Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22 261 Constants definition: 3 3 0 0 2 , 5 6 , , 12, 12 A bh k A A I h I bh I h          Axially-loaded beam, Bar or Rod problem Static analysis Dynamic analysis 2 2 0 d u EA pb dx   2 2 0 2 0 d U EA bI U dx    Boundary conditions Clamped-Clamped (CC)     0 0, 0 u u L   Clamped-Clamped (CC)     0 0, 0 U U L   Exact solutions     2 pb u x L x x EA   0 0 2 n n n EA n EA f L bI L bI      Euler-Bernoulli (EB) beam Static analysis Dynamic analysis 4 4 0 d w EI qb dx   4 2 0 4 0 d W EI bI W dx    Boundary conditions Simply Supported-Simply Supported (SS)         2 2 2 2 0 0, 0 0, 0, 0 d w d w w w L L dx dx     Simply Supported-Simply Supported (SS)         2 2 2 2 0 0, 0 0, 0, 0 d W d W W W L L dx dx     Exact solutions   4 4 3 4 3 2 24 qbL x x x w x EI L L L          2 2 2 2 2 0 0 2 n n n EI n EI f bI bI L L       Timoshenko (Tim) beam Static analysis Dynamic analysis 2 2 2 2 0 0 d w d G qb dx dx d dw EI G dx dx                        2 2 0 2 2 2 2 2 0 0 d W d G bI W dx dx d dW EI G bI dx dx                           Boundary conditions Clamped-Clamped (CC)         0 0, 0 0, 0, 0 w w L L       Simply Supported-Simply Supported (SS)         0 0, 0 0, 0, 0 d d W W L L dx dx       Exact solutions     2 4 2 2 2 2 2 3 3 2 3 2 24 2 2 3 12 qbL x x qbL x x w x EI L G L L L qbL x x x x EI L L L                             2 2 4 4 2 2 2 4 0 0 , 1 , bI bI EI n EI n P Q R G G bI bI L L               Table 3 : List of one dimensional static and dynamic exact solutions.