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SIMPLIFIED MODELING OF V-SHAPED BRIDGE PIERS
V 型桥墩的简化模型

HANS DE BACKER, AMELIE OUTTIER, and PHILIPPE VAN BOGAERT
HANS DE BACKER、AMELIE OUTTIER 和 PHILIPPE VAN BOGAERTDept of Civil Engineering, Ghent University, Ghent, Belgium
比利时根特根特大学土木工程系

Abstract  摘要

Designing a steel, V-shaped bridge pillar requires a complex and detailed finite element calculation model. The calculation can be simplified by using a simple one-dimensional beam model, but then the stress concentrations are not determined in detail. The use of a stress concentration factor, SCF, to make this possible. Before using this method, it is first verified whether both models behave correspondingly, by calculating the internal forces certain sections. Five basic load cases are used to perform this verification. Afterwards the stress concentration factors are determined for those same load cases. This method induces inconsistencies, some predictable but others unpredictable. The conclusion can however be that this simplified meth gives unreliable results. Therefore, a second method is used, where the high stresses in the curvatures of the pillar are related to the stresses in a nearby neutral section. Such a neutral section is defined as a section where the stress peaks due to the geometric effects caused by the actual geometry of the nodes of the V-shaped pilar are not present anymore. This method provides better results for the SCF; however, some issues remain, which are discuss in more detail in the paper. Finally, the impact of a varying radius of curvature on the stresses is also studied. Some clear trends are observed in this paper Overall, this paper discusses a possible simplified design method for V-shaped bridge pillars, at the same time listing the relevant design difficulties and possible pitfalls of a too simplified model.
设计钢制 V 型桥柱需要复杂而详细的有限元计算模型。使用简单的一维梁模型可以简化计算，但这样就无法详细确定应力集中。使用应力集中系数（SCF）可以实现这一点。在使用这种方法之前，首先要通过计算某些截面的内力来验证两种模型的行为是否一致。在验证过程中使用了五种基本载荷情况。然后再根据这些负载情况确定应力集中系数。这种方法会导致不一致，有些是可以预测的，有些则是无法预测的。但结论是，这种简化方法得出的结果并不可靠。因此，我们采用了第二种方法，即把支柱弯曲处的高应力与附近中性截面的应力联系起来。这种中性截面的定义是，由于 V 型支柱节点的实际几何形状所产生的几何效应而导致的应力峰值不再存在的截面。这种方法为 SCF 提供了更好的结果，但仍存在一些问题，本文将对此进行更详细的讨论。最后，本文还研究了曲率半径变化对应力的影响。总之，本文讨论了 V 型桥柱可能的简化设计方法，同时列出了相关的设计难点以及过于简化的模型可能存在的隐患。

Keywords: Stress concentration factors, Fatigue, Simplifications, Bridge supports, Steel.
关键词应力集中系数、疲劳、简化、桥梁支架、钢材。

1 INTRODUCTION  1 引言

V-shaped piers are becoming rather popular for long viaducts, from the 90 's on, up to date. However, most of these piers are concrete, the steel alternative being rare. The need for rapid construction increases the popularity among designers to consider steel V-shaped pillars. The latter show considerable horizontal resistance to braking and acceleration loads. In buildings they are sometimes valued for their smooth shape. The pillars for the new bridge across the Ourthe in Tilff are a steel example, however this one is placed orthogonal to the expected direction. Steel Vshaped pillars are more complex to design than the concrete equivalent. A detailed plate model is required to take the local effects of the curvatures at the corners into account (BAM Galere 2018).
从上世纪 90 年代至今，V 型桥墩在长高架桥上越来越受欢迎。然而，这些桥墩大多为混凝土结构，钢结构桥墩并不多见。由于需要快速施工，设计人员更倾向于考虑采用钢制 V 型支柱。后者对制动和加速荷载具有相当大的水平阻力。在建筑物中，它们有时因其平滑的形状而受到重视。蒂尔夫市横跨乌尔特河的新桥的支柱就是一个钢支柱的例子，不过它是与预期方向正交放置的。钢制 V 形支柱的设计比混凝土支柱更为复杂。需要一个详细的钢板模型来考虑四角曲率的局部影响（BAM Galere，2018 年）。

2 FINITE ELEMENT MODELLING - PLATE MODEL
2 有限元模型--板模型

The design of the plate model is based on plans of an existing V-column in a building and the method of Janssens (2020). The dimensions of this column are then scaled up to obtain realistic dimensions of a bridge pillar. The dimension in the frontal plane is shown in Figure 1, where no curvatures are added yet. Curvatures are added in all the corners, creating a smoother shape. This shape is then transformed into the 3D model of the pillar. The 2D view is tilted slightly forward,
钢板模型的设计基于一栋建筑中现有 V 型柱的平面图和 Janssens（2020 年）的方法。然后将该支柱的尺寸按比例放大，以获得桥梁支柱的实际尺寸。正面的尺寸如图 1 所示，其中尚未添加曲率。在所有角落都添加了曲率，使形状更加平滑。然后将此形状转化为支柱的三维模型。二维视图略微向前倾斜、
as the depth of the base is 1500 mm while at the top it is 1000 mm . Afterwards the top beam is added, which has a height of 714 mm .
因为底座深度为 1500 毫米，而顶部深度为 1000 毫米。然后再加上顶梁，其高度为 714 毫米。
Figure 1. The dimensions of the plate model in the front plane (left) and the detailed plate model (right).
图 1.平板模型前平面尺寸（左）和平板模型细节（右）。
In this paragraph the Von Mises stresses are analyzed, to observe the overall behavior of the pillar (The European Union 2003). Figure 2 (left) plots the Von-Mises stresses σ E + σ E + sigma_(E+)\sigma_{\mathrm{E}+} under the general load case. The subscript + indicates the outer fiber on which the stresses occur. The stresses are observed at the critical locations, so around the curvatures. The same method of obtaining the stresses is also used for the rest of this paper, which will focus on these critical locations. High stresses occur at the upper plate of the top beam, related to the application of the loads. This local effect is not studied as it is not the objective of this paper, but also because these are easily solved by locally increasing the plate thickness. The peaks in the curvatures are more interesting and will be the focus. Figure 2 (right) gives a closer look at one of these critical locations. A peak stress occurs at the connection of the curved plate with the front and the back plate. The peak stresses itself cannot be used, as these are unrealistic. An average stress is used over a distance equal to the plate thickness. Multiple lines are drawn starting in the stress peak, so at the plate edges, and are oriented away from it.
本段分析了 Von Mises 应力，以观察支柱的整体行为（欧洲联盟，2003 年）。图 2（左）绘制了一般载荷情况下的 Von Mises 应力 σ E + σ E + sigma_(E+)\sigma_{\mathrm{E}+} 。下标 + 表示发生应力的外层纤维。应力出现在临界位置，即曲率周围。本文其余部分也将采用同样的方法获取应力，并将重点关注这些关键位置。顶梁上板的应力较大，这与施加的荷载有关。本文不对这种局部效应进行研究，因为这不是本文的目的，而且通过局部增加板厚也很容易解决这些问题。曲率的峰值更为有趣，也将是研究的重点。图 2（右）对其中一个临界点进行了仔细观察。应力峰值出现在曲面板与前后板的连接处。峰值应力本身不能使用，因为这是不现实的。使用的是与板厚度相等距离内的平均应力。在应力峰值处绘制多条线，这些线位于板边缘，并远离应力峰值。
Figure 2. Von-Mises stresses under a general load case (left) and in one of the curvatures (right).
图 2.一般载荷情况下的 Von-Mises 应力（左图）和其中一个曲率下的 Von-Mises 应力（右图）。

3 FINITE ELEMENT MODELLING - BEAM MODEL
3 有限元模型 - 梁模型

The beam model consists out of four elements: a horizontal beam on top, two oblique elements on the side and a vertical element on the bottom. These are modelled in SCIA Engineer as lines with a cross-section corresponding to the plate model. These lines run through the center of the corresponding parts of the plate model, shown as the green lines in Figure 3. The oblique elements must be lengthened to connect to the top beam and to each other at the bottom. As a result, the vertical element at the bottom is required to be shorter. Determining the cross-sections of the top beam and the vertical element is straight forward, as these both have a rectangular cross-section. The oblique elements require some attention. The cross-section varies along the length, but these must continue further on than the plate model. The linear shape of the plates must be extended to find the cross-section at both ends. This is shown in red on Figure 3. External forces and moments on the pillar are shown in Figure 4.
梁模型由四个元素组成：顶部的水平梁、侧面的两个倾斜元素和底部的垂直元素。这些元素在 SCIA Engineer 中被模拟为与板模型截面相对应的线条。这些线穿过板模型相应部分的中心，如图 3 中的绿色线所示。斜向元素必须加长才能与顶部横梁相连，并在底部相互连接。因此，底部的垂直元素需要缩短。确定顶梁和垂直构件的横截面非常简单，因为它们都是矩形横截面。需要注意的是斜构件。横截面沿长度方向变化，但必须比板模型更长。必须扩展板的线性形状，以找到两端的横截面。这在图 3 中以红色显示。支柱上的外力和力矩如图 4 所示。
Figure 3. Geometry of the beam model.
图 3.横梁模型的几何形状。
Figure 4. External forces and moments on the pillar.
图 4.支柱上的外力和力矩

3 STRESS CONCENTRATION FACTORS
3 应力集中系数

Now that both models are modelled as corresponding, a comparison between them can be searched. This is done by means of stress concentration factors. These factors provide a number with which a stress value in the neutral section of the beam model must be multiplied to find the actual stress
既然两个模型都是相应的，那么就可以对它们进行比较。这可以通过应力集中系数来实现。这些系数提供了一个数字，梁模型中性截面上的应力值必须与这个数字相乘，才能得出实际应力
in the critical zones of the plate model. In this way, the designer only must use a simplified beam model to get to know the critical stresses.
在板模型的临界区。这样，设计人员只需使用简化的梁模型即可了解临界应力。
To find the stress concentration factors, the neutral sections as verified in Section III will be used. Because it was proven the plate model only corresponds to the beam model for the side on which the forces are applied, the following neutral sections shown in Figure 5 will be used. Section C is from the right beam, but this will be used to see which one of the sections B or C is best to compare with for curvature R 3 R 3 R_(3)\mathrm{R}_{3}.
为了找到应力集中系数，我们将使用第三节验证过的中性截面。由于已证明板模型只与受力侧的梁模型相对应，因此将使用图 5 所示的以下中性截面。C 截面来自右侧横梁，但这将用于查看 B 或 C 截面中的哪一个最适合与曲率 R 3 R 3 R_(3)\mathrm{R}_{3} 进行比较。
Figure 5. The neutral sections used in the beam model (left), the curvatures used in the plate model (middle), and the locations of the curvatures and neutral sections(right).
图 5.梁模型中使用的中性截面（左）、板模型中使用的曲率（中）以及曲率和中性截面的位置（右）。
Curvature R4 is split up to be able to compare them to section C or D. In the end, no good results are found for this SCF. The beam model gives an average stress over the whole cross section which results in a lot of sign changes in the stresses. Local compression in the plate model and tension as the average in the beam model cannot be trusted to compare with each other. A more accurate approach should be used to find an SCF.
曲率 R4 被分割开来，以便与 C 或 D 截面进行比较。梁模型给出了整个横截面的平均应力，这导致应力的符号变化很大。板模型中的局部压缩和梁模型中的平均拉伸不能相互比较。应该使用更精确的方法来找到 SCF。
Table 1. The SCFs of the plate model.
表 1.板块模型的 SCFs。
Curvature  曲率 Stress  压力 Average SCF  平均 SCF Diff.
R1 σ x σ x sigma_(x)\sigma_{\mathrm{x}} 2.55 1.41
σ y σ y sigma_(y)\sigma_{\mathrm{y}} 4.66 3.46
τ xy τ xy tau_(xy)\tau_{\mathrm{xy}} 3.99 0.25
R2 σ x σ x sigma_(x)\sigma_{\mathrm{x}} 1.00 0.25
σ y σ y sigma_(y)\sigma_{\mathrm{y}} 4.17 6.27
τ xy τ xy tau_(xy)\tau_{\mathrm{xy}} 1.72 1.42
R4 σ x σ x sigma_(x)\sigma_{\mathrm{x}} 1.88 1.48
σ y σ y sigma_(y)\sigma_{\mathrm{y}} 8.04 1.08
τ xy τ xy tau_(xy)\tau_{\mathrm{xy}} 3.99 0.25
Curvature Stress Average SCF Diff. R1 sigma_(x) 2.55 1.41 sigma_(y) 4.66 3.46 tau_(xy) 3.99 0.25 R2 sigma_(x) 1.00 0.25 sigma_(y) 4.17 6.27 tau_(xy) 1.72 1.42 R4 sigma_(x) 1.88 1.48 sigma_(y) 8.04 1.08 tau_(xy) 3.99 0.25| Curvature | Stress | Average SCF | Diff. | | :--- | :---: | :---: | :---: | | R1 | $\sigma_{\mathrm{x}}$ | 2.55 | 1.41 | | | $\sigma_{\mathrm{y}}$ | 4.66 | 3.46 | | | $\tau_{\mathrm{xy}}$ | 3.99 | 0.25 | | R2 | $\sigma_{\mathrm{x}}$ | 1.00 | 0.25 | | | $\sigma_{\mathrm{y}}$ | 4.17 | 6.27 | | | $\tau_{\mathrm{xy}}$ | 1.72 | 1.42 | | R4 | $\sigma_{\mathrm{x}}$ | 1.88 | 1.48 | | | $\sigma_{\mathrm{y}}$ | 8.04 | 1.08 | | | $\tau_{\mathrm{xy}}$ | 3.99 | 0.25 |
Therefore, the SCF of the plate model is searched. This has the advantage that it can use the maximum in the neutral section and not the average. This would lead to a more representable value as neutral stress. Because the beam model will be forgotten in this section, the curvatures R1 and R2 on the other side of the structure are used. These curvatures are located further from the application of the forces which could influence the results. The curvatures and neutral sections tested are shown in Figure 5. The SCF’s found using this method are much better than the ones found while comparing to the beam model. Neglecting some useless results, a constant SCF could be found except for curvature R3. The other results are added in Table 1, which gives the average value and the difference of the maximum and minimum value of the considered SCF.
因此，要搜索板模型的 SCF。这样做的好处是可以使用中性截面上的最大值而不是平均值。这将使中性应力值更具代表性。由于梁模型在本节中将被遗忘，因此使用了结构另一侧的曲率 R1 和 R2。这些曲率距离受力位置较远，可能会影响结果。测试的曲率和中性截面如图 5 所示。使用这种方法得出的 SCF 值比梁模型得出的 SCF 值要好得多。忽略一些无用的结果，除了曲率 R3 外，都能找到恒定的 SCF。其他结果见表 1，其中给出了所考虑的 SCF 的平均值以及最大值和最小值之差。
At last, the radii of the curvatures are changed to see what effect this has on the SCF. For every curvature, four other magnitudes are tested, two larger and two smaller than the original radius.
最后，改变曲率的半径，看看这对 SCF 有什么影响。对于每一个曲率，都测试了其他四个量级，其中两个比原来的半径大，两个比原来的半径小。
Due to the different radii per curvature, it can be easily seen on a graph at which location the best neutral section occurs by looking for stresses over the whole length of the beam. The best location for a neutral section will then be the point where the stresses in function of the length of the beam stop being linear. Figure 6 shows the results for section A near curvature R1 as an illustration. The different curves are the maximum Von-Mises stresses for the different radii of curvature, over the length of the top beam due to external load F x F x F_(x)\mathrm{F}_{\mathrm{x}}. The location of the neutral section is shown in red.
由于曲率半径不同，通过观察横梁整个长度上的应力，可以很容易地从图表上看出最佳中性截面的位置。因此，中性截面的最佳位置将是应力与梁长度的函数关系不再呈线性关系的点。图 6 显示了靠近曲率 R1 的 A 截面的结果。不同的曲线是外部荷载 F x F x F_(x)\mathrm{F}_{\mathrm{x}} 作用在梁顶长度上不同曲率半径的最大 Von-Mises 应力。中性截面的位置用红色表示。
Figure 6. The determination of the location of a neutral section.
图 6.中性截面位置的确定。
This is done, but it turned out the SCF is worse at these optimized neutral sections. Therefore, it is chosen to continue with the maximum of the stresses instead of the SCF’s from this point on. They will give the exact same curves in function of the radius and represent the absolute values.
但结果表明，在这些优化的中性截面上，SCF 更差。因此，从这一点出发，我们选择继续使用最大应力，而不是 SCF。它们将给出与半径函数完全相同的曲线，并代表绝对值。
The results of curvature R1 are used as an example to show this. Figure 7 shows the results in curvature R1 for respectively σ x σ x sigma_(x)\sigma_{x} and σ y σ y sigma_(y)\sigma_{y}. Only the stresses in curvature R1 are considered, as the stresses at the other locations are not influenced by a varying radius R1.
以曲率 R1 的结果为例说明这一点。图 7 分别显示了 σ x σ x sigma_(x)\sigma_{x} 和 σ y σ y sigma_(y)\sigma_{y} 时曲率 R1 的结果。只考虑了曲率 R1 的应力，因为其他位置的应力不受半径 R1 变化的影响。
Figure 7. Variation of the stresses σ x σ x sigma_(x)\sigma_{x} (left) and σ y σ y sigma_(y)\sigma_{y} (right) in R1.
图 7.R1 中应力 σ x σ x sigma_(x)\sigma_{x} （左）和 σ y σ y sigma_(y)\sigma_{y} （右）的变化。
The stress results generally tend to curve towards zero for an increasing radius, by following a parabolic shape. This is very clear for σ y σ y sigma_(y)\sigma_{y} under the external load F y F y F_(y)F_{y}. But the stresses σ x σ x sigma_(x)\sigma_{x} under the horizontal load F x F x F_(x)F_{x} remain constant in the considered interval. Also, the stress σ y σ y sigma_(y)\sigma_{y} under the external moment M z M z M_(z)\mathrm{M}_{\mathrm{z}} increases slightly when the radius of curvature is increased.
当半径增大时，应力结果一般呈抛物线形，趋向于零。在外部载荷 F y F y F_(y)F_{y} 作用下， σ y σ y sigma_(y)\sigma_{y} 的应力变化非常明显。但水平荷载 F x F x F_(x)F_{x} 作用下的应力 σ x σ x sigma_(x)\sigma_{x} 在所考虑的区间内保持不变。此外，当曲率半径增大时，外部力矩 M z M z M_(z)\mathrm{M}_{\mathrm{z}} 作用下的应力 σ y σ y sigma_(y)\sigma_{y} 也略有增加。
In most cases this results in a parabolic curve where the stresses decrease in function of a larger radius. In some cases, it is the other way around and the stresses increased for an increasing radius. Due to the small number of tests, it is not clear yet if this is an error or if this phenomenon is real.
在大多数情况下，这会形成一条抛物线，即半径越大，应力越小。在某些情况下，情况恰恰相反，半径越大，应力越大。由于测试次数较少，目前还不清楚这是误差还是真实现象。
At first sight, no errors occurred in these results, and it happened more than once. Overall, the tested radii are too large, and the range might be too small to see a clear curve. When looking to much smaller radii, a better approach can be used to make conclusions as of now, some cases seem to stay linear which seems not realistic.
乍一看，这些结果并没有出现错误，而且还不止一次。总的来说，测试的半径太大，范围可能太小，无法看到清晰的曲线。如果半径更小，可以采用更好的方法来得出结论。

4 CONCLUSIONS  4 结论

To use an SCF factor relating the plate model to the beam model, both are required to correspond to each other. This is only the case under some conditions, as shown in Section III. The determination of the SCF’s also led to reliability issues related to the obtained stresses. Thus, the use of this method does not seem as a safe and reliable method.
要使用与板模型和梁模型相关的 SCF 因子，需要两者相互对应。如第三节所示，这只是在某些条件下的情况。SCF 的确定也会导致与所获应力相关的可靠性问题。因此，使用这种方法似乎并不安全可靠。
If the SCF are related to the stresses in the plate model itself, the results improve. This method seems as a more stable and trustworthy method.
如果 SCF 与板模型本身的应力相关，结果会有所改善。这种方法似乎更稳定、更可靠。
Thirdly some conclusions can be made about the effect of a varying curvature on the stresses. Changing the curvature in one location does not influence the results at other curvatures. For most cases, the stresses decrease with an increasing radius. However, this could not always be proven for some specific scenarios where the stresses increase with an increasing curvature.
第三，可以就曲率变化对应力的影响得出一些结论。改变一个位置的曲率不会影响其他曲率的结果。在大多数情况下，应力会随着半径的增大而减小。然而，在某些特定情况下，应力会随着曲率的增大而增大，这一点并不总能得到证实。
This master thesis shed some light on the possibilities, but also the difficulties of the design of V-shaped pillars. Although there are some issues, the results are still very promising. With possible future research the design of such type of pillars can be facilitated significantly, which drastically reduces the required amount of work.
这篇硕士论文揭示了 V 型支柱设计的可能性和困难。虽然还存在一些问题，但研究结果还是很有希望的。通过今后可能开展的研究，此类支柱的设计可以大大简化，从而大幅减少所需的工作量。

References  参考资料

BAM Galere, Remplacement du Pont de Tilff, BAM Galere, 2018.
BAM Galere，Remplacement du Pont de Tilff，BAM Galere，2018。
Janssens, S., Parametric Study of an Existing Arch Bridge, Focussed on the Buckling Behaviour, Master’s Thesis, Faculty of Engineering and Architecture, Ghent University, Ghent, Belgium, May, 2020.
Janssens, S.，《现有拱桥的参数研究，重点关注屈曲行为》，比利时根特大学工程与建筑系硕士论文，根特，比利时，2020 年 5 月。
The European Union, EN 1991-2: Eurocode 1: Actions on Structures - Part 2: Traffic Loads on Bridges, September, 2003.
《欧洲联盟，EN 1991-2：欧洲规范 1：结构上的作用--第 2 部分：桥梁上的交通荷载》，2003 年 9 月。