Shakedown and collapse of elastic-Plastic structures under variable and cyclic loads

Vietnam Journal of Mechanics, VAST, Vol. 30, No. 4 (2008), pp. 233 – 240 Special Issue of the 30th Anniversary SHAKEDOWN AND COLLAPSE OF ELASTIC-PLASTIC STRUCTURES UNDER VARIABLE AND CYCLIC LOADS Pham Duc Chinh Institute of Mechanics, 264 Doi Can, Hanoi Abstract. The paper presents an introduction to the results obtained mainly by the author in shakedown analysis of elastic plastic structures, with the shakedown theorems for elastic plastic kinematic hardening materials playing the central role. Some illustrative examples of application are given. 1. INTRODUCTION Plastic collapse, along with Euler elastic instability of geometrically-thin structural members and Griffiths critical crack fracture of brittle-like solids, are among the out- standing phenomena limiting the load-bearing capacity of structures. Attainment of the ultimate yield state at a point within a structure does not imply plastic collapse but that at entire sections to make the structure a mechanism, which could not sustain the larger loads, does. Plastic instantaneous collapse of structures is determined by the static and kinematic theorems of plastic limit analysis. A structure subjected to loads varying ar- bitrarily within certain limits may fails even at lower load limits than those set by the instantaneous collapse criterion because the plastic deformations would accumulate unre- strictively or be bounded but vary cyclically and unceasingly. Otherwise, the total plastic dissipation in the structure would be bounded, hence its overall response to the exter- nal agencies should shake down to some purely elastic state due to a finite residual stress field developed inside the structure. Classical shakedown theory for elastic-perfectly plastic bodies has been well established in the literature. As for the plastic limit theory, the essen- tial contents of the classical shakedown theory are its dual static and kinematic theorems, which are path-independent (loading-history-independent): both theorems determine the shakedown boundary in the loading space under which a loaded structure should be safe regardless of the loading history, while it should fail if the boundary is allowed to be vi- olated unlimitedly. The shakedown design is not only safer its limiting case - the plastic limit one, but also applies to the larger class of dynamic problems, which lie outside the reach of the plastic limit theorems. In fact, under dynamic loading, a structure would never fail instantaneously, thank to the inertia effect, but fail incrementally (monotonically or nonmonotically) or plastic-cyclically with time. 234 Pham Duc Chinh 2. SHAKEDOWN THEOREMS Originally Melan-Koiter shakedown theorems have been constructed for idealistic elastic perfectly plastic material model, which is practical and suffice for plastic limit analy- sis. For shakedown analysis, much more realistic and practical material model is the elastic plastic kinematic hardening one involving Bauschinger effect. The hardening curve is gen- erally nonlinear, plastic-deformation-history dependent, and is bounded by the initial and ultimate yield stresses. Shakedown static and kinematic theorems have been constructed for the kinematic hardening material satisfying a realistic positive hysteresis postulate, which do not involve the plastic-deformation-history dependent hardening curve, except the initial yield stress and ultimate yield strength, keeping the path-independent spirit of classical Melan-Koiter theorems (Pham, 2007, 2008). Let σe(x, t) denote the fictitious elastic stress response of the body V (under the assumption of its perfectly elastic behaviour) to external agencies over a period of time (x ∈ V, t ∈ [0, T ]), called a loading history. The actions of all kinds of external agencies upon V can be expressed explicitly through σe. At every point x ∈ V , the elastic stress response σe(x, t) is confined to a bounded time-independent domain with prescribed limits in the stress space, called a local loading domain Lx. As a field over V , σe(x, t) belongs to the time-independent global loading domain L: L = {σe | σe(x, t) ∈ Lx, x ∈ V, t ∈ [0, T ]}. (1) In the spirit of shakedown theorems, the bounded loading domain L, instead of a particular loading history σe(x, t), is given a priori. Shakedown of a body in L means it shakes down for all possible loading histories σe(x, t) ∈ L . ks denotes the shakedown safety factor: at ks > 1 the structure will shake down, while it will not at ks < 1, and ks = 1 defines the boundary of the shakedown domain. Shakedown static theorem ks = min {U¯, C¯} , (2) where U¯ = sup ρ∈R { k | k(ρ+ σe) ∈ Yu , ∀σ e ∈ L} , (3) C¯ = sup ρ′ { k | k(ρ′ + σe) ∈ Yi , ∀σ e ∈ L} , (4) R is the set of admissible time-independent self-equilibrated residual stress fields ρ(x) that satisfy homogeneous equilibrium equations on V ; ρ′ is a time-independent stress field that is not requied to be self-equilibrated; Yu designates the elastic domain in the stress space that is bounded by the yield surface determined by the ultimate yield stress σuY , while Yi is the respective domain bounded by the yield surface determined by the initial yield stress σiY . Shakedown kinematic theorem k−1s = max {U, C} , (5) Shakedown and collapse of elastic-plastic structures under variable and cyclic loads 235 where U = sup e p∈A;σe∈L ∫ T 0 dt ∫ V σ e : epdV ∫ T 0 dt ∫ V Du(ep)dV , (6) C = sup x∈V ;σe∈L;eˆp;ρ′ (σe + ρ′) : eˆp Di(eˆp) , (7) A is the set of compatible-end-cycle plastic strain rate fields ep over the time cycles 0 ≤ t ≤ T : A = {ep | εp = ∫ T 0 e pdt ∈ C} ; (8) C is the set of compatible plastic strain increment fields on V ; eˆp and ρ′ are plastic strain rate and time-independent stress fields that are not required to satisfy any compatibility and equilibrium constraints; D(ep) is the dissipation function determined by the yield stress σY and the respective yield criterion, e.g. for a Mises material D(ep) = √ 2/3σY (e p : ep)1/2, (9) while for a Tresca material: D(ep) = 1 2 σY (| e p 1 | + | e p 2 | + | e p 3 |) = σY max{| e p 1 |, | e p 2 |, | e p 3 |}, (10) ep1, e p 2, e p 3 are the principal plastic strain rates; Du(e p) and Di(e p) are the dissipation func- tions with σuY and σ i Y taking the places of σY , respectively. At U¯ > C¯ of criterion (2) [or U < C of criterion (5)] the nonshakedown collapse mode is cyclic, otherwise the incremental collapse mode prevails. For broad classes of problems, the following reduced kinematic theorem is useful (Pham and Stumpf, 1994; Pham, 2008) Reduced kinematic theorem k−1s = max {I, A} , (11) where I = sup σe∈L;εp∈C ∫ V max tx [σe(x, tx) : ε p(x)]dV ∫ V Du(εp)dV , (12) A = sup x∈V ;σe∈L;εˆ p ;t1,t2 [σe(x, t1)− σ e(x, t2)] : εˆ p(x) 2Di(εˆ p) . (13) 236 Pham Duc Chinh 3. SOME ILLUSTRATIVE EXAMPLES Aa a first example of application, let us consider a beam of rectangular cross-section of the size b× 2h, which is bended up and down alternatively with the moment ±M (fig. 1). The outermost layers on the upper and lower sides of the beam begin to yield at the moment M = M iY = σ i Y 2 3 bh2, while the ultimate yielding moment (the plastic limit load) of the beam is M = MuY = σ u Y bh 2. Application of eqs. (11)-(13) gives the obvious result: k−1s = max {I, A} = max {M/M u Y ,M/M i Y } = M/M i Y ; (14) hence at M = M iY (ks = 1), the beam fails because of alternating plasticity started from the failure of the beam’s outermost layers. M b 2 h Fig. 1. A beam in bending Next, consider a beam clamped at the two ends A and D (no horizontal kinematic constraint). The vertical point load P is applied and removed slowly, but alternatively, atB and then C for un unlimited number of times. Application of criterion (12) with an optimal incremental collapse mechanism ABC shown in fig. 2 yields the nonshakedown limit load: PSD = 8.1M u Y /l (the respective plastic limit load is significantly larger: PP = 9M u Y /l). P P A B C D l/3 l/3 l A B C Fig. 2. A clamped beam under sequential loads A thick-walled hollow sphere of inner and outer radii a and b is subjected to qua- sistatic internal pressure q, which may vary arbitrarily from 0 to the limit qU (fig. 3). Application of criterion (11)-(13) yields k−1s = max {I, A} = q U max{ 1 2σuY ln(b/a) , 3b3 4σiY (b 3 − a3) } . (15) Shakedown and collapse of elastic-plastic structures under variable and cyclic loads 237 q a b Fig. 3. A hollow sphere under variable pressure In this case, the incremental collapse load coincides with the plastic limit one, while the alternating plasticity happens at the inner radius of the hollow sphere. At last consider a disk of radius a clamped at the edge and subjected to uniform quasiperiodic dynamic pressure q = q0 + q1sinωt , (16) where q0 =const, q1 and ω are quasistatic functions of time varying between the limits 0 ≤ q1 ≤ q U 1 , 0 ≤ ω ≤ ωU < ωI , (17) ωI is the principal natural frequency of the structure. Denote k4 = hm D ω2 , k4U = hm D ω2U , D = Eh3 12(1− ν2) , (18) where E is the Young’s modulus, ν - the Poisson ratio, m - the mass density, and h is the thickness of the disk. From the result for the elastic perfectly plastic material (Pham, 1997), we deduce the corresponding result for kinematic hardening material k−1s = max {I, A} , (19) where I = a2 12MuY + qU1 2MuY | Pr(kU , a) | + qU1 2MuY a a∫ 0 | Pθ(kU , r) | dr , (20) A = 3qU1 2M iY | Pr(kU , a) | (21) (with A = 1, the alternating plasticity collapse mode happens at the disk’s perimeter), MuY = h2 4 σuY , M i Y = h2 4 σiY , (22) 238 Pham Duc Chinh Pr(k, r) = [ kI1(ka)J0(kr)− kJ1(ka)I0(kr)− 1− ν r I1(ka)J1(kr) + 1− ν r J1(ka)I1(kr) ] k−3 [ J0(ka)I1(ka) + J1(ka)I0(ka) ]−1 , Pθ(k, r) = [ νkI1(ka)J0(kr)− νkJ1(ka)I0(kr) + 1− ν r I1(ka)J1(kr) − 1− ν r J1(ka)I1(kr) ] k−3 [ J0(ka)I1(ka) + J1(ka)I0(ka) ]−1 , (23) J0, J1, I0, I1 are Bessel functions. As an illustration, we fix the value Q0 = q0 a2 2Mu Y = 4.6. A graphical display of the incremental collapse curve I = 1 and the alternating plasticity collapse ones A = 1 in the plane of dimensionless coordinates Q1 = qU1 a2 2Mu Y against K = kua, at various values of σiY /σ u Y = 1/1.5, 1/2, 1/3 (corresponding in order to, presumably, increasing numbers of loading cycles; presume σuY is fixed for the material) are presented in Fig. 4. The domain under both the curves I = 1 and A = 1, is the shakedown domain. At high numbers of cycles, σiY should be lowered toward the fatigue limit and the shakedown domain would be reduced correspondingly. With σiY /σ u Y = 1/2 , the collapse mode changes from the alternating plasticity one to the incremental one as K increases (at about K = 2.2). With σiY /σ u Y = 1/1.5 , the disk fails only incrementally, while with σ i Y /σ u Y = 1/3, the alternating plasticity mode is the one that prevails. Fig. 4. Incremental (I = 1) and alternaning plasticity (A = 1) collapse curves at various values of σu Y /σi Y , in the plane of dimensionless load amplitude Q1 and frequency K Shakedown and collapse of elastic-plastic structures under variable and cyclic loads 239 4. CLOSURE For more details about our results and application of the theorems in solving many practical engineering problems (mainly on elastic perfectly plastic structures) see the Ref- erences. Future directions: - Develope methods to solve problems for practical engineering structures made of realistic kinematic hardening materials, especially those subjected to dynamic loading. - Improve further the base of shakedown theory, and extend the shakedown analysis to various complicated material models. ACKNOWLEDGEMENTS The work is supported by the National Science Foundation of Vietnam. 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[31] Pham D.C., Shakedown and Collapse of Elastic-plastic Kinematic-hardening Structures. VIII-National Conference on Mechanics Hanoi- 12/2007, pp. 69-78. Received May 5, 2009 THÍCH NGHI VÀ HỎNG CỦA CÁC KẾT CẤU ĐÀN DẺO CHỊU TẢI TRỌNG THAY ĐỔI VÀ LẶP LẠI Bài viết giới thiệu một số kết quả chính tác giả nhận được trong phân tích thích nghi các kết cấu đàn dẻo, với các định lý thích nghi cho vật liệu đàn dẻo tái bền động học đóng vai trò trung tâm. Một số ví dụ áp dụng được trình bầy.