 Today we will see the introduction to plastic analysis. Myself, Kaji Syed Shujat, working as assistant professor in civil engineering department, Walchen Institute of Technology, Solapur. The learning outcome of this today's session is, at the end of the session, student will be able to understand the concept of plastic analysis. We will start with introduction. First of all, ductility of steel. What is mean by ductility? It is a property of steel with the help of which the steel can be drawn into wires. This is a stress strain curve of mild steel. Here we are having stress on y-axis and on x-axis we are having strain. We will see important stages of the stress strain curve for this mild steel. First of all, the point A is the limit of proportionality. Point B is the upper yield point, point C is the lower yield point, point D is the range which is a plastic range, point D is the point at which the strain hardening starts, point E is the ultimate collapse or ultimate tension carrying capacity or compression carrying capacity, ultimate load carrying capacity and point F is the fracture point. Zone AB is the elastic range, zone CD is the plastic range, zone DE is the strain hardening range and here the stress reduces and zone F is the fracture point. Here are few of the properties of the mild steel. For mild steel, the yield stress is equal to 250 MPa, ultimate stress is equal to 410 MPa, elastic yield strain is equal to 0.12 percentage and epsilon ST which is nothing but the strain at which the strain hardening will start that is 1.5 percent. Young's modulus of elasticity is 2 into 10 to the power 5, 2 into 10 to the power Newton per mm square and epsilon ST is the strain at which Young's modulus at which the strain hardening will start which is equal to 6700 Newton per mm square. E ST is generally E by 30. Here we are having the graph of mild steel. Now for the plastic analysis we are having, first of all we will idealize the stress strain curve. Basically we are having the upper yield point and lower yield point and also the limit of proportionality. So, for the plastic analysis we will assume that proportionality limit, upper yield and as well as lower yield point will be regarded as the point 1 point. So, this is a point which is a first point, we will assume all the 3 point equal to as a 1 point. This range is the elastic range, basically AB is the plastic range zone. So, it is also known as plateau of yield and the effect of strain hardening is ignored in the plastic analysis. As per IS 800 2007, IS has given certain requirements for to satisfy the plasticity of steel. This is given in clause number 4.5.2 and the first clause is yield stress for the grade of steel shall not exceed 450 mega Pascal yield stress that is FY. Next is stress strain characteristic of steel shall be such as to ensure complete plastic movement redistribution. So, we have to have the plastic redistribution of the movement. Next is the stress strain diagram has a plateau at the yield stress extending for at least 6 times that of yield strain. So, yield stress should be equal to 6 times that of yield strength. Now, the ratio of tensile strength to the yield stress should not be less than 1.2 and the elongation on the gauge length should also be not less than 15 percent. Steel exhibits strain hardening capability and the member used for the plastic analysis should be shall be hot rolled or fabricated using hot rolled plates and section. These are the few requirements which are given in IS 800 2007 clause number 4.5.2. Before starting the plastic analysis, we will see few terms. First one is the shear factor. What is mean by shear factor? It is defined as the ratio of plastic movement to that of yield movement. Basically, it is a function of cross section. So, shear factor is equal to MP plastic movement divided by the yield movement MY. Basically, movement will be equal to stress multiplied by section modulus. So, for plastic, we are having Fy into Zp and for MY it is equal to Fy into Ze. Zp is the plastic section modulus and Ze is the elastic section modulus. So, Fy gets cancelled which is equal to Zp upon Zy. This is the basic definition of shear factor. It is a function of cross section and it is a ratio of plastic movement to that of yield movement. Next definition is load factor. So, it is defined as the ratio of collapsed load to that of working load. So, PU is the collapsed load and P is the working load which is equal to MP upon MY which is equal to Fy into Zp divided by Fy into Ze. F is the stress which is less than the yield stress. Now, Fy upon F which is equal to factor of safety and Zp upon Ze just now we have seen which is equal to the shape factor. So, the load factor will be equal to factor of safety multiplied by the shape factor. Now, we have to determine the collapsed load. Basically, we are having two methods. One is the statical method and another one is a kinematic method. So, in the first method we will take one a simple example of a simply supported beam subjected to a point load. Here, the working load is W and the collapsed load will be equal to Wp. So, the plastic movement is obtained by equating the maximum bending movement value equal to MP. So, MP will be equal to Wl by 4. Wp is the load at which the collapse will occur. So, MP will be equal to Wp into l by 4. Now, rearranging the terms, Wp will be equal to 4 MP by l. So, this is a statical method which uses the statically admissible bending movement diagram to obtain the collapsed load. In using statical method, it is necessary to know the correct bending movement diagram. So, this is a simple method in case of simple loading on determinate beam for which the bending movement diagram can be quickly visualized. Next one is nothing but here on our kinematic method which is also known as a mechanism method. So, we will have we will imagine that the load W has reached the value Wp and it will cause the collapse. So, now the external work done will be equal to Wp into delta. This is the work done which is done by the external load and this work will get absorbed by the plastic hinge. This is shown in the given figure. Now, as per this diagram, tan theta will be equal to delta divided by l by 2 half. Now, as delta is very small, theta will also be very small, tan theta will be equal to theta, rearranging the terms, delta will be equal to l by 2 into theta. Now, work done by load will be equal to work absorbed by plastic hinge. So, work done by load will be equal to Wp into delta and work absorbed by the plastic hinge will be equal to MP into theta. Force into displacement, force into displacement is the work done. So, delta in terms of ln theta, MP in terms of theta only. So, we will get the value of Wp which is equal to 4 MP buyer which is same as that of statical method which we have calculated by using the statical method. So, we will have some review questions. Whatever we have discussed till now, we will solve few questions. The first question is, a ductile structure is defined as one for which the plastic deformation before fracture is. The ductile structure is defined as one for which the plastic deformation before fracture is smaller than elastic deformation. Venetius is equal to the elastic deformation is much larger than the elastic deformation. Now, you can pause the video and answer this question. The answer to this question is, and the plastic deformation before fracture is much larger than the elastic deformation for a ductile structure. Another question is, plastic analysis of structure is applicable to the structures made of which of the following material, which of the following. The plastic analysis is applicable to which of the following structures. The first one is ductile and brittle material. Any structural member, brittle material only. D is ductile material only. Now, answer this question. So, the answer to this question is, the plastic analysis of structure is applicable to the structure made up of ductile materials only. So, these are my references. Thank you.