Module a Structural Analysis Suscos 2013 2014 l2 Wa

8/10/2019 Module a Structural Analysis Suscos 2013 2014 l2 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

Jean-Pierre Jaspart (University of Lige)


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L2 Global analysis of steel structures - Elastic analysis and plastic analysis.

European Erasmus Mundus

Master Course

Sustainable Constructions

under Natural Hazards

and Catastrophic Events

Conceptual Designof Buildings

2

Module A Structural Analysis

The structural analysis should take into account several effects, such as:

i) deformability and stiffness of the structure and supports;

ii) strength and stiffness of joints;iii) stability of the structure (global, members and local);

iv) behaviour of cross-sections (classif ication of sections);

v) imperfections (global and member imperfections).

Structural analysis Analytical process by which the response of

the structures (in terms stresses, internal forces and deformations) to

the acting loads is determined.

The understanding of the influence of all these effects is fundamental

for the designer, to make a suitable choice of the method of analysis

for a certain structure.

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples


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Sustainable Constructions

under Natural Hazards



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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

Global elastic (linear) analysis

Global plastic (non-linear) analysis

Structural analysis

Steel behaviour

- Elastic global analysis should be based on the assumption that the stress-strainbehaviour of the material is linear. As a consequence, to perform a global elastic

analysis, the stresses applied in any cross section of any member, must be lower

than the yield strength of the material (fy in steel structures).

- Plastic global analysis allows the plastification of some cross-sections (ingeneral forming plastic hinges) and consequent redistribution of forces for othersections (with less forces). In this type of analysis the material is modelled by

constitutive relationships non-linear: rigid-plastic; elastic-perfectly plastic (structural

steel), elastic-plastic.


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

- Plastic hinge is formed when in a bending section, all fibers reach the yieldstrength fy, in compression in one side of n.a. and in tension in the other side of

n.a.. The bending moment which can produce this stress diagram is called Plastic

Moment.

- The formation of a plastic hinge, and consequently the use of Structural Plastic

Analysis, require ductile materials and compact sections.

n.a.M

fy

fy

fy

fy

fy

fy

M=Mel Mel


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

oad Linear elastic analysis

Elasto-plastic analysis

isplacement

Rigid-plastic analysis

Elasto-perfectly plastic

Structural analysis

Load-displacement curve for elastic and plastic analysis


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples Isostatic structures

6

- Isostatic structure A structure with a number of external supports (or

internal restrictions) equal to the minimum necessary to be in equilibrium forany set of acting loads (e.g. 2D structure with 3 supports). The distribution of

forces and moments can be obtained through conditions of static equilibrium

only.

V1 V2

H1

Statically

determinate frame

Statically

determinated beam

In terms ofnumber of support reactions (or internal restrictions), structuresmay be classified in:

Hyperstatic structures


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

7

- Hyperstatic structure A structure with more than the minimum number of

external supports (or internal restrictions) necessary to be in equilibrium forany set of acting loads. In these structures, beyond the static conditions, it

must be used conditions of compatibility of deformations.

V1 V2

H1

M1M2

H1

2D indeterminate

frame =3

Indeterminated

beam =1

Global elastic analysis

Global plastic analysis

Isostatic structures

Hyperstatic structures


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

- How to perform a global linear elastic analysis of a structure?

- It is easy, even by hand calculation, in isostatic structures, because it is

needed only equilibrium equations.

- In hyperstatic structures, it is needed more complex methods such asthe well-know force method, displacement method, among others.However, in general we use automatic calculation. There are lots of

programs available in the market to perform elastic analysis of structures;

in this course we will use the software OSSA2D.

- The bending stiffness is a fundamental parameter in the analysis ofhyperstatic framed structures (axial stiffness and shear stiffness are in

general neglected in this type of structures). This is defined by EI, being E

the modulus of elasticity of the material (E = 210 GPa for steel) and I thesecond moment of area of cross section around the bending axis.

- In an elastic analysis the distribution of internal forces isproportional to the stiffness (in general the bending stiffness) of themembers. So, stiffer members support higher forces.


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

- Influence of bending stiffness in the elastic analysis of a hyperstatic frame

Bending moment diagram

Columns HEA 260 in both models (EI = 210x106x10450x10-8=21945 kN/m2)

Bending moment diagram

Beams IPE 300

(EI= 210x106x8356x10-8=17547.6 kN/m2)

Beams IPE 600

(EI= 210x106x92080x10-8=193368 kN/m2)


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

- How to perform an global plastic (non-linear) analysis of a structure?

- It is more complex. By hand calculation this is possible only for 2D

structures, with a low degree of indetermination. In more complex structures,this type of analysis requires the use of sophisticated softwares.

- Global plastic analysis may be performed by one of the two

following methods:

Static method (also designated by incremental method) the load isincremented stepwise until the collapse of the structure, which happen

(unless a partial mechanism is formed) when the number of sequential

plastic hinges reach +1, being the degree of indetermination of the

structure. This method is not suitable for perform plastic analysis by hand

calculations. Cinematic method (also designated by mechanism method) comprise three

conditions:

- Firstly, it is assumed a collapse mechanism which comprise +1 plastic

hinges (in general in beam-columns joints, mid-span sections of beams or

sections with point loads) mechanism condition;


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List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

Cinematic method (continuation):

- Then, it is established the equilibrium equations, considering that in cross

sections where were assumed the formation of a plastic hinge, the bendingmoment equals the plastic bending moment of the cross section

equilibrium condition, from which it is obtained the collapse load (if thestructure and consequently the plastic bending moment of cross section is

known) or the plastic bending moment for a given load (to be used in a pre-

design of a structure).- Finally it must be necessary to check if the bending moments in all other

sections are lower than the plastic moment evaluated for the present

mechanism - plasticity condition.

If this condition is not verified, this means that the present mechanism is not

the correct and it must be necessary to repeat the analysis for others, until to

find the correct one (which correspond the highest plastic bending moment

or the minimum collapse load, depending of which one are being evaluated).

Note: This type of analysis is valid provided that any significant decrease in the fullplastic moment due to axial forces are accounted for.


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EXAMPLE 1

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

Pre-design a cont inuous beam (S 275).

a) global elastic analysis;b) global plastic analysis.

42 kN/m 100 kN100 kN

3 m

A

BC E

F

G

3 m 3 m 3 m6 m

D

D

89.7 kNm

A

C

F

120.6 kNm

68.4 kNm

B

E

G

120.6 kNm

89.7 kNm

a) Elastic global analysis

Elastic bending moment diagram

Hyperstatic structure

= 2

12

Pre-design bending momentMEd = 120.6 kNm.


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EXAMPLE 1

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

VA

42 kN/m 100 kN100 kN

3 m

A

BC E

F

G

3 m 3 m 3 m6 m

D

Mpl

A

C

D F

Mpl


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EXAMPLE 2

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

14

50 kN

280 kN

4 m

4 m 4 m

A

B

C

D

E

189.9 kNm

A

B

C

D

E

247.0 kNm

341.6 kNm

142.9 kNm

Pre-design of a frame (S 275).

a) Elastic global analysis;b) Plastic global analysis.

a) Elastic global analysis

Elastic bending moment diagram

Pre-design bending moment

MEd = 341.6 kNm.


= 2


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50 kN

280 kN

pl

Mpl

Mpl

HE

VEVA

HA

MA

Plastic hinges

A

B

C

D

E



EXAMPLE 2

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

15

50 kN

280 kN

4 m

4 m 4 m

A

B

C

D

E

b) Plastic global analysis

Plastic bending moment diagram

pl

ri ght

B

pl

right

C

pl

right

D

MM

MM

MM

plEE

plEE

plE

MVH

MVH

M

428084

44

4

kNm280

kN140

kN70

pl

E

E

M

V

H

correct mechanism

Pre-design

bending moment

MEd = 280.0 kNm.

= 2, so it is required 3 plastic hinges

to form a collapse mechanism. It is

assumed a mechanism with plastic

hinges in sections B, C, D.

kNmMkNmM

kNV

kNH

VM

V

H

M

F

F

plA

A

A

AA

A

A

E

Vert

Horiz

280200

140

120

0.045042808

0.0280140

0.07050

0.0

0.0

0.0

.

.


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EXAMPLE 3

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

16

Calculate the collapse load (load multipl ierp)

of the frame with = 2 (S 275).

80 kN

5 m

6 m 6 m

1

3 4 5

5 m

40 kN

10 kN

20 kN2

Column 152x152 UC 37 (Mpl = 84.92 kNm);

Beam 305x165 UB 40 (Mpl = 171.35 kNm).


= 2


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EXAMPLE 3

List of contents

Structural analysis

Elastic analysis

Plastic analysis

Examples

17

px 80

Mpl

Mpl

H5

V5

V1

H1

Mpl

Plastic hinges

1

2

3

4

5

px 40

px 10

px 20

M5

M5

Mechanism 1 Hinges in section 2, 3 and 4.

35.1716

92.8420510

92.845

0.0101268010105200.0

0.010200.0

0.080400

554

13

12

5551

51

51

MVMM

HMM

xHMM

MHVM

HHF

VVF

pl

right

ppl

l eft

pll eft

ppp

gl obal

ppH

ppv

kNmkNmMkNHkNVkNVkNH plp 35.17194.794;43.59;05.161;59.144;98.16;547.2 55511

So, this is not the correct mechanism

80 kN

5 m

6 m 6 m

1

3 4 5

5 m

40 kN

10 kN

20 kN2

Column (Mpl = 84.92 kNm);

Beam (Mpl = 171.35 kNm).


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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 exclusiveuse by the SUSCOS 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 you

for your attention

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

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Module a Structural Analysis Suscos 2013 2014 l2 Wa

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