- Email: [email protected]
Application of the direct iteration method for non-linear analysis of steel frames in fire
Fire Safety Journal 35 (2000) 241}255
Application of the direct iteration method for non-linear analysis of steel frames in "re Jin-Cheng Zhao* College of Civil Engineering & Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China Received 22 October 1998; received in revised form 15 March 2000; accepted 20 April 2000
Abstract Based on "nite element method, a direct iteration method capable of predicting the nonlinear behavior of steel frames at elevated temperatures is proposed, in which the second-order e!ects of large de#ections, the progressive softening of materials with temperature rise, gradual penetration of inelastic zone both over the section and along the member, non-uniform distribution of temperature within steel members are all taken into consideration. The secant sti!ness matrix that relates the total element displacements to the corresponding total forces is used in deriving the basic "nite element equations. Di!erent from the commonly used Newton}Raphson method in which the deformations are calculated incrementally, the direct iteration method used in this paper allows the calculation of the total structural responses corresponding to a speci"ed load level and temperature distribution by a straight iteration process. During this process, strains are used to check whether a point on a section is in the elastic domain or not. A computer program was developed to apply this method for analyzing steel frames in "re. To examine the validity of the method and the program, comparisons are made between the program predictions and test results available for steel frames both at room temperature and under "re conditions, in most cases, a satisfactory agreement is obtained. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Steel frame; Non-linear analysis; Fire; Temperature; Inelastic zone; Finite element method
1. Introduction Because of the progressive degradation in the mechanical properties of steel with increasing temperatures, "re protection of the steelwork used in buildings is usually needed to meet the "re resistance requirements for building regulations. In order to * Corresponding author. E-mail address: [email protected] (J.-C. Zhao). 0379-7112/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 2 3 - 0
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Nomenclature A A A A(x) E E R E (x) R E @R F ,F ,F , F ,F f f f H I I I I(x) L l T t ,t
area of the cross section area of the elastic section core area of the softening parts of the section equivalent cross-sectional area modules of elasticity of steel at room temperature initial modules of elasticity of steel at temperature t. average modules of elasticity of steel at temperature t softening modules of elasticity of steel at temperature t
vertical or horizontal load proportional stress limit of steel at temperature t yield stress of steel at temperature t yield stress of steel at room temperature horizontal displacements of point A in Fig. 2 moment of inertia moment of inertia of the elastic section core moment of inertia of the softening parts of the section equivalent moment of inertia of the cross section beam span in Fig. 4 element length temperature of the steel temperature change at the upper and lower surface of the element section, respectively u, l displacements in the x and y directions u ,l ,h element nodal displacments at end 1 of the number u ,l ,h element nodal displacments at end 2 of the number V volume a coe$cient of thermal expansion of steel b reduction factor of the modules of elasticity of steel at temperature t e ,e ,e strain, thermal strain, stress-related strain F N e proportional strain limit of steel at temperature t p stress +, denotes a column vector [] denotes a matrix 12 denotes a row vector [K ], [K ] non-linear secant sti!ness matrices at room temperature analysis [K ] temperature-geometrical sti!ness matrix [K ] sti!ness matrix accounting for the e!ects of material non-linearity +P, vector of nodal forces +P , vector of nodal forces resulting from softening of an element +P , vector of nodal internal forces caused by thermal deformation 1D2 vector of nodal displacements
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243
make the "re protection design more rational and economical, it is necessary to have a sound understanding of the behavior of structures subject to a "re. For a long time in the past, this was based on standard "re test results on protected or unprotected specimens. Such method is time consuming and expensive as each test can only provide very limited amount of data, and in some cases, the scatter of results can be quite wide because of the inevitable variations in furnace characteristics and testing techniques. Moreover, the behavior of a furnace test specimen of small to medium size, subject to idealized load, boundary conditions and standard "re exposure, is not representative of the actual behavior of the structural member in a real "re. To overcome these drawbacks, a considerable amount of work has been done towards the development of alternative methods for predicting the behavior of building structures in "re, with an emphasis on the development of analytical methods based on computer simulations. The analytical method o!ers a cost-e!ective alternative to the traditional test method, and further more, it permits a more accurate prediction of the structural "re response by considering the signi"cance and severity of a real "re. It may therefore lead to a more rational and economical procedure with a more de"ned and uniform level of safety. This paper is mainly concerned with the analytical treatment of the structural response of steel frames at elevated temperatures. The behavior of steel structures in "re is very complicated because many factors are involved, such as the geometrical non-linearity caused by thermal de#ections, the complex material non-linearity resulting from the material softening under nonuniform temperature distribution and the redistribution of internal forces as a result of the thermal expansion and the formation of `inelastica zone. Recently, important achievements have been made in modeling the behavior of steel structures exposed to "re, a number of numerical methods based on "nite element technique have been proposed for "re resistance analysis of both 2D [1,2] and 3D [3,4] steel frames. Some methods can even permit the steel framed #oor systems to be analyzed [5]. The paper does not intend to summarize these current developments here, while it is worth mentioning that the Newton}Raphson method dealing with the incremental problems is widely adopted by most of researchers to calculate the non-linear structural response. In this paper, the direct iteration method is applied to predict the non-linear behavior of steel frames at a speci"ed load level and temperature distribution. The procedure can be repeated for increasing values of load or temperature, and thus can be used to calculate the whole structural response at room temperature or under "re conditions, using the secant sti!ness matrix. In addition to the e!ects of geometrical and material non-linearity, the presented method also permits an accurate consideration of the gradual penetration of inelastic zone in the restrained structural elements. These e!ects have been included by introducing additional sti!ness matrices and nodal force vectors in deriving the basic "nite element equations.
2. Mechanical properties of steel at high temperatures Proper models of stress}strain relationships at high temperatures are essential for an accurate prediction of the structural response under "re conditions. In the
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literature, a number of models exist based on "re test results under either transient or steady-state heating conditions. In this paper, a simpli"ed trilinear model based on the ECCS recommendations [6] is adopted, in which the creep strain is assumed to be implicitly included. At a speci"ed temperature state, this model of stress}strain curve depends on four material parameters, namely the initial modules of elasticity E , R proportional stress limit f , yield stress f and softening modules of elasticity E . It is @R considered that at high temperature T, the material is initially elastic with a modules of elasticity E , when the stress level equals f , softening of material occurs, the R stress}strain curve then follows another line with a slope E until the yielding stress @R f then assumes plastic #ow. Calculations of E , f , f , and E can be made by R @R following equations:
¹ f , f " 1# 767 ln(¹ /1750)
0(¹ )600C,
(1)
E "(1!17.2;10\¹#11.8;10\¹!34.5;10\¹ R #15.9;10\¹ )E , 0(¹ )6003C, (2) Q in which ¹ is the temperature of steel, f , E are yielding stress and modules of elasticity of steel at room temperature, respectively.
f "
f ,
¹ )2003C,
¹ !200 1! f , 200(¹ )3003C, 200 0.5f ¹ '3003C,
E "bE , @R R 0,
(3)
(4)
¹ (200 C, Q 1.76!1.04;10\¹ #2.13;10\¹ 200)¹ )600C, Q Q Q b" (5) !1.47;10\¹, Q 0, ¹ '6003C. The validity of this material model for use in "re resistance of steel frames is examined in another publication of the author, although the model shows somewhat conservative, especially for "re analysis of single elements (beams or columns) [7]. 3
3. Finite element analysis 3.1. Main assumptions The following assumptions are used to simplify the analysis: (1) steel begins to yield at the end of member and then propagates both across the section and along the length of the member;
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
(2) (3) (4) (5)
245
planes before deformation remain planes after deformation; no out of plane or torsion displacements occur; shear deformations are neglected; Temperature changes linearly across the section.
3.2. Basic theory A two-node beam}column element with six degrees of freedom is used for the "nite element analysis of 2D steel frames under "re actions. Under the assumption that residual stress is not considered, the total axial strain at any point of the element section can be expressed in terms of the thermal strain e and stress-related strain e as N follows: e"e #e . N
(6)
It should be noted here that the creep strain has been incorporated into the stress} strain relationships in a approximate way. According to Assumption 5, e , which is caused by temperature change t and t at the upper and lower surface of the element section, respectively, can be given by e "e #e ,
(7)
in which t #t a, e " 2 t !t ya, e " h
(8)
where h is the height of the cross section; y is the distance parallel to y direction from the point to the axis of symmetry of the cross section, and a is the coe$cient of thermal expansion of steel at high temperature. In this paper, a is assumed to be a constant, i.e. a"1.4;10\m/m 3C. The displacement functions of the element at high temperatures are assumed to be the same as at room temperature:
x x u" 1! u # u , l l
3x 2x 2x x v" 1! # v # x! # h l l l l #
3x 2x x x ! v # ! h . l l l l
(9)
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The only di!erence in Eq. (9) is that, the nodal displacements D" 1u , v , h , u , v ,h 2 include the displacements caused by thermal expansion. The strain}displacements relation for a beam}column element is written as e"u #v !yv .
(10)
Then the virtual strain within the element caused by nodal virtual displacements can be written as follows: de"du #v dv !ydv
(11)
According to the level of strain, the stress}strain relations can be expressed as follows:
p"
Ee , e )e , R N N f #bE (e!e !e ), e 'e , R N
(12)
where e is the proportional strain limit. Taking a steel frame at a speci"ed load and temperature level into consideration, the principle of virtual work results in the following equation:
1P2+dD,"
4
p de d<,
(13)
in which 1P2 is the vector of external loads, +dD, is the vector of nodal virtual displacements, and V is the volume of the element . For an element with biaxial symmetrical section, the element equilibrium equation can be written as +P,"[K #K !K #K ]+D,!+P ,#+P ,.
(14)
In Eq. (14), the secant sti!ness matrix consists of four distinct matrices. If an average elastic modules E (x), based on the average temperature over the element section, is R adopted for simpli"cation, the calculations of these matrices can be approximated by J J K " E (x)A(x)B2 B dx# E (x)I(x)B2 B dx, R R J J K " E (x)A(x)B2 B B dx#0.5 E (x)A(x)B2 B DB dx, R R (15) J K " E (x)A(x)B2 B dx, R J K "(1!b) f A (x)B2 B dx, in which A(x) and I(x) are the equivalent cross-sectional area and moment of inertia of the element section, respectively, with A(x)"A #bA , I(x)"I #bI , A is the area of elastic section core, A is the area of the softening parts of the section. I and
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247
I are the corresponding moment of inertia, and
1 1 B " ! , 0, 0, , 0, 0 , l l
B " 0,
6x 6x 4x 3x 6x 6x 3x 2x ! , 1! # , 0, ! , ! l l l l l l l l
B " 0,
12x 6 6x 4 6 6x 2 12x ! , ! , 0, ! , ! l l l l l l l l
,
.
The "rst two matrices, K and K , are the conventional secant sti!ness matrices for non-linear structural analysis at room temperature. K , which is the temperature geometrical sti!ness matrix, re#ects the e!ects of the thermal deformations of the element. For structural analysis at ambient temperature, K "0. K is the element sti!ness matrix accounting for the e!ects of material non-linearity. When only the geometrical non-linearity is considered, K "0. The remaining two terms, +P , and +P ,, can be calculated by the following equations: J a(t !t ) J +P ,"e E (x)A(x)B2 dx# E (x)I(x)B2 dx R R h (16) J A (x)B2 dx, +P ,"(1!b) f where +P , is the nodal internal force caused by thermal deformations of the element. When the temperature is uniformly distributed across the element (t "t ), only axial forces may be expected to be induced. Di!erent expansion will occur in non-uniformly heated cross sections, resulting in the development of bending moment as well as axial forces if such expansion cannot occur freely, this can be re#ected from Eq. (16). +P , is the nodal force resulting from the formation of softening part in the element, it also shows the e!ect of material non-linearity. The Gaussian integration method can be used to evaluate these matrices and force vectors. Once the structural equilibrium equations are assembled, the non-linear structural responses under a speci"ed load level and temperature distribution can be obtained by the direct iteration method, in which the secant sti!ness matrices, +P , and +P , are continuously updated in function of the former calculated deformations until the solution converges.
3.3. Extension of softening zone One main feature of the present method is the consideration of the progressive extension of the softening zones both over the section and along the member. Evaluation of the parameters A , A , I , I , and the softening length of the member is essential for the determination of the terms in Eq. (14). In this paper, each element is
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Fig. 1. The beam}column element.
divided into three segments, namely [0, x ], [x , x ] and [x , 1] (shown in Fig. 1). Based on Assumption 1, segments [0, x ] and [x , l] are partially in softening state, while segment [x , x ] is in elastic domain. Determination of x and x is based on the analysis of strain at the upper and lower edge of the cross section. When "e!e "*e at a point, this point is considered in softening state. For a partially inelastic section, the inelastic height can be evaluated by a similar strain analysis over the section to determine the values of A , A , I and I . A computer program NASFAF (Non-linear Analysis of Steel Frames Against Fire) is developed based on the above "nite element formulation. Taking the element temperature distribution and the stress}strain relations of steel at elevated temperatures as input data, the program predicts the responses of steel frames under any combination of loads and temperatures.
4. Comparisons with test results In order to check the validity and to show the potential of both the presented method and the program (NASFAF), a number of steel frame examples, previously tested by other researchers at either ambient or elevated temperatures, are analyzed in the following section and comparisons between the predictions and test results are presented. 4.1. Frame analysis at room temperature A one-bay, single-storey unbraced portal frame, with "xed bases as shown in Fig. 2, was tested by Arnold et al. [8]. The beam was a 10 I 25.4 stout shape of ASTM-A36 steel and the columns were 5 WF 18.50 stout shape of ASTM-A41 steel. Tension tests were also performed to determine the stress}strain characteristics of the A36 and A41
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
249
Fig. 2. Test frame and loads.
Fig. 3. Comparison with Ref. [8].
steels. Adopting an average set for material properties: yielding stress f , yielding strain e and modules of elasticity E as direct input data for the program NASFAF, the predicted load}de#ection relationship and the ultimate load were obtained. Fig. 3 shows the comparison of the horizontal displacement at point A (H ) under S di!erent load level between the computer predictions and the reported test results. From Fig. 3, we can see a satisfactory agreement between the two sets of results. Furthermore, the predicted ultimate load is close to the test value. This example gives an idea about the ability of the method to deal with the structural behavior at room temperature.
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J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
It is worth mentioning that in the present example, according to the calculation results, the "rst yielding occurred at the top of the right column (point A) when F reached a value of about 30 kN, followed by sections at the bottom of the right column (point B) and under the left beam load (point C). Once yield occurred at the bottom of the left column (point D), when F is about 63 kN, the lateral de#ection increases rapidly until the maximum load is reached. These results are in agreement with the analysis and test results provided by Arnold et al., despite some di!erence that may exist because of the use of the `plastic hingea theory in their analysis. In addition, the calculations also show that when F reaches about 65 kN, the upper and lower surface of the cross-section at point A are both in yielding state, the yielding height on the top surface of point A has been more than 40 mm, while the yielding length down the right column reaches about one-third of its length. 4.2. Analysis at elevated temperatures A series of tests on plane steel frames at elevated temperatures was performed in Germany and the test results were provided [9]. One of the tested frames (ZSR1) is shown in Fig. 4. The span L was 1200 mm, column height was 1170 mm, yielding stress and modules of elasticity were 355 N/mm and 210 kN/mm at room temperature. One bay of the frame was uniformly heated at a constant rate by electrical elements and the remaining two members were kept at room temperature. Comparison between the predicted de#ections and test results illustrated in Fig. 5 shows a satisfactory agreement. All the other tested frames, including series EHR, EGR and ZSR, were analyzed using the present method. In Fig. 6, we compare the predicted `critical temperaturesa with their measured results for all the frames, in each case, the error does not exceed 15 %. The `critical temperaturea is de"ned as the maximum temperature at which the deformations of the frame increased in an uncontrollable fashion. In the analysis, the `critical temperaturea is assumed to be the last converged solution from the program. In 1995, a series of "re tests on steel frames under di!erent load level and heating process was conducted by the author in China State Key Laboratory for Disaster
Fig. 4. Test frame ZSRI.
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
251
Fig. 5. Comparison between predictions and test results.
Fig. 6. Comparison of critical temperatures.
Prevention & Reduction in Civil Engineering [10]. All the test frames were twodimensional one-storey and one-bay unbraced portal frames. Heating was conducted by a specially constructed gas furnace. During each test, the applied loads on the frame were maintained constant while the frame temperature increase. The changes in temperature and de#ection of the frame were constantly recorded. Beam and columns of same cross section } H : 10, were made of Grade A3 steel, one of the commonly used structural steels in China. The typical frame height was 1450 mm, measured from the base to the centerline of the beam. The center-to-center span length was 1500 mm. Standard tension tests were also performed on specimens cut directly from the web of the cross section used to fabricate the test frames. The average test results were: f "293.5 N/mm, and E "2.0;10 N/mm
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J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
Fig. 7. Test frame 1 and thermocouples.
Fig. 8. Temperature}time curves of test 1.
In order to get uniform temperature distribution within steel members, di!erent from other tests (tests 2}4), test 1 was conducted in a closed furnace. Only four thermocouples were placed, one in the furnace near the top of the left column to measure the furnace temperature, and one on the web of each member (see Fig. 7). The measured temperature}time curves are shown in Fig. 8. Adopting these measured temperature values directly as input data, lateral de#ections are calculated using the program NASFAF. A comparison between the measured and calculated de#ections is shown in Fig. 9. Test 3 was conducted in an open furnace, the location of thermocouples is shown in Fig. 10. Temperature}time curves at the measured points are shown in Fig. 11. Comparison of deformations is shown in Fig. 12.
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253
Fig. 9. Comparison for test 1.
Fig. 10. Test frame 3 and thermocouples.
5. Conclusions Based on a "nite element formulation, a direct iteration method for non-linear analysis of two-dimensional steel frame at elevated temperatures is presented. The method includes the e!ects of geometric non-linearity, temperature-dependent material non-linearity and temperature variation across each element. Several experimental results are used to validate the approach and the numerical model. The comparisons with the experimental results con"rm that the model is able to predict with good manner the behavior of 2D steel frames at ambient temperature and during "re.
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J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
Fig. 11. Temperature}time curves of test 3.
Fig. 12. Comparison for test 3.
The model gives detailed description of the location of the plastic zone (local failure) within the compressed steel members and can detect the formation of a mechanism within the structure (global failure) during the "re. When the "re loading and temperature distribution are known, the advantage of the numerical model consists of its ability to solve the non-linear problem using realistic material properties and boundary conditions at any temperature with one direct iteration. It provides information about the structure stability at any instant during the "re, with minimum computational cost. Furthermore, due to its capacity to include large displacement
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
255
e!ects and any type of boundary conditions, the numerical model can be used as complementary tool to check the design of 2D steel frames under "re conditions. For a given temperature}time curve, it provides more accurate information about the internal stresses in complex structure, specially when real boundary conditions play a determinant role in the stability of the structure during "re.
References [1] Saab HA, Nethercot DA. Modeling steel frame behaviour under "re conditions. Engng Struct 1991;13:371}82. [2] Crozier DA, Wong MB. Elastoplatic analysis of steel frames at elevated temperatures. Proceedings of 13th Australian Conference on the Mechanics of Structures and Materials, University of Wollongong. [3] Wang YC, Moore DB. Steel frames in "re: analysis. Engng Struct 1995;17:462}72. [4] Najjar SR, Burgess IW. A non-linear analysis for three-dimensional steel frame in "re condition. Engng Struct 1996;18:77}89. [5] David C, Jeanes DE. Developing design concepts for structural endurance using computer models. Proceedings of the International. Conference on Design of Structures against Fire, Aston University. [6] ECCS-T3. European recommendations for the "re safety of steel structure. Amsterdam: Elsevier, 1983. [7] Zhao JC, Shen ZY. Material property model in the analysis of "re resistance of steel structure. Indust Construct (1995);26(9):3}6 (in Chinese). [8] Arnold P, Lu LW et al., Strength and behavior of an elastic hybrid frame. ASCE J. Struct Div (1968);94(ST1):2443}66. [9] Rubert A, Schaumann P. Tragverhalten stahlerner rahmensysteme bei brandbeanspruchung Stahlbau, 1985;54:280}7. [10] Zhao JC. Fire resistance of steel framed structures. Ph.D. Thesis, Tongji University (in Chinese), 1995.
Application of the direct iteration method for non-linear analysis of steel frames in "re Jin-Cheng Zhao* College of Civil Engineering & Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People's Republic of China Received 22 October 1998; received in revised form 15 March 2000; accepted 20 April 2000
Abstract Based on "nite element method, a direct iteration method capable of predicting the nonlinear behavior of steel frames at elevated temperatures is proposed, in which the second-order e!ects of large de#ections, the progressive softening of materials with temperature rise, gradual penetration of inelastic zone both over the section and along the member, non-uniform distribution of temperature within steel members are all taken into consideration. The secant sti!ness matrix that relates the total element displacements to the corresponding total forces is used in deriving the basic "nite element equations. Di!erent from the commonly used Newton}Raphson method in which the deformations are calculated incrementally, the direct iteration method used in this paper allows the calculation of the total structural responses corresponding to a speci"ed load level and temperature distribution by a straight iteration process. During this process, strains are used to check whether a point on a section is in the elastic domain or not. A computer program was developed to apply this method for analyzing steel frames in "re. To examine the validity of the method and the program, comparisons are made between the program predictions and test results available for steel frames both at room temperature and under "re conditions, in most cases, a satisfactory agreement is obtained. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Steel frame; Non-linear analysis; Fire; Temperature; Inelastic zone; Finite element method
1. Introduction Because of the progressive degradation in the mechanical properties of steel with increasing temperatures, "re protection of the steelwork used in buildings is usually needed to meet the "re resistance requirements for building regulations. In order to * Corresponding author. E-mail address: [email protected] (J.-C. Zhao). 0379-7112/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 3 7 9 - 7 1 1 2 ( 0 0 ) 0 0 0 2 3 - 0
242
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
Nomenclature A A A A(x) E E R E (x) R E @R F ,F ,F , F ,F f f f H I I I I(x) L l T t ,t
area of the cross section area of the elastic section core area of the softening parts of the section equivalent cross-sectional area modules of elasticity of steel at room temperature initial modules of elasticity of steel at temperature t. average modules of elasticity of steel at temperature t softening modules of elasticity of steel at temperature t
vertical or horizontal load proportional stress limit of steel at temperature t yield stress of steel at temperature t yield stress of steel at room temperature horizontal displacements of point A in Fig. 2 moment of inertia moment of inertia of the elastic section core moment of inertia of the softening parts of the section equivalent moment of inertia of the cross section beam span in Fig. 4 element length temperature of the steel temperature change at the upper and lower surface of the element section, respectively u, l displacements in the x and y directions u ,l ,h element nodal displacments at end 1 of the number u ,l ,h element nodal displacments at end 2 of the number V volume a coe$cient of thermal expansion of steel b reduction factor of the modules of elasticity of steel at temperature t e ,e ,e strain, thermal strain, stress-related strain F N e proportional strain limit of steel at temperature t p stress +, denotes a column vector [] denotes a matrix 12 denotes a row vector [K ], [K ] non-linear secant sti!ness matrices at room temperature analysis [K ] temperature-geometrical sti!ness matrix [K ] sti!ness matrix accounting for the e!ects of material non-linearity +P, vector of nodal forces +P , vector of nodal forces resulting from softening of an element +P , vector of nodal internal forces caused by thermal deformation 1D2 vector of nodal displacements
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
243
make the "re protection design more rational and economical, it is necessary to have a sound understanding of the behavior of structures subject to a "re. For a long time in the past, this was based on standard "re test results on protected or unprotected specimens. Such method is time consuming and expensive as each test can only provide very limited amount of data, and in some cases, the scatter of results can be quite wide because of the inevitable variations in furnace characteristics and testing techniques. Moreover, the behavior of a furnace test specimen of small to medium size, subject to idealized load, boundary conditions and standard "re exposure, is not representative of the actual behavior of the structural member in a real "re. To overcome these drawbacks, a considerable amount of work has been done towards the development of alternative methods for predicting the behavior of building structures in "re, with an emphasis on the development of analytical methods based on computer simulations. The analytical method o!ers a cost-e!ective alternative to the traditional test method, and further more, it permits a more accurate prediction of the structural "re response by considering the signi"cance and severity of a real "re. It may therefore lead to a more rational and economical procedure with a more de"ned and uniform level of safety. This paper is mainly concerned with the analytical treatment of the structural response of steel frames at elevated temperatures. The behavior of steel structures in "re is very complicated because many factors are involved, such as the geometrical non-linearity caused by thermal de#ections, the complex material non-linearity resulting from the material softening under nonuniform temperature distribution and the redistribution of internal forces as a result of the thermal expansion and the formation of `inelastica zone. Recently, important achievements have been made in modeling the behavior of steel structures exposed to "re, a number of numerical methods based on "nite element technique have been proposed for "re resistance analysis of both 2D [1,2] and 3D [3,4] steel frames. Some methods can even permit the steel framed #oor systems to be analyzed [5]. The paper does not intend to summarize these current developments here, while it is worth mentioning that the Newton}Raphson method dealing with the incremental problems is widely adopted by most of researchers to calculate the non-linear structural response. In this paper, the direct iteration method is applied to predict the non-linear behavior of steel frames at a speci"ed load level and temperature distribution. The procedure can be repeated for increasing values of load or temperature, and thus can be used to calculate the whole structural response at room temperature or under "re conditions, using the secant sti!ness matrix. In addition to the e!ects of geometrical and material non-linearity, the presented method also permits an accurate consideration of the gradual penetration of inelastic zone in the restrained structural elements. These e!ects have been included by introducing additional sti!ness matrices and nodal force vectors in deriving the basic "nite element equations.
2. Mechanical properties of steel at high temperatures Proper models of stress}strain relationships at high temperatures are essential for an accurate prediction of the structural response under "re conditions. In the
244
J.-C. Zhao / Fire Safety Journal 35 (2000) 241}255
literature, a number of models exist based on "re test results under either transient or steady-state heating conditions. In this paper, a simpli"ed trilinear model based on the ECCS recommendations [6] is adopted, in which the creep strain is assumed to be implicitly included. At a speci"ed temperature state, this model of stress}strain curve depends on four material parameters, namely the initial modules of elasticity E , R proportional stress limit f , yield stress f and softening modules of elasticity E . It is @R considered that at high temperature T, the material is initially elastic with a modules of elasticity E , when the stress level equals f , softening of material occurs, the R stress}strain curve then follows another line with a slope E until the yielding stress @R f then assumes plastic #ow. Calculations of E , f , f , and E can be made by R @R following equations:
¹ f , f " 1# 767 ln(¹ /1750)
0(¹ )600C,
(1)
E "(1!17.2;10\¹#11.8;10\¹!34.5;10\¹ R #15.9;10\¹ )E , 0(¹ )6003C, (2) Q in which ¹ is the temperature of steel, f , E are yielding stress and modules of elasticity of steel at room temperature, respectively.
f "
f ,
¹ )2003C,
¹ !200 1! f , 200(¹ )3003C, 200 0.5f ¹ '3003C,
E "bE , @R R 0,
(3)
(4)
¹ (200 C, Q 1.76!1.04;10\¹ #2.13;10\¹ 200)¹ )600C, Q Q Q b" (5) !1.47;10\¹, Q 0, ¹ '6003C. The validity of this material model for use in "re resistance of steel frames is examined in another publication of the author, although the model shows somewhat conservative, especially for "re analysis of single elements (beams or columns) [7]. 3
3. Finite element analysis 3.1. Main assumptions The following assumptions are used to simplify the analysis: (1) steel begins to yield at the end of member and then propagates both across the section and along the length of the member;
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(2) (3) (4) (5)
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planes before deformation remain planes after deformation; no out of plane or torsion displacements occur; shear deformations are neglected; Temperature changes linearly across the section.
3.2. Basic theory A two-node beam}column element with six degrees of freedom is used for the "nite element analysis of 2D steel frames under "re actions. Under the assumption that residual stress is not considered, the total axial strain at any point of the element section can be expressed in terms of the thermal strain e and stress-related strain e as N follows: e"e #e . N
(6)
It should be noted here that the creep strain has been incorporated into the stress} strain relationships in a approximate way. According to Assumption 5, e , which is caused by temperature change t and t at the upper and lower surface of the element section, respectively, can be given by e "e #e ,
(7)
in which t #t a, e " 2 t !t ya, e " h
(8)
where h is the height of the cross section; y is the distance parallel to y direction from the point to the axis of symmetry of the cross section, and a is the coe$cient of thermal expansion of steel at high temperature. In this paper, a is assumed to be a constant, i.e. a"1.4;10\m/m 3C. The displacement functions of the element at high temperatures are assumed to be the same as at room temperature:
x x u" 1! u # u , l l
3x 2x 2x x v" 1! # v # x! # h l l l l #
3x 2x x x ! v # ! h . l l l l
(9)
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The only di!erence in Eq. (9) is that, the nodal displacements D" 1u , v , h , u , v ,h 2 include the displacements caused by thermal expansion. The strain}displacements relation for a beam}column element is written as e"u #v !yv .
(10)
Then the virtual strain within the element caused by nodal virtual displacements can be written as follows: de"du #v dv !ydv
(11)
According to the level of strain, the stress}strain relations can be expressed as follows:
p"
Ee , e )e , R N N f #bE (e!e !e ), e 'e , R N
(12)
where e is the proportional strain limit. Taking a steel frame at a speci"ed load and temperature level into consideration, the principle of virtual work results in the following equation:
1P2+dD,"
4
p de d<,
(13)
in which 1P2 is the vector of external loads, +dD, is the vector of nodal virtual displacements, and V is the volume of the element . For an element with biaxial symmetrical section, the element equilibrium equation can be written as +P,"[K #K !K #K ]+D,!+P ,#+P ,.
(14)
In Eq. (14), the secant sti!ness matrix consists of four distinct matrices. If an average elastic modules E (x), based on the average temperature over the element section, is R adopted for simpli"cation, the calculations of these matrices can be approximated by J J K " E (x)A(x)B2 B dx# E (x)I(x)B2 B dx, R R J J K " E (x)A(x)B2 B B dx#0.5 E (x)A(x)B2 B DB dx, R R (15) J K " E (x)A(x)B2 B dx, R J K "(1!b) f A (x)B2 B dx, in which A(x) and I(x) are the equivalent cross-sectional area and moment of inertia of the element section, respectively, with A(x)"A #bA , I(x)"I #bI , A is the area of elastic section core, A is the area of the softening parts of the section. I and
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I are the corresponding moment of inertia, and
1 1 B " ! , 0, 0, , 0, 0 , l l
B " 0,
6x 6x 4x 3x 6x 6x 3x 2x ! , 1! # , 0, ! , ! l l l l l l l l
B " 0,
12x 6 6x 4 6 6x 2 12x ! , ! , 0, ! , ! l l l l l l l l
,
.
The "rst two matrices, K and K , are the conventional secant sti!ness matrices for non-linear structural analysis at room temperature. K , which is the temperature geometrical sti!ness matrix, re#ects the e!ects of the thermal deformations of the element. For structural analysis at ambient temperature, K "0. K is the element sti!ness matrix accounting for the e!ects of material non-linearity. When only the geometrical non-linearity is considered, K "0. The remaining two terms, +P , and +P ,, can be calculated by the following equations: J a(t !t ) J +P ,"e E (x)A(x)B2 dx# E (x)I(x)B2 dx R R h (16) J A (x)B2 dx, +P ,"(1!b) f where +P , is the nodal internal force caused by thermal deformations of the element. When the temperature is uniformly distributed across the element (t "t ), only axial forces may be expected to be induced. Di!erent expansion will occur in non-uniformly heated cross sections, resulting in the development of bending moment as well as axial forces if such expansion cannot occur freely, this can be re#ected from Eq. (16). +P , is the nodal force resulting from the formation of softening part in the element, it also shows the e!ect of material non-linearity. The Gaussian integration method can be used to evaluate these matrices and force vectors. Once the structural equilibrium equations are assembled, the non-linear structural responses under a speci"ed load level and temperature distribution can be obtained by the direct iteration method, in which the secant sti!ness matrices, +P , and +P , are continuously updated in function of the former calculated deformations until the solution converges.
3.3. Extension of softening zone One main feature of the present method is the consideration of the progressive extension of the softening zones both over the section and along the member. Evaluation of the parameters A , A , I , I , and the softening length of the member is essential for the determination of the terms in Eq. (14). In this paper, each element is
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Fig. 1. The beam}column element.
divided into three segments, namely [0, x ], [x , x ] and [x , 1] (shown in Fig. 1). Based on Assumption 1, segments [0, x ] and [x , l] are partially in softening state, while segment [x , x ] is in elastic domain. Determination of x and x is based on the analysis of strain at the upper and lower edge of the cross section. When "e!e "*e at a point, this point is considered in softening state. For a partially inelastic section, the inelastic height can be evaluated by a similar strain analysis over the section to determine the values of A , A , I and I . A computer program NASFAF (Non-linear Analysis of Steel Frames Against Fire) is developed based on the above "nite element formulation. Taking the element temperature distribution and the stress}strain relations of steel at elevated temperatures as input data, the program predicts the responses of steel frames under any combination of loads and temperatures.
4. Comparisons with test results In order to check the validity and to show the potential of both the presented method and the program (NASFAF), a number of steel frame examples, previously tested by other researchers at either ambient or elevated temperatures, are analyzed in the following section and comparisons between the predictions and test results are presented. 4.1. Frame analysis at room temperature A one-bay, single-storey unbraced portal frame, with "xed bases as shown in Fig. 2, was tested by Arnold et al. [8]. The beam was a 10 I 25.4 stout shape of ASTM-A36 steel and the columns were 5 WF 18.50 stout shape of ASTM-A41 steel. Tension tests were also performed to determine the stress}strain characteristics of the A36 and A41
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Fig. 2. Test frame and loads.
Fig. 3. Comparison with Ref. [8].
steels. Adopting an average set for material properties: yielding stress f , yielding strain e and modules of elasticity E as direct input data for the program NASFAF, the predicted load}de#ection relationship and the ultimate load were obtained. Fig. 3 shows the comparison of the horizontal displacement at point A (H ) under S di!erent load level between the computer predictions and the reported test results. From Fig. 3, we can see a satisfactory agreement between the two sets of results. Furthermore, the predicted ultimate load is close to the test value. This example gives an idea about the ability of the method to deal with the structural behavior at room temperature.
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It is worth mentioning that in the present example, according to the calculation results, the "rst yielding occurred at the top of the right column (point A) when F reached a value of about 30 kN, followed by sections at the bottom of the right column (point B) and under the left beam load (point C). Once yield occurred at the bottom of the left column (point D), when F is about 63 kN, the lateral de#ection increases rapidly until the maximum load is reached. These results are in agreement with the analysis and test results provided by Arnold et al., despite some di!erence that may exist because of the use of the `plastic hingea theory in their analysis. In addition, the calculations also show that when F reaches about 65 kN, the upper and lower surface of the cross-section at point A are both in yielding state, the yielding height on the top surface of point A has been more than 40 mm, while the yielding length down the right column reaches about one-third of its length. 4.2. Analysis at elevated temperatures A series of tests on plane steel frames at elevated temperatures was performed in Germany and the test results were provided [9]. One of the tested frames (ZSR1) is shown in Fig. 4. The span L was 1200 mm, column height was 1170 mm, yielding stress and modules of elasticity were 355 N/mm and 210 kN/mm at room temperature. One bay of the frame was uniformly heated at a constant rate by electrical elements and the remaining two members were kept at room temperature. Comparison between the predicted de#ections and test results illustrated in Fig. 5 shows a satisfactory agreement. All the other tested frames, including series EHR, EGR and ZSR, were analyzed using the present method. In Fig. 6, we compare the predicted `critical temperaturesa with their measured results for all the frames, in each case, the error does not exceed 15 %. The `critical temperaturea is de"ned as the maximum temperature at which the deformations of the frame increased in an uncontrollable fashion. In the analysis, the `critical temperaturea is assumed to be the last converged solution from the program. In 1995, a series of "re tests on steel frames under di!erent load level and heating process was conducted by the author in China State Key Laboratory for Disaster
Fig. 4. Test frame ZSRI.
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Fig. 5. Comparison between predictions and test results.
Fig. 6. Comparison of critical temperatures.
Prevention & Reduction in Civil Engineering [10]. All the test frames were twodimensional one-storey and one-bay unbraced portal frames. Heating was conducted by a specially constructed gas furnace. During each test, the applied loads on the frame were maintained constant while the frame temperature increase. The changes in temperature and de#ection of the frame were constantly recorded. Beam and columns of same cross section } H : 10, were made of Grade A3 steel, one of the commonly used structural steels in China. The typical frame height was 1450 mm, measured from the base to the centerline of the beam. The center-to-center span length was 1500 mm. Standard tension tests were also performed on specimens cut directly from the web of the cross section used to fabricate the test frames. The average test results were: f "293.5 N/mm, and E "2.0;10 N/mm
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Fig. 7. Test frame 1 and thermocouples.
Fig. 8. Temperature}time curves of test 1.
In order to get uniform temperature distribution within steel members, di!erent from other tests (tests 2}4), test 1 was conducted in a closed furnace. Only four thermocouples were placed, one in the furnace near the top of the left column to measure the furnace temperature, and one on the web of each member (see Fig. 7). The measured temperature}time curves are shown in Fig. 8. Adopting these measured temperature values directly as input data, lateral de#ections are calculated using the program NASFAF. A comparison between the measured and calculated de#ections is shown in Fig. 9. Test 3 was conducted in an open furnace, the location of thermocouples is shown in Fig. 10. Temperature}time curves at the measured points are shown in Fig. 11. Comparison of deformations is shown in Fig. 12.
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Fig. 9. Comparison for test 1.
Fig. 10. Test frame 3 and thermocouples.
5. Conclusions Based on a "nite element formulation, a direct iteration method for non-linear analysis of two-dimensional steel frame at elevated temperatures is presented. The method includes the e!ects of geometric non-linearity, temperature-dependent material non-linearity and temperature variation across each element. Several experimental results are used to validate the approach and the numerical model. The comparisons with the experimental results con"rm that the model is able to predict with good manner the behavior of 2D steel frames at ambient temperature and during "re.
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Fig. 11. Temperature}time curves of test 3.
Fig. 12. Comparison for test 3.
The model gives detailed description of the location of the plastic zone (local failure) within the compressed steel members and can detect the formation of a mechanism within the structure (global failure) during the "re. When the "re loading and temperature distribution are known, the advantage of the numerical model consists of its ability to solve the non-linear problem using realistic material properties and boundary conditions at any temperature with one direct iteration. It provides information about the structure stability at any instant during the "re, with minimum computational cost. Furthermore, due to its capacity to include large displacement
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e!ects and any type of boundary conditions, the numerical model can be used as complementary tool to check the design of 2D steel frames under "re conditions. For a given temperature}time curve, it provides more accurate information about the internal stresses in complex structure, specially when real boundary conditions play a determinant role in the stability of the structure during "re.
References [1] Saab HA, Nethercot DA. Modeling steel frame behaviour under "re conditions. Engng Struct 1991;13:371}82. [2] Crozier DA, Wong MB. Elastoplatic analysis of steel frames at elevated temperatures. Proceedings of 13th Australian Conference on the Mechanics of Structures and Materials, University of Wollongong. [3] Wang YC, Moore DB. Steel frames in "re: analysis. Engng Struct 1995;17:462}72. [4] Najjar SR, Burgess IW. A non-linear analysis for three-dimensional steel frame in "re condition. Engng Struct 1996;18:77}89. [5] David C, Jeanes DE. Developing design concepts for structural endurance using computer models. Proceedings of the International. Conference on Design of Structures against Fire, Aston University. [6] ECCS-T3. European recommendations for the "re safety of steel structure. Amsterdam: Elsevier, 1983. [7] Zhao JC, Shen ZY. Material property model in the analysis of "re resistance of steel structure. Indust Construct (1995);26(9):3}6 (in Chinese). [8] Arnold P, Lu LW et al., Strength and behavior of an elastic hybrid frame. ASCE J. Struct Div (1968);94(ST1):2443}66. [9] Rubert A, Schaumann P. Tragverhalten stahlerner rahmensysteme bei brandbeanspruchung Stahlbau, 1985;54:280}7. [10] Zhao JC. Fire resistance of steel framed structures. Ph.D. Thesis, Tongji University (in Chinese), 1995.