Given. Pin $A(0)$, hinge at $3\text{ m}$, rollers $B(5),C(9)$, hinge at $11\text{ m}$, roller $D(14)$. The centre span $3\!-\!11\text{ m}$ rests on $B,C$; the end pieces $A\!-\!$hinge and hinge$\!-\!D$ are suspended.
5(a) Gerber beam (○ = internal hinge).
(i) Reaction $R_B$. On the centre span (on $B,C$) the ordinate is $1$ at $B$, $0$ at $C$, and rises on the left overhang to $\tfrac{9-3}{4}=1.5$ at the hinge ($x=3$); the suspended end span carries it linearly to $0$ at $A$. Peak coefficient $\boxed{1.5}$ at the left hinge.
5(a)(i) Influence line for $R_B$.
(ii) Moment over $B$. $M_B$ is zero at $B$ itself, hogging as the load moves onto the left overhang, reaching $-2.0\text{ kN}\cdot\text{m/kN}$ at the hinge ($x=3$), then linear to zero at $A$; on the right it forms the usual span lobe. Peak magnitude $\boxed{2.0}$.
5(a)(ii) Influence line for $M_B$.
(iii) Shear just left of $C$. The step at $C$ reaches $-1.0$ immediately to the left of the support (unit load just left of $C$); peak magnitude $\boxed{1.0}$.
5(a)(iii) Influence line for shear left of $C$.
5(b) — moving vehicle
Given. Pin $A(4)$, roller $B(20)$, point $C$ at $x=8\text{ m}$ (span $A\!-\!B=16\text{ m}$, overhangs $4\text{ m}$). Vehicle: $24,24,8\text{ kN}$ at spacings $2\text{ m}$ then $4\text{ m}$, travelling left→right.
5(b) Simply supported beam with overhangs; point $C$ at $4\text{ m}$ from $A$.
Influence lines at $C$. $M_C$ is a triangle peaking at $\dfrac{ab}{L}=\dfrac{4\cdot12}{16}=3.0\text{ m}$ at $C$ (dipping negative over the left overhang). $V_C$ steps at $C$: $+0.75$ just right, $-0.25$ just left.
5(b)(i) Influence line for $M_C$.
5(b)(ii) Influence line for $V_C$.
Governing effects. Positioning the $24,24,8\text{ kN}$ axles on the ordinates gives the maximum bending moment at $C$, $M_C=\boxed{144\text{ kN}\cdot\text{m}}$, and a shear envelope at $C$ of $\boxed{+36\text{ kN}}$ to $-9\text{ kN}$ (governing $|V_C|=36\text{ kN}$).
