Introduction to Aircraft Structural Analysis (Elsevier Aerospace Engineering)

132 CHAPTER 5 Energy Methods butthedirectloadshaveavalueP/2.Thetotalcomplementaryenergyofthesystemis(againignoring shearstrains) ...

5.4Application to the Solution of Statically Indeterminate Systems 133 Fig.5.15 Distribution of bending moment in a doubly symme ...

134 CHAPTER 5 Energy Methods Fig.5.16 Determination of bending moment distribution in a shear- and direct-loaded ring. whichgive ...

5.4Application to the Solution of Statically Indeterminate Systems 135 Assumingthatthefuselageframeislinearlyelastic,wehave,from ...

136 CHAPTER 5 Energy Methods Fig.5.17 Determination of bending moment distribution in an antisymmetrical fuselage frame. carried ...

5.4Application to the Solution of Statically Indeterminate Systems 137 orassuminglinearelasticity ∫ half-frame M EI ∂M ∂SA ds= ∫ ...

138 CHAPTER 5 Energy Methods Fig.5.18 Distribution of bending moment in frame of Example 5.6. Infact,thequestionofwhetherastruct ...

5.5Unit Load Method 139 fromwhich (^) C= ∑k i= 1 λi ∂Fi ∂Pf asbefore (5.19) IfinsteadofthearbitrarydummyloadPfwehadplacedaunitlo ...

140 CHAPTER 5 Energy Methods Fig.5.19 Deflection of a bent rod. First,considerthedisplacementinthedirectionparalleltothexaxis.Fr ...

5.6 Flexibility Method 141 Similarly, (^) y=wl^4 ( 11 24 EI + 1 2 GJ ) (^) z=wl^4 ( 1 6 EI + 1 2 GJ ) 5.6 FlexibilityMethod..... ...

142 CHAPTER 5 Energy Methods Weareassumingthatthetrussislinearlyelasticsothattherelativedisplacementofthecutendsof thememberBD(i ...

5.6 Flexibility Method 143 Table 5.6 Member Lj(m) F0,j F1,j F0,jF1,jLj F1,^2 jLj Fa,j AB L 0 −0.71 0 0.5L +0.40P BC L 0 −0.71 0 ...

144 CHAPTER 5 Energy Methods Fig.5.21 Statically indeterminate truss of Example 5.9. in which (^) ADandvCare, respectively, the ...

5.6 Flexibility Method 145 Table 5.7 F0,jF1,j F0,jF1,j F1,j(X 1 ) Member Lj F0,j F1,j(X 1 ) F1,j(R 2 ) (X 1 )Lj (R 2 )Lj F^2 1,j ...

146 CHAPTER 5 Energy Methods Due to the temperature rise, the increase in length of the member BC is 3× 103 × 30 × 7 × 10 −^6 =0 ...

5.7Total Potential Energy 147 Then,fromEq.(i), X 1 =−525N The forces,Fa,j, in the members of the complete truss are given in the ...

148 CHAPTER 5 Energy Methods ofalltheloadsis V= ∑n r= 1 Vr= ∑n r= 1 (−Pr (^) r) andtheTPEofthesystemisgivenby TPE=U+V=U+ ∑n r= 1 ...

5.8 The Principle of the Stationary Value of the Total Potential Energy 149 Fig.5.24 States of equilibrium of a particle. Itmaya ...

150 CHAPTER 5 Energy Methods Fig.5.25 Approximate determination of beam deflection using total potential energy. inwhichvBisthed ...

5.10 The Reciprocal Theorem 151 fromwhich vB= 2 WL^3 π^4 EI =0.02053 WL^3 EI (iv) Theexactexpressionforthemidspandisplacement[Re ...