a simple static analysis of the ankle joint in stance and tip-toe, resulting joint reaction forces

1. Cardiff School of Engineering
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Student No: 1056984
Family Name: Divecha First Name: Hiren
Personal Tutor: Prof Sam Davies Discipline: MMM
Module Details
Module Name: Engineering Theory 1 Module No: ENT536
Coursework Title: Static analysis of the Ankle joint
Lecturer: Dr D M O’Doherty
Submission Deadline: 30/10/2010
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1

2. Static analysis of the Ankle joint
Hiren Maganlal Divecha
Candidate Number: 1056984
ENT536 – Engineering Theory 1
1. Abstract
The ankle joint plays an important role in supporting the body and allowing for propulsion. The
forces experienced vary greatly under different loading conditions and can reach up to 15 times
the body weight whilst sprinting. Under static conditions, free body analysis has been used in this
report to demonstrate the changes in joint reaction force from single leg stance (1.8 times body
weight) to single leg tip-toe stance (2.5 times body weight). Finally, the biomechanical reason for
maintaining non-weight-bearing status in conservatively managed small posterior malleolar
fractures is discussed after demonstrating that the joint reaction force with the foot suspended
horizontal in mid-air is only around 0.02 times the body weight acting in a near horizontal
direction. When compared to the near vertical direction and the magnitude of the joint reaction
force in single stance, it is clear to see how fracture displacement could occur if weight bearing
were allowed.
(word count = 2787)
Table of Contents
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3. 1. Abstract ...................................................................................................................................... 2
2. Introduction ............................................................................................................................... 4
3. Basic Anatomy of the Tibio-talar Joint ....................................................................................... 5
4. Static Analyses of the Tibio-talar Joint ....................................................................................... 6
a) Single Leg Stance .................................................................................................................... 7
b) Single Leg Tip-toe Stance ..................................................................................................... 10
c) Foot Suspended in Mid-Air................................................................................................... 13
5. Conclusion ................................................................................................................................ 15
6. References ................................................................................................................................ 17
3

4. 2. Introduction
The ankle joint is located between the leg and the foot. It is involved in supporting the body
during standing or the stance phase of gait, provides a lever arm for the push-off phase of gait
(thus resulting in propulsion) and provides some amount of “shock absorption” during walking
and running activities. The ankle joint experiences loading of half the body weight during bipedal
standing. This can rise to 3 times body weight during walking and up to 15 times body weight
during sprinting. The ankle joint will be considered in certain positions (single leg stance, single leg
tip-toe stance and suspended horizontally in mid-air) with static analysis to estimate the joint
reaction force as well as the muscle force exerted by either the gastrocnemius-soleus-plantaris
group or the tibialis anterior muscle.
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5. 3. Basic Anatomy of the Tibio-talar Joint
The ankle joint or, more specifically, the tibio-talar joint is formed by the articulation between the
distal tibia, distal fibula and the trochlea of the talus. The distal tibia and fibula form a mortise
into which the talus fits. This forms a uni-planar, hinged synovial joint. The joint is stabilized by its
bony congruency, which is tightest in dorsiflexion as the wider anterior part of the talus engages
within the mortise. This in part explains why injuries are more likely to occur in the plantar-flexed
foot when there is some relative “looseness” of the talus in the ankle mortise. Furthermore,
ligamentous structures stabilize the tibio-talar joint. These include the lateral ligamentous
complex (anterior talo-fibular, posterior talo-fibular and calcaneo-fibular ligaments) and the
stronger medial deltoid ligament, as well as the syndesmotic structures.
The axis of rotation of the tibio-talar joint has been determined to run from just inferior and
anterior to the tip of the lateral malleolus to just inferior to the tip of the medial malleolus (Isman
& Inman 1969). This allows dorsiflexion of 10° - 30° and plantar-flexion of 20° - 50°. Dorsi-flexion
is produced by contraction of the tibialis anterior muscle, which is weakly assisted by the extensor
digitorum longus, extensor hallucis longus and peroneus tertius muscles. Plantar-flexion is
produced by the contraction of the gastrocnemius, soleus and to a lesser extent the plantaris
muscle. These muscles have a common tendon, the Achilles tendon, which inserts into the
calcaneal tuberosity. It has been shown that in the normal tibio-talar joint during walking, the
average dorsi-flexion is 10.2° and plantar-flexion 14.2° (Stauffer et al 1977).
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6. 4. Static Analyses of the Tibio-talar Joint
For the purposes of the following static analyses of the tibio-talar joint, the following assumptions
have been made:
1. The ankle – foot is considered as a free body
2. The sagittal plane is only considered
3. The foot is taken as being a rigid structure. Only movements about the tibio-talar joint
centre of rotation are considered (i.e.: dorsi-plantar-flexion)
4. The tibio-talar joint is frictionless
5. The subject’s mass is 70 kg
6. The acceleration due to gravity (g) is 9.81 m.s-2
Electromyographic studies of standing subjects have shown that the gastrocnemius and soleus are
active whereas the tibialis anterior is not (Joseph & Nightingale 1952 & 1956). Thus, the tibialis
anterior shall be excluded from these analyses. More specifically, the tibialis anterior is active
during two specific points in the gait cycle. Firstly, for the first 10% of the stride at the point of
heel contact to mid-stance when its dorsiflexion action prevents foot slap and decelerates the
foot into stance. Secondly, for the latter 40% of the stride corresponding to the “swing phase”
from toe-off to heel strike when its function is to ensure clearance of the foot from the ground
throughout the swing phase. This pattern of activation has been confirmed by dynamic
electromyographic studies of the gait cycle in normal walking (Winter & Yack 1987; Ounpuu &
Winter 1989).
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7. A combination of sources have been referred to for specific measurements and angles in the foot
and ankle (Sammarco & Hockenbury 2001; Snijders 2001; Winter 2009). The length of the foot
from heel to metatarsals is 21 cm. The centre of rotation of the tibio-talar joint lies 5 cm anterior
and 4 cm superior to the point of action of the Achilles tendon (AT) on the calcaneum. The
Achilles tendon acts at an angle of 87° to the horizontal axis (Procter & Paul 1982). The ground
reaction force (GRF) is equivalent to the weight of the body minus the weight of the foot and acts
4 cm anterior to the centre of rotation of the tibio-talar joint (Hellebrandt et al 1938). The centre
of mass of the foot lies 6 cm anterior to and 2 cm below the centre of rotation of the tibio-talar
joint. The mass of the foot (mfoot) is taken as 1.5% of the total body mass i.e. 1.05 kg.
a) Single Leg Stance
The following static analysis will determine the force exerted by the Achilles tendon whilst a
subject performs a single leg stance with the foot flat on the floor. As the centre of gravity of the
body during standing acts in front of the tibio-talar axis of rotation, there is a moment acting to
rotate the body forwards around the tibio-talar joint (Smith 1957). This is balanced by the plantar
flexor muscle group (gastro-soleus-plantaris complex) acting via the Achilles tendon. Figure 1
shows the free body diagram of the ankle and foot with the relevant forces acting on it.
The joint reaction force is denoted by J and is shown to have orthogonal components Jx and Jy. It
acts at an angle β to the horizontal. The Achilles tendon force has orthogonal components ATx and
ATy and has a moment arm (p) from the centre of rotation of the tibio-talar joint.
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8. y
Jy
J
ATx
AT
ATy
β x
p
Jx
4 cm
87° α 6 cm 2 cm
5 cm 4 cm
mfoot x g
21 cm
GRF
Figure 1: Free body diagram of static forces acting on ankle joint in single leg flat stance
1.
2.
3.
8

9. 4.
5.
6.
Thus, when performing a single leg stance with the foot flat on the ground, the Achilles tendon
force in this simplified model is found to be 551 N with an overall joint reaction force of 1 217 N
(1.8 times the body weight) acting at an angle of 88.6° to the horizontal axis.
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10. b) Single Leg Tip-toe Stance
When performing a single leg tip-toe, the foot – ankle behaves as a class 2 lever i.e. the load is
located between the effort and the pivot. The ground reaction force now acts at the pivot point,
which is the plantar surface of the metatarsal heads. The plantar surface of the foot in this
example is inclined at 45° to the horizontal. The angle of action of the Achilles tendon is taken to
remain constant. The free body diagram of this scenario is shown in Figure 2 and the moment arm
of the weight of the foot (q) is calculated in Figure 3.
y
ATx
Jy J
AT
ATy p
x
β Jx
γ r
q
2 cm
87° α
6 cm
5 cm
mfoot x g
16 cm
45°
GRF
Figure 2: Free body diagram of static forces acting on ankle joint in single leg tip-toe stance
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11. q r
2 cm
ε
6 cm δ
μ
κ
mfoot x g
45°
Figure 3: Calculation of moment arm of weight of foot (q)
1.
2.
3.
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12. 4.
5.
6.
7.
Thus, when performing a single leg tip-toe stance, the Achilles tendon force in this simplified
model is found to be 1 192 N with a resulting joint reaction force of 1 711 N (2.5 times body
weight) acting at an angle of 58.8° to the horizontal axis.
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13. c) Foot Suspended in Mid-Air
With the foot suspended in mid-air in a horizontal position, the tibialis anterior (TA) muscle is
active in generating a dorsiflexion moment to counteract the plantar-flexing moment generated
by the weight of the foot around the centre of rotation of the tibio-talar joint. The gastrocnemius-
soleus-plantaris complex is presumed not be active in this situation, and this is certainly found to
be the case in electromyographic studies which show no activity during the latter 40% of the gait
cycle, corresponding to the “swing” phase when the foot is off the ground (Joseph & Nightingale
1952 & 1956; Winter & Yack 1987; Ounpuu & Winter 1989). The tibialis anterior muscle force is
taken to act at 30° to the horizontal (Procter & Paul 1982) with a perpendicular moment arm of
3.5 cm from the centre of rotation of the tibio-talar joint (Maganaris et al 1999). Figure 4
demonstrates the free body diagram for this scenario.
y
Jy
J
TAy
x
3.5 cm
β
TA
Jx
30°
4 cm TAx
6 cm
5 cm
mfoot x g
21 cm
Figure 4: Free body diagram of static forces acting on ankle joint in mid-air horizontal suspension
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14. 1.
2.
3.
4.
Thus in the suspended horizontal foot, the tibialis anterior force is calculated to be 18N with a
resulting joint reaction force of 15N (0.02 times body weight) acting at 5.5° to the horizontal.
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15. 5. Conclusion
Free body static analyses are useful in mathematically modelling joints and limbs under various
loading situations. However, because of the nature of the assumptions made, they only give us an
estimate of muscle and joint reaction forces. The in vivo situation is much more complicated. The
ankle – foot region has 33 articulations, which each experience some amount of friction and three
dimensional joint movement and muscle action. Furthermore, joints are never truly static (rather
quasi-static with very small torques experienced which are corrected under subconscious control
to maintain balance and posture). In the examples demonstrated above, the joint reaction force
has been shown to increase from 1.8 times body weight in single leg flat stance to 2.5 times body
weight when performing a single leg tip-toe stance with the foot at 45°. This explains why
patients with significant tibio-talar joint osteoarthritis experience increased pain when attempting
to stand up on their tip-toes.
The scenario of a small posterior malleolar (posterior segment of the distal tibial articular surface)
fracture is a good example of the clinical application of comparing free body analyses of the tibio-
talar joint in different weight-bearing situations. Undisplaced fractures involving less than 25% of
the distal tibial articular surface are usually managed without internal fixation and are treated in
cast immobilisation with a period of non weight-bearing (usually 6 – 8 weeks). The risk of post-
traumatic arthritis has been estimated at around 30% and initially this was felt to be related to
the increased overall contact stress on the joint surface due to a reduced contact area (Macko et
al 1991). However, recent studies have shown that the contact stresses with simulated posterior
malleolar fractures are not increased. Rather, the centre of stress distribution over the remaining
joint surface shifts more anteriorly. This anterior region does not normally experience much
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16. loading under physiological conditions, which may explain the higher incidence of degenerative
changes seen in patients with these types of injuries (Fitzpatrick et al 2004).
From the simplified model of a horizontal suspended foot, the joint reaction force is low (0.02
times body weight; this does not include the weight of a below knee cast) compared to the joint
reaction force expected with weight-bearing stance (1.8 times body weight). Furthermore, the
joint reaction force has been calculated to act at a near vertical angle (88°) in single leg stance
compared to the near horizontal joint reaction force (5.5°) when the foot is suspended
horizontally. The risk of fracture displacement is obvious if weight-bearing mobilisation in this
scenario were to be permitted before significant fracture healing had commenced.
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17. 6. References
Fitzpatrick, D. C., et al. 2004. Kinematic and Contact Stress Analysis of Posterior Malleolus
Fractures of The Ankle. Journal of Orthopaedic Trauma 18 (5), pp. 271-278.
Hellebrandt, F. A., et al. 1938. The Location of The Cardinal Anatomical Orientation Planes Passing
Through The Center of Weight in Young Adult Women. American Journal of Physiology, pp. 465-
470.
Isman, R. E., & Inman, V. T. 1969. Anthropometric Studies of The Human Foot and Ankle. Bulletin
of Prosthetics Research , pp. 97-129.
Joseph, J., & Nightingale, A. 1956. Electromyography of Muscles of Posture: Leg and Thigh
Muscles In Women, Including The Effects of High Heels. Journal of American Physiology 132 (3),
pp. 465-468.
Joseph, J., & Nightingale, A. 1952. Electromyography of Muscles of Posture: Leg Muscles in Males.
Journal of Physiology 117 (4), pp. 484-491.
Macko, V. W., et al. 1991. The Joint-Contact Area of The Ankle. The Contribution of the Posterior
Malleolus. The Journal of Bone and Joint Surgery 73, pp. 347-351.
Maganaris, C. M., et al. 1999. Changes in The Tibialis Anterior Tendon Moment Arm from Rest to
Maximum Isometric Dorsiflexion: In Vivo Observations in Man. Clinical Biomechanics 14, pp. 661-
666.
Ounpuu, S., & Winter, D. A. 1989. Bilateral Electromyographical Aanalysis of The Lower Limbs
During Walking in Normal Adults. Electroencephalography and Clinical Neurophysiology 72, pp.
429-438.
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18. Procter, P., & Paul, J. P. 1982. Ankle Joint Biomechanics. Journal of Biomechanics 15 (9), pp. 627-
634.
Sammarco, G. J., & Hockenbury, R. T. 2001. Biomechanics of the Foot and Ankle. In: M. Nordin, &
V. H. Frankel, Basic Biomechanics of The Musculoskeletal System. 3rd ed. Lippincott Williams &
Wilkins, pp. 223-255.
Smith, J. W. 1957. The Forces Operating at The Human Ankle Joint During Standing. Journal of
Anatomy 91 (4), pp. 545-564.
Snijders, C. J. 2001. Engineering Approaches to Standing, Sitting, and Lying. In: M. Nordin, & V. H.
Frankel, Basic Biomechanics of The Musculoskeletal System. 3rd ed. Lippincott Williams & Wilkins,
pp. 421-424.
Stauffer, R. N., et al. 1977. Force and Motion Analysis of the Normal, Diseased and Prosthetic
Ankle Joints. Clinical Orthopaedics and Related Research 127, pp. 189-196.
Winter, D. A., & Yack, H. J. 1987. EMG Profiles During Normal Human Walking: Stride-to-Stride
and Inter-subject Variability. Electroencephalography and Clinical Neurophysiology 67, pp. 402-
411.
Winter, D. A. 2009. Kinetics: Forces and Moments of Force. In: Biomechanics and Motor Control of
Human Movement. 4th ed. John Wiley & Sons, pp. 107-138.
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