It is often surprising to learn that forces transmitted through the knee, hip, and shoulder joints often exceed three times, two times, and one time the body weight, respectively, during activities of daily living. Why are these forces so high, and how are they known? What are the corresponding forces within bones, soft tissues, and implants? These internal forces are fundamental across the spectrum of orthopaedic care, such as deformity correction, overuse injuries, bone remodeling, implant wear, and fixation failure.

To understand internal forces, the musculoskeletal system can first be considered as a system of rigid links connected at the joints. Moments arise within the joints because of external loads, such as the ground reaction force during walking, or the force of a weight held by the hand. A moment is a turning or twisting load and is used somewhat interchangeably with the word torque. Joint moments are also caused by muscle forces. Because muscles are positioned anatomically close to joints, their pulling force must often be substantial to control joint positions, and this is further described in the next paragraphs. Statics analysis ignores inertial effects associated with body movements. Conversely, more advanced dynamics/kinetics analysis considers these inertial effects based on masses and accelerations. Static equilibrium means that the sum of forces and moments acting on a body segment must equal zero (Newton’s first law). If this sum is nonzero, motion results. For example, an initial approximation of the force in the Achilles tendon during push-off can be estimated by considering the foot as a free body in static equilibrium (Figure 1).

In reality, additional lesser muscles crossing the joints are active, some acting antagonistically, leading to additional joint compression. The aforementioned fundamental concepts have been greatly expanded with modern research, including computational modeling, and in vivo patient measurements with implants containing wireless force sensors.¹ Knowledge of joint forces, along with typical joint motions (kinematics), is
used for purposes such as wear testing of joint replacement components.

In addition to joint and muscle forces, it is often important to understand how these loads are transmitted across tissues and implant constructs, leading to stress (local force intensity, or force per unit area) and strain (local stretching). Failure in musculoskeletal tissues and implants results from excessive stress or strain where the tissues cannot adapt to these local stimuli. There are four basic types of loading usually considered, and during physiologic loading, all four types of loads may be present. Tendons and ligaments resist primarily only tension (axial) loads. For nonlong bones and portions of implants without a long axis, bending and torsion are less applicable. Each of these loads causes deformation and stresses in different ways, as shown Figure 2 and outlined in Table 1.

Area moment of inertia and polar moment of inertia properties depend on cross-sectional geometry of the tissue or implant. Because stresses and strains are higher at the outer surfaces in bending and torsion, these moment of inertia properties depend not only on how much material there is, but how far the material is distributed away from the center. Both normal and shear stresses (and strains) typically exist at a point in an object, and these vary in magnitude across different locations. In simplified scenarios, such as within the diaphyseal section of Figure 2, analytical estimates of stresses can be
made. As discussed in a 2021 study, for analysis of more complex geometries such as metaphyseal regions of bone or implanted constructs, a more sophisticated approach such as finite element computational modeling is needed.²

To experimentally measure the stiffness and strength of a long bone, it can be tested in one of the aforementioned loading modes, or a physiologically relevant combination of loads. In axial testing, the force and displacement of the actuator can be recorded from the testing machine. Analogously in torsion testing, the torque and rotation of the actuator can be recorded. Once the force versus displacement, or torque versus rotation, is plotted, the slope of the resulting linear portion of the curve indicates the axial rigidity or torsional rigidity, respectively. These rigidities depend not only on geometry of the object being tested (moment of inertia), but also on the inherent material properties (elastic moduli) of the object. Longer bones and implants will also displace overall more than shorter ones.

Bending tests involve additional considerations because pure bending moments are not as straightforward to apply. Instead, forces are applied in either of the following ways, and static equilibrium equations can be used to determine locations of maximum bending moment:

The stress-strain relationship, determined from the load-displacement relationship, is central when studying biomechanics of materials, including native tissues and orthopaedic implants. The stress-strain relationship is defined for a material, whereas the load-displacement relationship is assessed for an entire structure and thus depends on structure geometry, in addition to material. By characterizing how a material responds to loading, the stress-strain curve can provide insight into bone fractures or implant failures. Stress is defined as the amount of force applied per unit of cross-sectional area, whereas strain is defined as material lengthening over the original length in response to this stress. The linear elastic region, yield point, plastic region, ultimate strength, and failure are salient features of the stress-strain curve corresponding to elastic deformation, stress point at which the material becomes plastic, plastic deformation, the maximum amount of stress the material can withstand, and material failure. Although tensile loading is typically used to characterize material properties, compression or shear loading curves may also be generated (Figure 3).

Young modulus, or the elastic modulus, is the slope of the elastic region of the stress-strain curve, representing material stiffness. Young modulus is useful for comparing and selecting materials in orthopaedics. For example, bone-implant modulus mismatch is often cited as one of the causes of stress shielding and implant failure.

The plastic region of the stress-strain curve is the region of irreversible material deformation. The elastic and plastic regions are separated by the yield point, above which the material may sustain permanent damage. After the yield point, the material undergoes plastic elongation or yielding with little increase in stress. Strain hardening, which occurs because of the material undergoing changes in atomic and crystalline structure, is a phenomenon where plastic deformation increases material resistance to further deformation.

The ultimate tensile strength is the maximum stress that the material can withstand before failing. After the ultimate strength point, lengthening may continue for ductile materials, but with a reduction in stress. This is associated with necking of the material, whereby the cross-sectional area is reduced. Because orthopaedic implants typically fail through cyclic loading rather than acute loading, the ultimate tensile strength is less relevant than fatigue strength.

In vivo physiologic loading is cyclic with every cycle of gait, arm reach, etc. Fatigue strength, or fatigue limit, is defined as the highest stress the material can withstand for a given number of cycles. Incremental increases in loading cycles result in eventual failure that is brittle in nature. When repetitive cyclic loading below the yield strength results in failure, it is termed fatigue failure.