4 Principal Stresses
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This document discusses principal stresses, maximum shear stress, and how to calculate and visualize them. It describes principal stresses as the minimum and maximum normal stresses in a material, which occur when shear stress is zero. It presents equations to calculate principal stresses and maximum shear stress from the normal and shear stresses, and how to find the angles at which they act. The document also discusses sketching and visualizing these stresses on a stress element.
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This document discusses diarthrodial joints and joint motion. It identifies important connective tissue structures in diarthrodial joints like synovial fluid, articular cartilage, meniscus, ligaments, tendons, and muscles. These structures provide lubrication, load distribution, joint stability, and transmission of forces. Joints allow six degrees of freedom consisting of three translations and three rotations. Applying statics to joint biomechanics problems requires assumptions like known axes of rotation and muscle attachments, negligible friction, and consideration of only two-dimensional forces. Anthropometric data provides average human body measures used in the analysis.
10 hard and soft tissue biomechanics
This document discusses the viscoelastic properties of biological materials like bone and ligament, noting that the elastic modulus and strength of bone increases as the strain rate increases from slow walking to fast running. It also explains that bone is composed of mineralized collagen fibrils, giving it viscoplastic behavior where it will continue to deform under constant stress over time without fracturing. Graphs are presented showing the relationship between bone strain over time and increases in modulus and strength with higher loading rates.
8 beam deflection
This document discusses beam deflection. It begins by defining beam deflection and the factors that affect it, including bending moment, material properties, and shape properties. It then presents the general formula for calculating beam deflection using double integration of the bending moment equation. Examples are given of using boundary conditions to solve for deflection in simply supported beams, cantilever beams, and beams under various loading types. Common deflection formulas are also presented.
6 internal forces and moment
This document discusses internal forces and moments in bending structures. It defines shear forces and bending moments, and explains that when a structure is loaded, internal forces and moments keep it from being pulled apart. It outlines how to determine these internal forces and moments by drawing a free body diagram of the structure, then making imaginary cuts along the structure and analyzing the forces and moments at each cut section. Resources for learning about free body diagrams, support reactions, and shear forces and bending moments are also provided.
Biomechanical testing of clavicle fx devices
The document describes a study that tested various clavicle fracture fixation devices under torsion and bending loads. The objectives were to compare the mechanical strength, stiffness, and failure modes of different plate designs. Devices from Synthes and DePuy were tested, including dynamic compression plates, locking plates, and a pin. Results showed differences in torsional strength and stiffness between devices, as well as differences in failure mechanisms. Calculated torque values based on plate geometry and material properties were generally consistent with experimental torque measurements at yield, though assumptions introduce some error.
5 torsion
This document discusses torsion and stresses in structures subjected to twisting forces. It begins by describing the torsion formula which relates shear stress to radius and polar moment of inertia. Maximum shear stress occurs at the outer radius. Torque is dependent on position along the length. Principal stresses in torsion occur at 45 degrees to the centerline and are equal in magnitude but opposite in direction (tensile and compressive). The magnitude of the principal stresses equals the maximum shear stress.
3 strain transformations
Strain transformation equations and explanation; function, design and biomedical applications of strain gauges
3 stress transformations
This document discusses stress transformations, which are calculations of stresses in multiple planes within rigid bodies. It provides learning objectives about calculating stresses in different planes, defines stresses within rigid bodies and stress elements. It then recalls the definition of stresses in 3D and discusses summing forces on an inclined stress element. Finally, it presents equations to calculate the transformed stresses σxx', σyy' and τxy' in a new coordinate system defined by the angle θ.
2 three dimensional stress and strain
Brief summary of how strain is measured, and details on how strain and stress are described mathematically in 2D and 3D.
2 mechanical properties of bone
This document discusses the mechanical properties of bone. It begins by outlining the learning objectives, which are to conduct stress analysis on bone samples, identify how bone's anisotropic material properties arise, and analyze how changes in bone composition affect stress and strain. It then explains that mechanical testing of materials involves applying and measuring forces and deformations. The document discusses how bone is a composite material made of collagen and hydroxyapatite mineral, giving it anisotropic properties, and strongest in compression. It attributes bone's anisotropic properties to its cortical and cancellous bone structures. Finally, it notes bone is a dynamic material that repairs itself and changes with aging and weight-bearing levels.
2 stress and strain intro for slideshare
This document defines key concepts related to stress and strain in engineering. It introduces terminology like stress, strain, Young's modulus, and Poisson's ratio. It explains that internal forces develop in objects under load to prevent separation. Stress is defined as the internal force per unit area, and can be normal or shear. Strain is the deformation from loading, and can be normal changes in length or shear changes in angle. Constitutive equations relate stress and strain through material properties like Young's modulus and Poisson's ratio.
2 stress and strain intro
This document defines key concepts in stress and strain analysis:
1) It defines normal stress, shear stress, normal strain, shear strain, and material properties like Young's modulus and Poisson's ratio.
2) It explains that internal forces develop in materials under loading to prevent separation, and that stress is calculated as the limit of force over area as the area approaches zero.
3) It notes that stress is a tensor with directional components, while strain describes changes in size and shape from loading.
1 Bone Structure and Composition
Bone is a composite material made up of cells and an extracellular matrix. At the micro level, bone contains osteoblasts that build bone tissue, osteoclasts that resorb bone, and osteocytes embedded in the matrix. The matrix is made of collagen fibers mineralized with hydroxyapatite crystals. Bones can be classified by their macro structure as long, short, flat, or irregular. Long bones have a diaphysis and epiphyses. Bone structure is adapted to its mechanical functions according to Wolff's law.
Introduction to Active Learning
This document discusses active learning techniques that can be used in undergraduate engineering courses. It defines active learning as engaging students through activities that promote higher-order thinking skills. Examples of active learning include collaborative problem solving, brainstorming, and case studies. Research shows benefits of active learning include improved attendance, deeper questioning, greater interest, and higher grades. The document provides tips for incorporating active learning, such as starting with simple tasks and activities. It also addresses managing student resistance to active learning.
Engineering Knowledge, Skills, and Abilities
This document discusses key knowledge, skills, and abilities (KSAs) that engineering students need to develop for successful careers. It defines three ways of knowing - "knowing that" which involves declarative knowledge, "knowing how" which involves procedural knowledge, and "knowing why" which involves curiosity and continuous learning. The document proposes assessing students' problem-solving processes and provides examples of activities that develop different ways of knowing in engineering courses. It assigns reflection essays for students to discuss experiences with "knowing that, how, why" and how to develop important KSAs in future teaching.
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4 Principal Stresses
- 1. Principal Stresses BIOE 3200 - Fall 2015
- 2. Learning Objectives Describe what principal stresses and maximum shear stress are and why they are important Calculate principal stresses and maximum shear stress, and the angles at which they occur Visualize principal stresses and maximum shear stress on a stress element BIOE 3200 - Fall 2015
- 3. Review: Stresses within a rigid body BIOE 3200 - Fall 2015
- 4. Review: Stress Transformations 𝜎𝑥𝑥 ′ = 𝜎 𝑥𝑥+𝜎 𝑦𝑦 2 + 𝜎 𝑥𝑥−𝜎 𝑦𝑦 2 𝑐𝑜𝑠2𝜃 + 𝜏 𝑥𝑦 𝑠𝑖𝑛2𝜃 𝜎 𝑦𝑦 ′ = 𝜎 𝑥𝑥+𝜎 𝑦𝑦 2 + 𝜎 𝑦𝑦−𝜎 𝑥𝑥 2 𝑐𝑜𝑠2𝜃 − 𝜏 𝑥𝑦 𝑠𝑖𝑛2𝜃 𝜏 𝑥𝑦 ′ = 𝜎 𝑦𝑦−𝜎 𝑥𝑥 2 𝑠𝑖𝑛2𝜃 + 𝜏 𝑥𝑦 𝑐𝑜𝑠2𝜃 BIOE 3200 - Fall 2015
- 5. Principal Stresses: Minimum and maximum normal stresses Find by setting derivative of stress transformation equation to zero Also know that shear stress = 0 on principal plane ◦ Use shear stress transformation equation to develop equations for principal stresses BIOE 3200 - Fall 2015
- 6. Sketching principal stresses on a stress element BIOE 3200 - Fall 2015
- 7. Maximum Shear Stress Even if max normal stress is within limits of material strength, failure can occur in shear Find direction of max shear stress plane by setting derivative of shear stress transformation equation to zero BIOE 3200 - Fall 2015
- 8. Maximum Shear Stress Occurs on plane at which normal stresses are equal ◦ Can also solve by setting normal stress transformation equations equal to each other and solving for angle of max shear stress plane ◦ Plug angle into shear stress equation to determine equation for max shear stress BIOE 3200 - Fall 2015
- 9. Sketching Principal Stresses and Maximum Shear Stress Principal stresses occur at angles 1 and 2 from the x-y axes Maximum shear stress occurs at an angle s from the x-y axesBIOE 3200 - Fall 2015 s s
Editor's Notes
- To obtain a complete picture of the stresses in a bar, we must consider stresses acting in various orientations through the bar. Stresses on both normal and inclined sections are uniformly distributed. Only one intrinsic state of stress exists at a point in a stressed body, regardless of the orientation of the stress element used to portray the state of stress. We can rotate axes to represent stresses in a new coordinate system, oriented at any angle to the original x-y system.
- Stress transformation equations allow calculation of stresses (normal and shear) in any direction, given stresses defined along a known orientation (x-y axis system).
- Among an infinite number of planes that can be constructed about a point in a structure, there is one orientation at which normal stresses are at a maximum and minimum – principal stresses. That plane is the principal plane. (Mathematical description of how to prove that principal stresses exist at shear stress = 0 is given in textbook, section 2.4) Important in stress analysis – failure and fracture occur along planes of maximum stresses, so design must account for that. To find principal stresses, set derivative of normal stress wrt angle to zero to find angle at which stress is min/max; solve for angle (this is the angle of the principal plane, or principal angle); plug principal plane angle back into stress transformation equation. Principal stresses can also be determined by knowing that they occur on a plane where there is no shear stress; set shear stress transformation equation to zero and solve for principal angle, then plug into stress transformation equations.
- Derivation is mathematically demonstrated in textbook, by differentiating shear stress wrt angle, the maximum occurs where y and x normal stresses are equal. There are two values of 2θs in the range 0-360°, with values differing by 180°. There are two values of θs in the range 0-180°, with values differing by 90°. So, the planes on which the maximum shear stresses act are mutually perpendicular. Because shear stresses on perpendicular planes have equal magnitudes, the maximum positive and negative shear stresses differ only in sign.
- Maximum shear stress occurs when normal stresses are equal, so can solve for it by setting stress transformation equations x’ and y’ equal.
- The two principal stresses determined so far are the principal stresses in the xy plane. For stresses in 3D, there are always three principal stresses. Common terminology is that σ1 > σ2 > σ3. Principal stresses can be compressive or tensile.