Ch. 2: Fundamental of Structure Aircraft is a structure Structures provide aerodynamic properties Challenge: weight must minimum Structural strength vs weight Solid Mechanics How a solid object resists/supports load ? Solid ? Conservations ?
Elasticity Ability to bounce back Aircraft has limited elasticity Stress & Strain
Elasticity
Stress Strain Measure of force intensity = F/A Units !! Five Types Of Stress Tension Compression Bending force Torsion Shear force Strain Measure of deformation Units ? = /L
Stress & Strain Measure of force intensity
Example 2.1 A prismatic bar with rectangular cross section (20mmx40mm) and length, L=2.8m is subjected to an axial tensile force 70 kN. The measured elongation of the bar is =1.2mm. Calculate the tensile stress and strain in the bar.
Sign Conventions/Notation of Stresses Stress at a point (A) on a plane of a body has the components of; 1 direct stress, 2 shear stresses, x : direct stress in the direction of x axis xy : shear stress x: the plane, y: the direction On each plane, there are 2 equal; but opposite stresses In the x axis direction; +ve & tensile In the –ve x axis direction; -ve & compression
State of Equilibrium Stresses on opposites faces are differ & stated in Taylor’s series;
State of Equilibrium Stresses on opposites faces are differ & stated in Taylor’s series; X, Y, Z are force per unit area.
Plane Stress Aircraft structural components thin metal sheet Stress across thickness negligible
Stresses on Inclined Plane The chosen of axes system is arbitrary So need to examine stresses on other planes
Stresses on Inclined Plane
Example 2.2 A cylindrical pressure vessel has an internal diameter of 2m and is fabricated from 20mm thick plate. The pressure inside the vessel is 1.5N/mm2, and the vessel is also subjected to axial tensile load of 2500kN. Determine the direct and shear stresses on a plane inclined at an angle of 60o to the axis of the vessel. Determine also the maximum shear stress.
Example 2.3 A cantilever solid beam, circular cross section supports a compressive load of 50kN applied to its free end at a point 1.5m below a horizontal diameter in the vertical plane of symmetry together with a torque of 1200Nm. Calculate the direct and shear stresses on a plane inclined at 60o to the axis of the cantilever at a point on the lower edge of the vertical plane of symmetry. Sixma bending=My/I. I is 2nd moment of area, for circle=pai *d^4/64
Stop 20/09/2012
Principles Stress : direct (normal) The stress on the inclined plane is max/min when Differentiate n w.r.t , 2 solutions at and +/2 Principal stress (max) Sixma bending=My/I. I is 2nd moment of area, for circle=pai *d^4/64 Principal stress (min)
Principles Stress : shear The stress on the inclined plane is max/min when Differentiate w.r.t , 2 solutions at and +/2 The plane of max shear stress at 45o to principal plane Principal stress (max) Sixma bending=My/I. I is 2nd moment of area, for circle=pai *d^4/64 Principal stress (min)
Stress on Mohr’s Circle Stress on inclined plane; Sixma bending=My/I. I is 2nd moment of area, for circle=pai *d^4/64
Strain Force causes the body to deform : strain Direct strain : due to direct stress Shear strain: due to shear stress Points P & Q deformed to P’ & Q’ the normal strain are,
Shear Strain Shear strain components are; Rigid body motion ; a body undergoes displacement without inducing strains.
Example 2.3 Consider a 2-D body (a unit square ABCD) in xy plane. After deformation, four corners points move to A’, B’, C’ and D’ respectively. Assume the displacement in the x and y direction are given by u=0.01y and v=0.015x respectively. Obtain the deformed coordinates, normal strain and shear strain.
Principal Strain The principal strains are given by, The principal strains can be determined graphically by using Mohr’s circle for strain
Stress-Strain Relation 6 components stress; 6 components strains; For plane stress, we have E is Young’s modulus is Poisson’s ratio G is modulus of rigidity
Stress-Strain : Example 2.4
Stress-Strain : Example 2.5