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1、Chapter 10 Statically indeterminate beams Mechanics of MaterialsAnalysis by the differential equations of the deflection curveWelcome to Mechanics of Materials,this section discusses the most fundamental method for analyzing statically indeterminate beams by the differential equations of the deflect
2、ion curve.Anlysis by the differential equations of the deflection curveProcedure for statically indeterminate beams:-Method of integrationI.Free body diagram;-Report the degree of static indeterminacy,select redundants.II.Equations of equilibrium;-Relate the remaining reactions to the redundants and
3、 the load;III.Writing the differential equation and integrating;-Obtain its general solution.IV.Evaluating the unknown redundants;-Apply boundary,continuity,symmetry or other conditions.V.Answer the Question;-Remaining reactions,internal forces,stresses,displacements.To analyze statically indetermin
4、ate beams by the differential equations of the deflection curve is the most fundamental method,this method is also known as*method of integration.*The procedure is essentially the same as that for a statically determinate beam(see Sections 9.2,9.3,and 9.4),consists of the following steps:*First step
5、 is always constructing the Free body diagram;next writing Equilibrium of forces and moments;and then Writing the differential equation and integrating to obtain its general solution.Differential equations of the deflection curve could be any of the three*:(1)the second-order equation in terms of th
6、e bending moment,(2)the third-order equation in terms of the shear force,or(3)the fourth-order equation in terms of the intensity of distributed load.and then*Evaluating the unknown redundant reactions by applying boundary and other conditions.Last step,Answer the Question-Typically calculate desire
7、d internal stresses,relevant displacements.Example 1:A propped cantilever beam AB of a length L supports a uniform load of intensity q,as shown in the figure.Analyze this beam by solving the second-order differential equation of the deflection curve(the bending-moment equation).Determine the reactio
8、ns.Now lets look at one example to see how to analyze statically indeterminate beams by this method.A propped cantilever beam AB of a length L supports a uniform load of intensity q,as shown in the figure.Analyze this beam by solving the second-order differential equation of the deflection curve(the
9、 bending-moment equation),and to determine the reactions of this beam.Solution:Use a four-step problem-solving approach.Overall free body diagram1st degree indeterminate 1.Conceptualize:-Construct the free-body diagram of the entire beam.-Report the degree of indeterminancy.Here we still use the fou
10、r-step problem solving tehnique.*First,conceptualize,*construct the free-body diagram of the entire beam*and then report the degree of indeterminacy.Because the load on this beam acts in the vertical direction,there is no horizontal reaction at the fixed support.Therefore,the beam has three unknown
11、reactions(MA,RA,and RB).Only two equations of equilibrium are available for determining these reactions;therefore,the beam is*statically indeterminate to the first degree.2.Categorize:Redundant reaction:RB Bending moment M:The bending moment M at distance x from the fixed support is：-by considering
12、the equilibrium of the entire beam:(a)(b)The bending moment M in terms of load and redundant RB：Remaining reaction:RA and MA(c)xnext,categarize,To analyze this beam by solving the bending-moment equation,begin with a general expression for the moment.This expression is in terms of both the load and
13、the selected redundant.*Choose the reaction RB at the simple support as the redundant.*Then,by considering the equilibrium of the entire beam,*express the other two reactions,RA and MA in terms of RB.*The bending moment M at distance x from the fixed support is expressed in terms of the reactions as
14、*.This equation is obtained by constructing a free-body diagram of part of the beam and solving an equation of equilibrium*.*Substitute equation a into equation b to obtain*bending moment M in terms of the load and the redundant reaction.3.Analyze:Differential equation:The second-order differential
15、equation of the deflection curve:After two successive integrations:deflections of the beamslopes of the beam(e)(d)(f)next,analyze,*The second-order differential equation of the deflection curve is*.*After two successive integrations,the following equations are obtained for the*slopes and*deflections
16、 *of the beam.These equations contain three unknown quantities*C1,C2,and RB.3.Analyze:Boundary conditions:Apply these conditions to the equations for slopes(e)and deflections(f),Remaining reactions：Shear force;Bending moment;Internal stresses;Deflections.Equations of equilibriumnow let s look at the
17、*boundary conditions for this case:*Three boundary conditions pertaining to the deflections and slopes of the beam are apparent from an inspection of the beam.These conditions are(1)the deflection at the fixed support is zero,v(0)=0,(2)the slope at the fixed support is zero,v(0)=0 and(3)the deflecti
18、on at the simple support is zero,v(L)=0.*Apply these conditions to the equations for slopes and deflections,we can find that*C1=0,C2=0,and*RB=3ql/8 Thus,the redundant reaction RB is now known.With the value of the redundant RB established,*the remaining reactions RA and MA are*.As we can see from th
19、is example,once the redundant is known,all the remaining reactions can be found from equations of equilibrium.In effect,the structure has become statically determinate.*Therefore,internal forces,shear forces and bending moments,the stresses and deflections can be found by the methods described in pr
20、eceding chapters.4.FinalizeDifferential equationFourth-order differention equation in terms of qAlternative approaches：symbolically or numerically?This example presents the analysis of the beam by taking the reaction RB as the redundant reaction.An alternative approach is to take the reactive moment
21、*MA as the redundant.Then express the bending moment M in terms of MA,substitute the resulting expression into the second-order differential equation,and solve as before.*Still another approach is to begin with the fourth-order differential equation.The differential equation for a beam may be solved
22、 in symbolic terms only when the beam and its loading are relatively simple and uncomplicated,like this example.The resulting solutions are in the form of general-purpose formulas.However,in more complex situations,the differential equations must be solved numerically,using computer programs intended for that purpose.In such cases,the results apply only to specific numerical problems.EndThat is end of this video,next video is going to discuss the method of superposition that is applicable to a wide variety of structures.