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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.

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 F_x and F_y. The stresses produced in the X and Y directions are as follows:

The X-­direction stress (σ_x) ­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 σ_x:

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

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

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

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

The shear stresses (τ_xy, τ_yz, τ_zx) produced in triaxial loading are shown in figure 3.30b.

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