Modeling of steel frame structures in fire using OpenSees

Computers and Structures 118 (2013) 90–99

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Modeling of steel frame structures in fire using OpenSees Jian Jiang a,⇑, Asif Usmani b a b

College of Civil Engineering, Tongji University, Shanghai 200092, China School of Engineering, The University of Edinburgh, Edinburgh EH9 3JF, United Kingdom

a r t i c l e

i n f o

Article history: Received 21 July 2012 Accepted 29 July 2012 Available online 19 August 2012 Keywords: OpenSees Structures in fire Nonlinear structural analysis Benchmark tests

a b s t r a c t The OpenSees framework has been extended to deal with frame structures under fire conditions. OpenSees is an object-oriented, open source software framework developed at UC Berkeley and has so far been focused on providing an advanced computational tool for analyzing the non-linear response of structural frames subjected to seismic excitations. New classes defining time-dependent temperature distributions in the cross-section of members have been created and OpenSees material classes have been modified to include temperature dependent properties based on the Eurocode. New functions and interfaces have been added into existing element and section classes to calculate the member actions due to arbitrary thermal loading taking into account material degradation with increasing temperature for non-linear analyses. This paper reports on a number of benchmark tests to ascertain the performance of the new codes implemented in OpenSees for beams, frames, and plate structures. The analysis procedures being developed for structures exposed to fire in OpenSees will make it easier for users to define temperaturedependent material properties and allow for arbitrary non-uniform temperature distributions across and along an element by interfacing a fire and heat transfer analysis module also being developed for OpenSees. This work will also enable the modeling of earthquake damaged structural frames subjected to a subsequent fire. Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction For a number of largely historical reasons the design of structures under fire conditions has traditionally been dealt with in a rather simplistic manner where neither the load (fire) nor the structure (either single members, sub-frames or whole structure) is adequately characterized. It is assumed that passive fire protection measures (in terms of coatings on steel members and adequate cover in concrete members) based on isolated single element testing under a standard fire time-temperature curve (e.g. ISO834) will provide adequate structural fire resistance. This has been proved to be a reasonable assumption for the vast majority of real fire incidents where both steel and RC frame structures have performed much better than expected. This however has led to criticisms, particularly by the steel industry as it is steel framed structures that bear the brunt of this practice in terms of significant additional cost of fire protection. These criticisms are based on real events [1] and tests [2] which have shown that the level of passive fire protection in steel buildings based on the traditional approaches is excessive and wasteful. Since then there have been a number of fire related building collapses, most notably the complete collapse of three large steel frame buildings on September ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (J. Jiang).

11, 2001 in which fire is thought to be the main cause [3–5]. There have also been fire related lesser known reinforced concrete building collapses (Delft Architecure Faculty Buidling and an underground parking structure in Gretzenbach, Switzerland). These collapse events and the findings from the Broadgate and Cardington investigations lead to an inescapable conclusion. While traditional design approaches are adequate for most buildings but, despite being wasteful and expensive, they may not even be safe for a small number of unusual cases. The answer to this problem is of course developing new design approaches based on greater quantitative understanding of the response of structures subjected to fire. Considerable progress has been made in developing such understanding over the last two decades, particularly since the Building Research Establishment in UK (BRE) and British Steel (now Tata Steel) carried out the Cardington tests [2]. This was followed by a large modeling project at the University of Edinburgh [6,7] in collaboration with a number of other partners (British Steel BRE, Steel Construction Institute, Imperial College London) producing a great deal of new understandings based on models developed for the British Steel tests using the commercial finite element software package ABAQUS. Many other groups followed suit. The computer program ADAPTIC developed by Izzuddin [8] at Imperial College to study the non-linear dynamic behavior of framed structures at ambient temperature was extended to include fire and

0045-7949/$ - see front matter Crown Copyright Ó 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2012.07.013

J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99

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explosion effects on steel framed structures [9,10] and then to reinforced concrete floor slabs [11]. The program SAFIR developed by Franssen [12] at University of Liege (Belgium) is widely used by researchers and practitioners all over the world to model structures in fire. The program VULCAN also widely used in this field was developed by successive researchers since 1985 at the University of Sheffield, UK [13–17]. Although specialist programs are cost-effective to purchase and easy to use they lack generality and versatility, but more tellingly continuous development, quality, robustness and long term sustainability of such research group based software must remain in perpetual doubt because of a relatively small number of users and developers. ABAQUS, ANSYS and DIANA are among the most prominent commercially available general purpose finite element (FE) programs. They have a large library of finite elements and excellent GUIs to enable efficient and detailed modeling of almost all types of structural behavior involved in fire and also allow user subroutines for modeling special features of behavior. Despite obvious advantages commercial FE packages require substantial recurring investment for purchasing and maintenance and so are often not affordable for researchers. OpenSees [18] is an open source object oriented software framework developed at UC Berekeley and supported by PEER and Nees. OpenSees has so far been focussed on providing an advanced computational tool for analyzing the non-linear response of structural frames subjected to seismic excitations. Given that OpenSees is open source and has been available for best part of this decade it has spawned a rapidly growing community of users as well as developers who have added to it is capabilities over this period, to the extent that for the analysis of structural frames it has greater capabilities than that of many commercial codes. OpenSees offers the potential of a common community owned research code with large and growing modeling capability in many areas of structural engineering enabling researchers to collaborate freely across geographical boundaries with a much greater potential longevity of research and development efforts. Other strengths of the OpenSees framework is the inclusion of a high performance computing (or parallel) version and the adoption of the object oriented paradigm of software development using C++, which enforces a discipline on the developers and ensures that the framework will develop in a manner that is manageable and easy to maintain and most of the its components are ‘‘reusable’’ by other developers. This paper presents work currently underway in Edinburgh to add a ‘‘structures in fire’’ modeling capability in OpenSees which will be consistent with the ethos of the other components of OpenSees in terms of being object oriented and enabling the use of HPC hardware. We specifically present numerical models of a steel beam with varying boundary conditions, steel frames and plate

subjected to a linear through depth thermal gradient to test the performance of OpenSees against analytical, experimental and ABAQUS results. Furthermore, some key features of structural response to elevated temperatures are analyzed and highlighted.

Fig. 1. A general section divided into n fibers.

Fig. 2. Flow chart for thermal–mechanical analysis.

2. Theoretical model In an incremental-iterative nonlinear analysis, three phases can be identified: Predictor, corrector and convergence check. The predictor needs to predict an initial out of balance force and calculate the displacement increment due to this unbalanced force given the stiffness matrix at the previous step. For thermo-mechanical analysis, in addition to the general external load increment, the unbalance force should include the equivalent fixed end force due to thermal load and material softening. The corrector is concerned with the recovery of element force increment from the displacement increment obtained in the predictor phase. The total strain is updated for the new geometry of the structure and the stress state can be determined by subtracting thermal strain from the total strain. The resisting force can be obtained by integrating the resisting stress along the section and used to calculate the out of balance force for this iteration. Equilibrium of the structure is checked at the end of each iteration to ensure that convergence is achieved in the new deformed configuration. 2.1. Predictor The unbalance force resulting from thermal load and material softening should be calculated in the predictor phase. The thermal

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Fig. 3. New classes added to OpenSees for performing thermo-mechanical analysis.

2.6

1

1

ΔT ≠ 0

1m

2

2.4

2

ΔT = 0

3

Analytical OpenSees

2.2

0.1m

2.0

0.1m

1m

Fig. 4. Restrained beam subjected to uniform temperature rise over its left half.

load can be considered as elemental load derived from the temperature distribution along the section. In the finite element analysis, the elemental load should be transformed into equivalent nodal load. Fig. 1 shows a general fiber section, which is subdivided into longitudinal fibers, with the geometric properties and temperature conditions, as defined by a uniform temperature increment, DTr, and a through-depth thermal gradient, (T,z)r, for a given fiber, r. Thermal gradient has not been implemented in OpenSees, only mean temperature is used for simplicity, however this can conceivably be implemented in future to model very steep thermal gradients with fewer fibers. If the beam that the section belongs to is fully restrained, each fiber will have a force and moment associated with it. Integrating the forces in each fiber gives section force F sec ¼ ½F M, defined as in [19]

Didplacement (mm)

u

1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

100

200

300

400

500

600

700

800

900

o

Temperature ( C) Fig. 6. Horizontal displacement of midpoint of the beam against temperature.

F¼

X E r A r a r DT r

ð1Þ

r

M¼

X X F r ðzr  zÞ þ Er Ir ar ðT ;z Þr r

ð2Þ

r

Where z is the centroid of the section given by

z ¼ 5

Modulus of elasticity (MPa)

2.0x10

P r Ar Er  zr P r Ar Er

ð3Þ

The integration of section force along the element will gives thermally induced elemental load Fth 5

1.5x10

F th ¼

Z

l

BT ðxÞF sec ðxÞdx

ð4Þ

0 5

1.0x10

where B(x) is the strain-displacement transformation matrix [20].

4

5.0x10

Kr

Kr 0.0

0.2m

Kt 0

200

400

600

800

1000

o

Temperature ( C) Fig. 5. Modulus of elasticity at elevated temperature referred to Eurocode 3.

0.1m

6m Fig. 7. Beam with translational and rotational springs at the ends.

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J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99 Table 1 Various boundary conditions of beam in the test.

Kt Kr

1

2

3

4

5

6

0 0

>0 0

1 0

1 >0

1 1

>0 >0

Table 2 Input parameters of the single beam model. Area (m2) 0.02

Length (m) 6

E (0 °C) (N/m2) 2  1011

UDL (N/m) 1000

a (/°C)

Thermal gradient (°C/m) 500–5000

(a)

12  106

(b)

Kt (N/m) 6.7  108

Kr (N.m/rad) 3  106

(c)

Fig. 8. Schematic of beams with different translational end restraint: (a) free end; (b) spring end; (c) pinned end.

" BðxÞ ¼

1 L

0

0

0

6x4L L2

6x2L L2

# ð5Þ

Fth = thermal load; F 0re = updated resisting force due to material softening

Another source of unbalanced force is the Material softening or material degradation, which means the reduction of resisting capability of the material due to the increment of the temperature. The reduction is mainly due to the degradation of the elasticity modulus and yield stress of the material at elevated temperature. The imbalance between the applied external load and declined resisting force leads to further deformation of the structure. Therefore, at the beginning of each thermal load step, the temperature-dependent material properties should be updated given current temperature and then the resisting force should be calculated again given the converged deformation at last step using the updated material properties. The out of balance force Fu at the beginning of each load step is determined by

The initial displacement increment can then be determined by the updated out of balance force using the stiffness matrix at previous converged step.

F u ¼ F ex þ F th  F 0re

emechanical ¼ etotal  ethermal

ð6Þ

where Fex = external load including nodal load and elemental load;

Once the initial displacement increment is obtained, iterations are needed to determine the converged displacements for the nonlinear problem. In this case, when forming the out of balance force, there is no need to consider thermal load, i.e.

F u ¼ F ex  F 0re

ð8Þ

With these two modifications, the corrector phase of thermomechanical analysis can followed the general procedure of mechanical analysis of structures [21]. 350

ABAQUS - free end ABAQUS - Spring end ABAQUS - Pinned end OpenSees - free end OpenSees - Spring end OpenSees - Pinned end

300

Deflection at mid-span(mm)

10

ð7Þ

Also, the stress state depends only on the mechanical strain

ABAQUS-Free end ABAQUS-Spring end OpenSees-Free end OpenSees-Spring end

12

Horizontal displacement (mm)

2.2. Corrector

8

6

4

2

250 200 150 100 50

0

0

0

200

400

600

800

1000 o

Temperature at the bottom of the beam ( C) Fig. 9. Horizontal displacement of beams with different translational end restraints.

0

200

400

600

800

1000

o

Temperature ( C) Fig. 10. Vertical mid-span deflection of beams with different translational end restraints.

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250

220

-3

Rotation of the end (10 rad)

180 160 140

Moment at midspan (KN.m)

ABAQUS - free end ABAQUS - Spring end ABAQUS - Pinned end OpenSees - free end OpenSees - Spring end OpenSees - Pinned end

200

120 100 80 60 40

ABAQUS - free end ABAQUS - Spring end ABAQUS - Pinned end OpenSees - free end OpenSees - Spring end OpenSees - Pinned end

200

150

100

50

20 0

0 0

200

400

600

800

1000

400

600

800

1000

Fig. 14. Moment at mid-span of beams with different translational end restraints.

3. Thermo-mechanical analysis in OpenSees

300

Deflection at mid-span(mm)

200

Temperature at the bottom of the beam ( C)

Fig. 11. End rotation of beams with different translational end restraints.

UDL UDL + T,y UDL + T,y + P-δ

250

In order to apply the aforementioned solution algorithm in the OpenSees, new class was created to store the temperature distribution in the structure and the existing material classes were modified to include temperature dependent properties. New functions and interfaces are added into the existing element and section classes to calculate thermally induced unbalanced force.

200 150 100

3.1. Thermal load class 50 0 0

200

400

600

800

1000 o

Temperature at the bottom of the beam ( C) Fig. 12. Mid-span vertical deflection contributions from different causes.

0 -200 -400 -600

Axial force (KN)

0

o

o

Temperature at the bottom of the beam ( C)

-800

The thermal analysis and structural analysis is uncoupled in OpenSees so far which means that temperature distribution along the element should be provided as input before the mechanical analysis. Parallel work is progressing on automatically generating time varying structural temperature data from a heat transfer analysis within OpenSees however direct inputs will always be required such as for modeling of experiments. Therefore new thermal load class was created to store the temperature distribution through the depth of the section defined by coordinate and corresponding temperature. The temperature of each fiber in the section will be determined by the interpolation of the temperature at the nearest coordinate point according to its location. The class diagram of this thermal load class is shown in Fig. 3. The beam2dThermalAction is considered as an elemental load ranked with concentrated load and uniformly distributed load.

-1000

ABAQUS - Spring end ABAQUS - Pinned end OpenSees - Spring end OpenSees - Pinned end

-1200 -1400 -1600 -1800 0

200

400

600

800

1000 o

Temperature at the bottom of the beam ( C) Fig. 13. Axial force in beams with different translational end restraints.

Fig. 2 shows the flow chart of element state determination of thermo-mechanical analysis mentioned above.

3.2. Modified material class There are many types of material models available in OpenSees for steel, defining their mechanical constitutive relationships, however, some of these are needed to be modified to include temperature dependent properties. At this stage temperature dependence will only be added to the unixial steel models as this data is not reliably available for the multiaxial cases. The uniaxial properties for steel will be primarily based on Eurocode stipulations. A temperature dependent steel material class (shown in Fig. 3) was created based on existing steel material class , which has a bilinear stress–strain relationship. The yield stress and modulus of elasticity at elevated temperature were defined according to Eurocode 3 [22].

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J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99

(a)

(b)

(c)

Fig. 15. Schematic of beams with different rotational end restraint: (a) pinned end; (b) spring end; (c) fixed end.

300

10

ABAQUS - fixed end OpenSees - fixed end

8

Deflection at mid-span(mm)

Deflection at mid-span(mm)

250

200

150

ABAQUS - Pinned end ABAQUS - Spring end ABAQUS - Fixed end OpenSees - Pinned end OpenSees - Spring end OpenSees - Fixed end

100

50

6 4 2 0 0

200

400

600

800

1000

-2 -4

0 0

200

400

600

800

1000

1200

-6

o

Temperature at the bottom of beam ( C)

o

Temperature at the bottom of the beam ( C)

(b)

(a)

Fig. 16. Vertical deflection of beams with different rotational end restraints: (a) vertical deflection at mid-span; (b) detail of fixed-end beam in (a).

0

200

ABAQUS - Pinned end ABAQUS - Spring end OpenSees - Pinned end OpenSees - Spring end

160

-2000

140

Axial force (KN)

-3

Rotation of the end (10 rad)

180

120 100 80 60

ABAQUS - Pinned end ABAQUS - Spring end ABAQUS - Fixed end OpenSees - Pinned end OpenSees - Spring end OpenSees - Fixed end

-4000

-6000

-8000

40 20

-10000

0 0

200

400

600

800

1000

0

200

400

600

800

1000

o

Temperature ( C) Fig. 17. End rotation of beams with different rotational end restraint.

4. Validation Four examples are used to test the new code and comparisons have been made with analytical solutions, experimental data and ABAQUS analyses. These include a fully restrained beam, half of which is subjected to a uniform temperature increase; a beam with finite translational and rotational end restraints subjected to uniform distributed load and a thermal gradient through the section depth; a uniformly heated steel frame experiment; and simply supported plate subjected to thermal gradient. In nonlinear thermo-mechanical analysis of structures geometric nonlinearities play a much greater role right from the start of the analysis (particularly under conditions of restrained thermal expansion) when the geometric stiffness changes rapidly often leading to instability and bifurcation situations leading to convergence difficulties. Material nonlinearities are in fact much better behaved, it is for this reason

o

Temperature at the bottom of the beam ( C) Fig. 18. Axial force of beams with different rotational end restraint.

the paper focuses on examples of the former type. It is also believed that these examples represent an excellent set of benchmarks, never published before, for researchers undertaking nonlinear thermo-mechanical analysis of structures for first time or when using new or unfamiliar software. 4.1. Restrained beam under thermal expansion Fig. 4 shows a beam, only the left half of which is subjected to a uniform temperature increment from 0 °C to 800 °C. The right half of the beam acts as a translational spring to restrain the displacement of the left part. Two elements are used in the model. Temperature dependent elastic material was assumed and the properties are taken from Eurocode 3 [22] shown in Fig. 5. The initial modulus of elasticity at 0 °C is 200 GPa and a constant coefficient of thermal expansion a = 12  106/°C is assumed.

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J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99

Moment of beam at mid-span (KN.m)

350 300

Kr=0 Kr=3e6 Kr=1.5e7 Kr=3e7 Kr=3e8 Kr=3e9 Kr=Infinite

250 200 150 100 50 0 -50

0

200

400

600

800

1000

1200

-100 -150 -200 -250 o

Temperature at the bottom of the beam ( C) Fig. 19. Moment at mid-span of beams with various end rotational stiffness.

350

Moment at the end (KN.m)

300 250

ABAQUS - Spring end ABAQUS - Fixed end OpenSees - Spring end OpenSees - Fixed end

200 150 100 50 0 0

200

400

600

800

1000

-50 o

Temperature at the bottom of the beam ( C)

-100

Fig. 20. Reaction Moment at the left end of beam for fixed and spring end restraints.

0 0 -2000

200

400

600

800

1000 o

Temperature at the bottom of the beam ( C)

Axial force (KN)

-4000 -6000 -8000 -10000 -12000 -14000

ABAQUS OpenSees Euler Buckling

-16000

Fig. 21. Axial force and Euler buckling force with changing temperatures for fixed end.

The horizontal displacement of mid point 2 can be calculated analytically from [23]

u¼

EðTÞAaDT E0 A l

þ EðTÞA l

ð9Þ

where E0 and E(T) are the modulus of elasticity at ambient and elevated temperature, respectively. The horizontal displacement of node 2 against temperature is shown in Fig. 6. Node 2 displaces towards the right driven by thermal expansion until 500 °C and then begins to move back as the

decreasing of modulus of elasticity in the left element is unable to resist the stored strain energy and elastic rebound of the unheated right element. 4.2. Single beam with end restraints of finite stiffness Fig. 7 shows a 2D beam with finite stiffness end restraints which is subjected to a uniformly distributed load (UDL) and linear thermal gradient along the section height. The finite end restraints are represented by translational and rotational springs with constant stiffnesses Kt and Kr, respectively. Different boundary conditions can be achieved by setting values of Kt and Kr. Table 1 shows six most practical boundary conditions for beam elements in steel frames. OpenSees is used to analyses all six cases and the results are compared with ABAQUS. The temperature at top of the beam was assumed to be 0 °C and it varied linearly over the depth of the beam to temperatures of 100 °C to 1000 °C. The same material properties as in Section 4.1 are used, the details are listed in Table 2. Nonlinear analysis is carried out using OpenSees (using corotational transformation to conduct the geometric nonlinear analysis) and then compared to ABAQUS results. Case 1 Beams with translational end restraint The influence of various translational end restraints, as shown in Fig. 8, is presented here. Selected displacements of the beams are shown in Fig. 9–11 where the results from OpenSees agree well with those of ABAQUS. The horizontal displacement of the spring end follows a similar pattern to the example in Section 4.1 while the free end also returns after initial thermal expansion, but this is because of increasing thermal gradient which ‘‘pulls’’ the free end back. The vertical deflection at mid-span of beam arises from a combination of the UDL, which will produce additional deflection given degrading material, the thermal gradient and the additional P-d moment along the beam caused by the end restraints. Fig. 12 shows relative deflections at the mid-span for different combination of causes, illustrating that thermal gradient provides the greatest contribution to deflection. The axial force in the beams with end-restraints is plotted against temperature in Fig. 13. Produced by restrained thermal expansion, the axial force increases until approximately 200 °C before declining. At this stage there is very little change in material properties the thermal bowing induced curvature cancels some of the restrained thermal expansion and indirectly leads to the reduction in axial forces. The mid-span moment in all the beams is shown in Fig. 14. The restrained beams develop increasing P-d moment until 400 °C, after which it declines because of the reducing axial forces even though the deflection keeps increasing. At 1000 °C, the axial force drops down to nearly zero and the mid-span moment in restrained beam declines to nearly the same value as the UDL only induced moment in the free-end beam. It is therefore obvious that the translational restraint dominates the structural behavior over midrange temperatures (between 200 and 400 °C) resulting in high axial forces and mid-span moment in end-restrained beams. Case2 Beams with different rotational end restraint Models of beams with different rotational end restraint are shown in Fig. 15. Fig. 16 shows that the deflection at mid-span of finitely restrained beam was much larger than that in fixed-end beam. This is because, for the fully fix ended beam subjected to thermal gradient T,z, an equal and opposite curvature induced by the support moments cancels out the thermal curvature and, therefore, the fixed ended beam remains ‘straight’ with a constant moment M = EIaT,z along its length, hoowever unstable behavior of the beam can be seen between temperatures of 500 °C and 800 °C of

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J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99

F2

1170

F1

v4 F1=112kN F2=28kN

u2

1240 (a)

(b)

Fig. 22. Schematic of the tested steel frames (mm): (a) frame EHR3; (b) frame ZSR1.

a

400

x

350 o

100 C o 200 C o 300 C o 400 C o 500 C o 600 C o 700 C o 800 C

Stress (MPa)

300 250 200

b

150

y

100 Fig. 25. Simply supported rectangular plate with linear through thickness thermal gradient.

50 0 0.000

0.005

0.010

0.015

0.020

Strain Fig. 23. Temperature-dependent stress–strain curves of steel 37 used in test.

Fig. 16(b). At 600 °C the beam bends downward to its peak deflection (because of thermal bowing) and then snaps through to the opposite direction. This ‘‘thermal snap through’’ is driven by the additional hogging moment that occurs in the beam resulting from the center of stiffness of the beam section moving upwards (due to the greater material degradation in the bottom) as the temperature increases, which creates an ‘‘eccentricity’’ for the axial forces and

produces a moment opposite to the once caused by thermal bowing. Both ABAQUS and OpenSees results reproduce this effect, however as this represents an abrupt and unstable transition stage, the exact magnitudes of the displacements (which are of the order of a few mm compared to the beam depth of 200 mm) are not a measure of program accuracy. The end rotation for the spring-end beam in Fig. 17 again shows an increase in rotation followed by reduction due to material softening. Fig. 18 shows that higher axial forces result from higher rotational end restraints as expected. The fixed-end beam showed interesting behavior (Fig. 16(b)), deflection arising from the UDL

45

45 EHR Frame u2: Test v4: Test u2: OpenSees v4: OpenSees

35 30 25 20 15 10

35 30 25 20 15

5

10

0

5 0

100

200

ZSR Frame u1: Test u2: Test u1: OpenSees u2: OpenSees

40

Displacement (mm)

Displacement (mm)

40

300

400 o

500

600

0

100

200

300

400 o

Temperature ( C)

Temperature ( C)

(a)

(b)

Fig. 24. Comparison between predicted and test deflection results: (a) frame EHR3; (b) frame ZSR1.

500

600

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J. Jiang, A. Usmani / Computers and Structures 118 (2013) 90–99

Table 3 Input parameters of the plate model. Side length a = b (m)

Thickness t (m)

E (N/m2)

a (/°C)

m

T,z (°C/m)

2

0.025

2  1011

12  106

0.3

400–4000

Deflection at the center of plate (mm)

Theory_Constant E OpenSees_Constant E OpenSees_Changing E ABAQUS_Changing E

160

w¼

0:05 MT a2 ð1  mÞ D

ð11Þ

In this case, the temperature at top of the plate was assumed to be 0 °C and it varied linearly over the depth of the beam to temperatures of 100–800 °C at bottom. The material properties are listed in Table 3. The deflection at the center of plate is shown in Fig. 26. For constant E, the OpenSees results were compared with the analytical solution (Eq. (11)). For E varying with temperature, ABAQUS was used to compare against OpenSees results. Both comparisons show good agreement.

200 180

The expression of deflection above converges very rapidly. For a square plate (a = b), the numerical solution of deflection at the center can be determined from Eq. (10) in the form

140 120 100

5. Conclusions

80

The OpenSees framework has been extended to perform nonlinear thermo-mechanical analysis of structures in fire. The development was tested for a range of structural behaviors. The results agreed well with analytical solutions, ABAQUS as well as test data. A number of fundamental behaviors of structures in fire were also highlighted. Further work is being done to extend OpenSees shell formulations to deal with large displacements and to couple the thermo-mechanical analyses with heat transfer analyses.

60 40 20 0 0

200

400

600

800 o

Temperature at the bottom of plate ( C) Fig. 26. Deflection at the center of plate against temperature.

Acknowledgements and the P-d effect reached its downward peak at 600 °C, it then reversed sharply to an upward peak and then decreased nearly to zero. This curious buckling phenomenon occurred when the axial force exceeded the Euler limit as shown in Fig. 21. Figs. 19 and 20 show mid-span and end moments in beams with rotational restraint stiffnesses ranging from pinned end to fixed end.

The research was financially supported by the National Natural Science Foundation of China under Grant No. 51120185001. The authors thank J. Zhang and P. Kotsovinos for providing suggestions on the development of the thermomechanical analysis capability of OpenSees.

4.3. Steel frame test

References

A series of tests on plane steel frames at elevated temperatures were performed in Germany [24]. A schematic diagram of two steel frames EHR3 and ZSR1 are shown in Fig. 22. The braced two-bar frame (HER3) was subjected to a uniform temperature rise and only one bay of the two-portal frames (ZSR1) was uniformly heated. All structural elements were made of IPE80 I-shaped steel. The temperature dependent stress–strain relationship of Steel 37 used for the experiments is shown in Fig. 23. The yield stresses and modulus of elasticity are 382 N/mm2 and 210 N/mm2 at ambient temperature for EHR3 and 355 N/mm2 and 210 N/mm2 for ZSR1, respectively. Comparisons between the predicted deflections and the test results illustrated in Fig. 24 show satisfactory agreement.

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4.4. Bending of plate subjected to thermal gradient Now we consider a simply supported rectangular plate shown in Fig. 25 subjected to linear thermal gradient through the thickness only. The analytical solution for the deflection of the plate is given by [25]

w¼

1 X 1 16M T X sinðmp=2Þ sinðnp=2Þ h i 4 ð1  mÞDp m n mn ðm=aÞ2 þ ðn=bÞ2

ð10Þ

where MT = aT,zD(1 + m) is the equivalent thermal load, T,z is the thermal gradient per unit length, D = Et3/12(1  m2) is the bending stiffness.

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