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Chapter 1 Introduction and Mathematical Preliminaries

# Chapter 1 Introduction and Mathematical Preliminaries -...

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Chapter 1. Introduction and Mathematical Preliminaries 1. Scope Behavior of structures deals with (a) Micro state of stresses, strains, and displacements at a point (b) Macro global behavior, collapse mechanism, etc Theory of elasticity is concerned with (a) Equilibrium of forces (although equations may be expressed in terms of stress, a state of equilibrium must be established by forces) (b) Kinematics and compatibility examine strain-displacement relationship (c) Constitutive equations stress-strain relationship (d) Boundary conditions domain (e) Uniqueness applicability of solutions 2. Vector Algebra 1 1 2 2 3 3 x x y y z z i j k A A e A e A e Ae A e A e Ai A j A k = + + = + + = + + % % % % % % % % % % where 2 x k A ,A ,A are scalar components of the vector and 2 x e ,e ,k % % % are unit vectors. Unit vectors are mutually orthogonal only in the right-hand Cartesian coordinate system. Unless specifically mentioned the coordinate system adopted, the coordinate system is assumed to be the right-hand Cartesian coordinate system in all subsequent discussions herein. ( 29 1 2 2 2 2 1 2 3 / A A A A = + + length of a vector 1 1 2 2 3 3 cos AB C A B A B A B A B A B θ = = + + = % % scalar quantity termed to be dot product or scalar product 1

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1 0 0 cos x x x y z A x A A e A A A A A θ = = ⋅ + + = = % % vector projection on to the x-axis Particularly, if A % is a unit vector, x e % , then the dot product of two unit vectors is equal to the direction cosine of the angle between these two unit vectors. ( 29 cos , i j i j ij e e e e α = = % % % % These are procedures frequently used in elementary mechanics introduced in statics and mechanics of materials. 1 2 3 1 2 3 1 2 3 e e e C A B A A A B B B = × = % % % vector; referred to as vector product or cross product sin AB C A B θ = area of a parallelogram 3. Scalar and Vector Fields ( 29 1 2 3 f x ,x ,x temperature, potential, etc. ( 29 1 2 3 A x ,x ,x % vector fields, velocity, etc. 1 2 3 1 2 3 1 2 3 f f f f f f grad f f e e e , , x x x x x x = ∇ = + + = % % % % vector grad f f 3 1 2 1 2 3 A A A div A A x x x = ∇⋅ = + + % % scalar ( 29 ( 29 ( 29 1 2 3 1 2 3 i j k curl A A x x x A A A = ∇× = % % % % % % vector 2
\$ Integral Theorem Two integral theorems relating vector fields are particularly useful in structural mechanics for transforming between contour, area, and volume integral. Green’s theorem : Consider two functions ( 29 P x,y and ( 29 Q x,y which are continuous and have continuous first partial derivatives ( 1 C continuity) in a domain D , Green’s theorem states that ( 29 ( 29 ,x ,y C A Pdx Qdy Q P dxdy + = - Ñ where A is a closed region of D bounded by C . It should be noted that A should not have any holes in it. This Green’s theorem is the basis of an old instrument, planimeter, measuring the area enclosed by a closed contour.

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Chapter 1 Introduction and Mathematical Preliminaries -...

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