PSI - Issue 44

Francesca Barbagallo et al. / Procedia Structural Integrity 44 (2023) 363–370 Francesca Barbagallo et al. / Structural Integrity Procedia 00 (2022) 000 – 000

365

3

(a)

(b)

(c)

Fig. 1. Beam-column joint: geometry and reinforcements (a), A=shear zone, B=diagonal compression field for  <  (b) and for  >  (c)

When we talk of shear V Edj acting on a joint we usually indicate the horizontal component V Edj,h , but we might also refer to the vertical component V Edj,v which is related to the first one by the following relation V Edj,h = V Edj,v cot  (1) The shear strength V Rdj may be obtained from the horizontal equilibrium; this is here indicated by the symbol V Rdj ( eq.h ) . In the case  <  , the shear strength V Rdj ( eq.h ) is the sum of the two quantities V Rdj,h (A) and V Rdj,c , corresponding to the strength of the triangular portion A (equal to A sh  sh ) and to the horizontal component of the strength of the strut B (which depends on  f cd , concrete strength reduced to account for orthogonal tensile strains due to shear), respectively. When  >  V Rdj ( eq.h ) is given only by the contribution of A, which is an aliquot of A sh  sh because the vertical side of the triangle is shorter than the vertical side of the joint. In both cases an additional term V Rdj,min has to be added to account for a strength showed by test even when no horizontal reinforcement is present. We can thus write the shear resistance V Rdj ( eq.h ) given by horizontal equilibrium as

f



cot

cot  − 

if

cd

V

V

sh sh +  + A

A

  

=

( . )

,min

, j ef

Rdj eq h

Rdj

1.6

2 + 

1 cot

(2)

cot cot

 

if

V

V

A + 

  

=

( . )

,min

Rdj eq h

Rdj

sh sh

The shear strength V Rdj may also be obtained by multiplying the value given by vertical equilibrium for cot  ; this is here indicated by the symbol V Rdj ( eq.v ) . In the case  <  V Rdj ( eq.v ) is due only to the contribution of A, which corresponds to an aliquot of N Ed + A sv  sv because the horizontal side of the triangle is shorter than the horizontal side of the joint. When  >  V Rdj ( eq.v ) is given by the sum of two contributions, one is equal to the sum of the vertical force N Ed transmitted by the column and the confining force applied by the vertical reinforcement on the concrete of the joint A sv  sv , while the other one corresponding to the vertical component of the diagonal compression field. Adding also in this case the term V Rdj,min the shear resistance V Rdj ( eq.v ) , given by vertical equilibrium, can be written as

Ed N A

+ 

if

sv sv

V

V

  

=

+

( . )

,min

Rdj eq v

Rdj

cot  +   sv sv

(3)

Ed N A

f

2 cot (1 cot )   −   +  cot (cot cot )

if

cd

V

V

A

  

=

+

+

( . )

,min

, j ef

Rdj eq v

Rdj

cot

1.6



Note that the above formulations are slightly different from those reported in the papers of Fardis and in Eurocode 8, but this is just because original terms sin  , cos  , tan  tan  have been here expressed in function of cot  and cot  . For any value of  the shear strength V Rdj will be the lower between the values given by eq. (2) and eq. (3). The actual shear strength of the joint will be the maximum among the values obtained for all values of  . An important aspect of the Fardis model is the evaluation of the stresses in joint reinforcements  sh and  sv , which depend on horizontal (  h ) and vertical strain (  v ) at the center of the joint, with an upper limit given by f yd , but also from equilibrium conditions; e.g., when  <  we may start with a trial value of  v from which is possible to evaluate