 Myself R.C. Viradha, working as assistant professor in Valshan Institute of Technology, in this session we will study about strain energy due to gradual load. At the end of this session, students will be able to determine the strain energy stored in an object under gradual load and sudden load. Now, what is the strain energy? Strain energy is the energy stored or developed due to the external applied load on a body. Due to this load applied on a body, the molecules or atoms get distorted from its unstressed state and this will be going to transform into a strain energy. And this is in the form of potential energy. And the strain energy in the form of elastic energy can be recoverable in the form of mechanical work. The amount of energy will differ with gradual, sudden and impact type of loading conditions on a body. Now, let us know what is the strain energy in gradual load by plotting a graph of stress versus strain curve for a ductile material. Now, I will take stress on y axis, strain on x axis. For the ductile materials, we know this is a graph variation for stress and strain and this is the elastic limit for ductile material. This is for ductile material. The area under this curve is called as work done or strain energy stored. Now, we will derive the strain energy formula for gradual type of load applied on a body. So, for the gradual load, the load which is applied on a body in stepwise, it is from starting with zero value of load and it will increase in time with its full amount. So, that is shown on y axis and the change in length is seen on x axis and the area under this p and delta l curve is nothing but the work done in a material or body. And now, we are going to denote the strain energy by a capital letter u. And as I said, the strain energy is a form of potential energy and amount of strain energy can be utilized to do the mechanical work. So, we can equate the strain energy as work done and the work done from this graph p and delta l curve can be represented with area under the curve, area under the p delta l diagram. Now, as this looks like a right angle triangle, so I can write area of this diagram as half into height from the diagram and base of the diagram as strain energy. And as we know delta l is p l by a e replace delta l terms by p l by a e term. So, it becomes p square into l by a e. Now, multiply and divide both side by a by a to get term. This term I have multiplied and divide to get here a square. In order to get the term after simplification, it becomes p square by a square and here it remains a l by e will be the remaining term. And as we know this term p by a e is nothing but sigma. So, simplification finally we can get sigma square by 2 times e into a into l is nothing but volume and this equation is a strain energy in a ductile material due to gradual load of condition. Now, the strain energy equation contains the stress value and this stress value is different for gradual load, sudden load and impact type of load. So, in order to find the strain energy for all the three type of loading conditions. So, we can get the strain energy for different loads. Now, to find the stress due to gradual load third one. So, again we can use the same area as we seen earlier. So, again we know that strain energy is nothing but half into base into delta l. This is the area under this curve. So, here replace the delta l term by sigma l by e. And I can replace that term by p into sigma l by e. And also put this u value as we derived in earlier derivation. That is sigma square by 2 e into volume is nothing but half into p into sigma l by e. Now, on simplification, so I can get sigma into a into l equal to what p term will be remaining here into l. Because this term, this term and this term get cancelled. And one sigma here and one sigma get cancelled. Only remaining is sigma into volume that is a into l. And this side is p into l is remaining. Again, here l get cancelled here. So, remaining is what? p by a term as a sigma value for gradual load. So, now think a while. And similarly, as we derived here, so find the stress value for sudden type of load. For sudden type of load, the area under the curve we can write it as this. Here full load is taken like on y axis at a time. Because here full load is applied at a time. Here the step by step load is applied. Again same, we can go the area under this curve is a work done. And that is nothing but what? p into delta l. And if you replace this delta l term as this, we can get sigma value as 2 times p by a. And this value is a sigma for sudden load. And if you use this term of sigma and this term of sigma, put this value of sigma in this u equation that is strain energy equation. So, to get the strain energy for gradual load and sudden load. Thank you. Thank you.