326MAE (Stress and Dynamic Analysis) 340MAE (Extended Stress and Dynamic Analysis) 2017-18

3D Stress-Strain Analysis Week 1 - Session 2 Bill Dunn using slides by Saeid Hadidimoud

Session objectives To calculate 3D strains from 3D stresses To calculate 3D stresses from 3D strains To calculate the normal stress in a direction defined by its direction cosines for a given stress tensor To calculate the shear stress and its direction on a plane whose normal is defined by direction cosines, for a given stress tensor To calculate a new stress tensor for three new orthogonal directions To find the principal stresses for a given stress tensor, and their directions

Reminders: 1D and 2D Stress-Strain uniaxial stress (simple tension in a bar) If the only stress is the uniaxial component (with all other stress elements zero) then we have x = x /E ; x = E x However there are also two other components of strain!   y = - x = - x /E ; z = - x = - x /E Important! uniaxial stress does NOT mean uniaxial strain only! Example: A bar of metal follows linear elastic behaviour and is subjected to a load of 100kN uniformly distributed and acting on its cross section. The bar diameter is 50mm. Material is steel with a Young modulus of 200GPa and a Poisson’s ratio of 0.3. Find all the strain components. Show stress and strain in generic matrix form.

Sample Problem 1 A block of steel (E=200GPa and =0.3) is subjected to a uniformly distributed load of 500kN in the x direction. The block dimensions in x, y, and z are 200, 50, and 100 mm respectively. Find the change in dimensions of the block subjected to this load. If the material remains elastic up to a stress of 200MPa what is the maximum load under which your strain calculation approach remains valid? x = x /E ; x = E x y = - x = - x /E ; z = - x = - x /E

Reminders: 2D Stress-Strain relationships Note! The above equations simply demonstrate 2D strains in terms of 2D stresses. This does not mean that a 2D stress state produces a 2D strain state! Similarly stresses may be derived from strains: Note! The above equations simply demonstrate 2D stresses in terms of 2D strains. This does not mean that a 2D strain state produces a 2D stress state! A reminder that the three material constants are not independent! 𝐺= 𝐸 2(1+𝜈)

3D Strain-Stress relationships (strains in terms of stresses) As far as a material responds to loading in a linear elastic manner for any known stress state, strain state can be identified by simply knowing the material constants, E, G, and . Note! The above equations fully represent the state of strain at a location corresponding to its stress state and take account of all components of strain!

3D Stress-Strain relationships (stresses in terms of strains) As far as a material responds to loading in a linear elastic manner for any known strain state, strress state can be identified by simply knowing the material constants, E, G, and . Note! The above equations fully represent the state of stress at a location corresponding to its strain state and take account of all components of stress!

What is the state of “Plane Stress” Plane Stress conditions may be defined as: From which we find: Strain state shown below is a 3D state!

What is the state of “Plane Strain” Plane Strain conditions may be defined as: From which we find: Stress state shown below is a 3D state! Note! Stress component in z direction may also be written in terms of strains using the generic 3D stress strain relationships.

An example of plane stress? Really? Recalling thin wall pressurized cylinder For a thin walled pipe (wall thickness t) with a radius r, under internal pressure p, we only account for a hoop (tangential) stress. In a cylindrical coordinate this is seen as the only non-zero stress component! Can you suggest why the radial stress is usually overlooked?! A pressure inside means there is compressive stress acting on the inner layer in the radial direction! Why is this ignored? What if the pipe was thick walled? 𝜎 ℎ𝑜𝑜𝑝 = 𝑝𝑟 𝑡

An example of plane stress? Really? Recalling thin wall pressurized cylinder For a thin walled closed end vessel (wall thickness t) with a radius r, under internal pressure p, we account for hoop (tangential) and axial stresses. In a cylindrical coordinate this is still seen as a plane stress problem! 𝜎 ℎ𝑜𝑜𝑝 = 𝑝𝑟 𝑡 , 𝜎 𝑎 = 𝑝𝑟 2𝑡 Can you suggest why the compressive stress acting on the inner layer in the radial direction is ignored? What if the vessel was thick walled? What would be the difference between stress state at the inner and outer surfaces!? Think of real life problems such as beams, plates, shells, shafts,

Sample Problem 2 A closed end cylindrical pressure vessel of length 500 mm has in inner diameter of 150mm and outer diameter of 200mm. Two designers have attempted approximate approaches to evaluate the integrity of the vessel in service when subjected to an internal pressure of 50MPa. Designer 1: assumed the vessel as a thin walled structure and used the thin wall theory to estimate stresses. Designer 2: followed the similar approach in calculation but also considered the role of internal pressure as an acting stress in radial direction. Which design is more realistic and represents a more reliable design?

Sample Problem 2 length 500 mm inner diameter 150mm outer diameter 200mm. internal pressure 50MPa. Designer 1: used the thin wall theory

Sample Problem 2 length 500 mm inner diameter 150mm outer diameter 200mm. internal pressure 50MPa. Designer 2: used the thin wall theory and considered radial stress

Sample Problem 3 The state of stress on an infinitesimal element of a linear elastic component is given as a normal stress of 100MPa in one coordinate direction (normal to a plane “p”) and two shear components acting on that plane in the direction of the other two coordinates one positive and one negative, both of magnitude of 50MPa. Find all strain components for the coordinate system in which stresses are given. (E = 200 MPa, ν = 0.3) Write the strain and stress descriptions in matrix form.

Assumptions and limitations (conditions of validity) Is the material subjected to loading: Continuous? ……. Homogeneous? Isotropic? Elastic? Linear? Can you explain what these terms mean? And also what is: The nature of loads, method of application, shape and geometry, etc...

Summary: 3D Stress-Strain relationships Expected learning outcomes: Understand stress state (1D, 2D, 3D) and equivalent stress states? Understand strain state (1D, 2D, 3D) and equivalent strain states? Understand relationships of stress states with strain states? Appreciate interdependence of material constants E, G, and  Appreciate states of plane stress and plane strain Limits of validity of Hook’s Law The concept of experimental measurement (what can be measured?) (for further understanding refer to experimental strain gauging)