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A symmetrical three$-$hinged arch shown in Figure $5\mathrm{.}16$ is in the form of $\frac{y}{?}=\frac{4hx(L-x)}{L^2},$ where $L=40$ m and $h=10m.$ Support A is $3\mathrm{.}6$ m higher than support E $.$ It is designed to carry a uniformly distributed load of $2k\frac{N}{m}$ spanning $26$ m on span ABCD and a point load of $8$ kN and $5$ kN at point B and D$,$ respectively. Joint A$,$ C$,$ and E are hinged. Determine

a$.$ support reactions at A and E

b$.$ bending moment at B and D

c$.$ shear force, Q and thrust, N on the right hand side of point C and D $($with loading$)$

Figure $5\mathrm{.}16$

Answers:

a$.A_y=37\mathrm{.}22kN$;$A_x=40\mathrm{.}56kN$;$E_y=27\mathrm{.}78kN$;$E_x=40\mathrm{.}56kN$

b$.M_B=29\mathrm{.}14kNm($sagging$)$; $M_D=-26\mathrm{.}4kNm($hogging$)$

c$.$ At C: ${}_C=0$;$N_C=40\mathrm{.}56kN$;$Q_C=-2\mathrm{.}78kN$ At D: ${}_D=26\mathrm{.}57$;$N_Q=46\mathrm{.}47kN$;$Q_D=-2\mathrm{.}23kN$