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Simple Approach for Calculating Inelastic Deflections of Simply

Supported Steel Beams under Fire

Authors:

Shahid Iqbal, Michigan State University,East Lansing, MI 48824-1126, iqbalsha@msu.edu

Mahmud Dwaikat, Michigan State University,East Lansing, MI 48824-1126,dawaikat1@egr.msu.edu

Ronald S. Harichandran, Michigan State University, Phone: 517-355-5107, Fax: 517-432-1827

East Lansing, MI 48824-1126, harichan@egr.msu.edu

ABSTRACT

A simple approach is proposed for predicting inelastic deflections of simply supported steel

beams subjected to fire conditions. In this method, the same equations used for calculating elasticdeflections of beams at room temperature are used with an equivalent flexural rigidity.

Simplified equations for the equivalent flexural rigidity are developed using moment-curvature

relationships of steel beams. The effect of creep on deflection, which is significant under fire, isimplicitly accounted for in the equivalent flexural rigidity equations. The simplified method is

validated with deflections predicted using the ANSYS computer program for a range of load

ratios and heating rates. The proposed approach is simple enough to use in any fire-resistantdesign specification.

INTRODUCTION

Over the last decade, performance-based codes which allow more rational engineering

approaches for the fire design of steel members have been promoted. Consequently, the AISCSteel Construction Manual [AISC, 2005] now allows steel beams to be designed against fire

using room temperature design specifications and reduced material properties. The AISC Manual

specifies that for structural elements, the governing limit state is loss of load-bearing capacity.

However, the AISC Manual also specifies that excessive deformations are not acceptable if theydamage the slabs and walls of the compartment to the extent that these components cannot

control the horizontal and vertical spread of fire. Additionally, deformations need to be limited

because long beams may fail due to an abrupt increase of their deflection, despite the fact thatthey may have not reached their ultimate load bearing capacity [Skowronski, 1990].

Since design under fire may be governed either by the strength or deflection limit states, both

should be considered. While most design specifications have simple design equations for thestrength limit state of steel beams, no simple design equations are available for estimating

deflections of steel beams exposed to fire. Under fire, the stiffness of steel members reduces

significantly, deflections are likely to be in the inelastic regime, and equations available forestimating the elastic deflections are not applicable. While the deflections of steel beams exposed

to fire can be estimated using finite element analyses, it is desirable to have a simplified method

for use in routine fire resistant design.

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APPROACH FOR DEVELOPING SIMPLIFIED METHOD

Design specifications such as the AISC Manual allow steel beams exposed to fire to be designed

using room temperature design specifications with reduced material properties. According to this

concept, under fire conditions, the elastic deflections of steel beamscan be calculated using room

temperature deflection equations and reduced material properties. For example, the deflection ofa uniformly loaded simply supported beam can be calculated as:

IE

wL

Ts,

4

384

5= (1)

where L = length of the beam, w = load per unit length, I = moment of inertia of the cross-

section, Es,T = reduced elastic modulus calculated as kEEs,20C, Es,20C= elastic modulus used atroom temperature, and kE = stiffness reduction factor that accounts for reduction in stiffness of

steel at elevated temperatures.

At elevated temperatures, the stiffness of the steel beams degrades rapidly, the deflections arelikely to be beyond the elastic range, and (1) cannot be used in most cases. However, in the

inelastic range, (1) may be used for calculating deflections of steel beams by using an equivalentflexural rigidity,(Es,TI)eq, instead ofEs,TI. This concept is similar to the one used to calculate the

deflections of reinforced concrete beams by using an equivalent moment of inertia that accountsfor cracking. Therefore, it is proposed that in the inelastic range the deflection of a uniformly

loaded simply supported beam be calculated as:

eqTs IE

wL

)(384

5

,

4

= (2)

The moment curvature relationship of a typical steel section at any steel temperature, T, is

shown in Figure 1, whereinMis the applied moment, (Es,TI)1 is the initial flexural rigidity, and

(Es,TI)2 is the minimum secant flexural rigidity at the ultimate condition:

u

TuTs

MIE

,2, )( = (3)

where u = ultimate curvature, Mu,T= ZxFy,T = ultimate moment capacity, and My,T = SxFy,T =

yield moment capacity. Zx and Sx are the plastic and elastic section modulus, respectively, andFy,T = kyFy,20C = yield strength at steel temperature, T,Fy,20C = nominal yield strength at room

temperature, and ky = yield strength reduction factor which accounts for reduction in steel

strength at elevated temperatures.

The equivalent flexural rigidity, , will lie between (Es,TI)1 and (Es,TI)2 and can be

obtained from the deflection equations applicable in the elastic regime. For example, for a

uniformly loaded, simply supported beam, using (2):

eqTs IE )( ,

T

eqTs wLIE

=3845)(

4

, (4)

Using (4), the relationship between (Es,TI)eq and applied moment M can be obtained for

different values of applied load, w, provided the deflection, T, of the beam is known at

temperature T. For developing this relationship, the deflections of beams were obtained using theANSYS finite element software which is capable of handling geometric and material non-

linearity as well as thermal analysis [ANSYS Inc., 2007].

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IEkIE

CsETs o20,1,)( =

FIGURE 1 - TYPICAL MOMENT-CURVATURE DIAGRAM FOR A STEEL SECTION

The ANSYS finite element model comprised of two sub-models, namely the thermal model

and the structural model. The thermal model provided the temperature distribution in the steelmember which was applied as a thermal-body-load in the structural model of the steel beam. For

thermal analyses, two types of elements, namely PLANE55 and SURF151 were used. The cross-

section of the steel beam was meshed with PLANE55 elements and heat transferred throughconvection and radiation was applied on the exposed boundaries of the section using the

SURF151 element. A convective heat transfer coefficient ofc = 25 W/(m2.C) and a Stefan

Boltzmann radiation constant of= 5.6710-8

W/(m2

K4) were used. For structural analysis, the

beam was discretized using 90 BEAM189 elements which can account for material and

geometrical nonlinearities. Stress-strain curves developed by Poh (2001) and shown in Figure 2

were used in the ANSYS finite element model.Test data reported by Wainman [1987] was used to validate the structural finite element

model in ANSYS. The tested beam was a simply supported and uniformly loaded

UB35617167 section. Figure 3, which shows the deflections predicted by ANSYS and thoseobtained from tests, indicates that ANSYS predicts the mid-span deflection of the beam verywell.

The equivalent flexural rigidity versus applied moment curves were obtained for several

representative steel sections using the ANSYS model described above and (4). To ensure that theexpression for the equivalent flexural rigidity being developed is applicable for a range of steel

temperatures, the equivalent flexural rigidity curves were calculated at steel temperatures ranging

from 20C to 800C. The equivalent flexural rigidity curves will vary for different steel sectionsbecause each beam has different (Es,TI)1 and (Es,TI)2. Therefore, the equivalent flexural rigidityvs. applied moment curves were normalized using:

2,1,

2,,

)()(

)()(

IEIE

IEIE

TsTs

TseqTs

= (6)

and

TyTu

Ty

MM

MM

,,

,

= (7)

where = normalized flexural rigidity, and = normalized inelastic moment. The normalized

curves are shown in Figure 4.

u

TuTs

MIE

,2, )( =

u,T

y,T

uCurvature,

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25 30

Cy

Ts

F o20,

,

( )CsCy

Ts

EF oo20,20,

,

200

20 C

400 C

600 C

800 C

0

0.20.4

0.6

0.8

1

0 200 400 600 800 1000

Temperature, C

CsTsEE o

20,,/

CyTyFF o

20,,/

FIGURE 2 - MECHANICAL PROPERTIES FOR STRUCTURAL STEEL AT ELEVATED TEMPERATURE

0

200

400

600

800

0 5 10 15 20 25 30

Time, min

SteelTemperature,

C

0

20

40

60

80

100

120

140

160

Mid-spanDeflection,mm

Model

Test

Temp. of Top Flange

L =4500

Load = 35% of R.T ultimate capacity

Tem p. of Bottom Flange

FIGURE 3 - COMPARISON OF DEFLECTIONS PREDICTED BY ANSYS WITH TEST DATA

Through curve fitting, the following relationship was developed between the normalizedflexural rigidity and the normalized inelastic moment:

24.11

1

+

= (8)

Equation (8) is also shown in Figure 4. From (6), (7) and (8), the equivalent flexural rigidity maybe expressed as:

2

2,1,2,,

4.11

)()()()(

+

+=

IEIEIEIE

TsTsTseqTs (9)

Creep is insignificant in structural steel at room temperature, but it becomes very significantat temperatures above 400C, especially at higher loads. Despite the significant effect of creep

deformations in steel structures exposed to fire, creep is usually not explicitly included in thedesign process under fire because of the lack of data and difficulty of calculations. The usual

assumption made is that the stress-strain relationships used for fire design are effective

relationships which implicitly include creep deformations [Buchanan, 2001]. Other researchers[Anderberg, 1986 and Poh, 1995) have shown how creep deformations can be explicitly included

in computer models. Since the main focus of this study was to develop a simple approach for

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calculating deflections, instead of including creep deformations through a separate factor that islikely to be complex, the effect of creep deformations on the equivalent flexural rigidity is

accounted for by obtaining a fitted equation (8) that is a lower bound of the curves shown in

Figure 4. In the next section, while validating the simplified method, creep deformations are

explicitly included in the ANSYS analysis and it is shown that the equivalent flexural rigidity

expression obtained from (8) gives very reasonable results.

0.0

0.2

0.4

0.6

0.8

1.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Normalized Moment,

NormalizedRigidity,

T=20C

T=200C

T=400C

T=500C

T=600C

T=800C

Equation

FIGURE 4 - NORMALIZED FLEXURAL RIGIDITY VS. NORMALIZED INELASTIC MOMENT CURVES

VALIDATION OF SIMPLIFIED METHOD

For simplicity, most design specifications such as the AISC Manual and Eurocode 3, allow the

use of a uniform temperature distribution across the steel section. The uniform temperature can

be calculated by simple thermal analysis methods (e.g., lumped heat capacity method) instead ofsophisticated computer software. Therefore, the simple approach developed herein is based on

the assumption of a uniform temperature distribution.In performance based approaches currently being promoted, there is likely to be a limit on

deflections to ensure the integrity of the fire compartment and to provide safe conditions for firefighters. Until now, this deflection limit is not well defined or agreed upon. However, the British

Standard [BS 476-20, 1987] suggests a deflection limit ofL/20 that has been adopted by most

researchers. In validating the proposed approach, it is therefore appropriate to compare the fireresistance time given by the simple approach with that predicted by ANSYS when the mid-span

deflection of the beam reaches the limiting value ofL/20.

As mentioned earlier, creep is insignificant in structural steel at room temperature, but it

becomes very significant at elevated temperatures. Creep deformations are significant at slowerheating rates and higher loads. It is desirable that the proposed approach be applicable to varying

heating rates of steel (slow or fast) and all load ratios. Load ratio is the ratio of applied loadunder fire to the load that would cause collapse at room temperature. Therefore, the simple

approach is calibrated for different heating rates of steels and for different load ratios.

To determine whether the simple approach works for all heating rates, four fire scenariosproducing different heating rates of steel were selected. Slow and fast heating rates of steel

sections represent well insulated and poorly insulated (or unprotected) steel beams, respectively.

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The deflections predicted by the simple approach and ANSYS are shown in Figure 5. The simpleapproach gives deflections very close to those predicted by ANSYS and the maximum difference

in the fire resistance time (i.e., the time when the deflection reaches the limit ofL/20) from the

two computations is less than 2%.

0

75

150

225

300

375

450

0 20 40 60 80 100 1Time, min

Deflection,mm

20

Deflection

limit = L/20

HR = 5C/min.

HR = 10C/min.

HR = 30C/min.

HR = 15C/min.

FIGURE 5 - DEFLECTIONS PREDICTED BY ANSYS (BROKEN LINES) AND SIMPLE APPROACH (SOLID

LINES) FOR DIFFERENT HEATING RATES

The deflections predicted by the simple approach for five load ratios are compared with the

ANSYS predictions in Figure 6. The two methods give very similar deflections and the

maximum difference in the fire resistance times is less than 10% at the deflection limit ofL/20.Most buildings have a load ratio of 0.5 or less under fire [Buchanan 2001], and the simplified

method gives very accurate results in this range.

In all the preceding comparisons, in the elastic range (i.e., until the deflection reaches a valueof about 75 mm), the simple approach predicts deflections smaller than those predicted by

ANSYS. Elastic deflections were estimated using the reduced initial flexural rigidity, (Es,T)1 =

kEEs,20CI, where, kE accounts for reductions in the elastic modulus at elevated temperatures.

Under-prediction of deflections in the elastic range is most likely due to the reduction in the

elastic modulus being somewhat low in the beginning. However, deflections less than 75 mm arenot considered significant from a fire design point-of-view, and therefore the inaccuracy of thesimple approach in the elastic range is not important.

CONCLUSIONS

A simple approach is developed for predicting deflections of simply supported steel beamsexposed to fire. In this approach, deflections are predicted by the same equations used for

predicting elastic deflections of beams at room temperature with an equivalent flexural rigidity.

At the deflection limit ofL/20, the simple approach predicts deflections very close to thosepredicted by the ANSYS finite element program which accounts for thermal and creep effects.

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The proposed approach is very simple and can be included in design specifications for fire-resistant design of simply supported steel beams.

0

75

150

225

300

375

450

0 20 40 60 80 100 120 140Time, min.

Deflection,mm

Deflection

limit = L/20

LR = 0.80

LR = 0.70

LR = 0.60

LR = 0.50

LR = 0.35

FIGURE 6 - DEFLECTIONS PREDICTED BY ANSYS (BROKEN LINES) AND SIMPLE APPROACH (SOLID

LINES) FOR DIFFERENT LOAD RATIOS

REFERENCES

[1] AISC, Steel construction manual, 13th edition, American Institute of Steel Construction, Inc, 2005.

[2] Anderberg, Y. , Measured and predicted behavior of steel beams and columns in fire,Lund Institute of

Technology, Lund, Sweden, 1986.[3] ANSYS Inc., ANSYS software and manual,Release 11.0SP1 UP20070830, 2007.

[4] BS 476-20, Tests on building materials and structures part 20: method for determination of the fire resistance

of elements of construction (general principles), 1987.

[5] Buchanan, A. H. , Structural design for fire safety, John Wiley & Sons, Inc., New York, 2001.[6] Poh, K. W., Stress-strain-temperature relationship for structural steel,Journal of Materials in Civil

Engineering, ASCE, 13(5), 2001, pp.371-379.

[7] Poh, K. W. and Bennette, I. D., Analysis of structural member under elevated temperature conditions,Journal

of Structural Engineering, ASCE, 121(4), 1995, 664-675.

[8] Skowronski, W., Load capacity and deflection of fire-resistant steel beams,Fire Technology, 1990, pp. 310-

328.

[9] Wainman, D. E. and Kirby, B. R., Compendium of U.K. standard fire test data on unprotected structural steel,

Report No. RS/RSC/S10328/1/87/B, British Steel Corporation, 1987.

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