Lecture #6 Classification of structural analysis problems. Statical determinacy CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Structures Constrained (fixed) Unconstrained (free) External loads are balanced by constraint forces External loads are balanced by inertia forces 2 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Structures Constrained (fixed) a) stable (immovable) Unconstrained (free) a) stable (invariable) b) unstable (movable) b) unstable (variable) 3 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Kinematically unstable structures could not be analyzed by methods of structural mechanics. They represent mechanisms and are studied by engineering mechanics. Before starting the force analysis, one should check if the structure kinematically stable or not. The reason of instability could be internal or external. internally deficient externally deficient 4 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Instability could be instantaneous and permanent. Usually, structures which are unstable instantaneously, could be analyzed as geometrically nonlinear problems, but this is a special part of structural mechanics science. 5 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Three basic equations Equilibrium equations This is not only the sum of forces or moments, but applies for elementary volume as well Constitutive equations Physical law, expresses the relation between stress and strain Compatibility equations Solid body should remain continuous while being deformed 6 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Two basic nonlinearities Geometrical nonlinearity Equilibrium conditions depend on displacement values Physical nonlinearity Plastic effects are taken into account (nonlinear Physical law) --- Usually, only linear case of Compatibility conditions is studied 7 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Question #1: is problem stable or not? We must determine which science to use for analysis, and should we consider the geometrical nonlinearity. … and if structural analysis could be applied for a given problem, we get … Question #2: is structure statically determinate or not? The answer is required to choose the proper method of structural mechanics. 8 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS The structure is statically determinate if internal forces in all members and all constraint forces could be determined using equations of equilibrium only. statically determinate statically indeterminate 9 EXAMPLES OF TRUSSES USED IN BRIDGES 10 CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Statically determinate Statically indeterminate Equilibrium equations could Equilibrium equations could be directly solved, and thus be solved only when forces could be calculated coupled with physical law in an easy way and compatibility equations Stress state depends only Stress state depends on on geometry & loading rigidities Not survivable, moderately Survivable, widely used in modern aviation used in modern aviation (due to damage tolerance (due to damage tolerance requirement) property) Easy to manufacture Hard to manufacture 11 METHODS TO CLASSIFY THE PROBLEM To analyze the structure for kinematic stability and static determinacy, three methods are used: Is the structure simple? no Is the structure stable? yes Use structural analysis, to find both stability and determinacy Use kinematic analysis, to check for stability and suppose the determinacy. This is necessary but not sufficient method yes Use statical analysis, which is sufficient to confirm the results of kinematic analysis 12 BASIC DEFINITIONS • Rod (AC, CB, CD) – bar which works only in tesion/compression. Wires and columns are partial cases. • Disk (ABD) – any general bar, excluding rods. • Node (A, C, D) – joint of rods, including nodes at supports. • Hinge (none at this figure) – hinge between disks. 13 BASIC DEFINITIONS Degrees of freedom (DOF) – independent parameters which determine the position of the member. Disk has 3 DOFs in plane and 6 DOFs in space. Node has 2 DOFs in plane and 3 DOFs in space. Each type of support constrains certain number of DOFs. 14 STRUCTURAL ANALYSIS Two approaches are used: composition and decomposition. Members satisfying structural rules for planar systems: • node of two not collinear rods; • disk connected by three rods, not parrallel and not crossing in one point; • disk connected by a hinge and a rod which do not pass through the hinge. Members satisfying structural rules for spatial systems: • node of three rods not liying in one plane; • disk connected by six rods, neither two of them are collinear. 15 KINEMATICAL ANALYSIS Number of DOFs in system is calculated. Formulas for trusses: a) for 2d: i r c 2n b) for 3d: i r c 3n i – degree of indeterminacy; r – number of rods; c – number of constrained DOFs (or number of DOFs for free body if structure is free); n – number of nodes. 16 KINEMATICAL ANALYSIS Formulas for general structures: a) for 2d: i r c 2h 2n 3d b) for 3d: i r c 3h 3n 6d i – degree of indeterminacy; r – number of rods; c – number of constrained DOFs (or number of DOFs for free body if structure is free); h – number of hinges which are not nodes; n – number of nodes; d – number of disks. 17 KINEMATICAL ANALYSIS Results of kinematical analysis: i < 0 – unstable problem; i = 0 – statically determinate problem; i > 0 – statically indeterminate problem. If kinematical analysis shows that problem is stable, the result should be checked by statical analysis. 18 STATICAL ANALYSIS • Matrix of coefficients A(m,n) of static equilibrium equations is calculated. • The single condition is that rang(A)=min(m,n) • Despite the simplicity of formulation, statical analysis is most complex and comprehensive. • Statical analysis is sufficient by itself, but is usually used as a last step for complex problems. 19 STATICAL ANALYSIS - EXAMPLE Kinematical analysis supposes that structure is once statically indeterminate: i r c 2h 2n 3d 3 4 20 20 32 1 20 STATICAL ANALYSIS - EXAMPLE Statical analysis claim that structure is not stable! 21 STATICAL ANALYSIS - EXAMPLE 22 METHODS TO CLASSIFY THE PROBLEM To analyze the structure for kinematic stability and static determinacy, three methods are used: Is the structure simple? no Is the structure stable? yes Use structural analysis, to find both stability and determinacy Use kinematic analysis, to check for stability and suppose the determinacy. This is necessary but not sufficient method yes Use statical analysis, which is sufficient to confirm the results of kinematic analysis 23 TOPIC OF THE NEXT LECTURE Statically indeterminate structures. Method of forces All materials of our course are available at department website k102.khai.edu 1. Go to the page “Библиотека” 2. Press “Structural Mechanics (lecturer Vakulenko S.V.)” 24

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