Determining the moment-curvature diagram of aluminium
cross-sections in fire; a numerical code and practical
examples
Citation for published version (APA):
Meulen, van der, O. R., Maljaars, J., Soetens, F., & Twilt, L. (2010). Determining the moment-curvature diagram of aluminium cross-sections in fire; a numerical code and practical examples. In X. Kodur, & X. Franssen (Eds.), Proceedings of the Sixth International Conference on Structures in Fire (SIF'10) (pp. 1015-1022)
Document status and date: Published: 01/01/2010
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ABSTRACT
The use of aluminium as a construction material has been increasing since its first invention. At present, limited knowledge of the behaviour of aluminium beams in fire gives rise to excessively high insulation demands, decreasing its competitiveness. To study the behaviour of (statically undetermined) bending members in fire, it is required to know the moment-curvature diagram. For aluminium and stainless steel at both room and elevated temperatures, and for steel at elevated temperatures, this curve cannot be calculated analytically, but needs to be obtained numerically. A versatile numerical code is presented, capable of calculating the moment-curvature diagram for arbitrary geometries along any axis with arbitrary temperature and initial strain distributions.
INTRODUCTION
Aluminium alloys are a class of construction materials with many favourable properties for the construction industry; its excellent corrosion resistance, generally not requiring a coating system; a low self wight, its density being only a third of that of steel; and the possibility to produce highly optimized cross-sections through extrusion, often integrating load carrying with other functions such as fastening systems. These advantages outweigh its higher material costs in many instances, and the usage of aluminium in the construction industry has therefore been constantly increasing [8]. At present, the industry accounts for nearly 15% of the North-American aluminium consumption [13].
A less favourable property of aluminium and its alloys is their relatively low melting points of approximately 550 to 600◦C, and a decrease of load carrying potential at lower temperatures, as demonstrated in figure 1. This figure shows the majority of the reduction in strength to take place in a temperature range between 150 and 350◦C,
O. R. van der Meulen, MSc, PhD student∗,†,‡ (r.vandermeulen@m2i.nl) J. Maljaars, MSc PhD, Senior researcher ‡
F. Soetens, Prof. MSc †, ‡
L. Twilt, MSc, Senior researcher (ret.)‡
∗ Materials innovation institute (M2i), Delft, the Netherlands † Eindhoven University of Technology, the Netherlands
20 150 250 350 450 550 0 0.2 0.4 0.6 0.8 1 Temperature θ (◦C) 0.2% proof stress f 0.2, θ f 0.2 3004-H34 5005-0 5005-H14 5052-H34 5083-0 5083-H12 5454-0 5454-H34 6061-T6 6063-T5 6060-T66 6082-T4 6082-T6
Figure 1. The reduction of 0.2% proof stress of common aluminium alloys at elevated tempera-tures. Its value at elevated temperatures ( f0.2,θ) is normalized to the room temperature value ( f0.2).
Source:[3], after: [7].
150 200 250 300 350 400 0
0.5 1
Maximum temperature θmax (◦C)
Iinsulation thic kness t t150 A m/V = 1020 m−1 Am/V = 136 m−1
(a) Survival of structure
150 225 300 400 0
0.5 1
Max. temperature θmax (◦C)
Insulation thic kness t t150 (b) 30 minutes
Figure 2. The required normalized, insulation thickness as a function of the maximum allowable temperature in the member. Two conditions are considered, the first (a) the structure surviving the fire, and the second (b) the structure withstanding the fire for at least 30 minutes. A dwelling with a fire load density of 948 MJ/m2and an area of 200 m2is assumed. The insulation thickness t is normalized to the thickness (t150) for a maximum member temperature of 150◦C. The opening
conditions are varied between a partial opening in the 10 m short fac¸ade of the structure and two fully open, 20 m long fac¸ades. Two area-over-volume ratios Am/V are considered as are three active
measure factors (Πδn=∈ {0.78, 0.50, 0.26}). The insulation material is a type of rockwool. It is
noted that for an office of 800m2, the results are nearly identical. Source:[7], modified from: [5].
which is therefore the main temperature range of interest for fire design.
As the temperatures associated with fire easily exceed these values, insulation is generally required. The thickness of which decreases as the maximum allowable temperature is increased. Figure 2 shows the reduction in the required insulation thickness, for a fire according to the natural fire safety concept or NFSC [11], obtained from a study by Maljaars [7]. There is a 65 to 85% reduction in the required insulation thickness as the maximum allowable temperature in the member is increased from 150 to 350◦C. An accurate knowledge of the behaviour of aluminium members and their maximum allowable temperature thus results in significant savings in insulation costs, even allowing unprotected members in certain low fire load density applications, such as transport facilities [7].
In the study of the behaviour of bending members, having knowledge of the moment-curvature curve is required. An example of such a study is given in [9], where the stabilizing effects of elevated temperatures in aluminium alloys for plastic design are examined. For aluminium and stainless steel at both room and elevated temperatures, and steel at elevated temperatures, obtaining moment-curvature diagrams for all but the simplest, idealized, cross-sections is analytically impossible. This is
A B C D
(a) Rectangle (b) Complicated
Figure 3. Shapes with their subdivision into elements. Shapes are defined by their four corner coordinates, but these may overlap, creating triangles. Complicated shapes are possible as long as the corner coordinates define a convex area.
because their stress-strain diagrams are inelastic and are described by a Ramberg-Osgood equation [10], which does not permit the stress to be calculated as a function of the strain explicitly. In fire design, the situation is further complicated by effects as varying material properties as a result of uneven temperature distributions, and thermal expansion. It is for these reasons, numerical approaches where developed, for example [2] which separates the cross-section into smaller parts with equal material properties and thermal expansions and sums up their contributions to the moment resulting from a certain induced curvature. In this paper a more generally applicable code is presented, which allows the calculation of the moment-curvature diagram of any conceivable cross-sections, consisting of one or multiple different materials, subjected to a combination of normal and bending loads along an arbitrary axis, while taking into account thermal- and initial-stresses as desired.
SPECIFYING PROPERTIES
Geometry
The cross-section of a beams, prismatic in the out of plane direction, is performed by building it up from shapes. These shapes are polygons which may in principal have an arbitrary number of nodes, but four noded shapes are used here for ease of meshing. An example is shown in subfigure 3a, where a rectangle is defined by the coordinates ABCD. The shape does not need to be rectangular, any convex shape is permissible and even overlapping corner nodes are possible, the only non-trivial application being the definition of triangles. These examples are shown in subfigure 3b. In figure 3, all shapes are already subdivided into smaller entities known as elements. It is these elements that are used in the calculation. Properties as temperature, area, centroid, inertia, rotated coordinates and temperature dependent material models are assigned in pre-processing and are constant throughout the element. Their response to applied loads is calculated in the main algorithm and summed over all elements to produce the cross-sectional response.
Material models
All the elements belonging to a single shape share a single, temperature dependent, ma-terial model. Different mama-terial models or mama-terial models with different temperature dependent properties, may thus be combined in a single cross-section by dividing it into multiple shapes. This is useful when studying the effects of degenerated material properties in the heat affected zones near welds for example. It is possible to use an arbitrary material model, but because of the intended application to fire design, the Ramberg-Osgood [10] model is used throughout the paper.
PRE-PROCESSING
Rotated coordinates
The curvature can be specified by the user to operate about any line as specified by the angle α. A value of 0 indicating a curvature about the vertical axis, leading to compressive strains at the left hand side of the cross-section. The rotated coordinates are calculated by:
ˆ xi ˆ yi = cos (−α) − sin (−α) sin (−α) cos (−α) xi yi . (1) Geometric properties
The area of a non-self intersecting polygon is given by the surveyor’s formula.
Ai= 1 2 N
∑
j=1 (xjyj+ 1− xj+ 1yj) , (2)where i is the element number, and j number of the node along the perimeter of the polygon, the total number of which is N, equal to 4 in this case. xN+ 1= x1, and
yN+ 1= y1. The coordinates of the centroid of the polygon are equal to
Cx, i= 1 6Ai N
∑
j=1 (xj+ xj+ 1) (xjyj+ 1− xj+ 1yj), Cy, i= 1 6Ai N∑
j=1 (yj+ yj+ 1) (xjyj+ 1− xj+ 1yj). (3)The second moment of area (inertia) is dependent on the direction of loading and is thus a function of the rotated coordinates ( ˆx, ˆy). The expression was obtained from [12]
Ix, iˆ = 1 12 N
∑
j=1 ˜ y2j + ˜yjy˜j+ 1+ ˜y 2 j+ 1 dj, (4)where ˜xand ˜yare the coordinates in the rotated axis system relative to the centroid of the element; ˜xj= ˆxj−Cx, iand ˜yj = ˆyj−Cy, i. dj is equal to ˜xjy˜j+ 1− ˜xj+ 1y˜j.
Thermal expansion and initial stresses/strains
The free thermal expansion of each of the elements is obtained by multiplying the temperature Ti above the initial temperature T0, with a coefficient of (temperature
dependent) thermal expansion.
εth,i= αth,i(Ti) (Ti− T0) , (5)
The normal component of the cross-sectional thermal expansion may be restrained or not depending on the support conditions. For the unrestrained case, the normal component does not contribute to the stress, and needs to be subtracted. Initial stresses can be implemented by expressing them as strains
ε0,i=
σ0,i
Ei
, (6)
while taking care to make them self balancing, or subtracting a normal component afterwards.
begin
Calculate the stresses without curvature and its normal component
1 σκ 0,i←− f (εκ 0,i) ; Nκ 0←− ∑
N i=1σκ 0,iAi
Initial estimate for position of neutral axis
2 xˆna,0←− min ( ˆxi) +max ( ˆxi)−min ( ˆ2 xi)
3 while ( ˆxna,n6= ˆxna,min,n−1) ∧ ( ˆxna,n6= ˆxna,max,n−1) do
Calculate the strain and stress
4 εi←− εκ 0,i+ κ ( ˆxi− ˆxna,i,n − 1); σi←− f (εi)
Calculate change in normal force
5 ∆N ←− Nκ 0− ∑Ni=1σiAi
Determine the new boundaries for the location of the neutral axis
6 if ∆Nκ < 0 then ˆ xna,min,n←− ˆxna,n − 1 else ˆ xna,max,n←− ˆxna,n − 1
Make a new estimation for the neutral axis position
7 xˆna,n←− ˆxna,min,n+
ˆ
xna,max,n− ˆxna,min,n
2
Calculate elemental tangent stiffness and moment
8 Etan,i←− Ei 1+500 f0.2,iEini σi f0.2,i ni; Mi←− κEtan,iIxˆ Integrate moment 9 M←− ∑Ni=1[( ˆxi− ˆxna) σiAi+ Mi]
Figure 4. The main moment-curvature curve calculation algorithm.
THE MOMENT-CURVATURE CURVE CALCULATION ALGORITHM
The object of the algorithm is to find the strain and stress distribution in the cross-section for a number of independent calculation steps. Each calculation-step is charac-terized by a different curvature, all the calculation steps together forming a moment-curvature diagram. The algorithm is summarized in figure 4, the different steps are elaborated on hereafter, the number in the title corresponding to the number in the left hand margin of the algorithm.
1: Calculate stress without curvature The strain without curvature being is calcu-lated by
εκ 0,i= −εth− ε0+ εN. (7)
The stress without the applied curvature (εκ 0,i), and its normal component, are
then calculated and stored for efficiency. The implicit Ramberg-Osgood material model [10]is solved using a numerical, bi-section root finding algorithm [1]. 2: Initial estimate for the position of the neutral axis The neutral axis at the
cur-rent calculation step (n) is defined by the line ˆxna,n= ˆx. The initial estimate ˆxna,0
is determined by taking the average value of the minimum and maximum ˆx coordinate of all the centers of the elements in the cross-section.
3: Iterative procedure stop condition Finding the stress distribution in the cross-section is done in an iterative procedure because the location of the neutral axis is unknown a-priory. The procedure is using the bisection method, modified for floating point operations [1], and converges to the correct location of the neutral axis and corresponding stress-strain distribution until terminated by the stop condition.
4: Calculating stress and strains The stress is calculated by taking the strain without curvature εN,iand adding the strain due to the curvature using the trial solution for
100 mm 100 mm 10 10 (a) Cross-section 0 1 · 10−4 2 · 10−4 0 10 20 Analytical @ 20◦C Analytical @ 250◦C Simulations Curvature κ Moment M (K N m ) (b) M-κ diagrams
Figure 5. Verification case for the numerical code, a close to perfect bending member. The material used is AA6060-T66 with f0.2= 205, E = 7.0 · 104and n = 19 at 20◦C, and nθ= 9, Eθ= 5.46 · 10
4
and f0.2,θ= 79 at 250◦C.
5: Checking the equilibrium When the position of the neutral axis is chosen cor-rectly, the applied curvature should not induce a change in normal force. This condition is verified here, and a deviation is expressed as ∆N.
6: Updating the bounds Depending on the sign of the product of the applied curva-ture and ∆N, the bounds for the neutral axis position are adjusted.
7: Neutral axis position estimate A new estimate is made for the position of the neutral axis. This is the mean value of the two bounds.
8: Tangent stiffness and in-element moment because of the (small) bending stiff-ness of the elements themselves, a moment will be present in each element, which is denoted by Mi. To calculate the elemental moment, the tangent stiffness
in the element (Etan,i) is calculated, and multiplied with the second moment of
area as given in equation (4). By assuming a constant tangent moment through-out the element, a small error is introduced, as the real (tangent) stiffness is not constant.
9: Integrate moment The moment is integrated across the cross-section and is used to derive a moment–curvature (M–κ) diagram. The contribution from the in-element stiffnesses Mi becomes very small as the number of elements is
increased. For a mesh as shown in figure 6a, the term is completely irrelevant
VERIFICATION
To validate the numerical code, it is tested against analytical solutions for moment-curvature diagrams. As indicated before, such solutions are only obtainable for simple geometries. The case of a theoretical, perfect bending cross-sections with all mass in the outmost fibres, has its moment-curvature diagram expressed by
κ (M) = 2 h 2M EA + 0.002 2M f0.2A n , (8)
where A is the total area of the cross-section. Equation (8) is tested against the simulations for a cross-section as shown in figure 5a. The results of these simulations, and the analytically derived curves are presented together in figure 5b, demonstrating the proper functioning of the program.
12 12 100 mm 200 mm 12 12 8 mm (a) Cross-section 0 7.5 · 10−5 1.5 · 10−4 0 20 40 60 20 ◦C 175◦C 250◦C 350◦C Strong axis M (K N m ) 0 7.5 · 10−5 1.5 · 10−4 0 5 10 20◦C 175◦C 250◦C 350◦C Weak axis Curvature κ M (K N m ) (b) Moment-curvature
Figure 6. An I-section of AA6060-T66 subjected to a bending force along its weak and strong axis, at four different, constant temperatures. Material properties are taken from [6], thermal expansion is unrestrained.
PRACTICAL EXAMPLES
To illustrate the usability of the numerical code, an I-section with a geometry according to figure 6a is subjected to a bending force in its strong and weak direction for aluminium alloy 6060-T66 for a number of different temperatures. The resulting moment-curvature diagrams are given in figure 6b. At a curvature equal to 1.5 · 10−4, the moment at 250◦C is approximately 43% of the room temperature value for both bending directions. The value for the 0.2% proof stress at this temperature is only 38% of the room temperature proof stress [4], illustrating the potential savings of obtaining moment-curvature diagrams at elevated temperatures as oposed to scaling them by the 0.2% proof stress.
In a second example, the geometry as was given in figure 6a is subjected to a linear temperature gradient, ranging form the highest temperature at the lower flange of the cross-section, to the lowest temperature at the upper flange. This an arbitrary, but not unrealistic approximation of an insulated beam in fire conditions with a temperature drain such as a floor against its top flange. In figures 7a and 7b, we can see the moment-curvature curve for the beams subjected to a temperature gradient, compared to beams with a single, elevated temperature. From these figures it is clear that modeling a beam with a temperature gradient by a beam with a single temperature may be possible, but that taking the highest temperature in the gradient as the single temperature is over conservative. Taking a mean value for the temperature; 250◦C gives reasonable results in the case of figure 7a, but is unconservative for the case of figure 7b. It is noted that this result is only applicable for situations where thermal expansion and instability effects do not play a role. No generally valid approximation can be given, underlining the need for a numerical calculation technique as the one offered in this paper.
CONCLUSIONS
For the economical design of aluminium structures, an accurate understanding of the behaviour of beams at elevated temperatures is required. To study the behaviour of such members, the moment-curvature diagram is required, which is generally unobtainable analytically. A numerical solution technique is presented, capable of
0 7.5 · 10−5 1.5 · 10−4 0 20 40 225◦C 250◦C 275◦C Gradient 225-275◦C Curvature κ Moment M (K N m ) (a) Gradient 225-275◦C 0 7.5 · 10−5 1.5 · 10−4 0 20 40 275◦C 175◦C 250◦C 325◦C Gradient 175-325◦C Curvature κ Moment M (K N m ) (b) Gradient 175-325◦C
Figure 7. An I-section of AA6060-T66 subjected to a bending force along its strong axis with a temperature gradient along its height, the material properties are taken from [6], the thermal curvature is subtracted.
correctly calculating this curve for any cross-section, inelastic material, load direction and temperature distribution. The code was verified with the help of an analytical solution (which is available for certain idealized cross-sections) and shown to behave correctly. Practical examples are given for an I-section in bending, underlining the potential decrease in over-conservativeness as incurred by simpler approximations.
ACKNOWLEDGEMENT
This research was carried out under project number M81.1.108306 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl).
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