Lecture 14 Concluding lecture RELATIONSHIP OF STRUCTURAL ANALYSIS

Lecture #14 Concluding lecture

RELATIONSHIP OF STRUCTURAL ANALYSIS WITH OTHER SUBJECTS Mechanical engineering subjects General methods Aerospace engineering subjects Theoretical Mechanics, Mechanics of Materials Methods for structures and mechanisms Engineering Mechanics Structural Analysis Methods for specific objects Fundamentals of Machinery Design Strength of Aircraft 2

PLACE OF STRUCTURAL ANALYSIS IN THE ASSURANCE OF AIRCRAFT STRENGTH Mechanics of Materials Strength of local zones Structural Analysis Methods for analysis of general stress state Strength of Aircraft Methods for calculation of loading and strength 3

PLATES IN UNI-AXIAL COMPRESSION Object Buckling factor k 4

BUCKLING MODES OF AIRCRAFT PANEL 5

AIRPLANE STRUCTURES Stressed skin wing structure Figure: © Southwestern Illinois College 6

TYPES OF SHELL Vertical displacement plot Not stiffened shell Stiffened shell 7

TYPES OF SHELL Bending stress plot Not stiffened shell Stiffened shell 8

NORMAL STRESSES IN THIN-WALLED BEAMS The distribution of normal stresses obeys the hypothesis of planar cross sections: For the case of uniform linear material, it comes to be (for a right coordinate triad): 9

METHOD OF REDUCTION COEFFICIENTS The reduction coefficient is the ratio between the real stress in the member and the fictitious stress obeying the Hook law : The reduction coefficient is introduced to maintain the condition that fictitious strain is equal to real one: 10

THE CONCEPT OF SHEAR FLOW Both of possible stresses – normal and shear – usually act in the load-carrying structure. Instead of consideration of the shear stress t, it is convenient to introduce the shear flow q, using the thickness d : 11

SHEAR STRESSES RELATED QUESTIONS - shear flows due to the shear force, with no torsion; - shear center; - torsion of closed contour; - torsion of opened contour, restrained torsion and deplanation; - shear flows in the closed contour under combined action of bending and torsion; - twisting angles; - shear flows in multiple-closed contours. 12

CALCULATION OF THE SHEAR FLOW Formula used for shear flows in Structural Analysis: qf q 0 Shear center Opened contour SF = 0 q 0 = 0 SM = 0 Singly-closed SF = 0 SM = 0 Stat. ind. SF = 0 SM = 0 , Stat. ind. Multiple cell 13

RESISTANCE OF OPENED CONTOUR 14

RESISTANCE OF CLOSED CONTOUR 15

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RESISTANCE OF MULTI-CELL CONTOUR 17

LOADING OF FUSELAGE FRAME DUE TO BENDING OF FUSELAGE If the cross section is subjected to bending with moment Mx , the distributed load appears acting on the frame 18

SWEPTBACK WING ROOT TRIANGLE 19

DESIGN MODEL OF SWEPTBACK WING 20

EXAMPLE FOR A TOTAL STRESS STATE

MAIN METHODS COVERED Methods of structural analysis Analytical methods Method of joints Method of sections Numerical methods (finite element method) For statically determinate structures Force method Displacement method (slope-deflection method) Matrix methods For statically indeterminate structures 22

BASIC EQUATIONS OF SOLID MECHANICS From analytical point of view, every design model can be expressed as a set of three basic equations with corresponding boundary conditions. Equilibrium equations Constitutive equations Compatibility equations Boundary conditions: • geometrical features; • supports and other means of fixation; • applied displacements and forces. Design model 23

FLOWCHART OF STRUCTURAL ANALYSIS Real object Structural layout Design model Results of analysis Implementation on real object 24

FLOWCHART OF STRUCTURAL ANALYSIS Real object Structural layout Design model Results of analysis Implementation on real object 25

FLOWCHART OF STRUCTURAL ANALYSIS Real object Structural layout Depending on the kind of problem which is solved, the design model could be either as detailed as structural layout, or as generalized as below: Design model Results of analysis Implementation on real object 26

CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Structures Constrained (fixed) a) stable (immovable) Unconstrained (free) a) stable (invariable) b) unstable (movable) b) unstable (variable) 27

CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Stable structures Statically determinate Statically indeterminate Methods of classification Structural analysis Kinematical analysis Statical analysis 28

CLASSIFICATION OF STRUCTURAL ANALYSIS PROBLEMS Statically determinate Statically indeterminate Equilibrium equations could be directly solved, and thus be solved only when forces could be calculated in coupled with physical law 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 (due to damage tolerance requirement) property) Easy to manufacture Hard to manufacture 29

METHODS TO SOLVE INDETERMINATE PROBLEM Small degree of statical indeterminacy Force method Displacement methods Displacement method in matrix formulation Large degree of statical indeterminacy Numerical methods 30

METHODS TO SOLVE INDETERMINATE PROBLEM Stiffness method (displacement method) Flexibility method (force method) Displacements are set as unknowns Stresses are set as unknowns Equilibrium equations are solved Compatibility equations are solved 31

WAYS TO SOLVE A SOLID MECHANICS PROBLEM Displacements are set as unknowns Strains are derived Stresses are derived Equilibrium equations are solved Compatibility equations Constitutive equations Equilibrium equations Stresses are set as unknowns Strains are derived Constitutive equations Equilibrium equations Compatibility equations are solved 32

FINITE ELEMENT METHOD • very formalized and computer-friendly; • versatile, suitable for large problems; • applicable for any mechanical problem and for coupled field problems. 33

EXAMPLE OF FORCE METHOD APPLICATION 34

EXAMPLE OF FORCE METHOD APPLICATION (NUMERICAL SOLUTION) 35

POSSIBLE PROBLEMS IN AIRCRAFT STRUCTURE, SOLVABLE WITH HELP OF STRUCTURAL ANALYSIS The aircraft structure should be strong under complex loading and conditions occurring in service, during the certain service life. Aerospace structures always have very strict weight limitations, that’s why calculation methods should be extremely accurate and reliable. Failure under static and dynamic loads (maneuver, gust, landing loads, loads under vibration and flutter etc. ) Fatigue (cyclic loads) Buckling of compressed panels 36

THIS WAS THE VERY LAST LECTURE … All materials of our course are available at department website k 102. khai. edu 1. Go to the page “Библиотека” 2. Press “Structural Mechanics (lecturer Vakulenko S. V. )” 37