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1、Chap 7.Deflections of Statically Indeterminate Beams,2023-05-27,2,7.1 introduction,Equil.of force&moment known ext.force&reations“statically determinate”In cases of statically indet.prob.,ex.Fig.7.1 3 steps approach i)force&moment equil.ii)geometric compatibility iii)force-deform.relation,2023-05-27
2、,3,7.2 Deflection of Statically Indetermination Beams,M(x)includes unknown reactionsprinciple of superposition applicable,Ex 7.1),a)wall reactionsb)|M|maxc)v at x=a.Neglect weight.,2023-05-27,4,4 unknowns,2023-05-27,5,2023-05-27,6,L,-a,2023-05-27,7,2023-05-27,8,Which is the largest among MC,MA,MB?:d
3、ependent upon a,b,L,2023-05-27,9,Ex 7.2),A simply supported beam.reactions&S.F.D.,B.M.D?,2023-05-27,10,)Assume that the weight is included in q0.,Not indep.,2023-05-27,11,2023-05-27,12,B.C.at x=0,v=0 at x=L,v=0,v=0 at x=2L,v=0,L,sym,2023-05-27,13,Good!,S.F.D,B.M.D,4,24,3,2,2,2023-05-27,14,Ex 7.3),20
4、23-05-27,15,2023-05-27,16,Ex 7.4),Find end loads for given,2023-05-27,17,),2023-05-27,18,2023-05-27,19,L2,2023-05-27,20,2023-05-27,21,7.3 Method of Superposition,Ex 7.5),max.defl.and the location?,2023-05-27,22,)F.B.D,Table G.1.3,Table G.1.2,2023-05-27,23,From force&moment equil.,2023-05-27,24,2023-
5、05-27,25,2023-05-27,26,),Table G.1.3,Table G.1.2,2023-05-27,27,2023-05-27,28,Ex 7.6),The same as Ex7.2 with L=2L,with,2023-05-27,29,7.4 Displacement Method of Beams,For a beam basic knowledge-see Fig.7.9,2023-05-27,30,Equilibrium,Force-deformation,EI constant,v positive deflection of neutral axis of
6、 the beam,Basic equations,2023-05-27,31,For a beam made up of beam elements-see Fig.7.10,2023-05-27,32,In a general system,different cross-section&material but a single neutral axis,2023-05-27,33,3 steps approach,simultaneous linear algebraic equations for unknown node deflections&slopes,We need for
7、ce-deform.relation of a beam element,1.Applied concentrated loads,moments,and reaction loads from the supports act only at nodes,i.e.,at the junction point of two beam elements or at the endpoints.2.A constant distributed transverse load applied to an element is denoted by q(lb/in or N/m).3.The bend
8、ing modulus and the length of an element are denoted by EI(lbin2 or Nm2)and L(in or m),2023-05-27,34,2023-05-27,35,7.5 Derivation of Equations Relating Element End Shear Forces&Bending Moments to Element End Deflections&Slopes,valid along this beam element,?,2023-05-27,36,q=constant,B.C.,2023-05-27,
9、37,2023-05-27,38,2023-05-27,39,mean,Ele.End,Ele.End,Eq.,Eq.,(Fig.7.12),2023-05-27,40,7.6 Application to Single Element Problems,Ex 7.7),Neglect the weight.,2023-05-27,41,)A single element with B.C.,4 equations.,2023-05-27,42,2+L,2023-05-27,43,Ex 7.8),2023-05-27,44,Ex 7.9),Wall reactions&deflection a
10、t center?,),4 equations.,2023-05-27,45,2023-05-27,46,Ex 7.10),Reaction and slope at the right end of the load?,2023-05-27,47,2023-05-27,48,Ex 7.11),Wall reaction as specific and?,2023-05-27,49,7.7 Application of the Force-Deformation Relations-Two Element Beam Problems.,Ex 7.12),Reactions&Deflection
11、s at B?(neglect the weight.),2023-05-27,50,)statically indeterminate with concentrated force at mid point two element beam,With B.C.&from force&Moment equilibrium at nodes,Node element at node 2,2023-05-27,51,For element(1),For element(2),From,2023-05-27,52,For element(1),For element(2),From,2023-05
12、-27,53,And For element(2),From,2023-05-27,54,7.8 Use of BEAMMECH,Ex 7.13),Steel beam w828 with(a)deflection curve.(b)B.M.D.&S.F.D.(c)maximum bending stress,d,2023-05-27,55,)neglect the weight 3-element beam,5 unknowns,2023-05-27,56,From equilibrium of node 2,3,4,5 eqs.,2023-05-27,57,2023-05-27,58,20
13、23-05-27,59,2023-05-27,60,2023-05-27,61,2023-05-27,62,Moment,2023-05-27,63,Ex 7.14),Neglect the weight E=200GPa,I=3510-6m4maximum deflection&deflection curve S.F.D.,B.M.D.?,2023-05-27,64,),5 unknowns,5 eqs.,Solvable!,2023-05-27,65,Ex 7.15),maximum deflection&deflection curve at P=0,1,2,3 kips?,2023-05-27,66,),8 unknowns,8 eqs.,2023-05-27,67,Ex 7.16),With different EI,
