Triaxial State of Stress at any Critical Point in a Loaded Body

2. Triaxial State of Stress at any Critical Point in a Loaded Body • Cartesian stress components are found first in selected x-y-z coordinate axes (Fig. 4.1) • Three mutually perpendicular principal planes are found at unique orientations: • NO Shear stresses on these planes • Principal Normal Stresses: 1, 3, 2, one of which is maximum normal stress at the point • Three mutually perpendicular Principal Shearing Planes (Planes of max. shear) • Principal Shearing Stresses: 1, 2, 3, one of which is the maximum shear stress at the point • Normal stresses are NOT zero, NOT principal stresses and depend on the type of loading

3. 3-D Stress Transformations Equations • Relate Known Cartesian Stress Components at Any Point with Unknown Stress Components on Any Other Plane through the SAME Point • From equilibrium conditions of infinitesimal pyramid • Important for the Unique Orientations of the Principal Normal and Shearing Planes • Stress Cubic Equation- three real roots are the Principal Normal Stresses, 1, 2, 3. • Principal Shearing Stresses can be calculated from the Principal Normal Stresses as follows:

4. Mohr’s Circle Analogy for Stress – Graphical Transformation of 2-D Stress State • Principal stress solution of stress cube eq. for stresses in the x-y plane (Fig. 4.12) • Analogy with the equation of a circle plotted in the - plane leads to Mohr’s circle for biaxial stress: • Sign convention for plotting the Mohr’s circle (normal stress is positive for tension, shear is positive for clockwise (CW) couple) • Two additional Mohr’s circle for triaxial stress states • Find orientation of principal axes from the Mohr’s circle

6. Strain Cubic Equation and Principal Strains • STRAIN – a measure of loading severity, defining the intensity and direction of deformation at a point, w.r.t. specified planes through that point. • Strain state at a point is completely defined by: • Three normal and three shearing strain components in the selected x-y-z coordinate system, OR • Three PRINCIPAL strains and their directions from Strain Cubic Equation (similar to Stress Cubic Equation, where ’s are replaced by ’s, and shear stresses, ’s, are replaced by one-half of ’s.)

7. Summary of Example Problems • Example 4.8 – Principal Stresses in Beam • Hollow cylindrical member subjected to transverse forces (four-point bending), axial force and torque • Sketch state of stress at critical point (bottom edge) • Use “stress cubic equation” to find principal stresses from calculated Cartesian stresses at critical point • Find principal shear stresses from principal normal ones • Example 4.9 –Mohr’s Circle for Stress • Semi-graphical analysis of biaxial stress state at critical point of previous example cube is replaced by 2-D sketch in x-y plane • Principal normal and shear stresses are found graphically from the basic and the two additional Mohr circles, respectively

11. Summary of Textbook Problems – Problem 4.31, Principal Stresses • Identify critical points for each of the three types of loading applied on the bar • Locations where stresses are amplified by superposition of effects from different loads – top end of vertical diameter and left end of horizontal one • Sketch infinitesimal cube elements for the states of stress at critical points • Calculate stresses at critical points, in the given system of Cartesian coordinates • Use “stress cubic equation” to find the principal stresses at each of the two critical points • Top edge: 1=26,047 psi, 2=0, 3=-7683 psi • Left edge: 1=31,124 psi, 2=0, 3=-31,124 psi • Calculate the maximum shearing stress at each point • Top edge: max= 16,865 psi, while at the left edge: max= 31,124 psi