 In this problem, we have this cantilever beam AB, which has a distributed load with intensity W equal to 200 Nm. And we have this point load applied at A equal to 400 Nm. And we need to determine the span of the beam that requires strengthening given sigma max equal to 100 MPa. So basically what this problem is asking is we need to determine what is the profile, what is the distribution of the stress. So it could be something for example like this, right? So if this is the normal stress or bending stress, in this case we need to determine for which points of this tractor the bending stress is larger than 100 MPa. So for example in this case all this region will be above 100 MPa and then we would need to increase the strength of this section of the beam, right? So then of course in order to determine what is the distribution of stresses we need to use the flexural formula, this is equal to M times Y divided by the moment of inertia. So as usually let's calculate first what is the moment of inertia, what is this distance and what is the distribution of moments. So we can for example start calculating the moment of inertia. We need of course to estimate first what is the location of the neutral axis. This is the section of the structure and we define this to be the part number 2. This is part number 3 and this is part number 1. Then we know that the total area times the position of the centroid of the structure which it will be somewhere around there, so I call this distance H is equal to A1 Y1, this is the centroid of this area, 1 plus A2 Y2 plus A3 Y3, right? This is the reference, then we have that this equation H is equal to then A1 is equal to 80 times 8 and this centroid of the first area, this is equal to 80 over 2, 4 plus A2, this one this is 40 times 8 and the centroid of this part is here, so I'm considering this distance which is equal to 40 divided by 2 this is 20 plus 8 millimeters, 28 and A3 Y3 is exactly the same, so I multiply here times 2 divided by A1 plus A2 plus A3, this is equal to 16 millimeters. So we already know what is this distance, we know that the neutral axis is here and now we can calculate the moment of inertia of the whole section which of course is the moment of inertia of 1 plus the moment of inertia of 2 plus the moment of inertia of 3 Of course we need to use the parallel axis theorem, so for the first section we have that this is equal to the moment of inertia with respect to the centroid of 1 So in this case for a rectangular section we have that this is equal to BH cubed divided by 12 plus as I said the area of 1 times the distance from H1 to H then B is equal to 80 in this case times 8 cubed divided by 12 plus the area of this part is equal to 80 times 8 and the distance from Y1 to H is equal to 16 minus 4 and this is equal to we do the same for 2 The moment of inertia of the squared section is equal to 40 squared times the base 8 divided by 12 plus the area of 2 times the distance from the centroid of this section will be around here to the centroid of the section for the whole structure and of course the moment of inertia of 3 is equal to the moment of inertia of 2 Therefore if we sum these results we have that the moment of inertia of the whole section is equal to now we need to find the bending moment distribution this is our beam we have here the distributed load with intensity W equal to 200 N per meter and we have here an applied load B equal to 400 N so I define here my reference X and I need to calculate what is the distribution of moments so I actually don't need to calculate what is this reaction force RB or this reaction moment MB so I can just start analyzing the structure from this side of the structure so I don't care about these reaction forces in this case I will use VMW relationships so we have that dv dx in this case is equal to minus W so I defined positive upwards so dv dx is equal to minus W is equal to minus 200 so if I want to find what is the shear distribution I have to integrate this so it's the integral of minus 200 dx and this is equal to minus 200x plus a constant here so now in order to find this constant I apply the boundary condition because I know that at A at X equal to 0 this shear force must be exactly equal to this force B so at 0 V is equal to minus B so from here I find that A is exactly equal to minus B so this is minus 400 so therefore the shear force is equal to now I need to find what is the distribution of moments so we integrate here and now we need to find what is the value of this constant of integration B so again we know that at X equal to 0 we have a cantilever beam so the moment must be 0 there so from this condition we have that this is 0 this is 0 so B must be equal to 0 so finally the moment bending moment distribution M of X is equal to and from here we have that the larger dx the larger the bending moment so M max occurs at the maximum value of X so in this case the maximum value of X is equal to L which is equal to 2 meters so M max is equal to now we need to find if any point of this structure has a bending stress larger than 100 megapascals so to do this we are going to use the flexural formula and of course we know that the maximum stress in this case is equal to the maximum moment times Y max divided by the moment of inertia Y max so for this section we have that the neutral axis is located here at 16 millimeters and we have that this distance is 48 so of course Y max it's located at the bottom part so Y max is equal to 48 minus 16 and is equal to 32 millimeters then now we have all the information that we need and if we substitute the results that we have sigma maximum is equal to then from here we find that for X equal to L at the wall the maximum stress is larger than 100 megapascals so from the equation of moments we know that the distribution of stresses is something like this and we know of course that the distribution of bending moment is something which is proportional to this so if this is the limit of 100 megapascals I need to find what is the location of X for which the moment is creating a stress of 100 megapascals then I will know that I will need to reinforce all this part of the beam right so let's first find which this value of M which produces that sigma is equal to 100 megapascals so again we use this flexural formula and we have that 100 megapascals is equal to M0 times 0.032 divided by the moment of inertia then from here we find that M0 is equal to minus 853 millimeters and now I need to find what is the X at which this occurs so from the expression of the bending moment I have this is equal to so this should be equal to M0 then I can solve this equation so I have that this is a second order equation so X is equal to so from here we take the positive result then X is equal to 1.54 meters then in conclusion we need to reinforce our beam wherever the magnitude of the moment exceeds 853 meters and this is from X equal to 1.54 meters to B then this is X equal to 1.54 and we need to reinforce this part