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In Structural analysis of beams, it is necessary to determine the support reactions and internal forces from the effects of combining different types of loads such as point loads, distributed loads, and moment loads. Within an internal force, support reactions are the force of the moment exerted by the supports to stabilize the beam while internal forces are represented by shear force and bending moment diagrams which describe the distribution of the load over the beam. In a shearing force diagram, SFD is a graphical representation of the internal force caused by the bending moments as it varies from one point in the beam to another. A bending moment diagram, BMD on the other hand is a graphical representation of the moment that occurs because of the bending force.

For the said problem, it can be assumed that the beam has four sections with hinges located at the points,,, and. Different regions of the beam is examined with the influence of distributed and point loads. Applying the equilibrium conditions to the region we determine the support reactions for the structure of interest. Once these reactions are determined, it will now be possible to draw the shear force and bending moment diagrams as a function of location along the beam by observing how forces and moments change in the plane of each segment of the beam.

$1.$Total Heat Transfer Formula:

$[$Q$=$i$$Aq$($x$,$y$),$dA$]$

where $($Q$)$is the total heat transfer, $($q$($x$,$y$))$is the heat flux density, and $($dA$)$is the differential surface element.

$2.$Surface Element for a Rectangular Plate:

$[$dA$=$dx$,$dy$]$

For a rectangular plate, the bounds for $($x$)$and $($y$)$are determined by the plate's dimensions.

$3.$Given Heat Flux Function:

$[$q$($x$,$y$)=100+50$x$+20$y$]$

$4.$Limits of Integration:

$[0$x$2,0$y$1]$

$[$PLEASE EXPLAIN WITH FIGURE$]$