Shawn Kenny, Ph.D., P.Eng.Assistant ProfessorFaculty of Engineering and Applied ScienceMemorial University of Newfoundlandspkenny@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.

Lecture 22 Objective

to demonstrate by example equivalent force systemsto determine an equivalent force for a distributed load

3 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Example 22-01

Handle forces F1 and F2are 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.

a 0.15 m=

b 0.25 m=

c 0.3 m=

F1

6

3−

10−

⎛⎜⎜⎝

⎞⎟⎟⎠

N=

F2

0

2

4−

⎛⎜⎜⎝

⎞⎟⎟⎠

N=

4 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Equivalent Distributed Loads

Reduce to Single Equivalent LoadMagnitude and position

ApplicationsEnvironment• Wind• Fluid

Dead load• Weight• Snow, sand• Objects

5 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Distributed Loads

Load IntensityPressure (Pa)• Force per unit area• N/m2 or lb/ft2

Infinite # parallel vertical loads

Reduce from Area to Line Load

( ) ( ) ⎥⎦⎤

⎢⎣⎡=⎥⎦

⎤⎢⎣⎡=

mNxwm

mNaxpw 2

6 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

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

( )dxxwdF =

7 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Distributed Loads (cont.)

Magnitude Net Force (FR)Summation of all element forces

Area under line load curve

( ) dxxwdFFR ∑∑ ==

( ) AdAdxxwFAL

R === ∫∫

8 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Distributed Loads (cont.)

Resultant Moment (MRo)Element force moment

Total moment

Equivalent force

dFxMdFO =

( )∫∫∑ ===LL

ORo dxxwxdFxMMdF

( )∫==L

RRo dxxwxxFM

9 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Distributed Loads (cont.)

CentroidGeometric center of areaResultant force line of action

( )∫==L

RRo dxxwxxFM

( ) AdAdxxwFAL

R === ∫∫

( )

( ) ∫

∫

∫

∫==

A

A

L

L

dA

dAx

dxxw

dxxwxx

10 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Example 22-02

Determine equivalent resultant force and location for the following distributed load.

( )2Lxft5

lb4000

lbft2x400

lb4000

dxxwx

lb4000

dAx

dA

dAxx

10

0

210

0

10

0

A

A =⇒===== ∫∫∫

∫

( ) wLFlb4000ftlbx400dx400dAdxxwF R

10

0

10

0ALR =⇒===== ∫∫∫

L/2

11 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Example 22-03

Determine equivalent resultant force and location for the following distributed load.

( )3L,

3L2xm4

N1800

mN3x100

N1800

dxxwx

N1800

dAx

dA

dAxx

6

0

36

0

6

0

A

A =⇒===== ∫∫∫

∫

( )2

wLFN1800mNx50dxx100dAdxxwF R

6

0

26

0ALR =⇒===== ∫∫∫

L/3

12 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Comprehension Quiz 22-01

What is the location of FR or distance d?

A) 2 mB) 3 mC) 4 mD) 5 mE) 6 m

Answer: D

FR

3 m 3 m

d

13 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

Example 22-04

The beam is subjected to the distributed loading. Determine the length b of the uniform load and its position aon the beam such that the resultant force and couple moment acting on the beam are zero.

w1 40lbft

=

w2 60lbft

=

c 10ft=

d 6 ft=

14 ENGI 1313 Statics I – Lecture 22© 2007 S. Kenny, Ph.D., P.Eng.

References

Hibbeler (2007)http://wps.prenhall.com/esm_hibbeler_engmech_1

ENGI 1313 Mechanics I Faculty of Engineering and Applied Science

Shawn Kenny, Ph.D., P.Eng. Fall 2007

Page 1 of 2

EXAMPLE 22-01 (PROBLEM 4-131)

Handle forces F1 and F2 are applied to the electric drill. Replace this system by an equivalent resultant force and couple moment acting at point O. Express the results in Cartesian vector form.

F1

6

3−

10−

⎛⎜⎜⎝

⎞⎟⎟⎠

N=

F2

0

2

4−

⎛⎜⎜⎝

⎞⎟⎟⎠

N=

a 0.15 m=

b 0.25 m=

c 0.3 m=

ENGI 1313 Mechanics I Faculty of Engineering and Applied Science

Shawn Kenny, Ph.D., P.Eng. Fall 2007

Page 1 of 2

EXAMPLE 22-04 (PROBLEM 4-150)

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.

w1 40lbft

=

w2 60lbft

=

c 10ft=

d 6 ft=