[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx1
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Engineering 36
Chp09: Distributed
Loads
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx2
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads
The Load on an Object may be Spread out, or Distributed over the surface.
Load Profile, w(x)
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx3
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads
If the Load Profile, w(x), is known then the distributed load can be replaced with at POINT Load at a SPECIFIC Location
Magnitude of thePoint Load, W, is Determined by Area Under the Profile Curve
span
dxxwW
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx4
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads
To Determine the Point Load Location employ Moments (1st Moment of Force)
Recall: Moment = [LeverArm]•[Intensity] In This Case
• LeverArm = The distance from the Baseline Origin, xn
• Intensity = The Increment of Load, dWn, which is that load, w(xn) covering a distance dx located at xn
– That is: dWn = w(xn)•dx
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx5
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads
Now Use Centroidal Methodology
span
nn
span
x dxxwxIntensityLeverArm
And Recall:
Location Centroid theis xWxx Equating the
Ω Expressionsfind
W
dxxwx
x span
nn
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx6
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Distributed Loads on Beams
• A distributed load is represented by plotting the load per unit length, w (N/m). The total load is equal to the area under the load curve.
AdAdxwWL
0
AxdAxAOP
dWxWOP
L
0
• A distributed load can be REPLACED by a concentrated load with a magnitude equal to the area under the load curve and a line of action passing through the areal centroid.
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx7
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx8
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example:Trapezoidal Load Profile
A beam supports a distributed load as shown. Determine the equivalent concentrated load and the reactions at the supports.
Solution Plan• The magnitude of the
concentrated load is equal to the total load (the area under the curve)
• The line of action of the concentrated load passes through the centroid of the area under the Load curve.
• Determine the support reactions by summing moments about the beam ends
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx9
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example:Trapezoidal Load ProfileSOLUTION:
• The magnitude of the concentrated load is equal to the total load, or the area under the curve.
kN 0.18F
• The line of action of the concentrated load passes through the area centroid of the curve.
kN 18
mkN 63 X m5.3X
m6m
N
2
45001500
F
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx10
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Example:Trapezoidal Load Profile
0m .53kN 18m 6:0 yA BM
kN 5.10yB
0m .53m 6kN 18m 6:0 yB AM
kN 5.7yA
Determine the support reactions by summing moments about the beam ends After Replacing the Dist-Load with the Equivalent POINT-Load
ByAy
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx11
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
3D Distributed Loads
The Previous 2D Dist Load Profile had units of Force per Unit-Length (e.g., lb/in or N/m)
If 3D The Force acts over an AREA and the units become Force per Unit Area, or PRESSURE (e.g., psi or Pa)
Knowledge of the Pressure Profile allows calculation of an Equivalent Point Load and its Location
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx12
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading
Consider an Area Subject to a Pressure Load
Uniform Pressure Profile
The incremental Force, dFmn, Results from pressure p(xm,yn) acting on the incremental area dAmn= (dxm) (dyn)
Then the Total Force, F, on the Area
areaarea
p dAyxpdFF ,
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx13
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading: Total Force
The Differential Geometry is shown below
Then the Total Pressure Force
dA
dF
y all x,all
,
,
dxdyyxp
dAyxpdFF
nm
area
mnnm
area
mnp
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx14
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading – Pressure Ctr
Use MOMENT Methodology in 2-Dimensions to find the Location for the Point Force Fp
Then the Moment about the y-axis due to intensity dFmn and LeverArm xm
Then the Total y-axis Moment
dxdyyxpxd nmmx ,
pdxdy
dF
mx ny
surface
nmm
surface
xx
dxdyyxpx
d
,
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx15
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading – Pressure Ctr
Recall also Ωx = XC•Fp
Equating the two Ω expressions
The Similar Expression for YC
pdxdy
dF
mx my
surface
nmmpC dxdyyxpx FX ,
Isolating XC
p
surface
nmm
C F
dxdyyxpx
X
,
p
surface
nmn
C F
dxdyyxpy
Y
,
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx16
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading Summarized
Given a surface with Pressure Profile The Equivalent Force, Fp, Exerted on
the Surface due to the Pressure
yall x, all
dydxyxpF nmp ,
Fp is located at the Center of Pressure at CoOrds (XC,YC)
p
surface
nmm
C F
dxdyyxpx
X
,
p
surface
nmn
C F
dxdyyxpy
Y
,
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx17
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
WhiteBoard Work
Lets WorkThese NiceProblems
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx18
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
Engineering 36
Appendix 00
sinhT
µs
T
µx
dx
dy
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx19
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Beam Problem
For the Negligible-Wt Beam Find• Equivalent POINT-Load and it’s Location
(Point of Application, PoA)• The RCNs at Pt-A
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx20
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Problem
Find the Equivalent POINT-LOAD and its Point of Application (Location) For the Given Pressure Distribution
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx21
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx22
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx23
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx24
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx25
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics
Pressure Loading
The Differential Geometry is shown belwo
Then the Total Pressure Force
dA
dF
yall x, all
dydxyxp
dAyxpdFF
nm
area
mnnm
area
mnp
,
,
[email protected] • ENGR-36_Lec-24_Dist_Loads.pptx26
Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics