英國倫敦帝國理工學院南肯辛頓校區土木與環境工程系，SW7 2AZ

2006 年 12 月 14 日收到；2007 年 4 月 13 日收到修訂版；2007 年 4 月 23 日接受

2007 年 5 月 30 日可在線獲取

(2008) Chan and Gradner EHS.pdf

近年來，熱軋橢圓空心型材因其美觀和結構效率的互補性而受到工程師和建筑師的廣泛關注。然而，目前缺乏橢圓空心型材的設計指導，阻礙了其在建筑中的更廣泛應用。本文針對軸向壓縮的基本載荷條件解決了這一缺點。本文介紹了實驗室測試、數值建模和設計規則的制定。實驗計劃包括 25 個拉伸試樣試驗和 25 個短柱試驗。所有測試的橢圓空心型材的長寬比均為 2，截面尺寸范圍從 150 × 75 到 500 × 250 毫米。結果包括幾何缺陷測量和滿載端部縮短曲線。根據生成的測試數據開發并驗證了非線性有限元模型。使用經過驗證的數值模型進行參數研究，以研究不同細長和不同長寬比的橢圓空心型材。所得到的結構性能數據已用于建立橫截面細長和橫截面抗壓強度之間的關系，這表明，根據所提出的橫截面細長參數，歐洲規范 3 中圓形空心型材的 3 級細長極限 90 可以安全地用于橢圓形空心型材。BS 5950-1 中給出的等效半緊湊細長極限、AISC 360-05 中的非緊湊極限細長和 AS 4100 中給出的屈服細長極限也是有效的。BS 5950-1 中的改進有效面積公式也可以安全采用。目前正在進一步研究細長（4 級）橢圓形空心型材的有效面積公式。

使用有限元（Fe）軟件ABAQUS [9]的數值建模研究與實驗程序并行進行。該程序的主要目的是復制實驗壓縮測試，并驗證了模型，以進行參數研究。為Fe模型選擇的元素是四個節點的，減少的集成殼元素，每個節點的自由度六個自由度，在Abaqus元素庫中指定為S4R，適用于薄或厚的殼應用[9]。這些元素已被證明在類似的應用中表現良好[10-12]。通過基于彈性特征值預測進行網格收斂研究，仔細選擇了均勻的網格密度，以實現準確的結果，同時最大程度地減少計算工作。發現合適的網格尺寸為2A/10（A/B）×2A/10（A/B）毫米，上限為20×20 mm。

使用測量的試件尺寸和測量的材料應力-應變數據對短柱試驗進行建模。幾何缺陷的形式被認為是最低彈性特征模式模式，通常形狀對稱，圖 15 顯示了一個例子。除了測量的缺陷值外，缺陷幅度 w0 被認為是材料厚度 t 的三個固定分數（t/10、t/100 和 t/500）。沒有測量殘余應力數據，但在從橢圓形試件加工材料拉伸試樣時觀察到的可忽略不計的變形表明殘余應力很低。因此，殘余應力未納入本研究中的數值模型。真實應力-應變關系是從拉伸試樣試驗獲得的工程應力-應變曲線生成的，材料非線性通過分段線性應力-應變模型納入數值模型，以模擬應變硬化區域。對固定端模型施加了邊界條件，這是通過限制短柱底部的所有位移和旋轉以及短柱受力端除垂直位移以外的所有自由度來實現的；在整個分析過程中，都對垂直位移進行了監測。采用改進的 Riks 方法 [9] 來求解幾何和材料非線性短柱模型，從而可以追蹤卸載行為。

圖 16 顯示了 EHS 150 × 75 × 4-SC2 的數值失效模式，并與相應的變形試件進行了比較。表 4 列出了數值模擬的結果，其中顯示了不同缺陷水平下 FE 極限荷載與實驗極限荷載之間的比率，并進行了比較。試驗結果的復制令人滿意，數值模型能夠成功捕捉觀察到的剛度、極限荷載、一般荷載端部縮短響應和失效模式。圖 17 和 18 分別顯示了 EHS 150 × 75 × 4-SC2 和 EHS 150×75×5-SC1 的測試結果與 FE 結果之間的比較。無論缺陷幅度如何，數值模型始終低估了三個截面尺寸為 300 × 150 × 8.0 的短柱的極限荷載。這種低估的可能解釋包括橫截面周圍和短柱長度上的材料厚度變化以及材料屈服強度的變化（橫截面周圍或拉伸和壓縮性能之間）。

對缺陷的敏感性通常相對較低，較粗的截面顯示出最大的響應變化。例如，對于 EHS 150 × 75 × 8 模型，隨著缺陷幅度從 t/100 增加到 t/10，極限荷載降低了 20%。這種敏感性是由于在發生局部屈曲之前組成元件所達到的應變硬化水平。較不粗壯的截面位于屈服平臺之上或略低于屈服平臺，因此對局部屈曲點的變化不太敏感（就極限載荷而言）。對于屈服載荷和彈性屈曲載荷值相近的細長橢圓形空心截面，預計敏感性會增加。

在驗證了 FE 模型能夠復制長寬比為 2 的 EHS 測試行為的一般能力后，進行了一系列參數研究。參數研究的主要目的是調查橫截面細長和長寬比對極限承載能力的影響。根據對 150 × 75 × 6.3 截面進行的拉伸試樣試驗，開發了一個分段線性材料應力-應變模型，如圖 19 所示。非線性參數分析中的初始幾何缺陷采用最低彈性特征模態的形式，振幅 w0 為 t/100，這與測試結果最一致（表 4）。參數研究中考慮的截面尺寸為 150 × 150、150 × 100、150 × 75 和 150 × 50，厚度各不相同，以覆蓋橫截面細長范圍。該結果已用于驗證橢圓形空心截面所提出的細長參數和橫截面分類極限，并將在下一節中詳細討論。

A numerical modelling study, using the finite element (FE) package ABAQUS [9], was carried out in parallel with the experimental programme. The primary aims of the programme were to replicate the experimental compression tests and, having validated the models, to perform parametric studies. The elements chosen for the FE models were four-noded, reduced integration shell elements with six degrees of freedom per node, designated as S4R in the ABAQUS element library, and suitable for thin or thick shell applications [9]. These elements have been shown to perform well in similar applications [10–12]. A uniform mesh density was carefully chosen by carrying out a mesh convergence study based on elastic eigenvalue predictions with the aim of achieving accurate results whilst minimising computational effort. A suitable mesh size was found to be 2a/10(a/b) × 2a/10(a/b) mm with the upper bound of 20 × 20 mm.

The stub column tests were modelled using the measured dimensions of the test specimens and measured material stress–strain data. The form of geometric imperfections was taken to be the lowest elastic eigenmode pattern, typically symmetrical in shape, an example of which is shown in Fig. 15. The imperfection amplitude w0 was considered as three fixed fractions of the material thickness t (t/10, t/100 and t/500) in addition to the measured imperfection values. No residual stress data were measured, but the negligible deformation observed when the material tensile coupons were machined from the elliptical specimens indicated that the residual stresses were low. Therefore, residual stresses were not incorporated into the numerical models in this study. The true stress–strain relations were generated from the engineering stress–strain curves obtained from the tensile coupon tests and material non-linearity was incorporated into the numerical models by means of a piecewise linear stress–strain model to mimic, in particular, the strain-hardening region. Boundary conditions were applied to model fixed ends and this was achieved by restraining all displacements and rotations at the base of the stub columns, and all degrees of freedom except vertical displacement at the loaded end of the stub columns; this vertical displacement was monitored throughout the analysis. The modified Riks method [9] was employed to solve the geometrically and materially non-linear stub column models, which enabled the unloading behaviour to be traced. The numerical failure mode of EHS 150 × 75 × 4-SC2 is illustrated in Fig. 16 and compared with the corresponding deformed test specimen. Results of the numerical simulations are tabulated in Table 4, in which, the ratios between the FE ultimate load and the experimental ultimate load are shown and compared for different imperfection levels.

Replication of test results has been found to be satisfactory with the numerical models able to successfully capture the observed stiffness, ultimate load, general load–end shortening response and failure patterns. Comparison between test and FE results are shown for EHS 150 × 75 × 4-SC2 and EHS 150×75×5-SC1 in Figs. 17 and 18, respectively. The ultimate loads of the three stub columns of section size 300 × 150 × 8.0 are consistently under-predicted by the numerical models, regardless of the imperfection amplitude. Possible explanations for this under-prediction include variation of the material thickness around the cross-section and along the length of the stub columns and variation in material yield strength (either around the cross-section or between tensile and compressive properties).

Sensitivity to imperfections is generally relatively low, with the stockier sections showing the greatest variation in response. For example, in the case of the EHS 150 × 75 × 8 models, the ultimate load reduces by 20% with an increase of imperfection amplitude from t/100 to t/10. This sensitivity is due to the level of strain hardening achieved by the constituent elements before local buckling occurs. The less stocky sections lie on or marginally below the yield plateau and are therefore less sensitive (in terms of ultimate load) to variation in the point of local buckling. Increased sensitivity would be anticipated for slender elliptical hollow sections where the yield load and elastic buckling load were of similar value.

Having verified the general ability of the FE models to replicate test behaviour for EHS with aspect ratio of 2, a series of parametric studies were conducted. The primary aim of the parametric studies was to investigate the influence of cross-section slenderness and aspect ratio on the ultimate loadcarrying capacity. A piecewise linear material stress–strain model was developed from the tensile coupon tests conducted on the 150 × 75 × 6.3 sections, and is shown in Fig. 19. Initial geometric imperfections in the non-linear parametric analyses adopted the form of the lowest elastic eigenmode with an amplitude w0 of t/100, which has demonstrated the best agreement with the test results (Table 4). The section sizes considered in the parametric studies were 150 × 150, 150 × 100, 150 × 75 and 150 × 50 with varying thickness to cover a spectrum of cross-section slenderness. The results have been utilised for the validation of proposed slenderness parameters and cross-section classification limits for elliptical hollow sections and are discussed in detail in the following section.

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