Consider the beam represented in Figure $1,$ whose bending stiffness is constant and equal to $EI.$ This beam is subjected to a uniformly distributed moving load of $10k\frac{N}{m}$ along $1\mathrm{.}5$ m $.$

$11$

Figure $1$

$($a$)$ Represent the following influence lines and, for each case, determine the maximum and minimum values of those quantities, identifying the exact location of the corresponding uniformly distributed moving load:

i$.$ Bending moment in section $A,M_A$;

ii$.$ Shear force in the section immediately before section $B,V_{B^{-}}$;

iii. Shear force in the section immediately after section $B,V_{B^{+}}$;

iv$.$ Bending moment in section $S,M_S$;

v$.$ Vertical reaction force in support $B,R_B^v$;

vi$.$ Vertical reaction force in support $D,R_D^v.$

$($b$)$ What is the relation between the influence lines $V_{B^{-}},V_{B^{+}}$and $R_B^v?$

$($c$)$ Represent the influence line of the vertical displacement in section $S,{}_S^v,$ identifying the degree of the influence function in each interval and determining the maximum and minimum values. Determine the function that expresses the influence line of ${}_S^v(x)$ for $0.$