1

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five

rigid body

equilibriumRigid Body Equilibrium 1

Lecture 5

Architectural Structures

ARCH 331

lecture

ARCHITECTURAL STRUCTURES:

FORM, BEHAVIOR, AND DESIGN

ARCH 331

DR. ANNE NICHOLS

FALL 2013

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0 xx FR

0 yy FR

0MM

Equilibrium

• rigid body

– doesn’t deform

– coplanar force systems

• static:

Rigid Body Equilibrium 2

Lecture 5

Foundations Structures

ARCH 331

A

CB

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Free Body Diagram

• FBD (sketch)

• tool to see all forces on a body or a

point including

– external forces

– weights

– force reactions

– external moments

– moment reactions

– internal forces

Rigid Body Equilibrium 3

Lecture 5

Foundations Structures

ARCH 331

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Free Body Diagram

• determine body

• FREE it from:

– ground

– supports & connections

• draw all external forcesacting ON the body

– reactions

– applied forces

– gravity

Rigid Body Equilibrium 4

Lecture 5

Foundations Structures

ARCH 331

mg

+ weight

100 lb

100 lb

2

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Free Body Diagram• sketch FBD with relevant geometry

• resolve each force into components– known & unknown angles – name them

– known & unknown forces – name them

– known & unknown moments – name them

• are any forces related to other forces?

• for the unknowns

• write only as many equilibrium equations as needed

• solve up to 3 equations

Rigid Body Equilibrium 5

Lecture 5

Foundations Structures

ARCH 331

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Free Body Diagram

• solve equations

– most times 1 unknown easily solved

– plug into other equation(s)

• common to have unknowns of

– force magnitudes

– force angles

– moment magnitudes

Rigid Body Equilibrium 6

Lecture 5

Foundations Structures

ARCH 331

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Reactions on Rigid Bodies

• result of applying force

• unknown size

• connection or support type

– known direction

– related to motion prevented

Rigid Body Equilibrium 7

Lecture 5

Foundations Structures

ARCH 331

no vertical motion no translation

no translation

no rotation

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Supports and Connections

Rigid Body Equilibrium 8

Lecture 5

Foundations Structures

ARCH 331

3

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Supports and Connections

Rigid Body Equilibrium 9

Lecture 5

Foundations Structures

ARCH 331

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FBD Example

• 500 lb known

• pin – Ax, Ay

• smooth surface –

B at 4:3

• 3 equations

• sum moments at

– A?

– B?

Rigid Body Equilibrium 10

Lecture 5

Foundations Structures

ARCH 331

(Bx)

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0Fx 0Fy 0M

1

0F 0M1 0M

2

0M1 0M2 0M

3

Moment Equations

• sum moments at intersection where the

most forces intersect

• multiple moment equations may not be

useful

• combos:

Rigid Body Equilibrium 11

Lecture 5

Foundations Structures

ARCH 331

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Recognizing Reactions

Rigid Body Equilibrium 12

Lecture 5

Foundations Structures

ARCH 331

F F

unknowns

3

weight mgunknowns

3

4

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Recognizing Reactions

Rigid Body Equilibrium 13

Lecture 5

Foundations Structures

ARCH 331

unknowns

3

unknowns for

2 bodies

6

unknowns

2

mg

weight

F1 F2

weight

F1F2

mg

not independent

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Constraints

• completely constrained

– doesn’t move

– may not be statically determinate

• improperly or partially constrained

– has unknowns

– can’t solve

Rigid Body Equilibrium 14

Lecture 5

Foundations Structures

ARCH 331

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Constraints

• overconstrained

– won’t move

– can’t be solved with statics

– statically indeterminate to nth degree

Rigid Body Equilibrium 15

Lecture 5

Foundations Structures

ARCH 331

A C

B

200 lb-ft

60 lb

55 A

5’ 9’

CB

200 lb-ft

55

60 lb

Ax

Ay Cy

MRA

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Partial Constraints

Rigid Body Equilibrium 16

Lecture 5

Foundations Structures

ARCH 331

100 N 1 m

0.75 m

30 A

B

100 N 1 m

0.75 m

30 A

B

A B

W

500 mm

200 mm

B

W

5

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Cable Reactions

• equilibrium:

– more reactions (4) than equations

– but, we have slope relationships

– x component the same everywhere

Rigid Body Equilibrium 17

Lecture 5

Foundations Structures

ARCH 331

A

C

45 kN

4 m

2 m

B 6 m

45 kN

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Two Force Rigid Bodies

• equilibrium:

– forces in line, equal and opposite

Rigid Body Equilibrium 18

Lecture 5

Foundations Structures

ARCH 331

A B C

A

F2

BF1

dAF2

BF1

dA

F2

BF1

a

(no) (no)

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Three Force Rigid Bodies

• equilibrium:

– concurrent or parallel forces

Rigid Body Equilibrium 19

Lecture 5

Foundations Structures

ARCH 331

A B C

F1

F2

A

B

C

F3

F2

A

F1

B

C

F3

d1

d2

F2

A

F1

B

C

F3

a

(no)

beams!

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Concentrated Loads

Rigid Body Equilibrium 20

Lecture 5

Foundations Structures

ARCH 331

6

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Distributed Loads

Rigid Body Equilibrium 21

Lecture 5

Foundations Structures

ARCH 331

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Beam Supports

• statically determinate

• statically indeterminate

Rigid Body Equilibrium 22

Lecture 5

Foundations Structures

ARCH 331

L L L

simply supported

(most common)

overhang cantilever

L

continuous

(most common case when L1=L2)

L

L L

Propped Restrained

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loadingloading

L

0AdAwdxW

Equivalent Force Systems

• replace forces by resultant

• place resultant where M = 0

• using calculus and area centroids

Rigid Body Equilibrium 23

Lecture 5

Foundations Structures

ARCH 331

dx

w(x)

x

L

dx

y

x

elx

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Load Areas

• area is width x “height” of load

• w is load per unit length

• W is total load

Rigid Body Equilibrium 24

Lecture 5

Foundations Structures

ARCH 331

x

x/2

W

x/2

x

2x/3

W/2

x/3

x

x/2

W

x/6 x/3

W/2

0

Wxw w w 22

Wxw

w

2w

7

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Method of Sections

• relies on internal forces being in

equilibrium on a section

• cut to expose 3 or less members

• coplanar forces M = 0 too

Rigid Body Equilibrium 25

Lecture 5

Foundations Structures

ARCH 331

A B

C

P

F

E

D

P

.

A

By

AC

AB

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Method of Sections

• joints on or off the section are good to

sum moments

• quick for few members

• not always obvious where to cut or sum

Rigid Body Equilibrium 26

Lecture 5

Foundations Structures

ARCH 331

A B

C

P

F

E

D

P

. A

By

AC

AB

B

.

2p