ECIV 520 A Structural Analysis II Lecture 4 – Basic Relationships

Process of Matrix/FEM Analysis Reliability of Solution depends on choice of Mathematical Model Accurate Approximations of Solutions

Process of Matrix/FEM Analysis Reliability of Solution depends on choice of Mathematical Model

Process of Matrix/FEM Analysis Accurate Approximations of Solutions

Process of Matrix/FEM Analysis

Process of Matrix/FEM Analysis Reliability of Solution depends on choice of Mathematical Model Accurate Approximations of Solutions

Stress Resultant Force and Moment represent the resultant effects of the actual distribution of force acting over sectioned area

Stress Assumptions Consider a finite but very small area Material is continuous Material is cohesive Force can be replaced by the three components DFx, DFy (tangent) DFz (normal) Quotient of force and area is constant Indication of intensity of force

Normal & Shear Stress Normal Stress Shear Stress Intensity of force acting normal to DA Shear Stress Intensity of force acting tangent to DA

General State of Stress Set of stress components depend on orientation of cube

Basic Relationships of Elasticity Theory Concentrated Distributed on Surface Distributed in Volume

Equilibrium Equilibrium SFx=0 SFy=0 SFz=0 Write Equations of Equilibrium SFx=0 SFy=0 SFz=0

Equilibrium Equilibrium - X X

Equilibrium Equilibrium SFx=0 SFy=0 SFz=0 Write Equations of Equilibrium SFx=0 SFy=0 SFz=0

Boundary Conditions Prescribed Displacements

Boundary Conditions Equilibrium at Surface

Deformation Forces applied on bodies tend to change the body’s Intensity of Internal Loads is specified using the concept of Normal and Shear STRESS Forces applied on bodies tend to change the body’s SHAPE and SIZE Body Deforms

Deformation Deformation of body is not uniform throughout volume To study deformational changes in a uniform manner consider very short line segments within the body (almost straight) Deformation is described by changes in length of short line segments and the changes in angles between them

Deformation is specified using the concept of Normal and Shear STRAIN

Normal Strain - Definition Normal Strain: Elongation or Contraction of a line segment per unit of length

Normal Strain - Units Dimensionless Quantity: Ratio of Length Units Common Practice US in/in SI m/m mm/m (micrometer/meter) Experimental Work: Percent 0.001 m/m = 0.1% e=480x10-6: 480x10-6 in/in 480 mm/m 480 m (micros)

Shear Strain – Definition Shear Strain: Change in angle that occurs between two line segments that were originally perpendicular to one another

Cartesian Strain Components Normal Strains: Change Volume Shear Strains: Change Size

Small Strain Analysis Most engineering design involves application for which only small deformations are allowed DO NOT CONFUSE Small Deformations with Small Deflections Small Deformations => e<<1 Small Strain Analysis: First order approximations are made about size

Strain-Displacement Relations For each face of the cube Assumption Small Deformations

Stress-Strain (Constitutive)Relations Isotropic Material: E, n Generalized Hooke’s Law

Stress-Strain (Constitutive)Relations Note that: Equations (a) can be solved for s...

Stress-Strain (Constitutive)Relations Or in matrix form

Stress-Strain (Constitutive)Relations

Stress-Strain: Material Matrix

Special Cases s = E e One Dimensional: v=0 No Poisson Effect Reduces to: s = E e

Special Cases Two Dimensional – Plane Stress Thin Planar Bodies subjected to in plane loading

Special Cases Two Dimensional – Plane Strain Long Bodies Uniform Cross Section subjected to transverse loading

Special Cases Two Dimensional – Plane Stress Orthotropic Material Dm= For other situations such as inostropy obtain the appropriate material matrix

Strain Energy During material deformation energy is stored (strain energy) e.g. Normal Stress Strain Energy Density

Strain Energy In the general state of stress for conservative systems

Principle of Virtual Work Load Applied Gradually Due to another Force

PVW Concept Apply Virtual Load Apply Real Loads Internal Real Virtual Forces Real Deformns

Principle of Virtual Work A body is in equilibrium if the internal virtual work equals the external virtual work for every kinematically admissible displacement field

Principle of Virtual Work