1. Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca ENGI 1313 Mechanics I Lecture 22:Equivalent Force Systems and Distributed Loading

2. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 2 Lecture 22 Objective to demonstrate by example equivalent force systems to determine an equivalent force for a distributed load

3. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 3 Example 22-01 Handle forces F 1 and F 2 are applied to the drill. Replace this system by an equivalent resultant force and couple moment acting at point O. Express the results in Cartesian vector form.

4. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 4 Equivalent Distributed Loads Reduce to Single Equivalent Load  Magnitude and position Applications  Environment Wind Fluid  Dead load Weight Snow, sand Objects

5. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 5 Distributed Loads Load Intensity  Pressure (Pa) Force per unit area N/m 2 or lb/ft 2  Infinite # parallel vertical loads Reduce from Area to Line Load

6. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 6 Distributed Loads (cont.) Consider Element Length (dx)  Element force (dF) acts on element length (dx)  The line load w(x) represents force per unit length

7. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 7 Distributed Loads (cont.) Magnitude Net Force (F R )  Summation of all element forces  Area under line load curve

8. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 8 Distributed Loads (cont.) Resultant Moment (M Ro )  Element force moment  Total moment  Equivalent force

9. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 9 Distributed Loads (cont.) Centroid  Geometric center of area  Resultant force line of action

10. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 10 Example 22-02 Determine equivalent resultant force and location for the following distributed load. L/2

11. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 11 Example 22-03 Determine equivalent resultant force and location for the following distributed load. L/3

12. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 12 Comprehension Quiz 22-01 What is the location of F R or distance d?  A) 2 m  B) 3 m  C) 4 m  D) 5 m  E) 6 m Answer: D FRFR 3 m d

13. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 13 Example 22-04 The beam is subjected to the distributed loading. Determine the length b of the uniform load and its position a on the beam such that the resultant force and couple moment acting on the beam are zero.

14. ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng. 14 References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1