A Simplified Model for the Effect of Weld-Induced Residual Stresses on the Axial Ultimate Strength of Stiffened Plates

1 Introduction

The strength of plate elements and stiffened plates has been an important issue in the design of marine structures for a long time.The collapse strength of plate elements depends on several factors,including the boundary conditions and the weld induced initial distortions and the residual stresses.

Residual stresses occur through mechanisms inducing plastic deformations due to temperature gradients during thermal cycle and phase transformation in the material.High heat input introduced by welding to the material being welded may result in localized expansion,which is taken up during welding by either the molten metal or the placement of parts being welded.As a result of heating,non-uniform heat distributions,plastic deformations,and phase transformations occur on the material and generate different residual stresses patterns for weld region and in the heat-affected zone(HAZ).Welding typically produces large tensile stresses in the weld,balanced by lower compressive residual stresses elsewhere in the structure.Residual stresses induced by shrinkage o f the molten region are usually tensile.Transformation-induced residual stresses occur at the parts of the HAZ where the temperature exceeds the critical values for phase transformations.When the effect of phase transformations is dominant,the compressive residual stresses are formed in the transformed areas.

Unintended residual stress in a designed structure may cause failure mechanisms such as fatigue,brittle fracture,stress corrosion cracking,and creep cracking.It is revealed that the welding-induced residual stresses reduced the ultimate strength of the stiffened plates with a consequent reduction in hull girder ultimate moment(Gannon et al.2012;Chen and Guedes Soares 2016a).Hence,it is of great importance to have accurate predictions of the structural responses of the welding operations,in order to evaluate the strength capacity of the welded structures.

It has also been noted that the shape of initial geometric imperfections can significantly affect the ultimate strength of plated structures.In this regard,considerable research efforts have been devoted to the related investigations.Smith et al.(1992)suggested three levels of typical welding-induced plate initial imperfection:slight,average,and severe,based on surveys of naval ship plates.The average level is defined as the mean value of all the data of initial imperfection measurements.The mean of the 95th and above percentile distortion measurements defines the severe level of the imperfection,while the slight level is described as the 5%and below band data.It was reported that the initial imperfections in plates can be assumed to be proportional to the plate slenderness.The maximum initial deformation has been suggested as 0.025β2 t,0.1β2 t,and 0.3β2 t for the slight,average,and severe level,respectively.

The residual stresses affect the strength of plates(Cui and Mansour 1998;Paik and Sohn 2012)and thus the reduced strength of plates will also decrease the capability of the stiffened panels.This strength reduction has been shown to be a function of plate slenderness(Guedes Soares 1988),although in several design expressions this effect is taken into account implicitly(Guedes Soares 1992).Besides performing the direct measurements of the residual stresses existing in structures,researchers have also looked for empirical formula representing the residual stresses distribution,and to study the effects of the residual stresses on the strength of structures.Gordo et al.(1996)proposed an approximate procedure to describe the load-shortening curves for stiffened plates directly from mathematical expressions and showed the effect of residual stresses.The residual stresses can be classified by levels,η,each of them corresponding to a tensile zone with a width of the level times the thickness of the plate.Using this method and assuming that η depends on the welding process,and then plates with different thicknesses will have different residual stresses to the same level of weld tension block.Guedes Soares and Gordo(1996,1997)derived the equations to assess the strength of plates subjected to uniaxial and biaxial compressive and lateral pressure loads,including both effects of initial distortions and residual stresses.

On the other hand,numerical approaches including the finite element method(FEM)became widely used for predicting the structural responses to the welding process.To calculate the residual stress induced in ship plates,Chen et al.(2014)developed models and techniques to implement the finite element analysis(FEA)of the welding process.The effects of welding conditions and parameters were discussed by Mondal et al.(2015)and Chen and Guedes Soares(2016b).Based on studied cases with various welding sequences and plate thicknesses,a two-dimensional(2D)parametric relationship was proposed by Chen et al.(2015)to represent the residual stress distributed in stiffened plates.

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Based on the previous research on the residual stress,the objective of the present work is to propose an accurate model to predict the residual stresses and thus the compressive ultimate strength of the stiffened plates under uniaxial load.Considering the weld-induced initial geometrical imperfections and residual stresses,the axial ultimate strengths of stiffened plates are investigated by the nonlinear finite element method(FEM),in which the influences of initial imperfection and residual stress on the compressive strength of the stiffened plate are included.The residual stresses in the welded plates are calculated by a common idealized model as well as by an equation fitting to the result of thermo-elastoplastic finite element analysis.To verify the numerical approach,comparisons are performed between the current FEM and other simplified methods including IACS CSR.The results of ultimate strength from different methods are compared and discussed.

2 Ultimate Strength of Stiffened Plate Under Uniaxial Load

Given the difficulty of strength tests on actual ships,collapse tests were carried out on stiffened box girder models representing the parallel middle body of the ship’s hull(i.e.,Gordo and Guedes Soares 2008;Garbatov et al.2015;Saad-Eldeen et al.2013).Smith(1977)divided the cross section into small elements composed of a stiffener and the attached plating to calculate the strength capacity of the structure.More recently,many new approaches were proposed to obtain the average stress-average strain relationship of structures like stiffened plates and stiffened panels.

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2.1 Finite Element Model Validation

Table 1 lists the dimensions and properties of the studied plates which had been analyzed in the benchmark study of ISSC Committee VI.2(2000).Two groups of plates are studied,with 2400 and 4000-mm plate length,respectively.The flat-bar type stiffener is considered,with 150-mm height and 17-mm thickness.In the calculation of the ultimate strength of the plates,the yield stress and Young’s modulus of the material are considered as313.6 MPa and 205.8 GPa,respectively.When the stiffened is welded to the associated plate,one side of the stiffener is weldedfirstly;after a short interval,the other side is welded in the same direction.The plate slenderness is defined as

Table 1 Specimen configurations

a plate length,b plate width,t plate thickness,β plate slenderness,t p stiffener thickness,a/b aspect ratio

No.a/b/t/β tp/a/b-mm mm mm - mm -F31015 2400 800 10 3.123 17 3 F31315 2400 800 13 2.402 17 3 F31515 2400 800 15 2.082 17 3 F32015 2400 800 20 1.561 17 3 F32515 2400 800 25 1.249 17 3 F51015 4000 800 10 3.123 17 5 F51315 4000 800 13 2.402 17 5 F51515 4000 800 15 2.082 17 5 F52015 4000 800 20 1.561 17 5 F52515 4000 800 25 1.249 17 5

Fig.1 Configuration of the stiffened plate(1+1 spans)

To verify the current FEA,the results of current numerical approach are compared with IACS CSR(2006),Yao’s method(Yao and Nikolov 1991)based on a beam-column approach,Johnson-Ostenfeld’s formulation(Guedes Soares and Gordo 1997),and Masaoka’s method(Ueda and Masaoka 1995)using idealized structural unit method(ISUM).The IACS CSR equation described the load-end shortening curve for the beam-column buckling of ordinary stiffeners composing the hull girder transverse section.Yao considered the elastic large deflection analysis and the rigid plastic mechanism analysis in analytical forms.In the J-O type of formulation,the elasto-plastic behavior was considered together with the effective width approach for a plate.Masaoka used plate ISUM elements both for panels and stiffeners.Buckling collapse as a stiffened plate was not considered in his approach.

Figure 1 illustrates the configuration of the stiffened plate with 1+1 spans.A uniaxial compressive load is applied in one end section in the longitudinal direction.The constraints in all edges are listed in Table 2.

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The initial imperfections in the plate and stiffener are prescribed in the following modes identical to the ones considered in ISSC(2000),respectively:

Table 2 Boundary conditions in stiffened plate

The label U and R stand for the displacement and the rotation degree of freedom,respectively.The status Y represents the line is fixed in the corresponding degree of freedom,N means no constraint,and Const.defines a set of nodes(all the nodes in the line)coupled in the corresponding degree of freedom

Line UX UY UZ RX RY RZ AC Const. N Y Y N Y BD Const. N Y Y N Y EG Y N Y Y N Y FH Y N Y Y N Y AE N Y Y N Y Y CG N Y Y N Y Y

Fig.2 Finite element models of the stiffened plates with initial imperfections(1+1 spans;a/b=5;scale factor 1000)

where ω and υ are the vertical and lateral imperfections,respectively,t p is the thickness of the associated plate,x and y are the coordinates in the longitudinal and transversal axis,respectively,a and b are the length and width of the plate,and mis a constant equal to the aspect ratio of the plate.

Figure 2 shows the initial imperfections of the 4000-mm long stiffened plates in the finite element models.

Without considering the effect of residual stress,the load shortening curves of the plates are plotted in Figs.3 and 4.It is observed that the ultimate strength increases with the plate thickness.

Fig.3 Load-shortening curves of plates with different thicknesses(no welding residual stress;a/b=3)

Fig.4 Load-shortening curves of plates with different thicknesses(no welding residual stress;a/b=5)

The ultimate strengths of the studied stiffened plates follow power-law relationships with respect to the plate slenderness(see Fig.5).Between the two groups of plates with different slenderness(3 and 5),very similar results are observed in higher slenderness cases(β＞2.5).The aspect ratio of the plate only has slightly effect on the ultimate strength in lower slenderness cases.

where b is the plate width,t p is the thickness of the plate,σy is the yield stress of the plate,and E is Young’s modulus of the plate material.

Fig.5 Results of longitudinal ultimate strength with to plate slenderness

Fig.6 Ultimate strength without considering the residual stress,comparing with published results(a/b=3)

As shown in Fig.6,the present numerical results by ANSYS agree well with Yao and Nikolov(1991),except in the lowest slenderness case.The IACS CSR results in relatively small changes with respect to the plate slenderness,while the Masaoka method overestimates the ultimate strength in a certain level in all cases.The validation here is a comparison of numerical models for ultimate strength analysis of stiffened plates,with no consideration of the residual stress effect.Such comparison to experimental data might not be possible,since if the stiffeners are welded to the plate,there will always be the residual stress effect physically.

Since the analytical model of the residual stresses with rectangular blocks is acceptable,Eq.(9)can be used to define the width of the tensile rectangular block of the residual stress model.

2.2 Finite Element Model Validation

Table 3 lists the application of Smith’s classification on the current study.The maximum initial deformation has been suggested as 0.025β2 t,0.1β2 t,and 0.3β2 t for the slight,average,and severe level,respectively.The load-shortening curves of the 25-mm plate with different levels of initial imperfections are plotted in Fig.7.As the initial imperfections increase from slight to average level,the ultimate strength is reduced by 4.8%,whereas the severe imperfections reduce the ultimate strength by 18.0%(compared with the case of initial level).

Table 3 Levels of initial imperfections(mm)

Specimen No. Slight imperfection Average imperfection Severe imperfection F31015 2.44 9.75 29.26 F31315 1.88 7.50 22.50 F31515 1.63 6.50 19.51 F32015 1.22 4.87 14.62 F32515 0.98 3.90 11.70 F51015 2.44 9.75 29.26 F51315 1.88 7.50 22.50 F51515 1.63 6.50 19.51 F52015 1.22 4.87 14.62 F52515 0.98 3.90 11.70

Fig.7 Load-shortening curves of 25-mm plate with three different levels of initial imperfections(a/b=3)

The values of the ultimate strength of the stiffened plates with different levels of initial imperfections are plotted in Fig.8.It is observed that the initial imperfection has more significant effect on the plate with lower slenderness.

3 Welding-Induced Residual Stress

3.1 Idealized Model by ISSC

Figures 15,16,17,18,and 19 plot the load-shortening curves in plates with different slenderness,considering 0%,5%,10%,15%,20%,and 25%levels of residual tensile stress.The percentage is calculated as the ratio between the maximum compressive stress and maximum tensile stress.

Fig.8 Ultimate strength in the cases different levels of initial imperfections with respect to the plate slenderness(a/b=3)

Fig.9 Idealized welding residual stress distribution(ISSC 2000)

The tensile stress can be assumed as the yield stress of the material,and the magnitude of compressive residual stress can be obtained by the equilibrium requirement for the tensile and compressive parts as follows(Gannon et al.2010)

Between the idealized models and thermo-elasto-plastic calculations,there is a difference in the values of the compressive residual stress in the stiffened plate,as shown in Fig.13.The maximum compressive stress has a linear behavior with respect to the plate slenderness.

Table 4 List of widths of tensile stress zones(a/b=3)

β plate slenderness,2b t tension zone width,ηtensile width coefficient

Specimenβ 2b t/mmη F31015 3.123 71.3 3.56 F31315 2.402 63.7 2.45 F31515 2.082 59.7 1.99 F32015 1.561 52.2 1.31 F32515 1.249 47.0 0.94

Fig.10 Flow diagram of transient thermo-mechanical analysis model

According to Eqs.(5)and(6),the determination of the width of the tension zone mainly depends on the thicknesses of the plate and stiffener,while the aspect ratio of the plate has no influence on the distribution of the residual stresses.Table 4 lists the widths of the tensile stress blocks and the coefficient η of the tensile width in the studied stiffened plates.The coefficient η is defined as the ratio between the half-width of the tensile block and the thickness of the plate.

3.2 Residual Stress Calculation by FEM

The finite element analysis of the welding process can be defined as a three-dimensional(3D)coupled thermomechanical analysis.In some cases,the effect of the structural result on the thermal analysis is very small and can be even neglected.Thus,the indirect method,in which the thermal analysis and structural analysis are performed separately,can be utilized for these single directional coupled issues.In the thermal analysis in ANSYS®,the eight-noded,3D brick thermal element Solid 70 is used.Whereas,in the mechanical analysis,the analyzed element type is converted from Solid 70 to Solid 185 that has plasticity,hyper-elasticity,stress stiffening,creep,large deflection,and large strain capabilities.

Fig.12 Idealized welding residual stress distributions of stiffened plates.(a)Full distribution(b)Detailed distribution between transverse location 350-450 mm(a/b=3)

During the welding process,both the thermal properties and mechanical properties of the material are temperature-dependent.The heat exchange between the welded plate and its surroundings during welding and subsequently cooling takes place by both convection and radiation.The material properties of the metal can affect the distortion with the application of the heat.

Fig.11 Residual stress distribution by thermo-elastoplastic method(a/b=3,Unit:Pa)

Fig.13 Comparison of compressive residual stress in stiffened plates(a/b=3)

The stiffened plates analyzed in the present study are made from SM 400A shipbuilding steel.The thermal and mechanical properties refer to Deng et al.(2007).It is noted that as the temperature increases,the specific heat and the coefficient of thermal expansion increase whereas the yield strength,modulus of elasticity,and thermal conductivity of the steel decrease.The 3D double ellipsoidal heat source proposed by Goldak et al.(1984)is used to represent the heat input during the welding process.

Figure 10 illustrates a three-step procedure of the thermoelasto-plastic approach to predict the ultimate strength of welded plates.The three analyses are performed separately with the same geometrical model but different boundary conditions.Convection and radiation are applied as thermal boundary conditions in the first thermal analysis.The temperature distribution of all nodes can be obtained from the thermal analysis solved in the first step,and be used as body loads applying in the some geometric model to do the mechanical analysis.

In the second step,one node in a corner of the plate is fixed in three translational directions,the node in the other corner of the same transverse section is fixed in the vertical and longitudinal directions,and one more node in a corner of the end section is fixed in the vertical direction.The boundary condition is applied such that the plate could deform freely in any direction and rigid body motions are prevented.

Fig.14 Longitudinal residual stress distribution at mid-span of the plate

Fig.15 Load-shortening curves in 10-mm stiffened plates considering different levels of residual stress(a/b=3)

In the final step,the ultimate strength of the structure is calculated considering the welding-induced geometrical deflection and residual stress obtained in the previous analysis.In the strength analysis,a uniaxial compressive load is applied in one end section in the longitudinal direction,as shown in Fig.1.The constraints in all edges are listed in Table 2.The displacement and residual stress calculated in the second step will considered as the initial conditions.

In general,reductions in ultimate strength are observed when considering higher levels of residual stress,whereas in the high levels of 20%and 25%levels,the difference becomes very small.In higher residual stress cases,the stress-strain curve reduces less rapidly in the post-collapse stage.

Fig.16 Load-shortening curves in 13-mm stiffened plates considering different levels of residual stress(a/b=3)

Fig.17 Load-shortening curves in 15-mm stiffened plates considering different levels of residual stress(a/b=3)

Figure 12 plots the idealized model applied in a studied plate,comparing with the thermo-elasto-plastic results.The biggest compressive stress is observed in the thinnest(10 mm)plate in both groups of curves.In the cases of plates with different thicknesses,the distributions are similar while it is observed that the thinner plate has slightly smaller tensile stress and slightly bigger compressive stress.

where b is the width of the plate,h is the height of the stiffener,σcp andσcs are the compressive stress in the plate and stiffener,2b t and b s are the width of the tension zone in the plate and stiffener,σYp and σYs are the yield stress of the plate and stiffener,and t p and t w are the thicknesses of the plate and stiffener,respectively.

3.3 A Simplified Model

Fig.18 Load-shortening curves in 20-mm stiffened plates considering different levels of residual stress(a/b=3)

Fig.19 Load-shortening curves in 25-mm stiffened plates considering different levels of residual stress(a/b=3)

Figure 14 plots the residual stress distribution suggested by ISSC(2000),comparing with the numerical result of Chen et al.(2015).It is concluded that the Engineering curve is able to predict the basic characteristics of the residual stress distribution,such as the peak values of tensile and compressive stresses,and the width of the tension region.However,as can be seen in Fig.14,the idealized model is limited when more details about the residual stress distribution are required in specific region.For example,the ISSC curve neglects the transformations of the compressive residual stress along the associated plate width.

Hence,the following equation was proposed by Chen et al.(2015)to predict the real residual stress distribution by considering various parameters representing different details in the true curve.

where k is a scale factor,σy is the yield stress of the material,x is the longitudinal coordinate,τx stands for a half of the width in the tensile region,m is the number of half waves in the compression part of the curve in x direction,b is the half of the breadth,andαx,βx are controlling factors defined from the regression analysis.

Fig.20 Effect of residual stress on ultimate strength of stiffened plate(a/b=3)

4 Effect o f Residual Stress on Ultimate Strength

4.1 Ultimate Strength by Idealized Residual Stresses

Welding is a vital production process for industry and generates residual stresses at a remarkable level.When the finished weldment cools to ambient temperature,some areas cool and contract more than others,leaving residual stresses.Since the measurement of residual stress in welded plates requires costly equipment and qualified technicians,many researchers chose idealized models to represent the residual stress distribution in plates and stiffened panels(Gannon et al.2010;Cui and Mansour 1998).Figure 9 shows the idealized model suggested by ISSC(2000)for use in a benchmark study on the behavior of stiffened plates under axial compressive load where b t is the half-width of the tensile residual stress zone in the plate and b s is the width of the tensile residual stress zone in the stiffener web.The distribution of welding-induced residual stress was idealized as the combination of both tensile and compressive stresses.

According to the FEA calculations,Fig.11 plots the residual stresses existing in the mid-section of the plates after the welding process.Significant tensile residual stresses(342 MPa)approximately equal to the yield stress of the material(313.6 MPa)are observed in the weld.Outside the HAZ,the longitudinal residual stresses become compressive and remain constant at the edges of the plate.Since the welding was started in one side of the plate before the other,the longitudinal residual stresses distributed in the two sides are not exactly symmetric,whereas the difference is relatively small.

Considering the obtained tension zones in different cases,Fig.20 plots the ultimate strengths of the studied plates with respect to the plate slenderness.It is observed that 5%-7%reductions occur due to the presence of the residual stress.In plates with lower slenderness(β＜2.5),the strength of the plate reduces with a quasi-linear behavior when the plate slenderness increases.The current FEM result has a relatively good agreement with the results by Yao and Nikolov(1991),especially in higher slenderness cases.

Besides the plate slenderness,the column slenderness is one of the principle parameters affecting the ultimate strength of plate and stiffened panels subjected to compressive load.The column slenderness is defined as follows(Khan and Zhang 2011)

where a is the length of the stiffened plate,r is the radius of gyration of the cross sectional area,t p is the thickness of the plate,σy is the yield stress of the plate,and E is Young’s modulus of the plate material.

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Fig.21 Ultimate strength with respect to column slenderness

To study the effect of the column slenderness,two more configurations of the stiffener(250×19 and 350×35 mm2)from ISSC(2000)benchmark calculation are considered.The resultant ultimate strengths are plotted in Fig.21.

It is concluded that the strength of the stiffened plate reduces as the column slenderness increases.The reduction rate is more significant in the case of low column slenderness.

4.2 Ultimate Strength by Simplified Residual Stresses

According to the simplified expression in Eq.(7),Fig.22 displays the residual stress distribution in the cross section of a stiffened plate.By fitting to the present FEM result,the controlling factors in Eq.(7)are as follows:k=3.6,τx=12.1,m=2.4,b=84,αx=-4.1,βx=1.6.The calculated stress is applied to the finite element model as initial stress,as shown in Fig.23.

Considering the calculated residual stress,the load shortening curve is plotted in Fig 24.The idealized residual stress model results in 4.9%reduction in ultimate strength,while 3.5%reduction is observed in the model calculated by the simplified equation.The deviation between the results of the two methods is 1.42%.Given the less computational time comparing with thermo-elasto-plastic FEA,both methods are recommended to be used to represent the initial residual stress distribution in ultimate strength predictions.

采用香农-维纳指数（Shannon-Wiener index）分析金沙江淡水丝孢菌的多样性〔25〕。香农-维纳指数为：

4.3 Coefficient of the Tensile Width as a Function of the Plate Slenderness

Based on the studied stiffened plate,Fig.25 plots the coefficient of the tensile width as a function of the plate slenderness.The coefficient is defined as the ratio between the width of the tensile block and the thickness of the plate.A quasi-linear relationship is observed between the coefficientηand the plate slenderness β:

Fig.22 Comparison of longitudinal residual stress distribution in 10-mm plate(a/b=3)

Fig.23 Initial stress applied to the plate according to the simplified model(a/b=3)

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It is noted that the thickness of the stiffener in the stiffened plate will also affect the width of the tensile block.Plates with other geometries may show different sensitivity to the residual stresses.

5 Conclusions

Fig.24 Load-shortening curves of 10-mm stiffened plate(a/b=3)

A simplified model has been used to estimate the residual stress distributed in welded stiffened plates with various aspect ratios and slenderness,in order to investigate the compressive ultimate strength of the structures.The residual stress was calculated by a proposed equation fitting to the thermoelasto-plastic FEA result.The simplified model was verified by the existing ISSC idealized model and can be very helpful in engineering use.

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Without the consideration of the residual stresses,the ultimate strength of the studied stiffened plates follow power-law relationships with respect to the plate slenderness.The aspect ratio only has slightly influence on the ultimate strength in plates with lower plate slenderness.

As the initial imperfections increases from slight to average level,the ultimate strength is reduced by 4.8%,whereas the severe imperfections reduce the ultimate strength by 18.0%(compared with the case of initial level).It is concluded that the initial imperfection has a significant effect on the ultimate strengths of stiffened plates,especially in the plates with lower slenderness.

Fig.25 Coefficient of the tensile width as a function of the plate slenderness

The presence of the residual stress results in about 5%-7%reductions in axial ultimate strength of the studied plates.The strength of the stiffened plate reduces when the plate slenderness increases.

2.1 新疆汉族、维吾尔族、哈萨克族老年男性研究对象临床特征 在汉族研究对象中，病例组、对照组间高血压患病率的差异有统计学意义（P＜0.05）；在哈萨克族研究对象中，病例组、对照组间的高血压患病率、肥胖患病率的差异有统计学意义（P＜0.05）。见表1。

The strength of the stiffened plate reduces as the column slenderness increases,while the reduction rate is more significant in the case of low column slenderness.

A quasi-linear relationship is observed between the coefficient η and the plate slenderness.A linear equation has been proposed for defining the width of the tensile rectangular block of the residual stress model.

Funding Information This work was performed within the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering,which is financed by Portuguese Foundation for Science and Technology(Fundação para a Ciência e Tecnologia-FCT).The first author has been funded by a PhD scholarship from ABS,the American Bureau of Shipping.

（二）产业发展与新疆农产品区域品牌竞争力的关系。农产品区域品牌由产业集群而形成，依托产业发展而发展，区域品牌又能促进产业发展，两者之间呈现相互依存、相互促进的关系[4][7]。产业带动能力强，产业的可持续发展能力就越强，发展前景越广阔，越能吸引投资和参与，从而产业规模不断增加，又促进了农产品区域品牌竞争力的提升[14]80-86；[15]120-130。新疆农产品逐步形成区域产业带动效应，促进农产品品牌竞争力的提升。因此，提出如下假设：

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