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A biomechanical model for estimating moments of force at hip and knee joints in the barbell squat

by Bruce Ross, CEO, MyoQuip Pty Ltd (October 2006)


The barbell squat is a complex, mass load bearing multi-articular exercise movement. It is the basic lower body exercise prescribed in training programs for many sports even though it is unpopular with most athletes and is often performed inexpertly. One of the major problems when performing a full squat with heavy weights is that there appears to be excessive loading in the bottom part of the movement. At the same time loading through the top range of the movement seems inadequate.

This study examines the extent to which these effects may be attributable to changing values of resistive torque in moving from deep flexion to full extension of the hip and knee joints, i.e., to changes in limb geometry. A basic biomechanical model of the squat has been developed to calculate moments of force or torque applied about the axes of the hip and knee joints at various angles of those joints. I am not aware of any previous comparable study of the free weight squat.

The Model

A mathematically scaled model of a person of 180cm height and 100kg body weight was created consisting of four linked segments. These were the upper body or HAT (head, arms and trunk) assumed to be a rigid member; the thighs; the shanks; and the feet. The lengths of the segments as a percentage of total height were 50, 24, 22, and 4 respectively. Centres of gravity for the thighs and shanks were assumed to be both at 43.3% of segment length measured proximally. The proportion of body weight for the upper body, thighs and shanks was estimated as 68.6%, 20.0% and 8.6% respectively.

In order for stability to be maintained in squatting, the centre of gravity of the system (exerciser's body plus weight bar) must remain directly over the feet. Unless the centre of mass is constantly positioned directly above the ground reaction force vector, a moment would exist and the system would rotate, i.e., tip forward or backward.

To provide a determinate model and to facilitate calculation, a number of simplifying assumptions were used, Firstly, throughout the exercise movement the hip and knee joints move synchronously, i.e., at any point their angles are equal. Secondly, the force vector of the weight bar (FWB) was assumed to be located directly above that of the upper body (cgUB). Thirdly, it was assumed that the centre of gravity of the system remains directly above the ankle joint rather than at the midpoint of the foot as is usually and more correctly assumed. Figure 1 shows a simplified free body diagram incorporating the assumptions.

At each observation point throughout the exercise the body is evaluated in a static or constant velocity state and therefore can be treated as rigid. Moments of force were calculated for the knee and hip joints using a link-segment model of the form described in Winter (1990).

Other than its contribution to total body mass the weight of the exerciser's feet was ignored. For the present calculations the mass of the loaded weight bar was assumed to be 100kg. Its force (FWB) contributes to moments about the joints. The vertical reaction force (FGR) from the floor to the exerciser's feet also provides a force of flexion about the hip and knee joints. The constant velocity assumption means that the ground reaction force is simply the sum of the body mass and the mass of the weight bar, i.e., 200kg in this application of the model.

The range of motion investigated was from deep flexion of 40 for both hip and knee joints to lock-out or full extension at 180.


Figure 1

Free body diagram of the barbell squat


Figure 2 shows the moments of force about the hip and knee joints calculated using the model. It can be seen that very high moment values occur in deep squat positions. In fact at 60 flexion of both joints, torque values are 470N.m and 333N.m for the hip and knee joints respectively. In this model the parallel position for the thigh occurs at joint angles of 62.5. This is the position where the hip and knee joints are furthest from the force vectors of the weight bar and upper body, with the result that torque values for hip and knee joints reach their maxima here at 471N.m and 334N.m respectively.

Below this point it can be seen that torque values are declining, but this effect is counteracted by the fact that the leg extensor muscles are lengthening and therefore increasingly less able to deliver force.

It can also be seen that as the exerciser rises above joint angles of around 90 the torque values decline markedly and approach zero with full extension or lock-out.

I am unaware of any published studies of strength curves for complex exercises like the barbell squat but it can be expected that the leg extensor muscles function most efficiently in the mid range of the exercise movement. The conjunction of such a muscle strength profile with the torque curves shown above means that a heavy load would place the exerciser in a biomechanically disadvantageous position in the deep range of the movement. At the same time there would be inadequate effective activation of the leg extensor muscles through the top range.

Figure 2

Barbell squat - moment of force by hip and knee joint angle

It should be noted that the torque values were calculated with the exerciser stationary at each joint position, so they are isometrically determined. Different results would be obtained if measurements were made of actual dynamic movement. Results would also vary if the assumption of synchronised joint angles did not apply. However in both situations similar extreme variations in torque between bottom end and top end positions could be anticipated.

Correcting for variations in joint torque

A number of methods have been developed to improve the efficacy of the squat exercise. The most well known involve the addition of metal chains or rubber bands to the squat apparatus. With the former sections of chain are hung from each end of the weight bar. As the lifter descends links begin to pile on the floor, lessening the effective load and consequently the joint torque.

The usual method of using bands when squatting is to attach one or more heavy rubber bands to each end of the weight bar and anchor them to hooks on the floor. As the lifter rises tension in the bands increases adding to the effective load and the joint torque. However this system has no effect on the torque at the bottom end of the movement. To correct this a reverse band technique is employed. Here the bands from the weight bar are attached to the top of the squat rack or the ceiling. As the lifter descends tension in the bands increases, thereby compensating for the increasing torque in the bottom range.

The MyoQuip ScrumTruk has been developed to overcome the deficiencies in the conventional squat. It solves the problem of excessive variation in torque in two ways. Firstly it is operated in a horizontal body position thus greatly reducing the contribution of the user's own body weight to torque generation. Secondly its use of QuadTorq variable resistance technology compensates for torque variation at both ends of the movement. The ability to make adjustments to the rate of change of load means that the user can experience appropriate load and effective muscle activation through the whole range of movement.

Why tall people can't squat

It is generally recognised that people with long limbs are poor squatters. They often look awkward performing the exercise and the poundages they lift are usually unimpressive. The present study sheds light on why this is so.

Figure 3

Free body diagram of the barbell squat

Figures 3 and 4 compare the joint moment forces generated in the squat by three lifters of different height. In each case we assume that the lifter weighs 100kg and is squatting a weight bar loaded to 100kg. The assumed body heights are 160cm, 180cm and 200cm. Inspection of the two charts indicates that torque values vary directly with body height. In fact it can be seen that the moments of force at any joint angle are 25% higher for an athlete of 200cm than for one of 160cm. Therefore in the bottom range of the movement they are much more subjected to excessive loading.

Figure 4

Free body diagram of the barbell squat

There is an additional effect. Given that work can be measured as force times distance, it is obvious that a tall person will rise further and therefore perform more work than a shorter person. Again our 200cm subject is performing 25% more work than their 160cm counterpart.

Thus there are logical reasons for the perceived poor performance of tall people in the barbell squat.


This study has demonstrated that throughout a deep squat movement with heavy loading the moments of force experienced at the hip and knee joints typically vary from excessive to inconsequential. Because of this the leg extensor muscles are likely to be effectively activated for only a minor part of the exercise movement.

It therefore seems appropriate to question the efficacy of the squat as a general exercise for developing leg strength. In particular the wisdom of its use in preparing athletes for participation in sports that themselves have high incidence of back and knee injury must be doubted.


Abelbeck, K.G. Biomechanical model and evaluation of a linear motion squat type exercise. J. Strength Conditioning Res. 16: 516-524. 2002.

Robertson, D.G.E., G.E. Caldwell, J. Hamill, G Kamen and S.N. Whittlesey. Research Methods in Biomechanics. Champaign, IL: Human Kinetics, 2004.

Winter, D.A. Biomechanics and Motor Control of Human Movement. New York: John Wiley and Sons, Inc. 2nd Edn. 1990.

This article also appears on the MyoQuip Blog website

Comments are welcomed. For inclusion please email to Bruce Ross.

(This article may be reproduced so long as full acknowledgement of sources is provided.)

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