14:125:208: Introduction to Biomechanics Spring 2013 Homework Assignment #6

Due 4/01/13

Problem 1:

Above two identical beams are subjected to either 3pt bending (top) or 4pt bending (bottom). Both beams are subjected to a net load of 2P. To the right is a cross section of the beam. It is a hollow beam with a wall thickness of b and h as shown. Compare the two bending schemes. Derive expressions for the internal shear force and bending moment in these two cases, and draw shear and bending moment diagrams. Determine the locations (x and y) of the maximum tensile and compressive stresses and derive the equations that describe these stresses at the maximum locations. These bending tests are frequently used to characterize the material properties and/or mechanical response of materials, such as in the bone bending lab in the BME curriculum. Which test (3 or 4pt bending) do you think provides a better, more straightforward means of testing materials? Explain.

14:125:208: Introduction to Biomechanics Spring 2013 Homework Assignment #6

Due 4/01/13

Problem 2: (Tibia Stress Analysis of the Sun Salutation)

You are still taking Yoga classes and your instructor is still teaching you the Surya Namaskara or the Sun Salutation. As the instructor takes the Hasta Uttanasana or Raised Hands pose you are suddenly enlightened and you realize that you can analyze this posture biomechanically to determine stresses within the tibia bone.

Your Yoga Instructor Whole Body Lower Legs

You are enlightened with the following information:

Your Yoga Instructor weighs 700 N in total

The instructor’s COM is now directly over his feet (he is not leaning back as far as he was previously).

His legs makes contact with the floor an angle of 75° with respect to the horizontal and he keeps his knees extended.

The lower legs are 45 cm long to the knee joint. This results in a horizontal distance from the feet to the Tibiofemoral joint of 11.65 cm. We will ignore the weight of the lower legs.

The dashed line represents the patella tendon. It is at a 15 degrees angle with respect to the tibia, and attaches 5.8 cm distal to the tibiofemoral joint along the tibia bone. The perpendicular moment arm between the Tibiofemoral Joint and the patella tendon is 1.5 cm

The instructor’s two legs are idealized as a single hollow member (the Tibia), noting that since the feet are placed together any forces calculated would be distributed equally between the two legs.

a) Draw a free body diagram of the entire body in this pose. Determine any reactions at the instructor’s feet.

b) If we assume that the quadriceps are the only muscles supporting the legs then draw FBDs of the lower legs/feet and determine the magnitude of the quadriceps force, and the magnitude and direction of any reactions at the knee. You may wish to use a rotated coordinate system as shown such that the x-coordinate is tangent to the Tibia and the y-coordinate is perpendicular to it as shown. When using the rotated coordinate system remember to account for the rotation of the ground reactions. You can assume that all external loads act at the center of the cross-section of the bone.

75°

x

y W=700 N

11.65 cm

Tibiofemoral Joint

15° 1.5 cm

14:125:208: Introduction to Biomechanics Spring 2013 Homework Assignment #6

Due 4/01/13

c) Using the loading determined in part (b) draw a shearing

force and bending moment diagram over the entire length of the Tibia

The bone can be modeled as an annulus beam made up of a concentric hollow cylinder about a common origin. The inner and outer radius, cross sectional area, elastic modulus and moment of inertia of the bone (subscript b) are shown under the diagram to the right.

d) Determine the location (relative to the length along the axis of the leg and position in the

cross-section) and magnitude of the maximum stress on the Tibia.

RbO

Rbi

RbO=2.0 cm Rbi=1.33 cm Ab=7.0 × 10-4 m2 Ib=1.0 × 10-7 m4

Eb=18 GPa

Cortical bone

14:125:208: Introduction to Biomechanics Spring 2013 Homework Assignment #6

Due 4/01/13

Problem 3: (Curious George Breaks His Leg)

Curious George is caught defacing private property. While trying to escape he climbs down a fire escape and drops to the ground where he breaks his femur. He is taken to the hospital and his leg is casted and placed in a Russell traction. A Russell traction is used to hold the set bone straight and apply a known axial stress to strengthen the healing bone.

The Russell Traction is set up as follows: Curious George’s femur is 25 cm long and weighs 50 N. The center of mass occurs ½ way down the upper leg length distal to the hip joint and is shown with a ×. It is elevated to an angle of 16.26° with respect to the horizontal. A pulley is attached to the knee, also at an angle of 16.26° w.r.t. the horizontal, with a 10 N weight attached. A support rope is attached perpendicularly to the leg 2/3 of the upper leg length distal to the hip joint.

(a) Using the above information draw a free body diagram and determine the magnitude of any support rope tension and hip joint reaction forces. For simplicity assume all forces act in the x-y plane so the hip joint acts like a hinge joint to permit motion only in the x-y plane and that the femur is a continuous beam from the hip to the knee. You may also ignore the weight and any contribution from the lower leg. (Crummy assumptions but it simplifies the problem dramatically). Also, you may wish to use a rotated coordinate system such that the x-coordinate is tangent to the leg and the y-coordinate

L/2

10 N θ=16.26°

x y

×

2L/3

L/2

14:125:208: Introduction to Biomechanics Spring 2013 Homework Assignment #6

Due 4/01/13

is perpendicular to it as shown. You can assume that all external loads act at the center of the cross-section of the bone.

(b) Using the loading determined in part (a) draw a shearing force and bending moment diagram over the entire length of the upper leg

(a) The bone-cast system can be modeled as a composite beam made up of two concentric cylinders about a common origin. The bone is surrounded by soft tissue which is also surrounded by a fiberglass cast. The inner and outer radius, cross sectional area, elastic modulus and moment of inertia of the bone (subscript b) and cast (subscript c) are shown under the diagram to the right. Determine the location (relative to the length along the axis of the leg and position in the cross-section) and magnitude of the maximum stress on the bone. Compare this maximum stress to that of a bone without a cast subjected to the same loading.

Remember for a composite beam the axial stress in part

i is ∑

=

iii

ii AE

FEσ and the bending stress is ∑

=

iii

ii IE

yMEσ .

RbO

Rbi

RcO

Rci

RbO=2.0 cm Rbi=1.33 cm Ab=7.0 × 10-4 m2 Ib=1.0 × 10-7 m4

Eb=18 GPa

RcO=4.0 cm Rci=3.5 cm Ac=11.78 × 10-4 m2 Ic=8.32 × 10-7 m4 Ec=5 GPa

Cortical bone

Soft Tissue

Fiberglass Cast