Soft Lattice Truss Static Polynomial Response Using Energy Methods

A deterministic methodology is presented for developing closed-form dee ection equations for two-dimensional and three-dimensional lattice structures. Two types of lattice structures are studied: beams and soft lattices. Castigliano' s second theorem, which entails the total strain energy of a structure, is utilized to generate highly accurate results. Derived dee ection equations provide new insight into the bending and shear behavior of the two types of lattices, in contrast to classic solutions of similiar lattice truss structures. RUSSESare rigid skeletalframeworks utilized to provide sup- port for structures or equipment. They are generally composed of long slender members. Typically these members are joined with pin-connectors. Some attractive structural features of trusses are their low material-to-load-carrying characteristics relative to solid beams, ease of construction, and predictable behavior while incur- ring load. Truss designers rely on geometry, redundancy, and/or arch action to tailor and to optimize trusses for various load appli- cations. These and other design parameters play a crucial role in the performance of cranes, bridges, domes, and space-based structures. Various truss applications require designer evaluation of behavior foroperationalloading,vibrationalexcitations,orloadsduringtruss construction.Forpreliminaryanalysisorconceptualstudies,design- ers often study single-or double-layer planar latticetruss structures, which have regular and patterned geometries, to gain insight into theirstructuralbehavior.Latticestructuresareattractiveforstiffness and vibration analysis methods because their repetitive geometries are represented by mathematical models and numerical programsin a more accurate manner than trusses with curvature or local varia- tions.However, even the study of lattices encompasses a wide range ofanalysistechniques.Manyreportsandbookshavebeenpublished onlatticestiffnessanalysis,tailoringforload-transferefe ciency,and vibration response prediction. 1-8 One of themostcommon typesof analysis for lattice structures is the continuum approach in which a lattice' s stiffnessproperties are represented byan equivalent contin- uum model. However, continuum analysis is most useful for large lattices with many repeating cells. In general, transverse shear ef- fects are not included in the analysis of lattices; however, shear effects can be included with additional mathematical terms. This and other lattice analysis drawbacks have prompted this study. Objectives and Scope This paper provides a fundamental and integral approach to the study of lattice truss structures. First, a review of current lattice analysis methods is presented. Next, lattice geometry, symmetry, topology, and design are examined. Then lattice behavior or me- chanics are examined, and a new methodology for the analysis of Presented as Paper 97-1376 at the AIAA/ASME/ASCE/AHS/ASC 38th Structures,StructuralDynamics,andMaterialsConference,Kissimmee,FL, lattice structures is presented. The methodology provides insight into lattice design for strength or stiffness. Additionally, emphasis is placed on exact solutions of various lattice parameters such as nodal displacements and member loads. This limits the scope of the study but allows for greater insight into the behavior of selected lat- ticegeometries.TheinterestedreadershouldseeRef.9forcomplete details,validation using thee nite elementcode EAL,and additional lattice geometries. Specie cobjectivesofthisresearchare1 )todeveloptrussbeamge- ometries that under uniform loading exhibit classic fourth-, unique sixth-, and eighth-order behaviors for dee ection and 2 ) to develop simple closed-form, exact dee ection equations using Castigliano' s second theorem over the nodal domain of the uniform lattice struc- tures mentioned in objective 1.