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Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS BFC 21403 www.uthm.edu.my With Wisdom, We Explore

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Page 1: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Statically Indeterminate Beam & Frame

STRUCTURAL ANALYSISBFC 21403

www.uthm.edu.my With Wisdom, We Explore

Page 2: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Introductionwww.uthm.edu.my With Wisdom, We Explore

2

Analysis for indeterminate structure of beam and frame:

1. Slope-deflection method

2. Moment distribution method

3. Modified stiffness

Shear force

Moment

The main aim is to calculate

reaction forces

Beam Frame (non-sway/sway)

Displacement

Methods

Page 3: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Introductionwww.uthm.edu.my With Wisdom, We Explore

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The main problem of indeterminate beam is todeterminate support reactions.

A static equilibrium is not enough to solve this problem.

Page 4: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Introductionwww.uthm.edu.my With Wisdom, We Explore

4

In reinforced concrete building:

Page 5: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Introductionwww.uthm.edu.my With Wisdom, We Explore

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In steel building:

Page 6: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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In frame:

Page 7: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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In bridge:

Page 8: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Introductionwww.uthm.edu.my With Wisdom, We Explore

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Two method that can be used:

Slope deflection method Moment-distribution method

1. Fixed end moment

2. Moment result deflection and moment deposit (support shift)

3. Slope of support

4. Moment of support

1. Stiffness member

2. Distribution factor

3. Cary over factor

4. Fixed end moment

5. Distribution process

6. Moment of support

Page 9: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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For indeterminate structure, moment of member happenfrom:

i. Fixed end moment

ii. Deflection slope or rotation

iii. Support shift (support settlement)

To form equation of state, member must have uniformand homogeneous among two support.

Redundancy create the value of unknown that related toforce method, e.g. flexibility method.

Deformation can also contribute to the value of unknownthat related to deformation method, e.g. slope-deflectionmethod.

Page 10: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Consider typical beam BC from continuous beam:

Moment resultant at the end B and C can be identified as:

1. Fixed end moment

(FEM)

Page 11: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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2. Moment of Slope

(MS)

3. Moment of Support Displacement

(MSD)

Page 12: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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FEM: moment resultant at end to end outside taxincidence member that imposed to stated member whenboth supports are assumed as fixed, therefore therotation is zero.

Page 13: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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MS: Moment that occur at different direction of rotation.End gradient member would be positive if the rotation isclockwise. MS can be phased as following:

- End gradient B if C’s end control/fixed

- End gradient C if B’s end control/fixed

∴ The results are:

L

2EIθMS

2

1MS ;

L

4EIθMS

BBCCB

BBC

L

2EIθMS

2

1MS ;

L

4EIθMS

CCBBC

CCB

L

θ2θ2EI

L

2EIθ

L

4EIθMS

L

θ2θ2EI

L

2EIθ

L

4EIθMS

BCBCcerCB

CBCBcerBC

Page 14: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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MSD: Moment that occur due to deformation(displacement) at one end to another end, e.g. B’s endand C constrained.

- Moment of support displacement is generally calculated

using:

where

2DCBC

L

6EIΔMSDMSD

L

Δδ

L

6EIδMSDMSD DCBC

Page 15: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Moment resultant for beam BC is:

3δθ2θL

2EI

L

PabM

L

6EIδθ2θ

L

EI2

L

PabM

MSDMSFEMM

CB2

2

BC

CB2

2

BC

BCBCBCBC

3δθ2θL

2EI

L

PabM

L

6EIδθ2θ

L

EI2

L

PabM

MSDMSFEMM

BC2

2

CB

BC2

2

CB

CBCBCBCB

Page 16: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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Principle of analysis:

- Equilibrium equation: - Boundary condition:

Fixed end slope, ∆=0

0MM

0MMM

0MM

CBC

BCBAB

ABA

Page 17: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Example 4.1:

Determine the moment value and shear force to eachsupport and draw shear force diagram (SFD) andbending moment diagram (BMD) for structure beambelow. Assume EI is constant.

Page 18: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Procedure:

1. Calculate fixed end moment (FEM)2. Determine boundary conditions at support3. Calculate moment resultant for both ends4. Calculate rotation based on equilibrium equation5. Calculate end moments, MAB and MBA

6. Calculate reaction forces7. Calculate shear forces8. Draw shear forces diagram9. Calculate moment from end moments and shear forces

diagram10. Draw bending moment diagram

Page 19: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

Page 23: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Example 4.2:

Determine the moment value and shear force to eachsupport and draw shear force diagram (SFD) andbending moment diagram (BMD) for structure beambelow. Assume EI is constant.

Page 24: Statically Indeterminate Beam & Frame - AUTHORauthor.uthm.edu.my/uthm/www/content/lessons/601/Slides - Chapter 4... · Statically Indeterminate Beam & Frame STRUCTURAL ANALYSIS

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E

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E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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E

E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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E

E

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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E

E

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Tutorial 4.1

Determine the reaction to each support for thecontinuous beam in the figure below. The support at Caccidently constructed 10mm below its intendedposition. Given E=210e6kN/m4 and I=180e-6m4.

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Slope-Deflection Methodwww.uthm.edu.my With Wisdom, We Explore

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Tutorial 4.2

Determine the moment value and shear force to eachsupport and draw shear force diagram (SFD) andbending moment diagram (BMD) for structure beambelow. Assume EI is constant