Definition and Formula of Section Modulus – Direct and Bending Stresses – Strength of Materials


Hello friends here in this video what we will see what is the definition of section modulus and even its formula so section modulus it is defined as the ratio of moment of inertia of a section about its centroidal axis and to the distance of extreme layer from neutral axis so here I have written the definition of section modulus that it is defined as a ratio of moment of inertia of a section about its centroidal axis to the distance of extreme layer from neutral axis next mathematically section modulus is given by Z is equal to I upon Y so now here as I have written section modulus is given by Z is equal to I upon Y where I is the moment of inertia of the section about any of the axis it can be about xx or it can be about Y Y and Y is the distance from that axis to the extreme most layer now unit of Z that is section modulus is since it is I upon Y moment of inertia if suppose mm raised to 4 y is it of mmm so here this will be if mm raised to 4 y is in mm so 1 mm 1 mm gets cancelled so the unit is either mm cube centimeter cube or meter cube next I am taking an example in my first example if we have a rectangle now suppose this rectangular section we have and considering it to be a solid rectangular section now if I say that B is the width of the rectangle and D is the depth so if we have B and D here then first I will write the mi of the rectangle about x axis that is IX X will be equal to we know that for a rectangle moment of inertia about its horizontal centroidal axis X X that will be BD cube by 12 now Y value will be since we are considering moment of inertia about x axis so from x axis up to the topmost fiber or the bottom most fiber that will give us the distance Y so this is the distance Y and here it can be understood that y is nothing but D by 2 so therefore section modulus for the rectangle about the x axis will be IX 6 upon y next therefore ZX x will be IX X’s BD cube by 12 y is nothing but D by 2 so here we have B DQ by 12 into 2 by D so after the cancellation we are left with the formula that ZX x is equal to BD square by 6 so this is the section modulus for the rectangular section about x axis similarly if I want to write it about Y axis then iy y it will be moment of inertia for the rectangle about Y axis is d B cube by 12 now what will be distance Y Y is the distance between the axis about which we have taken moment of inertia to the topmost fiber here the topmost fiber is up till this edge or it can be this edge so this distance is y and it is clear that it is B by 2 so therefore the section modulus about Y axis will be iy by one why and here we have therefore z YY as DB cube by 12 it is divided by Y which is B by 2 so if I write multiplication that will be 2 by B so after the cancellation therefore z YY I’ll write down and zy y will be equal to DB square by 6 so here I have written the section modulus formula for a rectangle similarly we can show the section modulus for a circular section another example so here we have a circular section I will see that the outer diameter is capital D now I want to write section modulus formula for this circular section so first of all I x6 and now I will also be same because it is a circular section and ixx iyy it will be PI by 64 diameter is 2 4 this is the formula of moment of inertia next y value Y is the distance from if suppose I am taking moment of inertia about x axis so from x axis up to the topmost fiber or bottom most fiber in this case both are same so here this distance is y and it is very much clear that it is equal to the radius which is diameter by 2 even for the y axis if I can show you if I take mi about the moment of inertia about Y axis then from the y axis to the topmost fiber this distance is y so this is also D by 2 so therefore section modulus will also be same ZX X is equal to Z by Y and that will be either you take ixx iyy it will be 1 and the same here even if I have written iy y then also it would have been correct so therefore ZX x is equal to 0 iy IX X is PI by 64 diameter is 2 4 divided by Y Y is d by 2 so therefore this is PI by 64 D raised to 4 into 2 by D D and D gets cancelled so here it is d cube so the section modulus formula Z X X is equal to z YY that is pi by 32 d cube this is for a circular section now similarly we can add on writing the formula either for a hollow rectangle or a hollow circular section so I will take the next example this is of a hollow rectangle so here we have a hollow rectangular section I say that let the outer width B capital B inner with small B next let the outer depth B capital D and the inner depth be small D now I want to write down section modulus for this hollow rectangular section so now after I have drawn the hollow rectangle now I will write down the moment of inertia for the hollow rectangle about x axis will be it will be capital BD cube minus small BD cube divided by 12 now here since I have taken x-axis so the distance Y will be from the x-axis up to the top most fiber this distance will be Y and it is clear that it will be capital D by 2 so I will write down and Y is equal to capital D by 2 so therefore the section modulus about x-axis will be IX X upon y I xx is capital BD cube minus small V D cube divided by 12 divided by Y so it is divided by D by 2 so therefore ZX x is equal to capital BD cube minus small BD cube divided by 12 divided so here it will be multiplied by 2 by D so next this 2 and 12 gets cancelled so here I have the section modulus formula for a hollow rectangle about x axis that will be capital BD cube minus small BD cube upon 6 d now in a similar manner we can write for the y-axis I will write down here as I have written ixx iyy will be capital D be cube because I am calculating about y-axis – small D be cube divided by 12 and Y will be equal to now since I am taking moment of inertia about the y axis so the extremos fiber is either right right or towards left so any one we have to take the distance from the axis up to the extreme most fiber that is y and here it will be capital B by 2 next after writing iy y and y so therefore the section modulus about Y axis z YY will be iy y upon Y so therefore zy y will be iy y we have written here capital D V cube minus small DB cube divided by 12 it is divided by Y and Y is B by 2 so if I multiply it will become reciprocal that is 2 by B after the cancellation we have zy y is equal to capital D B cube minus small D B cube divided by 6 B so here is the section modulus for the hollow rectangular section about Y axis now let me it is take the last example that is of a hollow circular section now here is the hollow rectangular section here I will say that the inner diameter this is small D outer diameter is capital D now for this hollow circular section the moment of inertia about x axis will be same as about Y axis and the value will be PI by 64 outer diameter is 2 4 minus inner diameter is 2 4 then Y value will also be same y is the distance if I am taking mi about moment of inertia about x axis then from the x axis to the topmost fiber this gives us distance Y it is clear from the diagram that it is capital D by 2 so Y will be same either you take it from X or if you take it from Y excess the answer of Y will remain same that is it is outer diameter by 2 so therefore the section modulus about x axis and about y axis both are equal and that will be suppose if I take I xx or I’ve ever any one value so PI by 64 D raise to 4 minus small D raise to 4 divided by Y which is D by 2 so therefore Z X X is equal to Z by Y and that is PI by 64 capital D raise to 4 minus small T raise to 4 I will make this reciprocal if I write multiplication so that will be 2 by D after the cancellation here we have ZX x is equal to z YY and that will be pie by 32 capital e raise to 4 minus small R is 2 4 upon capital D so here is the section modulus formula for a hollow rectangular section so here in this video we have seen the definition of section modulus its unit its formula and we have seen four different examples that is cases for the section modulus

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Comments

  1. Hi, isn't this the formula for the elastic section modulus 'S'? How come we are using the elastic section modulus formula but calling it 'Z'? Genuinely want to know.

  2. ye adds wale bi kitnaa emotional drama krta hain …….ads dene se hyundai ki gaadiya zada thodi na bikengi….sala pareshan kr diya ads ne ….hr 2min baad ad daal rkhi ha….thts too bad

  3. Plzz sir aur kuch topic aur kuch subject start kijiye bahat help mil raha hey sir mere dosto bhi ap ki channel ko hi follow kar rahe hey plzz sir kuch new subject start kijiye bt thoda request sir thoda hindi boliye jyada comfortable hoga…

    Thanku sir…

  4. I in this context is the moment of Area not inertia , you are even measuring it in mm^4 while moment of inertia is in mm^2

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