232 Calculus and Structures

Transcription

1 3 Calculus and Structures

2 CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt

3 Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE SLOPE METHOD Te following teorem provides te justification for Metod 3, te slope metod. Teorem 1: If V ) is continuous at and a continuous force of density ) acts on te beam d M ) ten V ) or V ) M ) c Proof: Consider te portion of te beam between an arbitrary value of = and = + for small wit sear forces and bending moments sown in Fig. 1. V ) ) M + ) M ) Te sum of te moments about is, V + ) Fig. 1 M ) M ) V ) *) k 1) Note from te mean value teorem for Integrals in Section 3.7 Te total force acting on te interval [, + ] equals ). From te Mean Value Teorem for Integrals derived in Teorem of Capter 3, *) = ) were * is a point on [, +] and k is te distance from te center of gravity of te force distribution to point. Te magnitude of k is clearly less tan. Algebraically rearranging tis equation we get, M ) M ) V ) V ) V ) *) k) ) But since, we ave assumed V ) is continuous at, V ) V )) o ) and *) k o ) wy?) and Eq. reduces after eliminating te o) term to, 34 Calculus and Structures

4 Section 17.1 M ) M ) V ) It follows from te derivative macine tat V ) M ' ) and tus, 17. THE AREA METHOD d M ) V ) for any value of. Te net teorem provides te justification for Metod, te Area Metod wic we ave renamed te Calculus Metod. Teorem : M ) M ) = [signed area under te V) curve from 1 to ] or 1 M ) M ) V ). 1 1 Proof: Consider te grap of V ) from 1 to sown in Fig.. Let te signed area under te curve from 1 to be A 1,), anoter function of. y Let us find te derivative of A 1,) at in te usual way, Let = + and rewrite Eq. 3 to get, + 1 * A, ) A, ) m o ) 3) 1 1 Fig. or, A 1, ) A 1, ) m o ) A 1, ) A 1, ) o ) m 4) But A 1, ) A 1, ) is te area under te V ) curve from to + as sown in te detail of Fig.. Note tat tere is a value * were * suc tat te area between, and + equals te area of te rectangle, V *), i.e., Calculus and Structures 35

5 Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS A, ) A, ) V *) 1 1 Replacing tis in Eq. 4 and canceling, o ) V *) m 5) o ) If we now let in a limit sense, ten *, and, and Eq. 5 becomes, Since is arbitrary, da 1, ) V ) m V da, ) ) 1 In oter words, V ) is te slope derivative) of te signed area function, A 1, ), at =. So da, ) we see from Teorem 1 tat bot 1 d M ) and equal V ). But it can be sown tat if two functions ave te same derivative i.e., slope) ten tey must be eiter equal or differ by only a constant. So we ave found tat, M ) A 1, ) c 5) Setting = 1 and noting from Fig. 1 tat A 1, 1 ), it follows from Eq. 5 tat M 1 ) = + c or c = M 1 ). Terefore, M ) M ) A, ) 1 1 were A 1,) is te signed area under te V ) curve from 1 to. Letting = we ave te result of our teorem, M ) M ) A, ) = V ) THE RELATIONSHIP BETWEEN SHEAR FROCE AND DENSITY Teorem 3: If a beam eperiences a continuous force wit density ), ten dv ) ). Proof: Consider te portion of te beam from to + as sown in Fig.1. Te balance of forces is: 1 36 Calculus and Structures

6 Section 17.3 V ) V ) *) were * is some point in te interval, +) suc tat, again using te Mean Value Teorem for Integrals Teorem 1 from Capter 3), *) is te weigt of te beam area under te ) curve) on tis interval. Suc a point is guaranteed to eist by a more advanced teorem of calculus. Terefore, V ) V ) *) )) ) But since *) )) o ) wy? Terefore, V ) V ) )) From te derivative macine, dv ) ) Let us now apply tis to finding te bending moment of te beam under a continuous but constant load in Capter 8 by te slope metod, i.e., Metod 3. d M ) Since V ) it follows tat M ) V c ) In Capter 7 we found tat V = 5 1 so tat, M ) 5 1) c 5 1 c 5 5 c But M )= so tat = + c or c = and agreeing wit our result. M ) = Calculus and Structures 37

7 Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS We now ave te startling result tat we can find te area under te V ) curve from to 1 by merely evaluating te bending moment, M ) M 1 ) witout needing to compute te area directly. In oter words if V ) is a continuous and smoot function, M 1 ) M ) 1 V ) were M ) is any anti-derivative of V ). Since te constants for te slope metod c will be te same for bot M ) and M 1 ), tey will cancel out. Tis will be very elpful in applying te Calculus Metod to solving structures problems and for finding areas under curves witout aving to add up rectangles or trapezoids. In te net Capter we use tese ideas to find areas under curves witout aving to add rectangles. 38 Calculus and Structures

8 Section 17.3 Calculus and Structures 39