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Biomechanical analysis of femoral stress response during squatting: A combined multibody dynamics and finite element approach
⁎Corresponding author: Yafeng Li. liyf@tiangong.edu.cn
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Received: ,
Accepted: ,
This article was originally published by Reed Elsevier India Pvt. Ltd. and was migrated to Scientific Scholar after the change of Publisher.
Abstract
Abstract
Inappropriate squat postures are prone to induce femoral musculoskeletal injuries. Knee flexion angle (α) and stance width (L) are critical governing parameters, yet their specific impacts on femoral mechanical responses during squatting have not been fully elucidated. This study aims to analyze the influence mechanisms of these kinematic factors on femoral stress distribution.
This study employed a combined multibody dynamics and finite element analysis (MBD-FEA) method. Hip joint reaction forces during squats under varying α and L conditions were computed via the AnyBody Modeling System and subsequently applied as boundary conditions to a femoral finite element model, to clarify the mechanisms by which α and L influence femoral mechanical responses.
The results demonstrated that during squatting, tensile stress on the anterior femoral shaft was consistently greater than compressive stress on the posterior shaft, while tensile stress in Ward's triangle was persistently lower than compressive stress in the posterior femoral neck. As α increased, femoral stress displayed a nonlinear growth pattern characterized by “a rapid initial rise followed by slowing growth.” The effect of L on femoral stress was dependent on α: when α < 105°, stress increased progressively with increasing L; when α > 105°, this trend was reversed.
From a biomechanical standpoint, this study provides a theoretical foundation for the optimization of squat postures and the prevention of associated injuries.
Keywords
Femur
Squat movement
Knee flexion
Stance width
1 Introduction
The squat is a canonical strength and fitness training movement widely integrated into athletic performance enhancement and clinical rehabilitation regimens.1 Beyond promoting lower limb muscle hypertrophy and augmenting explosive power, it also acts as a key intervention for restoring lower limb motor function after musculoskeletal injury.1 However, improper squat execution can elicit a range of musculoskeletal injuries. Pereira et al.2 conducted a systematic review and demonstrated that inappropriate squat kinematics precipitate knee joint tensile overload, which increases the risk of patellofemoral pain syndrome and associated pathologies like muscle strains and ligamentous sprains. Kothurkar et al.3 utilized finite element analysis and revealed that non-standard postures induce marked articular cartilage stress elevation, which directly amplifies the risk of chondral injury. Escamilla et al.4 established that recurrent improper squatting causes aberrant patellofemoral joint load accumulation; under chronic loading, this pathological accumulation accelerates articular structure degeneration, heightening the likelihood of osteoarthritis onset. These adverse outcomes are directly linked to the kinematic parameters governing squat form. Among these parameters, knee flexion angle and stance width profoundly influence the lower extremities’ mechanical loading pattern, and are thus closely associated with exercise-related musculoskeletal injury incidence.
However, most existing relevant studies have focused primarily on the dynamic characteristics, joint reaction forces, and muscle activation profiles of the knee, hip, and ankle joints, while systematic research into the internal mechanical response mechanisms of the femur under varying squat postures remains notably scarce. Straub et al.5 examined the effects of adjustable squat parameters (including trunk inclination and stance width) on hip and knee joint moments and muscle demands, clarifying these parameters’ regulatory role in joint moment distribution. For weightlifters, Pürzel et al.6 investigated contact force changes across the hip, knee, ankle, and patellofemoral joints during squats at 70 %–90 % 1-repetition maximum (1-RM) intensity, quantifying the pattern of joint contact force elevation with rising intensity. Nevertheless, none of the studies above have addressed the femoral mechanical response mechanisms.
Therefore, this study adopts a combined multibody dynamics and finite element analysis (MBD-FEA) approach to elucidate the influence mechanisms of knee flexion angle and stance width on femoral mechanical responses during squatting. The results of this work will provide a theoretical basis for athletic performance enhancement, squat posture optimization, and prevention of exercise-related injuries from a biomechanical standpoint.
2 Methods
2.1 Model establishment
A 3D model of the right femur was segmented and extracted from computed tomography (CT) data of a male volunteer (height: 175 cm, body weight: 75 kg; data provided by Tianjin Hospital) using Mimics software, based on threshold ranges (cortical bone: 1344–2516 HU; cancellous bone: 1125–1344 HU). The model surface was then subjected to post-processing via Geomagic Studio software, including denoising, remeshing and spike removal, for geometric quality optimization (as shown in Fig. 1).
To obtain hip joint reaction forces during squatting, a musculoskeletal multibody dynamics model matching the volunteer's anthropometric data was established in AnyBody software, and squat tempo parameters were defined. Relevant studies have demonstrated that a slow eccentric (descending) phase tempo (e.g., 4 s) effectively promotes strength gains and quadriceps hypertrophy, with a marked effect on the vastus lateralis.7 Moderate tempos (2–4 s per repetition) yield comparable maximum strength gains to fast tempos, while being more conducive to movement control and injury prevention.8 Proper regulation of descending tempo and load reduces joint impact, lowering injury risk.9 Drawing on these findings, this study set the full squat cycle (including eccentric descending and concentric ascending phases) to 3 s (1.5 s per phase), with a simulation output frequency of 30 Hz. This configuration simulates a moderate-speed, controlled exercise mode, mitigating injury risks from overly rapid movements.
2.2 Squat kinematic parameters and hip joint coordinate system
Previous studies have classified squat movements into four categories based on knee flexion angle: deep squat (femorotibial angle: 40°–45°), parallel squat (femorotibial angle: 60°–70°), half squat (femorotibial angle: 80°–100°), and quarter squat (femorotibial angle: 110°–140°).10 Another classification divides squats into three types: half squat (femorotibial angle: 140°), parallel squat (femorotibial angle: 80°–110°), and deep squat (femorotibial angle <80°).11 Additional research has indicated that classifying squats into shallow squat (supplementary angle of femorotibial angle: 0°–90°), moderate squat (supplementary angle of femorotibial angle: 90°–100°), and deep squat (supplementary angle of femorotibial angle >110°) facilitates distinguishing differences in joint stress and muscle recruitment patterns.5 The lack of a unified squat classification standard has led to inconsistent terminology in relevant studies. Thus, this study integrated these perspectives and established a squat classification framework suitable for the present analysis. Taking α (defined as the knee flexion angle in Fig. 2(a), where 0° represents full extension) as the calibration index, squats were defined as follows: shallow squat (α < 70°), half squat (70° ≤ α ≤ 110°), and deep squat (α > 110°). To comprehensively cover these ranges, simulations were conducted at α values of 60°, 75°, 90°, 105°, and 120°.
Additionally, studies have shown that varying stance widths (L in Fig. 2(a)) impact muscle recruitment patterns and force generation focus: a narrow stance increases quadriceps activation,12 a wide stance enhances adductor and gluteus maximus involvement,5,12 and a shoulder-width stance achieves balanced force production.5,13 Accordingly, this study investigated three stance widths to determine their effect on femoral loading: narrow (80 % of shoulder width), normal (100 % of shoulder width), and wide (120 % of shoulder width).
To accurately quantify the mechanical response of the hip joint, a local coordinate system was established based on femoral anatomy, as shown in Fig. 2(b). The coordinate origin was located at the center of the femoral head. The Z-axis was defined parallel to the line connecting P1 (the intersection of the femoral shaft and neck axes) and P2 (the midpoint of the intercondylar fossa), representing the proximal-distal direction. The X-axis was defined tangent to the superolateral margin of the greater trochanter. Consequently, Fx, Fy, and Fz denote the medial-lateral, anterior-posterior, and proximal-distal hip joint forces, respectively.
2.3 Finite element simulation setup
2.3.1 Mesh generation
Tetrahedral elements offer strong geometric adaptability, accommodating complex curved surfaces and irregular structures, and feature high automatic meshing efficiency.14 Consequently, 10-node tetrahedral elements (C3D10) were used to mesh the femoral model. To balance computational accuracy and efficiency, the optimal element size was determined via mesh convergence analysis. Simulation results for three element sizes (2 mm, 2.5 mm, and 3 mm) were compared under narrow stance (80 % shoulder width) and 60° flexion load conditions. The 2.5 mm mesh solution met the mesh independence criteria, exhibiting a stress deviation of less than 5 % compared to the 2 mm mesh,15 thus confirming sufficient accuracy. The element size was ultimately set to 2.5 mm, resulting in a final model consisting of 204,165 elements and 299,794 nodes.
2.3.2 Material property assignment
Femoral cortical and cancellous bone were assigned a linear elastic material model.16,17 Though femoral material properties are inherently inhomogeneous, finite element analyses typically assume linear elasticity and isotropy for model simplification and computational efficiency.18 Prior work has shown this assumption has negligible impact on femoral biomechanical analysis outcomes,19 with specific material parameters listed in Table 1.
| Tissue Type | Density (kg/m3) | Elastic Modulus (MPa) | Poisson's Ratio |
| Cortical Bone | 1800 | 16800 | 0.3 |
| Cancellous Bone | 800 | 840 | 0.2 |
2.3.3 Loading and boundary conditions
Loads were applied via coordinate system transformation to simulate femoral forces under different squat postures. Using the upright coordinate system established in Fig. 2(b) as the reference, flexion and abduction angles were adjusted sequentially to adapt the coordinate system to the target squat posture (Fig. 2(c)). Note that the red resultant force vector in Fig. 2(c) is for illustrative purposes only. The actual loads consisted of the orthogonal components of the hip joint reaction forces derived from the AnyBody simulation. These forces were applied to the weight-bearing surface of the femoral head within the adapted coordinate system, with magnitudes corresponding to the peak resultant forces observed during squatting.
Regarding boundary conditions, the distal femur was fully constrained (restricting all three translational and three rotational degrees of freedom) to simulate the stabilizing effect of knee joint soft tissues, including ligaments, muscles, and the joint capsule.
2.3.4 Finite element model validation
To ensure the reliability of subsequent analyses, the constructed femoral finite element model was validated by simulating femoral forces under bipedal standing. Full constraints were applied to the distal femur, and an axial load of 367.5 N (≈50 % of body weight) was applied to the weight-bearing region of the femoral head, aligned with the coordinate system defined in Fig. 2(b). As illustrated in Fig. 3, the principal stress distribution showed strong agreement with the findings reported by Shah et al..20 Specifically, tensile stresses (maximum principal stress) were concentrated in the lateral femoral shaft, the superior aspect of the femoral neck (anterolateral/posterolateral), and the greater trochanter; conversely, compressive stresses (minimum principal stress) were predominantly observed in the medial femoral shaft and the compressive zone of the femoral neck. The peak maximum and minimum principal stresses were calculated as 10.17 MPa and −12.11 MPa, respectively. Quantitative comparisons with the scaled results from Shah et al. are presented in Table 2. Numerical discrepancies ranging from 28.9 % to 58.8 % were observed. These differences were primarily attributed to inter-subject variability in femoral geometry, material properties, and mesh strategies. However, the high consistency in stress distribution patterns confirms the validity of the current model.
| Stress Indicator (MPa) | Current Study (367.5N) | Shah et al.20(Scaled) | Deviation |
| Maximum Principal Stress | 10.17 | 6.4 | 58.9 % |
| Minimum Principal Stress | −12.11 | −9.7 | 28.9 % |
3 Results
3.1 Analysis of AnyBody simulation results
Fig. 4 illustrates the temporal profiles of hip joint reaction forces during the squat descent. Generally, Fx and Fz exhibited an increasing trend over time. When α ≤ 90°, the peak values of Fx and Fz both appeared at the lowest squat position (t = 1.5 s). However, for squats where α > 90°, these peaks occurred prior to reaching the lowest position. In contrast, Fy shifted gradually from positive to negative, and its peak value consistently occurred at the lowest squat position (t = 1.5 s) regardless of α variation. Based on the resultant force calculation (Equation (1)), the peak resultant force F appeared at the lowest position (t = 1.5 s) when α ≤ 90°. Conversely, when α > 90°, the peak F values occurred earlier, specifically at α = 96.7° (for the 105° squat) and α = 103.9° (for the 120° squat), respectively (as shown in Fig. 2(c)).(1)F=Fx2+Fy2+Fz2
Fig. 5 indicates that α exerted a much stronger influence on peak hip joint reaction forces compared to the stance width L. Taking the narrow stance as an example (Table 3), although the peak Fx generally increased with increasing α, its incremental rate diminished notably (from 41.9 % to 16.8 %). Furthermore, when α = 120°, the peak Fx exhibited a slight 0.9 % decrease compared with that at the previous angle. In contrast, the growth rate of peak Fy showed no obvious regularity and fluctuated considerably. Compared with the other two force components, the growth rate of Fz presented a distinct decreasing trend (from 25.8 % to 9.8 %).
| Flexion Angle α | Fx (N) | Growth Rate | Fy (N) | Growth Rate | Fz (N) | Growth Rate |
| 60° | 260.51 | – | 83.72 | – | 1256.16 | – |
| 75° | 369.63 | +41.9 % | 138.99 | +66.0 % | 1580.17 | +25.8 % |
| 90° | 495.26 | +34.0 % | 222.28 | +59.9 % | 2196.89 | +40.1 % |
| 105° | 578.59 | +16.8 % | 272.51 | +22.6 % | 2615.41 | +19.1 % |
| 120° | 573.63 | −0.9 % | 389.79 | +43.0 % | 2898.20 | +9.8 % |
Compared with α, stance width (L) exerted a weaker effect on hip joint reaction forces. As shown in Table 4, as L increased, the reduction rate of peak Fx became slightly more pronounced (from 2.68 % to 3.81 %), the sequential growth rate of peak Fy gradually rose (from 0.02 % to 0.11 %), and the sequential growth rate of peak Fz decreased slightly (from 1.52 % to 1.45 %).
| Stance width L | Fx (N) | Growth Rate | Fy (N) | Growth Rate | Fz (N) | Growth Rate |
| Narrow stance (80 %) | 495.26 | – | 222.28 | – | 2196.89 | – |
| Normal stance (100 %) | 482.01 | −2.68 % | 222.32 | +0.02 % | 2230.35 | +1.52 % |
| Wide stance (120 %) | 463.63 | −3.81 % | 222.57 | +0.11 % | 2262.77 | +1.45 % |
3.2 Finite element analysis results
Fig. 6 illustrates the equivalent stress distribution of the femur under different squat postures. Stress concentrations were primarily observed in the mid-to-distal segments of the femoral shaft, exhibiting a symmetric tension-compression pattern. Specifically, the anterior shaft was dominated by maximum principal stress σ1 (tensile), while the posterior shaft was governed by minimum principal stress σ3 (compressive). Additionally, notable stress concentrations were observed in Ward's triangle (anterior femoral neck) and the posterior femoral neck. Increasing α intensified the stress concentrations in these regions, whereas stance width (L) exerted a minor effect on the stress distribution patterns.
Quantitatively, as shown in Fig. 7, all stress indicators exhibited a biphasic growth pattern characterized by a rapid initial rise followed by a deceleration. In the small-angle stage (60°–90°), the stress growth amplitude ranged from 79.8 % to 106.5 %, whereas in the large-angle stage (90°–120°), this amplitude decreased to 18.4 %–30.5 %.
Throughout the squatting process, the absolute magnitude of σ1 consistently exceeded that of σ3. Specifically, tensile stress on the anterior femur ranged from 136.57 MPa to 361.74 MPa, which was higher than the compressive stress on the posterior femur (129.10 MPa–346.83 MPa). In the femoral neck region, stress in Ward's triangle (27.73 MPa–70.71 MPa) was consistently lower than that in the posterior femoral neck (35.71 MPa–83.01 MPa). Furthermore, the stress states differed significantly: Ward's triangle was dominated by tensile stress, while the posterior femoral neck was dominated by compressive stress.
Regarding the influence of stance width (L), a critical inflection point was identified at α = 105°. Taking σ1 as an example, when α < 105°, the stress magnitude followed the order: Narrow < Normal < Wide. However, when α > 105°, this order reversed to: Wide < Normal < Narrow. This reversal indicates that the effect of stance width on femoral stress is dependent on the knee flexion angle.
4 Discussion
In this study, we used a combined approach of MBD and FEA to investigate how knee flexion angle (α) and stance width (L) affect femoral stress.
The results indicate that α exerts a more significant influence on femoral stress than L. As α increased, the stress showed a trend of “rapid rise followed by slowing growth.” This trend matches the input loads calculated by our MBD simulation. The reason stress grows slower at deep squats (α > 90°) is likely because the body posture becomes more stable (optimized muscle moment arms and lower center of gravity), so the muscle forces do not need to increase as much.21
Besides the input loads, the shape of the bone also determined the stress distribution. We found that tensile stress on the anterior femoral shaft was always higher than the compressive stress on the posterior shaft. This is primarily attributed to the difference in bone thickness: the anterior cortical bone is thinner (≈1.2–1.6 mm) than the posterior one (≈1.6–2.2 mm).22,23 Therefore, under the same bending force, the thinner anterior bone suffers more stress. Additionally, the posterior femoral neck serves as the primary load transfer pathway due to its dense compressive trabeculae aligned with the mechanical axis, whereas Ward's triangle is a sparse region that naturally bears less load.20,24
Unlike flexion angle, stance width (L) had a smaller effect, but it showed an interesting reversal at α = 105°. Specifically, when α < 105°, a wide stance caused higher stress (Narrow < Normal < Wide); however, when α > 105°, this trend reversed (Wide < Normal < Narrow). This agrees with Larsen et al.,25 who suggested that stance width mainly changes how load is transferred. At deep squats (>105°), a wider stance likely changes the direction of the force line, helping to reduce the bending moment on the femur compared to a narrow stance.
Finally, our method has a clear advantage over previous studies. Many earlier studies used simple, static loads (like a constant weight).26,27 Our study used dynamic loads calculated from MBD. This allowed us to see the realistic, non-linear stress changes and the “crossover” effect at deep angles, which static models would miss.
While this study used a combined MBD-FEA framework to quantify femoral force-bearing characteristics across squat postures, it has several limitations. During modeling, femoral morphological differences from gender and individual variations were not fully incorporated. The assumption that bone is isotropic and homogeneous may cause deviations between local stress results and actual mechanical properties. In addition, the study only focused on young, healthy populations and excluded the elderly. These simplifications have somewhat restricted the direct generalization of conclusions to populations of different genders and ages.
5 Conclusion
Using a combined MBD-FEA approach, we quantitatively investigated the effects of knee flexion angle (α) and stance width (L) on femoral mechanical responses during squatting. Key findings are as follows:1.Femoral stress distribution exhibits significant regional asymmetry. During squatting, tensile stress in the anterior femoral shaft is consistently higher than compressive stress in the posterior shaft, while posterior femoral neck stress (compressive-dominated) is greater than that in Ward's triangle (tensile-dominated).2.Stress levels in key femoral regions (middle-distal femoral shaft, Ward's triangle, and posterior femoral neck) exhibit a nonlinear increasing trend of “rapid rise followed by slowing growth” with increasing α.3.The effect of stance width on stress is strongly dependent on flexion angle. When α < 105°, stress rises with a wider stance; when α > 105°, this trend reverses.
This study provides a quantitative biomechanical basis for squat training posture optimization and injury prevention, and offers a reference for sports rehabilitation and clinical osteoarticular disease intervention.
Informed consent
The patients agreed that their data could be used in the study.
Author contributions
Yafeng Li conceived and designed the overall technical and theoretical framework. Jiang Liu constructed and optimized the femoral model, conducted the multibody dynamics and finite element simulations, and wrote the manuscript. Bopeng Zhang and Feng Zhang provided technical support. Jing Zhang established the theoretical framework for the joint force acquisition method. Zhifeng Tian provided the femoral CT data and guidance on the biomechanical structural characteristics of the femur.
Ethical statement
The study was approved by Tianjin Hospital Clinical Trial Ethics Committee with the approval number SL20180047.
Research funding
The Tianjin Applied Basic Research Multiple Investment Fund (21JCYBJC01400).
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