东南大学：《建筑力学 Architectural Mechanics》课程教学课件（英文讲稿）A15 Indeterminate Problem（Statically Indeterminate Structures）

Statically Indeterminate Structuresmi@se.eol.cn

Statically Indeterminate Structures mi@seu.edu.cn

Contents·StaticallyDeterminateProblems（静定问题）·StaticallyIndeterminateProblems（超静定问题）·DegreesofIndeterminacy（超静定次数）: Advantages & Disadvantages of Indeterminate Structures（超静定结构的优缺点）·Redundancy&BasicDeterminateSystem（余约束与基本静定系）: General Ideas for Analyzing Indeterminate Structures（超静定问题一般解法）2

• Statically Determinate Problems（静定问题） • Statically Indeterminate Problems（超静定问题） • Degrees of Indeterminacy（超静定次数） • Advantages & Disadvantages of Indeterminate Structures （超静定结构的优缺点） • Redundancy & Basic Determinate System（冗余约束与 基本静定系） • General Ideas for Analyzing Indeterminate Structures （超静定问题一般解法） Contents 2

Contents·Force Method Solution Procedure （力 法）·StaticallyIndeterminateBars（超静定拉压杆）·AssemblyStress（装配应力）·Thermo Stress（热应力）·StaticallyIndeterminateShafts（超静定扭转轴）·StaticallyIndeterminateBeams（超静定弯曲梁）· Moment-Area Theorems with Indeterminate Beams （图 乘法求解超静定梁）·CombinedIndeterminateStructures（组合超静定结构）3

• Force Method Solution Procedure（力法） • Statically Indeterminate Bars（超静定拉压杆） • Assembly Stress（装配应力） • Thermo Stress（热应力） • Statically Indeterminate Shafts（超静定扭转轴） • Statically Indeterminate Beams（超静定弯曲梁） • Moment-Area Theorems with Indeterminate Beams（图乘 法求解超静定梁） • Combined Indeterminate Structures （组合超静定结构） Contents 3

Statically Determinate Problems: Problems can be completely solved via static equilibrium: Number of unknowns (forces/moments) = number ofindependent equations from static equilibrationxF07F2xAxFF=?=0AyyF=?M=0B1P4

• Problems can be completely solved via static equilibrium • Number of unknowns (forces/moments) = number of independent equations from static equilibration 0 ? 0 ? 0 ? x Ax y Ay A By F F F F M F                     Statically Determinate Problems 4 A B

Statically Indeterminate Problems: Problems cannot be solved from static equilibrium alone: Number of unknowns (forces/moments) ≥> number ofindependent equations from static equilibrationCLxF2F0F=?XAyF.= 0AR?yKC=?M>=02Fcy5

A B C • Problems cannot be solved from static equilibrium alone • Number of unknowns (forces/moments) > number of independent equations from static equilibration ? 0 ? 0 ? 0 ? ? Ax x Ay y By A Cx Cy F F F F F M F F                         Statically Indeterminate Problems 5

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 Műller￾Breslau. Christian Otto Mohr (1835 – 1918) James Clerk Maxwell (1831 – 1879) Heinrich Műller-Breslau (1851 - 1925) Force Method (Method of Consistent Deformations) 10