Recitation 1: Vector/Tensor Analysis and Review of Static Equilibrium

Structural Mechanics 2.080 Recitation 1 Semester Yr Recitation 1: Vector/Tensor Analysis and Review of Static Equilibrium 1.1 1.1.1 Scalars, Vectors, and Tensors Scalars Physical quantities that are described by a single real number are called scalars. Examples: density, energy, temperature, pressure 1.1.2 Vectors Vectors are physical quantities that are completely characterized by a magnitude and di­ rection. Examples: force, velocity, displacement 1.1.3 Tensors A tensor can be thought of as a linear operator that acts on one vector to generate a new vector. Example: Cauchy’s stress theorem t = (n (1.1) where t is the teaction vector, n is the normal vector, and ( is the stress tensor. 1.1.4 Indicial Notation Range Convention Wherever a subscript appears only once in a term (called a free or live index), the subscript takes on all the values of the coordinate space (i.e., 1,2,3 for a 3D space). Examples: Ai = (A1 , A2 , A3 ) 1-1 (3D vector) (1.2) Structural Mechanics 2.080 Recitation 1 3 (11 (12 (13 6 7 (ij = 4 (21 (22 (23 5 (31 (32 (33 2 (3X3 tensor) Semester Yr (1.3) Typically, indices i, j represent 3D space. Indices ↵, ( represent 2D space (e.g., plane strain or plane stress). A scalar quantity has 0 free indices, a vector has 1 free index, and a tensor has 2 (or more) free indices. Summation Convention (Einstein Notation) If an index appears twice in a term (called a dummy index), summation over the range of the index is implied. Examples: aii = a11 + a22 + a33 ai ai = a21 + a22 + a23 (1.4) (1.5) Comma convention A subscript comma followed by an index i indicates partial di↵erentiation with respect to each coordinate xi . The summation and range conventions apply to indices following the comma as well. Examples: ui,i = @u1 @u2 @u3 + + @z2 @z3 @z1 (1.6) Kronecker Delta, 6ij 6ij = ei · ej ⌘ ( 1, if i = j 0, if i 6= j (1.7) Examples: ui 6ij = uj (1.8) 6ij Ajk = Aik (1.9) Permutation Symbol, "ijk "123 = "231 = "312 = 1 (1.10) "132 = "213 = "321 = -1 (1.11) 1-2 Structural Mechanics 2.080 Recitation 1 Semester Yr "ink = 0 if there is a repeated index (1.12) Scalar Products (dot products) The scalar (dot) product of two vectors is defined by: u · v = |u||v| cos(u, v) (1.13) where (u, v) is the angle between the two vectors. u · v = u1 v1 + u2 v2 + u3 v3 = ui vi (1.14) Work, W = F · u (1.15) Examples: Vector Products (cross products) The vector (cross) product of two vectors is defined by: w =u⇥v (1.16) where w is a new vector and orthogonal to u and v. Its magnitude is given by |w| = |u||v| sin(u, v) (1.17) Moment, M = r ⇥ F (1.18) Examples: The cross product can be calculated as 2 3 e1 e 2 e 3 6 7 u ⇥ v = det 4 u1 u2 u3 5 v1 v2 v3 (1.19) u ⇥ v = "ijk uj vk (1.20) In terms of the permutation symbol, the cross product can be written as Example putting it all together: 1. Gradient of a scalar: r = @ @ @ i+ j+ k= @x @y @z 1-3 ,i (1.21) Structural Mechanics 2.080 Recitation 1 Semester Yr 2. Divergence of a vector: r·v = @v1 @v2 @v3 + + = vi,i @x @y @z (1.22) 3. Hooke’s Law: (ij = A"kk 6ij + 2µ"ij How many equations does this represent? 1.2 1.2.1 Static Equilibrium (of rigid structures) Framework for Structural Analysis Structural actions (i.e., loads): • Tension • Compression • Bending (combined tension + compression) • Shear • Torsion Structural elements: • Rods • Beams • Columns • Plates • Shells Structural systems: • Buildings • Bridges • Ships, etc. What influences how a structural behaves? 1-4 (1.23) Structural Mechanics 1.2.2 2.080 Recitation 1 Semester Yr Static Equilibrium ⌃F = 0 (1.24) ⌃M = 0 (1.25) Example: Find the minimum required cross-sectional area of each member to avoid yield. Example: Find the minimum required cross-sectional area of each member to avoi 1-5