1 - Complex Stress and Transformations

Question 1

Plane Stress

Answer 1

• Stress element – infinitesimal in size
• xyz axes are parallel to the edges of the element
• Faces of element are designated by the directions of their outward normal
• Normal stress is indicated by the symbol sigma, subscript indicates the face on which it acts
• Equal normal stresses act on the opposite sides
• Sign convention is positive (+ve) for tension and negative (-ve) for compression

Question 2

Shear stress

Answer 2

• Indicated by the symbol tau and has two subscripts
  
  • First denotes face on which stress acts
  • Second gives direction on that face
• Sign convention:
  
  • Positive: Positive face in the positive direction of the axis. Negative face in a negative direction of the axis
  • Negative: Positive face of the element in the negative direction
• tau(xy) = tau(yx)

Question 3

Stress on inclined sections

Answer 3

• Centre of element in same place
• The new element has axes x1 , y1 and z1 .
• The z1 axis coincides with the z axis whilst the x1 and y1 axes are rotated anticlockwise through an angle theta wrt the xy axis

Question 4

Transformation equations for plane stress - Procedure

Answer 4

• Construct a free body diagram showing the forces acting on the faces
• Denote the area of the angled face as A
• Normal and shear forces acting on the face are sigma_x1A and tau_xyA
• The area of the bottom (negative y face) is Asin(theta) and the area of the left hand side face is Acos(theta)
• Normal and shear forces acting on these faces have magnitudes and directions shown

Question 5

Transformation equations for plane stress

Answer 5

Question 6

Uniaxial stress:

Answer 6

If all stresses acting on the xy element are zero except for the normal stress x then the element is in uniaxial stress

Question 7

Pure shear stress

Answer 7

Obtained when sigma(x)=0 and sigma(y)=0 is substituted

Question 8

Biaxial stress:

Answer 8

The xy element is subjected to normal stresses in both the x and y directions but without any shear stresses. i.e. tau(xy)=0.