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Multiaxial loading and its effects

This is an excerpt from Biomechanics of Injury-3rd Edition by Ronald F. Zernicke,Steven Broglio & William C. Whiting.

We have so far considered only the ­simple case of uniaxial loading. In most real-­life situations, however, the forces applied to a body are multidimensional, and hence an understanding of multiaxial loading and its effects is essential. An analy­sis of multiaxial loading uses the same stress and strain concepts just discussed and extends them into two-­ and three-­dimensional space. Although the biaxial and triaxial responses are illustrated for tensile loading only, the concepts are equally applicable to compressive loading and to force vectors with reversed orientation. The following formulations are for linearly elastic materials.

Figure 3.29 Biaxial loading. (a) Applied forces in the X and Y direction (Fx and Fy, respectively) load the material biaxially. (b) Elongation caused by Fx and the resulting perpendicular contraction in the Y and Z directions. Dashed lines indicate conformation prior to loading. Solid lines indicate conformation while loaded.
FIGURE 3.29 Biaxial loading. (a) Applied forces in the X and Y direction (Fx and Fy, respectively) load the material biaxially. (b) Elongation caused by Fx and the resulting perpendicular contraction in the Y and Z directions. Dashed lines indicate conformation prior to loading. Solid lines indicate conformation while loaded.

Biaxial (Two-­Dimensional) Loading Responses Consider a three-­dimensional body (figure 3.29) with sides of length X′, Y′, and Z′ that is subjected to axial forces Fx and Fy. The stresses produced in the X and Y directions are as follows:

(3.36) (3.37)

The X-­direction stress (σx) ­will, according to Poisson’s effect, cause deformation in all three directions. Elongation in the X direction and contraction in the Y and Z directions are shown in figure 3.29b. By applying equations 3.34 and 3.35, we obtain X-­ and Y-­direction strains attributable to σx:

(3.38) (3.39)

We similarly obtain Y-­ and X-­direction strains attributable to σy:

(3.40) (3.41)

To obtain the combined effect of σx and σy, we add the strain effects just calculated. The net strains in the X and Y directions are then

(3.42) (3.43)

We have presented two normal stresses, σx and σy. As previously described, a tangential, or shear, stress (τ) is also created. In the case of biaxial loading, shear stress (τxy) is created as shown in figure 3.30a.

Figure 3.30 Shear stresses in response to biaxial and triaxial loading. (a) Shear stress (τxy) created by biaxial loading. ­Because of equilibrium conditions, τxy = τyx. (b) Triaxial tensile stresses (σx, σy, σz), shown as solid vectors, result in shear stresses (τxy, τyz, τzx), depicted as broken-­line vectors. Equilibrium constraints dictate that τxy = τyx, τyz = τzy, and τzx = τxz.
FIGURE 3.30 Shear stresses in response to biaxial and triaxial loading. (a) Shear stress (τxy) created by biaxial loading. ­Because of equilibrium conditions, τxy = τyx. (b) Triaxial tensile stresses (σx, σy, σz), shown as solid vectors, result in shear stresses (τxy, τyz, τzx), depicted as broken-­line vectors. Equilibrium constraints dictate that τxy = τyx, τyz = τzy, and τzx = τxz.
Figure 3.31 Triaxial loading. Applied forces in the X, Y, and Z directions (Fx, Fy, and Fz, respectively) load the material triaxially.
FIGURE 3.31 Triaxial loading. Applied forces in the X, Y, and Z directions (Fx, Fy, and Fz, respectively) load the material triaxially.

Triaxial (Three-­Dimensional) Loading Res­ponses The addition of a third axial force in the Z direction complicates the conceptual model only slightly, whereas the mathematical aspects of the model become quite involved. Focusing on the conceptual application (figure 3.31), we now have

  • Fx, which creates σx = Fx/(y′ · z′) and produces elongation in the X direction and contraction in the Y and Z directions
  • Fy, which creates σy = Fy/(x′ · z′) and produces elongation in the Y direction and contraction in the X and Z directions
  • Fz, which creates σz = Fz/(x′ · y′) and produces elongation in the Z direction and contraction in the X and Y directions

The equations for resultant strains seen in triaxial loading are

(3.44) (3.45) (3.46)

The shear stresses (τxy, τyz, τzx) produced in triaxial loading are shown in figure 3.30b.