ENGR-36_Lec-24_Dist_Loads.pptx 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Engineering 36 Chp09: Distributed Loads

ENGR-36_Lec-24_Dist_Loads.pptx 2 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)

ENGR-36_Lec-24_Dist_Loads.pptx 3 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 the Point Load, W, is Determined by Area Under the Profile Curve

ENGR-36_Lec-24_Dist_Loads.pptx 4 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Distributed Loads  To Determine the Point Load Location employ Moments (1 st Moment of Force)  Recall: Moment = [LeverArm][Intensity]  In This Case LeverArm = The distance from the Baseline Origin, x n Intensity = The Increment of Load, dW n, which is that load, w(x n ) covering a distance dx located at x n –That is: dW n = w(x n )dx

ENGR-36_Lec-24_Dist_Loads.pptx 5 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Distributed Loads  Now Use Centroidal Methodology  And Recall:  Equating the Ω Expressions find

ENGR-36_Lec-24_Dist_Loads.pptx 6 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. 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.

ENGR-36_Lec-24_Dist_Loads.pptx 7 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-24_Dist_Loads.pptx 8 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

ENGR-36_Lec-24_Dist_Loads.pptx 9 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example:Trapezoidal Load Profile SOLUTION: The magnitude of the concentrated load is equal to the total load, or the area under the curve. The line of action of the concentrated load passes through the area centroid of the curve.

ENGR-36_Lec-24_Dist_Loads.pptx 10 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Example:Trapezoidal Load Profile  Determine the support reactions by summing moments about the beam ends After Replacing the Dist-Load with the Equivalent POINT-Load ByBy AyAy

ENGR-36_Lec-24_Dist_Loads.pptx 11 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

ENGR-36_Lec-24_Dist_Loads.pptx 12 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Loading  Consider an Area Subject to a Pressure Load Uniform Pressure Profile  The incremental Force, dF mn, Results from pressure p(x m,y n ) acting on the incremental area dA mn = (dx m ) (dy n )  Then the Total Force, F, on the Area

ENGR-36_Lec-24_Dist_Loads.pptx 13 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Loading: Total Force  The Differential Geometry is shown below  Then the Total Pressure Force

ENGR-36_Lec-24_Dist_Loads.pptx 14 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 F p  Then the Moment about the y-axis due to intensity dF mn and LeverArm x m  Then the Total y-axis Moment

ENGR-36_Lec-24_Dist_Loads.pptx 15 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Loading – Pressure Ctr  Recall also Ω x = X C F p  Equating the two Ω expressions  The Similar Expression for Y C  Isolating X C

ENGR-36_Lec-24_Dist_Loads.pptx 16 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Loading Summarized  Given a surface with Pressure Profile  The Equivalent Force, F p, Exerted on the Surface due to the Pressure  F p is located at the Center of Pressure at CoOrds (X C,Y C )

ENGR-36_Lec-24_Dist_Loads.pptx 17 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics WhiteBoard Work Lets Work These Nice Problems

ENGR-36_Lec-24_Dist_Loads.pptx 18 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Registered Electrical & Mechanical Engineer Engineering 36 Appendix

ENGR-36_Lec-24_Dist_Loads.pptx 19 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

ENGR-36_Lec-24_Dist_Loads.pptx 20 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

ENGR-36_Lec-24_Dist_Loads.pptx 21 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-24_Dist_Loads.pptx 22 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-24_Dist_Loads.pptx 23 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-24_Dist_Loads.pptx 24 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

ENGR-36_Lec-24_Dist_Loads.pptx 25 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Pressure Loading  The Differential Geometry is shown belwo  Then the Total Pressure Force

ENGR-36_Lec-24_Dist_Loads.pptx 26 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics