Engineering 36
Chp 2: Force
Resultants (1)
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Resultant of Two Forces
 Force: Action Of One Body On
Another; Characterized By Its
• Point Of Application
• Magnitude (Intensity)
• Direction
 Experimental Evidence Shows That The
Combined Effect Of Two Forces May Be
Represented By A Single Resultant Force
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Concurrent Force Resultant
R  PQS
 CONCURRENT FORCES
 Set Of Forces Which
All Pass Through
The Same Point
• A Set Of Concurrent Forces Applied To A body
May Be Replaced By A Single Resultant Force
Which Is The Vector Sum Of The Applied Forces
 VECTOR FORCE COMPONENTS
 Two Or More Force Vectors
Which, Together, Have
PQ  F
The Same Effect As An
Original, Single Force Vector
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Resultant cont.
 The Resultant Is Equivalent To The Diagonal
Of A Parallelogram Which Contains The Two
Forces In Adjacent Legs
• As Forces are VECTOR Quantities and they
Add as Such
3D Vector
Engineering-36: Engineering Mechanics - Statics
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2D Vector
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Find the Force Resultant
 Find the Resultant of multiple Forces
by Vector Addition
 The two basic forms of Vector addition
• Decomposition
– Decompose all vectors into Axes Components
– Combine Like Components to Obtain Resultant
• Graphical → Mag & Dir to SCALE
– Tip-to-Tail (a.k.a. Head-to-Tail)
– Parallelogram
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
 Consider a Vector
Set in the XY Plane:
 All vector lengths are
SCALED relative to
their magnitudes
1. Place V1 at ANY
convenient location
2. Place V2 at the
TIP of V1
 Add in Tip-to-Tail
Fashion
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3. Place V3 at the
TIP of V2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
4. Connect the TAIL
of V1 to the TIP of
V3 to reveal the
RESULTANT, VR
3
4
2
1
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
 Consider a Vector
Set in the XY Plane:
 Add by
Parallelogram Rule
Engineering-36: Engineering Mechanics - Statics
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 Addition Of Three Or
More Vectors
proceeds Through
Repeated
Application Of The
Parallelogram Rule
 Note that
Parallelogram vector
addition proceeds in
TAIL-to-TAIL fashion
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
1. Layout scaled
vectors V1 & V2 in
Tail-to-Tail Fashion
2. Draw a
“Construction Line”
(a.k.a. “XL”) from
Tip of V1 that is
Parallel (a.k.a. ||)
to V2
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3. Draw an XL from
the tip of V2 that
is || to V1.
•
The Two XL’s will
intersect if V1 & V2
are NOT Parallel
4. Connect the tail-pt of
V1 & V2 to the XL
intersection to reveal
the intermediate,
Vector, Vinter
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
 Construction of Vinter
4
3
1
2
1
5. Start a NEW dwg
and LayOut Vinter &
V3 Tail-to-Tail
Engineering-36: Engineering Mechanics - Statics
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6. Draw an XL from the
tip of Vinter that is ||
to V3
7. Draw an XL from the
tip of V3 that is || to
Vinter
8. Connect the tail-pt
of Vinter & V3 to the
XL intersection to
reveal the Resultant
Vector, VR
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Vector Addition
7
5
6
8
5
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Parallelogram Vector Addition
 Any Number of
Vectors may be
added by the
parallelogram rule
by the repeated
Construction of
Intermediate Vectors
 Generally parallelogram addition is
more cumbersome than Tip-to-Tail
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Example: Graphical Force Add
 Consider Spring & Cable Supported Wt
 The Weight is
not moving;
i.e., it’s in
Static
Equilibrium
Spring Supports
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Spring Example Notes
 Springs
• Free-Length for
BOTH = 450 mm
• kAB = 1.5 N/mm
• kAD = 0.5 N/mm
 Find
• Tension in Cord, TAC
• Weight of Block, W
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Example: Solution Plan
 Use Spring Constants and Extension to
Find TAB & TAD
 Draw Vector Force PolyGon Noting that the
PolyGon Must Close for a system in
Equilibrium
 Draw Force/Vector PolyGon in Tip-to-Tail
form to reveal TAC & W
• Note that the Directions are known for W & TAC;
i.e., the Force LoA is CoIncident with Geometry
 Solve by Hand & AutoCAD Scaling
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Spring Digression: Hooke’s Law
 Robert Hooke (1635-1703)
formulated the relationship
between the force applied
to, and extension of, a
Linear Elastic structural
member. For a Spring:
FS  k  L
• Where
– Fs ≡ Spring Force (N or lb)
– k ≡ Spring Constant (N/m or lb/in)
– ΔL ≡ Spring Extension from Free-Length (m or in)
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Example: FBD on Ring/Eye
• Angles by ATAN
16.26  arctan 140 / 320  160
36.87  arctan 330 /580  140
28.07  arctan 32 /14  46
• LAB & LAD by
Pythagoras
LAB 
LAD 
330
320

 600 mm
2
 440 2 mm2
2
2
Engineering-36: Engineering Mechanics - Statics
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2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Calc Spring-Cable Tensions
 Recall Stretched
Lengths
• LAB = 550 mm
• LAD = 680 mm
 Use Hook’s law to
Calc the Tension
Fspring  k L 
 Free Length for Both  Thus
Springs is 450mm
 And the Spring
Constants
• kAB = 1.5 N/mm
• kAD = 0.5 N/mm
Engineering-36: Engineering Mechanics - Statics
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TAB  1.5 N mm550mm  450mm
TAB  150 N
TAD  0.5 N mm680mm  450mm
TAD  115 N
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Solution Hand
1. Select Scaling
Factor of 4in/150N
• XL for TAC LoA
from Tip of TAD
2. Draw Known Force • XL for W LoA
from Tail of TAB
TAB with Direction &
Scaled-Mag
3. From Tip of TAB Draw
Known
Force TAD
4. Make XL’s for Known
LoA’s For
W and TAC
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Solution → Hand Scaled
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Graphical Solution → AutoCAD
 Let’s use AutoCAD (c.f. EGNR22) to
GREATLY improve the accuracy of our
graphical Solution
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
AutoCAD Graphical Solution
 Scaling Up using
4” = 150N
TAC
TAC
W
TAC
150N
 1.75in 
4in
 65.6N
150N
W  5.52in 
4in
TAC  207.0 N
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Compare: Hand vs. ACAD
 Check the Pencil &
Paper Solution to
mathematically
precise ACAD soln
• TAC: 61/65.6 →
Hand Soln
9.3% Low
• W: 197/207 → 4.83% low
 Not Bad for Engr Comp-Pad, Ruler,
and Protractor
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
WhiteBoard Work
Let’s Work
This Nice
Problem

 Do:
• Graphically
• Analytically
Engineering-36: Engineering Mechanics - Statics
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Determine the design angle φ
(0° ≤ φ ≤ 90°) between struts AB
and AC so that the 400-lb
horizontal force has a component
of 600 lb which acts up to the left,
in the same direction BA. Also find
the magnitude of the force directed
along line AC. Take θ = 30°.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Strut-City
 Note the structure is
composed to
SLENDER RODS in
in tension, with
connecting pts at
the ends of the rods
 In this case member
forces are
CoIncident with
Geometry
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
WhiteBoard Work
Let’s Work
This Nice
Problem
 Do:
• Graphically
• Analytically
Engineering-36: Engineering Mechanics - Statics
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 Resolve F1 into
components along the u &
v axes and determine the
magnitudes of these
components.
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering 36
Appendix
Bruce Mayer, PE
Registered Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
WhiteBoard Work
Let’s Work
The Spring
Problem by
DeComp
Engineering-36: Engineering Mechanics - Statics
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y
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt
Engineering-36: Engineering Mechanics - Statics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • ENGR-36_Lec-04_Force_Resultants-1.ppt