26.2: The Engineering of Marathon Running

With 50,000 runners stepping with 475-lb force
3 times per second, it’ll be time to fix the roads!

Huff…owww…puff…ouch…pant.  Just nine miles to go this morning…that’s just 18,000 steps…


I am writing this entry from New York City, where the 45th NYC Marathon will take place this Sunday.  On that date, 50,000 hardy souls will run 26.2 miles through the streets of NYC’s 5 boroughs.  For some of these 50,000, a marathon is a regular event (for example, 69 year old Larry Macon set a record by running 239 marathons in 2013).  For others, it will be a first time try – which may end up being the first of many (who knows, maybe there’s a future Larry Macon in the bunch), or it might simply be a onetime only thing, meant to check off one more item on a bucket list.

I can totally relate to those bucket-listers.  To the average Joe, there doesn’t seem to be a more challenging feat, a greater example of self-discipline and personal accomplishment than completing a 26.2 mile run along with umpteen thousand others, while coming through with your head held high (remember to smile for the finish line picture!).  Suffer only once — and you can forever reap the admiration of all those who sit in traffic behind you looking at the 26.2 sticker on the back of your car.

Impress all your friends (plus cover up that ding)
with the marathon badge of honor

So a while back I decided to join the crowd.  Not here in New York, but home in Houston, where the marathon falls on a date (Sunday, January 17, 2016) that is much more compatible with the late training schedule dictated by the warm Houston summers.  Not really being much of a runner (or any other sort of exerciser) during my life, and having just recently passed a milestone birthday, I figured this would be a challenge worth tackling.

Six weeks ago I started training, using the popular Hal Higdon marathon program, recommended by my daughter who just completed her first marathon in Providence.  So far, so good: I haven’t missed a day yet, having covered 104 miles over that period.  I’ve listened to the same 45 songs on my iPhone over and over and over again.  I’ve come to intimately know every crack in the neighborhood sidewalks and every pebble in the local park’s running trails.  And I’ve woken up with aches and pains in my feet and legs every morning.

Running for a 9-mile stretch in the early morning before it gets light, there isn’t much else to do but listen to music and think.  For some reason, lately I’ve found my thoughts focusing on the derivation of the formula for calculating force generated on the leg while running.  If I recall correctly, it requires equating the net acceleration on the body’s center of mass during the time a foot is on the ground (upward acceleration) with that while both feet are off the ground (downward acceleration).

Semi-Sinusoidal Distribution of upward force on foot

In other words, assuming a half-sine distribution of the upward force during the time τ while one foot is on the ground, the upward acceleration on the body would be ∫[(F0/M)sin(π t /τ)-g] dt evaluated over the interval 0<t<=τ (with F0 being the maximum force in the foot/leg)..  The downward acceleration is –g, occurring during the rest of the time τ<t<=T, where T represents the interval between footfalls.

Integrating the former, the upward acceleration = 2(F0/M) τ/π -g τ; the downward acceleration is –gT+g τ.  Summing the acceleration over the entire time T and setting to zero (otherwise I would starting floating off the ground):

2(F0/M) τ/π – gT = 0, or:

F0 = Mg T π/(2 τ)

I remember reading that the time a runner’s foot spends on the ground is somewhat under 0.2 seconds, so I use 0.18 seconds (τ).  I take approximately 2000 steps per mile, and on my longer runs I maintain a pace of around 11 minutes to the mile (yes, I know that’s slow, but I still have 12 weeks more of training to get my speed up).  So that means that a foot hits the ground every 0.33 seconds (T).

My weight (Mg) is 165 pounds, so the maximum force in my leg should peak at around:

F0 = 165 * 0.33 π/(2 * 0.18) = 475 pounds!  Ouch!  No wonder my legs hurt all the time!

After arriving at that revelation, I look up and realize that I’ve already completed one 3-mile lap around the park without even noticing it.  But then I start to think that, as for a structural member, the total load on a bone is not the concern, but rather the stress.  So what would the stress on the leg and foot bones be?

Vertical force acting on the tibia, foot

Thinking about the arrangement of the leg bones, it seems that the tibia (the larger of the two leg bones compared to the fibula) would absorb most of the applied force, so conservatively I’ll assume it takes 100%.  Now what would the cross-sectional area be?  Reaching down to scratch my shin while I run, it seems like the tibia diameter might be somewhere around 1-1/2 inches, for a cross sectional area A = π 1.5*1.5/4 = 1.75 square inches.  But bones have a hollow interior for the marrow, so in reality there probably isn’t much more structural area than half of that, or 0.8 square inches.

So stress on the leg bone (tibia) should be something around 475/0.8 = 600 psi.  Is that good or bad?  My curiosity gets the best of me, so I sneak a look at my iPhone while I keep running.  Trying to make it look like I’m just checking my vital signs or changing my music, in reality I google the yield strength of bone and find that it is around 104 MPa https://en.wikipedia.org/wiki/Ultimate_tensile_strength .  Googling again, I convert that to 15,000 psi, which even with a large factor of safety is well above my applied stress of 600 psi.

But this is a very rough calculation, which doesn’t considering horizontal loads, developed moments due to the fact that the leg is not straight, joint capacities, etc.  So I google again while pretending to stop for a drink at a water fountain and am relieved to learn that the tibia is sufficiently strong to handle up to 4.7 times one’s bodyweight when walking https://en.wikipedia.org/wiki/Tibia#Strength.  So that would imply an equivalent allowable stress of 4.7 * 165 / 0.8 = 970 psi (or a factor of safety around 15) for axial stresses.

Yes, that’s where it hurts

But how about bending stresses?  It seems that feet experience the most flexure, so those bones probably suffer the maximum bending stress.  Estimating the bone (what’s that called, the cuboid bone?) above the metatarsals to be a 2” by ½” block, it should have a section modulus of approximately 0.0833 inches cubed.  If it is located 3” away from the ball of my foot, it would see a moment M = 3” x 475 pounds = 1425 in-lbs.  So bending stress S would be approximately:

S = 1425 / 0.0833 = 17,100 psi!  Ow!  No wonder my feet hurt all the time!

After scientifically proving why my feet hurt all the time, I realize that I have just completed my second 3-mile lap, again without noticing the drudgery.  Only one more to go, hopefully I can get all the way through it without stopping.

OK, now I understand why my legs and feet hurt so much during training.  But the calculations that I have done are based on single impact loads – each isolated step in the course of running.  There must be an additional effect due to the fact that I will be taking many of these single steps over the course of 26.2 miles on January 19 (plus the 422 extra miles I will have to put in to complete the training program).  Some sort of gradual loss of strength due to the cyclic loading of the repetitive steps…some sort of consequence of the fatigue loading…

So how do I calculate the loss of fatigue life?  Normally for a steel component I would do a cumulative damage calculation, based on summing various N-S combinations falling on the material’s fatigue curve, but I don’t know what the fatigue curve for bone looks like (nor do I want to fake tying my shoe while I google it right now).  But maybe I can estimate it in some way…I know that the tibia can handle 4.7 times a human’s body weight while walking…the average American walks what, around 3,000 steps per day?  If we live an average of 75 years, that would imply that one point on the fatigue curve for leg bones would be approximately 82,000,000 cycles at 1,000 pounds.  If I recall the construction of fatigue curves for steel and other metals correctly, I might guess that the fatigue curve follows a relationship where the log of the cycles corresponds to the square root of the ratio of the loads.  Or in other words, the legs should be able to resist a load of 475 pounds for 300,000,000,000 cycles.  (Yes, that’s 300 trillion steps!  And that’s without a hip transplant.)

So running 2,000 steps per mile for the 450 miles involved in the marathon and its training, I would use up about 1,000,000 of my allocated 300 trillion steps, or (1,000,000/300,000,000,000 ) * 75 years = only about 2 hours of my total lifetime running/walking capacity.  “Hey that’s nothing, I can do this!”, I realize, as I suddenly forget my pain and pick up my pace, finishing the third and last lap with a new personal best.  “And maybe after I complete my first marathon,” I think, “I’ll really train hard for my next one.  Or my next 239!”

MT-5 Larry Macon, I’m coming after you!

PS — If you can’t quite work out these formulas for yourself while you’re training or running your next marathon, maybe you should ask the runner beside you for help: it might be Freya Murray (famed structural engineer and occasional Olympic marathoner).

Do your feet hurt?  Maybe they are overloaded.  Why not analyze them, using CloudCalc, the scalable, collaborative, cloud-based engineering software.  www.cloudcalc.com – Structural Analysis in the Cloud.

By Tom Van Laan

Copyright © CloudCalc, Inc. 2015


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