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Structural Theory

Structural Theory 7.1 Introduction Basic Equations: Equilibrium, Compatibility, and Co nstitutive Law • Three Levels: Continuous Mechanics, Finite-Element Method, Beam–Column Theory • Theoretical Structural Mechanics, Computational Structural Mechanics, and Qualitative Structural Mechanics • Matrix Analysis of Structures: Force Method and Displacement Method 7.2 Equilibrium Equations Equilibrium Equation and Virtual Work Equation • Equilibrium Equation for Elements • Coordinate Transformation • Equilibrium Equation for Structures • Influence Lines and Surfaces 7.3 Compatibility Equations Large Deformation and Large Strain • Compatibility Equation for Elements • Compatibility Equation for Structures • Contragredient Law. | Liu X. Zhang L. Structural Theory. Bridge Engineering Handbook. Ed. Wai-Fah Chen and Lian Duan Boca Raton CRC Press 2000 7 Structural Theory Xila Liu Tsinghua University China Leiming Zhang Tsinghua University China 7.1 Introduction Basic Equations Equilibrium Compatibility and Constitutive Law Three Levels Continuous Mechanics Finite-Element Method Beam-Column Theory Theoretical Structural Mechanics Computational Structural Mechanics and Qualitative Structural Mechanics Matrix Analysis of Structures Force Method and Displacement Method 7.2 Equilibrium Equations Equilibrium Equation and Virtual Work Equation Equilibrium Equation for Elements Coordinate Transformation Equilibrium Equation for Structures Influence Lines and Surfaces 7.3 Compatibility Equations Large Deformation and Large Strain Compatibility Equation for Elements Compatibility Equation for Structures Contragredient Law 7.4 Constitutive Equations Elasticity and Plasticity Linear Elastic and Nonlinear Elastic Behavior Geometric Nonlinearity 7.5 Displacement Method Stiffness Matrix for Elements Stiffness Matrix for Structures Matrix Inversion Special Consideration 7.6 Substructuring and Symmetry Consideration 7.1 Introduction In this chapter general forms of three sets of equations required in solving a solid mechanics problem and their extensions into structural theory are presented. In particular a more generally used method displacement method is expressed in detail. 7.1.1 Basic Equations Equilibrium Compatibility and Constitutive Law In general solving a solid mechanics problem must satisfy equations of equilibrium static or dynamic conditions of compatibility between strains and displacements and stress-strain relations or material constitutive law see Figure 7.1 . The initial and boundary conditions on forces and displacements are naturally included. From consideration of equilibrium equations one can relate the stresses inside a body to external excitations including body and surface forces. .