Introduction to Advanced Structural Mechanics Prepared by: Antonio Palermo, PhD Student Revised by: Prof. Alessandro Marzani and Prof. Christian Carloni Aim of the Introductory Course The aim of these slides is to help students review some basic concepts of structural mechanics that will be exploited during the course of Advanced Structural Mechanics. Outline of the Introductory Course • PART I: Cross-sectional Properties. • PART II: Solid Mechanics: Displacements and Strains, Stress and Equilibrium, Constitutive Equations. • PART III: Internal Forces in Beams: Axial Force, Bending Moment, Shear Force, and Torque. The Euler- Bernoulli beam model. • PART IV Analysis of Statically determinate and indeterminate Structures. Links and resources Contacts: [email protected] [email protected] [email protected] Suggested reading: • Beer, Johnston, DeWolf, Mechanics of Materials. • Gere and Timoshenko, Mechanics of Materials. PART I Cross-sectional Properties Outline of PART I • Beam: Geometric Model. • Cross-Sectional Properties: • Area • First Moments of Area • Centroid • Second Moments of Area • Transfer of Axis Theorem and Rotation of Axes • Principal Axes and Central Ellipse of Inertia. Beam: Geometric Model A beam is a structural element generated by a planar figure Ω (i.e. cross section) that moves in the space remaining normal to the trajectory described by its centroid. Ω Beam: Geometric Model Geometric Requirements: Ω=const YES! • Ω(s) constant or can vary continuously: BEAM YES! Ω(z) NO! ℎ ≪ 푙 • 푏 ≪ 푙 h 푙 b Cross-sectional Properties For any cross-section Ω, it is possible to define some quantities that are related only to the cross-section geometry. Ω • Area A • Static Moment of Area • Centroid C C • Moments and Product of Inertia Cross-sectional Properties Area: 퐴 = 푑퐴 Ω 퐴 = [푚2] 푑퐴: Cross-sectional Properties First Moment of Area: Static Moments 푆푥 = 푦푑퐴 Ω 푆푦 = 푥푑퐴 Ω 푆 = [퐿3] Cross-sectional Properties Centroid: The centroid C of a plane figure or two- dimensional shape is the arithmetic mean position of all the points in the shape. The centroid C of an area is the point of intersection of all the straight lines that subdivide the plane figure in equal parts Cross-sectional Properties Centroid: 푆 푦 푥퐶 = 퐴 푆푥 푦 = 퐶 퐴 푥퐶, 푦퐶 = [퐿] Cross-sectional Properties Static Moment & Centroid: Properties • 푆푥 푎푛푑 푆푦 can be ⋛ 0 • The static moment 푆 is zero if calculated with respect to a centroidal axis (i.e. the centroid lies on the axis) • The Static Moment calculated with respect to an axis of symmetry = 0. • If an area has an axis of symmetry, the centroid C lies on the axis. • If an area has two axes of symmetry, the centroid C is located at the intersection of the axes . • 푆 = 푖 푆푖 (domain of integration can be added: geometric decomposition) Cross-sectional Properties Why is it called first moment? Let’s assume to apply a vector at C whose magnitude is the area A of the region W (the dimensions of this vector are [L2]). The moment of this vector with respect to y is 퐴푥퐶 , which is the static moment 푆푦 Cross-sectional Properties Second Moment of Area: Moment of Inertia: 2 퐼푥 = 푦 푑퐴 Ω 2 퐼푦 = 푥 푑퐴 Ω 퐼 = [퐿4] Cross-sectional Properties Second Moment of Area: Product of Inertia and Polar Moment of Area 퐼푥푦 = Ω 푥푦푑퐴 Product o Inertia 2 퐼푂 = Ω 푟 푑퐴 Polar Moment of Area 4 퐼푥푦, 퐼푂 = [퐿 ] Cross-sectional Properties Second Moments of Area: Properties • 퐼푥 푎푛푑 퐼푦 > 0 • 퐼푂 > 0 • 퐼푂 = 퐼푥 + 퐼푦 when O is the origin of the x and y axes. • 퐼푥푦 ⋚ 0 • 퐼푥푦 = 0 if either x or y is an axis of symmetry • 퐼 = 푖 퐼푖 (valid for all the Second Moments of Area) Cross-sectional Properties Change of coordinates (퐶) 푥′ = 푥 + 푑푦′푦(퐶) (퐶) 푦′ = 푦 + 푑푥′푥(퐶) Note that 푑푦′푦(퐶) and 푑푥′푥(퐶) are the coordinates of the centroid C with respect to the Cartesian system ′ Ox’y’. Thus 푑푦′푦(퐶) = 푥퐶 ′ and 푑푥′푥(퐶) = 푦퐶 Cross-sectional Properties Parallel Axes: Static Moment 푆푥′ = 푆푥(퐶) + 퐴푑푥′푥(퐶) = 퐴푑푥′푥(퐶) 푆푦′ = 푆푦(퐶) + 퐴푑푦′푦(퐶) = 퐴푑푦′푦(퐶) Note that 푆푥(퐶) = 푆푦(퐶) = 0 ′ (퐶) 푆푥′ = 푦 푑퐴 = 푦 + 푑푥′푥(퐶) 푑퐴 = 푆푥(퐶) + 퐴푑푥′푥(퐶) = 퐴푑푥′푥(퐶) Ω Ω Cross-sectional Properties Transfer-of-axis Theorem: Second Moments of Area Transfer-of-axis Theorem 2 퐼푥′ = 퐼푥(퐶) + 퐴 푑푥′푥(퐶) 2 퐼푦′ = 퐼푦(퐶) + 퐴 푑푦′푦(퐶) 퐼푥′푦′ = 퐼푥(퐶)푦(퐶) + 퐴푑푥′푥(퐶)푑푦′푦(퐶) Cross-sectional Properties Rotation of the Axes: 푥′ = 푥푐표푠훼 + 푦푠푛훼 푦′ = 푦푐표푠훼 − 푥푠푛훼 Cross-sectional Properties Rotation of the axes: Second Moment of Area 퐼 + 퐼 퐼 − 퐼 푥 푦 푥 푦 퐼푥′ = + cos 2훼 − 퐼푥푦푠푛2훼 2 2 퐼푥 + 퐼푦 퐼푥 − 퐼푦 퐼 = − cos 2훼 + 퐼 푠푛2훼 푦′ 2 2 푥푦 퐼푥 − 퐼푦 퐼 ′ ′ = 푠푛2훼 + 퐼 푐표푠2훼 푥 푦 2 푥푦 Cross-sectional Properties Principal axes (1/3) Goal: determine the value of 훼0 for which 퐼푥′ and 퐼푦′ are the maximum and minimum moments of inertia for the cross section (or viceversa) 푑퐼푦′ 푑퐼푥′ = 0 = 0 푑훼 푑훼 2퐼푥푦 푡푔2훼0 = − 퐼푥−퐼푦 Note that if we enforce 퐼푥′푦′ = 0 we obtain the expression above. Thus, when 훼 = 훼0 퐼푥′ and 퐼푦′ attain the maximum and minimum values (or vice versa) and simultaneously 퐼푥′푦′ = 0 Cross-sectional Properties Principal axes (2/3) With 퐶 ≡ 푂 we define 휉, 휂 as the centroidal principal axes: • 퐼휉, 퐼휂 principal moments of inertia (minimum/maximum moment of inertia or viceversa ) with: • 퐼휉휂 = 0 and: 퐼 +퐼 퐼 −퐼 2 푥(퐶) 푦(퐶) 푥(퐶) 푦(퐶) 2 • 퐼 , 퐼 = ± + 퐼 (퐶) (퐶) 휉 휂 2 2 푥 푦 Cross-sectional Properties Principal axes (3/3): Properties • If a figure has an axis of symmetry, one of the principal axis is the axis of symmetry. • Any other axis perpendicular to the first one (of symmetry) is the second principal axis. Cross-sectional Properties Mohr circle: Given the centroidal principal axes 휉, 휂, with 퐼휉 > 퐼휂 퐼휉 + 퐼휂 퐼휉 − 퐼휂 퐼 = + 푐표푠2훼 푥 2 2 퐼휉 + 퐼휂 퐼휉 − 퐼휂 퐼 = − 푐표푠2훼 푦 2 2 퐼휉 − 퐼휂 퐼 = 푠푛2훼 푥푦 2 Parametric equations of a circle on the plane 퐼푥, 퐼푥푦. Note that the expressions above are written as though x and y are two axes that rotate of an angle 훼 with respect to the princial axes Cross-sectional Properties Mohr circle: Parametric equations of a circle in the plane 퐼푥, 퐼푦. 퐼휉−퐼휂 퐼휉+퐼휂 푅 = ; 퐶 = , 0 2 2 *A. Di Tommaso. Geometria delle Masse Cross-sectional Properties Radii of Gyration & Ellipse of Inertia 퐼휉 = 퐼 = 퐴2 휉 퐴 휉 휉 퐼휂 = 퐼 = 퐴2 휂 퐴 휂 휂 Analytical expression: 휉2 휂2 2 + 2 = 1 휂 휂 Cross-sectional Properties Radii of Gyration & Ellipse of Inertia The Ellipse of Inertia provides a graphical representation of the inertia properties of the cross-section. 퐼푥 = 퐼 = 퐴2 푥 퐴 푥 푥 퐼푦 = 퐼 = 퐴2 푦 퐴 푦 푦 *A. Di Tommaso. Geometria delle Masse PART II Solid Mechanics: Displacements and Strains, Strains and Stresses, Stresses and Forces Outline of PART II • Displacements and Strains: compatibility equations • Strains and Stresses: constitutive equations • Stresses and Forces: equilibrium equations Displacements and Strains (1D) • displacement: 푢(푥) • axial strain: Phillips, Wadee. Pre course Reading Solid Mechanics [1] 퐴′퐵′ − 퐴퐵 푑푢(푥) 휖(푥) = = 퐴퐵 푑푥 different notation Note: the strain is assumed to be positive if the material/solid 푑푢1 휖11 = elongates and negative viceversa 푑푥1 Displacements and Strains (2D) • displacement: 푢1(푥1, 푥2) and 푢2(푥1, 푥2) • axial strain: 휕푢1 휖11 = 휕푥1 휕푢2 휖22 = 휕푥2 • shear strain: [1] 휋 휕푢2 휕푢1 훾12 = − 훽 = 휃 − 휆 = + with: 훾12 = 2휖12 2 휕푥1 휕푥2 Displacements and Strains (3D) • displacement: 푢1(푥1, 푥2, 푥3), 푢2(푥1, 푥2, 푥3) and 푢3(푥1, 푥2, 푥3) • axial strain: 휕푢1 휕푢2 휕푢3 휖11 = , 휖22 = , 휖33 = 휕푥1 휕푥2 휕푥3 • shear strain: 휕푢2 휕푢1 훾12 = + with: 훾12 = 2휖12 휕푥1 휕푥2 휕푢3 휕푢1 훾13 = + with: 훾13 = 2휖13 휕푥1 휕푥3 휕푢3 휕푢2 훾23 = + with: 훾23 = 2휖23 휕푥2 휕푥3 Displacements and Strains (3D) The compatibility equations link displacements and strains in 1D: in 2D: in 3D: 0 0 푑푢1 휕 0 휕 휖11 = 휖 휕 휕푥 휕 0 푑푥1 11 휕푥1 휖 1 푢1 11 휕푥 휕 휖22 = 0 휕푥2 0 2 푢 휖22 2 0 0 휕푥3 푢1 2휖12 휕 휕 휖33 = 휕 휕 0 푢2 휕푥2 휕푥1 2휖12 휕푥 휕푥 휕 푢3 2휖13 2 1 2휖 휕 0 휕푥1 23 휕 휕 휕푥3 0 휕푥3 휕푥2 In compact notation: 흐=Du Displacements and Strains In 3D for a given strain tensor 휖11 휖12 휖13 휖 휖 휖 흐 = 12 22 23 휖13 휖23 휖33 the principal strains 휖푎 and principal directions of strain 풂 ≡ [푎1, 푎2, 푎3], i.e. those directions in which there exist only axial strain and no distorsions, can be found solving the eigenvalue problem: 흐 − 휖푎푰 풂 = 0 흐 2nd order strain tensor 휖푎 principal strain 풂 principal directions 푰 identity matrix quantities to be detrmined Displacements and Strains In 2D the principal strains and the principal directions, in the plane 푥1 − 푥2, can be determined also as • principal strains: 2 휖11 + 휖22 휖11 − 휖22 휖 , 휖 = ± + 휖 2 1 2 2 2 12 • the angle 훼 of the principal directions 푎1, 푎2 w.r.t. the 푥1, 푥2 axes: 2휖12 푡푔2훼 = − 휖11 − 휖22 휖 −휖 휖 +휖 Note: Mohr circle of radius R = 1 2 and centre 퐶 = 1 2 2 2 Strains and Stresses The constitutive equations are the relations between kinetics (stress, stress-rate) quantities and kinematics (strain, strain-rate) quantities for a material.