PSI - Issue 72

Albena Doicheva / Procedia Structural Integrity 72 (2025) 235–242

239

E 1 = 1700 kN/cm

2 -the modulus of elasticity for normal concrete and E

1 = 4000 kN/cm

2 for high-strength concrete;

ζ 1 - multiplier for reduced tensile/compressive stiffness of the concrete section; ζ 2 = 1 and ζ 3 = 3 - multiplier for reduced tensile/compressive stiffness of the reinforcing bars; q = 0.05 kN/cm 2 - for all numerical results. The distance b [cm] varies in the interval [ h , 2.0] and is monitored by the ratio h/b .

h=30cm

h=75 cm

-0,50 0,50 1,50 2,50 3,50

-0,20 0,00 0,20 0,40 0,60 0,80 1,00 1,20

H1/(qL/2) H2/(qL/2) H3/(qL/2)

H1/(qL/2) H2/(qL/2) H3/(qL/2)

2,0 H1/qL/2; H2/qL/2; H3/qL/2

2,0 H1/qL/2; H2/qL/2; H3/qL/2

2,5

3,3 h/b

5,0

2,4

2,9

3,8 h/b

5,6

10,0

10,0

50,0

a)

b)

Figure 3. The parameters of the three support reactions for a beam with symmetrical cross-section and E 1 = 1700 kN/cm

2 ; and a) 75/75 cm; b)

30/30 cm, calculated by Equations (5) – (7).

Figure 3 shows the variation in the parameters of the three support reactions, calculated by Equations (5) – (7), while the crack between the beam and the column grows, and E 1 = 1700 kN/cm 2 . The three support reactions increase as the dimensions on the cross section of the beam decreases. For section 75/75cm, Fig. 3a), before the appearance of a crack, H 2 = 1.09 qL /2 , H 3 = 0.84 qL /2and H 1 = -0.09 qL /2 . Therefore H 1 is on a tensile state before the appearance of a crack. After the initiation of a crack H 2 decreases to 0.78 qL/2 , while H 3 slightly increases to b/h = 2.4 ( H 3 =0.86 qL /2) and then decreases to 0.68 qL/2 . The support reaction parameter H 1 /( qL /2) increases to a value of 0.45 . For section 30/30cm, Fig. 3b), before the appearance of a crack, H 2 =3.17 qL /2 , H 3 =2.39 qL /2 and H 1 =-0.22 qL /2 . After the appearance of a crack H 2 decreases to 2.63 qL/2 , while H 3 slightly increases to b/h = 2.7 ( H 3 =2.43 qL /2) and then decreases to 2.17 qL/2 . The support reaction parameter 1 /2 H qL increases to a value of 0.64. For all sections H 1 there are values smaller than those of the other two support reactions, H 2 and H 3 .

h=30cm

h=75 cm

-0,30 -0,10 0,10 0,30 0,50 0,70 0,90 1,10

-0,25 0,25 0,75 1,25 1,75 2,25 2,75 3,25

H1/(qL/2) H2/(qL/2) H3/(qL/2)

H1/(qL/2) H2/(qL/2) H3/(qL/2)

H1/qL/2; H2/qL/2; H3/qL/2

H1/qL/2; H2/qL/2; H3/qL/2

2,0

2,5

3,3

5,0

2,0

2,4

2,9

3,8 h/b

5,6

10,0

h/b

10,0

50,0

a)

b)

2 1 4000kN/cm E  ; and a) 75/75 cm; b)

Figure 4. The parameters of the three support reactions for a beam with symmetrical cross-section and

30/30 cm, calculated by Equations (5) – (7).

Figure 4 shows the variation in the parameters of the three support reactions, calculated by Equations (5) – (7), while the crack between the beam and the column grows, and 2 1 4000kN/cm E  . A comparison of the results of Fig.3 and Fig. 4 shows that as the modulus of elasticity of concrete increases, the support response H 1 increases while H 2 decreases. The support response H 3 is larger for higher concrete modulus until the crack opening around b/h=2.1 to 3, where there is an extremum and then decreases rapidly to values smaller than those before the crack opening.