PSI - Issue 66

Albena Doicheva et al. / Procedia Structural Integrity 66 (2024) 433–448 Author name / Structural Integrity Procedia 00 (2025) 000–000

442 10

h=25 cm

h=75 cm

-1,00 1,00 3,00 5,00 7,00

-0,70 -0,20 0,30 0,80 1,30 1,80

H1/qL H2/qL H3/qL

H1/qL H2/qL H3/qL

H1/qL; H2/qL; H3/qL

1,9 H1/qL; H2/qL; H3/qL 2,2 2,7

3,5

4,8

7,8

1,9

2,2

2,7

3,5

4,8

7,7

20,7

h/b

h/b

20,1

a)

b)

2 1 3700kN/cm E = ; and a) 75/75 cm; b)

Figure 7. The parameters of the three support reactions for a beam with asymmetrical cross-section and

25 /25 cm, calculated by Equations (22)–(24).

Figure 7 shows the variation in the parameters of the three support reactions, calculated by Equations (22)–(24), while the crack between the beam and the column grows, and 2 1 3700kN/cm E = . For section 75/75cm, Fig. 7a), before the appearance of a crack, 1 0.17 H qL =− , 2 1.64 H qL = and 3 1.26 H qL = . After the appearance of a crack 2 H decreases to 0.95 qL , while 3 H slightly increases to ( ) 3 / 2.1 1.28 b h H qL = = and then decreases to 0.79 qL . The support reaction parameter 1 H qL increases to a value of 1.05 . For section 25/25cm, Fig. 7b), before the appearance of a crack, 1 0.54 H qL =− , 2 6.07 H qL = and 3 4.50 H qL = . After the appearance of a crack 2 H decreases to 4.42 qL , while 3 H slightly increases to ( ) 3 / 2.2 4.59 b h H qL = = and then decreases to 3.57 qL . The support reaction parameter 1 H qL increases to a value of 1.87. A comparison of the results of Fig.6 and Fig. 7 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=2.4 to 2.9 where there is an extremum and then decreases rapidly.

h=75 cm

h=25 cm

-0,50 0,00 0,50 1,00 1,50 2,00

-1,00 1,00 3,00 5,00 7,00

H1/qL H2/qL H3/qL

H1/qL H2/qL H3/qL

H1(M)/qL H2(M)/qL H3(M)/qL

H1(M)/qL H2(M)/qL H3(M)/qL

H1/qL; H2/qL; H3/qL

H1/qL; H2/qL; H3/qL

2,0

2,6

3,8 h/b

7,1

2,0

2,4

2,9

3,8 h/b

5,6

50,0

10,0

50,0

a)

b)

Figure 8. The parameters of the three support reactions for a beam, neglecting the normal force in the strain potential energy expression, with symmetrical cross-section and 2 1 1700kN/cm E = ; and a) 75/75 cm; b) 25 /25 cm, calculated by Equations (12)–(14) and Equations (15)–(17). Changing the parameters of the three support reactions, with crack, between the beam and the column,, growth and neglecting the normal force in the strain potential energy expression (dashed lines), calculated by Equations (15)–(17) and E 1 =1700kN/cm 2 is shown on Fig. 8. The solid line shows the graphs from Fig. 4 calculated by Equations (12)– (14). Not considering the axial force results in a value of H 1 =0 until a crack occurs. Accounting for the axial force in the strain potential energy expression results in smaller values for H 1 and significantly larger values for H 3 . The support reaction H2 is smaller for the 25/25 cm beam. For the 75/75 cm beam, H 2 is smaller at the beginning of the crack formation and opening and becomes larger at the end of its development.