PSI - Issue 8

Francesco Penta et al. / Procedia Structural Integrity 8 (2018) 399–409 Francesco Penta et al./ Structural Integrity Procedia 00 (2017) 000 – 000

405

7

Table I - Equivalent stiffnesses for Pratt girder

Bending stiffnesses:

1 2 2

1 2

2

l

c c t E A l

 





h

c c

c c    l

l

E I

E I

2

  







p

d d

c c

d d

6 tan 2 c c d E I  

E I

with

=

cos



d 

t t

E I

6 t c c E I

E I

s n i

an

2

d 

d 





d d

t t

Axial stiffness:

l

E I

c 

 

1  c c E A E A  d d

2

,

2

a 

d  

2 



 





d

d d

c

1

2

2

2

2

l

l

l

l

t

t

d

d

where

I E I 

  

E

E I

288

3

2

cos

1 

d 





t t

d d

c c



l  

l

l

a

d

c

t

t t E I E I l l c c

6912

2

cos

sin

2 

d 

d 





a

t

c



 

 



2

2

4

2   24 24 12 24 12 sin       

cos      1 2 sin

24

t 

 





a

d

d

d

d

t

c

d

t

d

d

Hence, the equivalent shear stiffness will be:

ˆ M m

b     b

V

(11)

2

V 

 





l

2



c

v

b

The eq. (8) - (11) completely define the elastic behaviour of the equivalent Timoshenko beam. The range of validity of these homogenized equations is analysed in the sec. 5 based on the numerical results of a sensitivity analysis. 4. Validation analysis The equivalent beam model defined in Section 3, has been validated with a sensitivity analysis involving geometrical and mechanical parameters. The girder responses are deducted via f.e. analysis on cantilevered girders engendered by assembling Bernoulli-Euler beams. Unit vertical load is applied respectively at the free end and at the midpoint. The accuracy of the theoretical predictions has been quantified by the following dimensionless measure of the homogenization error:

v



e

% 100 

,

v

FE

being FE v the vector of the vertical displacements of the girder nodal sections obtained by f.e. analysis and hom FE    v v v with hom v the nodal vertical displacements vector of the homogenised beam. Furthermore, to have an additional measure of model accuracy and to get also direct indications about the influence exerted on the model equilibrium shapes by the couple-stress bending stiffness, for each examined girder geometry the maximum displacement f of the equivalent model is compared with that FE f of the corresponding f.e. model and the one ˆ f of the Timoshenko (Cauchy) beam having bending and shear stiffness equal respectively to the primary and shear stiffness of the couple-stress equivalent beam.