Find the analytical expression for of the spring constant as

Find the analytical expression for of the spring constant associated with case (a), (b), and (c).

Solution

y=FL33EIy=FL³/3EI

where

I=h³w/12

Together they comprise

F=Eh³w/AL³y

As you can see you have zero curvature (max slope) at the load in the first case, and max curvature (zero slope) at the load in the second case.

So in the first case if the deflection is =F33EI=F33EI for length , then the deflection on the second case (with only one row of beams) is

=F3192EI=F3192EI

because it is twice the deformation of a quarter length shared between two sides. Split the load in two (one half towards the left side and one half on the right side) and apply =2((F/2)(/4)33EI)=2((F/2)(/4)33EI).

With two rows, the deformation is half of this because the load is shared between the two rows.

=F3384EI=F3384EI

The proper way to handle these problems is to come up with the internal moment as a function of position

M(x)=MAF2(2x)M(x)=MAF2(2x)

where MAMA is the support moment. The find the total strain energy for one row of the structure considering bending only.

U=202M22EIdx=(F22+12FMa+48M2A)96EIU=202M22EIdx=(F22+12FMa+48MA2)96EI

Now impose the no rotation constraint on the support by solving

UMA=0}MA=F8UMA=0}MA=F8

and finding the displacement at the load from the principle of virtual work

=UF=F3192EI