The paper proposes a novel approach for the interval limit analysis of rigid-perfectly plastic structures with (nonprobabilistic) uncertain but bounded forces and yield capacities that vary within given continuous ranges. The discrete model is constructed within a polygon-scaled boundary finite element framework, which advantageously provides coarse mesh accuracy even in the presence of stress singularities and complex geometry. The interval analysis proposed is based on a so-called convex model for the direct determination of both maximum and minimum collapse load limits of the structures involved. The formulation for this interval limit analysis takes the form of a pair of optimization problems, known as linear programs with interval coefficients (LPICs). This paper proposes a robust and efficient reformulation of the original LPICs into standard nonlinear programming (NLP) problems with bounded constraints that can be solved using any NLP code. The proposed NLP approach can capture, within a single step, the maximum collapse load limit in one case and the minimum collapse load limit in the other, and thus eliminates the need for any combinatorial search schemes.

References

References
1.
Kamenjarzh
,
J. A.
,
1996
,
Limit Analysis of Solids and Structures
,
CRC Press
,
Boca Raton
.
2.
Elishakoff
,
I.
, and
Soize
,
C.
,
2012
,
Nondeterministic Mechanics
,
Springer
,
New York
.
3.
Möller
,
B.
, and
Beer
,
M.
,
2008
, “
Engineering Computation Under Uncertainty—Capabilities of Non-Traditional Models
,”
Comput. Struct.
,
86
(
10
), pp.
1024
1041
.10.1016/j.compstruc.2007.05.041
4.
Schuëller
,
G. I.
,
2007
, “
On the Treatment of Uncertainties in Structural Mechanics and Analysis
,”
Comput. Struct.
,
85
, pp.
235
243
.10.1016/j.compstruc.2006.10.009
5.
Alibrandi
,
U.
, and
Ricciardi
,
G.
,
2005
, “
Bounds of the Probability of Collapse of Rigid-Plastic Structures by Means of Stochastic Limit Analysis
,”
Proceedings of the 9th International Conference on Structural Safety and Reliability
,
Rome, Italy
, June 19–23.
6.
Alibrandi
,
U.
, and
Ricciardi
,
G.
,
2008
, “
The Use of Stochastic Stresses in the Static Approach of Probabilistic Limit Analysis
,”
Int. J. Numer. Methods Eng.
,
73
(
6
), pp.
747
782
.10.1002/nme.v73:6
7.
Augusti
,
G.
,
Baratta
,
A.
, and
Casciati
,
F.
,
1984
,
Probabilistic Methods in Structural Engineering
,
Chapman & Hall
,
London
.
8.
Staat
,
M.
, and
Heitzer
,
M.
,
2001
, “
LISA—A European Project for FEM-Based Limit and Shakedown Analysis
,”
Nucl. Eng. Des.
,
206
(
2–3
), pp.
151
166
.10.1016/S0029-5493(00)00415-5
9.
Staat
,
M.
, and
Heitzer
,
M.
,
2003
, “Part VII. Probabilistic Limit and Shakedown Problems,”
Numerical Methods for Limit and Shakedown Analysis—Deterministic and Probabilistic Problems
,
M.
Staat
, and
M.
Heitzer
, eds., (NIC Series, Vol. 
15
), pp.
217
268
.
10.
Ben-Haim
,
Y.
, and
Elishakoff
,
I.
,
1990
,
Convex Models of Uncertainty in Applied Mechanics
,
Elsevier Science
,
Amsterdam
.
11.
Song
,
Ch.
, and
Wolf
,
J. P.
,
1997
, “
The Scaled Boundary Finite-Element Method—Alias Consistent Infinitesimal Finite-Element Cell Method—For Elastodynamics
,”
Comput. Methods Appl. Mech. Eng.
,
147
(
1–2
), pp.
329
355
.10.1016/S0045-7825(97)00021-2
12.
Ooi
,
E. T.
,
Song
,
Ch.
,
Tin-Loi
,
F.
, and
Yang
,
Z.
,
2012
, “
Polygon Scaled Boundary Finite Elements for Crack Propagation Modelling
,”
Int. J. Numer. Methods Eng.
,
91
(
3
), pp.
319
342
.10.1002/nme.v91.3
13.
Chiong
,
I.
,
Ooi
,
E. T.
,
Song
,
Ch.
, and
Tin-Loi
,
F.
,
2014
, “
Scaled Boundary Polygons With Application to Fracture Analysis of Functionally Graded Materials
,”
Int. J. Numer. Methods Eng.
,
98
(
8
), pp.
562
589
.10.1002/nme.v98.8
14.
Maier
,
G.
,
1970
, “
A Matrix Structural Theory of Piecewise Linear Elastoplasticity With Interacting Yield Planes
,”
Meccanica
,
5
(
1
), pp.
54
66
.10.1007/BF02133524
15.
Chinneck
,
J. W.
, and
Ramadan
,
K.
,
2000
, “
Linear Programming With Interval Coefficients
,”
J. Oper. Res. Soc.
,
51
(
2
), pp.
209
220
.10.1057/palgrave.jors.2600891
16.
Deeks
,
A. J.
, and
Wolf
,
J. P.
,
2002
, “
A Virtual Work Derivation of the Scaled Boundary Finite-Element Method for Elastostatics
,”
Comput. Mech.
,
28
(
6
), pp.
489
504
.
17.
Tangaramvong
,
S.
,
Tin-Loi
,
F.
, and
Senjuntichai
,
T.
,
2011
, “
An MPEC Approach for the Critical Post-Collapse Behavior of Rigid-Plastic Structures
,”
Int. J. Solids Struct.
,
48
(
19
), pp.
2732
2742
.10.1016/j.ijsolstr.2011.05.022
18.
Tangaramvong
,
S.
,
Tin-Loi
,
F.
,
Wu
,
D.
, and
Gao
,
W.
,
2013
, “
Mathematical Programming Approaches for Obtaining Sharp Collapse Load Bounds in Interval Limit Analysis
,”
Comput. Struct.
,
125
, pp.
114
126
.10.1016/j.compstruc.2013.04.028
19.
Shaocheng
,
T.
,
1994
, “
Interval Number and Fuzzy Number Linear Programmings
,”
Fuzzy Sets Syst.
,
66
(
3
), pp.
301
306
.10.1016/0165-0114(94)90097-3
20.
Drud
,
A. S.
,
1994
, “
CONOPT—A Large-Scale GRG Code
,”
ORSA J. Comput.
,
6
, pp.
207
216
.10.1287/ijoc.6.2.207
21.
Brooke
,
A.
,
Kendrick
,
D.
,
Meeraus
,
A.
, and
Raman
,
R.
,
1998
,
GAMS: A User’s Guide
,
GAMS Development Corporation
,
Washington, DC
.
22.
Ferris
,
M. C.
,
1998
, “
MATLAB and GAMS: Interfacing Optimization and Visualization Software
,”
Computer Sciences Department, University of Wisconsin
, Madison, WI, Technical Report TR98-19.
23.
Sloan
,
S. W.
, and
Kleeman
,
P. W.
,
1995
, “
Upper Bound Limit Analysis Using Discontinuous Velocity Fields
,”
Comput. Methods Appl. Mech. Eng.
,
127
(
1–4
), pp.
293
314
.10.1016/0045-7825(95)00868-1
24.
Vicente da Silva
,
M.
, and
Antão
,
A. N.
,
2007
, “
A Non-Linear Programming Method Approach for Upper Bound Limit Analysis
,”
Int. J. Numer. Methods Eng.
,
72
(
10
), pp.
1192
1218
.10.1002/(ISSN)1097-0207
25.
Tangaramvong
,
S.
,
Tin-Loi
,
F.
, and
Song
,
C.
,
2012
, “
A Direct Complementarity Approach for the Elastoplastic Analysis of Plane Stress and Plane Strain Structures
,”
Int. J. Numer. Methods Eng.
,
90
(
7
), pp.
838
866
.10.1002/nme.v90.7
26.
Simo
,
J. C.
, and
Rifai
,
M. S.
,
1990
, “
A Class of Mixed Assumed Strain Methods and the Method of Incompatible Modes
,”
Int. J. Numer. Methods Eng.
,
29
(
8
), pp.
1595
1638
.10.1002/(ISSN)1097-0207
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