9.5 Resultant of a General Distributed Loading

9.5 Resultant of a General Distributed Loading

Pressure Distribution over a Surface� Consider the flat plate subjected to the loading

function ρ = ρ(x, y) Pa� Determine the force dF acting on the differential

area dA m2 of the plate, located at the differential area dA m2 of the plate, located at the differential point (x, y)dF = [ρ(x, y) N/m2](d A m2)

= [ρ(x, y) d A]N� Entire loading represented as

infinite parallel forces acting on separate differential area dA

9.5 Resultant of a General Distributed Loading

9.5 Resultant of a General Distributed Loading

Pressure Distribution over a Surface

� This system will be simplified to a single resultant force FR acting through a unique point on the platea unique point on the plate

Pressure Distribution over a Surface

Magnitude of Resultant Force

� To determine magnitude of FR, sum the differential forces dF acting over the plate’s

9.5 Resultant of a General Distributed Loading

9.5 Resultant of a General Distributed Loading

differential forces dF acting over the plate’s entire surface area dA

� Magnitude of resultant

force = total volume under

the distributed loading

diagram

∫ ∫=∑=A VR dVdAyxFF ),(; ρ

9.5 Resultant of a General Distributed Loading

9.5 Resultant of a General Distributed Loading

Pressure Distribution over a Surface

Location of Resultant Force

∫∫

∫∫

== VA

dV

xdV

dAyx

dAyxxx

),(

),(

ρ

ρ

� Line of action of action of the resultant force passes through the geometric center or centroid of the volume under the distributed loading diagram

∫∫

∫∫

∫∫

==

V

V

A

A

VA

dV

ydV

dAyx

dAyxyy

dVdAyx

),(

),(

),(

ρ

ρ

ρ