stress and through-thickness bending

Accepted Manuscript A global limit load solution for plates containing embedded off-set rectangular cracks under combined biaxial force/stress and through-thickness bending Yuanjun Lv, Jianguo Yang, Yuebao Lei, Zenglinag Gao PII:

S0308-0161(16)30428-8

DOI:

10.1016/j.ijpvp.2016.12.006

Reference:

IPVP 3587

To appear in:

International Journal of Pressure Vessels and Piping

Received Date: 12 January 2015 Revised Date:

18 December 2016

Accepted Date: 19 December 2016

Please cite this article as: Lv Y, Yang J, Lei Y, Gao Z, A global limit load solution for plates containing embedded off-set rectangular cracks under combined biaxial force/stress and through-thickness bending, International Journal of Pressure Vessels and Piping (2017), doi: 10.1016/j.ijpvp.2016.12.006. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT A global limit load solution for plates containing embedded off-set rectangular cracks under combined biaxial force/stress and through-thickness bending Yuanjun Lv1, 3, Jianguo Yang1, Yuebao Lei2,1 and Zenglinag Gao1*

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1. Institute of Process Equipment & Control Engineering, Zhejiang University of Technology, Hangzhou 310032, China 2. EDF Energy Nuclear Generation Ltd., Barnett Way, Barnwood, Gloucester, GL4 3RS, UK 3. Zhejiang Industry Polytechnic College, Shaoxing, Zhejiang 312000, China * Corresponding author, Phone: +8657188320763, Fax: +8657188320842, E-mail: [email protected]

Abstract

A global limit load solution for plates containing embedded off-set rectangular cracks under combined biaxial

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force/stress and through-thickness bending moment is derived based on the lower bound limit load theorem and the Mises criterion under a plane stress assumption. The limit load solution is validated using 3D elastic-perfectly plastic finite element (FE) analyses. The results show that the predictions using the solution

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developed in this paper are close to the FE results and slightly conservative. From the solution, limit load solutions for some other defect types in plates, such as extended surface cracks, through-thickness cracks, off-set extended embedded cracks and surface cracks under combined bi-axial force/stress and through-thickness bending are obtained.

Nomenclature

a

half depth of crack

A , B , C , A′ , B ′ , C ′

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Keywords: embedded off-set crack, limit load; combined loading; plate; finite element

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parameters used to define limit load solutions

area of part of the rectangular crack in the region of σ 1−

c

half-length of crack

L

half-length of plate

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Ae

LPF load proportionality factor LPFc critical LPF value for limit load LPFcΓ, LPFcФ

LPFc values obtained from LPF-Γ and LPF-Φ curves

m1L normalised limit through-thickness bending moment M 1 applied through-thickness bending moment M 1L limit through-thickness bending moment 1

ACCEPTED MANUSCRIPT n1L normalised limit end force n1L1 , n1L 4 normalised limit end force for δ = 1 and δ = 0 , respectively n2 L normalised limit stress parallel to the crack plane N1 applied end-force perpendicular to the crack plane

S1+ , S1−

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N1L limit end-force normalised σ 1+ and σ 1− , respectively

thickness of the plate

W

half width of the plate

y

distance between the neutral axis and the front surface of the plate

y∗

distance between the centroid of Ae and symmetry axis of rectangular crack

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t

yoff off-set of the centre of the crack measured from the plate symmetry plane

α normalised crack depth β normalised crack length axial displacement

δ

normalised distance between the neutral axis and the front surface of the plate

δ 01 , δ 02 , δ 03 , δ 04

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Γ

critical normalized neutral axis locations used to define the valid region of solutions

= δ + κ −1 2

θ

angle between the elastic part of the LPF-Γ curve and LPF axis

κ

normalised crack off-set, = yoff t

λ1

ratio between M 1 and N1t

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ε

λ11 , λ12 , λ13 , λ14

characteristic λ1 values used to define the valid region of solutions

λ2

ratio between σ 2 and σ m

λ∗

relating to crack geometric parameters and λ1

λ3

ratio between σ 2 and σ b

2

2

ACCEPTED MANUSCRIPT λ31 , λ32 , λ33 , λ34 σ 1+ , σ 1−

characteristic λ3 values used to define the valid region of solutions

possible maximum and minimum stresses perpendicular to the crack plane, respectively

σ 2 applied stress parallel to crack plane

σ m membrane stress σ y yield stress of material plate end surface rotation angle

χ

angle between the line of TES and LPF axis, = 2θ

ψ

solution parameter

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Ф

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σ b maximum elastic bending stress along the plate thickness

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σ 2 L limit stress of σ 2

1 Introduction

In structural integrity assessments of components containing defects using R6 [1] type procedures, the limit load of the defective component is one of the most important input parameters. It may be used to assess the

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component against plastic collapse and is also used in a fracture assessment using the failure assessment diagram (FAD) approach [1] in which the elastic-plastic fracture parameter J is estimated via the reference stress method [2]. In a J-estimation scheme using the reference stress method, the limit load of the defective component is a very important input in addition to the stress intensity factor (SIF) and the stress-strain relationship of the

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material. Therefore, many of the most commonly-used structural integrity assessment procedures, such as R6 [1], BS7910 [3] and API579 [4], contain appendices of recommended limit load solutions for the most common geometries under various load combinations.

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In pressure vessels and piping, a subsurface flaw is one of the common types of defects found in welds. In structural integrity assessments, components with subsurface defects are sometimes simplified to plates with embedded cracks. In the current version of R6 [1], a global limit load solution for embedded rectangular cracks in plates under combined tension and bending due to Lei and Budden [5] is recommended. A similar solution for embedded elliptical cracks was later developed by Li et al. [6]. However, both solutions ignored stress parallel to the crack plane. Some limit load solutions for defects in plates have considered the effect of stress parallel to the crack plane, such as Miller [7], Lei and Budden [8, 9], but no such solutions for embedded defects are currently available. In this paper, general lower bound limit load solutions for embedded rectangular cracks in plates under combined biaxial tensile/compressive force/stress and cross-thickness positive/negative bending are derived based on the lower bound limit load theorem [10] and the net-section collapse principle. The new limit load 3

ACCEPTED MANUSCRIPT solutions not only extend the Lei and Budden [5] solution to any combination of applied force and through-thickness bending moment but also include the effect of stress parallel to the crack plane. The reasons for considering a rectangular crack instead of an elliptical crack are as follows. For most structural integrity assessment standards or procedures, such as [1, 3, 4], flaws obtained from non-destructive testing need to be characterised before assessment. At first, the whole defect is enveloped by a rectangle and an ellipse (for embedded defects) inscribing the rectangle is then defined as an equivalent crack. This equivalent crack is then

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used in the fracture assessment for the calculation of parameters such as the J-integral and SIF. The limit load may be evaluated based on the equivalent elliptical crack. However the area of the equivalent elliptical crack may be less than that of the real defect. In this case, the limit load of the characterised elliptical crack may be higher than that of the real defect. Therefore, instead of an elliptical crack, the rectangular crack which

consideration for the purposes of structural integrity assessment.

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circumscribes the actual defect is often used in the limit analysis of structures. This is a conservative

The layout of this paper is as follows. Section 2 defines the geometric and loading parameters and normalisations

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of the limit loads. Section 3 addresses the theory and basic assumptions used in the development of the limit load solutions. The limit load solutions are then derived and summarised in Section 4. Section 5 provides 3D finite element (FE) validation. The newly-developed limit load solutions are discussed in Section 6 and conclusions are given in Section 7. Appendix A gives a detailed proof for the correct choice of limit load expression from various possible solutions of the governing equation. Appendices B and C contains details of the derivation of parameters used to define the validity ranges for the solutions. Solutions for the particular case of combined

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through-thickness bending moment and stress parallel to the crack plane are given in Appendix D.

2 Geometry and load parameters A plate of width 2W and thickness t containing an embedded rectangular crack of depth 2a and length

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2c is considered in this paper (Fig. 1). The crack is located in a plane perpendicular to the length direction of the plate, with an off-set yoff from the centre of the plane along the plate thickness (see Fig. 1). For convenience, the normalized crack depth, α , normalized crack length, β , and normalized crack off-set

α=

AC C

parameter, κ , will be used in the derivation of the limit solutions and are defined as

y a c , β= , κ = off t W t

(1)

Note that the crack off-set, yoff , is measured from the centroid of the rectangular crack to the symmetry axis of the cross section of the plate and its positive direction is indicated in Fig. 1. Only cases for κ ≥ 0 are considered due to symmetry of the problem. Therefore, the values of α , β and κ should be taken in the following ranges: 4

ACCEPTED MANUSCRIPT 0 ≤α ≤

1 1 , 0 ≤ β ≤1, 0 ≤ κ ≤ −α 2 2

(2)

A concentrated force, N1 , and a cross-thickness bending moment, M 1 , are applied at the centroid of the end sections of the plate. A tensile/compressive stress, σ 2 , is uniformly applied at the side surfaces along the

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x-direction. A positive value of M 1 corresponds to a bending moment which produces tensile stress in the front surface of the plate and a positive value for N1 and σ 2 indicates a tensile force/stress. The positive directions of all the three loads are shown in Fig. 1. The length of the plate, 2 L , is assumed to be much larger than both the plate width and thickness so that the stress distribution in the crack plane is not affected by the loading conditions at the ends of the plate under plastic-collapse conditions. For an elastic-perfectly plastic

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material with a yield stress σ y , the limit loads corresponding to N1 , σ 2 and M 1 are represented by N1L ,

solutions and are defined as

n1L =

N1L 2 M 1L σ 2L , m1L = , n2 L = 2 σy Wt σ y 2Wtσ y

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σ 2 L and M 1L , respectively. The normalised limit loads, n1L , m1L and n2 L , will be used in the limit load

(3)

and force is defined as

M 1 M 1L m1L σ = = = b tN1 tN1L 4n1L 6σ m

(4)

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λ1 =

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In order to define proportional loading, a dimensionless load ratio, λ1 , between the applied bending moment

σm =

AC C

where, σ m and σ b are the membrane and bending stresses, respectively, given by

N1 3M 1 , σb = 2Wt Wt 2

(5)

A second load ratio, λ2 ,between σ 2 and σ m is defined as

λ2 =

σ 2 n2 L = σ m n1L

(6)

5

ACCEPTED MANUSCRIPT Note that for N1 = 0 , λ1 → ∞ and λ2 → ∞ but the value of λ1 / λ2 may be finite. For this case, a dimensionless load ratio, λ3 , between σ 2 and M 1 is required to describe proportional loading and can be defined as

σ 2 2 n2 L λ 2 = = σ b 3m1L 6λ1

(7)

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λ3 =

3 Theory and basic assumptions

The limit load solution derived in this section is based on the lower bound limit load theorem [10] and the

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following basic assumptions: (1) only stresses along the directions of the plate length (z-direction) and width (x-direction) are considered and the stress along the plate thickness is ignored (plane stress assumption); (2) stress concentration in the near crack tip region is ignored; and (3) crack face contact is ignored when all or part

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of the crack is located in compressive stress zone. The assumption (3) leads to lower limit loads and, therefore, is conservative for structural integrity assessment.

The limit load solutions which will be derived below in Section 4 are for plates under plane stress conditions because the stress along the plate thickness is not considered. Limit load solutions for plane strain conditions may be obtained using the same methodology by including the y-direction stress due to the plate surface boundary conditions. However, the solution expressions are expected to be very complex.

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3.1 Stress assumption and crack type classification

The assumed possible z-direction stress distributions in the crack ligament at plastic collapse are shown in Figs. 2 and 3. For each stress distribution, the crack ligament may be divided into two stress zones which are separated by a “neutral axis” parallel to the front surface of the plate. The z-direction stress in each of the zones

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is assumed to be constant and equal to the maximum stress, σ 1+ , in one zone and the minimum stress, σ 1− , in the other zone, with σ 1+ ≥ σ 1− . There are two possible stress distribution types according to the direction of the

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applied bending moment. The first type is where the zone with the maximum stress σ 1+ is located in the front part of the plate cross section and the zone with the minimum stress σ 1− is in the rear part of the plate (Fig. 2). This distribution type is referred as “stress distribution A” in the remainder of this paper. The second stress distribution type is where the zone with the minimum stress σ 1− is located in the front part of the plate and that with the maximum stress σ 1+ is in the rear part of the plate (Fig. 3). This distribution type will be referred as “stress distribution B” in the remainder of this paper. The location of the neutral axial is defined by y , which is the distance between the neutral axis and the front surface of the plate (see Figs. 2 and 3). The normalised form of the neutral axis location is defined as

6

ACCEPTED MANUSCRIPT δ=

y t

(8)

There are four special values of δ indicating the four special locations of the neutral axis in the crack plane (see Fig. 4), which can be expressed as

=

1 −κ +α 2

1 = −κ −α 2 =0

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(y = t )

=1

t    y = − yoff + a  2   t    y = − yoff − a  2   ( y = 0)

(9)

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δ 01  δ 02   δ  03  δ 04

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Cracks can be classified as three types based on the location of the neutral axis in the crack plane. The cracks are classified as “crack type I” when δ 01 ≥ δ ≥ δ 02 (the neutral axis is located in the rear (large) ligament) and as “crack type III” when δ 03 ≥ δ ≥ δ 04 (the neutral axis is located in the front (small) ligament). Otherwise, the cracks are classified as “crack type II”, where conditions δ 02 > δ > δ 03 are satisfied. For crack type II, the neutral axis cuts through the crack, i.e., part of the crack is located in the zone with maximum stress σ 1+ and

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the other part of the crack is in the zone with the minimum stress σ 1− .

Limit load solutions will be obtained for each of the three crack types under each of the two stress distributions. 3.2 Yield criterion and z-direction stresses

Assuming that the x-direction stress parallel to the crack plane, σ 2 , is constant along the direction of the plate

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width and the y-direction stress is zero, the stresses in the crack ligament follow a 2-D stress state. Applying the Von Mises yield criterion to each point at the crack ligament, the z-direction stresses can be obtained and the

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normalized form may be expressed as

S1+ =

1 2  λ2 n1L + 4 − 3(λ2 n1L )   2

(10)

S1− =

1 2  λ2 n1L − 4 − 3(λ2 n1L )   2

(11)

where Eqn. (6) has been adopted and

7

ACCEPTED MANUSCRIPT S1+ =

σ 1+ σ 1− − S = , 1 σy σy

(12)

4 Limit load solutions Based on the lower bound limit load theorem, the assumed stress distributions at plastic collapse and the

equilibrium conditions are the corresponding limit loads. 4.1 Limit load solutions for stress distribution A

4.1.1 Cases for “crack type I”

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external loads should satisfy force and moment equilibrium conditions. The applied loads which satisfy the

Referring to Fig. 2(a) and taking force equilibrium along the z-direction and moment equilibrium about the

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x-axis, the following two equations can be obtained, respectively.

M1L = [Wy(t − y ) − 4acyoff ]σ 1+ − W (t − y ) yσ 1−

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N1L = (2Wy − 4ac )σ 1+ + 2W (t − y )σ 1−

(13)

(14)

Using Eqns. (1), (3), (8) and (12), Eqns. (13) and (14) can be simplified and expressed as

(

)

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n1L = (δ − 2αβ )S1+ + (1 − δ )S1− m1 L = 2δ (1 − δ ) S 1+ − S 1− − 8αβκ S 1+

(15)

(16)

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expressed as

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Further eliminating δ and m1L from Eqns. (15) and (16), an equation for n1L can be obtained and

An12L + Bn1 L 4 − 3λ 22 n12L − C = 0

(17)

where Eqns. (4), (10) and (11) have been adopted and

 A = 1 − (1 − 2αβ )λ2 + [1 − αβ (1 + 6κ + 2αβ )]λ22   B = 2(λ1 + αβ ) − αβ [1 − 2κ − 2αβ ]λ2  2 2 C = 1 − 4α β − 8αβκ

(18)

8

ACCEPTED MANUSCRIPT The valid region of Eqn. (17) is δ 02 ≤ δ ≤ δ 01 and its roots can be expressed, by re-arranging and squaring Eqn.(17), as

) (

)

2

A + 3B λ 2

(19)

2 2 2

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n1L = ±

(

AC + 2B2 ± AC + 2B2 − A2 + 3B2λ22 C2

It can be proved that Eqn. (17) has two real roots whereas Eqn. (19) has four. However, only one of the roots represents the real limit load. The sign before the outer square root of Eqn. (19) should be the same as the sign of N1L and, therefore, of N1 , that is, “+” when N1 > 0 and “−” when N1 < 0 . The “±” sign before the

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inner square root depends on the signs of N 1 and the coefficient B. It can be shown that the “−” sign should be chosen when BN1 > 0 and “+” when BN1 < 0 (see Appendix A). Therefore, the normalised limit force

(AC + 2B ) − (A 2 2

2

)

+ 3B2λ22 C2

A + 3B λ 2

2 2 2

(AC + 2B ) − (A 2 2

2

)

+ 3B2λ22 C2

A2 + 3B2λ22

for BN1 ≥ 0

(20)

for BN1 < 0

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 2 ± AC + 2B −  n1L =   AC + 2B2 + ± 

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can be expressed as

The corresponding normalised limit moment, m1L , can be obtained from Eqn. (4) with known λ1 and n1L . The limit stress along x-direction, σ 2 L , can be obtained from Eqn. (6) with known λ2 and n1L . 4.1.2 Cases for “crack type II”

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Assume that Ae is the area of the part of the rectangular crack in the region with σ 1− (the area of the cross

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section beyond the neutral axis), and y ∗ is the distance between the centroid of Ae and the symmetry axis of the rectangular crack (see Fig. 5). Referring to Fig. 2(b), the force and bending moment equilibrium equations can then be obtained as follows.

(

)

N1L = (2Wy + Ae ) σ 1+ − σ 1− − 4acσ 1+ + 2Wtσ 1−

[

(

)](

(21)

)

M1L = Wy(t − y ) − Ae y* − yoff σ1+ − σ1− − 4acyoffσ1+

(22)

Using Eqns. (1), (3), (8) and (12), Eqns. (21) and (22) can be simplified to

n1L = [δ + β (α − ε )]( S1+ − S1− ) − 2αβ S1+ + S1−

(23) 9

ACCEPTED MANUSCRIPT

[

(

]

)

m1 L = 2 δ (1 − δ ) − β α 2 − ε 2 + 2 (α − ε )βκ ( S1+ − S1− ) − 8αβκ S1+

(24)

where

 ε Ae = 2ac1 −   α

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(25)

and

y∗ =

1  ε a1 +  2  α

1 2

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ε = δ +κ −

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have been adopted and

(26)

(27)

Eliminating δ , ε and m1L from Eqns. (23) and (24) using Eqns. (4), (10), (11) and (27), the obtained equation for n1L is the same as Eqn. (17) but the coefficients A, B and C in Eqn. (17) become

(28)

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  3  2   2 2 2 2  A = 1 − (1 − 2αβ )λ2 + 1 + 4α β −  3α + 3κ + + α  β  λ2 4        B = (2α − 1)βκλ2 + 2(1 − β )λ1 + 2 βκ  2 C = (1 − β )1 − 4 βκ − 4α 2 β   1− β    

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For the cases of crack type II, Eqn. (17) and, therefore, Eqn. (19), are valid for δ 02 ≤ δ ≤ δ 03 and the normalised limit force, n1L , can still be expressed as Eqn. (20), where the sign before the outer square root of Eqn. (20) should be the same as the sign of N1 , that is, “+” when N 1 > 0 and “−” when N1 < 0 . 4.1.3 Cases for “crack type III”

Referring to Fig. 2(c), the force and bending moment equilibrium equations can be obtained and expressed as

(

)

N1L = 2Wy σ 1+ − σ 1− + (2Wt − 4ac )σ 1−

(29)

M 1L = W (t − y ) y (σ 1+ − σ 1− ) − 4acyoff σ 1−

(30)

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ACCEPTED MANUSCRIPT Using Eqns. (1), (3), (8) and (12), Eqns. (29) and (30) can be simplified to

(

)

n1L = δ S1+ − S1− + (1 − 2αβ )S1−

(31)

(

(32)

)

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m1L = 2δ (1 − δ ) S1+ − S1− − 8αβκS1−

Eliminating δ and m1L from Eqns. (31) and (32) using Eqns. (4), (10) and (11), the obtained equation for

n1L is the same as Eqn. (17) but the coefficients A, B and C in Eqn. (17) become

(33)

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 A = 1 − (1 − 2αβ )λ2 + [1 − αβ (1 − 6κ + 2αβ )]λ22   B = 2(λ1 − αβ ) + αβ (1 + 2κ − 2αβ )λ2  2 2 C = 1 − 4α β + 8αβκ

For the cases of crack type III, Eqn. (17) and, therefore, Eqn. (19), are valid for δ 04 ≤ δ ≤ δ 03 and the normalised limit force, n1L , can still be expressed as Eqn. (20), where the sign before the outer square root of Eqn. (20) should be the same as the sign of N 1 .

4.2.1 Cases for “crack type I”

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4.2 Limit load solutions for stress distribution B

Referring to Fig. 3(a), the normalised force and bending moment equilibrium equations can be obtained and expressed as

(

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n1L = (1 − δ )S1+ + (δ − 2αβ )S1−

)

m1L = −2δ (1 − δ ) S1+ − S1− − 8αβκS1−

(34)

AC C

(35)

Eliminating δ and m1L from Eqns. (34) and (35), using Eqns. (4), (10) and (11), an equation for n1L can be obtained and expressed as

An1L − Bn1L 4 − 3λ22 n1L − C = 0 2

2

(36)

where the coefficients A, B and C are the same as those defined by Eqn. (18). The roots of Eqn. (36) may be expressed by Eqn. (19).

11

ACCEPTED MANUSCRIPT The valid region of Eqn. (36) and, therefore, Eqn. (19) is δ 02 ≤ δ ≤ δ 01 when the coefficients A, B and C are defined by Eqn. (18). It can be proved that “+” should be chosen before the inner square root of Eqn. (19) when

BN 1 > 0 and “−” when BN 1 < 0 (see Appendix B). Therefore, the normalised limit force, n1L , for this case

(AC + 2B ) − (A + 3B λ )C

2

(AC + 2B ) − (A + 3B λ )C

2

2 2

2

2 2 2

A2 + 3B2λ22 2 2

2

2 2 2

A2 + 3B2λ22

for BN1 ≥ 0 for BN1 < 0

(37)

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 2 ± AC + 2B +  n1L =   AC + 2B2 − ± 

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can be expressed as

Note that the sign before the outer square root in Eqn. (37) should be “+” when N1 > 0 and “−” when

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N1 < 0 . 4.2.2 Cases for “crack type II”

Assume that Ae is the area of the part of the rectangular crack in the region of σ 1+ (beyond the neutral axial) and y ∗ is the distance between the centroid of Ae and the symmetry axis of the rectangular crack (see Fig. 5). Referring to Fig. 3(b), the normalised force and bending moment equilibrium equations can be obtained and

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expressed as n1L = −[δ + β (α − ε )]( S1+ − S1− ) − 2αβ S1− + S1+

[

(

)

](

)

(39)

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m1L = −2 δ (1 − δ ) − β α 2 − ε 2 + 2(α − ε )βκ S1+ − S1− − 8αβκS1−

(38)

where Eqns. (25)-(27) for Ae , y * and ε , respectively, have been adopted. Similarly, eliminating δ , ε

AC C

and m1L from Eqns. (38) and (39) using Eqns. (4), (10), (11) and (27), the same equations as Eqns. (36) and (19) can be obtained but the coefficients A, B and C in the two equations are defined by Eqn. (28). The valid region of Eqn. (36) for crack type II is δ 02 ≤ δ ≤ δ 03 when the coefficients A, B and C are defined by Eqn. (28) and the normalised limit force, n1L , for this case can still be expressed as Eqn. (37). Note that the sign before the outer square root in Eqn. (37) should be “+” when N1 > 0 and “−” when N1 < 0 . 4.2.3 Cases for “crack type III” Referring to Fig. 3(c), the normalized force and bending moment equilibrium equations can be obtained and may be expressed as

12

ACCEPTED MANUSCRIPT

(

)

n1L = −δ S1+ − S1− + (1 − 2αβ )S1+

(

(40)

)

m1L = −2δ (1 − δ ) S1+ − S1− − 8αβκS1+

(41)

Similarly, eliminating δ and m1L from Eqns. (40) and (41) using Eqns. (4), (10) and (11), the same

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equations as Eqns. (36) and (19) can be obtained but the coefficients A, B and C in the two equations are defined by Eqn. (33).

The valid region of Eqn. (36) is δ 04 ≤ δ ≤ δ 03 when the coefficients A, B and C are defined by Eqn. (33) and the normalised limit force, n1L , for this case can still be expressed as Eqn. (37). Note that the sign before the

4.3 Redefinition of the valid regions of the limit load solutions

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outer square root in Eqn. (37) should be “+” when N 1 > 0 and “−” when N1 < 0 .

Six limit load solutions have been obtained in Sections 4.1 and 4.2 and the validity range for each solution

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depends on the assumed stress distribution type and the position of the neutral axis, i.e. the value of δ , which may vary between 0 and 1. However, δ is an unknown before the equations (for example Eqns. (15) and (16)) are solved. Therefore, the valid ranges of the solutions should be re-defined using known parameters. The validity regions indicated by the neutral axis location parameter, δ , can alternatively be indicated by the load ratio, λ1 , as λ1 = λ1 (δ , α , β , κ , λ2 ) follows from Eqns. (4) and (10)-(12) and the two equilibrium

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equations for each solution, Eqns. (15) and (16) for example, where the geometric parameters α , β , κ and the load ratio λ2 are known. The four special neutral axis locations corresponding to δ 01 , δ 02 , δ 03 and

δ 04 in Eqn. (9) now can be mapped into the n1L ~ m1L space and indicated by four special λ1 values as λ11 ,

EP

λ12 , λ13 and λ14 , respectively, which have been obtained in Appendix B and may be expressed as

AC C

 − 2αβκ λ11 = λ14 = 1 − 2αβ   1  2 −  − 2(κ − α ) − 4αβκ ψ  2  − αβκλ λ12 = 2 4(κ − α ) + 4αβ   1  2 −  − 2(κ + α ) + 4αβκ ψ   − αβκλ λ =  2 2  13 4(κ + α ) − 4αβ

(42)

where

1 2

 

ψ = 1 −  − αβ λ2

(43) 13

ACCEPTED MANUSCRIPT The effect of ψ on defining the validity ranges of the limit load solutions is discussed in Appendix C. Note that the four special λ1 values depend only on known geometric and loading parameters and can be used to choose the correct limit load expressions before solving the equations. The validity ranges of the six limit load solutions obtained in Sections 4.1 and 4.2 can be clearly defined in the n1L ~ m1L space using ψ and

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λ11 ~ λ14 (see Fig. 6). Note that, in Fig. 6, λ12 can be positive or negative for a given ψ value but it can be proved that λ13 − λ11 < 0 when ψ > 0 and λ13 − λ11 > 0 when ψ < 0 . Referring to Fig. 6 for the validity ranges of the solutions, the full limit load solution can be summarised as follows in Section 4.4. 4.4 Summary of the limit load solutions

Based on the re-defined validity ranges, the limit load solutions can be summarised in the following two

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subsections, Section 4.4.1 for N1 ≠ 0 and Section 4.4.2 for N1 = 0 .

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4.4.1 The limit load solutions for N1 ≠ 0 The normalised limit force, n1L , may be expressed as

C

(AC + 2B ) − (A 2 2

2

)

+ 3B2λ22 C 2

C

(AC + 2B ) − (A 2 2

2

)

+ 3B λ C 2 2 2

B(λ1 − λ11 ) > 0

B(λ1 − λ11 ) < 0

(44)

2

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 ±  AC + 2B2 + n1L =  ±  2  AC + 2B −

where “+” corresponds to N1 > 0 and “−” to N1 < 0 . The normalised limit moment, m1L , and the

EP

normalised limit x-direction stress, n2 L , can then be obtained from Eqns. (4) and (6), respectively, using the known n1L (Eqn. (44)), λ1 and λ2 , and may be expressed as

AC C

m1L = 4λ1n1L , n2 L = λ2 n1L

(45)

The coefficients A, B and C in Eqn. (44) can be determined as follows.

λ12 ≥ λ1 ≥ λ11  For ∞ > λ1 ≥ λ11 λ ≥ λ > −∞  12 1

(λ12 − λ11 > 0)

(λ12 − λ11 < 0)

(ψ > 0)

14

ACCEPTED MANUSCRIPT λ11 ≥ λ1 > −∞  or ∞ > λ1 ≥ λ12 λ ≥ λ ≥ λ  11 1 12

(λ12 − λ11 > 0) (λ12 − λ11 < 0)

(ψ < 0)

(λ12 − λ13 > 0)

λ12 ≥ λ1 ≥ λ13  or λ12 ≥ λ1 > −∞ ∞ > λ ≥ λ 1 13 

(λ12 − λ13 > 0)

(λ12 − λ13 < 0)

(ψ < 0)

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(λ12 − λ13 < 0)

(ψ > 0)

Equation (28) should be used.

(ψ > 0) (ψ < 0)

Equation (33) should be used.

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λ14 ≥ λ1 ≥ λ13 λ13 ≥ λ1 ≥ λ14

For 

SC

∞ > λ1 ≥ λ12  For λ13 ≥ λ1 > −∞ λ ≥ λ ≥ λ  13 1 12

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Equation (18) should be used.

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4.4.2 The limit load solutions for N1 = 0

For combined bending moment, M 1 , and σ 2 , N1 = 0 and, therefore, n1L = 0 . For this case, solutions can

AC C

still be derived from the general solutions given in Section 4.4.1 (see Appendix D). The normalised limit moment, m1L , can be expressed as

 ±  m1L =  ±  

A′C ′ + 2 B′ + 2

4C ′

(A′C′ + 2 B′ ) − (A′ 2 2

B′ > 0

)

2

)

2

2

+ 108 B′ λ C ′

2

+ 108 B′ λ C ′

2 2 3

4C ′

A′C ′ + 2 B′ − 2

(A′C′ + 2B′ ) − (A′ 2 2

2 2 3

15

(46)

B′ < 0

ACCEPTED MANUSCRIPT where “+” corresponds to M 1 > 0 and “−” to M 1 < 0 . The coefficients A′ , B′ and C ′ in Eqn. (46) are given in Appendix D. The normalised limit x-direction stress, n2 L , can be evaluated from Eqn. (7) with known

m1L and λ3 : 3 λ3m1L 2

(47)

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n2 L =

5 Finite element validation

3D elastic-perfectly plastic FE analyses are performed for plates with embedded rectangular cracks under combined biaxial positive/negative force/stress and positive/negative through-thickness bending moment to

5.1 Geometry, material properties and loading cases

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validate the obtained limit load solutions. The commercial FE code ABAQUS [11] is used in this study.

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A plate of half-width W = 100 mm, thickness t = 60 mm and half-length L = 4W containing an embedded rectangular crack of depth 2a =15 mm and length 2c =50 mm (see Fig. 1) is used in all the cases analysed, corresponding to α = 0.125 , β = 0.25 and a / c = 0.3 . Only one quarter of the plate is modelled because of the symmetry in geometry and applied loads. Symmetry constraints are applied on the two symmetry planes. The y-direction constraint is applied at a node in the x-y symmetry plane to prevent rigid body displacement (see Fig. 7). Crack surface contact is not considered in the analyses when the crack (full or part) is located in a

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compressive stress zone. In other words, crack face overlap is allowed. The 8-noded element with reduced integration (ABAQUS type C3D8R) is adopted. A refined mesh is used in the vicinity of the crack tip to improve calculation accuracy. Three models were created to cover κ = 0, 0.1 and 0.2 . A typical mesh for

κ = 0.1 is shown in Fig. 7, which contains 35880 elements. The loading cases considered are λ1 = -5, -1,

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-0.2, 0, 0.2, 1 and λ2 = -1, 0, 1. The material properties used in the analyses are shown in Table 1. Note that the yield stress used in the analyses does not affect the normalized limit loads. Sensitivity to element type has been investigated by comparing the limit load results for meshes using element

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types C3D8R and C3D20R for α = 0.25 , β = 0.25 , κ = 0.1 and λ2 = 1 . Four values of λ1 have been selected and the results are shown in Table 2. From Table 2, the limit load values obtained using the 20-noded element type C3D20R are always lower than those obtained using element type C3D8R but the maximum relevant difference in the corresponding limit load values obtained using the two different element types is less than 0.66%. Therefore, the accuracy of the FE analysis can be assured by adopting the element type C3D8R. 5.2 Analyses A reference point (shown in Fig. 7) has been created in the midpoint of the intersection line between the end surface of the plate (parallel to the crack plane) and the symmetry plane. All the nodes in the end surface are coupled to the reference point. A concentrated force, N1 , and a cross-thickness bending moment, M 1 , are applied at the reference point. A tensile/compressive stress, σ 2 , is uniformly applied at the side surface along 16

ACCEPTED MANUSCRIPT the x-direction as a distributed load. All loads are applied proportionally. To increase the analysis efficiency and ensure computing precision, for all cases, the loading process is divided into two steps. In step 1, 70% of the theoretical limit load values are applied with a large increment size and in step 2, a very small increment size is used to increase the total applied load values (normally less than 1.2 times the theoretical limit load values) until plastic collapse occurs in order to obtain more accurate load-deformation curves. 5.3 Determination of limit load

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The elastic-perfectly plastic FE analysis does not directly give the limit load and further post-processing and judgement should be used to determine the load values at plastic collapse. There are several methods which could be used to determine the limit load of structure, for example, the twice elastic slope (TES) method, zero curvature method, tangent-intersection method and plastic strain method. The TES method is recommended by

SC

ASME [12] and is widely adopted in engineering designs and assessments because it is easy to use and conservative. The TES method is adopted in this paper to determine the FE limit loads. To determine the FE limit load values using the TES method, the load-displacement relationships need to be

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extracted from the FE analyses. In ABAQUS, the load proportionality factor (LPF), which is defined as the ratio between the current load and the planned maximum load, is used as a loading parameter. For proportional loading, this factor applies to all the load components for combined loading. The displacement should normally be the load-point displacement. In this study, N1 and M 1 are all applied at the reference point and the z-direction displacement, Γ , and the rotation angle, Φ , about the x-axis at the reference point are measured, i.e., the LPF-Γ and LPF-Ф curves are used to indicate the load-displacement behaviour of the models. A critical

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LPF value, LPFc, may then be determined using the TES method based on the LPF-Γ and/or LPF-Ф curves. The load values corresponding to the LPFc are taken as the limit loads. In the TES method, a straight line with an angle χ between line and the LPF axis is drawn from the origin and the critical LPF is determined from its intersection point with the load-displacement curve, where χ

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should satisfy tan χ = 2 tan θ and θ represents the angle between the LPF axis and the elastic part of the load-displacement curve (see Fig. 8(a)). Two critical limit values LPFcФ (shown in Fig. 8(a)) and LPFcΓ (shown

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in Fig. 8(b)) may be obtained according to the curve of LPF-Ф and LPF-Γ, respectively. Under general loading conditions, it is conservative to choose the value of LPFc =min(LPFcФ, LPFcΓ). For some special cases, only one LPF-displacement curve is used to determine the critical LPFc. When the structure is mainly subjected to axial force N1 , such as the case of λ1 = 0 , the rotational deformation is not reliable and LPFc = LPFcΓ. Similarly, when the structure is mainly subjected to moment M 1 , such as λ1 = 5 , LPFc = LPFcФ is adopted. 5.4 Comparison of theoretical predictions and FE results The obtained FE limit loads are properly normalized and plotted in Fig. 9 for comparison with the results predicted using the closed-form solutions of Section 4. From Fig. 9, it can be seen that all FE data are located outside the relevant yield surface predicted using the analytical limit load solutions. This indicates that the solutions derived in Section 4 are conservative when compared with the elastic-perfectly plastic FE results. For 17

ACCEPTED MANUSCRIPT all the FE cases in this paper, the maximum relevant difference between the FE results and the theoretical predictions is less than 12%.

6 Discussion 6.1. Comparison of solutions between this paper and R6 procedure In the current R6 procedure [1], a global limit load solution for plates with embedded rectangular cracks under combined tension and bending developed by Lei and Budden [5] is recommended. This corresponds to the limit

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load solution of this paper for N1 , M 1 ≥ 0 and λ2 = 0 , due to σ 2 = 0 . The coefficients A, B and C defined by Eqns. (18) and (28) reduce to

(48)

for “shallow cracks” in [1] and

(49)

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  A = 1   B = 2[(1 − β )λ1 + βκ ]  2 C = (1 − β )1 − 4 βκ − 4α 2 β   1− β    

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SC

A = 1   B = 2λ1 + 2αβ  2 2 C = 1 − 8αβκ − 4α β

for “deep cracks” in [1]. It is seen that λ1 − λ11 > 0 , because n1L > 0 and m1L > 0 , and B > 0 in its

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defined ranges. Therefore, B (λ1 − λ11 ) > 0 and the first equation in Eqn. (44) applies with “+” for the

n1L =

AC C

“ ± ”sign. Simplifying the equation obtains

C

(50)

B + B2 + C

Equation (50), together with Eqns. (48) and (49), reproduces the global limit load solution in [5]. 6.2 Limit load solutions for particular cases The limit load solutions of this paper are general and the limit load solutions for particular crack geometries can be obtained directly from the solutions in Section 4. 6.2.1 Plates with extended surface cracks For a plate with an extended surface crack, β ≡ 1 and κ ≡ 0.5 − α (shown in Fig. 10(a)). Therefore, only the “crack type I” conditions apply and Eqn. (18) reduces to 18

ACCEPTED MANUSCRIPT  A = 1 − (1 − 2α )λ2 + (1 − 2α )2 λ22  B = 2λ1 + 2α  2 C = (1 − 2α )

(51)

Combining Eqns. (44) and (51), the Mises limit load solution obtained by Lei and Budden [8] can be

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reproduced. 6.2.2 Plates with through-thickness cracks

For a plate with a through-thickness crack, α ≡ 1 2 and κ ≡ 0 (shown in Fig. 10(b)). Therefore, only “crack

SC

type II” conditions apply and Eqn. (28) reduces to

(52)

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 A = 1 − (1 − β )λ2 + (1 − β ) 2 λ22   B = 2(1 − β ) λ1  2 C = (1 − β )

Combining Eqn. (44) and (52), the limit load solution for plates with through-thickness cracks under combined biaxial force/stress and through-thickness bending obtained in [9] can be reproduced. 6.2.3 Plates with surface cracks

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For a plate with a surface crack, κ ≡ 0.5 − α (shown in Fig. 10(c)). This case satisfies the conditions of “crack type I” or “crack type II” and Eqns. (18) and (28) reduce to.

[

]

(53)

EP

 A = 1 − 2α 2 β 2 + 2(2 − 3α )αβ − 1 λ22 + (2αβ − 1)λ2  2 B = 2(β − 1)α βλ2 + 2λ1 + 2αβ C = 1 − 4(1 + αβ − 2α )αβ 

AC C

for cases of “crack type I” in and

  2 2  2  2 3  A = 4α β −  6α − 2α +  β + 1 λ2 + (2αβ − 1)λ2 + 1 2      1  2  B = − ( 2α − 1) βλ2 + 2(1 − β )λ1 + β (1 − 2α ) 2    β ( 2α − 1)2  − 4α 2 β  C = (1 − β ) 1 − 1− β   

(54)

19

ACCEPTED MANUSCRIPT for cases of “crack type II”. Combining Eqn. (44) with Eqns. (53) and (54), the global limit load solution for plates with surface cracks under combined biaxial force/stress and through-thickness bending developed by Lei and Budden [9] can be reproduced. 6.2.4 Plates with off-set extended embedded cracks For a plate with an off-set through-width crack, β ≡ 1 (shown in Fig. 10(d)). This case satisfies the conditions

[

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of “crack type I” or “crack type III” and Eqns. (18) and (33) reduce to

]

[

]

[

]

[

]

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for “crack type I” and.  A = 1 − 2α 2 − (6κ − 1)α − 1 λ22 + (2α − 1)λ2  2  B = − 2α + (2κ + 1)α λ2 + 2λ1 − 2α  2 C = 1 − 4α + 8ακ

(55)

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 A = 1 − 2α 2 + (6κ + 1)α − 1 λ22 + (2α − 1)λ2  2 B = 2α + (2κ − 1)α λ2 + 2λ1 + 2α  2 C = 1 − 4α − 8ακ

(56)

for “crack type III”. Combining Eqn. (44) with Eqns. (55) and (56), the global limit load solution for plates

be obtained.

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with off-set extended embedded cracks under combined bi-axial force/stress and through-thickness bending can

6.3 Effect of stress parallel to the crack plane on limit load The stress which is parallel to the crack plane has an influence on plastic collapse of plates with off-set

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rectangular cracks. Figure 11 shows a closed yield surface in n1L ~ m1L space for α = 0.25 , β = 0.25 and

κ = 0.1 with various λ1 values. It can be seen from Fig. 11 that for the cases with σ 2σ m < 0 , ignoring may significantly overestimate limit loads and, for the cases with σ 2σ m > 0 , ignoring

σ2

may

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σ2

underestimate the limit loads when λ2 is less than a certain value λ*2 , which depends on crack geometry and

λ 1 , and may significantly overestimate the limit loads when λ2 is greater than λ*2 . 6.4 Use of the solutions developed in this paper in structural integrity assessment The limit load solutions developed in Section 4 of this paper may be used in a structural integrity assessment against global plastic collapse of a plate containing an embedded defect. The assessment should be conservative because the solutions developed in this paper correspond to lower bound limit loads. This has been demonstrated by a range of elastic-perfectly plastic FE analyses in Section 5. The solutions are based on rectangular cracks but can be conservatively used for plates containing elliptical cracks of the same crack depth and length. A lower bound global limit load solution may also be used in structural integrity assessment against fracture. For example, the global limit load solution for surface cracks in plates [13] may well predict the J-integral via the 20

ACCEPTED MANUSCRIPT reference stress method [2]. However, a plate containing an off-set embedded defect has two crack ligaments which may behave very differently under the same applied loads. The smaller crack ligament may yield at a load much lower than the global collapse load. The global limit load for this case may lead to a good prediction of J for the crack tip corresponding to the larger ligament but may significantly underestimate J for the crack tip associated with the smaller ligament. Therefore, the use of the global limit load solutions developed in this paper for fracture assessment needs further validation using FE J data. Elastic-plastic FE analyses for plates containing

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embedded off-set elliptical cracks under combined loading, including stress parallel to crack plane, to create a J database are being carried out by the authors. The FE J results and further validation for the use of the limit loads in fracture assessment of plates containing off-set embedded defects will be published later.

7. Conclusions

(1) A global limit load solution for plates with embedded off-set rectangular cracks under combined biaxial

theorem and the Mises criterion under plane stress assumptions.

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force/stress and through-thickness bending moment has been derived based on the lower bound limit load

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(2) 3D elastic-perfectly plastic finite element (FE) analyses have shown that the limit load solutions developed in this paper are close to the FE results and slightly conservative.

(3) Limit load solutions for some other defect types in plates, such as extended surface cracks, through-thickness cracks, off-set extended embedded cracks and surface cracks, under combined bi-axial force/stress have been obtained from the global limit load solutions developed in this paper. (4) The limit load solutions developed in this paper may be used in structural integrity assessment against global

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plastic collapse but the use of the solutions in fracture assessment needs further validation using elastic-plastic FE J data.

Acknowledgement

The authors wish to acknowledge Dr Peter Budden of EDF Energy for his valuable comments on this paper.

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The research reported in this paper is supported by the National Natural Science Foundation of China (Grant

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No. 51475426) and the Science & Technology Program of Zhejiang Province (Grant No. 2014C23001).

References [1] R6, Assessment of the integrity of structures containing defects, Revision 4, Amendment 11. Gloucester, UK: EDF Energy Nuclear Generation Ltd; 2015.

[2] Ainsworth RA. The assessment of defects in structures of strain hardening material. Engineering Fracture Mechanics 1984; 19:633-642.

[3] BS7910:2013. Guide to methods for assessing the acceptability of flaws in metallic structures, BSI Standards Publication; 2013.

[4] API 579-1/ASME FFS-1 2016. Fitness-For-Service, the American Petroleum Institute and The American Society of Mechanical Engineers, Washington DC.

[5] Lei Y and Budden PJ. Limit load solutions for plates with embedded cracks under combined tension and 21

ACCEPTED MANUSCRIPT bending. International Journal of Pressure Vessels and Piping 2004; 81:589-597.

[6] Li RS, Gao, ZL and Lei Y. A global limit load solution for plates with embedded off-set elliptical cracks under combined tension and bending. Journal of Pressure Vessel Technology – Transaction of the ASME, 2012; 134: 011204-1.

[7] Miller AG. Review of limit loads of structures containing defects. International Journal of Pressure Vessels and Piping 1988; 32:197–327.

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[8] Lei Y and Budden PJ. Limit load solutions of plates with extended surface cracks under combined biaxial forces and cross-thickness bending. The Journal of Strain Analysis for Engineering Design 2014; 49: 533-546.

[9] Lei Y and Budden PJ. Global limit load solutions for plates with surface cracks under combined biaxial

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forces and cross-thickness bending. International Journal of Pressure Vessels and Piping 2015; 132-133:10-26.

[10] Chakrabarty J. Theory of plasticity, 3rd ed. Oxford: Elsevier Butterworth-Heinemann, 2006.

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[11] ABAQUS Version 6.10 User’s manual. Inc. and Dassault Systems Corp, 2010.

[12] ASME Boiler and Pressure Vessel Code, Section III, Rules for Construction of Nuclear Facility Components, Division 1-Subsection NB Class 1 Components, Article NB-3200, Design, American Society of Mechanical Engineers, New York, 2011.

[13] Lei Y. J-integral and limit load analysis of semi-elliptical surface cracks in plates under combined tension

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and bending. International Journal of Pressure Vessels and Piping 2004; 81:43-56.

22

ACCEPTED MANUSCRIPT Appendix A

Determining “±” sign before the inner square root in Eqn. (19)

Equation (19) has four roots but only two of them can satisfy Eqn. (17) and only one of them leads to the physically correct normalized limit force. This appendix gives the details for choosing the correct sign before the inner square root in Eqn. (19).

A.1 Cases for stress distribution A

n1L 4 − 3λ22 n1L = − 2

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Re-write Eqn. (17) as

A 2 C n1L + B B

A 2 C n1L + > 0 B B

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−

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For n1L > 0 , the right-hand side of Eqn. (A1) satisfies

(A1)

(A2)

For B < 0 , the following equation can be obtained from Eqn. (A2), noting that A > 0 from Eqns. (18), (28) and (33),

2

C A

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n1L >

(A3)

n1L <

C A

(A4)

AC C

2

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Similarly, for B > 0 , the following equation should be satisfied.

Repeating the above analysis for n1L < 0 , Eqn. (A3) can be obtained again for B > 0 and Eqn. (A4) for

B < 0 . The four cases can be summarized as  2 C n1L > A for n1L B < 0  n 2 < C for n B > 0 1L  1L A

(A5)

Equation (19) can be rewritten as 23

ACCEPTED MANUSCRIPT n1L =

C2

2

AC + 2 B 2 ±

(AC + 2 B ) − (A 2 2

2

=

)

+ 3B 2 λ22 C 2

C2 AC + 2 B 2 ± 4 B 2 + [2 − (1 − 2αβ )λ2 ]

2

(A6)

where the expressions for coefficients A and C in Eqn. (18) have been adopted. Inserting Eqn. (A6) into the first equation in Eqn. (A5), the following equation can be obtained because A > 0

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and C > 0 from Eqn. (18):

2 B 2 ± 4 B 4 + B 2 C [2 + (2αβ − 1)λ2 ] < 0 2

(A7)

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In Eqn. (A7), only taking the “−” sign can satisfy the inequality, that is, for n1L B < 0 the sign before the inner

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square root of Eqn. (19) must be “+”.

Similarly, inserting Eqn. (A6) into the second equation in Eqn. (A5) the following equation can be obtained ( A > 0 and C > 0 have been used again):

2 B 2 ± 4 B 4 + B 2 C [2 + (2αβ − 1)λ2 ] > 0 2

(A8)

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In Eqn. (A8), only taking the “+” sign can satisfy the inequality, that is, for n1L B > 0 the sign before the inner square root of Eqn. (19) must be “−”.

The above conclusions hold when the coefficients A and C for “crack type II” (Eqn. (28)) and “crack type III”

EP

(Eqn. (33)) are used. Note that the sign for n1L is the same as for N1 . Therefore, the following roles for the choice of the sign before the inner square root hold for cases obtained under the assumption of stress

AC C

distribution A.

+ for N1 B < 0  − for N1 B > 0

(A9)

A.2 Cases for stress distribution B Repeating the analyses performed in Section A.1 for stress distribution A, the results show that for stress distribution B, Eqns. (A5) and (A9) become Eqns. (A10) and (A11) below, respectively.

24

ACCEPTED MANUSCRIPT  2 C n < for n1L B < 0  1L A  n 2 > C for n B > 0 1L  1L A

(A10)

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and

− for N1 B < 0  + for N1 B > 0

(A11)

Definitions of special λ1 and λ3 values corresponding to δ 01 ~ δ 04 B.1 Definitions of λ11 ~ λ14

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Appendix B

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λ11 corresponds to the neutral axis location δ 01 . Inserting δ = δ 01 = 1 (see Eqn. (9)) into Eqns. (15) and (16) and then using Eqn. (4), λ11 can be obtained and may be expressed as

λ11 =

− 2αβκ 1 − 2αβ

(B1)

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λ12 corresponds to the neutral axis location δ 02 . Inserting δ = 1 2 − κ + α (see Eqn. (9)) and Eqns. (10) and (11) into Eqns. (15), (16) and then using Eqn. (4), λ12 can be obtained and may be expressed as

1  2 −  − 2(κ − α ) − 4αβκ ψ 2  − αβκλ λ12 =  2 4(κ − α ) + 4αβ where

1 2

AC C

EP

(B2)

 

ψ = 1 −  − αβ λ2

(B3)

Note that the special value of n1L for δ = 1 2 − κ + α should be used in the above derivation.

λ13 corresponds to δ 03 . Inserting δ = 1 2 − κ − α (see Eqn. (9)) and Eqns. (10) and (11) into Eqns. (31) and (32) and using Eqn. (4), λ13 can be obtained and may be expressed as 25

ACCEPTED MANUSCRIPT 1  2 −  − 2(κ + α ) + 4αβκ ψ 2  − αβκλ λ13 =  2 4(κ + α ) − 4αβ

(B4)

Note that the special value of n1L for δ = 1 2 − κ − α should be used in the above derivation.

Eqn. (4), λ14 can be obtained and found to be the same as Eqn. (B1). B.2 Definitions of λ31 ~ λ34

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λ14 corresponds to the neutral axis location δ 04 . Inserting δ = δ 04 = 0 into Eqns. (31), (32) and then using

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For combined σ 2 and M 1 , n1L = 0 . The four special neutral axis locations indicated by δ 01 ~ δ 04 can be indicated by four special λ3 values, that is, λ31 ~ λ34 , respectively. Equations (10) and (11) can be expressed

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as

S1− =

1 2  n2 L − 4 − 3(n2 L )   2

(B5)

(B6)

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1 2 S1+ =  n2 L + 4 − 3(n2 L )   2

λ31 corresponds to the neutral axis location δ 01 . Inserting δ = δ 01 = 1 (see Eqn. (9)) and n1L = 0 into

λ31 = −∞

(B7)

AC C

from Eqn. (7) as

EP

Eqns. (15), (16) and solving for n2 L and m1L from Eqns. (15), (16), (B5) and (B6), λ31 can be obtained

λ32 corresponds to the neutral axis location δ 02 . Inserting δ = 1 2 − κ + α (see Eqn. (9)) and n1L = 0 into Eqns. (15), (16) and solving for n2L and m1L from Eqns. (15), (16), (B5) and (B6), λ32 can be obtained from Eqn. (7) as

λ32 =

2(κ − α + αβ ) (6αβ − 3)(κ − α )2 − 12κ 2 − 12ακ + 6κ + 3 αβ + 3 2 4 

26

(B8)

ACCEPTED MANUSCRIPT λ33 corresponds to the neutral axis location δ 03 . Inserting δ = 1 2 − κ − α (see Eqn. (9)) and n1L = 0 into Eqns. (31), (32) and solving for n2L and m1L from Eqns. (31), (32), (B5) and (B6), λ33 can be obtained from Eqn. (7) as

(6αβ − 3)(κ + α )2

2(κ + α − αβ ) 3 3  − 12κ 2 + 12ακ − 6κ + αβ + 2 4  

(B9)

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λ33 =

λ34 corresponds to the neutral axis location δ 04 . Inserting δ = δ 04 = 0 (see Eqn. (9)) and n1L = 0 into Eqns. (31), (32) and solving for n2L and m1L from Eqns. (31), (32), (B5) and (B6), λ34 can be obtained

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from Eqn. (7) as

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λ34 = ∞

(B10)

Appendix C Effect of ψ on the validity ranges of the limit load solution Inserting δ = 1 and Eqns. (10) and (11) into Eqn. (15), the equation may be re-arranged and expressed as ψ 1 − αβ 2

n1L1 = 4 − 3(λ2 n1L1 )

2

(C1)

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where Eqn. (B3) has been adopted and n1L1 represents a special value of n1L for δ = 1 . From Eqn. (C1), it

can be seen that n1L1 > 0 when ψ > 0 and n1L1 < 0 when ψ < 0 , noting that 0 ≤ α ≤ 1 2 and 0 ≤ β ≤ 1.

EP

Similarly, inserting δ = 0 and Eqns. (10) and (11) into Eqn. (31), the equation may be re-arranged and expressed as

−ψ 2 n1L 4 = 4 − 3(λ2 n1L 4 ) 1 − αβ 2

AC C

(C2)

where Eqn. (B3) has been adopted and n1L 4 represents a special value of n1L for δ = 0 . From Eqn. (C2), it can be seen that n1L 4 < 0 when ψ > 0 and n1L 4 > 0 when ψ < 0 , noting that 0 ≤ α ≤ 1 2 and 0 ≤ β ≤1.

This indicates that the radial line for λ1 = λ11 is located in the right-hand half-space and the one for λ1 = λ14

is located in the left-hand half-space in the n1L ~ m1L plane when ψ > 0 . However, the radial line for

λ1 = λ11 is located in the left-hand half-space and the one for λ1 = λ14 is located in the right-hand half-space when ψ < 0 (see Fig. 6).

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ACCEPTED MANUSCRIPT Limit load solutions for N1 = 0 For combined bending moment and σ 2 , n1L = 0 and the limit loads are defined in m1L ~ n2 L space. The Appendix D

validity regions indicated by the neutral axis location parameter, δ , can alternatively be indicated by the load ratio, λ3 , as λ3 = λ3 (δ , α , β , κ ) follows from Eqns. (7) and (10)-(12) and the two equilibrium equations for each solution. The four special neutral axis locations corresponding to δ 01 , δ 02 , δ 03 and δ 04 in Eqn. (9)

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can now be mapped into the m1L ~ n2 L space and indicated by four special λ3 values, that is, λ31 , λ32 ,

λ33 and λ34 , respectively, which are defined in Appendix B.

For combined bending moment and σ 2 , the expression of m1L can obtained from

λ1 →∞

SC

m1L = lim (4λ1n1L (λ1 , λ2 )) = lim (4λ1n1L (λ1 , 6λ1λ3 )) λ1 → ∞

(D1)

be obtained as follows.

4C ′ A′C ′ + 2 B′ + 2

(A′C ′ + 2 B′ ) − (A′ 2 2

4C ′ A′C ′ + 2 B′ − 2

(A′C ′ + 2B′ ) − (A′

2

2 2

2

)

)

2

+ 108 B′ λ C ′ 2 2 3

B′ > 0

2

+ 108 B′ λ C ′ 2 2 3

TE D

 ±  m1L =  ±  

M AN U

where Eqn. (7) has been adopted. Inserting each equation in Eqn. (44) into Eqn. (D1), the m1L expression can

(D2)

B′ < 0

where “+” is for M 1 > 0 and the “−” sign corresponds to M 1 < 0 . The coefficients A′ , B′ and C ′ in

EP

Eqn. (D2) are defined, following Eqn. (D1), as

AC C

 A B A′ = lim  2  , B′ = lim   , C ′ = C λ1 → +∞ λ λ1 → +∞ λ  1  1

(D3)

Inserting Eqns. (18), (28) and (33) into Eqn. (D3) and taking the limit for λ1 → +∞ , the expressions for A′ ,

B′ and C ′ can be obtained and may be expressed as follows. For

λ31 ≤ λ3 ≤ λ32 ,

 A′ = 36[1 − αβ (1 + 6κ + 2αβ )]λ23   B′ = 2[1 − 3αβ (1 − 2κ − 2αβ )λ3 ]  ′ 2 2 C = 1 − 4α β − 8καβ

(D4)

28

ACCEPTED MANUSCRIPT For

λ32 < λ3 < λ33 ,

  3  2   2 2 2 2  A′ = 36 1 + 4α β −  3α + 3κ + + α  β  λ3 4        B′ = 2[(1 − β ) − 3(1 − 2α )βκλ3 ]  ′ 2 2 C = 1 − 4α β (1 − β ) − 4 βκ 

For

)

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(

(D5)

λ33 ≤ λ3 ≤ λ34 ,

AC C

EP

TE D

M AN U

SC

 A′ = 36[1 − αβ (1 − 6κ + 2αβ )]λ32   B′ = 2[1 + 3αβ (1 + 2κ − 2αβ )λ3 ]  ′ 2 2 C = 1 − 4α β + 8αβκ

29

(D6)

ACCEPTED MANUSCRIPT Tables Table 1 Material properties Young’s modulus E (GPa)

Yield stress σ y (MPa)

Poisson’s ratio

200

200

0.3

Table 2 Comparison of FE results using different element types Error (%)

C3D20R 0.9946 0.7305 0.2283 0.0476

SC

0.25% 0.6% 0.6% 0.4%

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C3D8R 0.9971 0.7351 0.2298 0.0478

EP

0 0.2 1 5

n1L

AC C

λ1

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ν

30

ACCEPTED MANUSCRIPT Figure list

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Figure 1 Schematic illustration of an embedded rectangular crack in a plate under combined biaxial force/stress and through-thickness bending Figure 2 Assumed stress distribution A along the z-direction at plastic collapse, (a) Crack type I, (b) Crack type II and (c) Crack type III Figure 3 Assumed stress distribution B along the z-direction at plastic collapse, (a) Crack type I, (b) Crack type II and (c) Crack type III Figure 4 Critical locations of the neutral axis Figure 5 Definition of Ae and y ∗ Figure 6 Validity ranges of the six limit load expressions, (a) ψ > 0 and (b) ψ < 0 Figure 7 A typical FE model used in the analyses ( α = 0.125 , β = 0.25 , κ = 0.1 ) Figure 8 Determination of limit loads from LPF-deformation curves using the TES method, (a) LPF-Φ curve and (b) LPF-Γ curve

Figure 9 Comparison of limit load solutions between FE results and predictions, (a) κ = 0 , (b) κ = 0.1 and (c)

SC

κ = 0 .2

Influence of λ 2 on the limit load ( α = 0.25 , β = 0.25 and κ = 0.1 ), (a)

EP

TE D

λ2 ≥ 0

AC C

Figure 11

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Figure 10 Special cases of an embedded crack in a plate, (a) An extended surface crack in a plate ( β ≡ 1 , κ ≡ 0.5 − α ), (b) A through-thickness crack in a plate ( α ≡ 1 2 , κ ≡ 0 ), (c) A surface crack in a plate ( κ ≡ 0.5 − α ) and (d) An extended embedded crack in a plate ( β ≡ 1 )

31

λ2 ≤ 0

and (b)

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SC

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ACCEPTED MANUSCRIPT

AC C

EP

TE D

Fig.1 Schematic illustration of an embedded rectangular crack in a plate under combined biaxial force/stress and through-thickness bending

32

M AN U

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(a) Crack type I

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ACCEPTED MANUSCRIPT

(c) Crack type III

AC C

EP

TE D

(b) Crack type II

Fig. 2 Assumed stress distribution A along the z-direction at plastic collapse

33

M AN U

SC

(a) Crack type I

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ACCEPTED MANUSCRIPT

EP

TE D

(b) Crack type II

(c) Crack type III

AC C

Fig.3 Assumed stress distribution B along the z-direction at plastic collapse

34

Critical locations of the neutral axis

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Fig.4

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ACCEPTED MANUSCRIPT

AC C

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TE D

Fig. 5 Definition of Ae and y ∗

35

EP

TE D

M AN U

(a) ψ > 0

SC

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ACCEPTED MANUSCRIPT

(b)

ψ <0

AC C

Fig. 6 Validity ranges of the six limit load expressions

36

EP

TE D

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SC

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ACCEPTED MANUSCRIPT

AC C

Fig. 7 A typical FE model used in the analyses ( α = 0.125 , β = 0.25 , κ = 0.1 )

37

EP

TE D

M AN U

SC

(a) LPF-Φ curve

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ACCEPTED MANUSCRIPT

(b) LPF-Γ curve

AC C

Fig. 8 Determination of limit loads from LPF-deformation curves using the TES method

38

κ =0

TE D

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(a)

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ACCEPTED MANUSCRIPT

κ = 0 .1

AC C

EP

(b)

(c)

κ = 0 .2

Fig. 9 Comparison of limit load solutions between FE results and predictions

39

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ACCEPTED MANUSCRIPT

M AN U

SC

(a) An extended surface crack in a plate ( β ≡ 1 , κ ≡ 0.5 − α )

AC C

EP

TE D

(b) A through-thickness crack in a plate ( α ≡ 1 2 , κ ≡ 0 )

(c) A surface crack in a plate ( κ ≡ 0.5 − α )

(d) An extended embedded crack in a plate ( β ≡ 1 ) Fig.10 Special cases of an embedded crack in a plate 40

SC

λ2 ≤ 0

TE D

M AN U

(a)

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ACCEPTED MANUSCRIPT

(b) λ 2 ≥ 0

AC C

EP

Fig. 11 Influence of λ 2 on the limit load ( α = 0.25 , β = 0.25 and κ = 0.1 )

41

ACCEPTED MANUSCRIPT

Highlights (for review) 1. A limit load solution for plates containing rectangular cracks is derived. 2. Combined biaxial stress and bending moment are loaded in the plates. 3. Predictions using the solution are close to FE results and slightly conservative.

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4. The limit load solutions for some other defect types in plates are obtained.

AC C

EP

TE D

M AN U

SC

5. Effect of stress parallel to the crack plane on limit load is discussed.