Free Body Diagrams

2

Visual representation of force and object interactions

Individual objects or members are isolated from their environment or system, illustrating all external forces acting upon it

• Assumption of Rigid Body• Newton’s Law• Conditions of Equilibrium

3

• The necessary and sufficient condition for the static equilibrium of a body are that the resultant force and couple from all external forces form a system equivalent to zero,

00 FrMF O

000

000

zyx

zyx

MMM

FFF

• Resolving each force and moment into its rectangular components leads to 6 scalar equations which also express the conditions for static equilibrium,

• For a rigid body in static equilibrium, the external forces and moments are balanced and will impart no translational or rotational motion to the body

• For every action there is an equal and opposite reaction

• If the resultant force on a particle is zero, the particle will remain at rest or will continue at constant speed in a straight line

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• Select the extent of the free-body and detach it from the ground and all other bodies.

• Indicate point of application and assumed direction of unknown applied forces.

• Unknown magnitude but known direction of force: Show vector arrows with magnitude assumed as positive

• Unknown magnitude and direction of force: Show x- and y- components of the vector with unknown magnitudes.

• Indicate point of application, magnitude, and direction of external forces, including the rigid body weight.

5

A stack of three books, each weighing 5 lb, is sitting on top of a table

Examples

Applied Force

Norm Reaction – Reaction force pushing up on the book

PLTW – DE book

W=5 lbf

N=5 lbf

y

x

6

W

Fapp

Ff

FNy

x

7

W W

Fapp Fapp

F f

F f

θ

θ

y

x

8

A fixed crane has a mass of 1000 kg and is used to lift a 2400 kg crate. It is held in place by a pin at A and a rocker at B. The center of gravity of the crane is located at G.

Determine the components of the reactions at A and B.

Free Body Diagram Reactions

• Different types of support reactions:• Cable, rope, or chain• Pin• Roller• Built-in end – Cantilever

Cable Support

Cable, rope, chain – Replace with a tension force only.

y

x

Cable Support

A sign with weight W is hung by two cables as shown

Cable Support

FBD of sign and cables

y

x

Pin Support

Pin – Replaced with TWO reaction forces, one vertical (y) and one horizontal (x).

A

Reaction

Force Joint / Pin A

Y Direction

Reaction Force

Joint / Pin A

X Direction

RFAY

RFAXA

y

x

Roller Support

Roller – Replaced with ONE reaction force, perpendicular to surface

A A

RFAY

Y Direction

Reaction Force

Joint / Roller A

y

x

Common Support Reactions

Beams and truss bridges are usually supported with one pin support and one roller support. This is called a simply supported object.

A BRFAX

RFAY RFBY

y

x

Built-in End Support

Built-in-end (cantilever) – Replaced with TWO forces: one horizontal and one vertical, and ONE moment

A RFAX

RFAY

MAccw

y

x

17

18

P

mg

Ay

Ax 450

B

y

x

19

y

x

• Determine B by solving the equation for the sum of the moments of all forces about A.

0m6kN5.23

m2kN81.9m5.1:0

BM A

kN1.107B

• Determine the reactions at A by solving the equations for the sum of all horizontal forces and all vertical forces.

0:0 BAF xx

kN1.107xA

0kN5.23kN81.9:0 yy AF

kN 3.33yA

20

y

x

21

Vectors

Parallelogram law

Triangle law

Trapezoid law

Polygon law

QPR

BPQQPR

cos2222

• Law of cosines,

• Law of sines,

Q

C

R

B

P

A sinsinsin

BC

B

C

22

A barge is pulled by two tugboats. If the resultant of the forces exerted by the tugboats is 5000 lbf directed along the axis of the barge, determine the tension in each of the ropes for a = 45o

y

x

• Graphical solution - Parallelogram Rule with known resultant direction and magnitude, known directions for sides.

lbf2600lbf3700 21 TT

• Trigonometric solution - Triangle Rule with Law of Sines

105sin

lbf5000

30sin45sin21 TT

lbf2590lbf3660 21 TT

23

636.1414.1

313.2tan

m 2.313m 515.0828.2

m 515.020tanm 414.1)2045cot(

m 414.1

m828.245cosm445cos

21

AE

CE

BDBFCE

CDBD

AFAECD

ABAF

6.58

38.6sin

N 1.98

110sin4.31sin

RT

N 8.147

N9.81

R

T

y

x

Thank You