BEAM-COLUMNS 6 THEORY OF ELASTIC BTABILITY 7 To simplify this equation the following additional notation will be used: Again,the first factor is the slope produced by the lateral lond Q acting alone at the center of the beam and the second factor represents the effect 以1P (1-13) of the axial load P.Values of the factor A(u)are given in Table A-2 “=2=2VE7 of the Appendix. Then Eq.(g)becomes By using Eq.(1-11)we obtain the maximum bending moment as follows: 8贺7--%7x侧 &=481 (1-140 M--EI dy八 k以QI tan u dr* 2Ptan2-4 (1-18) The first factor on the right-hand side of this equation represents the deflection which is obtained if the lateral load Q acts alone.The second The maximum bending moment is obtained in this case by multiplying factor,x(u),gives the influence of the longitudinal force P on the deflec- the bending moment produced by the lateral load by the factor (tan u)/u. tion 8.Numerical values of the factor x(u)for various values of the The value of this factor,as well as the previous trigonometric factors quantity tare given in Table A-1 in the Appendix.By using this table, A()and x(u),approaches unity as the compressive force becomes the deflections of the bar can be calculated readily in each particular case smaller and smaller and increases indefinitely when the quantity u from Eq.(1-14). approaches r/2,that is,when the compressive foroe approaches the When P is small,the quantity u is also small [see Eq.(1-13)]and the critical value given by Eq.(1-15). factor x(u)approaches unity.This can be shown by using the series 1.4.Several Concentrated Loads.The results of the previous article will now be used in the more general case of several lateral loads acting on 如=+号+答+… the compressed beam.Equations (1-7)and (1-8)show that for a given longitudinal force the deflections of the bar are proportional to the and retaining only the first two terms of this series.It is seen alao that lateral load Q.At the same time the relation between defections and x(u)becomes infinite when u approaches x/2.When u=x/2,we find the longitudinal force P is more complicated,sinee this force enters into from Eq.(1-13) the trigonometric functions containing k.The fact that deflections are p linear functions of Q indicates that the principle of superposition,which (1-15) is widely used when lateral loads act alone on a beam,can also be applied in the case of the combined action of lateral and axial loads,but in a some- Thus it can be concluded that when the axial compressive force what modified form.It is seen from Eqs.(1-7)and (1-8)that,if we approaches the limiting value given by Eq.(1-15),even the smallest increase the lateral load Q by an amount Q,the resultant deflection is lateral load will produce considerable lateral deflection.This limiting obtained by superposing on the deflections produced by the load Q the value of the compressive force is called the critical load and is denoted deflections produced by the load Q,provided the same axial force acta by P By using Eq.(1-15)for the critical value of the longitudinal on the bar. force,the quantity u[see Eq.(1-13)]can be represented in the following It can be shown that the method of superposition can be used also if form: several lateral loads are acting on the compressed bar.The resultant (1-16) deflection is obtained by using Eqs.(1-7)and (1-8)and superposing the separate deflections produced by each lateral load aeting in combination with the total axial force.Take the case of two lateral loads Q1 Thus u depends only on the magnitude of the ratio P/P To find the slope of the deflection curve at the end of the beam,we and Q at the distances c and cs from the right support (Fig.1-4). Proceeding as in the previous article,we find that the differential equation substitute c =1/2 and z=0 into Eq.(1-9),which gives of the deflection curve for the left portion of the beam(sI-ca)is Q1 r路=-9z-9a-Py (a) Q2(1-c0s4) Q =16E7w2co8“ =16E7A (1-170 Now consider the loads Q,and Q:acting separately on the compressed