Q$4.$

The structure shown in Figure $Q4$ is to he desioned elastically

Figure Q$4$ A two$-$span continuous beams with loadings

$($a$)$ If $V_A=4\mathrm{.}4kN,V_B=57\mathrm{.}6kN,$ and $V_C=41\mathrm{.}4kN,$ and $V_D=1\mathrm{.}6kN,$ write down the $2$nd order differential equation which could be used to solve for the deflection of the beam.

$[8$ marks$]$

$($b$)$ Write down the boundary conditions which would be used to find the constants of integration.

$[4$ marks$]$

$($c$)$ Using assumptions of the position of point of contraflexure estimate the value of the bending moment at B $.$

$[4$ marks$]$

$($d$)$ Using assumptions of the position of point of contrafexture estimate the value of the maximum deflection on $AB$ if $E=210k\frac{N}{m}{m}^2$ and lyy$=4400c{m}^4.$

$[4$ marks$]$

$($e$)$ Calculate the plastic moment Mp$,$ which would be used for the design. Provide sufficient calculation to determine your answer, including a sketch of the failure mechanism. You can assume for span $AB$ that the plastic hinge forms at the mid$-$span.

$[6$ marks$]$

$($f$)$ Elastic analysis of the beam gives the bending moments at B and C are $34$ kNm and $22$ kNm respectively. If the whole beam was designed using a plastic moment value of $22$ kNm by mistake, discuss which parts of the beam might fail. Sketch the order of failure including the sequence of each hinge formation.

$[5$ marks$]$

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