《ch4拉压超静定w》PPT课件.ppt

Indeterminate Stress Systems Mechanics of Materials 材料力学 Professor Shibin WANG （王世斌） Indeterminate Stress Systems Mechanics of Materials 关于超静定的基本概念关于超静定的基本概念 静定问题与静定结构未知力（内力或外力）个数未知力（内力或外力）个数 等于独立的平衡方程数等于独立的平衡方程数 超静定问题与超静定结构未知力个数多于独立未知力个数多于独立 的平衡方程数的平衡方程数 超静定次数未知力个数与独立平衡方程数之差未知力个数与独立平衡方程数之差 多余约束保持结构静定保持结构静定多余的约束多余的约束 简单的超静定问题 Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials 静定与超静定的辩证关系多余约束的两种作用：多余约束的两种作用： 增加了未知力个数，同时增加对变形增加了未知力个数，同时增加对变形 限制与约束，前者使问题变为不可解，限制与约束，前者使问题变为不可解， 后者使问题变为可解。后者使问题变为可解。 求解超静定问题的基本方法平衡、变形协调、平衡、变形协调、 物性关系。现在的物性关系体现为力与物性关系。现在的物性关系体现为力与 变形关系。变形关系。 求解超静定问题的基本方法求解超静定问题的基本方法 简单的超静定问题 Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials 拉压超静定问题 E2A2 l2 E3A3 l3=E2A2 l2 E1A1 l1 y x A BCD 例题5 FP FP FN3FN2 FN1 A Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials 平衡方程平衡方程 超静定次数：3-2=1 y x FP FN3FN2 FN1 Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials l1 l3 变形协调方程变形协调方程: :各各 杆变形的几何关系杆变形的几何关系 E2A2 l2 E3A3 l3=E2A2 l2 E1A1 l1 BCD A FP l2 Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials 平衡方程平衡方程: : 变形协调方程：变形协调方程： 物性关系物性关系: : Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials 结果：结果：由平衡方程、变形协调方程、物性关系 联立解出 例题5 Ch.4 Indeterminate Stress System Indeterminate Stress Systems Mechanics of Materials x y Ch.4 Indeterminate Stress Systems Big Question What is an Indeterminate System? Big Answer One in which unknown forces can not be determined from equilibrium alone. This is a DETERMINATE system For example: P1 P2 Rx Since:from We have ONE unknown, and ONE available equilibrium relationship. Indeterminate Stress Systems Mechanics of Materials x y This is an INDETERMINATE system P RAx Since RAx & RBx CAN NOT be found from equilibrium: 1 equation, 2 unknowns RBx Therefore, we need MORE INFORMATION! 4.1 Consider the following problem: Indeterminate Stress Systems Mechanics of Materials Examples of Statically Determinate Systems: Beams: Trusses: Other: 10 kN VAVB HA VA HA MAz VAVB HA VA HA MA No more than 2 unknowns at a joint Indeterminate Stress Systems Mechanics of Materials Examples of Statically Indeterminate Systems: Beams: Trusses: VB VA HA MA Other: How much load carried by concrete, how much by the steel? 10 kN - Too many reactions - Too many unknowns at a joint VA HA MAMB HB VB Indeterminate Stress Systems Mechanics of Materials x y e.g. We need more tools! 4.2 The Direct Method of Analysis Equilibrium of Forces Geometric Compatibility of Deformations Hookes Law P RAx RBx AB C u ab 12 E, A Equilibrium: 1 Indeterminate Stress Systems Mechanics of Materials Compatibility: 2 i.e. Contraction of part = Extension of part12 Hookes Law: 3 x y e.g. P RAx RBx AB C u ab 12 E, A Indeterminate Stress Systems Mechanics of Materials RAx RBx ab x y P P RAx RAx sx1 sx1 sx1 sx2 sx2 Fu1 Fu2 Indeterminate Stress Systems Mechanics of Materials 1(From ) From Compatibility : 2 Indeterminate Stress Systems Mechanics of Materials ab x y P sx1sx2 E, A 0 u -ve +ve Compression Tension NOTE: And from :2 Indeterminate Stress Systems Mechanics of Materials Consider some numbers P=20 kN AB C u 200 mm100 mm 12 E=70 GPa, A=100 mm2 Indeterminate Stress Systems Mechanics of Materials A Scottish engineer, Rankine made observations about the expansion and contraction of materials due to changes in temperature. 4.3 Thermal Strains (William Rankine, 1870) x y a=Coefficient of Linear Expansion (A material property) He noted that these deflections were proportional to the change in temperature the material experienced . Indeterminate Stress Systems Mechanics of Materials From Hookes Law: (Due to Forces and Temperature Changes) Mechanical Strain Thermal Strain Coef Th expan a x 10-6/oC Mild Steel 12 Aluminium 23 Concrete 10.8 Wood - Nylon 0.9 Rubber 130-200 Indeterminate Stress Systems Mechanics of Materials x y Example: Consider a bar constrained between two walls. From Hookes Law: RxRx E, n, a L (fixed) b d u=0 But, Indeterminate Stress Systems Mechanics of Materials i.e. And, Indeterminate Stress Systems Mechanics of Materials If a steel bar has a maximum axial stress of 300 MPa, what is the greatest allowable temperature increase? E, n, a, sMax E=200 GPa n=0.25 a=12x10-6 1/oC +ve (T) i.e. HOTCompression -ve (T) i.e. COLDTension Indeterminate Stress Systems Mechanics of Materials Reduce an indeterminate system into a series of determinate parts, and sum the parts in a way which would satisfy (i) Equilibrium & (ii) Compatibility. 4.4 The Method of Superposition RAx RBx u P AB C ab x y e.g. 12 RAx P AB C uu RAx” AB u” RBx =P = -RBx Indeterminate Stress Systems Mechanics of Materials Superposition: (i) At end A,RAx= RAx + RAx” (ii) At end B,u + u” = 0 From (ii), u + u” = 0 Indeterminate Stress Systems Mechanics of Materials From (i), RAx= RAx + RAx” (Since RAx = P & RAx”= -RBx) Etc See (a) The Direct Method of Analysis for the normal force diagram, sx1, sx2 and u. Indeterminate Stress Systems Mechanics of Materials Example: Composite Column. A short composite column is constructed by filling a round steel pipe with concrete as shown. If the column is placed between two rigid plates and loaded in compression, we would like to find the load-deflection relation for the column. Also, we wish to find the maximum allowable load if the allowable stresses for the steel and concrete are 100 MPa and 8 MPa. Take L=2m, d=0.5m, t=13mm, ES=200 GPa, and EC=14 GPa. Neglect the weight of the materials. Equilibrium Compatibility Hookes Law ? u vs P, and PMax ? Indeterminate Stress Systems Mechanics of Materials Draw FBDs of the steel pipe, concrete column, and an end plate. Equilibrium Compatibility Hookes Law FS+FC P FS FS uS FC FC uC From the FBD of the top plate: 1 2 Indeterminate Stress Systems Mechanics of Materials Consider FC and FS: 3 into :31 Inserting numbers: L=2 m, AS=20.95x10-3 m2, AC=196.3x10-3 m2 i.e. Stiffness! Indeterminate Stress Systems Mechanics of Materials Maximum Load? Assume Max Stress reached in steel first: Check: OK! So finally: Indeterminate Stress Systems Mechanics of Materials Example: Suspended Bar. A rigid bar DC is attached to two elastic wires AD and BC, having diameters d and 2d, modulus of elasticity E, and length L1. A load P is applied at point H. We wish to find the specific distance x from D at which the load P should be applied so that the rigid bar remains horizontal after the load is applied. Neglect the weight of the rigid bar and wires. Equilibrium Compatibility Hookes Law ? x ? Indeterminate Stress Systems Mechanics of Materials DC H Draw FBDs of the cables, and rigid bar. FADFBC uCuD P xL-x FAD FAD FBC FBC L1 uDuC Equilibrium From bar:1 2 Compatibility 3 Hookes Law or, x y +ve Indeterminate Stress Systems Mechanics of Materials Considering both cables: 4 Subbing into :34 (NOTE: ABC=4AAD) 5 Subbing into :25 Indeterminate Stress Systems Mechanics of Materials 4.5 Summary Statically indeterminate problems require more than equilibrium conditions to solve them. The following tools are required to determine unknown forces and deformations. Equilibrium Compatibility (of geometry) Hookes Law (stress vs strain behav