Radial head fractures are becoming a major public health problem and are an increasingly important target for both clinical and mechanical researchers. In this work, comparative biomechanical analyses of the stress-strain state of a healthy elbow joint and elbow joints with radial head compression from 2 to 5 mm due to injury are performed. Three-dimensional models of the elbow joint with cartilage surfaces and ligaments were constructed based on the results of computed tomography. This study is focused on an elbow joint range of motion ranging from 0° to 120° flexion. Analysis of the stress-strain state of cartilage and ligaments under the influence of functional loads is conducted using a finite element method (FEM) and the ABAQUS software package. The results show that with increasing compression of the radial head, contact stress increases at the olecranon, which can lead to cartilage damage. Analysis of displacement shows that compression of the radial head during full extension of the elbow joint leads to an increased humeral shift from 1.14° ± 0.22 in the healthy joint to 10.3° ± 2.13 during 5-mm compression of the radial head. Mathematical modeling performed in this study proved that reducing the height of the radial head and the contact area between the radial head and the humeral head led to increased medial collateral ligament stresses of up to 36 ± 3.8 MPa. This work confirmed that the head of the radius is the main stabilizing structure of the elbow joint and that the medial collateral ligament is the second structure responsible for valgus stability of the elbow joint.