two dimensional and 3d loading

ICBT CAMPUS-MTHND in Engineering

Module : Mechanical PrinciplesLecturer : Mihiran GalagedaraBatch: ME_03; AU_01; MT_01Copies : 65Date : 02/04/2013Session : 01 0f 15

Two- and three-Dimensional

LoadingBy: Mihiran GalagedaraB.Sc. Eng (Hons)UOM, AMIESL

Faculty of Engineering and Construction, ICBT Campus-MT,Sri Lanka.

DefinitionsIsotropic MaterialsMaterials are considered to be isotropic if the properties are not dependent on the direction.

- Mass and density- Modulus of Elasticity- Poisson's Ratio- Yield strength etc.

Orthotropic Materials• In this type of material, the elastic

constant have different values in the x, y and z directions so the results obtained in a test depend upon the direction in which the specimen was cut from the bulk material.

• This applies to materials with grain structures such as wood or rolled materials

Orthotropic Materials

Non Isotropic Materials• In this type of material, the elastic

constants are unpredictable and the results from any two tests are never the same.• This applies to materials such as glass

and other ceramics

Two Dimensional Loading

• All the examples considered so far in this chapter have dealt with slender members subjected to axial loads.• i.e., to forces directed along a single

axis.

Two Dimensional Loading contd…

• A two-dimensional stress system is one in which all the stresses lie within one plane such as the X-Y plane.


• The total strain in the X direction will therefore be given by:

• and the total strain in the Y direction will be:


• If any stress is, in fact, compressive its value must be substituted in the above equations together with a negative sign following the normal sign convention.

Quiz1. A material has stress of 2 MPa in the

x direction and 3 MPa in the y direction. Given the elastic constants E = 205 GPa and v = 0.27,calculate the strains in both directions.

Quiz2. A material has stresses of -2 Mpa in

the x direction and 3 Mpa in the y direction. Given the Elastic constant E = 205 Gpa and v = 0.27, calculate the strains in both directions

Three Dimensional Loading


• For an element subjected to multi-axial loading, the normal strain components resulting from the stress components may be determined from the principle of superposition. This requires:

1) Strain is linearly related to stress2) Deformations are small


EEE

EEE

EEE

zyxz

zyxy

zyxx

• With these restrictions:

Quiz3. The steel block shown is subjected to a

uniform pressure on all its faces. Knowing that the change in length of edge AB is -1.2 × 10-3 in., determine

a. The change in length of the other two edges,b. The pressure p applied to the faces of the

block. Assume E = 29 × 106 psi and v = 0.29.

References1. Hearn E.J, (2000), Mechanics of Materials, ISBN

0 7506 3265 82. Nash.W, Potter M.C, (2011), Strength of

Materials, ISBN: 978-0-07-163507-33. Johnston R, De Wolf J,(2000) Statics and

Mechanics of Materials, McGraw-Hill , ISBN 978-0-07-338015-5

two dimensional and 3d loading

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