Module a Structural Analysis Suscos 2013 2014 l5 Wa

8/10/2019 Module a Structural Analysis Suscos 2013 2014 l5 Wa

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European Erasmus Mundus Master Course

Sustainable Constructionsunder Natural Hazards and Catastrophic Events520121-1-2011-1-CZ-ERA MUNDUS-EMMC

Conceptual Design of Buildings(Course unit code 1C2)

Module AStructural Analysis

J.P. Jaspart (University of Lige)


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L5 Approximate methods for second order analysis.

European Erasmus Mundus

Master Course

Sustainable Constructions

under Natural Hazards

and Catastrophic Events

Conceptual Designof Buildings

2

Module A Structural Analysis

List of contents

Approximate

methods for

second order

analysis

Examples

- Amplified sway moment method (clause 5.2.2(4) of EC3-1-1)

(applied to framed structures with similar load distributions at all floors; the

first buckling mode is a sway mode; cr> 3.0);

- Sway-mode buckling length method (clause 5.2.2(8) of EC3-1-1).

For certain structures it is possible to use simplified methods of analysis,

which approximate non-linear effects.

They allow the analysis of a structure based on a first order elastic analysis

only.

Eurocode 3, Part 1.1 provides two simplif ied methods which allow to carry

out an approximate second order analysis in the scope of steel framed

structures.


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Master Course

Sustainable Constructions

under Natural Hazards



3


List of contents

Approximate

methods for

second order

analysis

Examples

The amplified sway-moment method uses linear elastic analysis coupled with the

amplification of the so-called sway moments by a sway factor, which depends

of the elastic buckling load of the structure (sensitivity to 2nd order effects). Themethod comprises the following steps:

i) a l inear elastic analysis is carried out for a modified frame with horizontalsupports at all floor levels (figure below on left); it results in a distribution ofbending moments in the frame and reactions at the horizontal supports;

ii) a second l inear elastic analysis is carried out for the original f ramesubjected to the horizontal reactions obtained in the first step (figure bellowon the right); the resulting bending moments are the sway moments;

Amplified sway moment method (clause 5.2.2(4) of EC3-1-1)

R3

R2

R1

R3

R2

R1

Modified no-sway framewith horizontal supports (NS)

Original sway frame subjected tothe horizontal reactions (S)


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Sustainable Constructionsunder Natural Hazards



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List of contents

Approximate

methods for

second order

analysisExamples

Amplified sway moment method (clause 5.2.2(4) of EC3-1-1)

I

S

Scr

I

NS

II

ap MMM

.

11

1

I

S

Scr

I

NS

II

ap ddd

.

11

1

IS

Scr

I

NS

II

ap NNN

.

11

1

iii) approximate values of the second-order internal forces M, V and N and

displacements d are obtained by adding the results from the two elasticanalyses (previous steps), according to equations:

I

S

Scr

I

NS

II

ap VVV

.

11

1

where cr.S is the critical load multiplier for the lowest sway buckling mode;

iv) finally, the maximum elastic resistance of the frame is reached with the

formation of the first plastic hinge.

In normative terms, this approach is summarized in clause 5.2.2(4) of EC3-1-1.


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Master Course





EXAMPLE 1

5

List of contents

Approximate

methods for second

order analysis

Examples

Consider the fol lowing steel frame with fixed bases, composed by columns HEA 260 and

beams IPE 400 submitted to the shown load combination.

Calculate the design internal forces

for the verification of the Ultimate

Limit State (ULS) by a 2nd order

elastic analysis (required because

cr= 7.82 calculated in Example 1 Lesson 4), using the s impl ified

amplified sway-moment method.

33.6 kN/m

204.0 kN 204.0 kN

10 m

5 m

5 m

45.0 kN/m

18.2 kN

14.8 kN

253.5 kN 253.5 kN

10 m

5 m

5 m

3

45

5

6

6

7

8

8

33

1 2

5

Selected sections to be used in the example


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Master Course





EXAMPLE 1

6

List of contents

Approximate

methods for second

order analysis

Examples

- the first step of the method consists of a linear elastic analysis of a modified

frame with horizontal supports at each floor level, as shown in the figure below;

- in this case, the horizontal reactions are equal to the horizontal loads as the

vertical loads have no horizontal effects. The bending and axial forces on

selected sections are indicated in the table.

33.6 kN/m

204.0 kN 204.0 kN

45.0 kN/m

18.2 kN

14.8 kN

253.5 kN 253.5 kN

18.2 kN

14.8 kN

Modified no-sway frame and load arrangement

No-sway

My(kNm) Nx(kN)1 60.3 -850.5

2 -60.3 -850.5

3 -309.0 41.9

3 -119.9 -850.5

3 189.0 -372.0

4 253.5 41.9

5 -309.0 41.95 119.9 -850.5

5 -189.0 -372.0

6 -200.9 -78.0

6 -200.9 -372.0

7 219.1 -78.0

8 -200.9 -78.0

8 200.9 -372.0


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Master Course





EXAMPLE 1

7

List of contents

Approximate

methods for second

order analysis

Examples

- In a second step, the sway moments (and other internal forces) are calculatedby performing a linear elastic analysis on the original (sway) frame loaded by the

horizontal reactions obtained in the previous step (figure below).

- The bending and axial forces on selected sections are indicated in the table.

No-sway Sway 2ndOrder

My(kNm) Nx(kN) My(kNm) Nx(kN) My(kNm) Nx(kN)

1 60.3 -850.5 -49.2 15.8 3.7 -832.3

2 -60.3 -850.5 -49.1 -15.8 -116.8 -868.7

3 -309.0 41.9 52.3 -7.4 -248.9 33.4

3 -119.9 -850.5 33.4 15.8 -81.5 -832.3

3 189.0 -372.0 -18.9 5.3 167.3 -365.9

4 253.5 41.9 0 -7.4 253.5 33.4

5 -309.0 41.9 -52.2 -7.4 -369.0 33.4

5 119.9 -850.5 33.3 -15.8 158.2 -868.75 -189.0 -372.0 -18.9 -5.3 -210.7 -378.1

6 -200.9 -78.0 26.6 -9.1 -170.3 -88.5

6 -200.9 -372.0 26.6 5.3 -170.3 -365.9

7 219.1 -78.0 0 -9.1 219.1 -88.5

8 -200.9 -78.0 -26.6 -9.1 -231.5 -88.5

8 200.9 -372.0 26.6 -5.3 231.5 -378.1

18.2 kN

14.8 kN

Original sway frame subjected

to the horizontal reactions


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Master Course





EXAMPLE 1

8

List of contents

Approximate

methods for second

order analysis

Examples

- The approximate values of the second-order internal forces are obtained by

adding the results from the first step with the results of the second step,

amplified by the factor:15.1

82.711

1

11

1

cr

- The following table summarize the results (My and Nx).

Bending moments and axial forces (ASM)

2nd

OrderAmplified sway

method

My

(kNm)My(kNm) (%)

1 2.9 3.7 +27.6

2 -115.1 -116.8 +1.5

3 -244.0 -248.9 +2.0

3 -78.3 -81.5 +4.1

3 165.8 167.3 +0.94 257.2 253.5 -0.8

5 -363.6 -369.0 +1.5

5 156.4 158.2 +1.2

5 -207.2 -210.7 +1.7

6 -168.7 -170.3 +0.9

6 -168.7 -170.3 +0.9

7 225.3 219.1 -2.5

8 -227.9 -231.5 +1.6

8 227.9 231.5 +1.6

2nd order moments determined

in Example 2 Lesson 3

No-sway Sway 2ndOrder

My(kNm) Nx(kN) My(kNm) Nx(kN) My(kNm) Nx(kN)

1 60.3 -850.5 -49.2 15.8 3.7 -832.3

2 -60.3 -850.5 -49.1 -15.8 -116.8 -868.7

3 -309.0 41.9 52.3 -7.4 -248.9 33.4

3 -119.9 -850.5 33.4 15.8 -81.5 -832.3

3 189.0 -372.0 -18.9 5.3 167.3 -365.9

4 253.5 41.9 0 -7.4 253.5 33.45 -309.0 41.9 -52.2 -7.4 -369.0 33.4

5 119.9 -850.5 33.3 -15.8 158.2 -868.7

5 -189.0 -372.0 -18.9 -5.3 -210.7 -378.1

6 -200.9 -78.0 26.6 -9.1 -170.3 -88.5

6 -200.9 -372.0 26.6 5.3 -170.3 -365.9

7 219.1 -78.0 0 -9.1 219.1 -88.5

8 -200.9 -78.0 -26.6 -9.1 -231.5 -88.5

8 200.9 -372.0 26.6 -5.3 231.5 -378.1


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Master Course





EXAMPLE 1

9

List of contents

Approximate

methods for second

order analysis

Examples

170.3 kNm

369.0kNm

248.9kNm

3.7kNm

81.5 kNm

167.3 kNm

170.3 kNm231.5 kNm

158.2 kNm

231.5 kNm

116.8 kNm

210.7 kNm

219.1kNm

253.5kNm

Bending moment diagram (ASM)

-365.9kN

-832.3 kN

-88.5kN

33.4kN

-378.1kN

-868.7 kN

Axial force diagrams (ASM)

- The following figure shows the internal force diagrams (bending moment

and axial force) obtained by the ASM


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This lecture was prepared for the Edition 1 of SUSCOS

(2012/14) by RUI SIMES (UC) and FLOREA DINU (UPT).

Adaptations brought by J.P. Jaspart (Ulg)

for Edition 2 of SUSCOS

The SUSCOS powerpoints are covered by copyright and are for the exclusive use by theSUSCOS teachers in the framework of this Erasmus Mundus Master. They may be

improved by the various teachers throughout the different editions.


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Thank youfor your attention

http://steel.fsv.cvut.cz/suscos

Module a Structural Analysis Suscos 2013 2014 l5 Wa

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