[email protected] ENGR-36_Lec-04_Force_Resultants-1.ppt 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed

[email protected] • ENGR-36_Lec-04_Force_Resultants-1.ppt1

Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Engineering 36

Chp 2: ForceResultants

(1)



Resultant of Two Forces

Force: Action Of One Body OnAnother; 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



Concurrent Force Resultant

CONCURRENT FORCES Set Of Forces WhichAll 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

SQPR

FQP

VECTOR FORCE COMPONENTS Two Or More Force VectorsWhich, Together, HaveThe Same Effect As An Original, Single Force Vector



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

2D Vector3D Vector



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



Graphical Vector Addition

Consider a Vector Set in the XY Plane:

All vector lengths are SCALED relative to their magnitudes

Add in Tip-to-Tail Fashion

1. Place V1 at ANY convenient location

2. Place V2 at the TIP of V1

3. Place V3 at the TIP of V2




4. Connect the TAIL of V1 to the TIP of V3 to reveal the RESULTANT, VR

1 2

34




Consider a Vector Set in the XY Plane:

Addition Of Three Or More Vectors proceeds Through RepeatedApplication Of The Parallelogram Rule

Note that Parallelogram vector addition proceeds in TAIL-to-TAIL fashion Add by

Parallelogram Rule




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

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




Construction of Vinter

5. Start a NEW dwg and LayOut Vinter & V3 Tail-to-Tail

12

3

4

1

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




5

6

7

85



Parallelogram Vector Addition

Any Number of Vectors may be added by the parallelogram ruleby the repeatedConstruction of Intermediate Vectors

Generally parallelogram addition is more cumbersome than Tip-to-Tail



Example: Graphical Force Add

Consider Spring & Cable Supported Wt

Spring Supports

The Weight is not moving; i.e., it’s in Static Equilibrium



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



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



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:

LkFS • Where

– Fs ≡ Spring Force (N or lb)

– k ≡ Spring Constant (N/m or lb/in)– ΔL ≡ Spring Extension from Free-Length (m or in)



Example: FBD on Ring/Eye

222

222

600320

440330

mmL

mmL

AD

AB

• LAB & LAD by Pythagoras

• Angles by ATAN

4614/32arctan07.28

140580/330arctan87.36

160320/140arctan26.16



Calc Spring-Cable Tensions

Recall Stretched Lengths• LAB = 550 mm

• LAD = 680 mm

Free Length for Both Springs is 450mm And the Spring Constants

• kAB = 1.5 N/mm

• kAD = 0.5 N/mm

Use Hook’s law to Calc the Tension

N 150

450550N 5.1

AB

AB

T

mmmmmmT

LkFspring Thus

N 115

450680N 5.0

AD

AD

T

mmmmmmT



Graphical Solution Hand1. Select Scaling

Factor of 4in/150N

2. Draw Known Force 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

• XL for TAC LoA from Tip of TAD

• XL for W LoA from Tail of TAB



Graphical Solution → Hand Scaled



Graphical Solution → AutoCAD

Let’s use AutoCAD (c.f. EGNR22) to GREATLY improve the accuracy of our graphical Solution



AutoCAD Graphical Solution

W

TAB

T AD

TAC

Scaling Up using 4” = 150N

65.6N4in

150N1.75in

AC

AC

T

T

N00724in

150Nin525

.

.

ACT

W



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



WhiteBoard Work

Let’s WorkThis NiceProblem

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°.

Do:• Graphically• Analytically



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







WhiteBoard Work

Let’s WorkThis NiceProblem

Resolve F1 into components along the u & v axes and determine the magnitudes of these components.

Do:• Graphically• Analytically







Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engineering 36

Appendix



WhiteBoard Work

Let’s WorkThe SpringProblem by

DeCompx

y



Documents

[email protected] ENGR-36_Lec-04_Force_Resultants-1.ppt 1 Bruce Mayer, PE Engineering-36: Engineering Mechanics - Statics Bruce Mayer, PE Licensed