Theoretical analysis on pullout of anchor from anchor–mortar–concrete anchorage system

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Engineering Fracture Mechanics 75 (2008) 961–985 www.elsevier.com/locate/engfracmech

Theoretical analysis on pullout of anchor from anchor–mortar–concrete anchorage system Shutong Yang

a,*

, Zhimin Wu a, Xiaozhi Hu b, Jianjun Zheng

c

a

b

State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, PR China Department of Mechanical and Materials Engineering, University of Western Australia, Nedlands, Perth, WA 6907, Australia c School of Civil Engineering and Architecture, Zhengjiang University of Technology, Hangzhou 310014, PR China Received 11 August 2006; received in revised form 13 March 2007; accepted 3 May 2007 Available online 13 May 2007

Abstract Theoretical studies on pullout of an anchor from an anchor–adhesive/mortar–concrete anchorage system have been carried out with the wide application of this anchorage system in civil engineering. Most of them focused on one interfacial debonding crack propagation from the loading end similar to the theoretical method on fiber pullout from matrix, or adopting a uniform interfacial shear stress model to calculate the tensile capacity of the anchor but with a limitary embedment length. Moreover, shear stresses were generally assumed uniformly along the thickness of the adhesive/mortar layer. Actually, the distributions of the shear stresses would be varied as both the thickness of the adhesive/mortar layer and the embedment length increase. Taking into account of this variation, the present study addressed the pullout of an anchor from an anchor–mortar–concrete anchorage system with two different boundary conditions for various embedment lengths analytically combining with the compatibility conditions and a simplified shear stress–slip relationship at the anchor– mortar interface. In the proposed analytical model, the variation of the shear stresses along the thickness of the mortar was gained. Additionally, the distributions of the tensile stresses in the anchor and the interfacial shear stresses along the embedment length were obtained under different loading conditions. After interfacial debonding, the sequences and probabilities of two interfacial debonding cracks developing from both ends of the system were analyzed according to the boundary conditions and the axial rigidities of both the anchor and concrete. Besides, the pullout load was expressed as a function of one or two debonding crack lengths. Then the maximum load Pmax was gained correspondingly, as well as the critical crack lengths, by using the theories of extremum. Results show that the obtained solutions consist with the previous work from other literature if the elastic modulus of concrete is assumed to be infinite and the anchor–mortar interface is thought as rigid before debonding. Subsequently, several fundamental structural and interfacial parameters were introduced to study their influences on the calculated results using the proposed model, such as the critical crack lengths, the initial cracking load Pini and the maximum load Pmax. It was found that Pmax increases linearly with the embedment length L, however, Pini is irrelevant to L if L is very long. Pmax increases monotonously with the thickness t of the mortar layer if the concrete layer keeps a constant thickness b, however, there appears a peak value in the curve of Pmax varying with t if the sum of b and t is invariable. Moreover, the difference between the values of Pmax under the two boundary conditions is marginal. Ó 2007 Elsevier Ltd. All rights reserved.

*

Corresponding author. Tel.: +86 411 84709842. E-mail addresses: [email protected] (S. Yang), [email protected] (Z. Wu).

0013-7944/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2007.05.004

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Keywords: Pullout; Anchor; Mortar; Concrete; Anchorage; Debonding crack

Nomenclature As a ab abc ac at atc b D Ec Es G k k0 L Mc

P Pini Pmax R

r t u uc um us x d dm k rc rce rl rs ru ry s s0

area of anchor cross-section debonding crack length in boundary 2 debonding crack length developing from the non-loading end in boundary 1 critical debonding crack length developing from the non-loading end in boundary 1 critical debonding crack length in boundary 2 debonding crack length developing from the loading end in boundary 1 critical debonding crack length developing from the loading end in boundary 1 thickness of the concrete layer in the anchorage system diameter of the anchor elastic modulus of the concrete elastic modulus of the anchor shear modulus of the mortar interfacial shear modulus of the anchor–mortar interface a parameter related to shear deformation of mortar embedment length of the anchorage system resultant moment of uniformly distributed bending moments induced by the pullout load in the cross-section of the anchorage system along a representative circle where the strains of the concrete are equal to zero under pure bending condition pullout load initial cracking load maximum pullout load distance between the anchor–mortar interface and a representative circle where the strains of the concrete are equal to zero under pure bending condition in the cross-section of the anchorage system radial coordinate of the anchorage system thickness of the mortar layer longitudinal displacement of mortar in arbitrary position along radial direction longitudinal displacement of the concrete at the mortar–concrete interface longitudinal displacement of the mortar at the anchor–mortar interface longitudinal displacement of the anchor longitudinal coordinate of the anchorage system shear slip at the anchor–mortar interface interfacial shear slip corresponding to the interfacial shear strength a parameter to be solved in Lagrange function tensile stress of the concrete at the mortar–concrete interface in boundary 1 normal stress of the concrete at the outer edge of the cross-section in boundary 1 tensile stress of the concrete at the mortar–concrete interface under pure bending condition tensile stress of the anchor uniform tensile stresses in the cross-section of the concrete compressive stress of the concrete at the outer edge of the cross-section under pure bending condition shear stress at the anchor–mortar interface interfacial shear stress at the non-loading end

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

sr ss su U

963

shear stress in arbitrary position along the thickness of the mortar residual frictional stress at the anchor–mortar interface shear strength at the anchor–mortar interface a symbol of a Lagrange function

1. Introduction The investigations and applications of an anchor embedded in a hardened cementitious material such as concrete have been carried out for many decades. For the convenience of the anchors embedded more efficiently in the concrete, researchers began to adopt post-installation for bonding the reinforcements into the concrete. General operations are drilling a hole in hardened concrete and installing a steel rod into the concrete with adhesive or cement grout. Cook et al. [1–4] analyzed the behaviors of single adhesive anchors and single headed or unheaded grouted anchors under tensile load in concrete. General failure modes of these anchorage systems can be summarized as pullout of a concrete cone, debonding at anchor–adhesive/grout or concrete–adhesive/grout interface, fracture of anchor, and combination of some of these failure modes [1–3]. Then several design models were recommended taking into account of all the possible failure modes [1–5]. Moreover, for an adhesive anchor installed into a damp, wet and uncleaned hole, the bond strength between anchor and concrete was generally reduced [6]. Besides, Cook et al. [1] also developed a procedure for evaluating the strength of closely spaced adhesive anchors and pointed out that a full embedment length spacing can ensure these anchors to achieve their single anchor strengths. Similar phenomena were also found in the experimental and numerical simulation studies by Li et al. [7]. James et al. [8] developed an approximate expression to predict the ultimate tensile capacity of the epoxy-adhesive anchors based on the analysis of linear and nonlinear finite element method. Colak [9] found that the addition of filler into epoxy adhesive can reduce the shear strength of adhesive anchors. Bickel and Shaikh [10] utilized two methods with proper adjustments to predict the shear capacity of single adhesive anchors. Sakla and Ashour [11] introduced artificial neural networks into predicting the tensile capacity of single adhesive anchors and found that the tensile capacity is linearly proportional to the embedment length. Furthermore, Beard and Lowe [12] adopted the ultrasonic guided waves to successfully inspect the maximum anchor length for a grouted anchor. Furthermore, theoretical analyses of fiber pulled out from matrix provide potential guidance for analyzing the adhesive/grouted anchors due to the similar structural patterns between these two types of anchorages. Based on reviewing the theories developed to explain the fiber debonding process in the matrix, Gray [13] pointed out that there were two types of resistance to fiber–matrix interfacial debonding, i.e., elastic bonding and frictional resistance to slipping. Moreover, the contribution of the elastic bonding to the total resistance to debonding decreased with the increase of the fiber embedment length [13,14]. According to the characteristics of stresses and deformations in the fibers and matrices, a shear-lag model was applied widely into the theoretical analyses [15–20], which assumes that the axial stresses in the matrix are negligible compared with those in the fiber and that the shear stresses in the fiber are so small that they can be ignored relative to those in the matrix. Hsueh [15] obtained the distributions of tensile stresses in the fiber and shear stresses at the fiber– matrix interface along the embedment length during the elastic stage in virtue of the shear-lag model. Besides, the behaviors of the composites after debonding were extensively analyzed [16–21]. Due to the Poisson’s effect [17], however, the frictional stress became lower as the tensile stress in the fiber increased in the debonding region. The shear-lag model was also modified by Hsueh [18] considering the radial stresses and strains. Moreover, a fracture mechanics method known as the fracture energy-based criterion was introduced to analyze the interfacial debonding behaviors [19–23] for all kinds of problems including pullout of inclined fibers from matrix [22]. Gao et al. [24] derived a fracture mechanics-based debonding criterion considering fiber–matrix debonding as an interface crack process, and taking into account of friction at the debonding zone and Poisson’s contraction of the fiber. Besides, the whole pullout load–end slip relationship of fiber can be gained accurately by Naaman et al. [25,26], wherein the process of fiber pullout from the beginning of loading to the complete interfacial debonding was obtained using the strength criterion and the subsequent states were analyzed based on the dynamic mechanism which considered Poisson’s effect, shrink-fit and fiber–matrix misfit

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theory [25]. If the free end of the fiber was restricted by the matrix, the mechanical anchorage was modeled by a spring connecting the fiber to a similar fictitious fiber along its axis [27], which induced a second peak in the pullout load–slip curve. Furthermore, Sastry and Phoenix [28] thought that there existed a plastic zone between the elastic and frictional debonding zones and the three-zone model was verified by a composite microbundle pullout experiment. Zhou et al. [29] introduced three boundary conditions, namely the fixed matrix bottom, fixed fiber and matrix bottom and the restrained matrix top surfaces, to analyze the fiber pullout from the matrix, and found that the restrained matrix top condition induces a lower initial debond, maximum debond and frictional pullout stresses compared with the fixed bottom condition [29]. Moreover, for the fixed bottom condition, Zhang et al. [30] presented an improved model to gain the stress fields in both bonded and debonded regions at the fiber–matrix interface considering a pullout rate-dependent frictional coefficient in the debonded region [30]. Then the proposed model gained a good agreement with numerical results [30]. Subsequently, the whole single fiber pullout process was obtained based on computer simulation [31]. It is also found that most of the studies emphasize that the interfacial debonding crack only appears at the position where the fiber enters the matrix, i.e., loading end, and propagates towards the other end. Actually, under the fixed bottom condition presented by Zhou et al. [29], if the matrix at the non-loading end suffered so large deformation that the local interfacial shear stress exceeds the shear strength, there will be another debonding crack developing there. Bazˇant and Desmorat [32] analyzed a pull–pull geometry of fiber or bar pullout test and presented two interfacial debonding cracks developing from both ends with the same lengths assuming both the fiber and matrix have the same axial rigidity. Then distributions of the interfacial shear stresses and the tensile stresses in the fiber were gained analytically. Based on Bazˇant and Desmorat’s [32] pioneering work, this paper concentrates on the behaviors of doublecrack propagation in an anchorage system but with different axial rigidities for both the anchor and the matrix, and the two cracks may not develop at both the loading and the non-loading ends simultaneously. Moreover, another medium is incorporated between the rod and the matrix in the present study just as the grouted anchors [3]. Fig. 1 shows a simplified model of an anchor–mortar–concrete anchorage system, with embedment length L, anchor diameter D, thicknesses of mortar and concrete layers t and b, respectively. Two boundary conditions are shown in Fig. 2a and b, respectively. Wherein boundary 1 means fixing the bottom surface of concrete and boundary 2 allows uniform distributed forces to act on the top surface of concrete. Herein, it should be pointed out that Cook et al. [1] have presented a bond stress model for an adhesive anchor, wherein the adhesive was treated as a shear lag surrounded by a rigid matrix. Moreover, a uniform bond stress model was also proposed to predict the ultimate load and gained a good agreement with the experimental data [2,33], however, the ratio of the embedment length to the anchor diameter should be limited [34]. anchor

mortar

concrete

L

t

t

b

b D

b

tD t

b

a. Anchor-mortar-concrete anchorage system

b. Details of cross section

Fig. 1. Schematic illustration of the proposed simplified anchorage system.

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

P

P

πb( D + 2t + b)

x r

P

965

P

πb( D + 2t + b)

x r

0

a. Boundary 1

0

b. Boundary 2

Fig. 2. Two boundaries for the simplified anchorage model where r-axis means the radial direction of the cross-section in the anchorage system.

As the thickness of the grout or adhesive layer increases, the shear stresses will be varied along the radial direction. Additionally, the values of the shear stresses become variable greatly along the embedment length L if L is large enough. Considering the variations of the shear stresses along both L and the thickness t of the mortar, this paper presented an analytical model for the pullout of the anchor from the proposed anchorage system using the compatibility conditions of the anchor–mortar interface and the mortar layer. Then the shear stresses varying along the radial direction were obtained and the distributions of the interfacial shear stresses and the tensile stresses in the anchor along the embedment length were gained during different loading stages. Subsequently, the sequences and probabilities of two debonding cracks developing from both the loading and the non-loading ends of the system were analyzed according to the characteristics of each boundary condition. Besides, the pullout load P was constructed as a function of the debonding crack lengths in an analytical manner. Then the maximum load Pmax and the corresponding critical crack lengths were acquired using extremum theories. Finally, the influences of some structural and interfacial parameters on the calculated results were studied extensively, including the embedment length L, the thickness t of mortar layer and the interfacial parameters at the anchor–mortar interface. 2. Analytical derivation 2.1. Fundamental assumption In the present study, the mortar layer is considered as a shear-lag model due to its free top and bottom boundaries. Therefore, the mortar deforms only in shear with a shear modulus G. Both the anchor and the concrete are treated as linear elastic materials with elastic moduli Es and Ec, respectively. Besides, the normal stresses are assumed to distribute uniformly over the anchor cross-section and the linear distributions of the stresses and strains in the concrete along the radial direction are implemented in this paper. The bonding at mortar–concrete interface is considered to be perfect, i.e., there is no shear slip including elastic and debonding slips appearing at this interface. The embedment length L is long enough relative to the diameters of both the anchor and concrete in the present study. Poisson’s effects, radial and circumferential deformations of all the medium materials are neglected. Moreover, a simplified mechanical model for the anchor–mortar interface is adopted as shown in Fig. 3, i.e., a linear branch up to the shear strength plus a horizontal line with a suddenly drop from the peak point, which reads

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τ

τu

τs k 1

0

δm

δ

Fig. 3. Simplified shear–slip curve of anchor–mortar interface where su is the shear strength of the anchor–mortar interface with dm as the corresponding slip and ss means the residual frictional stress.

s ¼ kd ð0 6 d 6 dm Þ

ð1aÞ

s ¼ ss

ð1bÞ

ðd > dm Þ

where k represents the interfacial shear modulus equal to su/dm. It should be noted that the shear stress will keep constant as frictional resistance at the debonding interface. Actually, it is not the case because the frictional stress becomes variable at large interfacial slip and very sensitive to the radial deformations. However, the radial effects are neglected in the following derivations and the subject only concentrates on the process until the complete debonding of the interface when the interfacial slip is not large enough [25]. Therefore, this approximation for the anchor–mortar interface is rational in this paper. Moreover, all of the medium materials are assumed to have sufficient strengths so that the only failure mode of this system is the pullout of the anchor. 2.2. Theoretical derivation Two infinitesimal elements are taken out from Fig. 2a or b for both the anchor and the mortar as shown in Fig. 4. Herein, rs is the tensile stress in the anchor cross-section, s is the shear stress at the anchor–mortar interface and sr means the shear stress in the mortar layer at a distance r from the longitudinal axis of the anchor, i.e., x-axis. s is expressed in a differentiating form of rs by considering the equilibrium of the anchor cylindrical element in Fig. 4a, i.e., s¼

D drs 4 dx

ð2Þ

Moreover, the variation of the shear stresses in the mortar layer along the radial direction can be obtained through the analysis of the mortar cylindrical shell element in Fig. 4b and neglecting high-order infinitesimal terms, i.e., 1 1 dsr ¼  dr sr r

ð3Þ

Integrating the two sides of the equal sign in Eq. (3) results in another form of sr varying with r, i.e., sr ¼

Ds 2r

ð4Þ

Herein, u is denoted as the axial displacement of the mortar at a distance r from x-axis. Then the relationship between u and sr can be expressed as following: sr ¼ G

du dr

ð5Þ

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

967

σ s + dσ s

x axis

τ

τ

dx

τr

τ r + dτ r

τ r + dτ r

τr

dx

σs dr D

a. Anchor cylindrical element

r

r

dr

b. Mortar cylindrical shell element

Fig. 4. Analysis of anchor and mortar infinitesimal elements.

Substituting Eq. (4) into Eq. (5) and integrating the two sides of the equal sign, we have u ¼ um þ

Ds D ln 2G 2r

ð6Þ

where um is the magnitude of u at r = D/2. Additionally, um can also be considered as the displacement of the mortar at the anchor–mortar interface along the positive direction of x-axis at a distance x from the origin. Besides, uc is defined as the displacement of the concrete at the mortar–concrete interface at a distance x from the origin. Therefore, uc can be gained from Eq. (6) based on the fundamental assumptions as following: uc ¼ um þ

Ds D ln 2G D þ 2t

ð7Þ

As for the anchor, the displacement along the positive direction of x-axis at a distance x from the origin is denoted as us. Obviously, the difference between us and um directly results in a bonding slip at the anchor– mortar interface, i.e., us  u m ¼ d

ð8Þ

If the anchor–mortar interface behaves elastically, Eq. (1a) dominates the interfacial behaviors. Then substituting Eqs. (1a) and (7) into Eq. (8) produces s¼

2kG ðus  uc Þ 2G þ Dk ln Dþ2t D

ð9Þ

Subsequently, the proposed anchorage systems with two boundaries should be discussed respectively due to the results using one boundary condition different from those using the other boundary condition. 2.2.1. Boundary 1 2.2.1.1. Elastic stage. Differentiating s in Eq. (9) with respect to x generates     ds 2kG dus duc 2kG 2kG rs rc ¼  ðes  ec Þ ¼  ¼ dx 2G þ Dk ln Dþ2t dx dx Es E c 2G þ Dk ln Dþ2t 2G þ Dk ln Dþ2t D D D

ð10Þ

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Herein, rc means the tensile stress of the concrete at the mortar–concrete interface, which can be gained as a function of rs as following (seen in A): P D2 A A rs pb 4b

ð11Þ

A¼

1 1  þ 2  b3 4b2 2b D þ 2t þ b  2b þ 2 t þ D2 þ 2R 2  3R þ b R 3R2

ð12Þ

R¼

3t þ 1:5D þ 2b b 6t þ 3D þ 3b

ð13Þ

rc ¼ where

Substituting Eqs. (2) and (11) into Eq. (10) gives d2 rs  a21 rs ¼ N 1 P dx2

ð14Þ

where   8kG 1 D2 A þ Es 4bEc 2DG þ D2 k ln Dþ2t D 8kGA   N1 ¼ pbEc 2DG þ D2 k ln Dþ2t D a21 ¼

ð15Þ ð16Þ

Solving this differentiating equation and considering the boundary conditions, rs ðx ¼ 0Þ ¼ 0 4P rs ðx ¼ LÞ ¼ pD2

ð17aÞ ð17bÞ

we have rs ¼

4 pD2

 Na21 ð1  ea1 L Þ 1

ea1 L  ea1 L

Da1 s¼ 4

4 pD2

Pe

a1 x

 Na21 ð1  ea1 L Þ 1

ea1 L  ea1 L

þ

Pe

N1 a21

a1 x

ð1  ea1 L Þ  pD4 2 ea1 L  ea1 L

Da1  4

N1 a21

Pea1 x þ

ð1  ea1 L Þ  pD4 2 ea1 L  ea1 L

N1 P a21

ð18Þ

Pea1 x

ð19Þ

Through the comparison between the interfacial shear stresses at the two ends, i.e, x = 0 and x = L, from Eq. (19), it is found that (1) If 4bEc > D2AEs, interfacial debonding firstly appears at the loading end, and interfacial debonding crack initiates there and propagates towards the other end as the pullout load increases. (2) If 4bEc = D2AEs, interfacial debonding cracks will develop from both the loading and the non-loading ends and propagate towards each other with the same crack lengths. (3) If 4bEc < D2AEs, interfacial debonding crack will initiate from the non-loading end and develop towards the loading end with the increase of the pullout load. For convenience, only Case (1) is considered in the following derivations. When s(x = L) attains the shear strength su, the corresponding pullout load is defined as the initial cracking load Pini in this paper as following: P ini ¼

ea1 L  ea1 L  N1 4 1  ðea1 L þ ea1 L Þ þ 2N 2 2 a a2 pD

4su  Da1

1

1

ð20Þ

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

969

2.2.1.2. Single debonding crack propagation. As the pullout load increases, the crack gradually propagates towards the non-loading end just as shown in Fig. 5. In the region 0 6 x 6 L  at, solving Eq. (14) due to the elastic behaviors of the interface and combining with the stress states at x = 0 and x = L  at result in rs ¼

s¼

4su Da1

 Na21 Pea1 ðLat Þ 1

ea1 ðLat Þ

þ

ea1 ðLat Þ

e

a1 x



4su Da1

þ Na21 Pea1 ðLat Þ

ea1 ðLat Þ

1

þ

ea1 ðLat Þ

ea1 x þ

N1 P a21

ð21Þ

N1 N1 4su 4su a1 ðLat Þ a1 ðLat Þ Da1 Da1  a21 Pe Da1 Da1 þ a21 Pe a1 x e þ ea1 x 4 ea1 ðLat Þ þ ea1 ðLat Þ 4 ea1 ðLat Þ þ ea1 ðLat Þ

ð22Þ

Moreover, the interfacial stress will keep constant as residual frictional stress ss in the region Lat < x 6 L. Therefore, rs in this region can be gained by using Eq. (2) as rs ¼

4ss 4su ea1 ðLat Þ  ea1 ðLat Þ N 1 ea1 ðLat Þ þ ea1 ðLat Þ  2 4ss xþ ðL  at Þ þ 2 P a ðLa Þ a ðLa Þ a ðLa Þ a ðLa Þ t t t t 1 1 1 1 D Da1 e þe þe D a1 e

ð23Þ

The pullout load P becomes a function of the debonding crack length at by considering Eq. (17b), i.e., P¼

4ss D

4su at þ Da tanhða1 ðL  at ÞÞ 1

4 pD2

ð24Þ

 Na21 ð1  coshða11ðLat ÞÞÞ 1

Then the maximum pullout load Pmax and the corresponding critical crack length atc are obtained by solving dP/dat = 0. If the interfacial shear stress at the non-loading end exceeds su before ultimate state, an interfacial debonding crack will develop there. Therefore, two debonding crack propagations from both the loading and the nonloading ends should be evaluated further. 2.2.1.3. Double debonding crack propagation. The interfacial shear stress at the non-loading end tends to be larger with the development of the debonding crack from the loading end, as well as the increase of P. Once the shear stress exceeds su, another debonding crack initiating from the non-loading end will develop towards the loading end. A schematic illustration for the two debonding cracks’ relative propagations is shown in Fig. 6.

P

at

x

0

Fig. 5. Single debonding crack propagation where at represents the crack length adjacent to the loading end.

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S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985 P

at

x ab 0

Fig. 6. Two debonding cracks propagations where ab is the length of the crack adjacent to the non-loading end.

Obviously, there are three regions appearing at the anchor–mortar interface, including one elastic portion ab 6 x 6 L  at and two frictional debonding ones. The expressions of the anchor stress and interfacial stress in the elastic region are gained as follows: 4su a1 ab 4su a1 ab ðe  ea1 ðLat Þ Þ ðe  ea1 ðLat Þ Þ N1 Da1 Da1 a1 x rs ¼ a ðLat a Þ e þ a ðLat a Þ ea1 x þ 2 P b b a1  ea1 ðLat ab Þ  ea1 ðLat ab Þ e1 e1 a1 ab a1 ðLat Þ a1 ab a1 ðLat Þ su ðe e Þ a1 x su ðe e Þ s ¼ a ðLat a Þ e  a ðLat a Þ ea1 x b  ea1 ðLat ab Þ b  ea1 ðLat ab Þ e1 e1

ð25Þ ð26Þ

Additionally, the solutions in other two debonding zones are given by combining s = ss with the continuities at x = ab and x = L  at, i.e., for 0 6 x < ab: rs ¼

4ss x D

ð27Þ

for L  at < x 6 L: rs ¼

  4ss 4su 1 1 N1 4ss xþ  ðL  at Þ þ 2P D Da1 tanhða1 ðL  at  ab ÞÞ sinhða1 ðL  at  ab ÞÞ D a1

ð28Þ

Two expressions of P with respect to both ab and at will be obtained analytically based on Eqs. (17a) and (17b) as follows:   4a21 ss 4su a1 1 1  ab  P¼ ð29Þ DN 1 DN 1 sinhða1 ðL  at  ab ÞÞ tanhða1 ðL  at  ab ÞÞ   4ss 4su 1 1 at þ  D Da1 tanhða1 ðL  at  ab ÞÞ sinh ða1 ðL  at  ab ÞÞ ð30Þ P¼ 4 N1  pD2 a21 Correspondingly, an equation for ab and at is also established    4ss 8su 16su a1 1 1 16a21 ss ðat þ ab Þ þ   3 ab ¼ 0  tanhða1 ðL  at  ab ÞÞ sinhða1 ðL  at  ab ÞÞ D Da1 pD N 1 pD3 N 1

ð31Þ

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971

For the sake of calculating Pmax, a Lagrange function U(at, ab, k) is constructed using Eqs. (29) or (30), as well as Eq. (31), i.e., Uðat ; ab ; kÞ ¼ M 0 þ k  M 1 where   4a21 ss 4su a1 1 1  M0 ¼ ab  DN 1 DN 1 sinhða1 ðL  at  ab ÞÞ tanhða1 ðL  at  ab ÞÞ    4ss 8su 16su a1 1 1 16a21 ss ðat þ ab Þ þ  M1 ¼  3 ab  tanhða1 ðL  at  ab ÞÞ sinhða1 ðL  at  ab ÞÞ D Da1 pD N 1 pD3 N 1

ð32Þ

ð33Þ ð34Þ

kis an unknown parameter to be solved. Through solving the equations oU/oi = 0 (i = ab, at, k), Pmax is conveniently gained accompanied by the corresponding critical crack lengths abc and atc, respectively. After that, the pullout load P decreases with unstable propagations of the two cracks. When ab + at = L, the interface is completely debonded along the embedment length. Subsequently, a large end slip appears and the anchor is gradually pulled out from the mortar, which is certainly beyond the present subject. 2.2.2. Boundary 2 Herein, the analysis method for boundary 2 is very similar to that for boundary 1. Therefore, only the solutions are listed as follows. 2.2.2.1. Elastic stage 4P ea2 x  ea2 x pD2 ea2 L  ea2 L P a2 ea2 x þ ea2 x s¼ pD ea2 L  ea2 L su pD ea2 L  ea2 L P ini ¼ a2 ea2 L þ ea2 L where  8kG 1 D2 2 a2 ¼ þ Es 4bEc ðD þ 2t þ bÞ 2DG þ D2 k ln Dþ2t D rs ¼

ð35Þ ð36Þ ð37Þ

ð38Þ

2.2.2.2. Crack propagation. For 0 6 x 6 L  a: rs ¼ s¼

4su ea2 x  ea2 x Da2 ea2 ðLaÞ þ ea2 ðLaÞ

su ðea2 x þ ea2 x Þ a e 2 ðLaÞ þ ea2 ðLaÞ

ð39Þ ð40Þ

where a means the crack length adjacent to the loading end for boundary 2. For L  a < x 6 L: 4ss 4su 4ss xþ ðL  aÞ tanhða2 ðL  aÞÞ  D Da2 D su pD P ¼ ss pDa þ tanhða2 ðL  aÞÞ a2 rs ¼

ð41Þ ð42Þ

Similarly, solving dP/da = 0 generates the maximum load Pmax and critical crack length ac. Moreover, the interfacial shear stress s0 at the free end is obtained by inserting x = 0 into Eq. (40), i.e., 2su ð43Þ s0 ¼ a ðLaÞ e2 þ ea2 ðLaÞ

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Beyond all doubt, ea2 ðLaÞ þ ea2 ðLaÞ P 2

ð44Þ

Therefore, s0 6 su can be easily got, which implies that there will be no interfacial debonding crack appearing from the non-loading end. It is concluded that only one debonding crack from the loading end propagates towards the free end until the interface is completely debonded in this case. 2.3. Analysis 2.3.1. Ec ! 1 If Ec ! 1, Eqs. (15), (16) and (38) can be simplified as a21 ¼ a22 ¼

8kG   Es 2DG þ D2 k ln Dþ2t D

N1 ¼ 0

ð45Þ ð46Þ

Therefore, for both boundaries 1 and 2, the differentiating equation only has a single form and the solutions are the same regardless of the boundary conditions adopted. In other words, there is only one debonding crack propagation appearing at the interface if Ec ! 1. 2.3.2. k ! 1 k ! 1 accounts for the absence of the elastic bonding slip although the shear stress increases. When the shear stress attains su, it suddenly drops to a constant value known as ss. Moreover, if k ! 1, Eqs. (15), (16) and (38) can be simplified as follows:   8G 1 D2 A 2 a1 ¼ 2 Dþ2t þ ð47Þ D ln D Es 4bEc 8GA pbEc D2 ln Dþ2t D  8G 1 D2 a22 ¼ 2 Dþ2t þ D ln D Es 4bEc ðD þ 2t þ bÞ

N1 ¼

ð48Þ ð49Þ

Therefore, a new series of solutions can be gained correspondingly using the proposed model. Moreover, if the thickness of the mortar is thin enough, the distribution of the shear stresses can be assumed uniform along the thickness of the mortar. In other words, if t is very low, D þ 2t 2t ! D D Besides, if both k and Ec tend to be infinite, Eqs. (47)–(49) are changed as ln

a21 ¼ a22 ¼

8G 1 2pG k0 ¼  Dþ2t ¼ E A E ln Es D2 ln Dþ2t s s s As D D

N1 ¼ 0

ð50Þ

ð51Þ ð52Þ

where As is the area of the anchor cross-section and k0 is a parameter related to the shear deformation of the mortar. Then the pullout load P can be expressed as a function of interfacial crack length a as following: P ¼ ss pDa þ

su pD tanhðaðL  aÞÞ a

ð53Þ

where a = a1 = a2. It can be found that Eqs. (51) and (53) completely consist with the analytical results proposed by Stang et al. [20]. The latter aims at a fiber anchored in an elastic matrix surrounded by a rigid matrix.

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3. Distributions of tensile stresses in the anchor and interfacial stresses along the embedment length This section concentrates on the distributions of rs and s along the embedment length during different loading stages using the proposed model expressed intuitively in the form of figures. For the convenience of calculation, some fundamental structural, material and interfacial parameters are adopted randomly irrespective of the actual material and interfacial properties as D = 20 mm, t = 5 mm, b = 100 mm, L = 1500 mm, Ec = 25 GPa, G = 10 GPa, Es = 210 GPa, k = 1000 MPa/mm, su = 10 MPa and ss = 5 MPa. It should also be pointed out that the adopted parameters are only used to study the shapes of the distribution curves irrelevant to the magnitudes of rs and s. In other words, the only failure mode is the pullout of the anchor from the anchorage system even if the pullout load is very large. Then the calculated results for both boundaries 1 and 2 are shown in Table 1. Herein, the distributions of rs and s are investigated at four different states, respectively, i.e., P = 0.05Pmax, P = 0.3Pmax, P = 0.75Pmax and P = Pmax. Obviously, when P = 0.05Pmax, the interface still behaves elastically for boundary 1 and attains the initial cracking state for boundary 2. Besides, the debonding crack lengths corresponding to the latter three states are gained in Table 2. Subsequently, by introducing the obtained results in Tables 1 and 2 combining with the adopted fundamental structural, material and interfacial parameters into the analytical expressions for rs and s, the distributions of rs and s in the four different loading states are illustrated detailedly in the following figures.

Table 1 Calculated results using the proposed model Boundary conditions

abc (mm)

atc (mm)

ac (mm)

Pini (kN)

Pmax (kN)

Boundary 1 Boundary 2

427.8 –

1015.9 –

– 1467.2

29 24

481.9 477.5

Table 2 Interfacial debonding crack lengths in different loading states Loading state

ab (mm)

at (mm)

a (mm)

P = 0.3Pmax P = 0.75Pmax P = Pmax

78 290.9 427.8

254.4 731.9 1015.9

381.5 1065.4 1467.2

Fig. 7. Distributions of rs along the embedment length during different loading stages for boundary 1.

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S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

It can be seen that there exists a horizontal portion at a certain distance from either the loading or the nonloading end in each distribution curve of rs for boundary 1 during the elastic stage. Moreover, rs = 0 appears within a large scope initiating from the free end with a size less than the embedment length for boundary 2 in the elastic state. The two phenomena demonstrate that the tensile stresses of the anchor far away from the ends are free of the edge effects and kept as constants according to the equilibrium of the system during the elastic stage, which gains a complete agreement with St. Venant Principle. Besides, according to Eq. (2), s = 0 appears in the region where rs keeps invariable just as shown in Figs. 8 and 10. As the pullout load increases, however, the above mentioned characteristics vanish gradually especially when the debonding crack appears. During the interfacial debonding stage, the length of the horizontal portion becomes shorter until zero when the pullout load attains Pmax, which can be seen clearly in Figs. 7 and 9. Obviously, it implies that the edge effects become stronger with the crack propagation so that rs in each position of the interface along the embedment length is greatly influenced by the boundaries. Additionally, if the pullout load is relatively low, the interfacial shear stresses only distribute within a local region. When the load exceeds 0.75Pmax, however, the distribution range will be enlarged as wide as the total embedment length due to the stronger edge effects just as shown in Figs. 8 and 10.

Fig. 8. Distributions of s along the embedment length during different loading stages for boundary 1.

Fig. 9. Distributions of rs along the embedment length during different loading stages for boundary 2.

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

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Fig. 10. Distributions of s along the embedment length during different loading stages for boundary 2.

4. Influences of various structural and interfacial parameters on calculated results Herein, different structural and interfacial parameters, including the embedment length L, the thickness t of mortar layer and the interfacial properties at the anchor–mortar interface, i.e., k and ss, are introduced to evaluate their influences on the calculated results using the proposed model, such as the critical crack lengths abc, atc and ac, the initial cracking load Pini and the maximum pullout load Pmax. The fundamental parameters needed in the proposed model are adopted as D = 20 mm, t = 5 mm, b = 100 mm, L = 1000 mm, Ec = 25 GPa, G = 10 GPa, Es = 210 GPa, k = 1000 MPa/mm, su = 10 MPa and ss = 5 MPa when they are treated as invariables. It should also be pointed out that the present adopted values, as well as the values given subsequently, are selected randomly irrespective of certain material properties but only to study the total variations of the calculated results with the chosen structural and interfacial parameters. For example, the calculated Pmax would be as high as 1000 kN so that the anchor must be fractured if it is a steel bar. However, the anchor is not an actual material and is assumed to have a sufficient tensile strength. Therefore, Pmax can be as high as possible in the present subject. 4.1. Influences of L on the calculated results Herein, the embedment length L is allowed from 1000 mm to 2000 mm. The curves are shown in Figs. 11– 13 to describe the influences of L on the calculated results. When L is long enough, the expressions of Pini in Eqs. (20) and (37) can be simplified as P ini ¼ P ini ¼

4su Da1

4 pD2

su pD a2

1  Na21

ð54Þ

1

ð55Þ

Obviously, Eqs. (54) and (55) account for the irrelevancy of Pini to the embedment length if L is very long just as shown in Fig. 12. Furthermore, Pini in boundary 2 is found to be lower than that in boundary 1 mainly because the displacement of the anchor at the loading end has an inverse direction with that of the concrete in boundary 2 so that the shear slip at the loading end attains dm much earlier than in boundary 1, which correspondingly reduces the value of Pini in boundary 2. Additionally, the maximum pullout load increases linearly with L just as observed by Sakla and Ashour [11] using artificial neural networks. Besides, the two curves for Pmax approach each other very closely, which implies that Pmax is not sensitive to the boundary conditions.

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985 2600

abc in boundary 1 atc in boundary 1 ac in boundary 2

2400 2200

abc/atc/ac(mm)

2000 1800 1600 1400 1200 1000 800 600 400 200 1000

1200

1400

1600

1800

2000

L(mm) Fig. 11. Variations of critical crack lengths with L.

32

Boundary 1 Boundary 2

30

Pini(kN)

28

26

24

22

20 1000

1200

1400

1600

1800

2000

L(mm) Fig. 12. Variations of Pini with L.

650

Boundary 1 Boundary 2

600 550

Pmax(kN)

976

500 450 400 350 300 1000

1200

1400

1600

1800

L(mm) Fig. 13. Variations of Pmax with L.

2000

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

977

4.2. Influences of t on the calculated results Due to the speciality of the proposed model, the effects of the hole size on the behaviors of the anchorage system deserves to be studied detailedly. Herein, the thickness t of mortar layer represents the hole size due to the constant D. Moreover, two cases should be considered, respectively. Case 1 focuses on a constant b with a varied t, and Case 2 aims at the invariable sum of b and t and allows t to be changed. 4.2.1. Case 1 Herein, b is kept as 100 mm and t is allowed to be improved from zero. The variations of the calculated results with t are shown in Figs. 14–16, respectively. Both Pini and Pmax increase with t but the increased rates are gradually reduced, which accounts for the weaker effects of the surrounding concrete. Besides, the two representative pullout loads in boundary 1 are both larger than those in boundary 2, however, the difference in the values of Pmax is marginal for both of the two boundary conditions. 4.2.2. Case 2 If b + t is kept constant, the variations with t are very different from those in Case 1. Herein, b + t is equal to 105 mm and t is allowed to approach 105 mm but can not attain it due to some singularities appearing in the process of calculation at t = 105 mm or b = 0. Then the comparisons are seen schematically in Figs. 17–19. In Fig. 17, the intersection point of the convex curve for atc and the concave curve for abc means that interfacial debonding cracks appear at both the loading and the non-loading ends simultaneously and propagate towards each other with the same crack lengths. Besides, the left part of this point accounts for the earlier appearance of interfacial debonding crack at the loading end and the right part demonstrates that the interface is firstly debonded at the non-loading end. For Pini in both boundaries 1 and 2, the variations are similar to each other but with higher values of Pini in boundary 1 if t is not large enough. When t is varied around 100 mm, Pini in boundary 2 is gradually reduced, however, Pini in boundary 1 increases sharply and tends to be infinite as t approaches 105 mm. Actually, Pini in the present study refers to the load corresponding to the interfacial shear stress attaining su at the loading end. Just as discussed above, however, when t becomes large enough, i.e., around 90 mm, the debonding crack will firstly appear at the free end. It means that the actual initial cracking load must be lower than the calculated Pini for boundary 1 in Fig. 18. Therefore, the portion of the Pini–t curve beyond t = 90 mm does not reflect the real variation of the initial cracking load for boundary 1. Besides, Pini ! 1 only accounts for the impossible appearance of a debonding crack at the loading end. Moreover, as t increases, the axial rigidity of the concrete becomes lower due to the decease of b. After the point of peak value in the Pmax–t curve for either boundary 1 1600

abc atc ac

1400

abc/atc/ac(mm)

1200 1000 800 600 400 200 0 0

20

40

60

80

100

t(mm) Fig. 14. Variations of critical crack lengths with t if b keeps constant.

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985 44

Boundary 1 Boundary 2

42 40 38 36

Pini(kN)

34 32 30 28 26 24 22 20 18 0

20

40

60

80

t(mm) Fig. 15. Variation of Pini with t if b keeps constant. 492

Boundary 1 Boundary 2

490 488

Pmax(kN)

486 484 482 480 478 476 0

20

40

60

80

t(mm) Fig. 16. Variation of Pmax with t if b keeps constant. 1600

abc atc ac

1400 1200

abc/atc/ac(mm)

978

1000 800 600 400 200 0 0

20

40

60

80

100

120

t(mm) Fig. 17. Variations of critical crack lengths with t if b + t keeps constant.

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

979

160

Boundary 1 Boundary 2

140

Pini(kN)

120 100 80 60 40 20 0

20

40

60

80

100

120

t(mm) Fig. 18. Variation of Pini with t if b + t keeps constant.

488

Boundary 1 Boundary 2

486

Pmax(kN)

484 482 480 478 476 0

20

40

60

80

100

120

t(mm) Fig. 19. Variation of Pmax with t if b + t keeps constant.

or 2, the anchorage effect of the concrete on the anchor tends to be much weaker due to the apparent reduced axial rigidity of the concrete and the increased thickness of the mortar layer, which results in the decrease of Pmax with t. 4.3. Influences of the interfacial parameters on the calculated results The interfacial shear modulus k and the frictional stress ss are the major parameters dominating the properties at the anchor–mortar interface. When one of the two parameters is changed, the pullout behaviors of the system may generate some new phenomena. Herein, the influences of k and ss on the pullout behaviors will be discussed respectively as follows. 4.3.1. Effects of k The varying curves of the calculated results with k are shown schematically in Figs. 20–22 as follows. Obviously, the critical crack lengths are almost irrelevant to the variation of k. Actually, the critical crack lengths are improved with the increase of k, however, the increments become lower and tend to be zero as

abc/atc/ac(mm)

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985 2000 1900 1800 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300

abc in boundary 1 atc in boundary 1 ac in boundary 2

0

2000

4000

6000

8000

10000

k(MPa/mm) Fig. 20. Variations of critical crack lengths with k.

30

Boundary 1 Boundary 2

28 26

Pini(kN)

24 22 20 18 16 14 0

2000

4000

6000

8000

10000

k(MPa/mm) Fig. 21. Variation of Pini with k.

482

Boundary 1 Boundary 2

481 480

Pmax(kN)

980

479 478 477 476 475 0

2000

4000

6000

8000

k (MPa/mm) Fig. 22. Variation of Pmax with k.

10000

abc/atc/ac(mm)

S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985 1700 1600 1500 1400 1300 1200 1100 1000 900 800 700 600 500 400 300 200 100 0

981

abc atc ac

0

2

4

6

8

10

12

τs(MPa)

Fig. 23. Variations of critical crack lengths with ss.

k ! 1. Therefore, the variations of the critical crack lengths with k seem marginal when k is large enough just as shown in Fig. 20 mainly because k is allowed from 1000 MPa/mm. By contraries, the initial cracking loads for both boundaries 1 and 2 are reduced greatly with k due to the earlier attainment of dm at the loading end induced by the increased k but constant su. However, Pini tends to be a constant as k ! 1. Similar phenomena are observed in Pmax in Fig. 22. Moreover, if k ! 1, all the critical crack lengths, initial cracking and maximum pullout loads tend to be constants. Certainly, it has been discussed above. 4.3.2. Effects of ss The magnitude of frictional stress dominates the behaviors of the anchorage system during the post-cracking stages and generates direct effects on the calculated results especially for the critical crack lengths and the maximum pullout load. Moreover, Pini is irrelevant to ss because the former only describes the initial cracking state of the system. To study the variations of the calculated results with ss, this controlling parameter is changed from 0 to su, i.e., 10 MPa and the results are shown schematically in the following figures. Herein, it should be pointed out that ss = 0 represents an ideal linear elastic model for the relationship of s and d, which means there is no crack propagation before Pmax is reached. In other words, the pullout load will attain Pmax when the interfacial shear stress at the loading end increases to su and Pini = Pmax. Otherwise, the critical crack length will be improved to a certain value even ss is still very low just larger than 0. This variation Boundary 1 Boundary 2

1000

Pmax(kN)

800

600

400

200

0 0

2

4

6

τs (MPa)

Fig. 24. Variation of Pmax with ss.

8

10

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S. Yang et al. / Engineering Fracture Mechanics 75 (2008) 961–985

results in a suddenly increment of the crack length with a very large initial ascending slope as shown in Fig. 23. Moreover, when ss = su, the relationship of s and d is known as ideal elasto-plastic so that Pmax will attain supDL at abc + atc = L or ac = L. The maximum pullout load increases linearly with ss. Furthermore, the two lines in Fig. 24 approach each other very closely, which indicates again that the maximum pullout load is not sensitive to the boundary conditions. 5. Conclusions In the present study, an analytical model was proposed for the pullout of the anchor from the anchor–mortar–concrete anchorage system with two different boundary conditions based on the compatibility conditions at the anchor–mortar interface. By the use of the proposed model, the variation of the shear stresses along the thickness of the mortar layer was acquired analytically. Then the distributions of the tensile stresses in the anchor and the interfacial shear stresses along the embedment length were obtained in an analytical manner during different loading stages. The probabilities and sequences of the interfacial debonding cracks from both ends of the anchorage were determined according to the axial rigidities of both the anchor and concrete, as well as the boundary conditions. For boundary 1, it was found that: (a) If 4bEc > EsD2A, the interfacial debonding crack will firstly appears at the loading end. (b) If 4bEc = EsD2A, the interfacial debonding cracks will appear at both the loading and the non-loading ends simultaneously and propagate with the same lengths. (c) If 4bEc < EsD2A, the interfacial debonding crack will firstly appear at the non-loading end. Then the pullout load was also gained as a function of one debonding crack length initiating from the loading end or two debonding cracks lengths developing from both the loading and the non-loading ends. Then the maximum pullout load Pmax and the corresponding critical crack lengths were calculated using the theories of extremum. Moreover, if the elastic modulus of concrete tends to be infinite and the anchor–mortar interface is assumed rigid before debonding, the solutions using the proposed model show a complete agreement with the previous work by Stang et al. [20]. Subsequently, several fundamental structural and interfacial parameters, such as L, t, k and ss, were introduced to study their influences on the calculated results using the proposed model, including abc, atc, ac, Pini and Pmax. Some conclusions can be drawn as follows: 1. abc, atc, ac and Pmax all increase linearly with L but Pini is irrelevant to the embedment length if L is very large. 2. If the thickness of concrete layer keeps constant, as t increases, both Pini and Pmax tend to be larger. However, the varying rates of the variables are gradually reduced. 3. If b + t keeps constant, there appears a maximum region in the Pmax–t curve for either boundary 1 or 2. Moreover, when t is large enough, the debonding crack will firstly appear at the non-loading end rather than the loading end for boundary 1. 4. Both the initial cracking load and the maximum pullout load decrease gradually until constants with the increase of k. 5. Both atc and ac increase rapidly with high initial ascending slopes as ss becomes larger, however, Pmax increases linearly with ss. 6. Pini in boundary 1 is generally larger than that in boundary 2 due to the postponed attainment of su at the loading end in boundary 1, however, Pmax is not sensitive to the boundary conditions. Acknowledgement The authors gratefully acknowledge that the National Natural Science Foundation of China (Grant No. 50578025) has supported this subject.

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Appendix A The distribution of the tensile stresses in the cross-section of the system for boundary 1 is shown schematically for both the anchor and the concrete in Fig. A.1a. Besides, the stresses in the concrete can be equivalent to the superposition of two distributions of normal stresses in Fig. A.1b and c, respectively. Wherein, the distribution of the stresses in the concrete in Fig. A.1c is a result of uniformly distributed bending moments induced by the pullout load in the cross-section along a representative circle where the strains of the concrete are equal to zero. Moreover, R in Fig. A.1c refers to a radial distance between the representative circle and the anchor–mortar interface. The uniform tensile stress ru in the section in Fig. A.1b is equilibrated by the pullout load P on the anchor as following: ru ¼ p 4

P  p4 D2 rs 2

ðA:1Þ

2

½ðD þ 2t þ 2bÞ  ðD þ 2tÞ 

In Fig. A.1c, the strains are assumed to distribute linearly along the radial direction of the section. The resultant moment Mc of the uniformly distributed bending moments can be gained by   p M c ¼ P  D2 rs R ðA:2Þ 4 Considering the linear distribution of the strains in Fig. A.1c, we have ry Ec rl Ec

¼

bR bR ) ry ¼ rl R R

ðA:3Þ

Moreover, the equilibrium condition of the forces in the section is expressed as following: Z

tþD2 þR

tþD2

R þ t þ D2  r  2pr dr ¼ rl R

Z

tþD2 þb

ry tþD2 þR

r  t  D2  R  2pr dr bR

ðA:4Þ

R can be gained by substituting Eq. (A.3) into Eq. (A.4), i.e., R¼

3t þ 32 D þ 2b b 6t þ 3D þ 3b

ðA:5Þ

σl σl Ec Ec

σ ce

σc σc

σy

σs

σs

σ ce

⇒

σu

σu

Ec

σy σl σl

σy

σy

+ x

r t 0t

R

D

R

Fig. A.1. The distribution of the stresses in the section of the system.

Ec

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Moreover, the resultant bending moment Mc in the section can also be gained as   Z tþDþR Z tþDþb 2 2 R þ t þ D2  r r  t  D2  R D Mc ¼ rl ry  2pr R þ t þ  r dr þ R bR 2 tþD2 tþD2 þR   D  2pr r  t   R dr 2

ðA:6Þ

Substituting Eqs. (A.2) and (A.3) into Eq. (A.6) gives rl ¼ 

3

b 3R2

2

 bR

P  18 D2 rs 2p   b4 2b3 1 2 þ b t þ D2 þ 4R 2  3R þ 2 b

ðA:7Þ

Then rc can be calculated by the sum of ru in Eq. (A.1) and rl in Eq. (A.7) as following rc ¼ ru þ rl ¼

P D2 A A rs pb 4b

ðA:8Þ

where A¼

1 1  þ 2  b3 4b2 2b D þ 2t þ b  2b þ 2 t þ D2 þ 2R 2  3R þ b R 3R2

ðA:9Þ

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