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Engineering stress solutions for bolted and pressurized steel structures
Engineering stress solutions for bolted and pressurized steel structures Nelli Aleksandrova PII: DOI: Reference:
S2352-0124(14)00007-1 doi: 10.1016/j.istruc.2014.09.003 ISTRUC 6
To appear in: Received date: Revised date: Accepted date:
4 July 2014 22 September 2014 26 September 2014
Please cite this article as: Aleksandrova Nelli, Engineering stress solutions for bolted and pressurized steel structures, (2014), doi: 10.1016/j.istruc.2014.09.003
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ACCEPTED MANUSCRIPT Engineering stress solutions for bolted and pressurized steel structures Nelli Aleksandrova
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Centre of Exact Sciences and Engineering, Madeira University, 9020-105 Funchal, Portugal (Madeira);
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E-mail: [email protected]; Phone: (351) 291-705-285; Fax: (351) 291-705-249
Keywords: Plates; Plasticity; Analytical solution; Steel Structures
1 Introduction
Bolts, rivets and location pins are some of the most utilized mechanical fasteners used in civil and mechanical engineering as well as in airspace industry. These fasteners play a crucial role in safety, reliability and serviceability of the structure by provoking a dangerous stress concentration zone (surrounding fasteners). In practice, numerous specific standards and regulations have been produced to assist civil engineers in installation, inspection and maintenance of such structures. On the other hand, a lack of rigorous mathematical procedures based on reliable material models and failure criteria to predict the development of stress concentration zone near the bolts often lead to the overestimated safety factors and heavier components in order to assure safety precautions. In fact, a number of parameters 1
ACCEPTED MANUSCRIPT need to be considered in successful design of bolted connections to describe the area surrounding fasteners. Among the most important parameters are geometrical characteristics
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(dimensions), material model, type of installation (interference) and service environment effect (ambient or elevated temperatures). Nowadays, numerical modeling, primary based on
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the finite element method (FEM), which incorporates experimental data and standardization
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rules, has become a popular technique used for two- and especially three-dimensional
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simulations of the bolt/surrounding area problems. The extensive review of such FEM solutions is provided in [1]. The review of common experimental procedures and industrial
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codes for design of bolted connections is given in [2]. Contemporary discussion of various specifications in terms of nominal strength and failure modes for stainless steel bolted
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connections is provided in [3]. Theoretical stress analysis based on the strength of materials
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was used in [4] to demonstrate the failure modes obtained experimentally in bolted single lap
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joints taking into consideration edge effects. Recently, a complete study of bearing stress in single bolt connections was conducted by Moze and Beg [5]. Several theoretical models of bearing resistance were discussed in agreement with Eurocode. It was concluded that the resistance of such connections depend on the geometry and material properties. Elegant engineering solutions to improve bolt/plate contact assemblies were suggested by Pedersen’s [6] showing the importance of Poisson’s ratio and influence of the contact pressure distribution. To this end, analytical methods continue to provide powerful tool for the preliminary engineering design and FEM verification of the problem since they allow for its rigorous mathematical formulation and analysis as well as account for the broad range of parameters and material models in the description of the dangerous stress concentration zone surrounding fasteners. To describe material behavior of this zone, the model of a non-elastic open-hole
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ACCEPTED MANUSCRIPT annular plate subjected to internal pressure or a plate with a pressed in elastic inclusion are suitable choices. A similar geometrical model of an infinite metal sheet with a hole made of
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non-elastic or composite material has been extensively used in engineering to describe the cold expansion process. In the frame of this model, Hsu and Forman [7] obtained an exact
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elastic-plastic solution for stresses assuming the sheet to be orthotropic in the thickness
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direction and obeying the Hill’s yield criterion. Later on, Ball [8] used the same model but
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added to the stress solution the consideration of the displacement control on the hole boundary. Chakrabarty [9] generalized the problem comparing the stress solution obtained on
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the basis of the Tresca yield criterion with its associated flow rule and the Huber-MisesHencky (HMH) yield criterion combined with deformation theory of plasticity for hardening
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and non-hardening materials isotropic in the plane of the plate; also, the hyperbolic thickness
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variation was considered as well as the related problem of an open-hole metal sheet subjected
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to uniform tension at infinity. The residual stress fields due to expansion of hole in an infinite plate were derived in [10] using the Tresca yield criterion (closed-form analytical solution) and HMH yield criterion (finite difference numerical technique) assuming an elastic-perfectly plastic material. It was shown that the choice of yield criterion significantly affects the results. On the other hand, an assessment of the influence of hardening in popular analytical models (including those by Hsu and Forman [7] and Ball [8]) to predict residual stresses was carried out by Wang and Zhang [11]. A new semi-analytical method of calculating residual stresses in pressure vessels which takes into account variations in Young modulus and tangent modulus was proposed by Sedighi and Jabbari [12]. To this end, recently, Leu, Liao and Lin [13] also derived new exact analytical solutions for plastic limit pressure adopting nonlinear combined isotropic/kinematic hardening and compare the results with perfectly plastic material model.
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ACCEPTED MANUSCRIPT Entirely numerical stress/strain solution of the cold expansion problem based on an infinite sheet geometrical model was done in [14] by FEM.
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However, for most structural engineering applications [4-6], the model of an infinite sheet should be substituted by the model of a comparable size inner/outer diameter annular plate or,
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depending on application, by the model of an annular finite size plate embedded into a rigid
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container (as in high pressure boilers, pressure vessels or loose-material containers) which
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further complicates the problem due to the imposed boundary condition on displacements. A suitable hole to structure dimension ratio (scale factor) provides correct estimation of hole
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effect on steel structures and pressure vessels [15]. By analogy with the cold expansion problem, two distinct analytical approaches may be considered based on either the Tresca
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yield criterion with its associated flow rule [16, 17] or the HMH yield criterion (or Hill’s yield
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criterion in the case of transversely orthotropic material) combined with the deformation
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theory of plasticity [18, 19]. To this end, experimental results show that the strength of most metal materials is better described by the HMH yield criterion [20]. However, the use of the deformation theory of plasticity, especially with an additional assumption of elastic incompressibility and neglect of elastic strains in the plastic zone (which permits the derivation of analytical formulae for stresses and simplifies substantially the analysis), is not completely justified. In [19], other simplifications to the formulation of the problem were made to get the displacement solution such as an introduction of a unified variable Poisson’s ratio which is valid for a specific strain hardening law. In more general cases, when the HMH yield criterion is employed with its associated flow rule, only a few numerical solutions are available [21, 22] and no analytical solutions are known for practical engineering purposes. So, the objective of the present study is to develop a unified analytical procedure capable of obtain mathematically rigorous stress fields both in the area surrounding fasteners and in
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ACCEPTED MANUSCRIPT the confined area enclosed in a rigid container. Two simple geometrical models are suggested and compared with each other for practical engineering applications: (I) classical model of an
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annular finite size plate subjected to internal pressure at its inner radius and free from stresses at its outer radius; (II) alternative finite size model of an annular plate subjected to internal
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pressure at its inner radius and embedded into a rigid container at its outer radius. The
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influence of the bolt on the surrounding area is simplified by the applying internal pressure.
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The material of the plate is assumed to be elastic-perfectly plastic obeying the HMH yield criterion with its associated flow rule. However, only the stress analysis is performed here to
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make emphasis on the advantages of the proposed unified analytical procedure (valid for both geometrical models) to preliminary engineering design including uncoupled stress solutions.
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For the alternative geometrical model II, only elementary kinematic analysis is required
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arising from the boundary condition on displacement at the outer radius of the plate which
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does not influence the general algorithm as opposite to the more involved problem with boundary condition on displacement at the inner radius of the plate [23]. It is worth noting that the related strain analysis may be conducted straight after the stresses are determined using the same geometrical models and assumptions of axial symmetry, plane stress state and small strains. Compressibility of the material in the outer elastic zone of the plate and summation of elastic and plastic strains in the plastic inner zone should be also considered the same way as it was done in [23]. That is, the general line of the stress/strain solution is based on the HMH yield criterion with its associated flow rule and elastic-perfectly plastic material behavior. Besides the elastic/plastic and residual stress solutions, the load-carrying capacity of each model is also discussed in the present study since for the perfect plasticity two values of the limit external loading may be distinguished, namely, the elastic load-carrying capacity when the structure starts yielding and the limit load-carrying capacity which corresponds to
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ACCEPTED MANUSCRIPT the full plasticization of the component. As it is known, this material model is frequently used in many other structural and mechanical applications and shows specific patterns while
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performing calculations [23-26]. Hence, it seems important both from mathematical and engineering points of view to analyze the load-carrying capacity of the structures. Other than
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that, correct prediction of the load-carrying capacity in general is crucial for achieving an
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optimal structural design [20] and may be considered as a limiting criterion.
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2 Elastic load-carrying capacity and initiation of plasticity
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In cylindrical coordinate system rz with non-zero radial, ˆ rr , and tangential, ˆ , stress tensor components, consider a thin annular plate of inner radius a and outer radius b subjected
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to gradually increasing radial pressure pˆ around its inner edge (Fig. 1). That is, the following
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ˆ rr pˆ at r a
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boundary condition in stress should be satisfied (1)
Additionally, for the geometrical model I the outer radius is considered traction free, that is, subject to the stress boundary condition
ˆ rr 0 at r b
(2)
For the geometrical model II, the plate is placed into a rigid container, that is, subject to the displacement boundary condition
uˆ 0 at r b
(3)
where uˆ is the radial displacement. Due to the circular symmetry of the problem,
ˆ rr ˆ rr (r ) , ˆ ˆ (r ) and plane stress assumption, ˆ zz 0 , there is only one nontrivial equilibrium equation
dˆ rr ˆ rr ˆ 0 dr r
(4)
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ACCEPTED MANUSCRIPT which is valid both in elastic and plastic zones of the plate. In the plastic zone, the HMH yield criterion is adopted and has the form 2 2 ˆ rr ˆ ˆ rrˆ Y 2
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(5)
where Y is the yield stress of the material.
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For sufficiently small values of external loadings pˆ the whole plate is entirely elastic with
2 Bb 2 2 Bb 2 1 1 ˆ A , A , uˆ r 1 A 1 Bb 2 2 2 E r 3 3 r r
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ˆ rr
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the well-known Lamé’s solution
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where A and Bb 2 are two convenient constants of integration and the coefficient 2
(6)
3 is
introduced for the convenience of further calculations; E is the Young’s elasticity modulus
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and is the Poisson’s ratio. To write out the resulting stress formulae, the dimensionless
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parameters may be introduced rr ˆ rr Y , ˆ Y , p pˆ Y ; r b , a b . Then, for the model I, the boundary conditions (1) and (2) define stress field in the elastic plate
rr
2 p 1 2 p 1 , 1 1 2 2 2 2 1 1
(7)
Analogously, for the model II, the boundary conditions (1) and (3) define stress field as
rr
p 2 2 1 1 p 2 2 1 1 , 2 2 1 1 2 2 1 1
(8)
These stress solutions are valid up to the value of p at which the plastic zone starts to develop. This value of p is determined from Eqs. (7)-(8) and yield criterion (5). It follows from Eqs. (7)-(8) that in the elastic plate the left-hand side of Eq. (5) attains its maximum value at
. Therefore, the plastic zone starts to develop from the inner radius of the plate, and, in general case, the plate consists of inner plastic and outer elastic zones divided by an unknown
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ACCEPTED MANUSCRIPT elastic/plastic boundary, c, or in dimensionless notations c b . Substituting subsequently Eqs. (7)-(8) at into Eq. (5) yields the value of the elastic load-carrying capacity of the
3 2
For the model II, it is
q 2 q 1 , q 2 1 1
1 1
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pelc 1
(9)
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pelc 1 2
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plate. For the model I, it is
2
(10)
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As it follows from Eqs. (7)-(10), both the elastic limit-load capacity and the elastic stress field
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of the model II depend on the compressibility of the material as opposite to the classical model I for which neither the elastic limit-load capacity nor the elastic stress field depend on
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the Poisson’s ratio.
3 Stress distributions in the elastic-plastic areas
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In the inner plastic zone, , there are two available equations - equilibrium Eq. (4) and yield criterion (5) - for two unknown stress components rr , , hence, the problem is internally statically determinate, at least, for the model I. For the model II, due to the boundary condition in displacement (3), some kinematic analysis is also required, however, it can be reduced to the analysis of elastic zone which will be shown later on. The stress solution in the plastic zone for both models proceeds via introduction of an auxiliary variable
suggested by Nadai [27] such that the yield criterion is automatically satisfied
rr 2 3 cos , 2 3 cos 3
(11)
This way two unknown stresses are replaced by only one unknown function r , the equation for which is obtained upon the substitution of Eq.(11) into the equilibrium Eq. (4) r sin
sin 0 r 6
(12) 8
ACCEPTED MANUSCRIPT Taken into account that plastic zone starts do develop from the inner edge of the plate, the boundary condition (1) combined with Eq. (11) gives
cos a 3 2 p
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(13)
3 , and considering equations for stresses (7), (8), and (11), the value of a with correct
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2
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where a is the value of at . It is seen from Eq. (13) that the limiting value of p is
sign is calculated as
(14)
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a ArcCos 3p 2
attains the limiting value of 2
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As p increases from zero, a increases from 2 and eventually becomes equal to when p
3 . That is, for both geometrical models, a non-hardening
3 Y , then the plate must start
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material cannot sustain a stress of magnitude greater than 2
to thicken appreciably to support any further increase in the radial load. Also, as it follows
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from Eq. (12), is a singular point. So, the value of a for numerical calculations, especially those based on commercial codes, should be limited by 2 a . The solution to Eq. (12) with boundary condition (14) has analytical form
2 sin 6 exp 3 sin 6 2
(15)
Thus, implicitly depends on p. Let be the value of at the elastic-plastic boundary . Then,
2 sin 6 exp 3 2 sin 6
(16)
Obviously, the condition of continuity of stresses should be preserved at the elastic-plastic boundary. Hence, taken into account that the general solution (6) is valid in the elastic zone,
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ACCEPTED MANUSCRIPT 1, and at the plastic zone, , the stresses are described by Eq. (11), the stress continuity equations will be cos A B 2
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cos 3 A B 2
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3 2 cos 6
(18)
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B 1 2 2 sin
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A
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from which the constants A and B are defined
(17)
where follows from Eq. (16). It should be mentioned at this point that all above equations
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are valid for both geometrical models and may be considered as a unified algorithm of calculations in comparison with particular procedures applied for classical solutions of similar
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cold-expansion problem or rigid shaft/plate assemblies. 3.1 Bolted connections
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To make further calculations on model I, the boundary condition (2), which is applied at the elastic zone together with Eq. (6), should be fulfilled. So, A B 0 at 1 , and using Eq. (18) one gets 1
2
1 tg 6 3
(19)
Multiplying Eq. (19) by Eq. (16) yields
1
2
sin 6 1 exp 3 cos 6 3
where a is directly related to p by Eq. (14) with 0 p 2
(20)
3 . For a given value of external
loading p and geometrical ratio , Eq. (20) determines the value of . After that, the radius of elastic/plastic boundary follows from Eq. (19). 3.2 Pressurized containers 10
ACCEPTED MANUSCRIPT Similarly, for the model II, the boundary condition (3), which is applied at the elastic zone together with Eq. (6), should be fulfilled. So, 1 A 1 B 0 at 1 , and using Eq.
2
(21)
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1 1 tg 6 1 3
1
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(8) one gets
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Multiplying Eq. (21) by Eq. (16) yields sin 6 1 1 exp 3 cos 6 1 3 2
(22)
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1
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For a given value of external loading p, geometric ratio and Poisson’s ratio , Eq. (22) determines . After that, the radius of elastic/plastic boundary follows from Eq. (21).
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Then, for both models, the stresses in each zone are calculated by Eq. (6) with Eq. (18) in
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the outer elastic zone and by Eq. (11) in the inner plastic zone where the auxiliary variable
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is related to the dimensionless radius via Eq. (15). This way, a unified algorithm of stress filed determination is concluded leading to an effective approximate engineering stress solution for bolted and open-hole thin-walled structures. The residual stresses are obtained subtracting the purely elastic solutions (7) and (8), respectively, for model I and II from the plastic solution (11).
4 Limit load-carrying capacity Considered structures reach their limit load-carrying capacity if there exists an external loading p such that the elastic outer zone vanishes. This is equivalent to the condition that the elastic/plastic boundary coincides with the outer edge of the plate, 1 . 4.1 Bolted connections Hence, for the geometrical model I, as it follows from Eq. (19)
3 tg 6 , or
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ACCEPTED MANUSCRIPT 2 . Then, Eq. (20) transforms to 1
2
2 exp 3 2 sin 6 3
(23)
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From here, for specific , one gets , and then the corresponding limit load plload is
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calculated using Eq. (14). It is important at this point to verify in principle the existence of the
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solution of Eq. (23) for all possible range of taken into account the fact that numerically the
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root-finding procedure is extremely sensitive to the initial guess-solution. As soon as the limit load is found, the critical value of may be found as well based on the condition that the
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maximum external load p according to Eq. (13) equals pmax 2
3 since the trigonometric
function is bounded by 1 cos a 1 . That is, the maximum value of a with the
2
2 exp 3 2 sin5 6 3
(24)
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1
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appropriate sign equals a . Substitution of a into Eq. (23) yields
From Eq. (24), one concludes that the critical value of 0.3377 which implies that for
0.3377 the plate never reaches its limit load-carrying capacity, that is, some region of the plate near the outer radius is always in the elastic state. 4.2 Pressurized containers This algorithm of finding the limit load-carrying capacity works for both models, it is simple and straight forward. For the model II, it shows immediately that no solution ( ) exists for any value of 0 1 since the magnitude of corresponding to 1 in this case equals
6 Arctg 3 C1 with C1 1 1 due to Eq. (21). It implies that container never reaches its limit load-carrying capacity for permissible external loading.
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ACCEPTED MANUSCRIPT 5 Results and Discussion A unified analytical procedure of stress calculations is developed for two classes of structural
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mechanics problems: bolted connections and pressurized containers, which lays the
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foundation for further research in strain/displacement fields. The material model of elastic-
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perfectly plastic behavior of the plate has an advantage of getting an uncoupled stress solution which is important for rapid and easy preliminary engineering design and leads to an effective
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method of estimation of the load-carrying capacity of the structures. Another popular power strain hardening model may only be used with the initial consideration of plastic strains as it
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has been done, for example, in [9]. Moreover, an immediate simplification would be achieved in this case for the determination of the stresses if the deformation theory of plasticity is
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applied together with the HMH yield criterion [9]. A comprehensive analysis of different hardening models and their suitability to pressure vessel applications is presented in [13]. The
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incorporation of hardening parameter into the already existing closed form solutions is also discussed in [11] and out of the scope of this paper. For geometrical model I (suitable (among others) for bolted connection applications) the dependence of the geometrical parameter on the external loading p is shown in Fig. 2. It is seen that the minimum distance between centers of fasteners should be at least 2.96 times the fastener diameter (b = 2.96a) based on the limit load-carrying capacity criterion. However, the AISC specification permits a minimum spacing of 2.67 times the fastener diameter in structural steel buildings [28]. So, the values predicted by the procedure proposed are on the safe side. For comparison, the elastic load-carrying capacity curve based on the infinite plate model [10] is provided (Fig. 2, dashed line) as well. For very small ratios of the results are identical, however, starting from 0.1 , the size effect cannot be neglected. Additionally,
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ACCEPTED MANUSCRIPT due to numerical treatment of the HMH yield criterion in [10], no limit load-carrying capacity curve can be constructed in a simple manner as it is done by the present solution.
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Radial and tangential stress distributions including residual stresses in the zone surrounding a fastener are shown in Figs. 3 and 4, respectively. The radial stress is
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compressive and greater than the circumferential one. The highest stress is at the edge of the
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hole corresponding to the maximum external loading. The residual stresses are almost
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negligible for practical loads. As concerning circumferential stresses, the highest values are reached at the elastic/plastic boundary and not at the edge of the hole. Predominantly, the
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stresses are tensile and comparable in value with those at the elastic plastic boundary. To this end, it is worth noting that the residual stresses might be beneficial for serviceability of the
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structure since they introduce compressive fields around fasteners (Fig. 4). The results
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obtained for stresses are in complete qualitative agreement with those presented by Pinho
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et.al. [10] which also confirms the suitability of the HMH yield criterion for steel structural applications. Following related studies [19], it is quite likely that introducing of strain hardening (for example, Ramberg-Osgood relation) would decrease tangential residual stresses near the contour of the hole. As concerning pressure necessary to bring the elasticplastic boundary to a specified radius, it should be increased with increasing strain hardening parameter. However, for both material models (perfectly plastic and strain hardening), yielding will always start at the inner edge of the hole. Nevertheless, strain hardening modeling needs special care for each individual phenomenon such as crack growth life [11], onset of instability [13] and so on. For pressurized containers, the elastic load-carrying capacity depends on the Poisson’s ratio (Fig. 5) and material never reaches its limit load-carrying capacity up to the prescribed maximum loading after which the plate must start to thicken appreciably to support any
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ACCEPTED MANUSCRIPT further increase in load. It is worth mentioning at this point that mixed boundary conditions pose additional challenges for elastic-perfectly plastic material. Figure 6 shows comparison of
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results obtained in the current research with those presented by Szuwalski & Zyczkowski [23] for dependence of external loading corresponding to the initiation of plastic zone on the value
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of Poisson’s ratio. In [23], the opposite boundary conditions ( uˆ 0 at r a and ˆ rr pˆ at
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r ) were considered in the framework of infinite plate model, and the resulting curve
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pelc is retrieved in Figure 6 (dashed line) which does not depend on the size effect. It can be seen that the elastic load carrying capacity in the case of shaft/plate junction strongly
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depends on , however, for elevated values of , the elastic load-carrying capacity of the shaft/plate structure is in general less than the one for pressurized containers, especially, for
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thin-walled ones. Based on the strain analysis, Szuwalski & Zyczkowski [23] discovered that
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the rigid shaft/plate assemblies experience the decohesion phenomenon when the positive
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radial strains tend to infinity at the shaft/plate joint in spite of the fact that the propagation of elastic/plastic border is very small. The corresponding load was named the decohesive carrying capacity, especially relevant to the elastic perfectly plastic material and represents dangerous collapse mechanism. It was also observed, that for incompressible material, the decohesion occurs immediately without any development of plastic zone. The same conclusion was done in another research [25] were again the boundary condition on displacement was imposed on the rigid shaft. Since the decohesion phenomenon is very harmful from the engineering point of view, the propagation of the elastic/plastic border for
1 2 was checked in the present study for pressurized containers (were the displacement condition is imposed at the opposite outer boundary) and presented in Figure 7 for different geometrical ratios . It can be seen that for thick and medium wall containers the plastic zone still can be developed and only very thin containers experience almost zero 15
ACCEPTED MANUSCRIPT plasticization. However, in contrast to the shaft/plate problem, the radial stresses for pressurized containers are negative everywhere (Fig.8) and the state of compression is
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observed in the material which “cannot lead directly to the decohesion” following definition given in [23]. Meanwhile, the results obtained here clearly show that pressurized cylinders do
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not experience limit load carrying capacity and the possible collapse mechanism may include
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local loss of geometrical stability or some sort of decohesion in terms of non-permissible
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material discontinuities. It is also seen from Figs. 2 and 5 that the elastic load-carrying capacity is greater for pressurized containers than in the zone surrounding a fastener.
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Complete radial and tangential stress distributions including residual stresses are shown in Figs. 8 and 9, respectively. Again, the highest radial stresses are at the edge of the hole but do
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not tend to zero as in the case of model I (due to different boundary condition at the outer
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radius of the plate). However, the general curve profiles are similar for both geometrical
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models. As concerning circumferential stresses, it is seen from Figs. 4 and 9 that the general curve profiles are similar for both geometrical models but the plastic zone in pressurized containers is less pronounced than near a fastener for the same value of external loading, and the elastic unloading creates beneficial residual compressive stresses near the edge of the hole. From the analysis of these results it can be concluded that pressurized containers are mostly vulnerable to the local loss of geometric stability rather than the limit load-carrying failures. In contrast, bolted connections experience ductile failures due to an excessive external loading. When the elastic zone is sufficiently large, as in the case of pressurized containers, the compressibility of the material should be taken into account since stability analysis is sensitive to both stresses, radial and circumferential, and here circumferential stresses may significantly depend on the Poisson’s ratio (Fig. 10).
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General Conclusions
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Two geometrical models are suggested for steel bolted connections and pressurized containers which are based on a thin annular disk subjected to a pressure at the inner radius with either
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free from stresses or rigidly restricted boundary condition, respectively, at the outer radius. A
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unified analytical procedure for determination of stresses is developed for both geometrical
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models which is based on the plane-stress assumption, Hooke’s law in the elastic range, HMH yield criterion in the plastic range, and continuity of stresses at the elastic-plastic boundary.
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The HMH yield criterion and elastic-perfectly plastic material behavior of plate are suitable choices in obtaining the uncoupled set of equations for stress solutions since they lead to
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simple closed form solutions as opposite to a strain hardening model which requires
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evaluation of plastic strains even if only preliminary stress design is requested.
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Criterion of the limit load-carrying capacity is proposed to determine the minimum distance between centers of fasteners. It corresponds to the full plasticization of the surrounding plate material and makes part of the closed form solution due to the choice of elastic-perfectly plastic model. The distances obtained agree well with AISC specifications and on the safe side.
In general results presented are useful for the preliminary engineering design stage to assist real three-dimensional modeling of cold expansion processes, bolt assemblies and pressurized containers; to establish stress intensity factors or size-dependent scale factors and to better understand mechanical resistance of the plates in bearing and possible failure mechanisms. So, the size effect and analytical formulae for propagation of elastic-plastic border and assessment of limit load carrying capacity are important results obtained in this study toward engineering improvements in structural mechanics.
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ACCEPTED MANUSCRIPT References [1] Mackerle J. Finite element analysis of fastening and joining: a bibliography (1990-2002).
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Int J Pressure Vessels and Piping 2003; 80:253-71.
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ACCEPTED MANUSCRIPT [12] Sedighi M, Jabbari AH. Investigation of residual stresses in thick-walled vessels with combination of autofrettage and wire-winding. Int J Pressure Vessels and Piping 2013; 111-
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fastener holes. Engineering Failure Analysis 2003; 10:13-24. [15] Wu HC, Mu B. On stress concentrations for isotropic/orthotropic plates and cylinders
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with a circular hole. Composites: Part B 2003; 34:127-34. [16] Durban D. An exact solution for the internally pressurized, elastoplastic, strain-
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hardening, annular plate. Acta Mechanica 1987; 66:111-28.
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[17] Gamer U. A concise treatment of the shrink fit with elastic-plastic hub. Int J Solids
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Structures 1992; 29(20):2463-9.
[18] Wanlin G. Elastic-plastic analysis of a finite sheet with a cold-worked hole. Engineering Fracture Mechanics 1993; 46(3):465-72. [19] Arora PR, Simha KRY. Analytical and experimental evaluation of coldworking process for strain hardening materials. Engineering Fracture Mechanics 1996; 53(3):371-85. [20] Guowei M, Iwasaki S, Miyamoto Y, Deto H. Plastic limit analyses of circular plates with respect to unified yield criterion. Int J Mech Sci 1998; 40(10):963-76. [21] Bektas N, Altan G, Ergun E, Demirdal G. Elastic-plastic and residual stress analysis of an aluminum disc under internal pressures. J Engineering Science 2004; 10(2):201-6. [22] Bektas N. Elastic-plastic and residual stress analysis of a thermoplastic composite hollow disc under internal pressures. J Thermoplastic Composite Materials 2005; 18:363-75.
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ACCEPTED MANUSCRIPT [23] Szuwalski K, Zyczkowski M. On the phenomenon of decohesion in perfect plasticity. Int J Solids Structures 1973; 9:85-98.
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[24] Szuwalski K. Decohesive carrying capacity in perfect and asymptotically perfect plasticity. Mechanika Teoretyczna I Stosowana 1990; 28(1-2):243-54.
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plane torsion. Int J Solids Structures 2000; 37:1727-42.
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[25] Latas W, Zyczkowski M. Decohesive carrying capacity of a disk under tension and in-
[26] Debski R, Zyczkowski M. On decohesive carrying capacity of variable-thickness annular
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perfectly plastics disks. ZAMM – Z Angew Math Mech 2000; 82(10):655-69.
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[27] Nadai A. Plasticity. New York: McGraw-Hill; 1931. [28] Tamboli AR. Handbook of structural steel connection design and details. 2rd ed.
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McGraw-Hill; 2009.
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ACCEPTED MANUSCRIPT Figure captions
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Figure 1 Geometric representation of a circular plate in cylindrical coordinate system Figure 2 Limit external loading curves for bolted connections
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Figure 3 Radial stress distributions for different values of external loading for bolted
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connections (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 4 Circumferential stress distributions for different values of external loading for bolted
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connections (solid lines – elastic/plastic stresses; dashed lines – residual stresses)
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Figure 5 Initiation of plastic zone in pressurized containers depending on geometric ratio Figure 6 Comparison of elastic load-carrying capacity curves for pressurized containers and
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rigid shaft/plate connections as a function of Poisson’s ratio
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Figure 7 Propagation of elastic/plastic border in incompressible pressurized containers for
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different values of geometric ratio
Figure 8 Radial stress distributions for different values of external loading for pressurized containers (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 9 Circumferential stress distributions for different values of external loading for pressurized containers (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 10 Influence of Poisson’s ratio on radial and circumferential stress distributions for pressurized containers (solid lines – radial stresses; dashed lines – circumferential stresses)
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S2352-0124(14)00007-1 doi: 10.1016/j.istruc.2014.09.003 ISTRUC 6
To appear in: Received date: Revised date: Accepted date:
4 July 2014 22 September 2014 26 September 2014
Please cite this article as: Aleksandrova Nelli, Engineering stress solutions for bolted and pressurized steel structures, (2014), doi: 10.1016/j.istruc.2014.09.003
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 Engineering stress solutions for bolted and pressurized steel structures Nelli Aleksandrova
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Centre of Exact Sciences and Engineering, Madeira University, 9020-105 Funchal, Portugal (Madeira);
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E-mail: [email protected]; Phone: (351) 291-705-285; Fax: (351) 291-705-249
Keywords: Plates; Plasticity; Analytical solution; Steel Structures
1 Introduction
Bolts, rivets and location pins are some of the most utilized mechanical fasteners used in civil and mechanical engineering as well as in airspace industry. These fasteners play a crucial role in safety, reliability and serviceability of the structure by provoking a dangerous stress concentration zone (surrounding fasteners). In practice, numerous specific standards and regulations have been produced to assist civil engineers in installation, inspection and maintenance of such structures. On the other hand, a lack of rigorous mathematical procedures based on reliable material models and failure criteria to predict the development of stress concentration zone near the bolts often lead to the overestimated safety factors and heavier components in order to assure safety precautions. In fact, a number of parameters 1
ACCEPTED MANUSCRIPT need to be considered in successful design of bolted connections to describe the area surrounding fasteners. Among the most important parameters are geometrical characteristics
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(dimensions), material model, type of installation (interference) and service environment effect (ambient or elevated temperatures). Nowadays, numerical modeling, primary based on
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the finite element method (FEM), which incorporates experimental data and standardization
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rules, has become a popular technique used for two- and especially three-dimensional
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simulations of the bolt/surrounding area problems. The extensive review of such FEM solutions is provided in [1]. The review of common experimental procedures and industrial
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codes for design of bolted connections is given in [2]. Contemporary discussion of various specifications in terms of nominal strength and failure modes for stainless steel bolted
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connections is provided in [3]. Theoretical stress analysis based on the strength of materials
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was used in [4] to demonstrate the failure modes obtained experimentally in bolted single lap
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joints taking into consideration edge effects. Recently, a complete study of bearing stress in single bolt connections was conducted by Moze and Beg [5]. Several theoretical models of bearing resistance were discussed in agreement with Eurocode. It was concluded that the resistance of such connections depend on the geometry and material properties. Elegant engineering solutions to improve bolt/plate contact assemblies were suggested by Pedersen’s [6] showing the importance of Poisson’s ratio and influence of the contact pressure distribution. To this end, analytical methods continue to provide powerful tool for the preliminary engineering design and FEM verification of the problem since they allow for its rigorous mathematical formulation and analysis as well as account for the broad range of parameters and material models in the description of the dangerous stress concentration zone surrounding fasteners. To describe material behavior of this zone, the model of a non-elastic open-hole
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ACCEPTED MANUSCRIPT annular plate subjected to internal pressure or a plate with a pressed in elastic inclusion are suitable choices. A similar geometrical model of an infinite metal sheet with a hole made of
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non-elastic or composite material has been extensively used in engineering to describe the cold expansion process. In the frame of this model, Hsu and Forman [7] obtained an exact
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elastic-plastic solution for stresses assuming the sheet to be orthotropic in the thickness
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direction and obeying the Hill’s yield criterion. Later on, Ball [8] used the same model but
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added to the stress solution the consideration of the displacement control on the hole boundary. Chakrabarty [9] generalized the problem comparing the stress solution obtained on
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the basis of the Tresca yield criterion with its associated flow rule and the Huber-MisesHencky (HMH) yield criterion combined with deformation theory of plasticity for hardening
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and non-hardening materials isotropic in the plane of the plate; also, the hyperbolic thickness
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variation was considered as well as the related problem of an open-hole metal sheet subjected
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to uniform tension at infinity. The residual stress fields due to expansion of hole in an infinite plate were derived in [10] using the Tresca yield criterion (closed-form analytical solution) and HMH yield criterion (finite difference numerical technique) assuming an elastic-perfectly plastic material. It was shown that the choice of yield criterion significantly affects the results. On the other hand, an assessment of the influence of hardening in popular analytical models (including those by Hsu and Forman [7] and Ball [8]) to predict residual stresses was carried out by Wang and Zhang [11]. A new semi-analytical method of calculating residual stresses in pressure vessels which takes into account variations in Young modulus and tangent modulus was proposed by Sedighi and Jabbari [12]. To this end, recently, Leu, Liao and Lin [13] also derived new exact analytical solutions for plastic limit pressure adopting nonlinear combined isotropic/kinematic hardening and compare the results with perfectly plastic material model.
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ACCEPTED MANUSCRIPT Entirely numerical stress/strain solution of the cold expansion problem based on an infinite sheet geometrical model was done in [14] by FEM.
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However, for most structural engineering applications [4-6], the model of an infinite sheet should be substituted by the model of a comparable size inner/outer diameter annular plate or,
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depending on application, by the model of an annular finite size plate embedded into a rigid
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container (as in high pressure boilers, pressure vessels or loose-material containers) which
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further complicates the problem due to the imposed boundary condition on displacements. A suitable hole to structure dimension ratio (scale factor) provides correct estimation of hole
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effect on steel structures and pressure vessels [15]. By analogy with the cold expansion problem, two distinct analytical approaches may be considered based on either the Tresca
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yield criterion with its associated flow rule [16, 17] or the HMH yield criterion (or Hill’s yield
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criterion in the case of transversely orthotropic material) combined with the deformation
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theory of plasticity [18, 19]. To this end, experimental results show that the strength of most metal materials is better described by the HMH yield criterion [20]. However, the use of the deformation theory of plasticity, especially with an additional assumption of elastic incompressibility and neglect of elastic strains in the plastic zone (which permits the derivation of analytical formulae for stresses and simplifies substantially the analysis), is not completely justified. In [19], other simplifications to the formulation of the problem were made to get the displacement solution such as an introduction of a unified variable Poisson’s ratio which is valid for a specific strain hardening law. In more general cases, when the HMH yield criterion is employed with its associated flow rule, only a few numerical solutions are available [21, 22] and no analytical solutions are known for practical engineering purposes. So, the objective of the present study is to develop a unified analytical procedure capable of obtain mathematically rigorous stress fields both in the area surrounding fasteners and in
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ACCEPTED MANUSCRIPT the confined area enclosed in a rigid container. Two simple geometrical models are suggested and compared with each other for practical engineering applications: (I) classical model of an
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annular finite size plate subjected to internal pressure at its inner radius and free from stresses at its outer radius; (II) alternative finite size model of an annular plate subjected to internal
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pressure at its inner radius and embedded into a rigid container at its outer radius. The
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influence of the bolt on the surrounding area is simplified by the applying internal pressure.
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The material of the plate is assumed to be elastic-perfectly plastic obeying the HMH yield criterion with its associated flow rule. However, only the stress analysis is performed here to
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make emphasis on the advantages of the proposed unified analytical procedure (valid for both geometrical models) to preliminary engineering design including uncoupled stress solutions.
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For the alternative geometrical model II, only elementary kinematic analysis is required
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arising from the boundary condition on displacement at the outer radius of the plate which
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does not influence the general algorithm as opposite to the more involved problem with boundary condition on displacement at the inner radius of the plate [23]. It is worth noting that the related strain analysis may be conducted straight after the stresses are determined using the same geometrical models and assumptions of axial symmetry, plane stress state and small strains. Compressibility of the material in the outer elastic zone of the plate and summation of elastic and plastic strains in the plastic inner zone should be also considered the same way as it was done in [23]. That is, the general line of the stress/strain solution is based on the HMH yield criterion with its associated flow rule and elastic-perfectly plastic material behavior. Besides the elastic/plastic and residual stress solutions, the load-carrying capacity of each model is also discussed in the present study since for the perfect plasticity two values of the limit external loading may be distinguished, namely, the elastic load-carrying capacity when the structure starts yielding and the limit load-carrying capacity which corresponds to
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ACCEPTED MANUSCRIPT the full plasticization of the component. As it is known, this material model is frequently used in many other structural and mechanical applications and shows specific patterns while
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performing calculations [23-26]. Hence, it seems important both from mathematical and engineering points of view to analyze the load-carrying capacity of the structures. Other than
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that, correct prediction of the load-carrying capacity in general is crucial for achieving an
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optimal structural design [20] and may be considered as a limiting criterion.
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2 Elastic load-carrying capacity and initiation of plasticity
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In cylindrical coordinate system rz with non-zero radial, ˆ rr , and tangential, ˆ , stress tensor components, consider a thin annular plate of inner radius a and outer radius b subjected
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to gradually increasing radial pressure pˆ around its inner edge (Fig. 1). That is, the following
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ˆ rr pˆ at r a
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boundary condition in stress should be satisfied (1)
Additionally, for the geometrical model I the outer radius is considered traction free, that is, subject to the stress boundary condition
ˆ rr 0 at r b
(2)
For the geometrical model II, the plate is placed into a rigid container, that is, subject to the displacement boundary condition
uˆ 0 at r b
(3)
where uˆ is the radial displacement. Due to the circular symmetry of the problem,
ˆ rr ˆ rr (r ) , ˆ ˆ (r ) and plane stress assumption, ˆ zz 0 , there is only one nontrivial equilibrium equation
dˆ rr ˆ rr ˆ 0 dr r
(4)
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ACCEPTED MANUSCRIPT which is valid both in elastic and plastic zones of the plate. In the plastic zone, the HMH yield criterion is adopted and has the form 2 2 ˆ rr ˆ ˆ rrˆ Y 2
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(5)
where Y is the yield stress of the material.
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For sufficiently small values of external loadings pˆ the whole plate is entirely elastic with
2 Bb 2 2 Bb 2 1 1 ˆ A , A , uˆ r 1 A 1 Bb 2 2 2 E r 3 3 r r
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ˆ rr
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the well-known Lamé’s solution
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where A and Bb 2 are two convenient constants of integration and the coefficient 2
(6)
3 is
introduced for the convenience of further calculations; E is the Young’s elasticity modulus
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and is the Poisson’s ratio. To write out the resulting stress formulae, the dimensionless
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parameters may be introduced rr ˆ rr Y , ˆ Y , p pˆ Y ; r b , a b . Then, for the model I, the boundary conditions (1) and (2) define stress field in the elastic plate
rr
2 p 1 2 p 1 , 1 1 2 2 2 2 1 1
(7)
Analogously, for the model II, the boundary conditions (1) and (3) define stress field as
rr
p 2 2 1 1 p 2 2 1 1 , 2 2 1 1 2 2 1 1
(8)
These stress solutions are valid up to the value of p at which the plastic zone starts to develop. This value of p is determined from Eqs. (7)-(8) and yield criterion (5). It follows from Eqs. (7)-(8) that in the elastic plate the left-hand side of Eq. (5) attains its maximum value at
. Therefore, the plastic zone starts to develop from the inner radius of the plate, and, in general case, the plate consists of inner plastic and outer elastic zones divided by an unknown
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ACCEPTED MANUSCRIPT elastic/plastic boundary, c, or in dimensionless notations c b . Substituting subsequently Eqs. (7)-(8) at into Eq. (5) yields the value of the elastic load-carrying capacity of the
3 2
For the model II, it is
q 2 q 1 , q 2 1 1
1 1
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pelc 1
(9)
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pelc 1 2
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plate. For the model I, it is
2
(10)
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As it follows from Eqs. (7)-(10), both the elastic limit-load capacity and the elastic stress field
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of the model II depend on the compressibility of the material as opposite to the classical model I for which neither the elastic limit-load capacity nor the elastic stress field depend on
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the Poisson’s ratio.
3 Stress distributions in the elastic-plastic areas
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In the inner plastic zone, , there are two available equations - equilibrium Eq. (4) and yield criterion (5) - for two unknown stress components rr , , hence, the problem is internally statically determinate, at least, for the model I. For the model II, due to the boundary condition in displacement (3), some kinematic analysis is also required, however, it can be reduced to the analysis of elastic zone which will be shown later on. The stress solution in the plastic zone for both models proceeds via introduction of an auxiliary variable
suggested by Nadai [27] such that the yield criterion is automatically satisfied
rr 2 3 cos , 2 3 cos 3
(11)
This way two unknown stresses are replaced by only one unknown function r , the equation for which is obtained upon the substitution of Eq.(11) into the equilibrium Eq. (4) r sin
sin 0 r 6
(12) 8
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cos a 3 2 p
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(13)
3 , and considering equations for stresses (7), (8), and (11), the value of a with correct
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2
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where a is the value of at . It is seen from Eq. (13) that the limiting value of p is
sign is calculated as
(14)
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a ArcCos 3p 2
attains the limiting value of 2
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As p increases from zero, a increases from 2 and eventually becomes equal to when p
3 . That is, for both geometrical models, a non-hardening
3 Y , then the plate must start
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material cannot sustain a stress of magnitude greater than 2
to thicken appreciably to support any further increase in the radial load. Also, as it follows
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from Eq. (12), is a singular point. So, the value of a for numerical calculations, especially those based on commercial codes, should be limited by 2 a . The solution to Eq. (12) with boundary condition (14) has analytical form
2 sin 6 exp 3 sin 6 2
(15)
Thus, implicitly depends on p. Let be the value of at the elastic-plastic boundary . Then,
2 sin 6 exp 3 2 sin 6
(16)
Obviously, the condition of continuity of stresses should be preserved at the elastic-plastic boundary. Hence, taken into account that the general solution (6) is valid in the elastic zone,
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cos 3 A B 2
6
3 2 cos 6
(18)
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B 1 2 2 sin
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A
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from which the constants A and B are defined
(17)
where follows from Eq. (16). It should be mentioned at this point that all above equations
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are valid for both geometrical models and may be considered as a unified algorithm of calculations in comparison with particular procedures applied for classical solutions of similar
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cold-expansion problem or rigid shaft/plate assemblies. 3.1 Bolted connections
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To make further calculations on model I, the boundary condition (2), which is applied at the elastic zone together with Eq. (6), should be fulfilled. So, A B 0 at 1 , and using Eq. (18) one gets 1
2
1 tg 6 3
(19)
Multiplying Eq. (19) by Eq. (16) yields
1
2
sin 6 1 exp 3 cos 6 3
where a is directly related to p by Eq. (14) with 0 p 2
(20)
3 . For a given value of external
loading p and geometrical ratio , Eq. (20) determines the value of . After that, the radius of elastic/plastic boundary follows from Eq. (19). 3.2 Pressurized containers 10
ACCEPTED MANUSCRIPT Similarly, for the model II, the boundary condition (3), which is applied at the elastic zone together with Eq. (6), should be fulfilled. So, 1 A 1 B 0 at 1 , and using Eq.
2
(21)
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1 1 tg 6 1 3
1
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(8) one gets
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Multiplying Eq. (21) by Eq. (16) yields sin 6 1 1 exp 3 cos 6 1 3 2
(22)
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1
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For a given value of external loading p, geometric ratio and Poisson’s ratio , Eq. (22) determines . After that, the radius of elastic/plastic boundary follows from Eq. (21).
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Then, for both models, the stresses in each zone are calculated by Eq. (6) with Eq. (18) in
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the outer elastic zone and by Eq. (11) in the inner plastic zone where the auxiliary variable
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is related to the dimensionless radius via Eq. (15). This way, a unified algorithm of stress filed determination is concluded leading to an effective approximate engineering stress solution for bolted and open-hole thin-walled structures. The residual stresses are obtained subtracting the purely elastic solutions (7) and (8), respectively, for model I and II from the plastic solution (11).
4 Limit load-carrying capacity Considered structures reach their limit load-carrying capacity if there exists an external loading p such that the elastic outer zone vanishes. This is equivalent to the condition that the elastic/plastic boundary coincides with the outer edge of the plate, 1 . 4.1 Bolted connections Hence, for the geometrical model I, as it follows from Eq. (19)
3 tg 6 , or
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2
2 exp 3 2 sin 6 3
(23)
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From here, for specific , one gets , and then the corresponding limit load plload is
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calculated using Eq. (14). It is important at this point to verify in principle the existence of the
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solution of Eq. (23) for all possible range of taken into account the fact that numerically the
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root-finding procedure is extremely sensitive to the initial guess-solution. As soon as the limit load is found, the critical value of may be found as well based on the condition that the
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maximum external load p according to Eq. (13) equals pmax 2
3 since the trigonometric
function is bounded by 1 cos a 1 . That is, the maximum value of a with the
2
2 exp 3 2 sin5 6 3
(24)
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1
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appropriate sign equals a . Substitution of a into Eq. (23) yields
From Eq. (24), one concludes that the critical value of 0.3377 which implies that for
0.3377 the plate never reaches its limit load-carrying capacity, that is, some region of the plate near the outer radius is always in the elastic state. 4.2 Pressurized containers This algorithm of finding the limit load-carrying capacity works for both models, it is simple and straight forward. For the model II, it shows immediately that no solution ( ) exists for any value of 0 1 since the magnitude of corresponding to 1 in this case equals
6 Arctg 3 C1 with C1 1 1 due to Eq. (21). It implies that container never reaches its limit load-carrying capacity for permissible external loading.
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ACCEPTED MANUSCRIPT 5 Results and Discussion A unified analytical procedure of stress calculations is developed for two classes of structural
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mechanics problems: bolted connections and pressurized containers, which lays the
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foundation for further research in strain/displacement fields. The material model of elastic-
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perfectly plastic behavior of the plate has an advantage of getting an uncoupled stress solution which is important for rapid and easy preliminary engineering design and leads to an effective
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method of estimation of the load-carrying capacity of the structures. Another popular power strain hardening model may only be used with the initial consideration of plastic strains as it
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has been done, for example, in [9]. Moreover, an immediate simplification would be achieved in this case for the determination of the stresses if the deformation theory of plasticity is
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applied together with the HMH yield criterion [9]. A comprehensive analysis of different hardening models and their suitability to pressure vessel applications is presented in [13]. The
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incorporation of hardening parameter into the already existing closed form solutions is also discussed in [11] and out of the scope of this paper. For geometrical model I (suitable (among others) for bolted connection applications) the dependence of the geometrical parameter on the external loading p is shown in Fig. 2. It is seen that the minimum distance between centers of fasteners should be at least 2.96 times the fastener diameter (b = 2.96a) based on the limit load-carrying capacity criterion. However, the AISC specification permits a minimum spacing of 2.67 times the fastener diameter in structural steel buildings [28]. So, the values predicted by the procedure proposed are on the safe side. For comparison, the elastic load-carrying capacity curve based on the infinite plate model [10] is provided (Fig. 2, dashed line) as well. For very small ratios of the results are identical, however, starting from 0.1 , the size effect cannot be neglected. Additionally,
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Radial and tangential stress distributions including residual stresses in the zone surrounding a fastener are shown in Figs. 3 and 4, respectively. The radial stress is
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compressive and greater than the circumferential one. The highest stress is at the edge of the
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hole corresponding to the maximum external loading. The residual stresses are almost
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negligible for practical loads. As concerning circumferential stresses, the highest values are reached at the elastic/plastic boundary and not at the edge of the hole. Predominantly, the
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stresses are tensile and comparable in value with those at the elastic plastic boundary. To this end, it is worth noting that the residual stresses might be beneficial for serviceability of the
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structure since they introduce compressive fields around fasteners (Fig. 4). The results
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obtained for stresses are in complete qualitative agreement with those presented by Pinho
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et.al. [10] which also confirms the suitability of the HMH yield criterion for steel structural applications. Following related studies [19], it is quite likely that introducing of strain hardening (for example, Ramberg-Osgood relation) would decrease tangential residual stresses near the contour of the hole. As concerning pressure necessary to bring the elasticplastic boundary to a specified radius, it should be increased with increasing strain hardening parameter. However, for both material models (perfectly plastic and strain hardening), yielding will always start at the inner edge of the hole. Nevertheless, strain hardening modeling needs special care for each individual phenomenon such as crack growth life [11], onset of instability [13] and so on. For pressurized containers, the elastic load-carrying capacity depends on the Poisson’s ratio (Fig. 5) and material never reaches its limit load-carrying capacity up to the prescribed maximum loading after which the plate must start to thicken appreciably to support any
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results obtained in the current research with those presented by Szuwalski & Zyczkowski [23] for dependence of external loading corresponding to the initiation of plastic zone on the value
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of Poisson’s ratio. In [23], the opposite boundary conditions ( uˆ 0 at r a and ˆ rr pˆ at
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r ) were considered in the framework of infinite plate model, and the resulting curve
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pelc is retrieved in Figure 6 (dashed line) which does not depend on the size effect. It can be seen that the elastic load carrying capacity in the case of shaft/plate junction strongly
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depends on , however, for elevated values of , the elastic load-carrying capacity of the shaft/plate structure is in general less than the one for pressurized containers, especially, for
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thin-walled ones. Based on the strain analysis, Szuwalski & Zyczkowski [23] discovered that
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the rigid shaft/plate assemblies experience the decohesion phenomenon when the positive
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radial strains tend to infinity at the shaft/plate joint in spite of the fact that the propagation of elastic/plastic border is very small. The corresponding load was named the decohesive carrying capacity, especially relevant to the elastic perfectly plastic material and represents dangerous collapse mechanism. It was also observed, that for incompressible material, the decohesion occurs immediately without any development of plastic zone. The same conclusion was done in another research [25] were again the boundary condition on displacement was imposed on the rigid shaft. Since the decohesion phenomenon is very harmful from the engineering point of view, the propagation of the elastic/plastic border for
1 2 was checked in the present study for pressurized containers (were the displacement condition is imposed at the opposite outer boundary) and presented in Figure 7 for different geometrical ratios . It can be seen that for thick and medium wall containers the plastic zone still can be developed and only very thin containers experience almost zero 15
ACCEPTED MANUSCRIPT plasticization. However, in contrast to the shaft/plate problem, the radial stresses for pressurized containers are negative everywhere (Fig.8) and the state of compression is
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observed in the material which “cannot lead directly to the decohesion” following definition given in [23]. Meanwhile, the results obtained here clearly show that pressurized cylinders do
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not experience limit load carrying capacity and the possible collapse mechanism may include
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local loss of geometrical stability or some sort of decohesion in terms of non-permissible
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material discontinuities. It is also seen from Figs. 2 and 5 that the elastic load-carrying capacity is greater for pressurized containers than in the zone surrounding a fastener.
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Complete radial and tangential stress distributions including residual stresses are shown in Figs. 8 and 9, respectively. Again, the highest radial stresses are at the edge of the hole but do
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not tend to zero as in the case of model I (due to different boundary condition at the outer
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radius of the plate). However, the general curve profiles are similar for both geometrical
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models. As concerning circumferential stresses, it is seen from Figs. 4 and 9 that the general curve profiles are similar for both geometrical models but the plastic zone in pressurized containers is less pronounced than near a fastener for the same value of external loading, and the elastic unloading creates beneficial residual compressive stresses near the edge of the hole. From the analysis of these results it can be concluded that pressurized containers are mostly vulnerable to the local loss of geometric stability rather than the limit load-carrying failures. In contrast, bolted connections experience ductile failures due to an excessive external loading. When the elastic zone is sufficiently large, as in the case of pressurized containers, the compressibility of the material should be taken into account since stability analysis is sensitive to both stresses, radial and circumferential, and here circumferential stresses may significantly depend on the Poisson’s ratio (Fig. 10).
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General Conclusions
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Two geometrical models are suggested for steel bolted connections and pressurized containers which are based on a thin annular disk subjected to a pressure at the inner radius with either
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free from stresses or rigidly restricted boundary condition, respectively, at the outer radius. A
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unified analytical procedure for determination of stresses is developed for both geometrical
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models which is based on the plane-stress assumption, Hooke’s law in the elastic range, HMH yield criterion in the plastic range, and continuity of stresses at the elastic-plastic boundary.
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The HMH yield criterion and elastic-perfectly plastic material behavior of plate are suitable choices in obtaining the uncoupled set of equations for stress solutions since they lead to
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simple closed form solutions as opposite to a strain hardening model which requires
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evaluation of plastic strains even if only preliminary stress design is requested.
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Criterion of the limit load-carrying capacity is proposed to determine the minimum distance between centers of fasteners. It corresponds to the full plasticization of the surrounding plate material and makes part of the closed form solution due to the choice of elastic-perfectly plastic model. The distances obtained agree well with AISC specifications and on the safe side.
In general results presented are useful for the preliminary engineering design stage to assist real three-dimensional modeling of cold expansion processes, bolt assemblies and pressurized containers; to establish stress intensity factors or size-dependent scale factors and to better understand mechanical resistance of the plates in bearing and possible failure mechanisms. So, the size effect and analytical formulae for propagation of elastic-plastic border and assessment of limit load carrying capacity are important results obtained in this study toward engineering improvements in structural mechanics.
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ACCEPTED MANUSCRIPT References [1] Mackerle J. Finite element analysis of fastening and joining: a bibliography (1990-2002).
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ACCEPTED MANUSCRIPT [12] Sedighi M, Jabbari AH. Investigation of residual stresses in thick-walled vessels with combination of autofrettage and wire-winding. Int J Pressure Vessels and Piping 2013; 111-
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ACCEPTED MANUSCRIPT Figure captions
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Figure 1 Geometric representation of a circular plate in cylindrical coordinate system Figure 2 Limit external loading curves for bolted connections
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Figure 3 Radial stress distributions for different values of external loading for bolted
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connections (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 4 Circumferential stress distributions for different values of external loading for bolted
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connections (solid lines – elastic/plastic stresses; dashed lines – residual stresses)
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Figure 5 Initiation of plastic zone in pressurized containers depending on geometric ratio Figure 6 Comparison of elastic load-carrying capacity curves for pressurized containers and
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rigid shaft/plate connections as a function of Poisson’s ratio
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Figure 7 Propagation of elastic/plastic border in incompressible pressurized containers for
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different values of geometric ratio
Figure 8 Radial stress distributions for different values of external loading for pressurized containers (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 9 Circumferential stress distributions for different values of external loading for pressurized containers (solid lines – elastic/plastic stresses; dashed lines – residual stresses) Figure 10 Influence of Poisson’s ratio on radial and circumferential stress distributions for pressurized containers (solid lines – radial stresses; dashed lines – circumferential stresses)
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