This text provides a comprehensive guide on how to calculate stress in various directions in thick-walled cylinders or tubes with closed ends, using formulas, examples, and an online calculator for internal and external pressure. It also covers the calculation of radial and tangential stresses in a thick-walled cylinder with open ends using Lamé’s equations. The text also discusses the use of stress, strain, and Young’s modulus for any elastic material using the stress equation and the formula E = σ/ε.
The text also covers the calculation of shear and bending stresses in circular beams and tubes using formulas and examples. It also discusses the sign convention, shear and moment diagrams, and the basic equations for these calculations. The text also discusses the calculation of hoop stress for thin-walled and thick-walled cylindrical and spherical vessels under internal and external pressure.
The text also discusses the basics of linear elastic fracture mechanics (LEFM), the stress intensity factor K and the fracture toughness KC, and how to calculate the factor of safety for fracture. The radial stress in pressure vessels can be calculated using the formula σr = pr/t, where σr is the radial stress, p is the internal pressure, and t is the thickness of the vessel’s walls.
The text also discusses the construction of a grid with normal stress on the horizontal axis and shear stress on the vertical axis. It also discusses the calculation of stresses for the inside and outside fibers of a curved beam in pure bending, drawing Mohr’s circle and determining the normal and shear stresses.
📹 Thin-Walled PRESSURE VESSELS in 8 MINUTES – Mechanics of Materials
Hoop Stress (tangential, circumferential), Longitudinal Stress (axial), and more! 0:00 Pressure Vessels Stresses 0:40 Dimensions …
How to calculate stress?
The weight of a given segment can be determined by substituting the weight for force in the stress formula, σ = F/A, where F is the force and A is the cross-section area. The yield strength of a solid material represents its maximum tensile stress before the onset of permanent deformation. Ultimate strength, on the other hand, denotes the maximum stress that a material can withstand before failure occurs.
What is the formula for the radius of curvature?
The Radius of Curvature Formula is a mathematical formula that represents the approximate radius of a circle at any point. It is calculated by dividing the radius of curvature by the derivative of curvature, denoted as K. The radius of curvature is the magnitude of a curved shape, which derives from a flat shape to a curve and from a bender back to a line. It is a scalar quantity, and the radius of curvature is not an actual shape or illustration, but rather an imagined circle.
What is the formula for bending stress?
The bending stress of a rail is calculated using the equation Sb = Mc/I, where Sb is the bending stress in pounds per square inch, M is the maximum bending moment in pound-inches, I is the rail’s moment of inertia in inches, and c is the distance from the base of the rail to its neutral axis. This information is based on the work of Elsevier B. V. and its licensors, and is protected by copyright laws.
What is the formula for stress mechanics?
The calculation of stress is based on the principle that the force applied to a material is divided by its cross-sectional area, resulting in the equation stress = force/area.
What is the relationship between radius and tension?
The radius of a circle affects the required tension, with larger circles requiring less tension and smaller ones increasing it. In circular motion, tension is the force exerted by a string or rope when stretched, providing the necessary centripetal force to keep the object moving. The tension in the string is crucial in maintaining the object’s motion in the circular motion, like a ball swung in a circle. Understanding the role of tension in circular motion is essential for understanding the dynamics of circular motion.
What is the formula for radius of curvature stress?
The greatest stress occurs at the outer surfaces of a beam. If the radius of curvature is designated as r and the requisite moment to achieve this condition is designated as M, then r is given by the equation r = (EI/M), where I is the beam’s second moment of inertia and E is Young’s modulus.
What is the formula for normal stress?
Normal stress is calculated by dividing axial force by the material’s cross-sectional area, determining the tension or compression it experiences under axial loads. Bending stress is influenced by the material type, dimensions, and external forces, varying linearly from the neutral axis. Shear stress, on the other hand, acts parallel to the material’s surface, unlike normal stress which acts perpendicular to the surface. The maximum stress occurs at the outermost fibers of the beam.
How to calculate curve radius?
The distance is divided by the square of the distance, resulting in a total of 144.
What is the relationship between radius and shear stress?
The relationship between radius and shear stress is inversely proportional, with an increase in radius having a negative impact on shear stress and a decrease in radius having a positive impact on it.
How to calculate for stress?
The weight of a given segment can be determined by substituting the weight for force in the stress formula, σ = F/A, where F is the force and A is the cross-section area. The yield strength of a solid material represents its maximum tensile stress before the onset of permanent deformation. Ultimate strength, on the other hand, denotes the maximum stress that a material can withstand before failure occurs.
What is the formula for actual stress?
True stress (σt) is a measure of the material’s strength and deformation, which is proportional to the force applied to the specimen. It increases until the specimen ruptures. In engineering and materials science, the stress-strain curve is used to determine the relationship between stress and strain. This curve is obtained by applying load to a test coupon and measuring deformation from tensile testing. It reveals properties of materials like Young’s modulus, yield strength, and ultimate tensile strength.
The stress-strain curve is plotted by elongating the sample and recording the stress variation with strain until the sample fractures. The engineering stress-strain curve is based on the original cross-section and gauge length, while the true stress-strain curve is based on the instantaneous cross-section area and length.
📹 Understanding the Area Moment of Inertia
The area moment of inertia (also called the second moment of area) defines the resistance of a cross-section to bending, due to …
there is a saying of Albert Einstein that says ”if you cant explain it simply u dont understand it well enough” I hope u can understand why Teachers get hard time to explain sth very simple like these. Thanks for this article, I now understand things that I learnt 2 years ago just within some minutes with simply explanations.
Legit this should be shown to everybody that tackles the centroid and moment of inertia lessons of the statics course! I just now learned that seeing an animation or understanding conceptually and intuitively is important before you even try to use the formulas in the text books. Or maybe im just not able to visualize how tf will there be a moment in a cross section. But this article taught me that, i cant thank u enought
dear sir, your presentation and explanation were really awesome, this is what exactly students are expecting from you i request you to please make a article on thick cylinders by showing the variation of pressure, hoop stresses . from inside to outside as well as from of side to inside. take cases of both external pressure and internal pressure i hope you will be making article on thick cylinder also..
Thank you for providing such clear explanations of engineering topics. The comments are right: THIS IS HOW ENGINEERING SHOULD BE TAUGHT! I’ve had 3 hours of lectures and read 2 textbooks on this topic and I didn’t really understand it until I watched your 11-minute article. The same is true for your articles on Shear Force & Bending Moments and Understanding & Analyzing Trusses. These are clear, concise, and understandable. Just in time for summer finals, too! Honestly, thank you!
I am a British retired chartered engineer with two engineering degrees and have never heard of the term “area moment of inertia” before and have always known this quantity as the “second moment of area”. To me the term “area moment of inertia” is a highly confusing term because there is no mass (inertia) in the calculation. Unfortunately the same symbol is used for the “second moment of area” and “moment of inertia” that is termed “mass moment of inertia” by some people. This could rise from the calculation of the two terms being similar with: Second moment of area = integral of r^2 da, where “a” is the area. This quantity is a resistance to bending or torsion Moment of inertia = integral of r^2 dm, where “m” is the mass. This quantity is a resistance to rotational acceleration and is the rotational equivalent of mass. By looking at the formulation of the second moment of area, for a given area the component has a greater second moment of area when the area is distributed away from the neutral axis. That is why many beams are shaped like the letter “I”. The second moment of area can also be used to calculate where the centre of pressure of a dam is positioned and it is the second moment of area / first moment of area, where the first moment of area is given by the integral of r da. For more information see: en.wikipedia.org/wiki/Second_moment_of_area en.wikipedia.org/wiki/Bending en.wikipedia.org/wiki/Moment_of_inertia vdocuments.mx/hydrostatic-pressure.
Back in the day while taking my electrical engineering core courses we were required to take things like statics and dynamics. It’s really too bad that my professors didn’t provide such basic instructions and insight. I may not have taken so long to figure out what on earth they were talking about! Keep up the good work.
Thanks for your work, mate. I’m a master degree student and I wish had your articles when I was a undergrad student. I would like to support your website, but I’m not a fan of patreon, so please allow the option to become a member of your website here on youtube. The currency of my country had gone to shit lately, but Im more than happy to support you with a dollar/month
Ok Ok . The work put in this article and the explanation is quite good … You got my respect ! Even though I have no idea why I am perusal this now because I have learned these years ago … Is just so pleasing to watch these animations . I wish I had these when I first learn these concepts . I had to imagine everything in my mind … Good work !!! Keep doing what you do !
Ugh I hate to admit it that as a mechanical engineering student, I am very incompetent when it comes to solid mechanics and materials strength subjects such as this, let alone analysis on machine elements such as bearings, gears, and shafts. My professor taught these subjects very very poorly. Good thing about the professors at my university is that professors that taught thermodynamics, heat transfer, fluid mechanics, y’know, thermofluid professors are quite keen on teaching thus actually lessens the guilt of being a mechanical engineering without any deep understanding of solid mechanical objects lol might as well call it thermofluids engineering then
I’ve been wanting to understand this topic for quite some time now but no article came close to the visuals and explanations such as this one! At first, I was confused about the importance of the area moment of inertia and the meaning of the the radius of gyration (since no articles had great explanations about the topic) but after perusal your article, it makes it easy to understand! Thank you! I highly recommend this to anyone wanting to understand the topic!
Moments of inertia must include mass or density in their definition since inertia is due to mass. Moments not including mass are not moments of inertia, they are just moments. I think the confusion comes from tables of moments of inertia given for uniform density materials, since the form of the equation is the same as that for the 2nd moment of area (apart from the lack of mass or density). Moment of inertia is the integral of the areal density function times radius squared.
Hi this is a very good material.. first of all.. i want to say thank you for uploading this. and second i have some questions.. first: ¿ you get paid for this articles for an institution.. or you just doit for yourself? second: what program do you use to create this articles..i have been thinking in creating some usefull articles. so i just thought that maybe i could use the same program u are usuing to really create understadable material.. thanks!
Quick question pls. When dealing with composite shapes why do we use the centroid of the whole shape as the reference and not the axis on which it’s place . Put simply, if I should turn an I beam such that it’s like an H is the moment of inertia going to be the same, are we still going to be using the same centroidal axis. Don’t mind my inquisitiveness 🤭🤭
My understanding why Mass and Area Moment of inertia is Mr^2 and Ar^2 For this we need to go back to High School Physics We know Moment is Quantiy * perpendicular distance Now Linear Momentum is Mass * Velocity The Moment of Momentum is mass * Velocity * perpendicular distance We also know that Moment of Momentum is Angular Momentum And angular Momentum = Mass Moment of Inertia * Angular Velocity So we can say that Angular Momentum = Linear Momentum * perpendicular distance Mass Moment of Inetia * Angular Velocity = Linear Momentum * Perpendicular Distance I * w = m*v*r I = m*v*r/w We know v = w* r So I = m*w*r*r / w = m*r*r So Mass Moment of inertia for point mass is mr^2 Now extending this concept for Area Moment of Inertia we have Ar^2
Although I am paying my university to teach this shit like you do, they just teach me how to integrate some non-sense to find some values which I have no clue where to use them about. GREAT JOB! I hope you can make other courses’ articles because I desperately need someone like you during distance-education period.
Here is some explanation of ‘moment of inertia’ and ‘area moment of inertia’ this Quality’s will not be used directly like Area, Volume Or Density but this value involves in calculating Bending Moment, Shear Forces, Acceleration or Retardation. There is no logic in the ‘moment of inertia’ calculation formulas.
Is it possible to find the principal (rotated) area moments of inertia by diagonalizing the tensor ? (Like you would do to find the principal mass moments of inertia) Instead of using the formula at 9:57 . And if so, is it possible to do so with a 2×2 matrix (instead of 3×3) since the area moment of inertia associated to the z axis is 0? Sorry if my English isnt very good
So area moment of inertia quantifies the resistance to bending. Now we can calculate area moment of inertia with respect to any axis. If the total area is on one side of the axis, then how is the bending defined in that situation? We can calculate the “resistance”, but the bending itself is not clear!
your limit is a few bit tricking on deriving the formula of rectangular section for moment of inertia where -h/2 to +h/2. if you do the operation after integration your answer will turn everything to zero. how about if your limit is from “zero to h” on the other hand you must show your simplification process in that part of your presentation because that is one of the most important portion of your operation so that all viewers will not be confused or you just remind us that after integration just neglect the signs, sorry just saying sir.
not mentioned that the name ” moment of inertia comes from the initial purpose: caclculate the rotational inertia of any given shape. ” how much moment ( newtonmeters) do I need to start a rotation of one rad per sec within one sec? ” if combined with density. and if combined with E, it gives rigidity. by the way, imagine those integrals if rho or E are functions instead if constants.
I took this course in the year 2018. It was called maths 142 and even now I feel it was one of the hardest courses of the whole degree (mechanical engineering obviously). This article was uploaded in 2020. If I had it in 2018, it surely would have helped me so much but I’m just glad that this contribution to humanity is here now.
One thing I’ve found that makes explaining this topic more intuitive is showing the moment of inertia graphically. The quantity is the same as the volume under a parabola extruded along the axis of rotation. We’re just doing double integral x^2 right? It helps me as a tool for explanation at least. The polar moment of inertia can be found by rotating the parabola around a vertical axis at the centroid of the cross section as well.
Man, thanks for making this kind of articles, it helps a lot. This year in my college the Static and Material´s Resistance´s teacher say he can’t go to classes cause of covid and the only work he does is to send a mail telling us what to study from a book whose author is he. He doesn’t explain anything and his book is terrible. Thank you man because you are saving a lot of students who make their best effort but some teachers seem to hate teaching.
Good day @The Efficient Engineer. Is it correct to say (let’s say we have three different sections as beams) that a section having the biggest moment of Inertia, will give the smallest bending stress “fb” upon the application of the same bending moment to those three beam sections? Thus, the bigger the moment of inertia, the more resistant to bending? Thanks in advance for the reply.
These articles are very good to brush the concepts. Just a suggestion, solve a problem with a variable cross section beam or add a multiphysics problem like a shaft with transverse loading, variable cross sectional geometry and varying temperature through out the length. That will be interesting, it will bring several theories together and how to solve it.
In studying this subject I continue to come across concepts that at first at least seem to be the same thing, only to later find that there are subtle differences between them that I initially missed. With that in mind, is the “bending axis” the same as the “neutral axis” or are they different things? It’s surprisingly difficult to find an answer to this seemingly simple question.
I really liked the article but there’s 2 things that bug me: 1: subscript notation what does it mean for a beam to bend in/about the x-axis, and how does that relate to its 2nd moment of area? like if a beam whose cross-section is in the xy plane is bending up and down (ie in the y-direction) do I use I_x or I_y for that? 2: double integral It’s a small thing but why are moments of area rarely ever written as a double integral? It’s easier to imagine the calculation as an integral over the x-axis then an integral over the y-axis.
I want to calculate the properties of some single and double angles that are not in the AISC database, So I started with a profile that is in the manual to check my results but the value of the torsional moment of inertia J that appear on the manual is not equal to Ix+Iy. does anyone know the AISC criteria for calculating J ?
I like the article. I would like to ask a question regarding area of moment of interia i.e in the formula distance square and why it so, what it really means in realistic way. I learn about second moment of inertia in the same way as present in this article but I like to know why distance square what it means in realistic way