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

368

6

1

1

(7)

cot

1

,max v  = − +

+

cot

2



cot



that once again depends only on  . In order to evaluate the shear resistance of the joint it is necessary to determine the point of intersection  x of the two curves, by equating the values of V Rdj ( eq.h ) and V Rdj ( eq.v ) . We obtain the same equation in cot  , both for    and    which may be written as 3 2 cot cot cot 0 a b c d  +  +  + = (8) with sh yd a A f = , , , j ef c ef cy sv yd b A f N A f = − − , , , cot sh yd j ef c ef cy c A f A f = −  sv yd d N A f = − − (9) If A sh >0 we have a third-degree equation. Its solution is mathematically known, but we prefer to shortly recall it. It is necessary to define

2 4 27 q p  = + 3 4

2 2 3 b c a a

3 2 27 3 a a = − + 3 2

b bc d

q

p

= − +

(10)

a

If  4  0

2 q

2 q

b

cot

(11)

3 a  = − +  + − −  − 3 4 4 3 x

Otherwise, it will be necessary to define two further parameters

3

 

 

q

p  = −

(12)

arccos

 =

 −

2

27

 

to obtain

b

3 2 cos   =  −

cot

(13)

x

3 3

a

If A sh = 0 we have a second-degree equation, the solution of which is well known. Because of the trend of the curves, if  x is between  h,max and  v,max the design shear resistance of the joint is the value corresponding to  x . If  x <  h,max the design shear resistance of the joint is the value corresponding to  h,max . If  x >  v,max the design shear resistance of the joint is the value corresponding to  v,max . 4.2. Design of joint reinforcements The shear resistance of a joint depends on geometry, axial force and RC class, but also on the horizontal and vertical joint reinforcements which must be properly designed. Let us analyze a joint with beams 30  60 cm, columns 40  70 cm,  =0.16, V Edj = 1000 kN. Fig. 5(a) shows that, without reinforcement, its shear strength is V Rdj = 651.8 kN; it must be increased by adding reinforcements. In most cases vertical reinforcement is defined by the number and diameter of the bars it is possible to allocate in the column and we imagine in this case to have the possibility of using 4  16 (2 in the central part of each side). This increases the shear resistance given by vertical equilibrium, as shown in Fig. 5(b), but this has negligible effect on the global shear resistance of the joint ( V Rdj = 651.9 kN). It is necessary to increase the shear resistance given by horizontal equilibrium by adding horizontal reinforcement. The quantity of reinforcement may be evaluated in two steps: