Mechanics of Solids ESO 202 Instructor: Dr. P. Chakraborty Department of Aerospace Engineering Office Location: 210/F NWTF Building Office Phone: 0512-679-7951 email: cpritam@iitk.ac.in 4th January 2023 Outline ▪ Motivation & Objectives ▪ Topics ▪ Other Details Motivation & Objective Definition ▪ Deals with forces and motion of solid bodies ▪ In scope of this course: Deformable solid bodies in equilibrium ▪ Equilibrium: Body is in rest or constant velocity (zero acceleration) under the action of forces ▪ Deformable: Relative motion between points on the body Deformable Body 2 2 1 1 Membrane before application of force (pulling) Membrane after application of force (pulling) A B A B Engineering Applications ▪ Deformable bodies in equilibrium Cruising aircraft Lift on wings Wing bending & twisting Engineering Applications Steady rotation of shaft Transmission of torque Ignoring centrifugal force Equal & opposite torsion & twist Engineering Applications Transmission tower Truss members - light weight - carry axial load Pulled by power cables Engineering Applications Cables - axial load members Bridge Beams - bending members Columns Transverse loads - Carry compressive load - Design against buckling Analysis of Engineering Applications ▪ Analysis required to design/modify applications to meet requirements of load without failure, deflection limits (avoid interference), weight and cost savings ▪ Apply principles of mechanics to analyse deforming structures in equilibrium under the action of forces Analysis Steps ▪ Select system of interest ▪ Idealization and simplification of the real system that exhibit phenomena under consideration ▪ Apply principle of mechanics to the idealized model Shear members Actual Aircraft Wing Axial members Idealized cross-section for analysis Course Objectives ▪ Simplified representation of structural members such as truss, beams, pipes, shafts, columns, etc. ▪ Derivation of algebraic and differential equations involving force, displacement and force-deformation relations to represent the deformed state of structural members in equilibrium ▪ Solving these equation to find deformation, forces, etc. of the structural members Topics Topics ▪ Force; moments; idealization of joints/supports (pin, fixed, etc.); friction; free body diagram; equilibrium conditions ▪ Important relations to analyse deformable bodies: Equilibrium conditions; deformation & compatibility; force-deformation relations. Examples: Spring, bar, truss, thin walled pressure vessels ▪ Definition of stress; equilibrium of a differential element; relation between plane normal & stress components; plane stress; Mohr’s circle Topics ▪ Definition of strain; infinitesimal strain-displacement relation; strain transformation; plane strain; Mohr’s circle; strain rosette ▪ Tensile test; stress vs strain responses; uniaxial elastoplastic curves of different materials; linear elastic constitutive relation in 3D; effect of thermal strain; equations of elasticity; thick-walled pressurized cylinder ▪ Torsion of circular shaft: Twist and strain; stress in circular shaft with linear elastic constitutive model; equilibrium condition; hollow thin-walled shafts Topics ▪ Idealization of beams (slender member subjected to transverse loads); shear force & bending moment diagrams; shear force & bending moment differential relation ▪ Stresses due to bending: Stress resultant force and moment; curvature-strain relation; stress-curvature for linear elastic beam; stress in symmetrical beams; shear stress due to bending ▪ Deflection of beams: Moment curvature relation; load deflection relation; superposition. (Must not use singularity functions) Topics ▪ Energy: Strain energy in elastic body; examples – axial, torsion, bending; complementary energy; Castigliano’s theorem. ▪ Stability: Stable & unstable equilibrium; elastic instability of columns (elastic buckling). Other Details Course Material • An Introduction to Mechanics of Solids (3rd Edition), S. H. Crandall, N. C. Dahl, T. J. Lardner, M. S. Sivakumar (text book). • Engineering Mechanics of Solids, E. P. Popov • Mechanics of Materials, Gere, Timoshenko • Slides used during class will be shared on mookit Break-up of Marks 5 Quiz (Announced) Duration: 1 hour each 25% Biometric attendance in class & tutorial 5% Mid-of-semester exam Duration: 2 hours 30% End-of-semester exam Duration: 3 hours 40% Home Problems (submission not required) 0% Other Policies ▪ Biometric attendance will be taken during classes and tutorials. Attendance will be checked on a periodic basis. Any student with < 50% attendance may be deregistered ▪ No make-up quizzes (announced & surprise) & mid-of semester examination ▪ Relative grading ▪ Details of students found using unfair means during exams or quizzes will be reported to competent authority and appropriate action will be taken Mechanics of Solids ESO 202 Instructor: Dr. P. Chakraborty Department of Aerospace Engineering 4th January 2024 Force ▪ Force is a directed interaction between bodies ▪ It is a simplified representation of complex physical interactions that may exist between bodies ▪ Mathematically, force can be represented as a vector ▪ Classification: ▪ Contact forces – between bodies in physical contact Force transmitted via contact between hand and shaft of cart Pulling of a cart Force transmitted via contact between roller and cage Ball bearing Force ▪ Classification: ▪ Non-contact forces – between physically separated bodies Gravitational force Magnetic force Force ▪ Newton’s laws of motion (relate force & motion) ▪ 1st Law: Effect of force on motion of a body ▪ 2nd Law: Mathematical relation between force and motion ▪ 3rd Law: Relation of forces between interacting bodies ▪ Unit of force obtainable from 2nd law ▪ SI unit – Newton (N) ▪ A newton is defined as the force which when acts on a body of 1 kg generates an acceleration of 1 m/s2 ▪ 1 N = 1 kg-m/s2 Force ▪ The net effect of a system of forces acting at a point can be expressed in terms of the resultant force F1 F2 𝐹 = 𝐹1 + 𝐹2 + 𝐹3 + 𝐹4 + 𝐹5 F3 Y X Z Resultant Force: F5 5 𝐹 = 𝐹𝑖 F4 𝑖=1 F2 Can also be obtained from vector diagram F1 F3 F F4 F5 Force Components of a force in an orthogonal axis-system 𝐹1 = 𝐹1𝑥 𝑖Ƹ + 𝐹1𝑦 𝑗Ƹ + 𝐹1𝑧 𝑘 Components of the resultant force 5 𝐹𝑥 = 𝐹𝑖𝑥 𝑖=1 5 𝐹𝑦 = 𝐹𝑖𝑦 𝑖=1 5 𝐹𝑧 = 𝐹𝑖𝑧 𝑖=1 Moment of a Force ▪ Force 𝐹 is applied at point P ▪ O is a fixed point in space ▪ 𝑟 is the position vector of P with respect to O ▪ Moment of 𝐹 about O is defined as 𝑀 =𝑟×𝐹 Moment of a Force ▪ Moment is a vector with a magnitude and direction ▪ Construct a plane consisting of vectors r and F ▪ Fcos is along OP and doesn’t contribute to moment (definition of moment) ▪ Magnitude of moment 𝑀 = 𝐹𝑟𝑠𝑖𝑛𝜙 ▪ Direction of moment determined by right hand screw rule Moment of a Force ▪ Direction is in that of the thumb of the right hand - when the fingers curl in the direction that the force F tends to turn about O Direction of moment Moment of a Force ▪ Components of moment vector in an orthogonal coordinate system 𝐹𝑥 𝑖Ƹ + 𝐹𝑦 𝑗Ƹ 𝑀 =𝑟×𝐹 = 𝑟𝑥 𝑖Ƹ + 𝑟𝑦 𝑗Ƹ × 𝐹𝑥 𝑖Ƹ + 𝐹𝑦 𝑗Ƹ = 𝑟𝑥 𝐹𝑦 − 𝑟𝑦 𝐹𝑥 𝑘 𝑟𝑥 𝑖Ƹ + 𝑟𝑦 𝑗Ƹ Moment of Forces ▪ Moment about point O due to multiple forces (fixed vector) F1 F2 r1 r5 F5 r2 O r4 F3 r3 F4 𝑀 = 𝑟1 × 𝐹1 + 𝑟2 × 𝐹2 + 𝑟3 × 𝐹3 +𝑟4 × 𝐹4 +𝑟5 × 𝐹5 5 = 𝑟𝑖 × 𝐹𝑖 𝑖=1 Couple ▪ A couple is a system of 2 parallel non-collinear forces having equal magnitudes and opposite directions M 𝑟1 × 𝐹1 r1 𝑟2 × 𝐹2 r2 F2 a F1 𝑀 = 𝑟1 × 𝐹1 + 𝑟2 × 𝐹2 = 𝑟2 × 𝐹1 + 𝐹2 + 𝑎 × 𝐹1 = 𝑎 × 𝐹1 ▪ The moment due to couple is independent of location of point O
