>>11381313>is there a simple formula for such a relation?Amazingly, there is.
In general, the relationship between the deflection in an elastic, prismatic beam with length under a distributed load is governed by the following fourth order ODE.
The term is something called flexural rigidity and is the product of the modulus of elasticity (material property) and the second moment of area of the beam's cross section (geometric property), but nevermind that. Since your load is "evenly spread," it is constant. So . Solving the differential equation above is now easy. However, as you know, you get constant of integration whenever a separable diff'eq is solved. So we need four boundary conditions to eliminate those. Because you beam is simply supported on both ends, boundary conditions are and (this is because there is no bending moment at a simple support). Getting in bed with all that algebra gives us
The maximum deflection obvious occurs at the midpoint. So
and presto you have an explicit formula for the maximum deflection. Notice that deflection is proportional to load and to the fourth power of its length.
Therefore, if you cut the length in half you can increase load by sixteen times for the same deflection. (Hint: all of these formulas can be found in the appendix of Mechanic of Materials by Beers.)
