Issue 29

F. Tornabene et alii, Frattura ed Integrità Strutturale, 29 (2014) 251-265; DOI: 10.3221/IGF-ESIS.29.22

Constants definition:

3 h I bh I  12,

3

, A bh k A A I     5 6 ,

h

,  

12





0

0

2

Axially-loaded beam, Bar or Rod problem

Static analysis

Dynamic analysis

2 d u EA pb dx 2

2 d U EA

2 

0 bI U

0

0

 





2

dx

Boundary conditions

Clamped-Clamped (CC)     0 0, 0 u u L  

Clamped-Clamped (CC)     0 0, 0 U U L  

Exact solutions

n EA 

n EA

pb

 





f  

n  

u x

L x x 



n

L bI

L bI

2

EA

2

0

0

Euler-Bernoulli (EB) beam

Static analysis

Dynamic analysis

4 d w EI

4 d W EI

qb  

2 

0 bI W

0

0





4

4

dx

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 24 qbL x EI  

  

4

3 x x

2 2 

2 n EI 

n

EI

 

f  

w x

2  

n  



n

4 L L 3

2

2

L

bI

L

bI

L

2



0

0

Timoshenko (Tim) beam

Static analysis

Dynamic analysis

2 d W d    

  

2 d w d                    2 2 2 0 qb dx dx d dw G dx dx    

2 

G

0 bI W

0



G

 

2

dx

dx

2

d

2                2 2 dW G bI dx

EI

EI

0

0

dx

Boundary conditions

Simply Supported-Simply Supported (SS)         0 0, 0 0, 0, 0 d d W W L L dx dx      

Clamped-Clamped (CC)

    0 0, 0 0,   

 

  L

w

w L

0,

0







Exact solutions

2

  

  

  

  

4 qbL x x qbL x x 2 2

2

 

w x



 



2

2

EI

L

2 L G L 

L

24



  

bI

bI

2 2 

4 4 

EI

n

EI n

P

Q

R

,

1    

,





2

2

2

4

L

L

G

G bI 

bI



3 qbL x x x       3 2 3 L L 2 2 3 12 EI L

  

  x

0

0





Table 3 : List of one dimensional static and dynamic exact solutions.

261