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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
Notas del editor
- 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.