Using concrete with $f_{ck}=27$MPa and reinforcement steel with $f_y=400$MPa, a doubly

reinforced concrete beam was constructed as shown in the figure below. After sufficient curing, a

static loading test was conducted using servo$-$hydraulic equipment at the Extreme Performance

Testing Center $($EPTC$).($Reinforcement cross$-$sectional areas: D$13=127m{m}^2,$ D$25=507m{m}^2,$

D$32\{$:$=794m{m}^2)$

$(1)$ Calculate the nominal flexural strength of the beam.

$(2)$ Calculate the shear strength of the beam.

$(3)$ Predict the failure mode based on the results of $(1)$ and $(2)$ and analyze the failure pattern

observed during the experiment.