Problem $5\mathrm{.}9.$ Simply supported beam with mid$-$span spring

Consider the uniform beam with simply supported ends as depicted in fig. $5\mathrm{.}36.$ A spring

of stiffness constant k is acting at mid$-$span, $()/($bar$)($k$)=$k$($L$^(3))/($H$\_(33)^($c$)).$ A uniform load p$\_(0)$ is acting

over the beam span of length L$.$ The unstressed length of the spring is such that it fall a

distance $\backslash$Delta short of reaching the beam, $()/($bar$)(\backslash$Delta $)=\backslash$Delta $($H$\_(33)^($c$))/($p$\_(0)$L$^(4)).(1)$ Find and plot the transverse

displacement distribution for this beam. $(2)$ Find and plot the corresponding distribution of

bending moment. $(3)$ What value of $()/($bar$)(\backslash$Delta $)$ that will minimize the maximum bending moment in

the beam. Hint: replace the spring by an unknown force, F$,$ acting at mid$-$span. This force can

then be evaluated by equating the beam mid$-$span displacement with that of the spring. Use

the following values for the plots: $()/($bar$)($k$)=600(,)/($b$)$ar $(\backslash$Delta $)=2\mathrm{.}010^(-3).$

Fig. $5\mathrm{.}36.$ Simply supported beam with mid$-$

span spring featuring a clearance $\backslash$Delta $.$

USE Wolfram MATHEMATICA