38 Arrange reasonably external forces（include reactions of supports）and make the maximum bending moment M max as small as possible.PL/2L/2xM+PL/4PL/43L/4P=qLL/54L/5SymmetryMx3PL/16+MxqL2/10+39qLL/5qL/5qL/2L/2Mx+402qL502qL-Mx-+-322qL-Mx+-40RbhReasonable ration between the height and the width of the section of the wooden beam with rectangular sectionLi Jie of north-Song dynasty pointed out in the book of written in the year 1100 that the reasonable ratio of height and width(h/b)of a rectangular wooden beam is 1.5.TYoung pointed out in the book of written in 1807 that the strength reaches the maximum when the reasonable ratio of height to width of a rectangular woden beam is h/b=and the rigidity reachesthe maximum when h/b=.41Reasonable section in general case1 1、If areas of sections are the same,select the section with a larger modulus in bending.zD1zaa42zD0.8Da12a1z43For the I-shape section the method to determine the shearing stress is similar to that for the shape section.0.8a2a21.6a22a2zwhen444、Lateral buckling of the beam1).Critical load of the beam with rectangular section in pure bendingLMMxyz45Select the material of high strength to increase the permissible stress For the same kind of materials，differences of their“E”are small,but those of their“jx”are larger.So the strength can be improved by changing another same kind of material,but the rigidity can not be improved.For the different kind of materials，differences of their E and G are very large（steel:E=200GPa,copper:E=100GPa），），So the rigidity can be improved by selecting different Rind of materials.But This will cause an obvious change in the cost of materials!46