Degrees of Indeterminacy of Structures? For a co-planer structure, there are at most threeequilibrium equations for each portion of the structure. Ifthere is a total of n portions and m unknown reactionforces from supports:m = 3n = Statically determinatem > 3n = Statically indeterminatem - 3n = Degrees of indeterminacy福15r4Degrees of Indeterminacy = 2Statically determinate6
• For a co-planer structure, there are at most three equilibrium equations for each portion of the structure. If there is a total of n portions and m unknown reaction forces from supports: 3 Statically determinate 3 Statically - 3 Degrees of indetermi indet n erminate acy m n m n m n Statically determinate Degrees of Indeterminacy = 2 Degrees of Indeterminacy of Structures 6
Advantages & Disadvantages of Indeterminate Structures:Advantages:- Redistribution of reaction forces / internal forces- Smaller deformation- Greater stiffness as a whole structure: Disadvantages:- Thermal and residual stresses due to temperature changeand fabrication errors7
• Advantages: Advantages & Disadvantages of Indeterminate Structures - Redistribution of reaction forces / internal forces - Smaller deformation - Greater stiffness as a whole structure • Disadvantages: - Thermal and residual stresses due to temperature change and fabrication errors 7
Redundancy & Basic Determinate System: Redundancy: unnecessary restraints without which thestatic equilibrium of a structure still holds.7e?? Basic determinate system: the same structure as of astatically indeterminate system after replacing redundantrestraints with extra constraining loads8
• Redundancy: unnecessary restraints without which the static equilibrium of a structure still holds. • Basic determinate system: the same structure as of a statically indeterminate system after replacing redundant restraints with extra constraining loads Redundancy & Basic Determinate System 8
How to Analyze Statically Indeterminate Structures?: Basic determinate structure: obtained via replacingredundant restraints with extra constraining loads.: Equilibrium: is satisfied when the reaction forces atsupports hold the structure at rest, as the structure issubjected to external loads: Deformation compatibility: satisfied when the varioussegments of the structure fit together without intentionalbreaks or overlaps: Deformation-load relationship: depends on the mannerthe material of the structure responds to the applied loads.which can be linear/nonlinear/viscous and elastic/inelastic:for our study the behavior is assumed to be linearly elastic9
• Basic determinate structure: obtained via replacing redundant restraints with extra constraining loads. How to Analyze Statically Indeterminate Structures? • Equilibrium: is satisfied when the reaction forces at supports hold the structure at rest, as the structure is subjected to external loads • Deformation compatibility: satisfied when the various segments of the structure fit together without intentional breaks or overlaps • Deformation-load relationship: depends on the manner the material of the structure responds to the applied loads, which can be linear/nonlinear/viscous and elastic/inelastic; for our study the behavior is assumed to be linearly elastic 9
Force Method (Method of Consistent Deformations). The method of consistent deformations or force methodwas originally developed by James Clerk Maxwell in 1874and later refined by Otto Mohr and Heinrich Muller-Breslau.Christian Otto MohrHeinrichMuller-BreslauJamesClerkMaxwell(1831- 1879)(1851-1925)(1835-1918)10
• The method of consistent deformations or force method was originally developed by James Clerk Maxwell in 1874 and later refined by Otto Mohr and Heinrich MűllerBreslau. Christian Otto Mohr (1835 – 1918) James Clerk Maxwell (1831 – 1879) Heinrich Műller-Breslau (1851 - 1925) Force Method (Method of Consistent Deformations) 10