QUESTION $2(130$ marks approx $30$ min $)$

Fig. $2$ shows a part of a corbel that transfers the load to a bridge pier $($not shown in figure$).$

The width of the corbel $($normal to the plane of paper$)$ is $W=400mm.$ The factored concentrated loads acting on the corbel are $P^{*}=2000kN.$ Drawing is not to scale. The bearing plates at the loading points are of size $300mmWmm.$ Based on an optimum load path analysis, an equivalent strut$-$and$-$tie model ABC $($dotted$)$ is also shown in Fig. $2,$ with

$L1=3\mathrm{.}5m$ and $L2=3m.$ Assume that the two $P*$ loads are acting at nodes A and $B.$ The corbel is to be designed based on the strut$-$and$-$tie model. Assume, the depth of tie $D_t=350mm.$

The characteristic compressive strength of concrete $(f_c^{\text{'}})=32$MPa and the yield stress of steel $(f_{sy})=500$MPa.

$($a$)$ Calculate the forces in members $AB$ and $AC$ of the strut and tie model. Clearly indicate which is a strut and which is a tie. $(20$ marks$)$

$($b$)$ Calculate the tensile reinforcement required for the tie connected at node $A.$ Assume that the depth of tension tie is $D_t.$ How do you anchor this reinforcement? $(20$ marks$)$

$($c$)$ Check whether the strut at node $A$ is safe in carrying the load applied. Assume the strut to be prismatic and sufficient web reinforcement will be provided for crack control. $(30$ marks$)$

$($d$)$ Draw the node A showing all the forces acting on it and check whether the node is safe from the strength point of view?

Is the node $A$ a hydrostatic node $($normal stresses on all the nodal faces are equal$)?$ Explain with reasoning.