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Plastic limit analysis of cylindrically orthotropic circular plates
Composite Structures 55 (2002) 455–466 www.elsevier.com/locate/compstruct
Plastic limit analysis of cylindrically orthotropic circular plates Guowei Ma *, Bazle A. Gama, John W. Gillespie Jr. Center for Composite Materials, University of Delaware, Newark, DE 19711, USA
Abstract The plastic limit analysis of cylindrically orthotropic circular plates is developed using a piecewise linear orthotropic yield criterion. The yield criterion is a modification of an isotropic formulation that consists of a series of weighted piecewise linear components. The piecewise linear yield criterion enables an analytical solution for the plastic limit load of cylindrically orthotropic circular plates. Plastic limit analysis for both simply supported and clamped circular plates under uniformly distributed load are carried out. Parametric studies are conducted to investigate the sensitivity of the plastic limit loads to material orthotropy and influences of orthotropic ratio and chosen yield criteria on the plastic limit loads of the circular plates are discussed. It is found that the plastic limit loads of the orthotropic circular plates are affected significantly by the orthotropic ratio. Enhancement of the circumferential yield moment will increase dramatically the plastic limit load of the plates. Moment and velocity fields of the plates in plastic limit state are also derived and discussed. The results obtained from the present study are helpful in understanding the failure mechanism of orthotropic circular plates and is useful for design. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Circular plate; Plastic limit load; Orthotropy; Yield criterion; Moment field; Velocity field
1. Introduction Structures are sometimes designed with enhanced strength in a certain direction to resist external load. Traditionally, stiffeners are the most commonly used method to increase stiffness and strength at minimal weight. Mechanical orthotropy may also arise from cold forming process of material. Some single crystal materials such as BCC iron, and textured polycrystalline materials exhibit anisotropic properties due to their inherent atomic structures. Fiber reinforced composites offer much higher strength-to-weight and stiffness-toweight by combining structural fibers and a more flexible matrix (e.g. glass fibers in synthetic resins). It has been found that the strength in the fiber direction of unidirectional fiber reinforced composites can be an order of magnitude higher than in the transverse direction [1]. Some composites such as metal matrix reinforced by metal fibers, plastics reinforced by Kevlar or glass fibers, not only have high strength and high stiffness, but also undergo plastic flow and considerable ductility. Plastic limit analysis of these kinds of orthotropic plates should be updated based on the yield properties of the plates.
*
Corresponding author. Fax: +1-302-831-8525. E-mail address: [email protected] (G. Ma).
The yield behavior of an orthotropic plate is always represented by a proper criterion. Numerous orthotropic yield criteria have been published over the past decades. One typical orthotropic yield criterion is referred to as Tsai–Hill criterion, which is an extension of the Hill anisotropic criterion to predict failure in an orthotropic, transversely isotropic lamina [2]. The criterion simplifies to the von Mises criterion for isotropic materials. Unfortunately, these criteria as well as many other commonly used orthotropic criteria have quadratic or polynomial forms and cannot be used straightforwardly to analytically derive the plastic limit load of a plate. A pioneering work that investigated the plastic limit behavior of cylindrically orthotropic circular plate under various loading cases has been carried out by Markowitz and Hu [3] who employed a modified Tresca criterion. The criterion, similar to the Tresca criterion for isotropic materials, has a piecewise linear form. The plastic limit solution given in their study satisfies both statical and kinematical admissible requirements, and is an exact solution. Save [4] and Save et al. [5] summarized the plastic limit solutions obtained for orthotropic circular plates. It is noticed that all the derivations for the plastic limit load of orthotropic circular plate in the previous studies were based on the modified Tresca criterion. Unfortunately, the Tresca criterion has obvious limitations because the effect of the intermediate principal stress on material strength is
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 7 4 - X
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G. Ma et al. / Composite Structures 55 (2002) 455–466
ignored [6]. The modified version of the Tresca criterion also has these limitations. A piecewise linear yield criterion described as a unified yield criterion has been suggested by Yu and He [7]. The unified yield criterion has been applied successfully to plastic limit analysis for isotropic circular plates [8–10]. The solutions with Tresca criterion and Yu criterion [11] are the special cases and the lower and upper bounds, respectively, of the general solution with the unified yield criterion. The solution using the von Mises criterion can be approximated by the general solution. The influences of the yield criteria on the plastic limit load, velocity and moment fields have been examined by comparing the results with the three specific yield criteria, i.e. the Tresca criterion, the von Mises criterion and the Yu criterion. Based on the results, the effect of the yield criteria on the plastic limit load depends on the loading and boundary conditions of the circular plate. The maximum differences of the plastic limit load by different yield criteria could be more than 20% for clamped circular plate and about 14% for simply supported circular plate [8–10]. It is noticed that the traditional solutions with the Tresca criterion always give a lower prediction of the plastic limit load. The moment and velocity fields based on the Tresca criterion have unreasonable features, for example, the circumferential moment on the simply supported circular plate is a constant over the entire plate; the velocity fields of both the simply supported and the clamped circular plates have a singularity at the plate center, etc. In this study, an orthotropic yield criterion, which is the generalization of the unified yield criterion, will be explored. The criteria retain a piecewise linear form and the modified Tresca criterion employed by the previous studies appears again as a special case. The loading condition, boundary condition and the plate itself are assumed axisymmetrical. Plastic limit loads together with the moment and velocity distributions of simply supported and clamped cylindrically orthotropic circular plates are derived. The influence of yield criterion and orthotropic ratio on the plastic limit load will be discussed.
it is extended to orthotropic materials or orthotropic structures. For an axisymmetric plate, there have only two moment components in the equilibrium equations [12]. The piecewise linear mathematical formula of the unified yield criterion may take the form of [8] Mh Mr ¼ ai þ bi M0 M0
ði ¼ 1–12Þ
in which Mr and Mh are the actual radial and circumferential moments, respectively; M0 is the yield moment, and they are integrated by the stress Rcomponents along h=2 the Mr ¼ h=2 rr z dz, Mh ¼ R h=2 section of the plate, 2 r z dz and M0 ¼ r0 h =4, in which rr and rh are, h=2 h respectively, the radial and circumferential stresses in the plate; r0 is the yield stress; h is the plate thickness. The coefficients ai and bi , with i ranging from 1 to 12, are parameters corresponding to the 12 segments of the yield curve, which are determined by a weighted parameter b which is ranged from 0 to 1. Detail description on the unified yield criterion can be found in the literatures [8–10]. The yield curves with b ¼ 0, 0.25, 0.5, 0.75 and 1 of the unified yield criterion for isotropic circular plate is shown in Fig. 1. For a cylindrically orthotropic circular plate with the principal axes oriented along the radial and circumferential directions, the unified yield criterion can be modified as, Mh Mr ¼ ai þ bi Mh0 Mr0
ði ¼ 1–12Þ;
ð2Þ
where Mr0 and Mh0 are the yield moments in the radial and circumferential directions, respectively. Normalizing the yield criterion by defining j ¼ Mh0 =Mr0 , mh ¼ Mh =Mr0 and mr ¼ Mr =Mr0 , Eq. (2) can be rewritten as follows, mh ¼ Ai mr þ Bi
ði ¼ 1–12Þ
2. Orthotropic yield criteria The unified yield criterion was originally proposed by Yu and He [7] which may represent or approximate different yield criteria such as maximum shear stress criterion (Tresca criterion), octahedral shear stress criterion (von Mises criterion), and Yu twin shear stress criterion [11]. Plastic limit analysis of circular plates with various loading conditions and boundary conditions has been studied based on the unified yield criterion [8–10]. Proper modification of the criteria should be made when
ð1Þ
Fig. 1. Unified yield criterion (mr ¼ Mr =M0 , mh ¼ Mh =M0 ).
ð3Þ
G. Ma et al. / Composite Structures 55 (2002) 455–466
in which Ai ¼ ai j, Bi ¼ bi j ði ¼ 1–12Þ. The orthotropic ratio j is the ratio of the circumferential yield moment to the radial yield moment. j ¼ 1 corresponds to the special case of isotropy, while j > 1 and j < 1 represents, respectively, the circumferential and the radial strengthening of the plate. Similar to the unified yield criterion, the Mises yield criterion can be modified as Mh2 Mh Mr M2 þ r2 ¼ 1: 2 Mh0 Mh0 Mr0 Mr0
ð4Þ
In terms of the dimensionless variables, Eq. (4) becomes m2h jmh mr þ j2 m2r ¼ j2 :
ð5Þ
Eq. (5) is a nonlinear equation with respect to the two variables mr and mh , which make it inconvenient for plastic limit analysis of circular plates. The quadratic
457
form in Eq. (4) is very close to the one of the Tsai–Hill orthotropic criterion. The only difference is that they have different coefficient in the interaction term. The yield curves with different orthotropic ratios of the modified unified yield criterion defined in Eq. (3) are plotted in Figs. 2(a)–(c) with respected to b ¼ 0, b ¼ 1 and b ¼ 0:5. The yield curve in Fig. 2(a) is exactly the same as employed in the previous studies [3,5]. Fig. 2(d) illustrates the modified von Mises criterion defined in Eq. (5). The orthotropic yield criteria are compared in Fig. 3 with different orthotropic ratios, i.e., j ¼ 2, j ¼ 0:5 and the isotropic case of j ¼ 1. It shows that, even in orthotropic conditions, the modified von Mises criterion can be approximately represented by the modified unified yield criterion by b ¼ 0.5. The extension of the unified yield criterion makes it possible to obtain the plastic limit solutions of orthotropic circular plates
Fig. 2. Orthotropic yield criteria: (a) b ¼ 0, modified Tresca criterion; (b) b ¼ 1; (c) b ¼ 0:5; (d) modified von Mises criterion.
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G. Ma et al. / Composite Structures 55 (2002) 455–466
or mrðijÞ ¼
bi bj ; ai aj
mhðijÞ ¼
ai bj aj bi j ai aj
ði ¼ 1–11; j ¼ i þ 1; i ¼ 12; j ¼ 1Þ;
ð7Þ
where the line with i ¼ 1 corresponds to line AB and i increases from 1 to 12 along with anti-clockwise sequence of the lines. Thus, i ¼ 12 corresponds to line LA. The symbol ði jÞ denotes the corner intersected by line i and line j. The values of the parameters ai and bi ði ¼ 1–12Þ in the unified yield criterion are listed in Table 1. Table 2 lists the moments at the 12 corners. (a)
3. General solutions
(b)
(c) Fig. 3. Comparison of the modified yield criteria with different orthotropic ratios: (a) j ¼ 2; (b) j ¼ 0:5; (c) j ¼ 1.
in view of not only the modified Tresca criterion but also a modified von Mises criterion and a modified Yu criterion. The moments at the corners of the yield curves in Fig. 3 can be derived by solving the equations for the two intersecting lines, which have the forms of mrðijÞ
Bi Bj ¼ ; Ai Aj
mhðijÞ
Ai B j Aj Bi ¼ Ai Aj
ði ¼ 1–11; j ¼ i þ 1; i ¼ 12; j ¼ 1Þ
ð6Þ
For convenience, dimensionless variables p and r, i.e., p ¼ Pa2 =Mr0 and r ¼ R=a, are defined for the cylindrically orthotropic plate with uniform thickness, where P is the uniformly distributed load, a is the outer radius of the plate and R is the radius variable in the range of 0 to a. For an axisymmetric loading, any two orthogonal directions at the central point are principal directions for stresses and orthotropy. Hence, the plate is locally isotropic and Mr ¼ Mh or mr ¼ mh at the plate center [5]. The moments at the plate center, thus, correspond to point A0 in Fig. 3. The point A0 may locate on different portions, namely lines with i ¼ 11, 12, 1, or 2, of the curves depending on the orthotropic ratio j. If j ¼ 1, A0 coincides with the corner point A. When j > 1, A0 falls on line KL (i ¼ 11) or LA (i ¼ 12), while A0 falls on line AB (i ¼ 1) or BC ði ¼ 2Þ when j < 1. For a simply supported circular plate in plastic limit state, the moment at the outer edge satisfies the yield point C in Fig. 3. The moments in the whole plate corresponds to the yield segments from A0 to C with anti-clockwise order. The total number of the segments could be 4 when A0 is on the line KL. It can be reduced to only one when A0 is on the line BC. On the other hand, the edge of a clamped circular plate corresponds to a yield point on line EF, which has been discussed in a previous study [9]. In the present study, the moments at the edge are assumed exactly corresponding to the corner point F in Fig. 3. Thus, the total number of the segments for the clamped circular plate is from 4 to 7 with reference to the orthotropic ratio j. The equilibrium equation of the circular plate is [12] dðrmr Þ=dr mh ¼ pr2 =2:
ð8Þ
With the aid of the piecewise linear form in Eq. (3), the radial moment can be integrated from Eq. (8) as mr ¼
Bi pr2 þ Ci r1þAi ; 1 Ai 2ð3 Ai Þ
ð9Þ
G. Ma et al. / Composite Structures 55 (2002) 455–466
459
Table 1 Constants ai and bi in the modified unified yield criterion AB (i ¼ 1)
BC (i ¼ 2)
CD (i ¼ 3)
DE (i ¼ 4)
EF (i ¼ 5)
FG (i ¼ 6)
GH (i ¼ 7)
HI (i ¼ 8)
IJ (i ¼ 9)
JK (i ¼ 10)
KL (i ¼ 11)
LA (i ¼ 12)
ai
b
b 1þb
1 1þb
1þb
1þb b
1b
b
b 1þb
1 1þb
1þb
1þb b
1b
bi
1þb
1
1
1þb
1þb b
1þb b
1þb
1
1
1þb
1þb b
1þb b
Table 2 Moments at the corners Nodes
A
B
C
D
E
F
G
H
I
J
K
L
mh
j
2ð1þbÞ j 2þb
j
1þb j 2þb
0
1þb j 2þb
j
2ð1þbÞ j 2þb
j
1þb j 2þb
0
1þb j 2þb
mr
1
1þb 2þb
0
1þb 2þb
1
2ð1þbÞ 2þb
1
1þb 2þb
0
1þb 2þb
1
2ð1þbÞ 2þb
where i represents each valid segment of the yield curve; Ci are integral constants determined by continuity and boundary conditions. For a general case where the moments in the plate covers a total of n segments on the yield curve, the continuity and boundary conditions of the radial moment will yield Bð1Þ pr02 1þA þ C1 r0 ð1Þ ¼ mrð0Þ ; 1 Að1Þ 2ð3 Að1Þ Þ
ð10aÞ
Bð1Þ pr12 1þA þ C1 r1 ð1Þ ¼ mrð1Þ ; 1 Að1Þ 2ð3 Að1Þ Þ
ð10bÞ
Bð2Þ pr12 1þA þ C2 r1 ð2Þ ¼ mrð1Þ ; 1 Að2Þ 2ð3 Að2Þ Þ
ð10cÞ
Bð2Þ pr22 1þA þ C2 r2 ð2Þ ¼ mrð2Þ ; 1 Að2Þ 2ð3 Að2Þ Þ
ð10dÞ
2 BðiÞ pri1 1þA þ Ci ri1 ð1Þ ¼ mrði1Þ ; 1 AðiÞ 2ð3 AðiÞ Þ
ð10eÞ
BðiÞ pri2 1þAð1Þ þ Ci ri ¼ mrðiÞ ; 1 AðiÞ 2ð3 AðiÞ Þ
ð10fÞ
2 BðnÞ prn1 1þA þ Cn rn1 ðnÞ ¼ mrðn1Þ ; 1 AðnÞ 2ð3 AðnÞ Þ
ð10gÞ
BðnÞ prn2 1þA þ Cn rn ðnÞ ¼ mrðnÞ ; 1 AðnÞ 2ð3 AðnÞ Þ
ð10hÞ
where AðiÞ and BðiÞ on the left side denote parameters of the ith valid segment from A0 in anti-clockwise order on the yield curve of Fig. 3. mrðiÞ ði ¼ 0–nÞ represents the radial moment at the ith corner point from A0 , e.g., mrð0Þ corresponds to the yield moment on point A0 , while mrðnÞ corresponds to the yield moment at the corner C for simply supported plate or the corner F for clamped plate. r0 ; r1 ; . . . ; ri ; . . ., and rn denote the demarcating radii corresponding to the valid corners on the yield curve. In particular, r0 equals zero and is at the plate
center where the moments correspond to corner A0 , rn equals 1 with reference to the plate outer edge. Defining a1 ¼ r1 =r2 , a2 ¼ r2 =r3 , ai ¼ ri =riþ1 and an1 ¼ rn1 =rn , then the following relationships exist r1 ¼ a1 a2 an1 , r2 ¼ a2 a3 an1 , ri ¼ ai an1 and rn1 ¼ an1 . With the aid of the continuity and boundary conditions, Ci ði ¼ 1–nÞ in Eqs. (10a)–(10h) are determined as follows, C1 ¼ 0; C2 ¼
ð11aÞ
Bð2Þ mrð1Þ 1 Að2Þ Bð1Þ 3 Að1Þ 1A ðr1 Þ ð2Þ ; þ mrð1Þ 1 Að1Þ 3 Að2Þ BðiÞ Ci ¼ mrði1Þ 1 AðiÞ Bð1Þ 3 Að1Þ þ mrð1Þ 1 Að1Þ 3 AðiÞ 2 1A ða1 a2 ai2 Þ ðri1 Þ ðiÞ ;
BðnÞ Cn ¼ mrðn1Þ 1 AðnÞ Bð1Þ 3 Að1Þ þ mrð1Þ : 1 Að1Þ 3 AðnÞ 2 1A ða1 a2 an2 Þ ðrn1 Þ ðnÞ ;
ð11bÞ
ð11cÞ
ð11dÞ
in which ratios a1 ; a2 ; . . . ; ai ; . . . ; an1 are given by Bð2Þ Bð1Þ 3 Að1Þ 2 mrð2Þ mrð1Þ a 1 Að2Þ 1 Að1Þ 3 Að2Þ 1 Bð2Þ þ mrð1Þ 1 Að2Þ Bð1Þ 3 Að1Þ 1Að2Þ mrð1Þ ¼ 0; ð12aÞ þ a 1 Að1Þ 3 Að2Þ 1
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G. Ma et al. / Composite Structures 55 (2002) 455–466
BðiÞ Bð1Þ 3 Að1Þ mrðiÞ mrð1Þ ða1 a2 ai1 Þ2 1 AðiÞ 1 Að1Þ 3 AðiÞ BðiÞ þ mrði1Þ 1 AðiÞ Bð1Þ 3 Að1Þ 1A 2 þ mrð1Þ ða1 a2 ai2 Þ ai1 ðiÞ 1 Að1Þ 3 AðiÞ ¼ 0;
ð12bÞ
BðnÞ Bð1Þ mrðnÞ mrð1Þ 1 AðnÞ 1 Að1Þ 3 Að1Þ 2 ða1 a2 an1 Þ 3 AðnÞ BðnÞ þ mrðn1Þ 1 AðnÞ Bð1Þ 3 Að1Þ 1A 2 þ mrð1Þ ða1 a2 an2 Þ an1 ðnÞ 1 Að1Þ 3 AðnÞ ¼ 0:
ð12cÞ
Eqs. (12a)–(12c) can be solved one by one by numerical iteration method such as half interval search method. After ai ði ¼ 1 to n 1Þ are calculated, the demarcating radii ri ði ¼ 1 to n 1Þ and the integral constants in Eqs. (11a)–(11d) are then obtained. The plastic limit load is derived from Eq. (10b) as Bð1Þ 2ð3 Að1Þ Þ p¼ : ð13Þ mrð1Þ 1 Að1Þ r12 Substituting the integral constants Ci ði ¼ 1–nÞ, the demarcating radii ri ði ¼ 0–nÞ and the plastic limit load p into Eq. (9), the radial moment field mr over the entire plate is then obtained. The velocity field of the plate is integrated from the compatibility conditions and associated plastic flow rule, which has the same form as that in the isotropic case [8], w_ ¼ w_ 0 c1i r1AðiÞ þ c2i ; ri1 6 r 6 ri and ði ¼ 1–nÞ;
ð14Þ
where c1i and c2i ði ¼ 1–nÞ are the integral constants corresponding to the n valid segments on the yield curve. ri ði ¼ 0–nÞ are again the demarcating radii, where the moment variables are located at the corner points of the yield curve. w_ 0 is the flow velocity at the plate center. The continuity and boundary conditions of the velocity field are: (1) w_ ðr ¼ 0Þ ¼ w_ 0 , (2) w_ and dw_ =dr ðr ¼ ri ; i ¼ 1 to n 1Þ are continuous and (3) w_ ðr ¼ 1Þ ¼ 0. Considering the above conditions, the constants c1i and c2i in Eq. (14) satisfy the following relationships: c21 ¼ 1; 1A c11 r1 ð1Þ
ð1
ð15aÞ þ c21 ¼
A Að1Þ Þc11 r1 ð1Þ
1A c12 r1 ð2Þ
¼ ð1
þ c22 ;
A Að2Þ Þc12 r1 ð2Þ ;
ð15bÞ ð15cÞ
1AðiÞ
c1i ri
1Aðiþ1Þ
þ c2i ¼ c1ðiþ1Þ ri AðiÞ
ð1 AðiÞ Þc1i ri
þ c2ðiþ1Þ ; Aðiþ1Þ
¼ ð1 Aðiþ1Þ Þc1ðiþ1Þ ri
1A
1AðnÞ
c1ðn1Þ rn1 ðn1Þ þ c2ðn1Þ ¼ c1ðnÞ rn ð1
A Aðn1Þ Þc1ðn1Þ rn1ðn1Þ
1A c1n rn ðnÞ
¼ ð1
ð15dÞ ;
ð15eÞ
þ c2n ;
ð15fÞ
A AðnÞ Þc1n rn1ðnÞ ;
ð15gÞ
þ c2n ¼ 0:
ð15hÞ
The above equations are a series of linear equations with the unknown variables of c1i and c2i ði ¼ 1–nÞ. The 2n unknown variables can be solved directly from the 2n simultaneous linear equations as the demarcating radii ri , ði ¼ 1–nÞ are known. 4. Simply supported orthotropic circular plate For a simply supported circular plate, with a certain orthotropic ratio j, there are four possible cases, namely the point A0 is respectively on segments KL, LA, AB or BC. The following will first determine the yield moments at the plate center for the four cases depending on the orthotropic ratio. The valid segments on the yield curve can then be formalized. The plastic limit load, moment field and velocity field are obtained from the general solution as the valid segments are determined. Case 1. Point A0 locates on segment KL. When j > 1, the circumferential yield moment Mh0 is greater than the radial yield moment Mr0 . The point A0 may locate on either segment KL or segment LA. Considering the special case when point A0 coincides with the corner L, there exists mh ¼ ja11 mr þ jb11 ; mh ¼ ja12 mr þ jb12 ;
ð16Þ
mh ¼ mr at the yield point A0 . Thus, the orthotropic ratio j in this special case is derived as, b11 b12 j¼ : ð17Þ a12 b11 a11 b12 When j > ððb11 b12 Þ=ða12 b11 a11 b12 ÞÞ, point A0 locates on the segment KL. The four segments ðn ¼ 4Þ, namely A0 L, LA, AB, and BC are valid. The radial moment at r ¼ 0 corresponding to point A0 is derived as, B11 jb11 or mrð0Þ ¼ ; ð18Þ 1 A11 1 ja11 which is determined by the weighted parameter b and the orthotropic ratio j. mrð0Þ ¼
Case 2. Point A0 locates on segment LA. When ððb11 b12 Þ=ða12 b11 a11 b12 ÞÞ P j > 1, point A0 falls on the segment LA. The valid segments are A0 A, AB
G. Ma et al. / Composite Structures 55 (2002) 455–466
and BC with the radial moment mr varying from the point A0 to the point C on the yield curve when r increases from 0 to 1. The radial moment at point A0 have the form of mrð0Þ ¼
B12 1 A12
or
mrð0Þ ¼
jb12 : 1 ja12
ð19Þ
Case 3. Point A0 locates on segment AB. When 1 P j > ððb1 b2 Þ=ða2 b1 a1 b2 ÞÞ, the moments on the entire plate correspond to the two yield segments of A0 B and BC. The radial moment mr at the plate center is mrð0Þ ¼
B1 1 A1
or mrð0Þ ¼
jb1 : 1 ja1
ð20Þ
Case 4. Point A0 locates on segment BC. When j 6 ððb1 b2 Þ=ða2 b1 a1 b2 ÞÞ, it yields the case that point A0 locates on BC. The moments on the plate correspond to only the yield segment BC. The valid segment is A0 C and the radial moment at the plate center is mrð0Þ ¼
B2 1 A2
or mrð0Þ ¼
jb2 : 1 ja2
ð21Þ
The radial moments mrðiÞ ; i ¼ 1–n at other corners and the parameters AðiÞ and BðiÞ as used in Eqs. (10a)– (10h) are listed in Table 3, in which mrðijÞ is given by Eq. (6) or (7). Once the valid segments and relative parameters are determined, the plastic limit load, the moment fields and the velocity fields are then obtained from Eqs. (9), (13) and (14), respectively, with the integral constants and demarcating radii being determined by the continuity and boundary conditions. Figs. 4(a)–(c) illustrate the moment fields in terms of the orthotropic unified yield criterion with respect to
461
j ¼ 2, j ¼ 1 and j ¼ 0:5. The three plots correspond to Cases 2–4, respectively. It is seen from Fig. 4 that when j 6¼ 1 or the plate has different yield moments in the radial and circumferential directions, both the moment distribution profiles and the moments at the plate center in terms of different weighted parameter b, i.e., b ¼ 0, b ¼ 0:5 and b ¼ 1 of the modified unified yield criterion, are quite different. The yield curve with b ¼ 0 which is the modified Tresca criterion always gives the lowest yield moment at the plate center. When the modified Tresca criterion is applied, the circumferential moment mh keeps constant on the entire plate in Cases 3 and 4; while it is constant in the range of r1 6 r 6 1 corresponding to Case 2. This is unreasonable and results from the neglection of the influence of the intermediate principal stress in the Tresca criterion. Based on the modified unified yield criterion with b 6¼ 0, the moment distribution of the orthotropic plate becomes more reasonable. The velocity fields corresponding to different cases are shown in Fig. 5. The Tresca criterion (b ¼ 0) gives a linear line (Figs. 5(b) and (c)) or piecewise linear lines (Fig. 5(a)) of the velocity distributions with different orthotropic ratios. The criteria with b ¼ 0:5 and b ¼ 1, which represents approximately the modified Mises criterion and the modified Yu criterion, yield smooth and orthotropic ratio dependent velocity distributions. It is obviously true that the weighted parameter b is the significant factor in unifying different yield criteria. The curve of plastic limit load versus the parameter b is plotted in Fig. 6(a) for the three orthotropic ratios of j ¼ 2, j ¼ 1 and j ¼ 0:5. It is seen that, for all the three cases, the plastic limit load increases monotonically with the increase of b. The plastic limit load when j ¼ 2 is much larger than those corresponding to j ¼ 1 and j ¼ 0:5. This indicates that much higher plastic limit load can be achieved when the plate is stiffened along the circumferential direction (j > 1).
Table 3 Yield moments at corner points and the corresponding AðiÞ and BðiÞ for simply supported circular plate i
1
2
3
4
Case 1 (n ¼ 4)
mrðiÞ AðiÞ BðiÞ
mrð11–12Þ A11 B11
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 2 (n ¼ 3)
mrðiÞ AðiÞ BðiÞ
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 3 (n ¼ 2)
mrðiÞ AðiÞ BðiÞ
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 4 (n ¼ 1)
mrðiÞ AðiÞ BðiÞ
mrð2–3Þ A2 B2
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G. Ma et al. / Composite Structures 55 (2002) 455–466
(a) (a)
(b) (b)
(c) Fig. 5. Velocity fields of orthotropic simply supported circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
(c) Fig. 4. Moment fields of orthotropic simply supported circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
Defining a difference ratio q p pTresca 100% q¼ pTresca
ð22Þ
makes it convenient to compare the effect of different yield criteria on the plastic limit load. Fig. 6(b) gives the difference ratio versus the weighted parameter b. It is found that the difference ratio increases gradually with the increase of b. The difference ratio is up to 7.1% and 12.6% when j ¼ 2; 8.1% and 14.0% when j ¼ 1 and 13.3% and 22.2% when j ¼ 0:5 corresponding to b ¼ 0.5 and 1, respectively. This implies that the plastic limit load based on different yield criteria are very different, which is helpful for optimization of design of an orthotropic circular plate.
The effect of the orthotropic ratio on the plastic limit load is illustrated in Fig. 7(a). It is seen that the plastic limit load increases sharply with the increase of the orthotropic ratio j, which indicates again that the strengthening along the circumferential direction does improve dramatically the load carrying capacity of a plate. The curve of difference ratio versus the orthotropic ratio with respect to b ¼ 0.5 and b ¼ 1 is plotted in Fig. 7(b). The maximum differences occurs at j ¼ 0:56 and they are 13.8% and 22.9% with respect to b ¼ 0.5 and b ¼ 1. The difference ratio remains relatively constant when j > 1 and the ratio is approximately 7% and 13% corresponding to b ¼ 0.5 and b ¼ 1. It repeatedly indicates that different yield criterion predicts significantly different plastic limit load as a function of the ratio of the orthotropic ratio.
G. Ma et al. / Composite Structures 55 (2002) 455–466
(a)
463
(b)
Fig. 6. Effects of yield criteria for simply supported circular plate: (a) plastic limit load versus b; (b) difference ratio of plastic limit load.
(a)
(b)
Fig. 7. Effect of orthotropic ratio for simply supported circular plate: (a) plastic limit load versus j; (b) difference ratio of plastic limit load.
Table 4 Yield moments at corner points and the corresponding AðiÞ and BðiÞ for clamped circular plate i
1
2
3
4
5
6
7
Case 1 (n ¼ 7)
mrðiÞ AðiÞ BðiÞ
mrð11–12Þ A11 B11
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð45Þ A4 B4
mrð5–6Þ A5 B5
Case 2 (n ¼ 6)
mrðiÞ AðiÞ BðiÞ
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð56Þ A5 B5
Case 3 (n ¼ 5)
mrðiÞ AðiÞ BðiÞ
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð5–6Þ A5 B5
Case 4 (n ¼ 4)
mrðiÞ AðiÞ BðiÞ
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð5–6Þ A5 B5
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5. Clamped circular plate For the clamped circular plate, there also exist four possible cases, namely, point A0 locates on KL, LA, AB, or BC corresponding to the different orthotropic ratio j. The moments on the entire plate corresponds to the segments from A0 to F with an anti-clockwise order as the radius variable r increases from 0 to 1. Similar to the four cases of the simply supported plate, the radial moment at the plate center mrð0Þ is calculated according to the weighted parameter b and the orthotropic ratio j.
The moments at the corner points in different cases and the parameters AðiÞ and BðiÞ in Eqs. (10a)–(15h) are listed in Table 4. The moment fields of the clamped circular plate with different orthotropic ratios, namely, j ¼ 2, j ¼ 1 and j ¼ 0:5, are shown in Figs. 8(a)–(c), respectively. Fig. 8(a) includes six valid yield segments, and Figs. 8(b) and (c) involve, respectively, five and four yield segments. It is seen that when j 6¼ 1 or the plate has different yield moments in the radial and circumferential directions, the moment distribution profiles and the moments at the plate center with different values of b, i.e., b ¼ 0, b ¼ 0:5 and b ¼ 1, of the unified yield criterion are again quite different.
(a) (a)
(b)
(c) Fig. 8. Moment fields of orthotropic clamped circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
(b)
(c) Fig. 9. Velocity fields of orthotropic clamped circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
G. Ma et al. / Composite Structures 55 (2002) 455–466
The velocity fields of the clamped plate for different cases are shown in Fig. 9. The modified Tresca criterion (b ¼ 0) again gives a linear line (Figs. 9(b) and (c)) when j ¼ 1 and j ¼ 0:5. The criteria with b ¼ 0:5 and b ¼ 1, which represent approximately Mises criterion and the Yu criterion, yield smooth and orthotropic ratio dependent velocity distributions. The curves of plastic limit load and the difference ratio versus parameter b for the clamped circular plate is plotted in Fig. 10(a). The plastic limit loads again increase gradually with the increase of b and are much larger than the corresponding values of the simply supported circular plate. The orthotropic ratio influences the plastic limit load significantly since the plastic limit load when j ¼ 2 is much larger than those when j ¼ 1 and j ¼ 0:5. The difference ratio
(a)
defined in Eq. (22) is up to 10.8% and 18.3% when j ¼ 2; 13.0% and 21.7% when j ¼ 1 and 16.4% and 27.3% when j ¼ 0:5 corresponding to b ¼ 0.5 and 1, respectively. These differences are larger than the corresponding results for the simply supported circular plate. This means that the yield criterion influences the plastic limit load of clamped circular plates more significantly than that of simply supported circular plate. The effect of different orthotropic ratio on the plastic limit load is shown in Fig. 11. The plastic limit load increases again sharply with the increase of the orthotropic ratio j. Although the difference ratio decreases with the increase of the orthotropic ratio, they remain about 9% and 16% corresponding to b ¼ 0.5 and b ¼ 1 for a wide range of the orthotropic ratio.
(b)
Fig. 10. Effect of yield criteria for clamped circular plate: (a) plastic limit load versus b; (b) difference ratio.
(a)
465
(b)
Fig. 11. Effect of orthotropic ratio for clamped circular plate: (a) plastic limit load versus j; (b) difference ratio.
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6. Conclusions A piecewise linear orthotropic yield criterion is suggested. Analytical solution by employing the piecewise linear orthotropic yield criterion has been derived for cylindrically orthotropic circular plates with simply supported and clamped edge conditions. It is found that the plastic limit load increases significantly with the increase of the orthotropic ratio. The chosen yield criterion affects the plastic limit load, moment fields and velocity fields. For the simply supported orthotropic circular plate, the difference ratio is a function of the orthotropic ratio and it is 7.1% and 12.6% when j ¼ 2 and 13.3% and 22.2% when j ¼ 0:5 with respect to the modified Mises criterion and the modified Yu criterion, respectively. The different ratio remain about 7% and 13% with respect to the modified Mises criterion and the modified Yu when j > 1. For the clamped orthotropic circular plate, the difference ratio decreases with the increase of the orthotropic ratio. It is about 9% and 16% with respect to the modified Mises criterion and the modified Yu criterion, respectively, when the orthotropic ratio is greater than 1. References [1] Daniel IM, Ishai O. Engineering mechanics of composite materials. Oxford: University Press; 1994.
[2] Tsai SW. Strength theories of filamentary structures. In: Schwartz RT, Schwartz HS, editors. Fundamental aspects of fiber reinforced plastic composites. New York: Wiley/Interscience; 1968. p. 3–11. [3] Markowitz J, Hu LW. Plastic analysis of orthotropic circular plates. ASCE J Eng Mech 1965;EM5:251–5. [4] Save MA. Limit analysis of plates and shells: research over two decades. ASCE J Struct Mech 1985;13:343–70. [5] Save MA, Massonnet CE, G.de Saxce. Plastic limit analysis of plates, shells and disks. Amsterdam, New York: Elsevier; 1997. [6] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950. [7] Yu MH, He LN. A new model and theory on yield and failure of materials under complex stress state. In: Mechanical behavior of materials-6, vol. 3. Oxford: Pergamon Press; 1991. p. 841–6. [8] Ma GW, Iwasaki S, Miyamoto Y, Deto H. Plastic limit analysis of circular plates with respect to the unified yield criterion. Int J Mech Sci 1998;40(10):963–76. [9] Ma GW, Hao H, Iwasaki S. Plastic limit analysis of a clamped circular plate with unified yield criterion. Struct Eng Mech 1999;7(5):513–25. [10] Ma GW, Hao H, Iwasaki S. Unified plastic limit analysis of circular plates under arbitrary load. ASME J Appl Mech 1999;66(2):568–70. [11] Yu MH. Twin shear stress yield criterion. Int J Mech Sci 1983;25(1):71–4. [12] Hodge PG. Limit analysis of rotationally symmetric plates and shells. Englewood Cliffs, NJ: Prentice-Hall; 1963.
Plastic limit analysis of cylindrically orthotropic circular plates Guowei Ma *, Bazle A. Gama, John W. Gillespie Jr. Center for Composite Materials, University of Delaware, Newark, DE 19711, USA
Abstract The plastic limit analysis of cylindrically orthotropic circular plates is developed using a piecewise linear orthotropic yield criterion. The yield criterion is a modification of an isotropic formulation that consists of a series of weighted piecewise linear components. The piecewise linear yield criterion enables an analytical solution for the plastic limit load of cylindrically orthotropic circular plates. Plastic limit analysis for both simply supported and clamped circular plates under uniformly distributed load are carried out. Parametric studies are conducted to investigate the sensitivity of the plastic limit loads to material orthotropy and influences of orthotropic ratio and chosen yield criteria on the plastic limit loads of the circular plates are discussed. It is found that the plastic limit loads of the orthotropic circular plates are affected significantly by the orthotropic ratio. Enhancement of the circumferential yield moment will increase dramatically the plastic limit load of the plates. Moment and velocity fields of the plates in plastic limit state are also derived and discussed. The results obtained from the present study are helpful in understanding the failure mechanism of orthotropic circular plates and is useful for design. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Circular plate; Plastic limit load; Orthotropy; Yield criterion; Moment field; Velocity field
1. Introduction Structures are sometimes designed with enhanced strength in a certain direction to resist external load. Traditionally, stiffeners are the most commonly used method to increase stiffness and strength at minimal weight. Mechanical orthotropy may also arise from cold forming process of material. Some single crystal materials such as BCC iron, and textured polycrystalline materials exhibit anisotropic properties due to their inherent atomic structures. Fiber reinforced composites offer much higher strength-to-weight and stiffness-toweight by combining structural fibers and a more flexible matrix (e.g. glass fibers in synthetic resins). It has been found that the strength in the fiber direction of unidirectional fiber reinforced composites can be an order of magnitude higher than in the transverse direction [1]. Some composites such as metal matrix reinforced by metal fibers, plastics reinforced by Kevlar or glass fibers, not only have high strength and high stiffness, but also undergo plastic flow and considerable ductility. Plastic limit analysis of these kinds of orthotropic plates should be updated based on the yield properties of the plates.
*
Corresponding author. Fax: +1-302-831-8525. E-mail address: [email protected] (G. Ma).
The yield behavior of an orthotropic plate is always represented by a proper criterion. Numerous orthotropic yield criteria have been published over the past decades. One typical orthotropic yield criterion is referred to as Tsai–Hill criterion, which is an extension of the Hill anisotropic criterion to predict failure in an orthotropic, transversely isotropic lamina [2]. The criterion simplifies to the von Mises criterion for isotropic materials. Unfortunately, these criteria as well as many other commonly used orthotropic criteria have quadratic or polynomial forms and cannot be used straightforwardly to analytically derive the plastic limit load of a plate. A pioneering work that investigated the plastic limit behavior of cylindrically orthotropic circular plate under various loading cases has been carried out by Markowitz and Hu [3] who employed a modified Tresca criterion. The criterion, similar to the Tresca criterion for isotropic materials, has a piecewise linear form. The plastic limit solution given in their study satisfies both statical and kinematical admissible requirements, and is an exact solution. Save [4] and Save et al. [5] summarized the plastic limit solutions obtained for orthotropic circular plates. It is noticed that all the derivations for the plastic limit load of orthotropic circular plate in the previous studies were based on the modified Tresca criterion. Unfortunately, the Tresca criterion has obvious limitations because the effect of the intermediate principal stress on material strength is
0263-8223/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 2 3 ( 0 1 ) 0 0 1 7 4 - X
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G. Ma et al. / Composite Structures 55 (2002) 455–466
ignored [6]. The modified version of the Tresca criterion also has these limitations. A piecewise linear yield criterion described as a unified yield criterion has been suggested by Yu and He [7]. The unified yield criterion has been applied successfully to plastic limit analysis for isotropic circular plates [8–10]. The solutions with Tresca criterion and Yu criterion [11] are the special cases and the lower and upper bounds, respectively, of the general solution with the unified yield criterion. The solution using the von Mises criterion can be approximated by the general solution. The influences of the yield criteria on the plastic limit load, velocity and moment fields have been examined by comparing the results with the three specific yield criteria, i.e. the Tresca criterion, the von Mises criterion and the Yu criterion. Based on the results, the effect of the yield criteria on the plastic limit load depends on the loading and boundary conditions of the circular plate. The maximum differences of the plastic limit load by different yield criteria could be more than 20% for clamped circular plate and about 14% for simply supported circular plate [8–10]. It is noticed that the traditional solutions with the Tresca criterion always give a lower prediction of the plastic limit load. The moment and velocity fields based on the Tresca criterion have unreasonable features, for example, the circumferential moment on the simply supported circular plate is a constant over the entire plate; the velocity fields of both the simply supported and the clamped circular plates have a singularity at the plate center, etc. In this study, an orthotropic yield criterion, which is the generalization of the unified yield criterion, will be explored. The criteria retain a piecewise linear form and the modified Tresca criterion employed by the previous studies appears again as a special case. The loading condition, boundary condition and the plate itself are assumed axisymmetrical. Plastic limit loads together with the moment and velocity distributions of simply supported and clamped cylindrically orthotropic circular plates are derived. The influence of yield criterion and orthotropic ratio on the plastic limit load will be discussed.
it is extended to orthotropic materials or orthotropic structures. For an axisymmetric plate, there have only two moment components in the equilibrium equations [12]. The piecewise linear mathematical formula of the unified yield criterion may take the form of [8] Mh Mr ¼ ai þ bi M0 M0
ði ¼ 1–12Þ
in which Mr and Mh are the actual radial and circumferential moments, respectively; M0 is the yield moment, and they are integrated by the stress Rcomponents along h=2 the Mr ¼ h=2 rr z dz, Mh ¼ R h=2 section of the plate, 2 r z dz and M0 ¼ r0 h =4, in which rr and rh are, h=2 h respectively, the radial and circumferential stresses in the plate; r0 is the yield stress; h is the plate thickness. The coefficients ai and bi , with i ranging from 1 to 12, are parameters corresponding to the 12 segments of the yield curve, which are determined by a weighted parameter b which is ranged from 0 to 1. Detail description on the unified yield criterion can be found in the literatures [8–10]. The yield curves with b ¼ 0, 0.25, 0.5, 0.75 and 1 of the unified yield criterion for isotropic circular plate is shown in Fig. 1. For a cylindrically orthotropic circular plate with the principal axes oriented along the radial and circumferential directions, the unified yield criterion can be modified as, Mh Mr ¼ ai þ bi Mh0 Mr0
ði ¼ 1–12Þ;
ð2Þ
where Mr0 and Mh0 are the yield moments in the radial and circumferential directions, respectively. Normalizing the yield criterion by defining j ¼ Mh0 =Mr0 , mh ¼ Mh =Mr0 and mr ¼ Mr =Mr0 , Eq. (2) can be rewritten as follows, mh ¼ Ai mr þ Bi
ði ¼ 1–12Þ
2. Orthotropic yield criteria The unified yield criterion was originally proposed by Yu and He [7] which may represent or approximate different yield criteria such as maximum shear stress criterion (Tresca criterion), octahedral shear stress criterion (von Mises criterion), and Yu twin shear stress criterion [11]. Plastic limit analysis of circular plates with various loading conditions and boundary conditions has been studied based on the unified yield criterion [8–10]. Proper modification of the criteria should be made when
ð1Þ
Fig. 1. Unified yield criterion (mr ¼ Mr =M0 , mh ¼ Mh =M0 ).
ð3Þ
G. Ma et al. / Composite Structures 55 (2002) 455–466
in which Ai ¼ ai j, Bi ¼ bi j ði ¼ 1–12Þ. The orthotropic ratio j is the ratio of the circumferential yield moment to the radial yield moment. j ¼ 1 corresponds to the special case of isotropy, while j > 1 and j < 1 represents, respectively, the circumferential and the radial strengthening of the plate. Similar to the unified yield criterion, the Mises yield criterion can be modified as Mh2 Mh Mr M2 þ r2 ¼ 1: 2 Mh0 Mh0 Mr0 Mr0
ð4Þ
In terms of the dimensionless variables, Eq. (4) becomes m2h jmh mr þ j2 m2r ¼ j2 :
ð5Þ
Eq. (5) is a nonlinear equation with respect to the two variables mr and mh , which make it inconvenient for plastic limit analysis of circular plates. The quadratic
457
form in Eq. (4) is very close to the one of the Tsai–Hill orthotropic criterion. The only difference is that they have different coefficient in the interaction term. The yield curves with different orthotropic ratios of the modified unified yield criterion defined in Eq. (3) are plotted in Figs. 2(a)–(c) with respected to b ¼ 0, b ¼ 1 and b ¼ 0:5. The yield curve in Fig. 2(a) is exactly the same as employed in the previous studies [3,5]. Fig. 2(d) illustrates the modified von Mises criterion defined in Eq. (5). The orthotropic yield criteria are compared in Fig. 3 with different orthotropic ratios, i.e., j ¼ 2, j ¼ 0:5 and the isotropic case of j ¼ 1. It shows that, even in orthotropic conditions, the modified von Mises criterion can be approximately represented by the modified unified yield criterion by b ¼ 0.5. The extension of the unified yield criterion makes it possible to obtain the plastic limit solutions of orthotropic circular plates
Fig. 2. Orthotropic yield criteria: (a) b ¼ 0, modified Tresca criterion; (b) b ¼ 1; (c) b ¼ 0:5; (d) modified von Mises criterion.
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G. Ma et al. / Composite Structures 55 (2002) 455–466
or mrðijÞ ¼
bi bj ; ai aj
mhðijÞ ¼
ai bj aj bi j ai aj
ði ¼ 1–11; j ¼ i þ 1; i ¼ 12; j ¼ 1Þ;
ð7Þ
where the line with i ¼ 1 corresponds to line AB and i increases from 1 to 12 along with anti-clockwise sequence of the lines. Thus, i ¼ 12 corresponds to line LA. The symbol ði jÞ denotes the corner intersected by line i and line j. The values of the parameters ai and bi ði ¼ 1–12Þ in the unified yield criterion are listed in Table 1. Table 2 lists the moments at the 12 corners. (a)
3. General solutions
(b)
(c) Fig. 3. Comparison of the modified yield criteria with different orthotropic ratios: (a) j ¼ 2; (b) j ¼ 0:5; (c) j ¼ 1.
in view of not only the modified Tresca criterion but also a modified von Mises criterion and a modified Yu criterion. The moments at the corners of the yield curves in Fig. 3 can be derived by solving the equations for the two intersecting lines, which have the forms of mrðijÞ
Bi Bj ¼ ; Ai Aj
mhðijÞ
Ai B j Aj Bi ¼ Ai Aj
ði ¼ 1–11; j ¼ i þ 1; i ¼ 12; j ¼ 1Þ
ð6Þ
For convenience, dimensionless variables p and r, i.e., p ¼ Pa2 =Mr0 and r ¼ R=a, are defined for the cylindrically orthotropic plate with uniform thickness, where P is the uniformly distributed load, a is the outer radius of the plate and R is the radius variable in the range of 0 to a. For an axisymmetric loading, any two orthogonal directions at the central point are principal directions for stresses and orthotropy. Hence, the plate is locally isotropic and Mr ¼ Mh or mr ¼ mh at the plate center [5]. The moments at the plate center, thus, correspond to point A0 in Fig. 3. The point A0 may locate on different portions, namely lines with i ¼ 11, 12, 1, or 2, of the curves depending on the orthotropic ratio j. If j ¼ 1, A0 coincides with the corner point A. When j > 1, A0 falls on line KL (i ¼ 11) or LA (i ¼ 12), while A0 falls on line AB (i ¼ 1) or BC ði ¼ 2Þ when j < 1. For a simply supported circular plate in plastic limit state, the moment at the outer edge satisfies the yield point C in Fig. 3. The moments in the whole plate corresponds to the yield segments from A0 to C with anti-clockwise order. The total number of the segments could be 4 when A0 is on the line KL. It can be reduced to only one when A0 is on the line BC. On the other hand, the edge of a clamped circular plate corresponds to a yield point on line EF, which has been discussed in a previous study [9]. In the present study, the moments at the edge are assumed exactly corresponding to the corner point F in Fig. 3. Thus, the total number of the segments for the clamped circular plate is from 4 to 7 with reference to the orthotropic ratio j. The equilibrium equation of the circular plate is [12] dðrmr Þ=dr mh ¼ pr2 =2:
ð8Þ
With the aid of the piecewise linear form in Eq. (3), the radial moment can be integrated from Eq. (8) as mr ¼
Bi pr2 þ Ci r1þAi ; 1 Ai 2ð3 Ai Þ
ð9Þ
G. Ma et al. / Composite Structures 55 (2002) 455–466
459
Table 1 Constants ai and bi in the modified unified yield criterion AB (i ¼ 1)
BC (i ¼ 2)
CD (i ¼ 3)
DE (i ¼ 4)
EF (i ¼ 5)
FG (i ¼ 6)
GH (i ¼ 7)
HI (i ¼ 8)
IJ (i ¼ 9)
JK (i ¼ 10)
KL (i ¼ 11)
LA (i ¼ 12)
ai
b
b 1þb
1 1þb
1þb
1þb b
1b
b
b 1þb
1 1þb
1þb
1þb b
1b
bi
1þb
1
1
1þb
1þb b
1þb b
1þb
1
1
1þb
1þb b
1þb b
Table 2 Moments at the corners Nodes
A
B
C
D
E
F
G
H
I
J
K
L
mh
j
2ð1þbÞ j 2þb
j
1þb j 2þb
0
1þb j 2þb
j
2ð1þbÞ j 2þb
j
1þb j 2þb
0
1þb j 2þb
mr
1
1þb 2þb
0
1þb 2þb
1
2ð1þbÞ 2þb
1
1þb 2þb
0
1þb 2þb
1
2ð1þbÞ 2þb
where i represents each valid segment of the yield curve; Ci are integral constants determined by continuity and boundary conditions. For a general case where the moments in the plate covers a total of n segments on the yield curve, the continuity and boundary conditions of the radial moment will yield Bð1Þ pr02 1þA þ C1 r0 ð1Þ ¼ mrð0Þ ; 1 Að1Þ 2ð3 Að1Þ Þ
ð10aÞ
Bð1Þ pr12 1þA þ C1 r1 ð1Þ ¼ mrð1Þ ; 1 Að1Þ 2ð3 Að1Þ Þ
ð10bÞ
Bð2Þ pr12 1þA þ C2 r1 ð2Þ ¼ mrð1Þ ; 1 Að2Þ 2ð3 Að2Þ Þ
ð10cÞ
Bð2Þ pr22 1þA þ C2 r2 ð2Þ ¼ mrð2Þ ; 1 Að2Þ 2ð3 Að2Þ Þ
ð10dÞ
2 BðiÞ pri1 1þA þ Ci ri1 ð1Þ ¼ mrði1Þ ; 1 AðiÞ 2ð3 AðiÞ Þ
ð10eÞ
BðiÞ pri2 1þAð1Þ þ Ci ri ¼ mrðiÞ ; 1 AðiÞ 2ð3 AðiÞ Þ
ð10fÞ
2 BðnÞ prn1 1þA þ Cn rn1 ðnÞ ¼ mrðn1Þ ; 1 AðnÞ 2ð3 AðnÞ Þ
ð10gÞ
BðnÞ prn2 1þA þ Cn rn ðnÞ ¼ mrðnÞ ; 1 AðnÞ 2ð3 AðnÞ Þ
ð10hÞ
where AðiÞ and BðiÞ on the left side denote parameters of the ith valid segment from A0 in anti-clockwise order on the yield curve of Fig. 3. mrðiÞ ði ¼ 0–nÞ represents the radial moment at the ith corner point from A0 , e.g., mrð0Þ corresponds to the yield moment on point A0 , while mrðnÞ corresponds to the yield moment at the corner C for simply supported plate or the corner F for clamped plate. r0 ; r1 ; . . . ; ri ; . . ., and rn denote the demarcating radii corresponding to the valid corners on the yield curve. In particular, r0 equals zero and is at the plate
center where the moments correspond to corner A0 , rn equals 1 with reference to the plate outer edge. Defining a1 ¼ r1 =r2 , a2 ¼ r2 =r3 , ai ¼ ri =riþ1 and an1 ¼ rn1 =rn , then the following relationships exist r1 ¼ a1 a2 an1 , r2 ¼ a2 a3 an1 , ri ¼ ai an1 and rn1 ¼ an1 . With the aid of the continuity and boundary conditions, Ci ði ¼ 1–nÞ in Eqs. (10a)–(10h) are determined as follows, C1 ¼ 0; C2 ¼
ð11aÞ
Bð2Þ mrð1Þ 1 Að2Þ Bð1Þ 3 Að1Þ 1A ðr1 Þ ð2Þ ; þ mrð1Þ 1 Að1Þ 3 Að2Þ BðiÞ Ci ¼ mrði1Þ 1 AðiÞ Bð1Þ 3 Að1Þ þ mrð1Þ 1 Að1Þ 3 AðiÞ 2 1A ða1 a2 ai2 Þ ðri1 Þ ðiÞ ;
BðnÞ Cn ¼ mrðn1Þ 1 AðnÞ Bð1Þ 3 Að1Þ þ mrð1Þ : 1 Að1Þ 3 AðnÞ 2 1A ða1 a2 an2 Þ ðrn1 Þ ðnÞ ;
ð11bÞ
ð11cÞ
ð11dÞ
in which ratios a1 ; a2 ; . . . ; ai ; . . . ; an1 are given by Bð2Þ Bð1Þ 3 Að1Þ 2 mrð2Þ mrð1Þ a 1 Að2Þ 1 Að1Þ 3 Að2Þ 1 Bð2Þ þ mrð1Þ 1 Að2Þ Bð1Þ 3 Að1Þ 1Að2Þ mrð1Þ ¼ 0; ð12aÞ þ a 1 Að1Þ 3 Að2Þ 1
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BðiÞ Bð1Þ 3 Að1Þ mrðiÞ mrð1Þ ða1 a2 ai1 Þ2 1 AðiÞ 1 Að1Þ 3 AðiÞ BðiÞ þ mrði1Þ 1 AðiÞ Bð1Þ 3 Að1Þ 1A 2 þ mrð1Þ ða1 a2 ai2 Þ ai1 ðiÞ 1 Að1Þ 3 AðiÞ ¼ 0;
ð12bÞ
BðnÞ Bð1Þ mrðnÞ mrð1Þ 1 AðnÞ 1 Að1Þ 3 Að1Þ 2 ða1 a2 an1 Þ 3 AðnÞ BðnÞ þ mrðn1Þ 1 AðnÞ Bð1Þ 3 Að1Þ 1A 2 þ mrð1Þ ða1 a2 an2 Þ an1 ðnÞ 1 Að1Þ 3 AðnÞ ¼ 0:
ð12cÞ
Eqs. (12a)–(12c) can be solved one by one by numerical iteration method such as half interval search method. After ai ði ¼ 1 to n 1Þ are calculated, the demarcating radii ri ði ¼ 1 to n 1Þ and the integral constants in Eqs. (11a)–(11d) are then obtained. The plastic limit load is derived from Eq. (10b) as Bð1Þ 2ð3 Að1Þ Þ p¼ : ð13Þ mrð1Þ 1 Að1Þ r12 Substituting the integral constants Ci ði ¼ 1–nÞ, the demarcating radii ri ði ¼ 0–nÞ and the plastic limit load p into Eq. (9), the radial moment field mr over the entire plate is then obtained. The velocity field of the plate is integrated from the compatibility conditions and associated plastic flow rule, which has the same form as that in the isotropic case [8], w_ ¼ w_ 0 c1i r1AðiÞ þ c2i ; ri1 6 r 6 ri and ði ¼ 1–nÞ;
ð14Þ
where c1i and c2i ði ¼ 1–nÞ are the integral constants corresponding to the n valid segments on the yield curve. ri ði ¼ 0–nÞ are again the demarcating radii, where the moment variables are located at the corner points of the yield curve. w_ 0 is the flow velocity at the plate center. The continuity and boundary conditions of the velocity field are: (1) w_ ðr ¼ 0Þ ¼ w_ 0 , (2) w_ and dw_ =dr ðr ¼ ri ; i ¼ 1 to n 1Þ are continuous and (3) w_ ðr ¼ 1Þ ¼ 0. Considering the above conditions, the constants c1i and c2i in Eq. (14) satisfy the following relationships: c21 ¼ 1; 1A c11 r1 ð1Þ
ð1
ð15aÞ þ c21 ¼
A Að1Þ Þc11 r1 ð1Þ
1A c12 r1 ð2Þ
¼ ð1
þ c22 ;
A Að2Þ Þc12 r1 ð2Þ ;
ð15bÞ ð15cÞ
1AðiÞ
c1i ri
1Aðiþ1Þ
þ c2i ¼ c1ðiþ1Þ ri AðiÞ
ð1 AðiÞ Þc1i ri
þ c2ðiþ1Þ ; Aðiþ1Þ
¼ ð1 Aðiþ1Þ Þc1ðiþ1Þ ri
1A
1AðnÞ
c1ðn1Þ rn1 ðn1Þ þ c2ðn1Þ ¼ c1ðnÞ rn ð1
A Aðn1Þ Þc1ðn1Þ rn1ðn1Þ
1A c1n rn ðnÞ
¼ ð1
ð15dÞ ;
ð15eÞ
þ c2n ;
ð15fÞ
A AðnÞ Þc1n rn1ðnÞ ;
ð15gÞ
þ c2n ¼ 0:
ð15hÞ
The above equations are a series of linear equations with the unknown variables of c1i and c2i ði ¼ 1–nÞ. The 2n unknown variables can be solved directly from the 2n simultaneous linear equations as the demarcating radii ri , ði ¼ 1–nÞ are known. 4. Simply supported orthotropic circular plate For a simply supported circular plate, with a certain orthotropic ratio j, there are four possible cases, namely the point A0 is respectively on segments KL, LA, AB or BC. The following will first determine the yield moments at the plate center for the four cases depending on the orthotropic ratio. The valid segments on the yield curve can then be formalized. The plastic limit load, moment field and velocity field are obtained from the general solution as the valid segments are determined. Case 1. Point A0 locates on segment KL. When j > 1, the circumferential yield moment Mh0 is greater than the radial yield moment Mr0 . The point A0 may locate on either segment KL or segment LA. Considering the special case when point A0 coincides with the corner L, there exists mh ¼ ja11 mr þ jb11 ; mh ¼ ja12 mr þ jb12 ;
ð16Þ
mh ¼ mr at the yield point A0 . Thus, the orthotropic ratio j in this special case is derived as, b11 b12 j¼ : ð17Þ a12 b11 a11 b12 When j > ððb11 b12 Þ=ða12 b11 a11 b12 ÞÞ, point A0 locates on the segment KL. The four segments ðn ¼ 4Þ, namely A0 L, LA, AB, and BC are valid. The radial moment at r ¼ 0 corresponding to point A0 is derived as, B11 jb11 or mrð0Þ ¼ ; ð18Þ 1 A11 1 ja11 which is determined by the weighted parameter b and the orthotropic ratio j. mrð0Þ ¼
Case 2. Point A0 locates on segment LA. When ððb11 b12 Þ=ða12 b11 a11 b12 ÞÞ P j > 1, point A0 falls on the segment LA. The valid segments are A0 A, AB
G. Ma et al. / Composite Structures 55 (2002) 455–466
and BC with the radial moment mr varying from the point A0 to the point C on the yield curve when r increases from 0 to 1. The radial moment at point A0 have the form of mrð0Þ ¼
B12 1 A12
or
mrð0Þ ¼
jb12 : 1 ja12
ð19Þ
Case 3. Point A0 locates on segment AB. When 1 P j > ððb1 b2 Þ=ða2 b1 a1 b2 ÞÞ, the moments on the entire plate correspond to the two yield segments of A0 B and BC. The radial moment mr at the plate center is mrð0Þ ¼
B1 1 A1
or mrð0Þ ¼
jb1 : 1 ja1
ð20Þ
Case 4. Point A0 locates on segment BC. When j 6 ððb1 b2 Þ=ða2 b1 a1 b2 ÞÞ, it yields the case that point A0 locates on BC. The moments on the plate correspond to only the yield segment BC. The valid segment is A0 C and the radial moment at the plate center is mrð0Þ ¼
B2 1 A2
or mrð0Þ ¼
jb2 : 1 ja2
ð21Þ
The radial moments mrðiÞ ; i ¼ 1–n at other corners and the parameters AðiÞ and BðiÞ as used in Eqs. (10a)– (10h) are listed in Table 3, in which mrðijÞ is given by Eq. (6) or (7). Once the valid segments and relative parameters are determined, the plastic limit load, the moment fields and the velocity fields are then obtained from Eqs. (9), (13) and (14), respectively, with the integral constants and demarcating radii being determined by the continuity and boundary conditions. Figs. 4(a)–(c) illustrate the moment fields in terms of the orthotropic unified yield criterion with respect to
461
j ¼ 2, j ¼ 1 and j ¼ 0:5. The three plots correspond to Cases 2–4, respectively. It is seen from Fig. 4 that when j 6¼ 1 or the plate has different yield moments in the radial and circumferential directions, both the moment distribution profiles and the moments at the plate center in terms of different weighted parameter b, i.e., b ¼ 0, b ¼ 0:5 and b ¼ 1 of the modified unified yield criterion, are quite different. The yield curve with b ¼ 0 which is the modified Tresca criterion always gives the lowest yield moment at the plate center. When the modified Tresca criterion is applied, the circumferential moment mh keeps constant on the entire plate in Cases 3 and 4; while it is constant in the range of r1 6 r 6 1 corresponding to Case 2. This is unreasonable and results from the neglection of the influence of the intermediate principal stress in the Tresca criterion. Based on the modified unified yield criterion with b 6¼ 0, the moment distribution of the orthotropic plate becomes more reasonable. The velocity fields corresponding to different cases are shown in Fig. 5. The Tresca criterion (b ¼ 0) gives a linear line (Figs. 5(b) and (c)) or piecewise linear lines (Fig. 5(a)) of the velocity distributions with different orthotropic ratios. The criteria with b ¼ 0:5 and b ¼ 1, which represents approximately the modified Mises criterion and the modified Yu criterion, yield smooth and orthotropic ratio dependent velocity distributions. It is obviously true that the weighted parameter b is the significant factor in unifying different yield criteria. The curve of plastic limit load versus the parameter b is plotted in Fig. 6(a) for the three orthotropic ratios of j ¼ 2, j ¼ 1 and j ¼ 0:5. It is seen that, for all the three cases, the plastic limit load increases monotonically with the increase of b. The plastic limit load when j ¼ 2 is much larger than those corresponding to j ¼ 1 and j ¼ 0:5. This indicates that much higher plastic limit load can be achieved when the plate is stiffened along the circumferential direction (j > 1).
Table 3 Yield moments at corner points and the corresponding AðiÞ and BðiÞ for simply supported circular plate i
1
2
3
4
Case 1 (n ¼ 4)
mrðiÞ AðiÞ BðiÞ
mrð11–12Þ A11 B11
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 2 (n ¼ 3)
mrðiÞ AðiÞ BðiÞ
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 3 (n ¼ 2)
mrðiÞ AðiÞ BðiÞ
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
Case 4 (n ¼ 1)
mrðiÞ AðiÞ BðiÞ
mrð2–3Þ A2 B2
462
G. Ma et al. / Composite Structures 55 (2002) 455–466
(a) (a)
(b) (b)
(c) Fig. 5. Velocity fields of orthotropic simply supported circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
(c) Fig. 4. Moment fields of orthotropic simply supported circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
Defining a difference ratio q p pTresca 100% q¼ pTresca
ð22Þ
makes it convenient to compare the effect of different yield criteria on the plastic limit load. Fig. 6(b) gives the difference ratio versus the weighted parameter b. It is found that the difference ratio increases gradually with the increase of b. The difference ratio is up to 7.1% and 12.6% when j ¼ 2; 8.1% and 14.0% when j ¼ 1 and 13.3% and 22.2% when j ¼ 0:5 corresponding to b ¼ 0.5 and 1, respectively. This implies that the plastic limit load based on different yield criteria are very different, which is helpful for optimization of design of an orthotropic circular plate.
The effect of the orthotropic ratio on the plastic limit load is illustrated in Fig. 7(a). It is seen that the plastic limit load increases sharply with the increase of the orthotropic ratio j, which indicates again that the strengthening along the circumferential direction does improve dramatically the load carrying capacity of a plate. The curve of difference ratio versus the orthotropic ratio with respect to b ¼ 0.5 and b ¼ 1 is plotted in Fig. 7(b). The maximum differences occurs at j ¼ 0:56 and they are 13.8% and 22.9% with respect to b ¼ 0.5 and b ¼ 1. The difference ratio remains relatively constant when j > 1 and the ratio is approximately 7% and 13% corresponding to b ¼ 0.5 and b ¼ 1. It repeatedly indicates that different yield criterion predicts significantly different plastic limit load as a function of the ratio of the orthotropic ratio.
G. Ma et al. / Composite Structures 55 (2002) 455–466
(a)
463
(b)
Fig. 6. Effects of yield criteria for simply supported circular plate: (a) plastic limit load versus b; (b) difference ratio of plastic limit load.
(a)
(b)
Fig. 7. Effect of orthotropic ratio for simply supported circular plate: (a) plastic limit load versus j; (b) difference ratio of plastic limit load.
Table 4 Yield moments at corner points and the corresponding AðiÞ and BðiÞ for clamped circular plate i
1
2
3
4
5
6
7
Case 1 (n ¼ 7)
mrðiÞ AðiÞ BðiÞ
mrð11–12Þ A11 B11
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð45Þ A4 B4
mrð5–6Þ A5 B5
Case 2 (n ¼ 6)
mrðiÞ AðiÞ BðiÞ
mrð12–1Þ A12 B12
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð56Þ A5 B5
Case 3 (n ¼ 5)
mrðiÞ AðiÞ BðiÞ
mrð1–2Þ A1 B1
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð5–6Þ A5 B5
Case 4 (n ¼ 4)
mrðiÞ AðiÞ BðiÞ
mrð2–3Þ A2 B2
mrð3–4Þ A3 B3
mrð4–5Þ A4 B4
mrð5–6Þ A5 B5
464
G. Ma et al. / Composite Structures 55 (2002) 455–466
5. Clamped circular plate For the clamped circular plate, there also exist four possible cases, namely, point A0 locates on KL, LA, AB, or BC corresponding to the different orthotropic ratio j. The moments on the entire plate corresponds to the segments from A0 to F with an anti-clockwise order as the radius variable r increases from 0 to 1. Similar to the four cases of the simply supported plate, the radial moment at the plate center mrð0Þ is calculated according to the weighted parameter b and the orthotropic ratio j.
The moments at the corner points in different cases and the parameters AðiÞ and BðiÞ in Eqs. (10a)–(15h) are listed in Table 4. The moment fields of the clamped circular plate with different orthotropic ratios, namely, j ¼ 2, j ¼ 1 and j ¼ 0:5, are shown in Figs. 8(a)–(c), respectively. Fig. 8(a) includes six valid yield segments, and Figs. 8(b) and (c) involve, respectively, five and four yield segments. It is seen that when j 6¼ 1 or the plate has different yield moments in the radial and circumferential directions, the moment distribution profiles and the moments at the plate center with different values of b, i.e., b ¼ 0, b ¼ 0:5 and b ¼ 1, of the unified yield criterion are again quite different.
(a) (a)
(b)
(c) Fig. 8. Moment fields of orthotropic clamped circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
(b)
(c) Fig. 9. Velocity fields of orthotropic clamped circular plate: (a) j ¼ 2; (b) j ¼ 1; (c) j ¼ 0:5.
G. Ma et al. / Composite Structures 55 (2002) 455–466
The velocity fields of the clamped plate for different cases are shown in Fig. 9. The modified Tresca criterion (b ¼ 0) again gives a linear line (Figs. 9(b) and (c)) when j ¼ 1 and j ¼ 0:5. The criteria with b ¼ 0:5 and b ¼ 1, which represent approximately Mises criterion and the Yu criterion, yield smooth and orthotropic ratio dependent velocity distributions. The curves of plastic limit load and the difference ratio versus parameter b for the clamped circular plate is plotted in Fig. 10(a). The plastic limit loads again increase gradually with the increase of b and are much larger than the corresponding values of the simply supported circular plate. The orthotropic ratio influences the plastic limit load significantly since the plastic limit load when j ¼ 2 is much larger than those when j ¼ 1 and j ¼ 0:5. The difference ratio
(a)
defined in Eq. (22) is up to 10.8% and 18.3% when j ¼ 2; 13.0% and 21.7% when j ¼ 1 and 16.4% and 27.3% when j ¼ 0:5 corresponding to b ¼ 0.5 and 1, respectively. These differences are larger than the corresponding results for the simply supported circular plate. This means that the yield criterion influences the plastic limit load of clamped circular plates more significantly than that of simply supported circular plate. The effect of different orthotropic ratio on the plastic limit load is shown in Fig. 11. The plastic limit load increases again sharply with the increase of the orthotropic ratio j. Although the difference ratio decreases with the increase of the orthotropic ratio, they remain about 9% and 16% corresponding to b ¼ 0.5 and b ¼ 1 for a wide range of the orthotropic ratio.
(b)
Fig. 10. Effect of yield criteria for clamped circular plate: (a) plastic limit load versus b; (b) difference ratio.
(a)
465
(b)
Fig. 11. Effect of orthotropic ratio for clamped circular plate: (a) plastic limit load versus j; (b) difference ratio.
466
G. Ma et al. / Composite Structures 55 (2002) 455–466
6. Conclusions A piecewise linear orthotropic yield criterion is suggested. Analytical solution by employing the piecewise linear orthotropic yield criterion has been derived for cylindrically orthotropic circular plates with simply supported and clamped edge conditions. It is found that the plastic limit load increases significantly with the increase of the orthotropic ratio. The chosen yield criterion affects the plastic limit load, moment fields and velocity fields. For the simply supported orthotropic circular plate, the difference ratio is a function of the orthotropic ratio and it is 7.1% and 12.6% when j ¼ 2 and 13.3% and 22.2% when j ¼ 0:5 with respect to the modified Mises criterion and the modified Yu criterion, respectively. The different ratio remain about 7% and 13% with respect to the modified Mises criterion and the modified Yu when j > 1. For the clamped orthotropic circular plate, the difference ratio decreases with the increase of the orthotropic ratio. It is about 9% and 16% with respect to the modified Mises criterion and the modified Yu criterion, respectively, when the orthotropic ratio is greater than 1. References [1] Daniel IM, Ishai O. Engineering mechanics of composite materials. Oxford: University Press; 1994.
[2] Tsai SW. Strength theories of filamentary structures. In: Schwartz RT, Schwartz HS, editors. Fundamental aspects of fiber reinforced plastic composites. New York: Wiley/Interscience; 1968. p. 3–11. [3] Markowitz J, Hu LW. Plastic analysis of orthotropic circular plates. ASCE J Eng Mech 1965;EM5:251–5. [4] Save MA. Limit analysis of plates and shells: research over two decades. ASCE J Struct Mech 1985;13:343–70. [5] Save MA, Massonnet CE, G.de Saxce. Plastic limit analysis of plates, shells and disks. Amsterdam, New York: Elsevier; 1997. [6] Hill R. The mathematical theory of plasticity. Oxford: Clarendon Press; 1950. [7] Yu MH, He LN. A new model and theory on yield and failure of materials under complex stress state. In: Mechanical behavior of materials-6, vol. 3. Oxford: Pergamon Press; 1991. p. 841–6. [8] Ma GW, Iwasaki S, Miyamoto Y, Deto H. Plastic limit analysis of circular plates with respect to the unified yield criterion. Int J Mech Sci 1998;40(10):963–76. [9] Ma GW, Hao H, Iwasaki S. Plastic limit analysis of a clamped circular plate with unified yield criterion. Struct Eng Mech 1999;7(5):513–25. [10] Ma GW, Hao H, Iwasaki S. Unified plastic limit analysis of circular plates under arbitrary load. ASME J Appl Mech 1999;66(2):568–70. [11] Yu MH. Twin shear stress yield criterion. Int J Mech Sci 1983;25(1):71–4. [12] Hodge PG. Limit analysis of rotationally symmetric plates and shells. Englewood Cliffs, NJ: Prentice-Hall; 1963.