Lecture #6 Classification of structural analysis problems. Statical determinacy

CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS 2

CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS 3

internally deficient externally deficient 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 6

Two basic nonlinearities CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Two basic nonlinearities 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

statically determinate 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 be directly solved, and thus forces could be calculated in an easy way Equilibrium equations could be solved only when coupled with physical law and compatibility equations Stress state depends only on geometry & loading Stress state depends on rigidities Not survivable, moderately used in modern aviation (due to damage tolerance requirement) Survivable, widely used in modern aviation (due to damage tolerance 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: 12

Disk (ABD) – any general bar, excluding rods. 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

Disk has 3 DOFs in plane and 6 DOFs in space. 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

Two approaches are used: composition and decomposition. 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

i – degree of indeterminacy; r – number of rods; KINEMATICAL ANALYSIS Number of DOFs in system is calculated. Formulas for trusses: for 2d: for 3d: 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

i – degree of indeterminacy; r – number of rods; KINEMATICAL ANALYSIS Formulas for general structures: for 2d: for 3d: 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

i < 0 – unstable problem; i = 0 – statically determinate problem; 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

rang(A)=min(m,n) 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: 20

Statical analysis claim that structure is not stable! 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: 23

Statically indeterminate structures. Method of forces 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