ISSUES IN COMPOSITE STRUCTURAL DESIGNMAE4301/AE5339/ME5339 Spring 2014
MAE4301/AE5339/ME5339 Spring 2014
1.3 Special Stresses
1.7 Stress Transformation
1.9 Shear stress
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
1.1 Concept of Stress
Stress is the term defining the intensity and the direction of the internal forces acting at a given point on a specific plane.
The stress vector is defined as
3 stress vector components for a given plane
(1-1)
MAE4301/AE5339/ME5339 Spring 2014
Stress Components
If the plane is selected at the normal of the plane along the x-direction, then we have,
Normal Stress components: x, y, z
Shear Stress components: xy, yz, xz; yx, zy, zx
6 Components, Only 3 components are independent.
Normal Stress Shear Stress
MAE4301/AE5339/ME5339 Spring 2014
Refers to the direction
Refers to the plane
MAE4301/AE5339/ME5339 Spring 2014
MAE4301/AE5339/ME5339 Spring 2014
1.2 Sign Convention of Stress
“+” Stress – Stress points to the “+” direction of the axis on the “+” plane
(The normal of the plane is in the “+” direction)
“-” Stress – Stress points to the “-” direction of the axis of the “+” plane.
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
1.3 Special Stresses:
Pure Shear – No normal stresses.
Plane Stress –Stress on one of the three planes are equal to zero.
(2-D Stress)
Uniaxial stress– Normal stress acting on one direction only.
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
MAE4301/AE5339/ME5339 Spring 2014
1.4 Equation of Equilibrium
 
where Fx, Fy and Fz are so called body forces such as gravitational force, magnetic force, etc.
(1-3)
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MAE4301/AE5339/ME5339 Spring 2014
 
(1-5)
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MAE4301/AE5339/ME5339 Spring 2014
 
 
MAE4301/AE5339/ME5339 Spring 2014
 
 
(1-6)
or
MAE4301/AE5339/ME5339 Spring 2014
1.7 Stress Transformation
MAE4301/AE5339/ME5339 Spring 2014
 
 
MAE4301/AE5339/ME5339 Spring 2014
 
(1-8)
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MAE4301/AE5339/ME5339 Spring 2014
3-Dimensional Stress Transformation
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3-Dimensional Stress Transformation
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2-Dimensional Stress Transformation
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(1-11)
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
MAE4301/AE5339/ME5339 Spring 2014
(1-12)
MAE4301/AE5339/ME5339 Spring 2014
1.7.2 Plane Stress Transformation (Cont’d)
Consider the state of stress in the x’-y’-z’ coordinate system transferring a “-” to the x-y-z coordinate system. Then we can write
(1-13)
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
Equation (1-6) gives,
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
(1-13a)
Three roots of the above equation(1-13) are the principal stresses. The corresponding three sets of directional cosines are their directions.
Expanding the determinant, we have
where
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The stress invariants
Chapter 1-- *
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Absolute Shear Stress
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Solution:
ksi
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Using
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MAE4301/AE5339/ME5339 Spring 2014
MAE4301/AE5339/ME5339 Spring 2014
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
For max. shear on the x-y plane, i.e. n=0
(1-14)
MAE4301/AE5339/ME5339 Spring 2014
1.10 Octahedral Planes and Stresses
An octahedral plane is a plane cut across one of the corners of a principal stress element
The normal stresses acting on these planes are identical; so do the shear stresses. There are given as,
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
1.10 Octahedral Planes and Stresses (Cont’d)
The normal stresses acting on those planes are identical; so do the shear stresses. There are given as,
(Stress Invariant)
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
1.10 Octahedral Planes and Stresses(Cont’d)
The directional cosines of there surfaces l,m,n are the combination of
The octahedral normal stress tends to enlarge or compress the octahedron but not distort it.
The octahedral shear stress tends to distort it but not change its volume.
Octahedron
Putting all eight surfaces together, we obtain an octahedron as:
The interest in the octahedron surface is in a failure criterion that states “the failure of this element is strictly dependent on the shear stresses on these surfaces”.
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
From vector analysis, we have
Chapter 1-- *
MAE4301/AE5339/ME5339 Spring 2014
zx
xz
xy
yx
t
t
t
t