     # Structural Analysis Ch1...آ  2019-05-03آ  Analysis of Statically Indeterminate Frames 7. Influence Lines

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• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

Structural Analysis I

Spring Semester, 2018

Hae Sung Lee

Dept. of Civil and Environmental Engineering Seoul National University

y δ

y f

z δ zf

x δ

x f

y M yθ

z M zθ xM xθ

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

Contents

1. Introduction

2. Reactions & Internal Forces by Free Body Diagra

ms

3. Principle of Virtual Work

4. Analysis of Statically Indeterminate Beams

5. Analysis of Statically Indeterminate Trusses

6. Analysis of Statically Indeterminate Frames

7. Influence Lines for Determinate Structures

8. Influence Lines for Indeterminate Structures

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

1

Chapter 1

Introduction

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

2

1.1 Mechanics of Material - Structural Mechanics

 Problem

Calculate the reaction force at each support and draw the moment and shear force dia

gram for the two-span beam shown in the figure.

 Solution

 Equilibrium Equation

qLRRRF cbay 20 =++→=

qLRRLRLRLqLM cbcba 220220 =+→=×−×−×→=

00 22

0 =−→=×−×+×+×−→= cacab RRLRLR L

qL L

qLM

qLRRLRLRLqLM babac 220220 =+→=×+×+×−→=

Since there are three unknowns in two independent equations, we cannot determine a unique

solution for the given structure, and thus we need one more equation to solve this problem.

The main issue of this class is how to build additional equations to analyze statically inde-

terminate structures.

EI EI

q

Ra Rb Rc

L L

q

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

3

Fundamentals of Differential Equations

 Governing Equations

The governing equations of a (engineering) system are usually defined by a system of diffe-

rential equations, which governs behaviors of the given system within a domain. Notice that

the domain does not include boundaries.

11 qwEI =′′′′ for lx

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

4

1.2 Mechanics of Material

 Governing Equation

– Left span

11 2

1 3

1

4

1 ''''

1 24 dxcxbxa

EI

qx wqEIw ++++=→=

– Right Span

22 2

2 3

2

4

2 ''''

2 24 dxcxbxa

EI

qx wqEIw ++++=→=

 Boundary Conditions

– Left support

0)0()0( , 0)0( 111 =′′−== wEIMw

– Center support

)()( , )()( , 0)()( 212121 LwLwLwLwLwLw ′′=′′′−=′==

– Right support

0)0()0( , 0)0( 222 =′′−== wEIMw

Since there are eight unknowns with eight conditions, we can solve this problem.

 Determination of Integration Constant – Left Support

xcxa EI

qx wbwdw 1

3 1

4

11111 24 02)0( , 0)0( ++=→==′′==

– Right Support

xcxa EI

qx wbwdw 2

3 2

4

22222 24 02)0( , 0)0( ++=→==′′==

x x

y

z

z

y

q

w1 w2

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

5

– Center Support

  

 

==

−== →

    

   

+=+

−−−=++

=++=

=++=

EI

qL cc

EI

qL aa

La EI

qL La

EI

qL

cLa EI

qL cLa

EI

qL

LcLa EI

qL Lw

LcLa EI

qL Lw

48

48

3

6 2

6 2

3 6

3 6

0 24

)(

0 24

)(

3

21

21

2

2

1

2

2 2

2

3

1 2

1

3

2 3

2

4

2

1 3

1

4

1

)32( 48

334 21 xLLxxEI

q ww +−=≡

8 3

, 8

3 2 11

2 11

qL qxwEIVx

qL x

q wEIM +−=′′′−=+−=′′−=

 Moment Diagram

 Shear Diagram

 Reactions

0.375qL

L 8 3

+

-

+

0.625qL

-

0.375qL 0.375qL 1.25qL

0.125qL2

0.070qL2

L 8

3

+

-

+

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

6

1.3 Mechanics of Material + α

1.3.1 Main idea

 Original Problem

 Case I (Removal of the center support)

 Case II (Application of the reaction force)

 Original Problem = Case I + Case II (compatibility condition)

δ0+ δR=0

1.3.2 Calculation of δ0

 Bending Moment

qLx qx

M +−= 2

2

q

q

δ0

δR

Rb

q

qL2/2

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

7

 Governing Equation

bax qLxqx

EIwMwEI ++−=→−=′′ 624

34

00

 Boundary (support) Conditions

– Left Support : 0)0(0 =w

– Right Support : 0)2(0 =Lw

 Determination of Integration Constant

– Left Support

0 0)0(0 =→= bEIw

– Right Support

EI

Lq aLa

LqLq LEIw

24

)2( 0)2(

6

)2(

24

)2( )2(

334

0 =→=+−=

 Deflection

))2()2(2( 24

334 0 LxLxxEI

q w +−=

EI

Lq LLLLL

EI

q Lw

384 )2(5

))2()2(2( 24

)( 4

334 00 =+−==δ

1.3.3 Calculation of δR

 Bending Moment

22 ,

2 2

2 1

1

LRxR M

xR M bbb −=−=

 Governing Equation

  

 

+++−=

++= →

  

−=′′ −=′′

222

2 2

3 2

2

111

3 1

1

22

11

412

12

bxa LxRxR

EIw

bxa xR

EIw

MwEI

MwEI

bb R

b R

R

R

1x 2x

RbL/2

Rb

• Dept. of Civil and Environmental Eng., SNU

Structural Analysis Lab. Prof. Hae Sung Lee, http://strana.snu.ac.kr

8

 Boundary (support and mid-span) Conditions

– Left Support & Right Support

0)( , 0)0( 21 == Lww RR

– Mid-span

)( )0()()( , )0()( 221121 LwLwLwLw RRRRRR θ=′=′=θ=

 Determination of Integration Constant

– Left Support

0 0)0( 11 =→= bwR

– Right Support & Mid-span

  

 

−=

=

−=

  

  

′==+=′

==+=

=+++−=

6

0 4

(0) 4

)(

(0) 12

)(

0 412

)(

3

2

2

2

1

221

2

1

221

3

1

22

33

2

LR b

a

LR a

wEIaa LR

LwEI

EIwbLa LR

LEIw

bLa LRLR

LEIw

b

b

R b

R

R b

R

bb R

 Deflection

)23( 12

)3( 12

32 2

3 22

1 23

11

LLxx EI

R w

xLx EI

R w

b R

b R

+−−=

−=

EI

RL wLw bRRR 48

)2( )0()(

3

21 −===δ

 Compatibility Condition

δ0+ δR=0 → 0 48

)2(

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