- Email: [email protected]
Plane strain elastic–plastic bending of a strain-hardening curved beam
International Journal of Mechanical Sciences 43 (2001) 39}56
Plane strain elastic}plastic bending of a strain-hardening curved beam Parviz Dadras* Wright State University, Dayton, OH 45435, USA Received 4 November 1998; received in revised form 18 November 1999; accepted 23 November 1999
Abstract Elastic}plastic analysis of plane strain pure bending of a strain-hardening curved beam has been presented. Only a linear hardening case has been analyzed as the nonlinear equations for a general hardening case could not be solved analytically. A numerical scheme for the computation of stresses and displacements in di!erent stages of deformation has been given and limited FEM veri"cations have been presented. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Curved beams; Elastic}plastic bending
1. Introduction Elastic analysis of pure bending of a wide curved bar under plane strain deformation is given by Timoshenko and Goodier [1]. Rigid-perfectly plastic, and rigid-work hardening analyses are given by Hill [2] and Dadras and Majlessi [3], respectively. The elastic}plastic solution for perfectly plastic material is given by Sha!er and House [4]. Solutions for perfectly plastic incompressible material [5,6], and perfectly plastic compressible material [7] are also available. The analysis for an elastic strain-hardening material is given in this paper. A numerical scheme for solving the resulting equations is also presented and limited comparisons with FEM results are given. 2. Analysis A curved beam of inner radius a and outer radius b is subjected to a moment M, as shown in Fig. 1. It is assumed that the beam is "xed at point r"a and h"0 throughout the deformation.
* Present address: 1435 Meadow Lane, Yellow Springs, OH 45387, USA. 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 1 0 2 - 2
40
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Nomenclature a, b c, o r, h u, v e ,e ,c P F PF p ,p P F M > E, l p, eN e C, e . eC , e. F F
deformed inside and outside radii, respectively "rst and second elastic}plastic radii, respectively radial and tangential coordinates, respectively radial and tangential displacements, respectively strain components in polar coordinates radial and tangential stresses, respectively applied bending moment yield strength elastic constants e!ective stress and strain, respectively e!ective elastic and plastic strains, respectively elastic and plastic tangential strain components, respectively
Fig. 1. Geometry of the curved beam.
Analyses are performed for three stages of deformation: all elastic, "rst elastic}plastic deformation with a plastic zone formed within a(r)c, and second elastic}plastic where an additional plastic zone appeared at o(r(b.
3. All elastic The stress analysis is as given by Timoshenko and Goodier [1]. For simplicity the simpli"ed equations given by Chakrabarty [8] are used here:
b b !1 #B ln , p "!A P r r
(1)
b b #1 !B 1!ln , p "A F r r
(2)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
41
where
4M b 4M b A" ln , B" !1 Na a Na a
(3)
and
N"
b 4b b . !1 ! ln a a a
The development of the displacement equations are also given in Refs. [1,5,7]. However, a more general solution, also suitable for the later stages of deformation and based on a di!erent boundary condition assumed here, will be presented. The strain}displacement relations are *u e" , P *r u 1 *v , e " # F r r *h *v v 1 *u "0. c " ! # PF *r r r *h
(4) (5) (6)
The strains and the stresses are only functions of r. Therefore, from Eq. (5) it follows that *u *v ! " "g, *h *h
(7)
where g is only a function of h. Substitution in Eq. (6): r
*v !v"g. *r
(8)
The solution of Eq. (8) is v"rf !g, where f is another function of h. Di!erentiating Eq. (9) w.r.t. h twice, shows that df dg "0 and ! "g. dh dh
(9)
(10)
The solutions of Eq. (10) and considering Eq. (9) results in v"ra h!a sin h!a cos h. Since v "!v , it follows that a "0 and >F \F v"ra h!a sin h.
(11)
(12)
42
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
The above equation for the tangential displacement v is valid in both elastic and plastic zones. The radial displacement u will be found by considering the p}e relationships in plane strain (e "0): X e "m p !m p , (13) P P F e "m p !m p , (14) F F P where m "(1!l)/E, m "l(1#l)/E, and l is Poisson's ratio. An equation for u is obtained by substituting for p and p in Eq. (13) and integrating, P F b b (15) u"A(m #m ) #BKr 1#ln #ArK#Bm r#f, r r
where K"m !m , and f is only a function of h. From Eq. (7) it follows that f"a cos h!a sin h. (16) Once again a "0 since u "u . The constant a is found from Eq. (5) by using Eqs. (15), (16), >F \F (14), and the stress equations (1) and (2): a "!2Bm (17) # where the subscript 1E indicates a in an elastic "eld. The remaining constant a is found from the boundary condition r"a, h"0, u"v"0:
b b a "!A(m #m ) !BaK 1#ln !AaK!Bm a. # a a
(18)
The bending moment from
@
?
p r dr"!M F
(19)
is
M"
1 b 1 B Ba!Ab ln # !A (b!a). 2 a 2 2
(20)
Eqs. (3), (18), and (20) along with an equation for b are the system of equations for determining the unknowns A, B, a , M and b as the inside radius a is allowed to increase. The deformed # thickness of the beam is independent of h. Considering the plane of symmetry (h"0) for convenience, the deformed thickness is b!a"t #u !u , @ ? where t is the initial thickness, and u and u are radial displacements on the plane of symmetry at @ ? b and a, respectively. For the point at r"a and h"0, the radial and tangential displacements are zero, as required by the boundary condition at this point. Therefore, it follows that, db du "1# @ . da da
(21)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
The radial displacement u is determined from Eq. (15), @ u "A(m #m )b#Bbm #AbK#a @ # with a given in Eq. (18). #
43
(22)
4. First elastic}plastic The following e!ective stress (p) e!ective strain (e ) relation was "rst examined: p">#a(e !e )L, (23) W where > and e are the stress and strain at yield, respectively, and a is a material constant. The use W of Eq. (23) led to nonlinear di!erential equations for strains which could not be solved. Linear strain hardening behavior (n"1) was, therefore, used. p">#a(e !e ), (24) 7 where a is the slope of the plastic line. Eq. (24) is further modi"ed by using e "e .#e C and e C"p/E, where the superscripts p and e denote plastic and elastic, respectively. Further manipulations lead to Ea p"># e .. E!a
(25)
The e!ective stress and e!ective plastic strain for the "rst plastically deforming zone are (3 (p !p ), p" P 2 F
(26)
2 e ." e.. (3 F
(27)
Therefore, p !p "P#Qe., F P F where P"2>/(3 and Q"4Ea/3(E!a). The equilibrium equation [1] is thus, *p r P "p !p "P#Qe.. F P F *r
(28)
(29)
From integration of Eq. (29), and using Eq. (28),
e. p "P ln r#Q F dr#C , P r
(30)
e. p "P ln r#Q F dr#C #P#Qe., F F r
(31)
44
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
where C is a constant. Since e "e."0, it follows that X X e #e "eC #eC. (32) F P F P The elastic strain components in the plastic zone will be related to the stresses as given by Eqs. (13) and (14) and the stresses will still follow Eqs. (1) and (2) with new relations for A and B which will be developed later. Therefore, from Eqs. (13), (14), and (30)}(32),
e. e #e "K 2P ln r#2Q F dr#2C #P#Qe. . F F P r
(33)
a u 1 *v u " #a ! cos h e " # F r r *h r r
(34)
Also,
in view of Eq. (12) for v which is still valid in the plastic zone, where a and a are constants in the "rst plastic zone. From Eq. (34), *u *e "e "e #r F !a . P F *r *r
(35)
Upon substitution for e from Eq. (35) into Eq. (33) and using e "e.#eC , the following equation is P F F F obtained: 2e.#2eC #r F F
e. *(eC #e.) F F !KQe.!2KQ F dr"K(2P ln r#2C #P)#a . F r *r
(36)
eC and subsequently its derivative w.r.t. r are found from Eq. (13), with p and p given by Eqs. (30) F P F and (31). Therefore,
e. eC "KP ln r#KQ F dr#KC #m (P#Qe.). F J F r The following equation for e. is obtained by substituting from Eq. (37) into Eq. (36): F m de. 2 F # e." , r F r dr where m"(a !2m P)/(m Q#1). The solution of Eq. (38) is m C e." # , F 2 r
(37)
(38)
(39)
where C is a constant. The "nal equations for p and p are then obtained by using Eq. (39) in Eqs. P F (30) and (31) and applying the boundary condition r"a, p "0 to "nd the constant C : P Qm r QC 1 1 ln ! ! , (40) p " P# P 2 a 2 r a
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Qm p " P# F 2
1#ln
r QC 1 1 # # . a 2 r a
45
(41)
The radial displacement equation is determined by "nding e "eC #e., where eC "m p ! F F F F F m p , with p and p as given in Eqs. (40) and (41): P P F
Qm r C Q KQC m # (m Q#1)#m P. e "K P# ln # (m #m )#1 # F 2 a r 2 2a 2
(42)
Substituting e in Eq. (34) results in F
Qm r C Q KQC r u"Kr P# ln # (m #m )#1 # 2 a r 2 2a #
mr (m Q#1)#m Pr!ra #a cos h. 2
(43)
The above equation along with Eq. (12) are the displacement equations for the plastic zone at a(r(C. From the boundary condition r"a, h"0, u"0 the following relations are obtained: aa C (m Q#1) a " ! . 2 a
(44)
Therefore,
Qm r C Q KQC r a (m Q#1)C ! (r!a)! . (45) u"Kr P# ln # (m #m )#1 # 2 a r 2 2a 2 a Stress relations (1) and (2), and the displacement equations (12) and (15)}(17) are still applicable in the elastic zone with new parameters A, B, and a which will be determined from the continuity # considerations between the elastic and plastic zones. From, Eq. (19) the following moment equation is obtained:
M"!
c Qm QC c 1 P# # ln ! (c!a) 2 2 2 a 4
Qm QC P# # 2 a
b 1 A ! (Bc!A) ln ! (b!c) B! . c 2 b
(46)
Continuities of tangential and radial stresses and displacements at the elastic}plastic boundary are enforced in this solution. The following equations for A and B are obtained from p and P p continuities, respectively: F
c QC 1 1 Qm c b A" ! ! P# ln #B ln , b!c 2 c a 2 a c
(47)
46
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
b 1 Qm B" A #1 ! P# c (1!ln (b/c)) 2
1#ln
c QC 1 1 ! # a 2 c a
.
(48)
An equation identical to Eq. (47) is also obtained from the requirement of zero radial force,
@
p dr"0. F ? The continuity equation for the tangential displacement at r"c is
(49)
a ch!a sin h"ca h!a sin h, (50) # # where a is still !2Bm and a is a new constant yet to be determined. Substituting for a from # # Eq. (44), an expression for a is obtained which is indeterminate at h"0 (plane of symmetry) where computations for stresses and displacements will be carried out. The limit value of a as h approaches zero is found by di!erentiation as !2Bm c!a !(1#m Q)(C /a) # . a " c!(a/2)
(51)
The new a is determined from the continuity of radial displacements at r"c, # Qm c C Q KQC c a " P# Kc ln # (m #m )#1 # # 2 a c 2 2a
(c!a) (m Q#1)C A(m #m )b ! ! [(1#m Q)m#2m P]! 2 c a
!BcK 1#ln
b !AcK!Bm c. c
(52)
From m"(a !2m p)/(m Q#1), and substituting for a in Eq. (51), the following equation for # m is obtained:
2 a (m #m )b m" KcP ln !m Pc# cK# A c(m Q#1)#KQc ln(c/a) c c KQc b Q 1 # (m #m )# C . # cK ln !m c B! 2a c 2c c
(53)
Eqs. (47), (48), and (53), along with three equations for b, c, and C are the system of equations for the solution of the problem as bending proceeds to larger values of the internal radius a. The outer radius b is determined from Eq. (21), with u still given by Eq. (22). An equation for c is obtained by @ setting p"y in Eq. (26), and using p and p from Eqs. (1) and (2) at r"c: F P 2A c"b . (54) (2>/(3)#B
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
47
At r"c, e."0. Therefore, from Eq. (39) it follows that, F m C "! c. 2
(55)
5. Second elastic}plastic The second plastic zone occurs at o)r)b as deformation continues. The e!ective stress and e!ective plastic strain in this zone are (3 (p !p ), p" F 2 P
(56)
2 eN. e N"! (3 F
(57)
The following strain, stress and displacement equations are obtained by using a similar approach as in the "rst-plastic zone: g C eN" # , F 2 r
(58)
Qg b KQC C Q g # ln # (m #m )#1 !Pm # (Qm #1), e "K P! F 2 r 2b r 2 2
(59)
where 2m P#a g" m Q#1
(60)
and C are a are parameters yet to be determined.
Qg b QC 1 1 ln # ! , p " P! P 2 r 2 b r
Qg p "! P! F 2
1!ln
(61)
b QC 1 1 # # , r 2 b r
(62)
Qg b KQC C Q r# r ln # (m #m )#1 u"K P! 2 r 2b r 2 g ! Pm r# (m Q#1)r!ra #a cos h, 2 v"ra h!a sin h.
(63) (64)
48
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
In the elastic zone (c)r)o), the previous stress equations (1) and (2) can no longer be used, as the boundary condition p "0 at r"b is out of its domain (this boundary condition has already P been applied in the second plastic zone). The new equations [1] are A p " #B(1#2 ln r)#2D, P r
(65)
A p "! #B(3#2 ln r)#2D. F r
(66)
Based on these equations and using the approach of Section 3, the following relationships for displacements in the elastic zone are obtained: A u"!(m #m ) #2KBr(ln r!1)#B(m !3m )r#2DKr#a cos h, # r
(67)
v"4Bm rh!a sin h, #
(68)
where A, B, D, and a are new parameters yet to be determined. # The stresses and displacements in the "rst-plastic zone (a(r)c) are still given by Eqs. (40), (41), (43), and Eq. (12) with new equations for C and m given later. The following equations for A and B are obtained from the continuity of p at o and c, P respectively:
1 Qg b Q 1 o ln !Bo(1#2 ln o)!2Do# C o ! , A" P! b o 2 o 2 B"
1 1#2 ln c
Qm c A QC 1 1 P# ln ! !2D! ! 2 a c 2 c a
(69)
.
(70)
An equation for D is obtained from the condition of zero radial force (Eq. (49)):
QcC 1 1 1 1 1 !A ! #[o(1#2 ln o)!c(1#2 ln c)]B# ! D" 2 2(c!o) c o a c
Qm c Qg o QoC 1 1 # P# c ln # P! o ln ! ! 2 a 2 b 2 b o
.
(71)
The following equations for C and C are obtained from the continuity of p at c and o, F respectively:
2ca A Qm C " ! #B(3#2 ln c)#2D! P# Q(a#c) c 2
2bo A Qg C " ! #B(3#2 ln o)#2D# P! Q(b#o) o 2
1#ln
c a
1!ln
b o
,
.
(72)
(73)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
49
The procedures for "nding m and g equations are similar to that employed earlier (Eqs. (50)}(53)) and are based on u and v continuities at c and o: 4Bm c!a !(m Q#1)(C /a) # a " c!a/2
(74)
and a still given by Eq. (44).
Qm c C Q KQC c Kc ln # (m #m )#1 # a " P# # 2 a c 2 2a c!a C A ! [(m Q#1)m#2m P]!(m Q#1) # (m #m ) 2 a c !Bc(m !3m )!2KDc!2KBc(ln c!1),
(75)
c!p #4Bm o , a " o!p
(76)
a "c#a #p a !p , #
(77)
where b o KoQ ln # , p " 2(m Q#1) o 2
(78)
KoP b KQC o C Q # p " ln # (m #m )#1 , (m Q#1) o 2b o 2
(79)
A c"!(m #m ) #2KBo(ln o!1)#B(m !3m )o#2KDo. o
(80)
From Eqs. (74)}(80) and using (m, a ) and (g, a ) relationships, the following equations for m and g are obtained:
A(m #m ) 2 #c(2K ln c#3m !m )B ! m" c c(m Q#1)#KcQ ln(c/a)
KQc Q 1 c # (m #m )# C !m Pc!KcP ln , 2a 2c c a
# 2KcD!
(81)
A(m #m ) 2 #o(2K ln o#3m !m )B ! g" o o(m Q#1)!KoQ ln (b/o)
# 2KoD!
KQo Q 1 b # (m #m )# C #m Po!KoP ln . 2b 2o o o
(82)
50
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Eq. (21) is used for determining b, with u from Eq. (63) given by @ C b u " (m Q#1)! a #a . @ b 2
(83)
From yielding conditions at c and o the following equations for c and o are obtained:
c"
!2C , m
(84)
o"
!2C . g
(85)
Eqs. (69)}(73) and (81)}(85) are the relationships used in the solution of the problem during the second-plastic phase of deformation. Bending moment is determined from Eq. (19) and the following relationship is obtained: 1 M"! 4
Qm P# 2
Qg # P! 2
c!a#2c ln
o!b#2o ln
c QC c # c!a#2a ln a a a
o QC o ! o!b#2b ln b b b
o # A ln !B[o(ln o#1)!C(ln c#1)]!D(o!c). c
(86)
6. Numerical results and comparison with FEM 5ndings A curved beam with initial dimensions a"60 mm, b"80 mm, and the following mechanical properties was considered: Elastic modulus, E"200 Gpa. Yield strength, >"200 Mpa. Slope of plastic line, a"2020.202 Mpa. The same solution approach was used in all the three phases of deformation (all elastic, "rst-plastic, and second-plastic). In each phase, the initial values of the unknown parameters at the beginning of deformation (i"1) had to be determined. For phase one, the undeformed a and b, along with a small value for moment M"10 N m (&0.07% of M for start of plastic deformation), A"0, B"0, and a "0 were used. Singular matrices resulted when M"0, or 10 N m and # A and B values calculated from the appropriate relationships were used. It was also noticed that the results at the end of all-elastic phase were not appreciably a!ected by large variations in the value of the initial M. The "nal values of a and b at the onset of yielding at r"a were used as the initial values of a and b at the start of "rst-plastic phase, with an assumed c"a#1;10\ mm. The initial values of A, B, m, and C were calculated from Eqs. (47), (48), (53), and e continuity at r"c, respectively. P Similarly, the initial values of a, b, and c at the start of the second-plastic phase were set equal to the
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
51
values of these parameters at the end of the previous phase; and the initial values of the remaining unknown parameters A, B, D, C , C , m, and g were found from the system of linear equations (69)}(73), (81), and (82), respectively, with an initial o"b!3.8;10\ mm. The matrices of the coe$cients for both "rst- and second-plastic phases were very ill-conditioned. The estimates of the reciprocal of L1 condition number given by IMSL routine DLSARG was as small as 1;10\. The choice of initial c in "rst-plastic, and of initial o in the second-plastic phase, had very pronounced e!ects on the initial values of some of the parameters: m and C in the former, and g and C in the latter case. The selected values for initial c and o corresponded to the largest L1 condition number &10\ that were obtained by examining a limited number of solutions in each case. However, no formal procedure for L1 maximization was developed. In spite of this, and the persistent IMSL warning, the calculated stresses and displacements, based on the selected c and o and the results for the initial values of other parameters, looked reasonably accurate. For example, in the second-plastic case at i"1, the radial displacements at c and o, calculated at the corresponding elastic and plastic zones were identical. Also, the continuities of radial and tangential stresses at c and o were maintained to at least 11 decimal places in all cases. Stresses, strains, displacements, and the bending moment were determined based on the calculated initial values in the "rst increment of deformation (i"1) in each phase. Additionally, for any parameter x, its derivative with respect to the changing inside radius a was also determined. This involved formulation for dx/da based on the functional dependence of the parameters
Fig. 2. Distributions of radial stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) after n"1.49;10 increments of deformation in second elastic}plastic phase.
52
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
x, y, z,2 given by the governing equations in each phase, dx *x dy *x dz " # #2 . da *y da *z da
(87)
A system of nonhomogeneous linear equations with dx/da, dy/da, dz/da,2 as unknowns, the known coe$cients *x/*y, *x/*z,2 and *x/*a, *y/*a, *z/*a,2 as the nonzero right-hand side numbers was thus developed and solved by IMSL routine DLSARG. The current values of the solution parameters at the beginning of subsequent increments (i#1) were calculated from, a
"a #*a , (88) G G dx *a , (89) x "x # G G> G da G where the incremental increase of the inside radius *a "1;10\, 1;10\, and 2;10\ mm for G deformations in phases 1, 2, and 3, respectively. The solution parameters were A, B, M, b, a for # all-elastic, A, B, b, c, C , m for "rst-plastic, and A, B, D, b, c, o, C , C , m, g for the second-plastic phase. Stress and displacement distribution plots are shown in Figs. 2}4. ANSYS 5.3 FEM plots are given in Figs. 5}7 for comparison. Plots for variations at the ends of the all-elastic and "rst-plastic G>
Fig. 3. Distributions of tangential stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, (c) after n"5;10 increments of deformation in second elastic}plastic phase, and (d) n"1.49;10.
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
53
Fig. 4. Distributions of radial displacement (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) after n"1.49;10 increments of deformation in second elastic}plastic phase.
phases, as well as deformation into the second-plastic phase are given in each "gure. The same geometry and mechanical properties with linear hardening behavior in plane strain pure bending were used in the FEM solution. The point r"a on the symmetry plane (h"0) was "xed (u"0, v"0) and deformation was performed in 20 substeps. In Figs. 2}7, the abscissa for the FEM pathplots is (r!a), while for the analytical plots it is the radial position r. As a result, a shift to the right of the latter plots with increased a is noticed. Otherwise, the distribution patterns are generally similar. However, some di!erences need to be mentioned. The boundary conditions p "0 at r"a and b are not satis"ed by the FEM plots based on a nodal solution. Also, the yield P conditions at r"a at the end of the all-elastic phase, and at r"b at the end of the "rst-plastic phase are not satis"ed by the FEM solution. As a result, larger di!erences between p values at F r"a and b are observed for the FEM results (Figs. 3 and 6). The numerical results for radial and tangential stresses (Figs. 2 and 3) are generally in favorable agreement with the corresponding FEM results (Figs. 5 and 6). The largest deviations (17% and 12%) are for p and p values at the end of the elastic stage. The displacement results (Figs. 4 and 7) P F show larger discrepancies, up to 33% di!erence between the radial displacements at the end of the all-elastic stage. Better agreements are noticed in later stages of deformation in all cases. In spite of these, the accuracy of the numerical computations cannot be truly ascertained. A manifestation of possible error accumulation after a large number of incremental steps n can be seen in Fig. 3, where at n"1.49;10 after the start of the second-plastic phase a discontinuity of p at r"c becomes F
54
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Fig. 5. FEM plots of radial stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
Fig. 6. FEM plots of tangential stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
55
Fig. 7. FEM plots of radial displacement (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
apparent, while at an earlier stage n"5;10 a similar discontinuity is still not noticeable. The accumulated errors in parameters A, B, and D are the likely cause in this case, as p based on the F elastic-equation (Eq. (66)) ceases to satisfy the yield condition (and p continuity) at r"c for large F values of n. Surprisingly, the p values from the elastic and plastic equation (Eqs. (65) and (61)) P continue to agree to at least three decimal places, even though the same parameters A, B, and D are used in the computation of elastic p . P The error accumulation problem, and concerns over the uncertainties of the initial-stage computations at the start of each phase, indicate that a more skillful and robust computational scheme may be needed, so that the full potential of the analytical solution can be realized, and a more detailed examination of the results become warranted.
7. Conclusions (a) An analytical solution for plane strain elastic}plastic bending of a linear hardening material has been developed. (b) The case of a general nonlinear hardening material has resulted in nonlinear di!erential equations for which no solution has been found. (c) FEM veri"cations for stress and displacement distributions have been presented and reasonable agreements have been noticed.
56
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
References [1] Timoshenko SP, Goodier JN. Theory of elasticity, 3rd ed. New York: McGraw-Hill, 1970. [2] Hill R. The mathematical theory of plasticity. London: Oxford University Press, 1967. [3] Dadras P, Majlessi SA. Plastic bending of work hardening materials. ASME Transactions of the Journal of Engineering and Industry 1982;104:224. [4] Sha!er BW, House Jr. RN. The elastic}plastic stress distribution within a wide curved bar subjected to pure bending. ASME Transactions of the Journal of Applied Mechanics 1955;22:305. [5] Sha!er BW, House Jr. RN. The signi"cance of zero shear stress in the pure bending of a wide curved bar. Journal of Aerosol Science 1957;24:307. [6] Sha!er BW, House Jr. RN. Displacements in a wide curved bar subjected to pure elastic}plastic bending. ASME Transactions of the Journal of Applied Mechanics 1957;24:447. [7] Eason G. The elastic}plastic bending of a compressible curved bar. Applied Science Research 1960;A9:53. [8] Chakrabarty J. Theory of plasticity. New York: McGraw-Hill, 1987.
Plane strain elastic}plastic bending of a strain-hardening curved beam Parviz Dadras* Wright State University, Dayton, OH 45435, USA Received 4 November 1998; received in revised form 18 November 1999; accepted 23 November 1999
Abstract Elastic}plastic analysis of plane strain pure bending of a strain-hardening curved beam has been presented. Only a linear hardening case has been analyzed as the nonlinear equations for a general hardening case could not be solved analytically. A numerical scheme for the computation of stresses and displacements in di!erent stages of deformation has been given and limited FEM veri"cations have been presented. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Curved beams; Elastic}plastic bending
1. Introduction Elastic analysis of pure bending of a wide curved bar under plane strain deformation is given by Timoshenko and Goodier [1]. Rigid-perfectly plastic, and rigid-work hardening analyses are given by Hill [2] and Dadras and Majlessi [3], respectively. The elastic}plastic solution for perfectly plastic material is given by Sha!er and House [4]. Solutions for perfectly plastic incompressible material [5,6], and perfectly plastic compressible material [7] are also available. The analysis for an elastic strain-hardening material is given in this paper. A numerical scheme for solving the resulting equations is also presented and limited comparisons with FEM results are given. 2. Analysis A curved beam of inner radius a and outer radius b is subjected to a moment M, as shown in Fig. 1. It is assumed that the beam is "xed at point r"a and h"0 throughout the deformation.
* Present address: 1435 Meadow Lane, Yellow Springs, OH 45387, USA. 0020-7403/01/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 0 3 ( 9 9 ) 0 0 1 0 2 - 2
40
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Nomenclature a, b c, o r, h u, v e ,e ,c P F PF p ,p P F M > E, l p, eN e C, e . eC , e. F F
deformed inside and outside radii, respectively "rst and second elastic}plastic radii, respectively radial and tangential coordinates, respectively radial and tangential displacements, respectively strain components in polar coordinates radial and tangential stresses, respectively applied bending moment yield strength elastic constants e!ective stress and strain, respectively e!ective elastic and plastic strains, respectively elastic and plastic tangential strain components, respectively
Fig. 1. Geometry of the curved beam.
Analyses are performed for three stages of deformation: all elastic, "rst elastic}plastic deformation with a plastic zone formed within a(r)c, and second elastic}plastic where an additional plastic zone appeared at o(r(b.
3. All elastic The stress analysis is as given by Timoshenko and Goodier [1]. For simplicity the simpli"ed equations given by Chakrabarty [8] are used here:
b b !1 #B ln , p "!A P r r
(1)
b b #1 !B 1!ln , p "A F r r
(2)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
41
where
4M b 4M b A" ln , B" !1 Na a Na a
(3)
and
N"
b 4b b . !1 ! ln a a a
The development of the displacement equations are also given in Refs. [1,5,7]. However, a more general solution, also suitable for the later stages of deformation and based on a di!erent boundary condition assumed here, will be presented. The strain}displacement relations are *u e" , P *r u 1 *v , e " # F r r *h *v v 1 *u "0. c " ! # PF *r r r *h
(4) (5) (6)
The strains and the stresses are only functions of r. Therefore, from Eq. (5) it follows that *u *v ! " "g, *h *h
(7)
where g is only a function of h. Substitution in Eq. (6): r
*v !v"g. *r
(8)
The solution of Eq. (8) is v"rf !g, where f is another function of h. Di!erentiating Eq. (9) w.r.t. h twice, shows that df dg "0 and ! "g. dh dh
(9)
(10)
The solutions of Eq. (10) and considering Eq. (9) results in v"ra h!a sin h!a cos h. Since v "!v , it follows that a "0 and >F \F v"ra h!a sin h.
(11)
(12)
42
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
The above equation for the tangential displacement v is valid in both elastic and plastic zones. The radial displacement u will be found by considering the p}e relationships in plane strain (e "0): X e "m p !m p , (13) P P F e "m p !m p , (14) F F P where m "(1!l)/E, m "l(1#l)/E, and l is Poisson's ratio. An equation for u is obtained by substituting for p and p in Eq. (13) and integrating, P F b b (15) u"A(m #m ) #BKr 1#ln #ArK#Bm r#f, r r
where K"m !m , and f is only a function of h. From Eq. (7) it follows that f"a cos h!a sin h. (16) Once again a "0 since u "u . The constant a is found from Eq. (5) by using Eqs. (15), (16), >F \F (14), and the stress equations (1) and (2): a "!2Bm (17) # where the subscript 1E indicates a in an elastic "eld. The remaining constant a is found from the boundary condition r"a, h"0, u"v"0:
b b a "!A(m #m ) !BaK 1#ln !AaK!Bm a. # a a
(18)
The bending moment from
@
?
p r dr"!M F
(19)
is
M"
1 b 1 B Ba!Ab ln # !A (b!a). 2 a 2 2
(20)
Eqs. (3), (18), and (20) along with an equation for b are the system of equations for determining the unknowns A, B, a , M and b as the inside radius a is allowed to increase. The deformed # thickness of the beam is independent of h. Considering the plane of symmetry (h"0) for convenience, the deformed thickness is b!a"t #u !u , @ ? where t is the initial thickness, and u and u are radial displacements on the plane of symmetry at @ ? b and a, respectively. For the point at r"a and h"0, the radial and tangential displacements are zero, as required by the boundary condition at this point. Therefore, it follows that, db du "1# @ . da da
(21)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
The radial displacement u is determined from Eq. (15), @ u "A(m #m )b#Bbm #AbK#a @ # with a given in Eq. (18). #
43
(22)
4. First elastic}plastic The following e!ective stress (p) e!ective strain (e ) relation was "rst examined: p">#a(e !e )L, (23) W where > and e are the stress and strain at yield, respectively, and a is a material constant. The use W of Eq. (23) led to nonlinear di!erential equations for strains which could not be solved. Linear strain hardening behavior (n"1) was, therefore, used. p">#a(e !e ), (24) 7 where a is the slope of the plastic line. Eq. (24) is further modi"ed by using e "e .#e C and e C"p/E, where the superscripts p and e denote plastic and elastic, respectively. Further manipulations lead to Ea p"># e .. E!a
(25)
The e!ective stress and e!ective plastic strain for the "rst plastically deforming zone are (3 (p !p ), p" P 2 F
(26)
2 e ." e.. (3 F
(27)
Therefore, p !p "P#Qe., F P F where P"2>/(3 and Q"4Ea/3(E!a). The equilibrium equation [1] is thus, *p r P "p !p "P#Qe.. F P F *r
(28)
(29)
From integration of Eq. (29), and using Eq. (28),
e. p "P ln r#Q F dr#C , P r
(30)
e. p "P ln r#Q F dr#C #P#Qe., F F r
(31)
44
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
where C is a constant. Since e "e."0, it follows that X X e #e "eC #eC. (32) F P F P The elastic strain components in the plastic zone will be related to the stresses as given by Eqs. (13) and (14) and the stresses will still follow Eqs. (1) and (2) with new relations for A and B which will be developed later. Therefore, from Eqs. (13), (14), and (30)}(32),
e. e #e "K 2P ln r#2Q F dr#2C #P#Qe. . F F P r
(33)
a u 1 *v u " #a ! cos h e " # F r r *h r r
(34)
Also,
in view of Eq. (12) for v which is still valid in the plastic zone, where a and a are constants in the "rst plastic zone. From Eq. (34), *u *e "e "e #r F !a . P F *r *r
(35)
Upon substitution for e from Eq. (35) into Eq. (33) and using e "e.#eC , the following equation is P F F F obtained: 2e.#2eC #r F F
e. *(eC #e.) F F !KQe.!2KQ F dr"K(2P ln r#2C #P)#a . F r *r
(36)
eC and subsequently its derivative w.r.t. r are found from Eq. (13), with p and p given by Eqs. (30) F P F and (31). Therefore,
e. eC "KP ln r#KQ F dr#KC #m (P#Qe.). F J F r The following equation for e. is obtained by substituting from Eq. (37) into Eq. (36): F m de. 2 F # e." , r F r dr where m"(a !2m P)/(m Q#1). The solution of Eq. (38) is m C e." # , F 2 r
(37)
(38)
(39)
where C is a constant. The "nal equations for p and p are then obtained by using Eq. (39) in Eqs. P F (30) and (31) and applying the boundary condition r"a, p "0 to "nd the constant C : P Qm r QC 1 1 ln ! ! , (40) p " P# P 2 a 2 r a
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Qm p " P# F 2
1#ln
r QC 1 1 # # . a 2 r a
45
(41)
The radial displacement equation is determined by "nding e "eC #e., where eC "m p ! F F F F F m p , with p and p as given in Eqs. (40) and (41): P P F
Qm r C Q KQC m # (m Q#1)#m P. e "K P# ln # (m #m )#1 # F 2 a r 2 2a 2
(42)
Substituting e in Eq. (34) results in F
Qm r C Q KQC r u"Kr P# ln # (m #m )#1 # 2 a r 2 2a #
mr (m Q#1)#m Pr!ra #a cos h. 2
(43)
The above equation along with Eq. (12) are the displacement equations for the plastic zone at a(r(C. From the boundary condition r"a, h"0, u"0 the following relations are obtained: aa C (m Q#1) a " ! . 2 a
(44)
Therefore,
Qm r C Q KQC r a (m Q#1)C ! (r!a)! . (45) u"Kr P# ln # (m #m )#1 # 2 a r 2 2a 2 a Stress relations (1) and (2), and the displacement equations (12) and (15)}(17) are still applicable in the elastic zone with new parameters A, B, and a which will be determined from the continuity # considerations between the elastic and plastic zones. From, Eq. (19) the following moment equation is obtained:
M"!
c Qm QC c 1 P# # ln ! (c!a) 2 2 2 a 4
Qm QC P# # 2 a
b 1 A ! (Bc!A) ln ! (b!c) B! . c 2 b
(46)
Continuities of tangential and radial stresses and displacements at the elastic}plastic boundary are enforced in this solution. The following equations for A and B are obtained from p and P p continuities, respectively: F
c QC 1 1 Qm c b A" ! ! P# ln #B ln , b!c 2 c a 2 a c
(47)
46
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
b 1 Qm B" A #1 ! P# c (1!ln (b/c)) 2
1#ln
c QC 1 1 ! # a 2 c a
.
(48)
An equation identical to Eq. (47) is also obtained from the requirement of zero radial force,
@
p dr"0. F ? The continuity equation for the tangential displacement at r"c is
(49)
a ch!a sin h"ca h!a sin h, (50) # # where a is still !2Bm and a is a new constant yet to be determined. Substituting for a from # # Eq. (44), an expression for a is obtained which is indeterminate at h"0 (plane of symmetry) where computations for stresses and displacements will be carried out. The limit value of a as h approaches zero is found by di!erentiation as !2Bm c!a !(1#m Q)(C /a) # . a " c!(a/2)
(51)
The new a is determined from the continuity of radial displacements at r"c, # Qm c C Q KQC c a " P# Kc ln # (m #m )#1 # # 2 a c 2 2a
(c!a) (m Q#1)C A(m #m )b ! ! [(1#m Q)m#2m P]! 2 c a
!BcK 1#ln
b !AcK!Bm c. c
(52)
From m"(a !2m p)/(m Q#1), and substituting for a in Eq. (51), the following equation for # m is obtained:
2 a (m #m )b m" KcP ln !m Pc# cK# A c(m Q#1)#KQc ln(c/a) c c KQc b Q 1 # (m #m )# C . # cK ln !m c B! 2a c 2c c
(53)
Eqs. (47), (48), and (53), along with three equations for b, c, and C are the system of equations for the solution of the problem as bending proceeds to larger values of the internal radius a. The outer radius b is determined from Eq. (21), with u still given by Eq. (22). An equation for c is obtained by @ setting p"y in Eq. (26), and using p and p from Eqs. (1) and (2) at r"c: F P 2A c"b . (54) (2>/(3)#B
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
47
At r"c, e."0. Therefore, from Eq. (39) it follows that, F m C "! c. 2
(55)
5. Second elastic}plastic The second plastic zone occurs at o)r)b as deformation continues. The e!ective stress and e!ective plastic strain in this zone are (3 (p !p ), p" F 2 P
(56)
2 eN. e N"! (3 F
(57)
The following strain, stress and displacement equations are obtained by using a similar approach as in the "rst-plastic zone: g C eN" # , F 2 r
(58)
Qg b KQC C Q g # ln # (m #m )#1 !Pm # (Qm #1), e "K P! F 2 r 2b r 2 2
(59)
where 2m P#a g" m Q#1
(60)
and C are a are parameters yet to be determined.
Qg b QC 1 1 ln # ! , p " P! P 2 r 2 b r
Qg p "! P! F 2
1!ln
(61)
b QC 1 1 # # , r 2 b r
(62)
Qg b KQC C Q r# r ln # (m #m )#1 u"K P! 2 r 2b r 2 g ! Pm r# (m Q#1)r!ra #a cos h, 2 v"ra h!a sin h.
(63) (64)
48
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
In the elastic zone (c)r)o), the previous stress equations (1) and (2) can no longer be used, as the boundary condition p "0 at r"b is out of its domain (this boundary condition has already P been applied in the second plastic zone). The new equations [1] are A p " #B(1#2 ln r)#2D, P r
(65)
A p "! #B(3#2 ln r)#2D. F r
(66)
Based on these equations and using the approach of Section 3, the following relationships for displacements in the elastic zone are obtained: A u"!(m #m ) #2KBr(ln r!1)#B(m !3m )r#2DKr#a cos h, # r
(67)
v"4Bm rh!a sin h, #
(68)
where A, B, D, and a are new parameters yet to be determined. # The stresses and displacements in the "rst-plastic zone (a(r)c) are still given by Eqs. (40), (41), (43), and Eq. (12) with new equations for C and m given later. The following equations for A and B are obtained from the continuity of p at o and c, P respectively:
1 Qg b Q 1 o ln !Bo(1#2 ln o)!2Do# C o ! , A" P! b o 2 o 2 B"
1 1#2 ln c
Qm c A QC 1 1 P# ln ! !2D! ! 2 a c 2 c a
(69)
.
(70)
An equation for D is obtained from the condition of zero radial force (Eq. (49)):
QcC 1 1 1 1 1 !A ! #[o(1#2 ln o)!c(1#2 ln c)]B# ! D" 2 2(c!o) c o a c
Qm c Qg o QoC 1 1 # P# c ln # P! o ln ! ! 2 a 2 b 2 b o
.
(71)
The following equations for C and C are obtained from the continuity of p at c and o, F respectively:
2ca A Qm C " ! #B(3#2 ln c)#2D! P# Q(a#c) c 2
2bo A Qg C " ! #B(3#2 ln o)#2D# P! Q(b#o) o 2
1#ln
c a
1!ln
b o
,
.
(72)
(73)
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
49
The procedures for "nding m and g equations are similar to that employed earlier (Eqs. (50)}(53)) and are based on u and v continuities at c and o: 4Bm c!a !(m Q#1)(C /a) # a " c!a/2
(74)
and a still given by Eq. (44).
Qm c C Q KQC c Kc ln # (m #m )#1 # a " P# # 2 a c 2 2a c!a C A ! [(m Q#1)m#2m P]!(m Q#1) # (m #m ) 2 a c !Bc(m !3m )!2KDc!2KBc(ln c!1),
(75)
c!p #4Bm o , a " o!p
(76)
a "c#a #p a !p , #
(77)
where b o KoQ ln # , p " 2(m Q#1) o 2
(78)
KoP b KQC o C Q # p " ln # (m #m )#1 , (m Q#1) o 2b o 2
(79)
A c"!(m #m ) #2KBo(ln o!1)#B(m !3m )o#2KDo. o
(80)
From Eqs. (74)}(80) and using (m, a ) and (g, a ) relationships, the following equations for m and g are obtained:
A(m #m ) 2 #c(2K ln c#3m !m )B ! m" c c(m Q#1)#KcQ ln(c/a)
KQc Q 1 c # (m #m )# C !m Pc!KcP ln , 2a 2c c a
# 2KcD!
(81)
A(m #m ) 2 #o(2K ln o#3m !m )B ! g" o o(m Q#1)!KoQ ln (b/o)
# 2KoD!
KQo Q 1 b # (m #m )# C #m Po!KoP ln . 2b 2o o o
(82)
50
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Eq. (21) is used for determining b, with u from Eq. (63) given by @ C b u " (m Q#1)! a #a . @ b 2
(83)
From yielding conditions at c and o the following equations for c and o are obtained:
c"
!2C , m
(84)
o"
!2C . g
(85)
Eqs. (69)}(73) and (81)}(85) are the relationships used in the solution of the problem during the second-plastic phase of deformation. Bending moment is determined from Eq. (19) and the following relationship is obtained: 1 M"! 4
Qm P# 2
Qg # P! 2
c!a#2c ln
o!b#2o ln
c QC c # c!a#2a ln a a a
o QC o ! o!b#2b ln b b b
o # A ln !B[o(ln o#1)!C(ln c#1)]!D(o!c). c
(86)
6. Numerical results and comparison with FEM 5ndings A curved beam with initial dimensions a"60 mm, b"80 mm, and the following mechanical properties was considered: Elastic modulus, E"200 Gpa. Yield strength, >"200 Mpa. Slope of plastic line, a"2020.202 Mpa. The same solution approach was used in all the three phases of deformation (all elastic, "rst-plastic, and second-plastic). In each phase, the initial values of the unknown parameters at the beginning of deformation (i"1) had to be determined. For phase one, the undeformed a and b, along with a small value for moment M"10 N m (&0.07% of M for start of plastic deformation), A"0, B"0, and a "0 were used. Singular matrices resulted when M"0, or 10 N m and # A and B values calculated from the appropriate relationships were used. It was also noticed that the results at the end of all-elastic phase were not appreciably a!ected by large variations in the value of the initial M. The "nal values of a and b at the onset of yielding at r"a were used as the initial values of a and b at the start of "rst-plastic phase, with an assumed c"a#1;10\ mm. The initial values of A, B, m, and C were calculated from Eqs. (47), (48), (53), and e continuity at r"c, respectively. P Similarly, the initial values of a, b, and c at the start of the second-plastic phase were set equal to the
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
51
values of these parameters at the end of the previous phase; and the initial values of the remaining unknown parameters A, B, D, C , C , m, and g were found from the system of linear equations (69)}(73), (81), and (82), respectively, with an initial o"b!3.8;10\ mm. The matrices of the coe$cients for both "rst- and second-plastic phases were very ill-conditioned. The estimates of the reciprocal of L1 condition number given by IMSL routine DLSARG was as small as 1;10\. The choice of initial c in "rst-plastic, and of initial o in the second-plastic phase, had very pronounced e!ects on the initial values of some of the parameters: m and C in the former, and g and C in the latter case. The selected values for initial c and o corresponded to the largest L1 condition number &10\ that were obtained by examining a limited number of solutions in each case. However, no formal procedure for L1 maximization was developed. In spite of this, and the persistent IMSL warning, the calculated stresses and displacements, based on the selected c and o and the results for the initial values of other parameters, looked reasonably accurate. For example, in the second-plastic case at i"1, the radial displacements at c and o, calculated at the corresponding elastic and plastic zones were identical. Also, the continuities of radial and tangential stresses at c and o were maintained to at least 11 decimal places in all cases. Stresses, strains, displacements, and the bending moment were determined based on the calculated initial values in the "rst increment of deformation (i"1) in each phase. Additionally, for any parameter x, its derivative with respect to the changing inside radius a was also determined. This involved formulation for dx/da based on the functional dependence of the parameters
Fig. 2. Distributions of radial stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) after n"1.49;10 increments of deformation in second elastic}plastic phase.
52
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
x, y, z,2 given by the governing equations in each phase, dx *x dy *x dz " # #2 . da *y da *z da
(87)
A system of nonhomogeneous linear equations with dx/da, dy/da, dz/da,2 as unknowns, the known coe$cients *x/*y, *x/*z,2 and *x/*a, *y/*a, *z/*a,2 as the nonzero right-hand side numbers was thus developed and solved by IMSL routine DLSARG. The current values of the solution parameters at the beginning of subsequent increments (i#1) were calculated from, a
"a #*a , (88) G G dx *a , (89) x "x # G G> G da G where the incremental increase of the inside radius *a "1;10\, 1;10\, and 2;10\ mm for G deformations in phases 1, 2, and 3, respectively. The solution parameters were A, B, M, b, a for # all-elastic, A, B, b, c, C , m for "rst-plastic, and A, B, D, b, c, o, C , C , m, g for the second-plastic phase. Stress and displacement distribution plots are shown in Figs. 2}4. ANSYS 5.3 FEM plots are given in Figs. 5}7 for comparison. Plots for variations at the ends of the all-elastic and "rst-plastic G>
Fig. 3. Distributions of tangential stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, (c) after n"5;10 increments of deformation in second elastic}plastic phase, and (d) n"1.49;10.
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
53
Fig. 4. Distributions of radial displacement (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) after n"1.49;10 increments of deformation in second elastic}plastic phase.
phases, as well as deformation into the second-plastic phase are given in each "gure. The same geometry and mechanical properties with linear hardening behavior in plane strain pure bending were used in the FEM solution. The point r"a on the symmetry plane (h"0) was "xed (u"0, v"0) and deformation was performed in 20 substeps. In Figs. 2}7, the abscissa for the FEM pathplots is (r!a), while for the analytical plots it is the radial position r. As a result, a shift to the right of the latter plots with increased a is noticed. Otherwise, the distribution patterns are generally similar. However, some di!erences need to be mentioned. The boundary conditions p "0 at r"a and b are not satis"ed by the FEM plots based on a nodal solution. Also, the yield P conditions at r"a at the end of the all-elastic phase, and at r"b at the end of the "rst-plastic phase are not satis"ed by the FEM solution. As a result, larger di!erences between p values at F r"a and b are observed for the FEM results (Figs. 3 and 6). The numerical results for radial and tangential stresses (Figs. 2 and 3) are generally in favorable agreement with the corresponding FEM results (Figs. 5 and 6). The largest deviations (17% and 12%) are for p and p values at the end of the elastic stage. The displacement results (Figs. 4 and 7) P F show larger discrepancies, up to 33% di!erence between the radial displacements at the end of the all-elastic stage. Better agreements are noticed in later stages of deformation in all cases. In spite of these, the accuracy of the numerical computations cannot be truly ascertained. A manifestation of possible error accumulation after a large number of incremental steps n can be seen in Fig. 3, where at n"1.49;10 after the start of the second-plastic phase a discontinuity of p at r"c becomes F
54
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
Fig. 5. FEM plots of radial stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
Fig. 6. FEM plots of tangential stress (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
55
Fig. 7. FEM plots of radial displacement (a) at the end of all-elastic, (b) at the end of "rst elastic}plastic, and (c) during the second elastic}plastic phase.
apparent, while at an earlier stage n"5;10 a similar discontinuity is still not noticeable. The accumulated errors in parameters A, B, and D are the likely cause in this case, as p based on the F elastic-equation (Eq. (66)) ceases to satisfy the yield condition (and p continuity) at r"c for large F values of n. Surprisingly, the p values from the elastic and plastic equation (Eqs. (65) and (61)) P continue to agree to at least three decimal places, even though the same parameters A, B, and D are used in the computation of elastic p . P The error accumulation problem, and concerns over the uncertainties of the initial-stage computations at the start of each phase, indicate that a more skillful and robust computational scheme may be needed, so that the full potential of the analytical solution can be realized, and a more detailed examination of the results become warranted.
7. Conclusions (a) An analytical solution for plane strain elastic}plastic bending of a linear hardening material has been developed. (b) The case of a general nonlinear hardening material has resulted in nonlinear di!erential equations for which no solution has been found. (c) FEM veri"cations for stress and displacement distributions have been presented and reasonable agreements have been noticed.
56
P. Dadras / International Journal of Mechanical Sciences 43 (2001) 39}56
References [1] Timoshenko SP, Goodier JN. Theory of elasticity, 3rd ed. New York: McGraw-Hill, 1970. [2] Hill R. The mathematical theory of plasticity. London: Oxford University Press, 1967. [3] Dadras P, Majlessi SA. Plastic bending of work hardening materials. ASME Transactions of the Journal of Engineering and Industry 1982;104:224. [4] Sha!er BW, House Jr. RN. The elastic}plastic stress distribution within a wide curved bar subjected to pure bending. ASME Transactions of the Journal of Applied Mechanics 1955;22:305. [5] Sha!er BW, House Jr. RN. The signi"cance of zero shear stress in the pure bending of a wide curved bar. Journal of Aerosol Science 1957;24:307. [6] Sha!er BW, House Jr. RN. Displacements in a wide curved bar subjected to pure elastic}plastic bending. ASME Transactions of the Journal of Applied Mechanics 1957;24:447. [7] Eason G. The elastic}plastic bending of a compressible curved bar. Applied Science Research 1960;A9:53. [8] Chakrabarty J. Theory of plasticity. New York: McGraw-Hill, 1987.