Issue 51

G. Cocchetti et alii, Frattura ed Integrità Strutturale, 51 (2020) 356-375; DOI: 10.3221/IGF-ESIS.51.26

self-weight distribution along the arch (  CCR

= 1,  M

= 1), respectively leading to a “linear” , a “quadratic” and a

“cubic problem” in the algebraic solution for unknown triplet A (  ),  (  ), h (  described and discussed in [6]. All what will be considered in the following will stick to classical hypothesis  M = 0, i.e. by assuming a self-weight distribution along the geometrical centreline of the arch, in classical Heyman sense, though with correct evaluation  CCR = 1 of the tangency condition in Eqn. (1) c . An extensive discussion on the differences between the three arising solutions has been reported in [6], including about the spreading discrepancies appearing for over-complete (horseshoe) circular masonry arches. Known Heyman “linear” solution may finally be obtained from Eqns. (1), for  CCR = 0,  M = 0, and can be classically represented as: ). This is originally and extensively derived,

2

     

2 2 cos sin cos    +



+





sin

=



=





A

cot

cot

2

2 cos sin cos    +



−

sin cos 



(

)(

)



−

sin 1 cos  −



(3)

 

= =

2

H

(

)



1 cos +



H   = =   h h





cot

Going to properly correct CCR solution (  CCR = 0), for the complete semi-circular arch, i.e.  = A =  /2, system (1) renders the following characteristic CCR solution triplet for the purely-rotational collapse mode to be recorded: = 1,  M

r 

=

0.951141 54.4963 , rad = 

r 

=

=

  =

0.107426,

h

0.621772 (

/ 2)

(4)

r

More generally, at variable half-opening angle  of the circular masonry arch (thus at variable A =  cot (  /2)), the solution of system (1) for  CCR = 1,  M = 0 can be analytically represented in closed-form as follows [6]:



2

2

2 (2 )  − + − f S g S

f

gS S

 =

A g

f = =

( ) ( sin )'   

= +

S C 

f

    

2

( )  = = + = + g g S Cf  

SC

2

2

g S f −

2 − +

gS S



2 = 

with

(5)

+

f

g

= = = =

( ) sin ( ) cos  



S S C C

2

2

2 −  − + S f gS S

f

h  =  



2

S

Indeed, triplet A ( 

),  (  ), h (  ) becomes a double-valued function of inner hinge position  , coming from the solution of

the following “quadratic problem” [6]:

2

2

(2 ) S g S A fg A g  − − + =  2

0

2

) 4( + − − + − = ) 4( ) 0 g g S g f  

(

f

(6)

 

2

2( − − + − = ) S h g f f

2

S h

0



The above three quadratic equations in A ,  , ( A , h ) and ( A ,  ). Notice that solutions (5) b

h can be obtained from system (1) by eliminating in turn couples (  , h ), for  and h can be obtained from a 2×2 subsystem formed and (5) c

by Eqns. (1) a ) become single-valued at  =  s  = 1.12909 rad = 64.6918°, namely at the value of  setting to zero the term under square roots and (1) c , namely those independent on A (  ) in source system (1). Functions A (  ),  (  ), h ( 

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