钢结构基本原理课件 (7).ppt

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1、Strength and Stiffness of BeamBasic Principles of Steel Structures3.1.2 Stiffness of Flexural Members3.1.1 Strength of Flexural Members3.1.1 IntroductionStrength of Flexural Members1 1Beams are widely used as structural elements,in almost all kinds of buildings,bridges and other engineering structur

2、es.Beams are used mainly for carrying transverse loads,which mainly generate moments and shears in a beam sections.2 23 3Axial forces can also exist in beams.However,it can be neglected when transverse loads dominate.4 4Since beams are mainly used for carrying transverse loading,higher flexural stif

3、fness and capacity are desirable.As the material mechanics know:In the elastic stage,when the bending moment acting on the component section wraps around the spindle x axis,the maximum positive stress at the edge of the Member section is:Bending strength 1.1.3.1.1 Strength of Flexural Members1 1 Wnx

4、 is the net sectiona modulus about the x-axis2 2 Mx is moments about x and y axis.Fig.3.1 The normal stress distribution on the beam section in each load stage Bending strength 1.1.3.1.1 Strength of Flexural Members1 1When reaches the steel yield point fy,(Fig.3.1b)the member section is in the elast

5、ic limit state.My=Wx fy 3 3when the entire section of the member is completely plastic,the section reaches the maximum bending capacity,plastic bending moment Mp=Wp fy (Fig.3.1d)4 4Then the section forms plastic hinge and reaches the plastic limit state 2 2As Mx further increases,the section of the

6、member begins to develop plasticity inwards and enters the elastoplastic state(Fig.3.1c)Bending strength 1.1.3.1.1 Strength of Flexural Members Wp is the section plastic modulus of the section to the x-axis.Generally,xp=Mp /My .x and y are plasticity factors or shape factors for x and y axis(for H-s

7、hapes,x=1.05,y=1.20;for box sections,x=y=1.05;for other shape,see Table.3.1);f is the flexural design strength.Table.3.1 plasticity adaptation factors3.1.1 Strength of Flexural MembersThe bending strength of the beam shall be satisfied：For beams bent in both directions,the strength shall be satisfie

8、d：Bending strength 1.1.3.1.1 Strength of Flexural Members1 1 My:Moments about x and y axis,Wny:Net section modulus about y 2 2 plasticity factors is not considered for beams requiring fatigue failure calculation.x=1.0,y=1.0 The shear strength formula:V Shear force of section.Ix Moment of inertia of

9、the gross section around the strong axis.S Area moment.tw Web thickness.fv Design value of shear strength of steel.When the member has shear force in both directions of principal axis:At the same time,the checking conditions should be satisfied:Shearing strength 2.2.3.1.1 Strength of Flexural Member

10、sFig.3.2 Shear stress As shown in Fig.3.3,there is no bearing stiffener at the concentrated load of the beam,or there is a moving concentrated load.Fig.3.3 Local compressive stress distributionLocal compressive stress 3.3.3.1.1 Strength of Flexural Members3.1.1 Strength of Flexural Members1 1The web

11、 of the beam will bear the local compressive stress generated by the concentrated load.The local compressive stress is greatest at the junction of the beam web and the upper flange,and decreases to zero at the lower flange,as shown in Fig.3.3b.2 23 3 In fact,the local compressive stress is not unifo

12、rmly distributed along the longitudinal direction of the beam.However,to simplify the calculation,assume that the local compressive stress is evenly distributed in the range of lz.Local compressive stress 3.3.Calculation formula:3.1.1 Strength of Flexural MembersLocal compressive stress 3.3.1 1F Con

13、centrated load.Consider dynamic coefficients for dynamic loads.3 3lz lz is the assumed distribution length of the concentrated load on the upper edge of the calculated height of the web,calculated as follows:2 2 Concentrated load amplification factor,Crane beam =1.35,Other beams =1.0,Beam support =1

14、.0.Beam end:Concentrated load in midspan:3.1.1 Strength of Flexural MembersLocal compressive stress 3.3.1 1 a is the supporting length of the concentrated load along the span direction,the crane:a=50mmhy is the distance from the concentrated load to the edge of the calculated height of the web.hr is

15、 the height of the track.2 23 3b is the distance from the beam end to the outer edge of the support plate,if b is greater than 2.5hy,take 2.5hy.Reduced stress 4.4.3.1.1 Strength of Flexural Members1 1Shear and bending moments are generally acting on the beam at the same time.Sometimes localconcentra

16、tion forces are also acting.2 2Although the normal stress and the shear stress are not the maximum,it may be more dangerous under the simultaneous action of them.3 3According to the fourth strength theory,in the complex stress state,if the reduced stress at a certain point reaches the yield point of

17、 steel in one-way tension,the point will enter the plastic state.Reduced stress formula:,c are the normal stress,shear stress and local compressive stress simultaneously generated at the same point on the calculated height edge of the web.In Net section moment of inertia y1 Distance from calculation

18、 point to neutral axis of beam1 Increase coefficient of strength design value，When and c are same sign,take 1=1.2；When and c are opposite sign or c=0,take 1=1.1；3.1.1 Strength of Flexural MembersReduced stress 4.4.(3.10)The maximum deflection in the beam caused by the standard value of the load(with

19、out considering the partial factor and dynamic coefficient of the load).Allowable deflection value of beam.3.1.2 Stiffness of Flexural Members1 1The stiffness of the beam is measured by the deflection under standard load 2 2The insufficient rigidity of the beam will affect normal use or appearance 3 3The vibration of the beam under dynamic influence can also be controlled by limiting the deformation of the beam Checking formula of beam stiffness：:Stiffness 1.1.