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1、精选优质文档-倾情为你奉上4-6 Inadequate Lateral Support As the distance between points of lateral support on the compression flange (l) becomes larger, there is a tendency for the compression flange to buckle laterally. There is no upper limit for l. To guard against the buckling tendency as l becomes larger, h
2、owever, the ASDS provides that be reduced. This, in effect, reserves some of the beam strength to resist the lateral buckling. Figure 4-10b shows a beam that has deflected vertically with a compression flange that has buckled laterally. The result is a twisting of the member. This is called lateral-
3、torsional buckling. For simplicity we refer to this buckling mode of the beam as lateral buckling. Two general resistances are available to counteract lateral buckling: torsional resistance of the cross section and lateral bending resistance of the compression flange. The total resistance to lateral
4、 buckling is the sum of the two. The ASDS conservatively considers only the larger of the two in the determination of a reduced . The ASDS, Section F1.3, establishes empirical expressions forfor the inadequate lateral support situation. Tension and compression allowable bending stresses are treated
5、separately. The tensionis always 0.60. Only the compression is reduced. For typical rolled shapes, this is of no consequence because the shapes are symmetrical and the lowerof the two values will control. Note that the provisions of this section pertain to members having an axis of symmetry in, and
6、loaded in, the plane of their web. They also apply to compression on extreme fibers of channels bent about their mayjor axis. The ASDS provides three empirical equations for the reduced compression. The mathematical expressions that give an exact prediction of the buckling strength of beams are too
7、complex for general use. Therefore, the ASDS equations only approximate this strength for purposes of determining a reasonable. The that is finally used is the larger of the values determined from the applicable equations. The first two, ASDS Equations (F1-6) and (F1-7), give the value when the late
8、ral bending resistance of the compression flange provides the lateral buckling resistance. The third, ASDS Equation (F1-8), gives when the torsional resiastance of the beam section provides the primary resistance to lateral buckling. In no case should be greater than 0.6for beams that have inadequat
9、e lateral support. The equations that will be applicable will depend on the value of the ratio l/, wherel=distance between points of lateral support for the compression flange (in.)=radius of gyration of a section comprising the compression flange plus one-third of the compression web area taken abo
10、ut an axis in the pane of the web (in.), as shown in Figure 4-11Here is a tabulated quantity for rolled shapes (see the ASDM, Part 1), and may be considered a slenderness ratio of the compression portion of the beam with respect to the y-y axis. The equations for are as follows:Where= a liberalizing
11、 modifying factor whose value is between 1.0 and 2.3 that accounts for a moment gradient over the span and a decrease in the lateral buckling tendency; may be conservatively taken as 1.0; see ASDS, Section F1.3, for details d= depth of cross section (in.)=area of compression flange ()Figure 4-12 dep
12、icts the decision-making process for the calculation of. Note that one will use ASDS Equations (F1-6) and (F1-8) or ASDS Equations (F1-7) and (F1-8). The larger resulting is used. Note that Table 5 of the Numerical Values section of the ASDS provides the following numerical equivalents for A36 steel
13、 (=36ksi):4-7 Design of Beams for Moment The basis for moment design is to provide a beam that has a moment capacity() equal to or greater than the anticipated maximum applied moment M. The flexure formula is used to determine a required section modulus S:The section modulus on which the selection w
14、ill be based is assumed to be the strong-axis section modulus. The Allowable Stress Design Selection Table ( Table) in the ASDM, Part 2, can be used to make this selection. It lists common beam shapes in order of decreasing section modulus. This table also lists the resisting momentof each section.
15、The value ofis calculated using an allowable bending stressof 23.8 ksi (or 23.76 ksi) rather than the rounded value of 24.0 ksi. This may cause some small inconsistencies in calculations and results.4-8:Beam Design for MomentBased on the foregoing examples,a general procedure may be established for
16、the design of beams for moment .1. Establish the condition of load,span,and lateral support.This is best done with a sketch.Establish the steel type.2. Determine the design moment.If necessary,complete shear and moment diagrams should be drawn.An estimated beam weight may be included in the applied
17、load.3. The beam curves should be used to select an appropriate section when possible.As an alternative, must be estimated and the required section modulus determined.4. After the section has been selected,recompute the design moment,including the effect of the weight of the section.Check to ensure
18、that the section selected is still adequate.5. Check any assumption that may have been made concerning or.6. Be sure that the solution to the design problem is plainly stated.4-9:Shear in BeamsExcept under very special loading conditions,all beams are subjected to shear as well as moment.In the norm
19、al process of design,beams are selected on the basic of the moment to be resisted and then checked for shear.Shear rarely controls a design unless loads are very heavy(and,possibly,close to the supports)and/or spans are very short.From strength of material,the shear stress that exists within a beam
20、may be determined from the general shear formula Where shear stress on a horizontal plane located with reference to the neutral axis(ksi) V=vertical shear force at that particular section(kips) Q=statical moment of area between the plane under consideration and the outside of the section,about the n
21、eutral axis()I=moment of inertia of the section about the neutral axis()b=thickness of the section at the plane being considered(in.)This formula furnishes us with the horizontal shear stress at a point,which,as shown in any strength of material text,is equal in intensity to the vertical shear stres
22、s at the same point in a beam.Deflection When a beam is subjected to a load that creates bending,the beam must sag or deflect,as shown in Figure 4-22.Although a beam is safe for moment and shear,it may be unsatisfactory because it is too flexible.Therefore,the consideration of the deflection of beam
23、s is another part of the beam design process.Excessive deflections are to be avoided for many reasons.Among these are the effects on attached nonstructural elements such as windows and partitions,undesirable vibrations,and the proper functioning of roof drainage systems.Naturally,a visibly sagging b
24、eam tends to lesson ones confidence in both the strength of the structure and the skill of the designer.To counteract the sag in a beam,an upward bend or camber may be given to the beam.This is commonly done for longer beams to cancel out the dead load deflection and,sometimes,part of the live load
25、deflection.One production method involves cold bending of the beam by applying a point load with a hydraulic press or tam.For shorter beam,which are not intentionally cambered,the fabricator will process the beam so that any natural sweep within accepted tolerances will be placed so as to counteract
26、 expected deflection.Normally,deflection criteria are based on some maximum limit to which the deflection of the beam must be held.This is generally in terms of some fraction of the span length.For the designer this involves a calculation of the expected deflection for the beam in question,a determi
27、nation of the appropriate limit of deflection,and a comparison of the two.The calculation of deflections is based on principles treated in most strength-of-materials texts.Various methods are available.For common beams and loadings,the ASDM,part 2,Beam Diagrams and Formulas,contains deflection formu
28、las,The use of some of these will be illustrated in subsequent examples.The deflection limitations of specifications and codes are usually in the form of suggested guidelines because the strength adequacy of the beam is not at stake.Traditionally,beams that have supported plastered ceilings have bee
29、n limited to maximum live load deflections of span/360.This is a requirement of the ASDS,Section L3.1.The span/360 deflection limit is often used for live load deflections in other situation.It is common practice,and in accordance with some codes,to limit maximum total deflection(due to live load an
30、d dead load) to span/240 for roofs and floors that support other than plastered ceilings.The ASDS Commentary ,Section L3.1,contain guideline of another nature.It suggests:1. The depth of fully stressed beams and girders in floors should,if practicable,be not less than times the span.2. The depth of
31、fully stressed roof purlins should,if practicable,be not less than times the span,except in the case of flat roofs.Further,it recommends that where human comfort is the criterion for limiting motion,as in the case of vibrations,the depth of a steel beam supporting large,open floor areas free of part
32、itions or other sources of damping should be not less than 1/20 of the span.Since the moment of inertia increases with the square of the depth,the guidelines for minimum beam depth limit deflections in a general way.The ASDS Commentary,Section K2,also contains a method for checking the flexibility o
33、f roof systems when ponding,the retention of water on flat roofs,is a consideration.Holes in BeamsBeams are normally found as elements of a total structural system rather than as individual,isolated entities.They are penetrated by mechanical and electrical systems,are enveloped by nonstructural elem
34、ents,and must be connected to other structural members.They must sometimes be cut to provide clear areas.The problem of holes in beams is a common one. Among the more evident effects of holes in beams(or,generally,any decrease in cross-sectional area)is capacity reduction.Two such reductions may be
35、readily identified.Holes in beam webs reduce the shear capacity.Holes in beam flanges reduce the moment capacity.The ASDS is not specific concerning a recommended design procedure for beams with web holes.The common procedure of reducing shear capacity in direct proportion to web area reduction is a
36、n oversimplification for other than small holes.Where larger openings occur,it is common practice to reinforce the beam web by welding stiffening members around the perimeter of the hole. Some general rules may be stated with regard to web holes.They should be located away from area of high shear.Fo
37、r uniformly loaded beams,web holes near the center of the span will not be critical.Holes should be centered on the neutral axis to avoid high bending stressed.The holes should be round or have rounded corners(for rectangular holes)to avoid stress concentrations.The Cutting of web holes in the field
38、 should not be allowed without the approval of the designer.With respect to holes in the flange of beams,it is the moment capacity that is affected.The cross-sectional property that governs moment capacity is moment of inertia I.The web of a wide-flange beam contributes very little to the moment of
39、inertia,and the effect of web holes on moment capacity may be neglected.The effect of web holes on moment capacity may be neglected.The effect of flange hole,however,is to reduce the moment of inertia.The calculation of the reduction is accomplished by subtracting from the gross moment of inertia th
40、e quantity for each hole,where A=cross-sectional area of the hole(diameter*flange thickness)()d=distance from the neutral axis to the centroid of the hole(in.)The neutral axis shifts very little where holes exist in only one flange and may be assumed to remain at the centroid of the gross cross sect
41、ion.For our discussion no distinction will be made as to whether the flange is tension or compression,even though beams are usually controlled by the strength of the compression flange.Finally,it is the conservative practice of some designer to consider both flange to have holes（in a symmetrical pat
42、tern)even though only one does.It is generally agreed that flange holes for bolts do not reduce the moment capacity of beams to the extent indicated by the reduced moment of inertia as described in the preceding paragraph.The ASDS,in section B10,states that no reduction in moment of inertia shall be made for bolt holes in either flange provided that Where =net flange area =gross flange areaIf Then the flexural properties properties of the member shall be based on an effective tensive flange area ,where 专心-专注-专业