Laterally supported aluminum flexural members with symmetric cross sections

Rev. A (Progress Report 15) 5/6/2003

Paper 1 - 1

Paper 1

Laterally supported aluminum flexural members with symmetric cross sections

Yongwook Kim and Teoman Pekz

Abstract The Specification for Aluminum Structures published by the Aluminum

Association does not fully account for the plastic-ultimate bending capacity: For compressive component elements, the allowable stress is based on the yield stress.

In this study, possible modifications to the current specification are suggested so that rather compact extruded flexural members can be evaluated more precisely. As a first step, limit state stress equations are modified so that the plastic-ultimate capacity is fully incorporated. Closed form solution of the ultimate shape factor is provided to set a new cut-off corresponding to the ultimate stress. The ultimate shape factor is simplified for practical design purposes. A parametric study of component elements using the finite element analyses suggests a complete form of the modified limit state stress equations. Using the modified limit state stress equations, more accurate approaches are proposed to compute the plastic-ultimate moment capacity. A parametric study of doubly symmetric I-shaped sections using the finite element method shows that the approaches developed in this study are significantly better than the current specification provisions. Flexural tests conducted in this research also support the approaches.

To maintain a certain factor of safety against yielding in the allowable stress design equations, an approach to use only a potion of the ultimate-plastic capacity is suggested. Using this approach, a larger inelastic reserve capacity is recognized for the material with a larger margin between the ultimate and yield stresses.


Paper 1 - 2

Introduction: The AA Specification for component elements The Specification for Aluminum Structures published by the Aluminum

Association (2000a, abbreviated to the AA Specification) requires designers to check whether the computed stress based on the applied loads is less than both of the allowable stresses of a tension side component element as summarized in Table 1. The factors in front of the tensile yield and ultimate stresses for the web incorporate the shape of a non-linear stress distribution, which is often called the shape factor.

Table 1. Options to Compute Tension Limit State Stress of a Flexural Member by AA (2000a)

Component element Allowable stress based on limit state of yield stress Allowable stress based on

limit state of ultimate stressa

Tension flange of structural shapes Fty/ny Ftu/nu

Tension web of structural shapes 1.30Fty/ny 1.42Ftu/nu

aExcept for some of 2014-T6, 6066-T6 and 6070-T6 family materials, for which nu is replaced with ktnu due to notch sensitivity. See AA (2000a) for details.

Fty = tensile yield stress Ftu = tensile ultimate stress ny = factor of safety on yield stress nu = factor of safety on ultimate stress

However, the check against the limit state of the ultimate stress is not available when a limit state stress is computed for a compressive component element. Compressive limit state stress equations for a component element refer to yielding, inelastic buckling, and post buckling ranges as shown in Figure 1.

For this reason, the ultimate-plastic capacity of aluminum members has not been fully incorporated in the specifications, although aluminum alloys are hardening and ductile materials.

Fig. 1. Limit state stress for a component element in the AA Specification

The limit state stress equations in this figure are expressed in terms of the equivalent slenderness ratio (p) as defined in Equation (1), originating from Equation (2).


Paper 1 - 3

3.266p

p

b bt tk

= = for = 1/3 (1)

( )2 2

2 2212(1 ) /cr p

p

E EF kb t

= (2)

For a flange or web with junctions between component elements idealized as simply-supported boundary conditions, equivalent slenderness ratios depending on plate buckling coefficient kp are listed in the AA Specification as exampled in Table 2.

Table 2. Equivalent slenderness ratios in the AA Specification (S.S denotes simply-supported boundary condition)

(a) AA 3.4.16

(b) AA 3.4.15

(c) AA 3.4.18

1.6pbt

= 5.1pbt

= 0.67pbt

=

The post-buckling equation seen in Figure 1 was proposed by Jombock and Clark

(1968) to take account of the non-linear stress distribution of a buckled plate. B and D factors were determined by Clark and Rolf (1966) simplifying the inelastic plate buckling equation proposed by Stowell (1948) based on experimental studies.

The cut-off in the yielding range was set against the yield stress without a scientific basis. It can be assumed that the rather arbitrarily chosen cut-off for column members to prevent a catastrophic failure after yielding was also employed for the component elements according to Alcoa (1955). Due to the inelastic reserve capacity, however, component elements of a flexural member are not as critical as the ones of a column after yielding. For this reason, the cut-off equations for compressive component elements are to be investigated.

The ultimate cut-off and shape factor for the ultimate and plastic flexural capacity Rigorous analytic ultimate shape factor for rectangular web elements

The ultimate moment capacity of a member can be obtained if ultimate capacities of component elements are known. For example, in the case of a doubly symmetric I-shaped section, the ultimate capacity of the flange element is the ultimate stress of the used material, obviously. On the other hand, the ultimate capacity of the web is the ultimate bending capacity. Since the stress distribution on the web is non-linear, the moment capacity can be obtained through integration of the stress-distribution.

For the integration, the stress-strain relation is curve-fitted by the modified Ramberg-Osgood equation:

0.002n

y

f fE F

= + (3)


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In order to use this equation, exponent n should be determined. Since the AA Specification provides the yield stress, Youngs modulus and the ultimate stress, the strain at the ultimate stress (referred to as the ultimate strain hereinafter, u) should be obtained to determine exponent n as expressed in Equation (4). ( )log 500

log

u

u

y

Fu E

FF

n

= (4)

However, the statistical data of the ultimate strain is not available. Instead, the minimum percent elongation is only available in the Aluminum Standards and Data (2000c), abbreviated to ASD. Since the percent elongation is somewhat larger than the ultimate strain, in general, the ultimate strain may be assumed from the percent elongation.

As Eberwien et al. (2001) showed, a moment capacity for a rectangular block with non-linear stress distribution expressed by the modified Ramberg-Osgood equation can be obtained using the Bernoulli-hypothesis. Based on the idea, the ultimate moment capacity is computed and normalized by the yield moment capacity, resulting in a closed form solution of the ultimate shape factor for a web element of a doubly symmetric section:

222

2 2

1 1 1 12 3 500 2 1 500 2

n n

u u u u uu

u y y

bh F F F F Fn nME n F n F E

+ = + + + + (5)

2

6y

y

bh FM = (6)

22

2 2

3 1 1 1 13 500 2 1 500 2

n n

u u u u u uw

y u y y y

M F F F F Fn nM F E n F n F E

+ = = + + + +

(7)

As seen in DOD (1994) and tensile coupon tests in this study, the typical value of the ultimate strain (u) is approximately 6 to 8% for 6061-T6. For the same material, the typical value of percent elongation is 12% as tabulated in the Aluminum Design Manual, ADM in short, (2000b). Similarly, since the minimum value of the percent elongation is 8% for 6061-T6 as shown in ASD (2000c), the minimum value of the ultimate strain may be approximately 4 to 6%. To understand the effect of the ultimate strain variation, shape factors are computed for various ultimate strain values using Equation (7) in Figure 2. From the figure, it is found that the shape factor variation is less than 1% if the ultimate strain is set to either 50% larger or smaller than the minimum percent elongation.

Fig. 2. Variation of the shape factor for a rectangular web element with respect to the

ultimate strain (6061-T6)


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Thus, the conjecture that the ultimate strain may be assumed from the percent elongation is reasonable. In this study, the recommended value of the ultimate strain is between the half and the same amount of the minimum percent elongation. The ultimate shape factors are computed using Equation (7) for some 6000 series alloys in Table 3, which are more frequently used for extrusions.

The obtained shape factors in Table 3 can be directly used as the proposed cut-off for the limit state of the ultimate stress when those are multiplied by the yield stress. Since the compressive stress distribution is approximately the same as the tension one for doubly symmetric compact sections, the shape factors (w) in this study are also compared to those for the tension web available in AA: Equation (8). It can be concluded that the shape factor in AA (2000a) is quite close to the rigorous one proposed in this study for 6000 series alloys. Table 3. Material properties and ultimate shape factors for rectangular sections (webs) for

some 6000 series alloys (FIX DATA WITH 10000KSI E)

Alloy-temper Fy a

(MPa) E a

(MPa) Fu a

(MPa) u b n c w d 1.42e

w

tu yF F

6005-T5 241.15 69589 261.82 0.04 35.23 1.5976 1.0363 6061-T6, T6510, T6511 241.15 69589 261.82 0.04 35.23 1.5976 1.0363

6063-T5 f 110.24 69589 151.58 0.04 9.23 1.9468 0.9971 6063-T5 g 103.35 69589 144.69 0.04 8.74 1.9768 0.9944

6063-T6, T62 172.25 69589 206.70 0.04 16.01 1.7362 1.0189 6066-T6, T6510, T6511 310.05 69589 344.50 0.04 27.18 1.6227 1.0284

6070-T6, T62 310.05 69589 330.72 0.03 39.29 1.562 1.0313 6105-T5 241.15 69589 261.82 0.04 35.23 1.5976 1.0363 6351-T5 241.15 69589 261.82 0.04 35.23 1.5976 1.0363 6463-T6 172.25 69589 206.70 0.04 16.01 1.7362 1.0189

a. Minimum values from ADM (2000b) and ASD (2000c) b. 50% of the minimum percent elongation listed in ASD (2000c) c. Equation (4) d. Equation (7) e. 1.42Ftu/Fy is the ultimate shape factor for tension side web by AA: Table 1 f. Test coupon diameter or thickness up through 12.7mm g. Test coupon diameter or thickness between 12.7 and 25.4mm

Simplified ultimate shape factor for rectangular web elements The rigorous analytic expression for the ultimate shape factor of a rectangular

web element in Equation (7) is too complicated for practical design purposes. Thus, a simplified expression such as the one used in the AA Specification (2000a) for tension component elements, Equation (8), is necessary.

1.42 tuu wty

FF

= = (8) Although the performance of Equation (8) is found to be satisfactory for some

6000 series alloys in Table 3, it has never been investigated for a wide variety of alloys-tempers. In ASD (2000c), more than a thousand of materials are listed according to types of alloys, tempers, dimensions of tested specimens, orientations of tested coupons taken


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from specimens and product types of specimens (plates, pipes or extruded shapes). Among these data, 986 alloy-temper combinations are chosen for the investigation. The bases to choose the data are as follows: First, minimum material properties required for Equation (7) should be available. Second, the ultimate strain should exceed 1.5%, when it is assumed to be the half of the minimum percent elongation. Third, the orientation of the test coupon should be longitudinal or not specified.

1 1.5 2 2.5 3 3.50.5

1

1.5

2

2.5

3

3.5

4

4.5

5

total number of data = 986 = 0.5 x percent-elongationalloy-temper = no transverse direction, > 1.5%

Fig. 3. Comparison of the ultimate shape factor approximations using Equations (8) and (9)

for a plate under bending

For the selected alloy-temper combinations, the shape factors are computed using Equation (7) and plotted with respect to the Ftu/Fty ratio in Figure 3. From the figure, it is clear that Equation (8) becomes unconservative as the Ftu/Fty ratio increases. Thus, a more precise curve-fit is proposed for the ultimate shape factor:

1.25 0.2tuu wty

FF

= = + (9) For practical design purposes, Equation (9) could be used instead of Equation (7)

for the ultimate shape factor of a rectangular web element under bending.

Parametric study for component elements and proposed AA design equations

In order to show a possible modification to the design equation for the yielding range, a parametric study is conducted for component elements of doubly symmetric I-shaped sections using the finite element method.

The finite element program, ABAQUS developed by Hibbitt, Karlsson and Sorensen, Inc. is used for the analyses. Four-noded general-purpose shell elements with 4 integration points are used to take into account the large thickness variation in a series of

w

u

1.25 0.2tuwty

FF

= +

1.42 tuwty

FF

=

u

Equation (8)

(current AA)

Equation (9)

(proposed)

Equation (7)

tu tyF F


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models. The alloy and temper is assumed to be extruded 6061-T6 with minimum material properties in Table 3 except for the ultimate strain. To observe the variation of the analysis results, two ultimate strain values are used; 4% and 8%. According to the variation of the ultimate strain, the Ramberg-Osgood exponent (n) and the shape factor for the web (w) vary.

For material model, isotropic hardening is used, since this study deals with monotonic loading. In this case, once a stress reaches the ultimate stress, the stress remains constant as the plastic strain exceeds the ultimate strain. Due to uncertainty after this stage, it is assumed that the whole member reaches the failure when the von-Mises stress at a single point of a member reaches the ultimate stress. This occurs when the member is too compact to buckle. On the other hand, the failure of the member can also be initiated by buckling when the member is less compact. In this case, the peak load of the member is obtained before any point of the member reaches the ultimate stress. In this study, these two possibilities of failure are considered simultaneously to find an ultimate load factor.

Fig. 4. Boundary and loading conditions for finite element analyses

Fig. 5. Parametric study results for component elements (a) web (b) flange


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The boundary conditions for component elements of doubly symmetric I-shaped sections are idealized as shown in Figure 4. To avoid singularity of the stiffness matrix, one longitudinal degree of freedom is restrained at the span center. Equal and opposite loadings are applied at the loaded edges, at which rigid beam elements are attached.

Prior to the non-linear analyses, elastic eigen-value analyses are conducted to generate initial geometric imperfections. The maximum amplitude of imperfections is determined based on the industrial production limitation provided by ASD (2000c).

The results of the parametric study for component elements are shown in Figure 5. For most ranges, the finite element analyses are close to the current AA Specification. However, for yielding range (width/thickness S1), significant difference is detected. For very low width-to-thickness range, the ultimate shape factors are employed for the proposed cut-off: 1.086 (= Fu/Fy) for the flange element, 1.598 for the web element (if u = 4% is used). For the remaining part of the yielding range between the proposed and current cut-off values, it is found that a linear extension of the inelastic buckling range (S1 width/thickness S2) equation can be used. These modifications work well compared to both series of finite element analyses based on 4% and 8% of the ultimate strain.

The AA Specification equations modified in this study will be used as the limit state of the ultimate stress, while the current ones as the limit state of the yield stress. The symbolic design equations of component elements are summarized for the two limit states in Table 4.

Table 4. Design equations and borders of equations

limit state of component element

limit state stress

1b St

limits S1

limit state stress

1 2bS St

limits

S2

limit state stress

2bSt

Flange Ff = Fy yB F

D

yield stress

(current AA) Web Fw = 1.3 Fy

1.3 yB FD

Flange Ff =Fu uB F

D

ultimate stress (proposed AA)

Web Fw = w Fy w yB FD

bF B Dt

= 1k BD

2k BEFbt

=

Note. 1. F = Ff or Fw 2. B,D, k1 and k2 for a flange differ from those for a web. See AA (2000a) for details. 3. Equation (1) for (Table 2) 4. Equation (7) for w

The moment capacity evaluation approaches In the AA specification, to compute the moment capacity of a member the limit

state stress of each component element is multiplied by the elastic section modulus of the entire cross section. Among obtained moment capacities, the minimum one is chosen as the allowable moment capacity of the member. This is denoted MMCA (the Minimum Moment Capacity Approach) hereinafter. Since possible interactions and stress redistributions between component elements are disregarded, this approach is expected to be rather conservative.


Paper 1 - 9

As an alternative in AA, limit state stresses obtained from all component elements can be averaged according to contributory area. The averaged stress is multiplied by the section modulus to compute the moment capacity. This is called weighted average stress approach (denoted WASA).

The weighted average stress equation was first introduced by Jombock and Clark (1968) to compute the crippling strength of aluminum trapezoidal formed sheet members: Equation (10),

16

16

f f w wwt

f w

F A F AF

A A+= + (10)

where Ff = limit state stress for the flange, Fw = limit state stress for the web, Af = entire compression side flange area, and Aw = entire web area.

Although the WASA was verified through experiments by the researchers, a theoretical basis of the weighted average method has never been investigated. Thus, the accuracy of the method is questionable for various geometric shapes other than the formed sheet members.

The theoretical basis of the WASA is investigated for a doubly symmetric I-section shown in Figure 6. Multiplication of both the denominator and numerator of Equation (10) by (hc/2)2 and simplification assuming that the flange thickness is relatively small result in Equation (11):

2 2f w c u c

wtf w

M M h M hFI I I

+ = = + (11) where Mf = moment capacity of the flanges, Mw = moment capacity of the web, If = moment of inertia of the flange elements, Iw = moment of inertia of the web element, Mu = total moment capacity and I = total moment of inertia.

Fig. 6. Contributions made by component elements to the moment capacity of an -section

Equation (11) impies that the weighted average stress is an approximatly linearized bending stress mesured at the mid-thickness of the flange. Therefore, to obtain an appropriate total moment capacity, the corrections shown in Table 5 have to be made:

The correction in Table 5 is insignificant when the flange thickness is relatively small, such as the ones of the thin-walled formed sheet members used for experimental evidences by Jombock and Clark (1968). However, it is known that extrusion is more economical than cold-forming, in general. In addition, sections consisting of component elements with large width-to-thickness ratio are not suitable for extrusion due to production difficulty: For details, see Kissell et al. (1995). For these reasons, standard sections listed in the ADM (2000b) are mostly made of relatively thick component


Paper 1 - 10

elements, falling into the yielding or, at least, inelastic buckling ranges. Thus, the modifications shown in Table 5 as well as Table 4 are significant for extruded sections.

Table 5. Correction in the current WASA current WASA

(WASA) proposed WASA

(WASA2)

Mu = (Fwt)(S) Mu = (Fwt)(S)c

hh

Note. S = section modulus = I /(h/2)

As an alternative to the proposed WASA, the Total Moment Capacity Approach (TMCA) is also suggested in this study. In the TMCA, the limit state stress from each component element is multiplied by the contributory section modulus to compute the contributory moment capacity. All the moment capacities from component elements are added to obtain a member moment capacity. For example, the moment capacity of an I-shaped section shown in Figure 6 can be expressed as Equation (12) based on the TMCA.

16u f f c w w oM F A h F A h= + (12)

Since contributions by all component elements are made to a member capacity in both the WASA2 and TMCA, these are expected to be more accurate than MMCA.

Parametric Study of I-Shaped Sections To validate the improvements discussed above, a parametric study is conducted

for doubly symmetric I-shaped sections using the finite element analyses. The width of the entire flange ( w in Figure 6) and the depth between centerlines of flanges ( hc in Figure 6) are maintained as 254 mm, while uniform component element thicknesses vary so that wide ranges of width-to-thickness variations of component elements can cover most of the standard sections listed in the ADM (2000b). The length of the members is fixed as 2540 mm. The boundary conditions are determined as shown in Figure 7. Other details regarding the finite element models are the same as those for the parametric study for component elements.

Fig. 7. Model geometry of an I-shaped section for a parametric study

As a first step, the influence of the ultimate-plastic capacity consideration in the AA Specification is investigated as shown in Figure 8. The approach abbreviations used in Figure 8 are denoted as follows: For example, MAA-U-WASA is the moment capacity evaluated by AA using the conventional WASA, considering the limit state of the


Paper 1 - 11

Ultimate stress, while MAA-Y-WASA is the moment capacity considering the limit state of the Yield stress. The horizontal axis is the slenderness factor. The vertical axis is the moment capacity obtained from the finite element analysis normalized by the moment capacities using the approaches specified in the graph.

As seen in this figure, an approximately 7% improvement in the member capacity is observed by the consideration of the ultimate-plastic capacity. In addition, the variation of the data significantly decreases when the ultimate-plastic capacity is considered.

0.3 0.4 0.5 0.6 0.7 0.8

0.8

0.9

1

1.1

1.2

1.3

MFE

M /

Map

proa

ches

MFEM / Mapproaches mean c.o.vMFEM / MAA-Y-WASAMFEM / MAA-U-WASA

1.209 0.056

1.141 0.029

= y crF F

Fig. 8. Influence of the ultimate-plastic capacity consideration in AA (u = 4%)

0.3 0.4 0.5 0.6 0.7 0.8

0.8

0.9

1

1.1

1.2

1.3

MFE

M /

Map

proa

ches

MFEM / Mapproaches mean c.o.vMFEM / MAA-U-WASAMFEM / MAA-U-WASA2

1.141 0.029

1.04 0.024

= y crF F

Fig. 9. Influence of the modification in WASA (u = 4%) Comparison of models obtained by WASA and WASA2 defined in Table 5 is

shown in Figure 9. The AA Specification approach is improved approximately by 10% of the member capacity due to the modification of WASA. In addition, the difference between the AA specification and the finite element analysis is significantly decreased.

In Figure 10, the currently available two approaches in AA (2000a, MMCA and WASA) are compared to the ones developed in this study (WASA2 and TMCA). As seen in the figure, a significant improvement is made for a wide range of slenderness. Since the difference between the WASA2 and TMCA is insignificant, either approach could be


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employed. The moment capacities obtained from all four approaches as well as the finite element analyses are listed in Table 6.

Table 6. Parametric study results

tw (mm) tf (mm) AA-Y-MMCA

y

MM

AA-Y-WASA

y

MM

AA-U-WASA2

y

MM

AA-U-TMCA

y

MM

FEM-4%

y

MM

FEM-8%

y

MM

y

cr

F =

F

25.400 42.342 1.000 1.023 1.309 1.305 1.375 1.378 0.233 25.400 31.750 1.000 1.031 1.277 1.279 1.347 1.340 0.289 25.400 21.158 1.000 1.047 1.223 1.229 1.290 1.280 0.384 25.400 15.875 0.971 1.037 1.157 1.163 1.240 1.230 0.473 25.400 12.700 0.892 0.990 1.105 1.111 1.200 1.190 0.566 12.700 42.342 1.000 1.012 1.289 1.281 1.314 1.305 0.259 12.700 31.750 1.000 1.017 1.251 1.249 1.280 1.270 0.335 12.700 21.158 1.000 1.025 1.171 1.172 1.210 1.200 0.469 12.700 15.875 0.954 0.992 1.085 1.087 1.150 1.140 0.581 12.700 12.700 0.871 0.929 1.013 1.014 1.090 1.090 0.679 8.458 42.342 1.000 1.008 1.282 1.272 1.290 1.275 0.268 8.458 31.750 1.000 1.011 1.241 1.238 1.256 1.240 0.351 8.458 21.158 1.000 1.017 1.151 1.151 1.180 1.170 0.506 8.458 15.875 0.948 0.975 1.056 1.057 1.100 1.100 0.644 8.458 12.700 0.863 0.905 0.975 0.975 1.020 1.030 0.767 6.350 42.342 1.000 1.006 1.276 1.267 1.276 1.260 0.349 6.350 31.750 1.000 1.009 1.233 1.230 1.240 1.220 0.358 6.350 21.158 1.000 1.013 1.136 1.136 1.170 1.150 0.524 6.350 15.875 0.945 0.966 1.034 1.035 1.080 1.080 0.677 6.350 12.700 0.860 0.892 0.946 0.947 0.998 1.000 0.817

0.3 0.4 0.5 0.6 0.7 0.80.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

MFE

M /

Map

proa

ches

MFEM / Mapproaches mean c.o.v

MFEM / MAA-Y-MMCAMFEM / MAA-Y-WASAMFEM / MAA-U-WASA2MFEM / MAA-U-TMCA

1.248 0.055

1.209 0.056

1.04 0.024

1.04 0.021

MTEST / Mapproaches

Fig. 10. Comparison between the current and proposed approaches (u = 4%) All the finite element computations in Figures 8 to 10 are made when the ultimate

strain is 4%. The percent symbol in the subscript of FEM analyses in Table 6, 4% or 8%,

= y crF F


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represents the ultimate strain value used. When the ultimate strain is 8%, the results do not change significantly as compared in Table 6.

Experiments and FEM Simulation To support the approaches developed in this study, physical tests are conducted

for three doubly symmetric AA standard I-shaped sections; I-3x1.64. The alloy and temper of the specimens is 6063-T6, of which the minimum material properties are listed in Table 3. Since the study is based on the strength of component elements, continuous lateral supports are required. However, such supports are virtually impossible for practical tests. For this reason, a parametric study is conducted using the finite element method to find an appropriate lateral support spacing so that the ultimate load factor and the corresponding displacement are similar to those of continuous one: The study results in 304.8 mm (12 in.) for the lateral support spacing. The determined test setup is shown in Figure 11 together with the dimensions for the tested specimens.

Fig. 11. (a) Dimensions of section -3x1.64 (b) schematic test setup side-view

All dimensions are in mm and not to scale

The residual deformation is shown in Figure 12 when the specimens were removed from the test frame. A single ripple is formed near the span center in each specimen.

The bending test setup is simulated numerically using the finite element method. Bi-linear spring elements are attached between spreader plates and the specimen so that only compression can be transferred. This is to simulate the contact behavior of the actual test setup, in which the spreaders were simply placed on the specimen without any moment connections such as welding and bolting.

Two types of elements are used: Four noded linear shell elements with reduced integration are used for the SHELL model, while twenty noded quadratic hexahedral solid elements with reduced integration for the SOLID model. The SOLID model is composed of two layers of solid elements through the thickness.

Initial geometric imperfections are generated using elastic eigen-value analyses with a maximum amplitude based on either the actual measurement in this study (0.048 mm, model FEM 1) or the industrial production limitations (0.127 mm, model FEM 2 to 3) by ASD (2000c). In order to fairly compare the finite element simulations with the tests, the median of five tensile coupon test results obtained from one of the specimens is introduced into the finite element analyses.


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Fig. 12. Residual deformation of the tested specimens

Fig. 13. Deformed shape near failure using finite element method simulation (SOLID)

0 50 100 1500

0.2

0.4

0.6

0.8

1

1.2

1.4

DESCRIPTION IMP MAX-LF DISP-XL MODEL TEST 1TEST 2TEST 3 FEM 1 FEM 2 FEM 3

0.0480.1270.127

1.21.2751.3011.2381.2181.204

109.2120.7127.7121.2101.9131.1

SHELL SHELL SOLID

DISP-XL = Displacement at the maximum LF

SCVD (Span Center Vertical Displacement, mm)

LF =

Mu

/My

Fig. 14. Load factor-displacement result comparison for -3x1.64

A deformed shape near the failure from one of the finite element simulations exampled in Figure 13 is similar to the ones from the physical tests shown in Figure 12. The load factor-displacement curves obtained from both physical tests and the finite element simulations show close agreements each other as plotted in Figure 14. The variation in the test results would be mainly due to the variations of the material properties as well as initial geometric imperfections.


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The average of the maximum load factors obtained from the physical tests is compared to the current AA approaches (MMCA and WASA) and those developed in this study (WASA2 and TMCA) in a dashed oval of Figure 10. The test results follow the trend of the parametric study, which supports the validation of the proposed approaches in this study.

Application to the AA Specification It is shown in this study that the limit state stress based on the ultimate-plastic

capacity works well with both the finite element simulations and physical tests. However, it is desirable to maintain a certain factor of safety against yielding in actual structural designs.

As summarized in Table 7, the AA Specification (2000a) allows choosing the minimum of the allowable stresses based on the limit states of the yield and ultimate stresses for tension component elements. However, there is only one allowable stress based on the limit state of the yield stress for compression component elements.

Procedure I, which is one of the proposed approaches shown in Table 7, is almost the same as the current AA Specification except that allowable stresses for both limit states are available not only for the tension but also for compression component elements due to the development in this study. In this approach, the same factor of safety against yielding is maintained as the AA Specification, which sets a fixed number, 1.65 for building type structures. However, it does not seem to be reasonable to employ the uniform safety factor regardless of the margin between the yield and ultimate stresses for a wide variety of alloy-temper combinations.

Thus, an alternative approach to compute the allowable stress is proposed for an additional inelastic reserve capacity: Procedure II in Table 7. In this approach, 25% of the margin between the allowable stresses based on the yield and ultimate limit states is added to the allowable stress based on the yield limit state. For this reason, the safety factor against yielding varies depending on the ultimate to yield stress ratio. The safety factor against yielding can be defined as

y yy

a

Fn

F= (13)

Table 7. Allowable stress in the AA Specification and the proposed approaches

approaches allowable stress for tension component element allowable stress for compression

component element current AA

Specification ( )min ,a ay auF F F= Fa = Fay Procedure I ( )min ,a ay auF F F= (14) proposed

approaches Procedure II ( ) ( )0.25 min 1.25 ,a ay au ay ay auF F F F F F= + (15) Note: Fa = the (final) allowable stress

Fay = the allowable stress based on the yield limit state Fau = the allowable stress based on the ultimate limit state See Table 8 for details.


Paper 1 - 16

Table 8. Details of allowable stress equations for (a) tension element (b) compression element and (c) shape factors

(a)

AA Section allowable stresses

ay y ty yF F n= 3.4.2 3.4.4 ( )au u ty t uF F k n=

(b) AA

Section allowable stresses

1b t S limit S1

allowable stresses 1 2S b t S

limitS2

allowable stresses 2S b t

y cyay

y

FF

n= y cyB F

D

1

ayy

bF B Dn t

= 3.4.15 3.4.16 3.4.18 u cy

auu

FF

n= yu

nu cynB F

D

1

auy

bF B Dn t

= 1k BD

2ay au

y

k BEF F bnt

= =

(c) AA Section yield shape factor ultimate shape factor

3.4.2 1.0y = u tu tyF F = 3.4.4 1.3y = 1.25 0.2u tu tyF F = + a

3.4.15, 3.4.16 1.0y = u tu cyF F = b 3.4.18 1.3y = 1.25 0.2u tu cyF F = + b

Note a. In the AA Specification, = 1.42u tu tyF F . b. Not available in the AA Specification

c. For other coefficients, see the AA Specification

1 1.5 2 2.5 3 3.5 40.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

mean stdev min maxny~ 1.54 0.129 1.32 2.01

ny = 1.65~

ny = 1.32~

ny = 1.00~

kt = 1.00kt = 1.10kt = 1.25

0 200 400

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

(a) (b)

Fig. 15. Safety factor of the tensile allowable stress (AA 3.4.4) for a plate under bending

yn

tu tyF F numbers of alloy-temper


Paper 1 - 17

In Figure 15a, the safety factors against yielding of a tension side rectangular web element under bending are computed and plotted using solid circles for the previously selected 986 alloy-temper combinations when Procedure II is used (AA Section 3.4.4). It is found that the average safety factor (1.54) is only 6.7% smaller than the current fixed safety factor (1.65). Solid and dashed curves represent the analytic expressions of the safety factors for the materials with different notch sensitivities (kt). It is also noted that the majority of the alloy-temper combinations are concentrated near the average as shown in Figure 15b. In addition, the minimum safety factor is set to 1.32.

Conclusions The cut-off based on the yield stress in the AA Specification is replaced with the

ultimate shape factor developed in this study, which is eventually simplified for practical design purposes. With the shape factor, the compressive limit state stress of a component element can be evaluated more precisely. In addition, the modification is more consistent with the specification approach for tension elements.

Secondly, the empirically developed weighted average stress approach (WASA) is investigated. From the study, it is found that the weighted average stress equation is an approximation of a linearized ultimate bending stress at the centroid of the flange, while the current specification equation is considered to be the bending stress at the extreme fiber; a simple modification is proposed to the WASA.

As a result of the study, moment capacities of laterally supported flexural members are evaluated more accurately, when a series of parametric study is conducted for a wide range of slenderness using the finite element method. The study is also further validated by the flexural tests of standard sections.

In addition, two procedures are suggested for the final form of the future specification. In these approaches, a certain factor of safety against yielding is maintained. Procedure II is more reasonable, since the amount of additional inelastic reserve capacity used in the approach depends on the margin between the ultimate and yield stresses. In the proposed approaches, the current frame of the specification is maintained.

Acknowledgement This study is sponsored by the U.S. Department of Energy and the Aluminum Association. Their sponsorship is gratefully acknowledged.

References The Aluminum Association (2000a). The Specification for Aluminum Structures. The Aluminum Association (2000b). The Aluminum Design Manual. The Aluminum Association (2000c). Aluminum Standards and Data. Aluminum Company of America (Alcoa) (1955). Alcoa Structural Handbook, A Design

Manual for Aluminum. Clark, J.W., Rolf, R.L. (1966). Buckling of Aluminum Columns, Plates, and Beams.

Journal of the Structural Division, ASCE, Vol. 92, Proc. Paper 4838. Eberwien, U., Valtinat, G. (2001). The fullness method: A direct procedure for

calculation of the bending moment of a symmetrical aluminum cross section. The


Paper 1 - 18

8th International Conference in Aluminum (INALCO) in Munich, Germany, March 28-30

Jombock, J.R., Clark, J.W. (1968). Bending Strength of Aluminum Formed Sheet Members. J. of the Structural Div., ASCE, Vol.94, No. ST2, Proc. Paper 5816.

Kissell, J.R., Ferry, R.L. (1995), Aluminum structures, John Wiley & Sons, Inc. Stowell E.Z. (1948). A Unified Theory of Plastic Buckling of Columns and Plates.

NACA Technical Note 1556, National Advisory Committee for Aeronautics. U.S. Dept. of Defense (DOD) (1994), Metallic Materials and Elements for Aerospace

Vehicle Structures. Military Handbook Vol. 1.

Nomenclature f = the ultimate shape factor for flange u = the ultimate shape factor w = the ultimate shape factor for web y = the yield shape factor w8% = the ultimate shape factor for web when the ultimate strain is 8% = variable strain u = strain at the ultimate stress = the ultimate strain = constant for equivalent slenderness ratio depending on the plate buckling coefficient = slenderness factor = (Fy/Fcr)0.5 p = equivalent slenderness ratio = Poissons ratio Af = entire compression side flange area Aw = entire web area B = buckling formula constant intersecting vertical axis for zero width-to-thickness ratio b = the width of a plate element or a flange element D = buckling formula constant for slope of the inelastic buckling range E = Youngs modulus f = variable stress F = limit state stress for the flange or web Fa = the (final) allowable stress Fau = the allowable stress based on the ultimate limit state Fay = the allowable stress based on the yield limit state Fcr = minimum buckling stress Ff = limit state stress for the flange Fp = limit state stress of a component plate element Ftu = tensile ultimate stress Fty = tensile yield stress Fu = the ultimate stress Fw = limit state stress for the web Fy = the yield stress h = depth of the web element or the entire depth of an I-shaped section hc = depth of an I-shaped section between centerlines of flanges ho = depth of an I-shaped section between inner surfaces of flanges I = total moment of inertia


Paper 1 - 19

If = moment of inertia of the flange elements Iw = moment of inertia of the web element k1 = a coefficient to determine slenderness limit S2 k2 = a coefficient to determine limit state stress when width-to-thickness is larger than S2 kt = a notch sensitivity factor kp = plate buckling coefficient LF = load factor = Mu/My or Pu/Py Mf = moment capacity of the flanges Mn = moment capacity obtained from each approach Mu = total moment capacity or moment capacity from FEM or test Mw = moment capacity of the web n = an exponent used for Ramberg-Osgood equation nu = factor of safety on ultimate stress ny = factor of safety on yield

yn = variable factor of safety on yield S = section modulus S0 = width-to-thickness distinguishing proposed yielding and inelastic buckling range S1 = width-to-thickness distinguishing yielding and inelastic buckling range S2 = width-to-thickness distinguishing inelastic and elastic buckling range t = thickness of a plate element tf = flange thickness tw = web thickness w = width of an entire flange of an I-shaped section

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Laterally supported aluminum flexural members with symmetric cross sections