1 Introduction

Conventional numerical methods in the limit and shakedown analysis are mostly based on the mathematical programming such as interior-point method or second-order cone programming (SOCP) [1, 2]. Numerical methods based on the mathematical programming are successfully implemented, combining the finite element software with the optimization packages. For example, Simon and Weichert [2] used the commercial finite element software ANSYS for elastic stress computation as well as the software package IPSA for numerical optimization and compared obtained results with ones by various packages including LANCELOT, IPDCA and IPOPT.

The linear matching method (LMM) originally proposed by Ponter et al. [3, 4] makes it possible to conduct the limit analysis using only finite element analysis but not using the mathematical programming. The LMM of which the iterative process can be considered as the nonlinear mathematical programming in essence is a powerful method for limit analysis via the iteration of linear elastic FEAs. It has been proved theoretically that the LMM could offer a monotonically reducing sequence of upper bound.

Even though the LMM has been widely used for the limit analysis of metal structures [5,6,7] as well as composite materials [8,9,10,11] subjected to thermal and mechanical loading and implemented by using the finite element software such as ABAQUS [6, 12] and ADINA [9], it has an important disadvantage that the lower bound computed in iterative process could not be used for the evaluation of error of upper bound.

This is due to the fact that solution of lower bound may be seriously oscillated in iteration process in spite that lower bound converges to upper one as the iteration number increases. Moreover, the evaluation of lower bound needs much of iterations for its convergence. To overcome this difficulty, Pisano et al. [9,10,11] proposed to use the LMM for the evaluation of upper bound and the Elastic Compensation Method (ECM) for the computation of lower bound. Nevertheless, in author’s opinion, their method for obtaining the lower bound by the ECM and evaluating the upper one by LMM may be failed to compute lower bound and upper one simultaneously and to evaluate the error between lower bound and upper one in real-time in iterative process due to the difference in both algorithms. In addition, this method needs nearly the double size of computational cost.

In this paper, a simple method is proposed in order to improve the numerical solution of lower bound by the LMM. The Young’s modulus varying spatially is determined in every iteration such that not only stress state lies on the yielding surface but also strain state does not exceed a certain value of reference strain, leading to the evaluation of lower bound based on strain state but not stress one. The proposed method can improve the numerical solution of lower bound by LMM without any affection on the upper bound. The lower bound newly evaluated based on the strain is always smaller than the upper bound in the iterative process and finally equal to the upper bound after sufficient numbers of iteration go on. Some numerical examples are considered in order to confirm the effectiveness of proposed approach. Numerical examples show the validity and the improvement of numerical accuracy of our approach. It should be mentioned that our approach can predict lower bound and upper one simultaneously within the framework of only the LMM without using the ECM.

This paper is organized as follows: The basic formulation of LMM and its algorithm is briefly introduced in Sect. 2. Section 3 describes a modified algorithm of LMM and proofs of some lemmas. Several numerical problems including plane problem, axisymmetric problem and three dimensional problem are illustrated in Sect. 4 for the validation of proposed approach. Finally, concluding remarks are presented in Sect. 5.

2 Linear matching method

2.1 Upper and lower bound theorem

The material is assumed to be rigid/perfectly-plastic which follows the von Mises yielding condition. It is assumed that a body occupies the volume V with the surface S where a traction is given as zero or \(P\cdot p_i (\mathbf{x})\) on \(S_T \) and displacement \({\bar{u}}_i =0\) is specified on \(S_u (S=S_T +S_u)\). Here, P is a scalar parameter defining relative magnitude of applied load as compared with a reference load \(p_i \). The lower and upper bound theorem of limit load can be postulated as follows, respectively [3].

Lower bound theorem

If, for the external load \(P=P_{\mathrm{LB}}\), there exists a statically possible stress field \(\sigma _{{\textit{ij}}}^*\) such that

$$\begin{aligned} f\left( {\sigma _{{\textit{ij}}}^*} \right) \le \sigma _y \end{aligned}$$
(1)

at every point within V, then \(P_\mathrm{L} \ge P_{\mathrm{LB}}\). Here, f is a von Mises yield function and \(\sigma _y \) is the uniaxial yield stress. Thus, one can know that a maximum value of \(P_{\mathrm{LB}}\) becomes lower bound of limit load \(P_\mathrm{L} \).

Upper bound theorem

If, for the external load \(P=P_{\mathrm{UB}}\), there exists a kinematically possible displacement rate field \({\dot{u}}_i^*\) and its corresponding strain rate field \({\dot{\varepsilon }}_{{\textit{ij}}}^*\) such that

$$\begin{aligned} P_{\mathrm{UB}} \int _{S_T } {p_i \dot{u}_i^*=\int _V {\sigma _{{\textit{ij}}}^{p*}}} {\dot{\varepsilon }}_{{\textit{ij}}}^*\mathrm{d}V \end{aligned}$$
(2)

where \(\sigma _{{\textit{ij}}}^{p*}\) is a stress point at yield associated with \({\dot{\varepsilon }}_{{\textit{ij}}}^*\), then satisfies \(P_\mathrm{L} \le P_{\mathrm{UB}} \). Hence, one can know that a minimum value of \(P_{\mathrm{UB}} \) becomes upper bound of limit load \(P_\mathrm{L} \).

According to above theorems, the upper and lower bound of limit load can be extracted by reducing into the optimization problem, respectively.

2.2 Conventional linear matching method

For the LMM, the Young’s modulus is changed spatially such that the stress field corresponding to a certain kind of kinetically possible strain field is placed on the yielding surface at every point of material. The Poisson’s ratio is recommended to take the value close to 0.5 in order to account for the plastic incompressibility, in general [3, 4]. Thus, we adopt the Poisson’s ratio as 0.4999999 in this work.

The LMM algorithm could be formulated as follows [3].

  • Initialization: Set \(P_{\mathrm{UB}}^0 =1\) and \(E^{1}( \mathbf{x})=E\).

  • kth iteration:

The linear elastic analysis with the Young’s modulus of \(E^{k}(\mathbf{x})\) is performed under a load of p and as a result, \(\sigma _{{\textit{ij}}}^k \), \(\varepsilon _{{\textit{ij}}}^k \) and \(u_i^k \) is obtained, respectively. And then, lower and upper bounds of the limit load at kth iteration are evaluated as

$$\begin{aligned} P_{\mathrm{LB}}^k= & {} \frac{\sigma _y }{\max \left( {\sigma _{\mathrm{eq}} \left( {\sigma _{{\textit{ij}}}^k } \right) } \right) } \end{aligned}$$
(3)
$$\begin{aligned} P_{\mathrm{UB}}^k= & {} \frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}} \left( {\varepsilon _{{\textit{ij}}}^k } \right) } \mathrm{d}V}{\int _{S_T } {p_i u_i^k \mathrm{d}S} } \end{aligned}$$
(4)

where \(\sigma _{\mathrm{eq}} \) and \(\varepsilon _{\mathrm{eq}} \) denote the equivalent stress and equivalent strain, respectively.

The \(E^{k+1}\) at \(k+1\hbox {th}\) iteration is taken as follows.

$$\begin{aligned} E^{k+1}=\frac{\sigma _y }{\varepsilon _{\mathrm{eq}} \left( {\varepsilon _{{\textit{ij}}}^k } \right) } \end{aligned}$$
(5)

Equation (5) gives \(E^{k+1}\) at \(k+1\hbox {th}\) iteration such that the stress field corresponding to strain field \(\varepsilon _{{\textit{ij}}}^k\) obtained at \(k\hbox {th}\) iteration lies on the yielding surface.

3 A modified linear matching method

We update the Young’s modulus at \(k+1\hbox {th}\) iteration as

$$\begin{aligned} {\tilde{E}}^{k+1}=\frac{E_{{\mathrm{ref}}}}{E_{\min }^{k+1} }E^{k+1} \end{aligned}$$
(6)

where \({\tilde{E}}^{k+1}\) is the Young’s modulus, \(E^{k+1}\) the one obtained by Eq. (5), \(E_{\min }^{k+1} =\min ({E^{k+1}( \mathbf{x})})\). \(E_{{\mathrm{ref}}} \), being an arbitrary constant, is taken as actual Young’s modulus in this work, in convenience.

The lower bound at \(k\hbox {th}\) iteration is newly evaluated as follows but not Eq. (3).

$$\begin{aligned} \bar{{P}}_{\mathrm{LB}}^k =\frac{\varepsilon _y }{\max \left( {\varepsilon _{\mathrm{eq}} \left( {\varepsilon _{{\textit{ij}}}^k } \right) } \right) } \end{aligned}$$
(7)

Here, \(\varepsilon _y ={\sigma _y }/{E_{{\mathrm{ref}}} }\).

We denote the upper and lower bounds evaluated according to Eqs. (3) and (4) by \({\tilde{P}}_{\mathrm{UB}}^k \) and \({\tilde{P}}_{\mathrm{LB}}^k \), respectively, at kth iteration of the LMM with Young’s modulus computed by Eq. (6). To distinguish this LMM from the conventional one, we call \({\tilde{P}}_{\mathrm{UB}}^k \) and \({\tilde{P}}_{\mathrm{LB}}^k \) the modified upper and lower bound, respectively.

To make clear the meaning of Eq. (6), we assume two distributions of Young’s modulus with \(E_{1} ( \mathbf{x})\) and \(E_{2} ( \mathbf{x})=\alpha E_{1} ( \mathbf{x})\) for the same structure, respectively. Here, \(\alpha \) is a constant. Denoting the stress and strain field corresponding to \(E_{1} ( \mathbf{x})\) as \(\sigma _1\) and \(\varepsilon _1 \), and \(E_{2} ( \mathbf{x})=\alpha E_{1} ( \mathbf{x})\) as \(\sigma _{2} \) and \(\varepsilon _{2} \), it is self-evident that one can get following relation according to the elasticity theory.

$$\begin{aligned} \sigma _{2} =\sigma _{1} ,\quad \varepsilon _{2} =\frac{1}{\alpha }\varepsilon _2 \end{aligned}$$
(8)

Equation (8) implies that only the strain field decreases (or increases) by \(\alpha \) times without the change in stress field when the distribution of Young’s modulus increases (or decreases) by \(\alpha \) times. Equation (6) could be rewritten as follows.

$$\begin{aligned} {\tilde{E}}^{k+1}=\frac{E_{{\mathrm{ref}}} }{E_{\min }^{k+1} }E^{k+1}=\frac{\max \varepsilon _{\mathrm{eq}}^k }{\varepsilon _y }E^{k+1}=\alpha E^{k+1} \end{aligned}$$
(9)

Equation (9) implies that our modified LMM yields the elasticity modulus \({\tilde{E}}^{k+1}\), being the Young’s modulus \(E^{k+1}\) in the classical LMM multiplied by a certain constant. From Eqs. (9) and (8), considering expression (5) of the Young’s modulus in the classical LMM, one can know that \(\alpha \) scales the strain field obtained at \(k\hbox {th}\) iteration lest the equivalent strain exceeds a certain value of strain, \(\varepsilon _y\), and that \(E^{k+1}\) makes the stress field to lie on the yielding surface for such a scaled strain field. This is shown schematically from Fig. 1.

Fig. 1
figure 1

Modified LMM

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Furthermore, from \(E_{\min }^{k+1} \le E^{k+1}\), one can get following relation.

$$\begin{aligned} {\tilde{E}}^{k+1}\ge E_{{\mathrm{ref}}} \end{aligned}$$
(10)

Based on above discussions, we can get three comments as follows.

First comment The modified upper bound and lower bound, \({\tilde{P}}_{\mathrm{UB}}^k \) and \({\tilde{P}}_{\mathrm{LB}}^k \), are equal to the conventional upper bound and lower one, \(P_{\mathrm{UB}}^k \) and \(P_{\mathrm{LB}}^k \), respectively. Here, \({\tilde{P}}_{\mathrm{UB}}^k \) and \({\tilde{P}}_{\mathrm{LB}}^k \) means the upper bound and the lower one evaluated by Eqs. (4) and (3) in our modified LMM, respectively.

This comment can be proved as follows.

According to Eq. (9), the modified Young’s modulus, \({\tilde{E}}^{k}\), is expressed as

$$\begin{aligned} {\tilde{E}}^{k}=\alpha E^{k}. \end{aligned}$$
(11)

From Eq. (8), one can know that stress field, \({{\tilde{\sigma }}}_{{\textit{ij}}}^k \), strain field, \({\tilde{\varepsilon }}_{{\textit{ij}}}^k\), and displacement field, \({\tilde{u}}_i^k\), are related to the conventional \(\sigma _{{\textit{ij}}}^k \), \(\varepsilon _{{\textit{ij}}}^k\) and \(u_i^k \) as follows.

$$\begin{aligned} {\tilde{\sigma }}_{{\textit{ij}}}^k =\sigma _{{\textit{ij}}}^k , \quad \tilde{\varepsilon }_{{\textit{ij}}}^k =\frac{1}{\alpha }\varepsilon _{{\textit{ij}}}^k , \quad {\tilde{u}}_i^k =\frac{1}{\alpha }u_i^k \end{aligned}$$
(12)

Substituting the second and the third in Eq. (12) into Eq. (4), one can obtain the following expression.

$$\begin{aligned} {\tilde{P}}_{\mathrm{UB}}^k =\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}} \left( {\tilde{\varepsilon }_{{\textit{ij}}}^k } \right) } \mathrm{d}V}{\int _{S_T } {p_i {\tilde{u}}_i^k \mathrm{d}S} }=\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}} \left( {{\varepsilon _{{\textit{ij}}}^k }\big /\alpha } \right) } \mathrm{d}V}{\int _{S_T } {p_i {u_i^k }\big /\alpha \mathrm{d}S} }=\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}} \left( {\varepsilon _{{\textit{ij}}}^k } \right) } \mathrm{d}V}{\int _{S_T } {p_i u_i^k \mathrm{d}S} }=P_{\mathrm{UB}}^k \end{aligned}$$
(13)

Thus, the convergence of modified upper bound, \({\tilde{P}}_{\mathrm{UB}}^k \), can be proved naturally by the convergence of upper bound in the conventional LMM.

Likewise, substituting the first in Eq. (12) into Eq. (3) yields following relation.

$$\begin{aligned} {\tilde{P}}_{\mathrm{LB}}^k =P_{\mathrm{LB}}^k \end{aligned}$$
(14)

Second comment The newly evaluated lower bound \(\bar{{P}}_{\mathrm{LB}}^k \) is smaller than or equal to the upper bound \(P_{\mathrm{UB}}^k \).

This comment can be proved as follows.

According to Clapeyron’s theorem of the elasticity theory, the work done by external surface traction \(p_i \) is twice the strain potential energy as great as it is [13]. Namely,

$$\begin{aligned} \int _{S_T } {p_i u_i^k } =\int _V {\sigma _{{\textit{ij}}}^k \varepsilon _{{\textit{ij}}}^k \mathrm{d}V=} \int _V {\sigma _{\mathrm{eq}}^k \varepsilon _{\mathrm{eq}}^k \mathrm{d}V=} \int _V {{\tilde{E}}^{k}\varepsilon _{\mathrm{eq}}^k \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} \end{aligned}$$
(15)

Substituting Eq. (15) into Eq. (4), one gets the following.

$$\begin{aligned} P_{\mathrm{UB}}^k =\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {{\tilde{E}}^{k}\varepsilon _{\mathrm{eq}}^k \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} } \end{aligned}$$
(16)

Introducing \(\varepsilon _{\mathrm{eq}\max }^k =\max \left( {\varepsilon _{\mathrm{eq}}^k ( \mathbf{x})} \right) \) and considering \({\tilde{E}}^{k}\ge E_{{\mathrm{ref}}}\) at all points according to Eq. (10), following relation can be obtained.

$$\begin{aligned} P_{\mathrm{UB}}^k =\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {{\tilde{E}}^{k}\varepsilon _{\mathrm{eq}}^k \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }\ge \frac{\varepsilon _y }{\varepsilon _{\mathrm{eq}\max }^k }\frac{\int _V {E_{{\mathrm{ref}}} \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {{\tilde{E}}^{k}\varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }\ge \frac{\varepsilon _y }{\varepsilon _{\mathrm{eq}\max }^k }\frac{\int _V {\varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {\varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }\ge \frac{\varepsilon _y }{\varepsilon _{\mathrm{eq}\max }^k }=\bar{{P}}_{\mathrm{LB}}^k \end{aligned}$$
(17)

Namely, the newly evaluated lower bound \(\bar{{P}}_{\mathrm{LB}}^k \) is smaller than or equal to the upper bound \(P_{\mathrm{UB}}^k \).

In similar way, one can prove that the lower bound \(P_{\mathrm{LB}}^k \) evaluated by Eq. (3) is smaller than or equal to the upper bound \(P_{\mathrm{UB}}^k\).

In fact, combining Eq. (15) with Eq. (4), following relation is obtained.

$$\begin{aligned} P_{\mathrm{UB}}^k =\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {\sigma _{\mathrm{eq}}^k \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }\ge \frac{\sigma _y }{\max \left( {\sigma _{\mathrm{eq}}^k } \right) }\frac{\int _V {\varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {\varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }\ge \frac{\sigma _y }{\max \left( {\sigma _{\mathrm{eq}}^k } \right) }=P_{\mathrm{LB}}^k \end{aligned}$$
(18)

Third comment The newly evaluated lower bound evaluated by Eq. (7) converges to \(P_{\mathrm{LB}}^k \) and \(P_{\mathrm{UB}}^k \) as \(k\rightarrow \infty \). Namely, \(\bar{{P}}_{\mathrm{LB}}^k \rightarrow P_{\mathrm{LB}}^k \) and \(\bar{{P}}_{\mathrm{LB}}^k \rightarrow P_{\mathrm{UB}}^k \), \(k\rightarrow \infty \).

This comment can be proved as follows.

After performing sufficient numbers of iterations, let \(\varepsilon _{\mathrm{eq}}^{k-1} =\varepsilon _{\mathrm{eq}}^k\).

According to Eqs. (5) and (6),

$$\begin{aligned} {\tilde{E}}^{k}= & {} \frac{E_{{\mathrm{ref}}} }{\min \frac{\sigma _y }{\varepsilon _{\mathrm{eq}}^{k-1} }}\cdot \frac{\sigma _{\mathrm{y}} }{\varepsilon _{\mathrm{eq}}^{k-1} }=E_{{\mathrm{ref}}} \frac{\max \varepsilon _{\mathrm{eq}}^{k-1} }{\varepsilon _{\mathrm{eq}}^{k-1} }=E_{{\mathrm{ref}}} \frac{\max \varepsilon _{\mathrm{eq}}^k }{\varepsilon _{\mathrm{eq}}^k } \end{aligned}$$
(19)
$$\begin{aligned} \sigma _{\mathrm{eq}}^k= & {} {\tilde{E}}^{k}\varepsilon _{\mathrm{eq}}^k =E_{{\mathrm{ref}}} \frac{\max \varepsilon _{\mathrm{eq}}^k }{\varepsilon _{\mathrm{eq}}^k }\varepsilon _{\mathrm{eq}}^k =E_{{\mathrm{ref}}} \max \varepsilon _{\mathrm{eq}}^k . \end{aligned}$$
(20)

Substituting Eq. (20) into Eq. (3), one can obtain the following relation.

$$\begin{aligned} P_{\mathrm{LB}}^k =\frac{\sigma _y }{\max \sigma _{\mathrm{eq}}^k }=\frac{\sigma _y }{E_{{\mathrm{ref}}} \max \varepsilon _{\mathrm{eq}}^k }=\frac{\varepsilon _y }{\max \varepsilon _{\mathrm{eq}}^k }=\bar{{P}}_{\mathrm{LB}}^k \end{aligned}$$
(21)

This means that the lower bound evaluated by Eq. (7) satisfies the lower bound theorem after performing sufficient numbers of iterations.

Meanwhile, one can easily prove that the lower bound \(\bar{{P}}_{\mathrm{LB}}^k \) evaluated by Eq. (7) converges to the upper bound. In fact, combining Eqs. (20), (15) and (4), one can get the following relation.

$$\begin{aligned} P_{\mathrm{UB}}^k =\frac{\int _V {\sigma _y \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }{\int _V {E_{{\mathrm{ref}}} \max \left( {\varepsilon _{\mathrm{eq}}^k } \right) \varepsilon _{\mathrm{eq}}^k \mathrm{d}V} }=\frac{\sigma _y }{E_{{\mathrm{ref}}} \max \left( {\varepsilon _{\mathrm{eq}}^k } \right) }=\frac{\varepsilon _y }{\max \left( {\varepsilon _{\mathrm{eq}}^k } \right) }=\bar{{P}}_{\mathrm{LB}}^k \end{aligned}$$
(22)

Therefore, the lower bound \(\bar{{P}}_{\mathrm{LB}}^k \) satisfies the lower bound theorem and becomes the upper bound in the limit state of \(\varepsilon _{\mathrm{eq}}^{k-1} =\varepsilon _{\mathrm{eq}}^k \).

4 Numerical examples

As discussed above, the LMM is based on the analysis of elastic problem considering the incompressibility with the Young’s modulus varying spatially. In order to perform the elastic analysis which is necessary for the implementation of LMM, the commercially available FE code ANSYS was used in this paper. The incompressibility of material can be considered sufficiently by using the element integration technology and the element formulae in ANSYS which are appropriate for the incompressible property. The Young’s modulus varying spatially can be defined by using UserMat, ANSYS subroutine. The value of Young’s modulus to be used in current iteration is saved in the state variable of UserMat in the previous iteration. Once the current iteration for implementing the LMM is finished, the upper bound is evaluated by using Eq. (4) and the lower bound by using Eq. (3) or (7). Next, the analysis process is restarted in order to continue the next iteration.

4.1 Square plate with a hole in its center

The square plate with a hole in its center subjected to the uniform pressure \(p=\sigma _y \) is considered as shown from Fig. 2. The limit load factor is equal to 0.8 for plane stress condition with \(R/L=0.2\). The FE model consists of 5000 PLANE182 elements and 5151 nodal points. B-bar Method (Selective Reduced Integration) is used for the element integration. Figure 3 compares the upper bound and the lower one predicted using the modified LMM with the predictions evaluated by the ECM as the iteration number increases. In this figure, solid line denotes the upper bound, dashed line represents the lower bound evaluated using Eq. (3), dotted line shows the lower bound computed using Eq. (7), and dash-dot line indicates the lower bound predicted by ECM. As seen from Fig. 3, the evaluation of lower bound by modified LMM using Eq. (7) shows the best trend of convergence. Meanwhile, the prediction of lower bound by ECM also indicates the good trend of convergence, but the convergence rate in that case is smaller than in the case using Eq. (7). Moreover, the computation of lower bound by LMM using Eq. (3) shows the convergence with local oscillations.

Fig. 2
figure 2

Square plate with a hole in its center

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Fig. 3
figure 3

Convergence process of upper bound and the lower one versus iteration number

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Fig. 4
figure 4

Relative error of lower bound obtained using modified LMM and ECM as iterative process goes on

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Figure 4 shows the relative errors of lower bound evaluated using Eq. (7) and ECM against the analytical solution of 0.8, respectively.

Necessary parameters such as stress components, strain ones and Young’s modulus can be obtained at every iteration. Figure 5 shows the distribution of von Mises stress and equivalent strain evaluated at \(40{\mathrm{th}}\) iteration of modified LMM.

Fig. 5
figure 5

Distribution of von Mises stress and equivalent strain evaluated at \(40{\mathrm{th}}\) iteration of modified LMM: a von Mises stress; b equivalent strain

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4.2 Single-edge cracked plate subjected to tension (SE(T))

Figure 6 shows the geometry of single-edge cracked plate under tension. Assuming plane strain condition, the width b is equal to a half of the length 2L and \(a/b=0.6\) as well as \(b=1\) is applied. Here, a is a crack length. Figure 7 presents FE mesh applied. The FE model consists of 7301 PLANE182 elements and 7455 nodal points. Considering the incompressibility, B-bar Method (Selective Reduced Integration) and mixed u-p formulation are used.

Fig. 6
figure 6

Geometry of SE(T) specimen

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The analytical solution of limit load factor \(\lambda =p/{\sigma _y }\) in plane strain condition is expressed as

$$\begin{aligned}&\lambda =1.702\gamma \left\{ {\left[ {({0.206-x})^{2}+0.5876({1-x})^{2}} \right] ^{1/2}+( {0.206-x} )} \right\} \quad x>0.545 \end{aligned}$$
(23)
$$\begin{aligned}&\left. {\begin{array}{l} \lambda \ge \gamma ( {1-x-1.232x^{2}+x^{3}}) \\ \lambda \le \gamma \left( {1-x-1.232x^{2}+x^{3}+22x^{3}( {0.545-x})^{2}} \right) \\ \end{array}} \right\} \,\,x<0.545, \end{aligned}$$
(24)

where \(x=a/b\) [14, 15].

Here, \(\gamma \) is equals to \(2/{\sqrt{3}}\) for the von Mises yielding condition and \(2/{\sqrt{3}}\) for the Tresca yielding condition, respectively.

Fig. 7
figure 7

FE mesh

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According to Eq. (23), an analytical limit load factor is equals to 0.207 for this example. We computed the lower bound of limit load by using the LMM, the modified LMM and the ECM, respectively. Figure 8 compares the values of lower bound obtained using three different approaches as well as the upper bound as iterative process goes on.

Fig. 8
figure 8

Convergence process of upper and lower bound versus iteration number

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As seen from Fig. 8, the values of lower bound evaluated using the LMM show the some numerical oscillations, while the convergence rate of ECM is slower than that of modified LMM even though it converges to the upper bound.

Fig. 9
figure 9

Relative error of lower bound obtained using modified LMM and ECM versus iteration number

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Due to numerical error of finite element analysis, we assume that the ECM is the reliable approach for the evaluation of lower bound and thus choose a value of 0.2096 obtained via 500 iterations of the ECM as a reference value of lower bound. Figure 9 shows the relative error of lower bound evaluated using the modified LMM and the ECM against the reference value, respectively. As shown from this figure, the LMM always yields the smaller relative error than the ECM for the evaluation of lower bound of limit load.

4.3 Axisymmetric pressure vessel subjected to the internal pressure

The axisymmetric pressure vessel is studied, which has been investigated by many researchers [16, 17], as shown in Fig. 10. The geometrical dimension of \(R_b =4500\) mm, \(R_z =L=3000\) mm, \(l=658.2\) mm, \(R_k =360\) mm and \(s=225\) mm is assumed, respectively.

Fig. 10
figure 10

a Geometry of pressure vessel; b FE mesh for limit analysis

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For the internal pressure of \(p=\sigma _y \), the limit load factor is found as \(0.0746\sim 0.083\) in Ref [16] and as 0.078 in Ref [17], respectively.

Fig. 11
figure 11

Convergence process of lower bound and upper one versus iteration number

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Fig. 12
figure 12

Relative error of lower bound obtained using modified LMM and ECM versus iteration number

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The FE model consists of 384 PLANE182 elements and 485 nodal points. B-bar Method (Selective Reduced Integration) and mixed u-p formulation are used.

Figure 11 depicts the convergence process of lower bound evaluated using three different approaches as well as upper bound. As seen from this figure, the evaluation of lower bound by the modified LMM has the best trend of convergence. A lower bound of 0.0783 was predicted via 500 iterations using the ECM. This value was taken as a reference value, and the relative error was compared based on this value as shown in Fig. 12.

4.4 Pipe-junction under internal pressure

The 3D pipe-junction subjected to the internal pressure is considered for limit analysis as shown in Fig. 13. The geometrical dimension of \(D=39\) mm, \(d=15\) mm and \(s=t=3.4\) mm is assumed, respectively. For the internal pressure of \(p=\sigma _y \), a limit load factor is known as 0.1443 in Ref. [17] and as 0.134 in Ref. [18], respectively.

Fig. 13
figure 13

a FE mesh for the limit analysis; b geometry of 3D pipe-junction

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Fig. 14
figure 14

Convergence process of limit load factor for 3D pipe-junction

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Fig. 15
figure 15

Relative error of lower bound obtained using modified LMM and ECM versus iteration number

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The FE model consists of 2889 SOLID185 elements and 7037 nodal points. B-bar Method (Selective Reduced Integration) and mixed u-p formulation are used.

Figure 14 depicts the convergence process of upper bound and the lower one evaluated using three different approaches. As seen from Fig. 14, the evaluation of lower bound by the modified LMM has the best trend of convergence.

A lower bound of 0.1485 computed via 500 iterations using the ECM was used as the reference value for the comparison of relative error as shown in Fig. 15.

5 Conclusion

In this paper, we proposed a modified LMM and a newly defined lower bound for improving the convergence LMM. It was proved that this modified LMM never affects the numerical solution of lower and upper bounds of conventional LMM and that the newly evaluated lower bound satisfies the lower bound theorem and converges to the upper bound after sufficient numbers of iteration go on. Several numerical examples confirmed that the proposed algorithm is more accurate than other approaches. Moreover, our approach can effectively predict the upper bound as well as the lower one simultaneously only using LMM.

It should be noted that the current method can be applied effectively to the limit analysis of large scale problems in practice because it can be implemented using ANSYS UserMat and moreover the limit analysis is performed like the general elastic FEA in ANSYS.