구조역학 Structural Analysis - KINX

HYUNGCHUL YOON, PH.D.ASSOCIATE PROFESSOR

SCHOOL OF CIVIL ENGINEERING

1. Statics of Structure

[email protected]

구조역학Structural Analysis

mailto:[email protected]

Reactions (반력)

구조역학

Prof. Hyungchul Yoon ([email protected])

SIGN CONVENTION (방향과 부호)

• X-axis▪ (+) Positive: Right

▪ (−) Negative: Left

• Y-axis▪ (+) Positive: Up

▪ (-) Negative: Down

• Rotation▪ (+) Positive: Clockwise

▪ (−) Negative: Counter-Clockwise+

y

x

+

+

구조역학


ROLLER SUPPORT (롤러)

or

Sketch

• Prevents▪ Vertical Translation

• Allows▪ Horizontal Translation

▪ Rotation

• Reactions

▪ Ry

구조역학


PIN (HINGED) SUPPORT (힌지, 경첩)

Sketch

• Prevents▪ Horizontal Translation

▪ Vertical Translation

• Allows▪ Rotation

• Reactions▪ Rx

▪ Ry

구조역학


FIXED END SUPPORT (고정단)

Sketch


▪ Vertical Translation

▪ Rotation

• Allows▪ None

• Reactions▪ Rx

▪ Ry

▪ M

구조역학


GUIDE SUPPORT (가이드)

Sketch


▪ Rotation

• Allows▪ Vertical Translation

• Reactions▪ Rx

▪ M

구조역학


INTERNAL HINGE (내부힌지)

Rx

Ry Ry

Sketch

• Prevents▪ Relative Translation

• Allows▪ Absolute Translation

▪ Rotation

• Reactions▪ Rx

▪ Ry

구조역학


FREE BODY DIAGRAMS (FBD, 자유물체도)

• Simplified Sketch of Structure (all or in part) showing ▪ All External Forces

▪ Reactions (Forces & Moments)

▪ Internal Forces (for partial structure)

구조역학


EQUILIBRIUM (힘의 평형)

• Forces and Moments acting on a body are balanced

• 3 Equilibrium Equations for 2D Structures

→+ Fx = 0

↑+ Fy = 0

↻+ M = 0

구조역학


• Compute the reactions for the beam.

EXAMPLE 1.1

FBD

10 𝑚 5 𝑚 10 𝑘𝑁

구조역학


• Compute the reactions for the beam.

EXAMPLE 1.2

SPLIT! When there is an internal hinge:

→+ 𝐹𝑥 = 𝐸𝑥 = 0

10 𝑚 5 𝑚 5 𝑚 5 𝑚

24 𝑘𝑁

12 𝑘𝑁

𝑐𝑥

𝑐𝑦10 𝑚 5 𝑚 10 𝑚

24 𝑘𝑁

12 𝑘𝑁

5 𝑚

구조역학


EXAMPLE 1.2

→+ 𝐹𝑥 = −𝐶𝑥 = 0; 𝐶𝑥 = 0

↻+ 𝑀𝑐 = 24 5 − 𝐸𝑦 10 = 0; 𝐸𝑦 = 12 𝑘𝑁

↑+ 𝐹𝑦 = 𝐶𝑦 − 12 − 24 + 𝐸𝑦 = 0; 𝐶𝑦 = 24 𝑘𝑁

↻+ 𝑀𝐴 = −𝐵𝑦 10 + 𝐶𝑦 15 = 0; 𝐵𝑦 = 36 𝑘𝑁

↑+ 𝐹𝑦 = 𝐴𝑦 + 𝐵𝑦 − 𝐶𝑦 = 0; 𝐴𝑦 = −12 𝑘𝑁

Member CE

Member AC

Determinacy(정정, 부정정, 불안정)

구조역학


DETERMINACY

# of Unknown Reactions R = 3

FBD

# of Equilibrium Equations: 3

R = 3 (Statically Determinate)

구조역학


DETERMINACY


FBD

# of Equilibrium Equations = 3

R < 3 (Unstable)

구조역학


DETERMINACY


FBD

# of Equilibrium Equations = 3

R > 3 (Statically Indeterminate to 2nd degree)

구조역학


DETERMINACY

• Let 𝑹 = number of unknown reaction forces and moments

Condition Determinacy and Stability

# of Unknowns < # of Equations𝑹 < 𝟑

Unstable

# of Unknowns = # of Equations𝑹 = 𝟑

DeterminateStable (?)

# of Unknowns > # of Equations𝑹 > 𝟑

IndeterminateStable (?)

구조역학


DETERMINACY (INTERNAL HINGE)

FBD

Total # of Unknowns = Total # of Equations?

# of External Reactions : R=4 # of Free Body Diagrams 2

# of Internal Reactions : 2 Total # of Equations : 3*2=6

Total # of Unknowns : R+2=6

Statically Determinate

구조역학


DETERMINACY (INTERNAL HINGE)FBD

Total # of Unknowns < Total # of Equations


# of Internal Reactions : 2*2 Total # of Equations : 3*3=9

Total # of Unknowns : R+2*2=7

Unstable

?

구조역학



FBD

Total # of Unknowns > Total # of Equations


# of Internal Reactions : 2*2 Total # of Equations : 3*3=9

Total # of Unknowns : R+4=10

Statically Indeterminate to 1st Degree

?

구조역학




• Let 𝑪 = number of internal restraints

Total # of Unknowns ? Total # of Equations

# of External Reactions : R # of Free Body Diagrams C+1

# of Internal Reactions : 2C Total # of Equations : 3C+3

Total # of Unknowns : R+2C

R+2C ? 3C+3

R-C ? 3

구조역학






# of Unknowns < # of Equations𝑹 − 𝑪 < 𝟑

Unstable

# of Unknowns = # of Equations𝑹 − 𝑪 = 𝟑

DeterminateStable (?)

# of Unknowns > # of Equations𝑹 − 𝑪 > 𝟑

IndeterminateStable (?)

구조역학


PARALLEL REACTIONS

FBD

# of Unknown Reactions R = 3 # of Equilibrium Equations = 3


Is this beam stable?

No, the equation σ𝐅𝐱 = 𝟎 will not be satisfied

구조역학


CONCURRENT REACTIONS

FBD

# of Unknown Reactions R = 3 # of Equilibrium Equations = 3


Is this beam stable?

No, the equation σ𝐌𝐎 = 𝟎 will not be satisfied

구조역학


SUMMARY




# of Unknowns < # of Equations𝑹 − 𝑪 < 𝟑

Unstable

# of Unknowns = # of Equations𝑹 − 𝑪 = 𝟑

DeterminateBut unstable, if reactions form a

parallel or concurrent force system

# of Unknowns > # of Equations𝑹 − 𝑪 > 𝟑

IndeterminateBut unstable, if reactions form a

parallel or concurrent force system

구조역학


EXAMPLE 1.3

R = 4

C = 0

R – 3 – C = 1

Indeterminate to 1st Degree

구조역학


EXAMPLE 1.4

R = 5

C = 2

R – 3 – C = 0

Determinate

구조역학


EXAMPLE 1.5

R = 5

C = 0

R – 3 – C = 2

Indeterminate to 2nd Degree

구조역학


EXAMPLE 1.6

R = 6

C = 2

R – 3 – C = Indeterminate to 1st Degree

# of Equilibrium Equations = 3 + 3

# of Unknowns 6 + 1 = 7

구조역학 Structural Analysis - KINX

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