写在前面：附录K的翻译早就结束了，现在整理出来供大家参考。此外，有个小问题目前还没发解决，参考文献没办法实现点击以后在文本内跳转，试了几种方法都不行，如果有大神知道怎么回事儿欢迎留言！

附录K      Strut-and-tir section     拉压杆截面

K.1 General     概述

In the strut-and-tie design method, an idealized truss is designed to carry forces through a discontinuity (D)-region as illustrated for the design of the deep beam that is shown in Figure K.1 a) and b). A simple strut-and-tie (or truss) model is selected for the flow of forces that consists of compressive struts that run from the point of loading to the supports, and a tension tie between these two supports that carries the horizontal components of these diagonal compression struts. After calculating the factored loads in the strut “C” and tie “T” ，the design involves ensuring that the compressive capacity of the struts (area）×（stress limit) is greater than “C”, and that an adequate amount of reinforcement is provided so that its tie strength (area)× （stress limit) is greater than “T”. In addition, the capacity of the nodes shall be checked.

在拉压杆设计方法中，对于深梁的设计可以构造一个理想化的桁架，通过一个不连续的D形区域来承受外力，如图 K.1 a) 和 K.1 b)所示。选择一个简单的拉压杆(或桁架)模型进行力流分析，该模型由从加载点延伸到支撑的压杆和这两个支撑之间的拉杆组成，拉杆承载这些对角压杆的水平分量。在计算支柱“C”和拉杆“T”中的系数荷载后，设计包括确保支柱的抗压能力(面积×应力极限)大于“C”，并提供足够数量的钢筋，以使其拉杆强度(面积×应力极限)大于“T”。此外，应验算节点的承载力。

a) Strut-and-tie model

b) Beam

Figure K.1 - Example for the design of a deep beam using the strut-and-tie method

As illustrated in Figure K.1 and Figure K.2 a), b) and c), for many cases the shape of a suitable strut-and-tie model is easy to identify.

如图 K.1 和图 K.2 a)， b)和 c)所示，在很多情况下合适的拉压杆模型的形状很容易识别。

a)trut-and-tie model

b) Strut-and-tie model

c) Strut-and-tie model

Figure K.2 - Simple shapes of strut-and-tie models

Even for such relatively simple design problems, there can be multiple suitable strut-and-tie model shapes as shown in Figure K.2. The basis of the strut-and-tie method gives the designer freedom in the selection of model shape, and providing that the selection is reasonable the member will have an ultimate strength that is equal to or greater than the calculated strength as the strut-and-tie method is a lower-bound design methodology. The performance of the designed structures under service and overloads will depend in part on the shape of the selected model. Thereby, designers with limited expertise in the theory and use of the strut­ and-tie method are referred to as design examples and guidance in [5] to [17].

即使对于这样相对简单的设计问题，也可以有多个合适的拉压杆模型形状，如图 K.2 所示。基于拉压杆（的模型）给设计人员提供了选择模型形状的自由，在合理选择的前提下，由于拉压杆法是一种下界设计方法，构件的极限强度将等于或大于计算强度。在正常使用和过载情况下，设计结构的性能将部分取决于所选模型的形状。因此，文献 [5] to [17] 对于拉压杆法的理论和应用知识有限的设计师可以作为设计实例和指导。

a) Strut-and-tie model     b) Strut-and-tie model                    c)Strut-and-tie model

Figure K.3 - Three examples for carrying load in a deep beam

a) 拉压杆模型              b) 拉压杆模型                 c) 拉压杆模型

K.2 Example of a rock anchor foundation     岩石锚杆基础实例

An example is now given for a somewhat more complicated design problem in which a complex 3-dimensional strut-and-tie model is needed to complete a suitable design. This example is for the design of a rock-anchor foundation for a wind tower. The plan view of the footing for this wind tower is shown in Figure K.4 a). As shown, 16 rock anchors are used to post-tension this foundation to the rock. Figure K.4 b) shows that this anchoring force would flow down to the supporting rock as a fan-shaped strut that spreads in a radial inward direction; compressive stress trajectories are shown as dashed lines. The spreading of this force would create a radial tension at the base of the footing that would have to be carried by Tie A; tie forces are shown as solid lines. While not shown in this figure, this anchoring force would also spread from the top of the foundation in a circumferential direction. In accordance with the strut-and-tie modelling approach, it would be required to check the stresses in the concrete struts (or node) that is beneath the anchor plates (post-tensioning force)/(area of plate) is less than the compressive stress limit for concrete under this bearing condition. There is no need to check the conditions of the concrete at the base of the footing as this stress would be smaller than that at the top.

现在给出了一个比较复杂的设计问题的例子，其中需要一个复杂的三维拉压杆模型来完成适当的设计。这个例子是设计一个风塔的岩石锚杆基础。风塔的基础平面图如图 K.4 a)所示。如图所示， 16 个岩石锚杆用于将基础后张拉至岩石上。从图 K.4 b)可以看出，这种锚固力会以扇形支柱的形式向下流向支护岩石，并沿径向向内扩散;压应力轨迹如虚线所示。这种力的扩散会在基底产生径向张力，这种张力必须由 a 拉杆来承受;拉杆力以实线表示。虽然没有在这个图中显示，这种锚定力也将从基础的顶部在圆周方向上扩散。根据拉压杆建模方法，在这种承载条件下，需要检查锚板下方的混凝土支柱(或节点)的应力(后张拉力/板面积)是否小于混凝土的极限压应力。没有必要检查基础底部的混凝土状况，因为这种应力会比顶部的应力小。

The tower is then bolted to this foundation via a ring of “tower anchor bolts” that extend from the top of the footing to an embedded anchor plate near the base of the foundation. Since the dimensions of the concrete are greater between the loaded plates than at the loaded plates, this compression would spread out between the plates as also shown in Figure K.4 b). A strut­ and-tie model for the flow of forces is shown in Figure K.4 b) in which Tie B and Tie C are needed at the turning points (joints or nodes) of the compressive struts; the centre point of these nodes is denoted by solid dots. Reinforcement is required to equilibrate these tie forces, and this could be provided by radial-oriented reinforcing bars at the location of the ties shown in Figure K.4 b). Note that the left and right sides of Figure K.4 b) show a slightly different location for Tie B; this is a feature of the strut-and-tie method that the designer selects a load path, and then reinforces for the selected load path. Some designers find this flexibility to be disconcerting, but providing that reasonable selections are made, the strut-and-tie method has proven that this flexibility can be accommodated with good performance.

然后通过一圈“塔锚栓”将塔架固定在基础上，这些“锚栓”从基础顶部延伸到靠近基础基础的嵌入式锚板上。混凝土上的加载区域的间距大于加载区域的尺寸，加载区域之间的压力会如图 K.4 b)中所示分散。拉压杆模型力流如图 K.4 b)，其中在压杆的转折点（结点或节点）处需要有拉杆B和拉杆C;这些节点的中心点用实点表示。为了平衡这些拉杆力，需要配筋，如图 K.4 b)所示，这些力由位于拉杆处的径向钢筋提供。应注意图 K.4 b)的左右两侧显示的拉杆 b 的位置略有不同;这是拉压杆方法的一个特点，由设计师选择一个加载路径然后对所选的加载路径进行加固。一些设计者发现这种灵活性令人不安，但如果进行了合理的选择，拉压杆法已经证明这种灵活性可以提供良好的性能。

When the tower supports a lateral load due to wind effects, a significant downward compressive force will be transmitted from the ring on one side of the tower, and a significant upwards tensile force will be transmitted on the opposite side of the tower, as shown in Figure K.4 c). On the tension side (right side in figure) , this force may exceed the preload level and resuIt in a tension force in the tie that runs from the top of the foundation through to the embedded plate near the bottom of the footing as shown in this figure. This tie force would be balanced at the joint (or node) at the bottom of this tie by the vertical component of diagonal compressive struts that spreads out from the base of the tie in the direction of the topside rock-anchor bolts and the other half of the footing. A portion of the diagonal compression force that runs towards the rock anchor is expected to run directly from the lower tower anchor plate to the rock anchor plate (strut” d” for direct), with the remainder to be carried by a truss type mechanism. The higher the shear-span-to-depth ratio, the larger the fraction to be taken via the truss mechanism. The fip (1999) Bulletin on the practical design of structural concrete [10] provides guidance for what portion of the load should be taken directly, what portion of the load should be taken indirectly, and how to distribute the tie reinforcement. The horizontal component of the diagonal compression forces that run to the node under the rock anchor plate shall be equilibrated by the horizontal tie at the top of the footing that is shown in Figure K.4 c).

塔架承受风致横向荷载时,一个显著的向下压力将沿一侧塔壁传递,同时塔壁另一侧有显著的向上拉力传递,如图 K.4 c)所示。拉力侧(图中右侧)可能会超过预加载水平，导致从基础顶部到基础底部附近的预埋板的拉杆中产生拉力，如图所示。这种拉力将在该拉杆底部的结点(或节点)处通过对角压杆的垂直分量进行平衡，这些斜压杆从拉杆的基部向上部岩石锚栓和另一半基础的方向展开。一部分斜向岩石锚的压缩力从塔架下部锚板直接传递到岩石锚板(支柱“d”为直接)，其余由桁架承担。剪切跨度与深度之比（译注：剪跨比）越大，通过桁架机理得到的部分越大。 fip(1999)公报对结构混凝土 [10] 的实际设计提供了指导，什么部分的荷载应直接采用，什么部分的荷载应间接采采用，以及如何分配拉杆钢筋。传递到岩石锚板下节点的斜向压缩力的水平分量应由基础顶部的水平拉杆来平衡，如图 K.4 c)所示。

a) Plan view of rock anchor foundation

b) Strut-and-tie model and anchoring

c) Strut-and-tie model for overturning moment

Figure K.4 - Strut-and-tie models for a rock-anchor foundation

The footing shall be designed for the summation of the loadings in Figure K.4 b) and Figure K.4 c), which may be taken as a simple linear superposition of demands. An effective radial tie could be provided by reinforcement that extends from beneath one rock anchor towards its counterpart on the opposite side of the footing. Since the wind may blow from any direction, a radial tie would be needed beneath and between all eight pairs of anchor plates. It is impractical to provide all tie reinforcement in this way as it would require eight different layers of ties; a viable design solution is to transition from radial ties to a square grid of ties as shown in Figure K.5. The strength and amount of reinforcement should be calculated using the web­ shaped vertical segment as shown in Figure K.4 c), and the grid reinforcement shouId be provided, with adequate splice length, so that the radial force component of the grid reinforcement is equal to or greater than the calculated demand.

基础应按照图 K.4 b)和图 K.4 c)中荷载的总和进行设计，可将荷载的总和简化取为各要求的线性叠加。一个有效的径向拉杆可以通过钢筋提供，从一个岩石锚的下面延伸到其对应的另一侧的基础。由于风可以从任何方向吹来，在所有 8 对锚板的下方及其之间需要一个径向拉杆。以这种方式提供所有的钢筋拉杆加固是不切实际的，因为它将需要8层不同的拉杆;一种可行的设计方案是将径向拉杆过渡到如图 K.5 所示的方格拉杆。钢筋强度和数量应按照图 K.4 c)所示的网状垂直段进行计算，并且网格钢筋应有足够的接头长度从而使得其径向分力不小于计算要求。

Figure K.5 - Top tie reinforcement in a rock-anchor foundation

Another challenge to this strut-and-tie design is that there are circumferential actions created due to the overturning moment and anchorages. As shown in Figure K.4 c), when the overturning moment creates an uplift force that is transmitted down to the anchor point on the inner ring, a diagonal strut carries part of this load to the outside ring of rock anchors. A component of this upward diagonal compression is in the circumferential direction because the outside ring is larger in diameter and because the anchorages are not in a continuous ring. This will lead to a circumferential tension force at the top of the footing under the rock anchors for which circumferentially oriented bars should be placed to carry this tension, as denoted by the hollow circles in Figure K.4 c).

这种拉压杆设计的另一个挑战来自于倾覆力矩和锚固产生的环向作用。如图 K.4 c)所示，当倾覆力矩产生一个向上的力并向下传递到内环上的锚点时，一个对角支撑将该力的一部分传递到锚的外环上。这种向上的对角压缩的一个分量是环向的，因为外圈的直径更大， 而且锚不是一个连续的环。如图 K.4 c)中的空心圆所示，这将导致在岩石锚下的基脚顶部产生环向拉力，应配置环向钢筋来承受这种拉力。

K.3 Reference documents     参考文献

[3] fib Model Code (2012b) Final draft, Volume 2, fib Bulletin 66, Federation Internationale du Beton, Lausanne, Switzerland, 370 p., ISBN: 978-2-88394-106-9 [4] AASHTO (2012), AASHTO LRFD Bridge Design Specifications, Customary U.S. Units, 6th Edition, American Association of State Highway Transportation Officials, Washington, DC, 2012

[6] Marti, P. (1985），“Basic tools of reinforced concrete beam design，” ACI Journal, Proceedings, 82(1), pp. 45-56 [7] Schlaich, J., Schafer, K., and Jennewein, M.，“Toward a consistent design of structural concrete，” PCI Journal, 32(3), 1987, pp. 74-150 [8] Schlaich, M., and Anagnostou, G.，“Stress fields for nodes of strut-and-tie model，” Journal of Structural Engineering, ASCE , 116(1), 1990, pp. 13-23 [9] Nielsen, M. P., Limit Analysis and Concrete Plasticity, 2nd Ed., CRC Press LLC, Boca Raton, FL., 1999 [10] Practical Design of Structural Concrete , fip Bulletin September 1999, 113 p. ISBN: 978- 1-874266-48-8 [11] fib, Textbook on Behaviour , Design and Performance, Volume 3 : Durability - Design for Fire Resistance - Member, Design - Maintenance , Assessment and Repair - Practical Aspects , Structural Concrete Textbook , first edition , 292 p., 1999 [12] fib, Design examples for the 1996 FlP recommendations Practical design of structural concrete, fib Bulletin No. 16, 2002 , 198 p., ISBN: 978-2-88394-056-7 [13] Joint ACl/ASCE Committee 445 ，“Examples for the Design of Structural Concrete using Strut-and-Tie Models”，Special Publication of the American Concrete Institute, No. 208, October , 2003 [14] Mitchell, D., Collins, M.P., Bhide, S.B., and Rabbat, B.S.，“AASHTO LRFD Strut-and-Tie Model Design Examples”，Portland Cement Association [15] Martin, B.T., and Sanders, D.H., Verification and Implementation of Strut-and-Tie Model in LRFD Bridge Design Specifications ”，American Association of State Highway and Transportation Officials (AASHTO) , November, 2007 [16] Joint ACl/ASCE Committee 445，“Further Examples for the Design of Structural Concrete using Strut-and-Tie Models”，Special Publication of the American Concrete lnstitute , No. 273 , September, 2010 [17] fib Task Group 1.1，“Design Examples for Strut-and-Tie Models”，Bulletin No. 61 of the International Concrete Federation, 2011, 220 p. ISBN: 978-2-88394-101-4

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