Further development of Ko displacement theory for deformed shape predictions of nonuniform aerospace structures

Original publisher: Edwards, Calif. : National Aeronautics and Space Administration, Dryden Flight Research Center, [2009] OCLC Number: (OCoLC)672029577 Excerpt: ... case expressions for uniform beams, ( ). The modified slope and deflection equations for slightly c = c i − 1 i nonuniform cantilever beams are derived in the subsequent sections. SLIGHTLY NONUNIFORM CANTILEVER BEAMS When the cantilever beam is slightly tapered ( that is, ), the logarithmic terms in the slope ( c c ) → 1 i i − 1 and deflection equations ( eqs. ( 14 ) and ( 16 ) ) will approach zero ( that is, ). Therefore, the log ( c c ) → 0 e i i − 1 perturbation method must be used to expand the logarithmic term,, in the neighborhood of log ( c c ) e i i − 1 to obtain nonzero mathematical expressions, so that the slope and deflection equations can ( c c ) ≈ 1 i i − 1 degenerate into those for the uniform cantilever beam case,. ( c c ) = 1 i i − 1 Slope Equations For a slightly tapered beam, the logarithmic term,, in slope equation ( 11 ) can be log ( c c ) e i i − 1 expanded in the neighborhood of ( that is, ). Carrying out the expansion up to ( c c ) ≈ 1 log ( c c ) ≈ 0 i i − 1 e i i − 1 2 the second-order terms in causes the logarithmic term,, to take on the following log ( c c ) ( c − c ) e i i − 1 i − 1 i form as approaches [ ] ( see Appendix B for details of mathematical expansions; c ( c c ) → 1 c i − 1 i i − 1 i refs. 2, 6 ): c c c − c i i i − 1 i → 1; log ≈ ( c − 3c ) e i i − 1 ( 17 ) 2 c c 2c i − 1 i − 1 i − 1 Substitution of equation ( 17 ) into slope equation ( 11 ) yields the following slope equation for the slightly nonuniform cantilever beam ( ref. 2 ), with the ( ) factor in the denominators eliminated: c − c i − 1 i ⎡ ⎤ ⎛ ⎞ l c Δ i; c → c θ ε ε nθ tan = 2 − + + ta ( 18 ) ⎢ ⎥ ⎦ i − 1 i i i − 1 i i − 1 ⎜ ⎟ ⎠ 2c c ⎝ i − 1 i − 1 ⎣ Applying the descending recursion relationship causes slope equation ( 18 ) to become i ⎡ ⎤ ⎥ ⎛ ⎞ c Δl 1 j ⎢ tanθ = 2 − ε + ε + tanθ; c → c ( 19 ) ⎜ ⎟ ⎠ ∑...