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Biomechanics
Consider an artery with an inner radius of 2.5 mm and an outer radius of 3 mm. Input pressure to the artery is 101 mmHg and output pressure is 99 mmHg. Pressure changes linearly along the length of the vessel. Blood viscosity is 2cP and the vessel length is 3 cm. Assume blood flow is governed by Poiseulle's equation. a) Use Poiseuille's velocity profile equation to derive Poiseuille's formula for volumetric flow rate. b) Calculate flow, wall shear stress and mean circumferential wall stress for the artery. Use the pressure in the middle of the vessel to calculate sigma. c) The person begins to exercise, so blood flow increases by 50 percent. The pressure in the middle of the vessel, wall shear stress and mean circumferential wall stress must remain the same. Determine the new values of the inner and outer vessel radius. d) The person goes for a verly long hike in the desert and becomes dehydrated, so their blood viscosity increases by 25 percent. What are the new values of the inner vessel radius if blood flow and blood pressure must remain the same? e) The person is treated with a vasodilator that increases the inner radius of the vessel to 3 mm and the outer radius to 3.25 mm. If bloood flow Q in the vessel remains constant, what is teh new perssure drop along the vessel?
Consider an artery with internal radius a, external radius b, and luminal pressure P. Suppose that the distribution of circumferential wall stress in the artery wall created by lumenal pressure P can be expressed as a function of r based on the assumption of an Isotropic Hookean Elastic Solid. Now, suppose that a residual stress distribution is created by the cells in the vessel wall to alter this distribution of sigma such that it becomes uniform across teh vessel wall. a) Write a dimensionless expresson for the ratio of sigma/P solely in terms of a, b and r. b) Suppose the artery has dimensions a=2.1 mm, b=2.6 mm, and P=90 mmHg. At a point in the vessel wall where r=2.3 mm, what is the residual stress?
The stress tensor at a point in a continuum is given as follows: matirx. a) Find the constant sigma22 so that the stress vector on some plane at the point will be zero. b) Give the unit normal to this traction free plane.
You are designing a hip prosthesis, and want to characterize a new alloy that you are interested in using. In preliminary testing you have determined its Young's Modulus E to be 80 GPa, and its Shear Modulus G to be 30 GPa. you apply a stress loading condition as shown below, and model the alloy as a Hookean Elastic Solid. a) Calculate the Poisson's ratio for the alloy. b) Calculate teh strain tensor. c) You want to consider a second alloy for your design, but have not measured the Young's Modulus or Shear Modulus. You know that the Poisson's ratio is 0.25. From the shear stress loading and strain measurements shown below, calculate the Young's Modulus E and the Shear Modulus G.
A block of weight 900 N is resting on an incline plane. The plane is at an angle of 65 degrees. What is the minimum possible coefficient of static friction?
In class, we derived an expression for circumferential wall stress in an artery wall, sigma=Pr/h, where Pi is internal pressure, r, is internal radius, and h is wall thickness. To simplify, we considered Pi to be acting on a plane of luid in the vessel normal to the y axis and included this fluid in our free body diagram. Derive the expression for sigma again, but this time consider Pi to be acting on the vessel surface as shown below on the left.
Consider an artery with internal radius a, external radius b, and luminal pressure P. Suppose that the distribution of circumferential wall stress in the artery wall created by lumenal pressure P can be expressed as a function of r based on the assumption of an Isotropic Hookean Elastic Solid. Now, suppose that a residual stress distribution is created by the cells in the vessel wall to alter this distribution of sigma such that it becomes uniform across teh vessel wall. a) Write a dimensionless expresson for the ratio of sigma/P solely in terms of a, b and r. b) Suppose the artery has dimensions a=2.1 mm, b=2.6 mm, and P=90 mmHg. At a point in the vessel wall where r=2.3 mm, what is the residual stress?
The stress tensor at a point in a continuum is given as follows: matirx. a) Find the constant sigma22 so that the stress vector on some plane at the point will be zero. b) Give the unit normal to this traction free plane.
You are designing a hip prosthesis, and want to characterize a new alloy that you are interested in using. In preliminary testing you have determined its Young's Modulus E to be 80 GPa, and its Shear Modulus G to be 30 GPa. you apply a stress loading condition as shown below, and model the alloy as a Hookean Elastic Solid. a) Calculate the Poisson's ratio for the alloy. b) Calculate teh strain tensor. c) You want to consider a second alloy for your design, but have not measured the Young's Modulus or Shear Modulus. You know that the Poisson's ratio is 0.25. From the shear stress loading and strain measurements shown below, calculate the Young's Modulus E and the Shear Modulus G.
A block of weight 900 N is resting on an incline plane. The plane is at an angle of 65 degrees. What is the minimum possible coefficient of static friction?
In class, we derived an expression for circumferential wall stress in an artery wall, sigma=Pr/h, where Pi is internal pressure, r, is internal radius, and h is wall thickness. To simplify, we considered Pi to be acting on a plane of luid in the vessel normal to the y axis and included this fluid in our free body diagram. Derive the expression for sigma again, but this time consider Pi to be acting on the vessel surface as shown below on the left.
Your company has developed a specialized running prosthetic. It has been designed to act like a spring, bending under load and then straightening during the runner's stride to provide propulsion. The company is considering changing the current design, a solid rectangle, to a hollow cylinder to reduce weight. However, then do not want to sacrifice durability and ask you to do an analysis. a) Draw a FBD and solve for the internal moment along the length of the foot as a function of x. b) Calculate the maximum normal stress caused by bending of the foot. How far along the x axis does this occur? c) The company wants the outer radius of the new hollow cylinder design to be 1 cm. To ensure durability, they do not want to exceed the maximum normal stress produced in the old solid rectangular design. What should the inner radius of the new design be to meet these requirements?
A solid cylinder has a radius a and is oriented along a set of axes as shown in the figure. It is subjected to a complex loading condition yielding the Cauchy stress tensor shown below. All of the Greek characters in the tensor are constants. Verify that no traction exists on the curved surface of the cylinder.
The stress tensor in equilibrium is given by the following matirx. Find the principal stresses S1, S2, S3 at the pint P(a,0,2a^1/2).
You are testing a polymer to determine if it can be used to replace the achilles tendon, which is notoriously difficult to repair once it has been torn. The polymer is difficult to fabricate in the lab, so you would like to develop a Voigt model for thhe polymer that you can use to perform tests. You have springs of elasticity 10 N/cm, 15 N/cm, and 30 N/cm, as well as dashpots with viscosities of 4 Ns/cm, and 15 Ns/cm with which to build your model. You have collected the following data from your polymer: Constant 18 N of force produces a maximum displacement of 3 cm at long times. A steady state deformation rate of 5 cm/s produces a force at t=1s of 50 N. Draw the model that represents the polymer, and specify the elasticity or viscosity values for each component.
A composite specimen is attached to a wall and uniaxially loaded as shown below. The speciment is treated as two connected sections and the specimen's weight can be ignored. The left side of the specimen has a square cross section, while the right section has a circular ring cross section (shown in gray). Cross section dimensions are shown below at the right, with the outer ring having a diameter of 1 cm and the inner ring having a diameter of 0.5 cm. Assume that the left portion has a Young's Modulus of E=1500 KN/cm^2. After loading, the total length of the composite specimen is 12 cm. a) What is the Young's modulus E of the right section with the circular ring cross section? b) Note the 50 degree plane generated by making a cut through the right section. Calculate both the shear stress and normal stress on this plane.
Given the Cauchy stress tensor at a point in an arbitrary body, find the following for a surface defined by a unit normal vector n. a) The traction vector T acting on the surface that passes through the point. b) The total traction on the surface. c) The resultant normal stress on the surface. d) The resultant shear stress on the surface.
A solid cylinder has a radius a and is oriented along a set of axes as shown in the figure. It is subjected to a complex loading condition yielding the Cauchy stress tensor shown below. All of the Greek characters in the tensor are constants. Verify that no traction exists on the curved surface of the cylinder.
The stress tensor in equilibrium is given by the following matirx. Find the principal stresses S1, S2, S3 at the pint P(a,0,2a^1/2).
You are testing a polymer to determine if it can be used to replace the achilles tendon, which is notoriously difficult to repair once it has been torn. The polymer is difficult to fabricate in the lab, so you would like to develop a Voigt model for thhe polymer that you can use to perform tests. You have springs of elasticity 10 N/cm, 15 N/cm, and 30 N/cm, as well as dashpots with viscosities of 4 Ns/cm, and 15 Ns/cm with which to build your model. You have collected the following data from your polymer: Constant 18 N of force produces a maximum displacement of 3 cm at long times. A steady state deformation rate of 5 cm/s produces a force at t=1s of 50 N. Draw the model that represents the polymer, and specify the elasticity or viscosity values for each component.
A composite specimen is attached to a wall and uniaxially loaded as shown below. The speciment is treated as two connected sections and the specimen's weight can be ignored. The left side of the specimen has a square cross section, while the right section has a circular ring cross section (shown in gray). Cross section dimensions are shown below at the right, with the outer ring having a diameter of 1 cm and the inner ring having a diameter of 0.5 cm. Assume that the left portion has a Young's Modulus of E=1500 KN/cm^2. After loading, the total length of the composite specimen is 12 cm. a) What is the Young's modulus E of the right section with the circular ring cross section? b) Note the 50 degree plane generated by making a cut through the right section. Calculate both the shear stress and normal stress on this plane.
Given the Cauchy stress tensor at a point in an arbitrary body, find the following for a surface defined by a unit normal vector n. a) The traction vector T acting on the surface that passes through the point. b) The total traction on the surface. c) The resultant normal stress on the surface. d) The resultant shear stress on the surface.
Consider a solid beam of length L with a fixed support at A. The opposite end of the beam is subected to a force B, applied at angle theta. The beam has a weight W, applied at L/2. Calculate the reaction forrces and moments acting on the beam using the following values, L=0.5 m, Mass=10 kg, B=45 N, theta=135 degrees.
You have performed pressure-stretch tests on normotensive and hypertensive vessels, as shown below. For this question, assume you stopped your tests when the vessel reached a radiuus of 0.55 mm, as marked by the red line. What is the Pressure Strain modulus for each type of vessel under the control tone condition? b) What is the pressure strain modulus for each type of vessel when the vascular smooth muscle cells are maximally contracted using norepinephrine? c) Explain how the hypertensive vessels compare to normotensive. What phenomenon in the vessel wall is responsible for this difference?
You get a summer job working with a construction company. You're placing the rafters for the roof, and your boss wants to place them as shown in the figure because it will require less wood. You think this is risky and would like to rotate each beam by 90 degrees, even though this will be more costly. a) Draw a FBD and calculate the internal moment throughout the beam as a function of x. Plot Mint as a function of x. b) Calculate the maximum normal stress caused by bending of the beam. At what point along the x axis does this occur? c) You estimate that the maximum normal stress each rafter can take without breaking is 3.5E7 Pa. Write a sentence or two to your boss explaining why he should follow your suggestion.
A person holds a 30 N weight 35 cm from the elbow E. The weight of the arm, 10 N, acts 15 cm from the elbow. The biceps muscle, 25 cm in length, acts 3 cm from the elbow. Assume the elbow is a hinge joint and the biceps muscle is a cable connection. a) Determine the internal forces and moments in the forearm as a function of x. b) sketch the curves. c) Draw a FBD of the arm including any reaction forces and moments. Treat the elbow as a hinge and the muscle as a cable. d) Calculate the values of any reaction forces and moments, as well as the value of the biceps muscle force. e) With the arm lowered so that theta=50 degrees, the biceps muscle has now been lengthened. Determine the change in length and use the Young's Modulus of rubber to calculate the cross sectional area required to generate the biceps force calculated in d if the muscle is modeled as a solid cylinder.
For the exercise shown below, the quadriceps muscle works against the moment created by both the weight of the person's leg and the applied load. The weight of the leg, 7 kg, acts at A. The applied load, 10 kg, acts at B. a) Calculate the moment about the knee joint due to the weight and the applied load as a function of the angle theta. b) Pick a few points and plot M0 as a function of the angle from 0 to 90 degress.
John accepts a bet from his friends to jump off the second story of a building. John lands with all of his weight in his heels with a force of 1500 N. Assume his weight was evenly distributed between both legs. In the diagram to the right, the red square is the point of the calcaneus bone, where the force of the man's landing is acting. The green circle is the center of the ankle joint, where a moment is being produced by F. If the distance between the center of the ankle joint and the piont of the calcaneus, d, is 0.1 m and the angle theta is 45 degrees, calculate the moment generated about the ankle joint by F. If a human ankle can withstand 30 Nm without being damaged, was the bet a good idea? Consider F to be the only force.
A weightlifter performing a bench press exercise with dumbbells has lost control of the weight on the left side such that the weight has tipped forward as shown below. A free body diagram of the left arm is also shown below. Calculate how much force the left deltoid muscle must bear to maintain the weight in rotational equilibrium about the left shoulder joint, which is a hinge.
You have performed pressure-stretch tests on normotensive and hypertensive vessels, as shown below. For this question, assume you stopped your tests when the vessel reached a radiuus of 0.55 mm, as marked by the red line. What is the Pressure Strain modulus for each type of vessel under the control tone condition? b) What is the pressure strain modulus for each type of vessel when the vascular smooth muscle cells are maximally contracted using norepinephrine? c) Explain how the hypertensive vessels compare to normotensive. What phenomenon in the vessel wall is responsible for this difference?
You get a summer job working with a construction company. You're placing the rafters for the roof, and your boss wants to place them as shown in the figure because it will require less wood. You think this is risky and would like to rotate each beam by 90 degrees, even though this will be more costly. a) Draw a FBD and calculate the internal moment throughout the beam as a function of x. Plot Mint as a function of x. b) Calculate the maximum normal stress caused by bending of the beam. At what point along the x axis does this occur? c) You estimate that the maximum normal stress each rafter can take without breaking is 3.5E7 Pa. Write a sentence or two to your boss explaining why he should follow your suggestion.
A person holds a 30 N weight 35 cm from the elbow E. The weight of the arm, 10 N, acts 15 cm from the elbow. The biceps muscle, 25 cm in length, acts 3 cm from the elbow. Assume the elbow is a hinge joint and the biceps muscle is a cable connection. a) Determine the internal forces and moments in the forearm as a function of x. b) sketch the curves. c) Draw a FBD of the arm including any reaction forces and moments. Treat the elbow as a hinge and the muscle as a cable. d) Calculate the values of any reaction forces and moments, as well as the value of the biceps muscle force. e) With the arm lowered so that theta=50 degrees, the biceps muscle has now been lengthened. Determine the change in length and use the Young's Modulus of rubber to calculate the cross sectional area required to generate the biceps force calculated in d if the muscle is modeled as a solid cylinder.
For the exercise shown below, the quadriceps muscle works against the moment created by both the weight of the person's leg and the applied load. The weight of the leg, 7 kg, acts at A. The applied load, 10 kg, acts at B. a) Calculate the moment about the knee joint due to the weight and the applied load as a function of the angle theta. b) Pick a few points and plot M0 as a function of the angle from 0 to 90 degress.
John accepts a bet from his friends to jump off the second story of a building. John lands with all of his weight in his heels with a force of 1500 N. Assume his weight was evenly distributed between both legs. In the diagram to the right, the red square is the point of the calcaneus bone, where the force of the man's landing is acting. The green circle is the center of the ankle joint, where a moment is being produced by F. If the distance between the center of the ankle joint and the piont of the calcaneus, d, is 0.1 m and the angle theta is 45 degrees, calculate the moment generated about the ankle joint by F. If a human ankle can withstand 30 Nm without being damaged, was the bet a good idea? Consider F to be the only force.
A weightlifter performing a bench press exercise with dumbbells has lost control of the weight on the left side such that the weight has tipped forward as shown below. A free body diagram of the left arm is also shown below. Calculate how much force the left deltoid muscle must bear to maintain the weight in rotational equilibrium about the left shoulder joint, which is a hinge.
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