2.17 For the 0-45°-90° strain rosette of Example 2.6, find general expressions for the principal strains and maximum shear strains in terms of the measurable values 80, E45°, and E90. Example 2.6 Design an experimental set-up using strain gauges whereby one can measure a complete 2-D strain in a small region (i.e., averaged over a small region even though strain is, strictly speaking, defined at a point). Solution: From Eq. (2.57), we see that an extensional strain relative to a primed coordinate system is related to the 2-D components of strain relative to an original coordinate system. If an extensional strain exx is measurable by a strain gauge, then measuring three extensional strains would provide three equations for the three components Exx, Eyy, and Exy; that is, as illustrated in Fig. 2.22, " = Exx cos² α1+2exy cos a₁ sin a₁ + Eyy sin² a₁, Exx Exx cos2 a2 + 2exy cos a2 sin a2 + Eyy sin² a2, Exx = Exx cos² α3+2exy cos a3 sin α3 + Eyy sin² α3, x strain gauge placements FIGURE 2.22 Placement of three strain gauges to form a so-called strain rosette. It is assumed that each gauge is affixed to the surface at a known angle. Although small, strain gauges are obviously of finite, not infinitesimal, size, thus information from rosettes necessarily represent mean values of strain within the region of measurement. where the angles a₁, a2, and a3 relate the coordinate systems to a baseline x direction. Clearly then, these equations represent three equations in terms of three unknowns (Exx Eyy, Exy) provided that exx xx xx a1, a2, and a3 are measured. (Note: Whereas the strains come from the resistance changes, the angles are known because we are the ones who glue the gauges onto the surface). Although any values of a1, a2, and a3 are fine, certain values are preferred. For example, a₁ =0, a₂ =л/4, and α3=л/2 radians or a₁ = 0, α₂ =л/3, and α3=2/3 radians are common. For example, let us consider the former case: E = (α= 0) = Exx == x (a = 1) = exx (¥)² - E45 E +2exy (/)(/)+ex (4) 2 Eyy! E90 E + Ex (a = ½) = Eyy* " Hence, if, are known from the gauges, Exx=0, Eyy=E90°, and Exy E45-E0/2-E90/2 are thereby measurable. έχ = Exx cos 2a + 2exy sin a cos a + εyy sin²α, Eyy Exx sin²α-2exy sin a cosa + Eyy cos²a, = exy = 2 sin a cosa (Eyy Ex) + (cos ²a - sin²a)Exy, 2 (2.57) where a is again the angle that relates the (o; x, y, z) and (o; x, y, z) Cartesian coordinate systems. Note: If a=0, then the components relative to the two systems are equal, as they should be. Similarly, principal values for strain are