 Hello everyone, my name is Saurabh Deshmukh, working as an assistant professor in Department of Mechanical Engineering, Vulture Institute of Technology, Sallapur. In this video, we are going to learn how to find the field variables or how to calculate the field variables within a triangular element using shape functions. The learning outcome. At the end of this session, the learner will be able to calculate the field variable of triangular element using shape function. So this is our problem. The nodal displacements of the element are given below for the point i, j and k. Now we need to calculate the displacement at points 2, 2 and 3, 4. So now I will draw the diagram here first. So this is our Cartesian coordinates. This is our triangular element, this is our point i, this is j and this is k. The coordinates of point i are 1 comma 1. The coordinates of point j are 8 comma 2 and coordinates of point k are 4 comma 6. Now we need to calculate the displacement at point with Cartesian coordinates 2 comma 2 and 3 comma 4. So we know that u equals to n1 u1 plus n2 u2 plus n3 u3. So now we need to first calculate the values of n1, n2 and n3. Here u1, u2 and u3 are nothing but ui, uj and uk. We know their values. So I will write the values of ui, uj, uk here. It is ui, uj and uk equals to, it is 2, 1 and 3 into 10 raise to minus 3 mm. Now we need to calculate the displacement at point 2 comma 2 and 3 comma 4. So how to calculate the n1, n2 and n3 here? So we will calculate for the point with Cartesian coordinates 2 comma 2 for 2 comma 2. We need to calculate the value of n1. So n1 equals to. Now let's suppose the point 2 comma 2, we name it as a p. So let's assume that point p lies here, it is 2 comma 2. Now we will join pk, pi and pj. Now there are 3 triangles. First one is ipj, pjk and pik. So we will name it for the opposite of k, this is triangle k. For the opposite triangle to the i is i and opposite to the j is j. Now we need to calculate the shape functions of the point 2 comma 2 with respect to i. So how to find out it? It is area of, for calculating the shape functions with respect to point i we need to take the triangle opposite to it. So it is area of triangle pjk upon area of triangle, it is whole triangle. It is ijk. Similarly to calculate the value of n2, it is n2 equals to area of triangle. Now we need to calculate the value of a2 that is with respect to j, that is the shape functions of point 2 comma 2 with respect to j. So opposite to that it is triangle pik, pik upon area of triangle ijk. And also n3 equals to it is area of triangle, for the k take the opposite triangle to it. So it will be pij, it is pij upon area of triangle ijk. So how to find the area of triangle? It is very simple, we need to find the determinant of their coordinates. So we will first calculate the area of triangle ijk. So area of triangle ijk equals to it is nothing but 1 by 2 into determinant of, the first column is 1, 1, 1 and in the second and third column we feel the cartesian coordinates of that point of the vertices of that particular triangle that is ijk. So what is the cartesian coordinates of i? It is 1, 1, for j it is 8, 2 and for k it is 4, 6. So what is area of triangle ijk here? It is equals to 1 by 2 into bracket, it is 40 into minus 1 into 4 minus 2 it is minus 4 and 1 into 4 into minus 8 it is minus 4. So it will be equals to 60. So we will find the area of triangle pij, pik and pij. So we will find here the area of triangle pjk equals to it is 1 by 2 into determinant of it is 1, 1, 1 the coordinates of point p are 2, 2, 8, 2 and 4, 6. So after calculating this we will get 1 by 2 into it is 40 minus 8 and this is also minus 8. So it will be equals to it is 24 upon 2 it is 12. Similarly area of triangle pik equals to 1 by 2 into it is 1, 2, 2, 1, 1, 1, 1, 1, 4, 6. So it is 1 by 2 into you see it is 1 into 4 minus it is 2 minus 2 into 6 minus 1 it is 5 minus 10 2 this is plus 6. It is minus 2 upon 2 it is minus 1 but area cannot be negative so it will be 1. Area of triangle pij equals to it is 1 by 2 into determinant of 1, 2, 2, 1, 1, 1 and 4 j it is 182. So it is equals to it is 1 by 2 into this is minus 6 this is minus 2 and plus 2 into 7 it is 14. So it is equals to 6 by 2 it is 3. So we have calculated the area of triangles now we need to calculate n1 n2 n3. So we will calculate it here n1 equals to what is it n1 it is area of triangle pji pji is 12 upon area of triangle ijk it is 16 so 12 by 16 equals to now n2 equals to it is 1 by 16 and n3 it is 3 by 16. So if you calculate it by calc by using the calc we will get n1 equals to 12 by 16 it is 0.75 n2 it is 0.0625 and n3 it is 0.1875. So now we need to calculate the displacement at point 2 comma 2. So at u at 2 comma 2 equals to n1 what is the value of n1 it is 0.75 we need to substitute this value in this equation 0.75 into ui u1 is ui it is 2 into 10 raise to minus 3. So 2 into 10 raise to minus 3 plus n2 it is 0.0625 into 1 into 10 raise to minus 3 that is uj u2 is nothing but and plus n3 is 0.1875 into u3 is nothing but uk it is 3 into 10 raise to minus 3 equals to. So now we will substitute this value into the calc so we will solve it here it is 0.75 multiplied by 2 into 10 raise to minus 3 plus it is 0.0625 into 10 raise to minus 3 plus 0.1875 into 3 into 10 raise to minus 3. So it is 2.125 into 10 raise to minus 3 m m. So u at 2 comma 2 these are the references. Thank you.