Reference: Abbott, Richard. Analysis and Design of Composite and Metallic Flight Vehicle Structures 3 Edition, 2019

General use of failure envelopes; resulting stresses inside the envelope show adequate strength, those outside show inadequate strength:

Figure 3.4.3‑1: General Application of Two-Dimensional Failure Envelopes

The regions of the 2D principal stress plane correspond to the various possible modes of loading as follows (the Tresca Envelope is shown in this diagram for reference only):

Figure 3.4.3‑2: Regions of 2D Principal Stress Field

Most failure envelopes are plotted on a two-dimensional principal stress graph. This simplifies the approach as the shear stress is zero in the principal directions.

3.4.3.1. Mohr’s Circle

The definition of Mohr’s Circle is in part taken from  (NASA TM X-73305, 1975).

Figure 3.4.3‑3: Mohr’s Circle Definition

The following text is taken directly from  (NASA TM X-73305, 1975):

1. Make a sketch of an element for which the normal and shearing stresses are known and indicate on it the proper sense of those stresses.
2. Set up a rectangular coordinate system of axes where the horizontal axis is the normal stress axis, and the vertical axis is the shearing stress axis. Directions of positive axes are taken as usual, upwards and to the right.
3. Locate the center of the circle which is on the horizontal axis at a distance of (fx + fy)/ 2 from the origin. Tensile stresses are positive, compressive stresses are negative
4. From the right-hand face of the element prepared in step (1), read off the values for the f_x and f_s and plot the controlling point “a”. The coordinate distances to this point are measured from the origin. The sign of f_x is positive if tensile, negative if compressive; that of f_s is positive if upwards, negative if downward.
5. Draw the circle with the center found in step (3) through controlling point “a” found in step (4). The two points of intersection of the circle with the normal-stress axis given the magnitudes and sing of the two principal stresses. If an intercept is found to be positive, the principal stress is tensile, and conversely.
6. To find the direction of the principal stresses, connect point “a” located in step (4) with the intercepts found in step (5). The principal stress is given by the particular intercept found in step (5) acts normal to the line connecting this intercept point with the point “A” found in step (4)
7. The solution of the problem may then be reached by orienting an element with the sides parallel to the lines found in step (6) and by indicating the principal stresses on this element.

To determine the maximum or the principal shearing stress and the associated normal stress:

1. Determine the principal stresses and the planes on which they act per the previous procedure.
2. Prepare a sketch of an element with its corners located on the principal axes. The diagonals of this element will this coincide with the directions of the principal stresses.
3. The magnitude of the maximum (principal) shearing stresses acting on mutually perpendicular planes is equal to the radius of the circle. There shearing stresses act along the faces of the element prepared in step (2) towards the diagonal, which coincides with the direction of the algebraically normal stress.
4. The normal stresses acting on all the faces of the element are equal to the average of the principal stresses, considered algebraically. The magnitude and sign of these stresses are also given by the distance from the origin of the coordinate system to the center of Mohr’s circle.

AA-SM-041-030 Stress Analysis – Mohrs Circle

3.4.3.2. Maximum Principal Stress Envelope

The maximum principal stress envelope assumes that both the maximum and minimum principal stresses can occur at the same point simultaneously. This is likely to be optimistic and fail to predict material failure when it is likely to occur. For this reason, this envelope is not used for analysis purposes and is included here for information only.

Figure 3.4.3‑4: Maximum Principal Stress Envelope

In this envelope, the 45-degree line that illustrates the pure shear condition extends into a region beyond the typical shear strength of most materials.

3.4.3.3. Tresca Criterion

The Tresca Criterion is also called the maximum shear stress criteria.

Figure 3.4.3‑5: Tresca Envelope

For the Tresca Criterion, the allowable shear stress = the allowable tensile stress divided by two:

Or, assuming linear material behavior up to ultimate failure level:

This criterion regarding  shear stress is conservative for almost all metals as the shear strength is greater than half of the tensile strength.

The Tresca stress tensor (or the effective stress) should be compared to the allowable material strength. The Tresca stress tensor can be calculated from the principal stresses in the following way:

AA-SM-041-031 Stress Analysis – Tresca

The Tresca criterion is conservative compared to the more realistic Von Mises Criterion.

3.4.3.4. Von Mises Criterion

The Von Mises criterion is also called the octahedral shear stress criterion. When plotted for plane stress states the Von Mises stress envelope is an ellipse.

For the Von Mises Criterion, the allowable shear stress = the allowable tensile stress divided by the square root of three.

Or, assuming linear material behavior up to ultimate failure level.

Figure 3.4.3‑6: Von Mises Envelope

AA-SM-041-020 Stress Analysis – 2D Von Mises Stress

The Von Mises criterion is considered generally representative for ductile materials and the relationship between Fsu/Fsy and Ftu/Fty is close enough for most analysis purposes.