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INPUT   DATA

♥ EXAMPLE Of Input/Output

Title

Normal stress in x-direction, σx
Normal stress in y-direction, σy
Normal stress in z-direction, σz
Shear stress in xy-plane, τxy
Shear stress in yz-plane, τyz
Shear stress in xz-plane, τxz

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OUTPUT   VARIABLES   &   GRAPHS

VARIABLES	Values
♦ Principal stress, σ1		Graphs:  3D Mohr's circle
♦ Principal stress, σ2
♦ Principal stress, σ3
♦ Max Shear stress, τmax1
♦ Max Shear stress, τmax2
♦ Max Shear stress, τmax3

Principal Stresses And The Mohr's Circles

Consider a triaxial stress problem, with six known stress components: normal stresses σ_x, σ_y, and σ_y, and three shear stresses τ_xy, τ_xy and τ_xy. This combination results in 6 stress components: three principal stresses σ₁, σ₂ & σ₃, and three maximum shear stresses τ_max1, τ_max2 & τ_max3. The 3 principal stresses can be found by solving for the 3 roots of the cubic equation below. And then the 3 associated shear stresses can be calculated.

where
     σ_x = normal stress in x direction
     σ_y = normal stress in y direction
     σ_z = normal stress in y direction
     τ_xy = shear stress in xy-plane
     τ_yz = shear stress in yz-plane
     τ_xz = shear stress in xz-plane
     σ₁ = principal stress
     σ₂ = principal stress
     σ₃ = principal stress
     τ_max1 = maximum shear stress
     τ_max2 = maximum shear stress
     τ_max3 = maximum shear stress

All stress values are given here in the same units.

The Mohr's circle here is a 3-circle graphical representation of the analytical equations above. From it, the characteristics and extremas of the stresses on the element can be determined.

Tips

    ◊ Use link EXAMPLE Of Input/Output  to demo data entry expectations and results; you may edit & use it as starting point
    ◊ If the required Java plug-in not installed on your computer, an auto-download of this plug-in will be initiated before the plot is displayed.

BIBLIOGRAPHY

• Brown TH; Marks' Calculations for Machine Design; McGraw Hill, New York, 2005, Chapter 5.
• Young WC & Budynas RG; Roark's Formulas for Stress and Strain; 7th Edition, McGraw Hill, New York, 2002.
• Mohr's Circle for 2-D & 3-D stress analysis, www.engapplets.vt.edu/Mohr/java/nsfapplets/MohrCircles2-3D/Applets/applet.htm; Virginia Tech Department of Aerospace and Ocean Engineering, Blacksburg VA, USA.