Stationary and extremum variational formulations for the elastostatics of cable networks. (English) Zbl 0587.73026
The static analysis of elastic cable networks submitted to generic, conservative loads and prescribed dislocations is considered in this paper. The cables are assumed as stress-unilateral (only tensile stresses are admitted), and represented according to a Lagrangian standpoint, under the customary large displacements hypothesis. A variational formulation of the problem is given, as the stationary of a saddle functional with respect to displacements and (sign-constrained) tractions. Uniqueness properties for the solution are derived from this statement, together with two complementary (constrained) minimum formulations, which correspond to the well-known extremum principles of the total potential energy and the complementary energy. The case of a network loaded only at the nodes is exposed as a specialization.
MSC:
| 74S30 | Other numerical methods in solid mechanics (MSC2010) |
| 74G30 | Uniqueness of solutions of equilibrium problems in solid mechanics |
| 74H25 | Uniqueness of solutions of dynamical problems in solid mechanics |
Keywords:
static analysis; elastic cable networks; generic, conservative loads; dislocations; stress-unilateral; large displacements hypothesis; stationary of a saddle functional; tractions; complementary (constrained) minimum formulations; extremum principles; total potential energyReferences:
| [1] | Thornton C.H., Birnstiel C.,Three-dimensional suspension structures, J. Struct. Div. A.S.C.E. 93, 247–270, (1967). |
| [2] | Saafan S.A.,Theoretical analysis of suspension roofs, J. Struct. Div. A.S.C.E. 96, 393–405 (1970). |
| [3] | Knudson W.C., Scordelis A.C.,Cable forces for desired shapes in cable-net structures, Proc. 1971 IASS Pacific Symposium Part II on Tension Structures and Space Frames, Tokyo and Kyoto, 93–102. |
| [4] | Bucholdt H.A., Mc Millan B.R.,Iterative methods for the solution of pretensioned cable structures and pinjoined assemblies having significant geometrical displacements, Proc. 1971 IASS Pacific Symposium Part II on Tension Structures and Space Frames, Tokyo and Kyoto, 305–316. |
| [5] | Argyris J.H., Scharpf D.W.,Large deformation analysis of prestressed networks, J. Struct. Div. A.S.C.E. 48, 633–654, (1972). |
| [6] | Panagiotopoulos P.D.,Stress-unilateral analysis of discretized cable and membrane structures in the presence of large displacements, Ing. Arch. Bd. 44, H.5, 291–300, (1975). · Zbl 0356.73056 · doi:10.1007/BF00534488 |
| [7] | Greenberg D.P.,Inelastic analysis of suspension roofstructures, J. Struct. Div. A.S.C.E. 96, 905–930, (1970). |
| [8] | Jonatowski J.J., Birnstiel C.,Inelastic stiffened suspension space structures, J. Struct. Div. A.S.C.E. 96, 1143–1166, (1970). |
| [9] | Murray T.A., Willems N.,Analysis of inelastic suspension structures, J. Struct. Div. A.S.C.E. 97, 2791–2806, (1971). |
| [10] | Contro R., Maier G., Zavelani A. Inelastic analysis of suspension structures by nonlinear programming, Comp. Meth. in Applied Mech. and Eng. 5, 127–143, (1975). · Zbl 0299.73035 · doi:10.1016/0045-7825(75)90050-X |
| [11] | Panagiotopoulos P.D.,A variational inequality approach to the inelastic stress-unilateral analysis of cable structures, Comput. Structures 6, 133–139, (1976). · Zbl 0326.73068 · doi:10.1016/0045-7949(76)90063-8 |
| [12] | O’Brien W.T.,General solution of suspended cable problems, J. Struct. Div. A.S.C.E. 93, 1–26, (1967). |
| [13] | H.H., Robinson A.R.,Continuous method of suspension bridge analysis, J. Struct. Div. Proc. A.S.C.E. 94, (1968). |
| [14] | Peyrot A.H., Goulois A.M.,Analysis of flexible transmission lines, J. Struct. Div. A.S.C.E. 104, 763–779, (1978). |
| [15] | Peyrot A.H., Goulois A.M.,Analysis of cable structures, Comput. Structures 10 (5), 805–813, (1979). · Zbl 0417.73065 · doi:10.1016/0045-7949(79)90044-0 |
| [16] | Leonard J.W., Recker W.W.,Nonlinear dynamics of cables with initial tension, J. Engng. Mech. Div., Proc. A.S.C.E. 98 (1972). |
| [17] | Jayaraman H.B., Knudson W.C.,A curved element for the analysis of cable structures, Comput. Structures 14 (3–4), 325–333, (1981). · doi:10.1016/0045-7949(81)90016-X |
| [18] | Hengold W.H., Russell J.J.,Equilibrium and natural frequencies of cable structures (a nonlinear finite element approach), Comput. Structures 6, 267–271, (1976). · Zbl 0332.73081 · doi:10.1016/0045-7949(76)90001-8 |
| [19] | Odzemir H.,A finite element approach for cable problems, Int. J. Solids Structures 15, 427–437, (1979). · Zbl 0404.73069 · doi:10.1016/0020-7683(79)90063-5 |
| [20] | Lemaire J.P.,Etude de systemes de cables, Annales de l’Inst. Techn. du Bat. et des Travaux Publics, No. 361, May 1978, 111–134. |
| [21] | Noble B., Sewell M.J.,On dual extremum principles in applied mathematics, J. Inst. Maths. Applics 9, 123–193, (1972). · Zbl 0236.49002 · doi:10.1093/imamat/9.2.123-a |
| [22] | Reissner E.,On a variational theorem for finite elastic deformations, J. Math. Physics 32, 129–135, (1953). · Zbl 0051.16303 |
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