PSI - Issue 42

Yuebao Lei et al. / Procedia Structural Integrity 42 (2022) 80–87 Author name / Structural Integrity Procedia 00 (2019) 000–000

84

5

ignoring the parallel bending stress in the limit load evaluation may underestimate elastic-plastic J when the limit load is used in the reference stress J prediction scheme.

30

30

Prediction FE, NPBS=0

Prediction FE, NPBS=0

25

25

FE, NPBS=2.25 FE, NPBS=3.6 FE, NPBS=-2.25

20

20

FE, NPBS=2.25 FE, NPBS=3.6 FE, NPBS=-2.25 NPBS = �� � ⁄

15

15

J / J e

J / J e

NPBS = �� � ⁄

10

10

a/c=0.6 a/t=0.5

5

5

a/c=0.6 a/t=0.5

0

0

0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

(  ref ) 0 /  y

 ref /  y

(a) ( σ ref ) 0 based on Eqn. (5), ignoring σ 2b (b) σ ref based on Eqn. (11) considering σ 2b Fig. 3 Comparison of normalised FE J for applied bending stress σ b with/without σ 2b with the reference stress predictions without/with considering the effect of σ 2b in σ ref estimation ( a / c =0.6, a / t =0.5)

30

30

Prediction FE, NPBS=0

Prediction FE, NPBS=0

25

25

FE, NPBS=1.5 FE, NPBS=2.4 FE, NPBS=-1.5 FE, NPBS=-2.4 NPBS = �� � ⁄

FE, NPBS=1.5 FE, NPBS=2.4 FE, NPBS=-1.5 FE, NPBS=-2.4 NPBS = �� � ⁄

20

20

15

15

J / J e

J / J e

10

10

a/c=0.6 a/t=0.8

a/c=0.6 a/t=0.8

5

5

0

0

0

0.5

1

1.5

2

2.5

0

0.5

1

1.5

2

2.5

 ref /  y

(  ref ) 0 /  y

(a) ( σ ref ) 0 based on Eqn. (5), ignoring σ 2b (b) σ ref based on Eqn. (11) considering σ 2b Fig. 4 Comparison of normalised FE J for applied σ m + σ b + σ 2m with/without σ 2b with the reference stress predictions without/with considering the effect of σ 2b in σ ref estimation ( a / c =0.6, a / t =0.8)

5. Limit load/reference stress solutions considering parallel bending stress For a plate containing a surface crack, a global limit load solution for combined membrane and bending stresses normal to the crack plane and membrane stress parallel to the crack plane is developed by Lei and Budden (2015), which has been summarised in Section 2. However, there is no known limit load solution for this geometry including bending stress parallel to the crack plane. In this section, a reference stress estimation is developed for this geometry to include the effect of the bending stress parallel to the crack plane. 5.1. Limit load solution for plates without cracks A global limit load for defect-free thin plates/shells under combined biaxial membrane stresses and biaxial through thickness bending stresses has been developed by Rozenblium (1960), based on the lower bound limit load theorem and Von Mises yielding criterion. For a thin plate (the thickness is small compared with other dimensions of the plate) of thickness t , a Cartesian coordinate system (1, 2) is located in the middle of the plate and in the middle plane along the thickness (Fig. 5). The membrane and through-thickness bending stresses applied on the plane perpendicular to axis 1 are denoted by σ 1m and σ 1b , respectively, and the membrane and through-thickness bending stresses applied on the plane perpendicular to axis 2 are denoted by σ 2m and σ 2b , respectively (Fig. 5). The corresponding limit values for