Stress and strain analysis.pptx in medical mechanics

medical mechanics

1. Stress and strain analysis

2. Stress Analysis: Definition: Stress is a measure of the internal forces within a material or structure that arises from external loads or forces. It is expressed as force per unit area and helps determine how materials deform or fail under applied forces. Types of Stress: Normal Stress (σ): Acts perpendicular to the surface of the material. Tensile stress : Causes the material to elongate. Compressive stress : Causes the material to compress. Shear Stress (τ): Acts parallel to the surface of the material. Results from forces or moments that cause one part of the material to slide or deform relative to another part. Bending Stress : Occurs in beams subjected to bending moments. It varies across the cross-section of the beam and is highest at the outermost fibers. Torsional Stress : Occurs in shafts or structural members subjected to twisting moments (torque). It varies along the circumference of the shaft and is highest at the outermost surface.

3. Strain Analysis: Definition: Strain is a measure of the deformation experienced by a material in response to stress. It quantifies how much the material changes in length, shape, or volume due to applied forces. Types of Strain: Normal Strain (ε): Measures the change in length per unit length of the material. Tensile strain : Corresponds to elongation. Compressive strain : Corresponds to contraction. Shear Strain (γ): Measures the change in angle between two lines originally perpendicular in the material. It is related to shear stress and is important in analyzing materials under shear loading.

4. S tress-strain curve A stress-strain curve is a graphical representation of the relationship between stress (force per unit area) and strain (deformation) of a material undergoing mechanical loading. It provides valuable information about the material's mechanical properties and behavior under different conditions.

5. Stress-strain curve

6. Stress-strain curve

7. Stress-strain curve 2 . Yield Point: The point on the stress-strain curve where the material begins to deform plastically (non-linear deformation). Yielding occurs when the material no longer returns to its original shape after the stress is removed. Yield strength (or yield point) is the stress level at which this permanent deformation begins. 3. Plastic Region: Beyond the yield point, the material enters the plastic region where strain increases without a corresponding increase in stress. Plastic deformation is permanent and typically leads to necking (localized narrowing) in ductile materials.

8. Stress-strain curve 4. Ultimate Tensile Strength (UTS): The maximum stress that a material can withstand during stretching before it fails. It is the highest point on the stress-strain curve. 5. Fracture Point: The stress and strain at which the material fractures or breaks. This is the end of the stress-strain curve, where the material can no longer withstand the applied stress.

9. Key Features of a Stress-Strain Curve: Modulus of Elasticity (Young's Modulus, E): The slope of the linear elastic region of the stress-strain curve. It represents the stiffness of the material. Yield Strength: The stress at which plastic deformation begins. It indicates the point where the material transitions from elastic to plastic behavior. Ultimate Tensile Strength (UTS): The maximum stress the material can withstand before failure. It gives an indication of the material's strength. Ductility: The extent of plastic deformation a material can undergo before fracturing. Materials that exhibit large plastic deformation are considered ductile. Toughness: The ability of a material to absorb energy and plastically deform before fracturing. It is represented by the area under the stress-strain curve.

10. B eam theory Beam theory, also known as beam mechanics, is a branch of engineering mechanics that focuses on the behavior of beams under different types of loading and support conditions. It provides fundamental principles and equations for analyzing the stresses, strains, and deflections of beams, which are essential in designing and evaluating structures such as bridges, buildings, aircraft wings, and mechanical components.

11. Beam theory Types of Beams: Simply Supported Beams : Supported at both ends with pinned or roller supports. Fixed Beams : Supported at both ends with rigid connections that prevent rotation and translation. Cantilever Beams : Supported at only one end, with the other end free. Types of Loading: Point Loads : Concentrated forces applied at specific points along the beam. Distributed Loads : Forces applied continuously along a portion or the entire length of the beam. Moments : Couples or torques that cause rotational deformation.

12. Beam theory

13. Beam theory Assumptions: Beams are assumed to be slender structures with a length much greater than their cross-sectional dimensions. Deformations are small and occur primarily in the plane of the beam (plane sections remain plane before and after deformation). Material properties are uniform and isotropic.

14. Beam theory

15. Beam theory

16. Example 1: Simply Supported Beam with Uniformly Distributed Load

17. Example 1: Simply Supported Beam with Uniformly Distributed Load

18. Example 1: Simply Supported Beam with Uniformly Distributed Load

19. Example 1: Simply Supported Beam with Uniformly Distributed Load

20. Example 1: Simply Supported Beam with Uniformly Distributed Load

21. Example 2: Cantilever Beam with Point Load

22. Example 2: Cantilever Beam with Point Load

23. Example 2: Cantilever Beam with Point Load

24. Example 2: Cantilever Beam with Point Load

25. Example 2: Cantilever Beam with Point Load

26. T orsion theory Torsion theory in structural mechanics and engineering deals specifically with the behavior of solid objects subjected to twisting forces, known as torsion. Torsional Deformation : Torsion occurs when an object (often a shaft or a structural member) is subjected to twisting moments (torque) about its longitudinal axis . This twisting creates shear stresses and strains within the material, which can lead to deformation and failure if not properly accounted for in design . Torsional Shear Stress : Torsional shear stress, τ is the stress caused by torsional moments acting on a solid shaft or structural member . It varies along the cross-section of the member, being maximum at the outer surface (periphery) and zero at the axis . Torsional Equilibrium : For a member to be in torsional equilibrium, the sum of all torques (moments) applied at any cross-section must be zero . Torsional equilibrium is crucial in ensuring that the member rotates without angular acceleration under applied torsional loads.

27. Torsion theory

28. Torsion theory 3. Stress Distribution : The distribution of shear stress across the cross-section of a shaft varies linearly from zero at the center (axis) to a maximum at the outer surface, following Hooke's Law of shear stress distribution.

29. Example 1: Calculating Maximum Shear Stress

30. Example 1: Calculating Maximum Shear Stress

31. Example 1: Calculating Maximum Shear Stress

32. Example 2: Calculating Polar Moment of Inertia

33. Example 2: Calculating Polar Moment of Inertia

34. Example 2: Calculating Polar Moment of Inertia