건설안전역학

(1)

건설안전역학

(Constructional Safety Mechanics)

토목안전환경공학과 안전트랙

옥승용

LN09: Deflection Computation

by using Energy Method (1)

(2)

Class Schedule

Week Topics Remarks

01 Introduction to class (1) & (2)

02 Analysis of Truss Structures (1) Homework #01

03 Analysis of Truss Structures (2)

04 Analysis of Horizontal Beams (1) Quiz #01

05 Analysis of Horizontal Beams (2)

06 Analysis of Frame Structures (1) Quiz #02

07 Analysis of Frame Structures (2)

08 Mid-Term Exam

09 Deflection Computation by using Energy Method (1) 10 Deflection Computation by using Energy Method (2)

11 Deflection Computation by using Energy Method (3) Quiz #03 12 Analysis of Indeterminate Structures (1)

13 Analysis of Indeterminate Structures (2) Quiz #04

(3)

• Structural Analysis

– Equilibrium Equation: Compute the forces (reactions, member forces) – Compute the deformation of the members

• Truss, Beam and Frame Structures

– Deflection & Rotation

• Analysis Method for Deflection of Structures

– 기하학적 방법

• Moment-Area Theorem (모멘트 면적법)

• Method of Elastic Load (탄성하중법)

• Conjugate-Beam Method (공액보법)

• Double Integration Method & Differential Equation of Deflection (이중적 분 & 처짐의 미분방정식)

– 에너지방법

• Principle of Virtual Work (가상일의 원리)

Chapter Preview

(4)

• Structural Deformation

– Truss: axial deformation – Beam: deflection & slope

– Frame; axial deformation, deflection & slope

Energy Method

(5)

• Structural Deformation

– Truss: axial deformation – Beam: deflection & slope

– Frame; axial deformation, deflection & slope

Energy Method

수학적 방법

• Double integration method

기하학적 방법

• Moment-area method

• Conjugate-beam method

에너지 방법

• Virtual Work

• Castigliano’s Theorem

상대적으로 단순하중이 작용하는 보다 더 복잡한 하중조건이나

Loads Structure Deformation

(6)

Definition of Work

(7)

• Work is involved with force and displacement (movement)

Definition of Work

W   F r

(8)

Definition of Work

W   F r

F

q

D

cos W    F r F  D  q

F _W _{   D} _{F r} _F

For constant force

(9)

Definition of Work

W   F r

k

x₁

F

₁

F

₂

x₂

2

2 2

2 2 2

0

1 1 1

2 2 2

x

W   F dr    kxdx     kx     kx  F x 1

W   F dr   2 ^F D

F

x k

x₁ F₂=kx₂

F₁=kx₁

x₂

F=kx

For linearly-varying force

(10)

• The work done by the force F that remains constant along with the displacement

• The work done by the force F that varies in proportion with the

displacement

Definition of Work

 

1 1 1

1 0 0 0 1

W  

^D

F dr    F 

^D

dr   F r

^D

 ^FD

¹ ₀¹

¹

¹ ¹

2 W  

x

F dr   ^F D

F

x k

D₁ F₂

F₁

D₂ F

D x F

2

2 2 2

0

1 2 W  

x

F dr   ^F D

D

 

2 2 2

2 0 0 0 2

W  

^D

F dr    F 

^D

dr   F r

^D

 ^FD

(11)

• “The work done by all the external forces acting on a structure is transformed into internal work (or strain energy), which is developed when the structure deforms.”

The Law of Conservation of Energy

U _e = U _i

하중이 한 외적 일 내적 변형에너지

PD= ∫ u dl D

P

내력: u

부재변형: dl

여기서의 부재변형이라 함은

축변형, 전단변형, 회전각 등을 의미한다.

(12)

• Beam Structure subjected to Vertical Force

Applications: Rotational Deformation

P

L

x

P

V M

e i

U  U

D

 P U

_e

2 1

D

How about internal work?

(13)

Applications: Rotational Deformation

P

L

x

P

V M

e i

U  U

D

 P U

_e

2 1

D

▪ A work done by M(x) and d q is dU

_i

= M d q

▪ If the moment is gradually applied to a structure & the final rotation is q , then a total work done by M(x) is

1 0 2

U

i

 

^q

M d ^q  M ^q

▪ However, if the moment is already applied to the

structure & other loadings further distort the structure

q ’, then M rotates q   q ^

(14)

[Sup: dq] 휨을 받는 보 부재(1)

변형전 변형후

L

0

 dx   q d 보의 길이방향 변형률

1

( )

L y d dx yd

dx y dx

 q q



   

 

1 0

0 0

x

dx y dx dx

L L y

L L dx y



  



  

 

  

      

변형 전 (e-f)의 길이 변형 후 (e-f)의 길이

수직변형률







¹

곡률반지름 (radius of curvature)

곡률(curvature)

(15)

[Sup: dq] 휨을 받는 보 부재(2)

• 보의 수직응력

- 보의 수직응력

- 단면에 작용하는 수직응력은 중립면으로부터의 거리 y 에 따라 선형적으로 변함

y Ey E

E _x

x



 



  

x y



x

Negative(−)

Positive(+)

(16)

[Sup: dq] 휨을 받는 보 부재(3)

 보의 수직응력

• SF

_x

= 0

• Moment

     0



^x



A A

dA E y dA



x



dM    dA  y

 

2

A A x

A

M dM y dA

E y dA

E y dA EI





 

  

 

  

     

 



1 M

 

 

x

  E y  

   0

A

y dA

x

dA



 y

dM

x

dA y 

 dM

 I  

A

y dA

²



(17)

Applications: Rotational Deformation

P

L

x

P

V M

e i

U  U

D

 P U

_e

2 1

D

dx M

d dx dx

 q     EI



1

2

2 2

i

dU Md M dx q EI

 

▪ According to the Beam Theory,

L

0

 dx   q d

1 M

M EI

  EI

    

(18)

Applications: Rotational Deformation

P

L

x

P

V M





^L

i

EI

dx U M

0 2

2  ^ ^



^L

i

EI

L P EI

dx U Px

0

3 2 2

6 1 2

) (

2 3 3

1 1

2 6 3

P L PL

P EI EI

 D   D 

M   Px

e i

U  U

D

 P U

_e

2

1

(19)

• (1) External Work done by Force

– Suppose F' is already applied to the bar.

– Then, the bar deforms by an amount D'.

– Another force P is now applied.

– So, the bar deforms further by an amount D.

– A total deformation of the bar is D'+D.

The Law of Conservation of Energy

What is the work done by F'? W

₁

= ½ F' D'

What is the additional work done by P & D?

F'

W

₂

= ½ P D

Work done by F' when the bar

deforms by the further deformation D.

D' D

F'

P F'

W

₃

= F' D

(20)

• (1) External Work done by Force

– Suppose F' is already applied to the bar.

– Then, the bar deforms by an amount D'.

– Another force P is now applied.

– So, the bar deforms further by an amount D.

– A total deformation of the bar is D'+D.

The Law of Conservation of Energy

What is the work done by F'? W

₁

= ½ F' D'

What is the total work done by F' & P?

F'

W = ½ (F'+P)( D'+D)

D' D

P F'

F'

(21)

• If the virtual F' is unit load(=1), the

displacement (D) induced by the external load P can be computed to be PD'.

The Law of Conservation of Energy

F'

W = ½ (F'+P)( D ' +D)

= ½ F' D ' + ½ F' D+ ½ P D '+½ P D W

₁

+W

₂

+W

₃

= ½ F' D '+F' D+ ½ P D

½ F' D+ ½ P D '=F' D

∴ ½ P D '=½ F' D  P D '=F' D

F' D=1×D=D= P D '

D' D

F'

P F'

P ~ D : external load

F' ~ D' : virtual load

(22)

• 가상일의 원리

• Unit load method (단위하중법)

• 모든 구조물에 적용 가능한 비교적 가장 일반적인 방법

• 에너지 보존의 법칙을 확장한 이론

• Determine the displacement D of point A caused by the loads P

₁

, P

₂

and P

₃

.

– where L is the original length of any member in the system, and the dL is the deformation of the member.

Principle of Virtual Work

Principle of virtual work



u P D  



Work of external

loads

Work of internal

loads

(23)

Principle of Virtual Work

• Step 1.

– Place a virtual load on A in the same direction as D.

– Virtual load: P' = 1 (unit load)

– The unit load P' creates an internal (virtual) load u in the body.

• Step 2.

– Now, apply the loads P₁, P₂and P₃.

– Point A will deform by an amount D, causing an internal member in the body to deform by an amount dL.

– External work: U_e = 1 × D – Internal work: U_i = ∫ u dL

• Step 3.

– By the principle of conservation of energy,

u

P'=1

(24)

Principle of Virtual Work

U

_e

= 1 × D = U

_i

= ∫ u dL

Real deformation to compute when the structure is subjected to real loads P_i

Internal forces caused by virtual unit load that is applied at the same point as D & to the same direction as D

Real internal deformation caused by real loads

(25)

• Beam Structure subjected to Axial Force

Applications: Axial Deformation

P A

  ^D L

P E E

A L

     ^D

  E 

dL Pdl

 EA 1 u 

P

내력: u U

_e

= 1 × D = U

_i

= ∫ u dL

Unit load 1에 의한 내력

외력 P에 의한 내부변형

PL

D  EA

(26)

Principle of Virtual Work

U

_e

= 1 × D = U

_i

= ∫ u dL

Real deformation to compute when the structure is subjected to real loads P_i

Internal forces caused by virtual unit load that is applied at the same point as D & to the same direction as D

건설안전역학