 Deshmukh, working as an assistant professor in Department of Mechanical Engineering, Vulturen Institute of Technology, Solapur. In this video, we are going to learn the shape functions for two-noded 1D element, the learning outcome. At the end of this session, the learner will be able to find the shape functions for two-noded 1D element. Now, consider a two-noded 1D element, so this will be our first node and this is our element and this will be our second node. So, consider at first node, the displacement will be u1 and at second node, it will be u2. Let consider the length of this element as L. We will consider a reference plane here and from the reference plane, the first node lies at distance of x1 and the second node lies at distance of x2. Now, considering the linear distribution of field variable, so we will get the equation as u equals to a1 plus a2x, okay? So what is u here? u is field variable at any point within this element. What is u1? It is field variable at node 1 and u2 is the field variable at node. After applying the boundary conditions at node 1, what is the value of u here? That is u equals to u1 and x equals to x1. So equation 1 becomes u1 equals to a1 plus a2x1. This is our equation 2 and at node 2, u is u2 and x is x1. Therefore, u2 equals to a1 plus a2x2. This is equation 3. Now we need to find the values of a1 and a2. So for finding the values of a1 and a2, we need to solve the equations 2 and 3. So we will subtract equation 2 from 3, that is equation 3 minus equation. So we will get u2 minus u1 equals to a1 we will get cancel, okay? And these terms will be x2 minus x1 into a2. Therefore a2 equals to u2 minus u1 upon x2 minus x1. Therefore a2 equals to u2 minus u1. What is x2 minus x1 here? This is x2 minus x1 is nothing but the length of that element. So it will be a1. So now we need to find the value of constant a1. So we will substitute this value of a2 in equation 2. What we will get? It will be from equation 2 u1 equals to, we will substitute the value of a2 in equation 2. It will be a1 plus what is the value of a2 it is u2 minus u1 upon l into x1. Therefore a1 equals to, it is u1 will transfer this term to the left hand side. It will be minus u2 x1 by l plus u1 x1 by l which is equals to, we will multiply this by l. So it will be l u1 plus x1 u1 by l minus x1 u2 by l which is equals to, we will take u1 as a constant. So we will take u1 as a constant here. So what it will be? l plus x1 upon l into u1 minus x1 u2 by l. So we will substitute these values of u1 and u2 in equation 1. So putting values of a1 and a2 in equation 1. So the equation 1 becomes u equals to, what is the value of a1? This is l plus x1 upon l into u1 minus x1 u2 by l plus the value of u2 it is u2 minus u1 by l into x. So further simplifying this equation, so what is the value of l? What is l here? It is x2 minus x1. So we will substitute here x2 minus x1 x2 minus x1 plus x1 by l into u1 minus x1 u2 by l plus multiplying this x into the bracket u2x by l minus u1x by l. So this x1 x1 we will get cancelled. This will be x2 by l u1 minus x1 by l u2 plus x by l u2 minus x by l u1. So from first and last term we will take constant u1 equals to the equation becomes x2 minus x by l into u1 plus from the second and third term we will take u2 as a common. So what it will become minus x1 upon l plus x by l u2. So what is this? This term is known as the shape functions. We denote the shape functions by the letters n1 n2 etc. So this is our u. So therefore u equals to n1 u1 plus n2 u2 where n1 equals to x2 minus x by l and n2 equals to x minus x1 upon l. So for node 1 we will find the shape functions and for node 2 we will find the shape function. For node 1 the x equals to x1 therefore n1 equals to if you substitute the value of x as x1 what is the x2 minus x1 it is l so it will be x2 minus x1 by l which is equals to 1. And the n2 becomes if you substitute the value x value of x as x1 the x1 x1 will get cancelled so n2 becomes 0 this is for node 1 and for node 2 the n1 equals to for node 2 what is the value of x it is x equals to x2. So n1 becomes x2 minus x2 x2 minus x2 by l so it will be 0 and n2 equals to if you substitute the value of x as x2 it will be x2 minus x1 by l. So what is x2 minus x1 again it is l l by l it will be 1. So for the node 1 the shape values are 1 comma 0 and for node 2 the shape values are 0 comma 1. These are the references thank you.