What kind of supports should be given to a chair while designing the chair? Supposing a simplistic chair (stool) with four legs, What kind of supports should we give to the end of the legs in the chair? If this question was asked in mechanics, the answer would be simply supported. But in design, we should fix three legs in y direction and one leg in all directions. That is because the fixed leg acts as reference for the movement of the other three legs. In other words, we are restricting the velocity of the chair in all three directions, ie giving boundary conditions in all three directions. Then we get the solution as below. Now let us consider the following cases: The solution for case 3 is given below
Note that the left leg(as shown in figure 1) is the fourth leg. The bilinear equation with the essential boundary condition of u(0) = U0 and natural boundary condition of at x = L , and the domain 0 < x <L appears in several forms in engineering.
CABLE(ROPE) UNDER A TRANSVERSE LOAD u = deflection a= tension q(x)= transverse load (eg: weight of cable) Q = Axial force BAR IN TENSION u = deflection a= EA q(x)= Friction(Traction on the bar) Q = Axial force HEAT TRANSFER u = Temperature a= Thermal conductivity(k) q(x)= Heat generated Q = Heat flux ONE DIMENSIONAL LAMINAR INCOMPRESSIBLE FLOW (Gradient(p)=const) u = velocity a= viscosity (η) q(x)= Pressure gradient Q = Axial stress FLOW IN POROUS MEDIA IN ONE DIMENSION u = Fluid head a= Permeability const q(x)= flux Q = Flow ELECTROSTATICS u = Electrostatic potential a= dielectric constant q(x)= charge density Q = Electric Flux 
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