Mesh convergence study
The maximum stress and maximum displacement on the left SIJ surface of sacrum were analyzed for each of the meshes, under MOP-F1, F2, F3 and F4, which are shown in Fig. 3. The differences in maximum stress and maximum displacement between mesh 3 and mesh 4 under four MOPs were less than 5%, which was considered as reasonably close ranges. According to these results, mesh 3 with 458,867 elements was selected for further study.
Model validation
The stresses were located mainly in the upper and posterior areas of the acetabulum and extended to the iliac crest, the incisura ischiadica major, and the rear acetabulum. The area of stress concentration and maximum value of stress were consistent with those reported in a previous study [23]. Under eight loading conditions, the displacements were agreement not only with those in an experimental study but also with those in some computational studies [21, 24, 25], and these results are shown in Fig. 4.
Stress of pelvis and SIJ
The stress distributions of pelvis under four MOPs are shown in Fig. 5. Under MOP-F1, the stress of ventral pelvis was mainly concentrated on the left SIJ, extended to the arcuate line, the right SIJ, the right anterior inferior iliac spine, the upper part of right acetabulum, and the outer upper edge of right pubis. While the stress of dorsal pelvis was mainly concentrated around the left posterior inferior iliac spine and the greater ischial notch. The maximum stress value was 76.9 MPa. Under MOP-F2, the area of stress concentration was roughly the same as that under MOP-F1, but the maximum stress value was higher, at 85.0 MPa. Under MOP-F3, the stress of ventral pelvis was mainly concentrated on the left iliac crest, the left SIJ, the arcuate line, and the superior ramus of pubis, while the stress of dorsal pelvis was mainly concentrated on the left iliac crest and the greater sciatic notch. The maximum stress value was 52.6 MPa. Under MOP-F4, the area of stress concentration was roughly the same as that under MOP-F3. The maximum stress value was 80.0 MPa.
The distributions of stresses on the SIJ surface of sacrum are shown in Fig. 6. Under four MOPs, the principal stresses were concentrated on the anterior and inferior part of left SIJ. Higher stress was observed on the left SIJ for the four MOPs. Among them, MOP-F3 produced the highest stress on the left SIJ, at 6.6 MPa, while MOP-F1 produced the lowest stress on the left SIJ, at 5.6 MPa.
Displacement of SIJ
In MOP-F1, the displacements of the left SIJ were 1.088, 0.305 and 0.033 mm in the anterior-posterior (AP), superior-inferior (SI) and medial-lateral (MI) direction, respectively. In MOP-F2, the displacements were 1.211, 0.186 and 0.064 mm in the AP, SI and MI direction, respectively. In MOP-F3, the displacements were 0.962, 0.048 and 0.117 mm in the AP, SI and MI direction, respectively. In MOP-F4, the displacements were 1.105, 0.064 and 0.094 mm in the AP, SI and MI direction, respectively. The four MOPs mainly produced anterior-posterior displacement. The displacement of the left SIJ under four MOPs are shown in Fig. 7.
Strain of ligaments
The strains of six ligaments under four MOPs are shown in Fig. 8. For most of ligaments, the strain of the left ligament was greater than that of the right ligament under four MOPs. In MOP-F1, the left SS, ASL and ISL had a higher strain value, which were 3.71, 1.41 and 1.36%, respectively. In MOP-F2, the left SS, ST and ASL had a higher strain value, which were 4.29, 1.51 and 1.28%, respectively. In MOP-F3, the left SS, ASL and ST had higher strain value, which were 3.05, 1.61 and 1.09%, respectively. In MOP-F4, the left SS, SPSL and ASL had a higher strain value, which were 2.85, 1.90 and 1.04%, respectively.