# Shell Element Internal Stresses

The basic shell element stresses are identified as S11, S22, S12, S13, and S23. You might expect that there would also be an S21, but S21 is always equal to S12, so it is not actually necessary to report S21. Sij stresses (where i can be equal to 1 or 2 and j can be equal to 1, 2 or 3) are stresses that occur on face i of an element in direction j. Direction j refers to the local axis direction of the shell element. Thus S11 stresses occur on face 1 of the element (perpendicular to the local 1 axis) and are acting in the direction parallel to the local 1 axis (that is, the stresses act normal to face 1). As another example, S12 stresses occur on face 1 of the element (perpendicular to the local 1 axis) and are acting in the direction parallel to the local 2 axis (that is, the stresses act parallel to face 1, like shearing stresses). The figure below shows examples of each of these basic types of shell stresses. ETABS reports internal stresses for shell elements at the four corner points of the appropriate face of the element. For example, refer to Figure "a" below. On the positive 1 face internal stresses are reported by ETABS at points A, B, C and D. Shell internal stresses are reported for both the top and the bottom of the shell element. The top and bottom of the element are defined relative to the local 3-axis of the element. The positive 3-axis side of the element is considered to be the top of the element. Thus in Figure "a" above, internal stresses at the top of the element include stresses at the joints labeled A and C and internal stresses at the bottom of the element include stresses at the joints labeled B and D. The Figure below clearly illustrates the points where ETABS reports the shell element internal stress values. The transverse shear stresses calculated by ETABS (S13 and S23) are average values. The actual transverse shear stress distribution is approximately parabolic; it is zero at the top and bottom surfaces and has its maximum or minimum value at the midsurface of the element. ETABS reports the average transverse shear value. An approximation to the maximum (or minimum) transverse shear stress would be 1.5 times the average shear stress.

The figure below illustrates the positive directions for shell element internal stresses S11, S22, S12, S13 and S23. Also shown are the positive directions for the principal stresses, S-Max and S-Min, and the positive directions for the maximum transverse shear stresses, S-Max-V. For values of S13 and S23 at any angle, the maximum transverse shear stress, S-MaxV, can be calculated from: 