Let us now consider a linear bridging law of the following type to represent the bridging action of the through thickness reinforcement: p p w3 0 b = + (4) The linear law particularizes to the Dugdale law 0 p p = (for 0 3 = b ) and to the proportional linear law p w3b = (for p 0 0 = ). In the results that follow, we non-dimensionalize the variables by the laminate thickness h ( w hW º , u hU º and x X hº ). Thus, the transverse displacement obeys: b W d 0 W W 2 2 2 2 2 4 4 + + = ¶ ¶ + ¶ ¶ x b x (5) For Timoshenko beam: (R c )E h (1 c )( R c ) 12 c R 2 l 3 2 l 2 l 2 l - - - - = b b ; E 12 h (1 c )(R c ) R b 3 2 l 2 l b - - = ; E 12p (1 c )(R c ) R d 0 2 l 2 l - - = ; (6a) For Euler-Bernoulli (E-B) beam: 2 l 12c =b ; E h b 3 12b = ; E p d 0 12 = ; (6b) The general solution to Eqn. 5 is: x b b x b b x b b x b b x 2 4 2 2 4 2 2 4 2 2 4 2 4 2 1 2 4 4 2 1 2 3 4 2 1 2 2 4 2 1 2 1 2 2 ( ) b b b b K e K e K e K e b d W - + - + - - - + - + - - - - - - + + + = - + . (7) There are three regimes to the solution behaviour that are independent of the boundary conditions, and they have been identified below ( Note: S = b3 h / (12 k G) ): · Case 1: b2 < 0 and b4 > 4b2 => Exponential behavior - For Timoshenko beam, this is true provided: S S E v + £ 1 2 r and ) )( ) (1 ) 3( (1 2 2 2 2 2 l l l l S c R c S c c - - - ³ - (8a) - For the E-B beam, the above condition is never satisfied. · Case 2: b2 > 0 and b4 > 4b2 => Oscillatory and non-decaying behavior - For Timoshenko beam, this is true provided: v / E S /(1 S ) 2 > + r and ) )( ) (1 ) 3( (1 2 2 2 2 2 l l l l S c c S c R c - - ³ - - (8b.1) - For the E-B beam, this condition is satisfied when: v / E 2 S R 2 ³ r (8b.2) · Case 3: b4 < 4b2 => Oscillatory with exponential decay behavior - For the Timoshenko beam, this is true when: ) )( ) (1 ) 3( (1 2 2 2 2 2 l l l l S c c S c R c - - < - - (8c.1) - For the E-B beam, this is true when: v / E 2 S R 2 < r (8c.2)
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