在将钢结构锚固到混凝土时，施工项目中通常使用锚栓或化学锚栓以确保结构稳定性和承载能力。该模型在 IDEA StatiCa Detail 软件中进行分析。
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"value": "<p>CSFM（协调应力场法）在混凝土中考虑连续应力场（二维有限单元），并辅以离散\"杆\"单元表示钢筋（一维有限单元）。因此，钢筋并非弥散地嵌入混凝土二维有限单元中，而是被显式建模并与之连接。计算模型中采用平面应力状态。</p>\n<figure data-asset-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" data-image-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3f70008c-0c34-4dbe-8219-4d8aa7079bb5/Visualization%20of%20the%20calculation%20model.png\" data-asset-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" data-image-id=\"9e86fe68-36a5-433d-9451-40d2b5078b86\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 6\\qquad Visualization of the calculation model of a structural element (trimmed beam) in Idea StatiCa Detail.}}}\\]</em></p>\n<p>既可以对整体<a data-item-id=\"a11adc2d-9c84-4667-8061-600660e1ad87\" href=\"\">墙体</a>和梁进行建模，也可以对梁的局部（部分）（即孤立的非连续区，又称截断端）进行建模。对于墙体和整体梁，支座的定义方式须使结构形成（外部）静定或超静定体系。梁截断端处的荷载传递通过特殊的圣维南传递区引入，从而确保被分析局部区域内的应力分布符合实际。</p>"
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"value": "<p>使用 CSFM 对结构进行评估时，需分别针对正常使用极限状态和承载能力极限状态荷载组合进行两种不同的分析。正常使用极限状态分析假定构件的极限性能是令人满意的，且材料在正常使用荷载水平下不会达到屈服条件。这一方法使得正常使用极限状态分析可以采用简化的本构模型（混凝土应力-应变图采用线性段），从而提高数值稳定性和计算速度。因此，建议采用下文所述的工作流程，即首先进行承载能力极限状态分析。</p>\n<h3>承载能力极限状态分析</h3>\n<p>根据模型直接给出的结果，对特定设计规范所要求的各项验算进行评估。ULS 验算包括混凝土强度、钢筋强度以及锚固（粘结剪应力）验算。</p>\n<p>为确保结构构件具有高效的设计，强烈建议先进行一次初步分析，其中应考虑以下步骤：</p>\n<ul>\n <li>选取最不利的荷载组合。</li>\n <li>仅计算承载能力极限状态（ULS）荷载组合。</li>\n <li>使用较粗的网格（通过增大 Setup 中默认网格尺寸的倍数，见图 19）。</li>\n</ul>\n<figure data-asset-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" data-image-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aabe4d74-d599-4c9d-a62d-8e448a66360a/Mesh%20multiplier.PNG\" data-asset-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" data-image-id=\"8c27dc0f-1cfe-4026-bbf5-4b51604c3558\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 19\\qquad Mesh multiplier.}}}\\]</em></p>\n<p>这样的模型可以非常快速地完成计算，使设计人员能够高效地检查结构构件的配筋细部，并反复重新运行分析，直至满足最不利荷载组合下的所有验算要求。一旦此初步分析的所有验算要求均得到满足，建议纳入完整的极限荷载组合，并采用较细的网格尺寸（即程序推荐的网格尺寸）。用户可以通过倍数来调整网格尺寸，该倍数可在 0.5 到 5 之间取值（图 19）。</p>\n<p>基本结果和验算结果（应力、应变和承载比，即根据规范计算得到的“计算值/限值”，以及混凝土构件中主应力的方向）通过不同的图形显示，其中压力通常以红色表示，拉力以蓝色表示。既可以突出显示整个结构的全局最小值和最大值，也可以显示每个用户自定义部分的最小值和最大值。在程序的另一个选项卡中，可以显示更多高级结果，如张量值、结构变形，以及用于计算钢筋拉力刚化所采用的配筋率（有效配筋率和几何配筋率）。此外，还可以显示所选荷载组合或荷载工况下的荷载和反力。</p>\n<h3>正常使用极限状态分析</h3>\n<p>SLS 评估包括应力限制、裂缝宽度和挠度限值的验算。按照与 ULS 相似的方式，根据适用规范对混凝土和钢筋构件的应力进行校核。</p>\n<p>正常使用极限状态分析对承载能力极限状态分析所采用的本构模型进行了一定的简化。分析中假定为完全粘结，即在正常使用极限状态下不对锚固长度进行验算。此外，忽略混凝土受压应力-应变曲线的塑性段，其弹性段则为线性且无限延伸。这些简化提高了数值稳定性和计算速度，并且只要正常使用状态下材料的实际应力限值明显低于其屈服点（这也是规范所要求的），就不会降低解答的普遍适用性。因此，仅当所有验算要求均得到满足时，用于正常使用极限状态的简化模型才是有效的。</p>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"theoretical_background_detail___crack_width_calcul\"></object>"
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"value": "<p>非线性（非弹性）有限元分析模型由多种类型的有限单元组成，用于模拟混凝土、钢筋以及两者之间的粘结。混凝土单元和钢筋单元首先独立划分网格，然后通过多点约束（MPC 单元）相互连接。这样可使钢筋相对于混凝土占据任意的相对位置。如果需要计算锚固长度校核，则在钢筋与 MPC 单元之间插入粘结和锚固端弹簧单元。</p>\n<figure data-asset-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" data-image-id=\"03fd72f4-b362-492a-8885-349785eaa70a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/511cc4d5-618a-4542-ac53-52a29549070f/Finite%20element%20model.png\" data-asset-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" data-image-id=\"03fd72f4-b362-492a-8885-349785eaa70a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 13\\qquad Finite element model: reinforcement elements mapped to concrete mesh using MPC elements and bond elements.}}}\\]</em></p>\n<h3>混凝土</h3>\n<p>混凝土采用四边形和三角形壳单元（CQUAD4 和 CTRIA3）建模，分别由四个或三个节点定义。这些单元中仅考虑平面应力，即不考虑 z 方向的应力或应变。</p>\n<p>每个单元有四个或三个积分点，位于其尺寸约 1/4 处。在每个单元的每个积分点处，计算主应变方向 α<sub>1</sub>、α<sub>2</sub>。在这两个方向上，根据 Fig. 2 所规定的混凝土应力-应变关系曲线，分别计算主应力 σ<em><sub>c</sub></em><sub>1</sub>、σ<em><sub>c</sub></em><sub>2</sub> 和刚度 <em>E</em><sub>1</sub>、<em>E</em><sub>2</sub>。需要注意的是，压力软化效应的影响将主压方向的行为与另一主方向的实际状态耦合在一起。</p>\n<h3>钢筋</h3>\n<p>钢筋采用双节点一维“杆”单元（CROD）建模，仅具有轴向刚度。这些单元连接到专门开发的“粘结”单元，用于模拟钢筋与周围混凝土之间的滑移行为。这些粘结单元随后通过 MPC（多点约束）单元连接到代表混凝土的网格上。这种方法允许对钢筋和混凝土分别独立划分网格，同时在后续步骤中确保二者之间的相互连接。</p>\n<h3>粘结单元</h3>\n<p>锚固长度的校核是通过在有限元模型中实现混凝土单元（2D）与钢筋单元（1D）之间的粘结剪应力来完成的。为此，开发了一种“粘结”有限单元类型。</p>\n<p>粘结单元的定义与壳单元（CQUAD4）类似，同样由 4 个节点定义，但与壳单元不同的是，它仅在上、下两对节点之间具有非零的剪切刚度。在模型中，上部节点与代表钢筋的单元相连，下部节点与代表混凝土的单元相连。该单元的行为通过粘结应力 τ<em><sub>b</sub></em> 描述，其为上、下节点之间滑移量 δ<em><sub>u</sub></em> 的双线性函数，参见 Fig. 14。</p>\n<figure data-asset-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" data-image-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1cc20fd2-92d7-42dc-ac17-24f318cbd45c/Bond.PNG\" data-asset-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" data-image-id=\"a031a0ff-a5a7-4a37-b59f-cb1c408f080b\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 14\\qquad (a) conceptual illustration of the deformation of a bond element; (b) a stress-deformation function.}}}\\]</em></p>\n<p><br></p>\n<p>粘结-滑移关系的弹性刚度模量 <em>G</em><em><sub>b</sub></em> 定义如下：</p>\n<p>\\[G_b = k_g \\cdot \\frac{E_c}{Ø}\\]</p>\n<p>其中：</p>\n<p><em>k</em><em><sub>g</sub></em> 取决于钢筋表面特性的系数（默认 <em>k</em><em><sub>g</sub></em><sub> </sub>= 0.2）</p>\n<p><em>E</em><em><sub>c</sub></em> 混凝土的弹性模量（对于 EN 规范取 <em>E</em><em><sub>cm</sub></em>）</p>\n<p>Ø 钢筋直径</p>\n<p>用于校核锚固长度的极限粘结剪应力设计值（折减值）<em>f</em><em><sub>bd</sub></em>，取自所选相应设计规范 EN 1992-1-1 或 ACI 318-19。塑性段的强化默认按 <em>G</em><em><sub>b</sub></em>/10<sup>5</sup> 计算。</p>\n<h3>锚固弹簧</h3>\n<p>为钢筋端部设置锚固措施（如弯钩、弯折、环形锚固等），并满足设计规范的规定，可以将钢筋的基本锚固长度（<em>l</em><em><sub>b,net</sub></em>）按一定系数 β（以下称为“锚固系数”）予以折减。锚固长度设计值（<em>l</em><em><sub>b</sub></em>）计算如下：</p>\n<p>\\[l_b = \\left(1 - \\beta\\right)l_{b,net}\\]</p>\n<p><em>l</em><em><sub>b,net</sub></em> 的预期折减相当于钢筋端部在锚固折减系数所给出的最大承载力百分比处被激活，如 Fig. 15a 所示。</p>\n<figure data-asset-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" data-image-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/748b5346-4251-4154-b923-919c94d0c6d0/Model%20for%20the%20reduction%20of%20the%20anchorage%20length.PNG\" data-asset-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" data-image-id=\"6e05f6d3-2d4c-4c6c-90f0-89e34117415c\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 15\\qquad Model for the reduction of the anchorage length:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) anchorage force along the anchorage length of the reinforcing bar; (b) slip-anchorage force constitutive relationship.}}}\\]</em></p>\n<p>锚固长度的折减在有限元模型中通过在钢筋端部设置弹簧单元（Fig. 15）来实现，该弹簧单元由 Fig. 15b 所示的本构模型定义。该弹簧所能传递的最大力（<em>F</em><em><sub>au</sub></em>）为：</p>\n<p>\\[F_{au} = \\beta \\cdot A_s \\cdot f_{yd}\\]</p>\n<p>其中：</p>\n<p><em>β</em> 基于锚固类型确定的锚固系数，</p>\n<p><em>A</em><em><sub>s</sub></em> 钢筋的横截面积，</p>\n<p><em>f</em><em><sub>yd</sub></em><em> </em> 钢筋屈服强度的设计值（折减值）。</p>"
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"value": "<p><strong>CSFM 考虑裂缝处的混凝土最大主压应力（σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>2</sub></strong><em><strong><sub>r</sub></strong></em><strong>）和钢筋应力（σ</strong><em><strong><sub>sr</sub></strong></em><strong>），而忽略混凝土的抗拉强度（σ</strong><em><strong><sub>c</sub></strong></em><strong><sub>1</sub></strong><em><strong><sub>r</sub></strong></em><strong> = 0），但保留其对钢筋的拉力刚化效应。</strong> 考虑拉力刚化后，可以模拟钢筋的平均应变（ε<em><sub>m</sub></em>）。计算中考虑了不产生滑移而张开的虚拟旋转无应力裂缝（图 2a），并同时考虑了裂缝处的平衡以及钢筋的平均应变。 </p>\n<figure data-asset-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" data-image-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/70d687dc-a209-4d67-aeb9-c0bdabacd5c1/Fig.%202%20-%20Basic%20assumptions%20of%20CSFM.png\" data-asset-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" data-image-id=\"a5b4f7ac-3fc1-4050-9269-afdb9901a92e\" alt=\"\"></figure>\n<p><em>\\( \\textsf{\\textit{\\footnotesize{Fig. 2\\qquad Basic assumptions of the CSFM: (a) principal stresses in concrete; (b) stresses in the reinforcement direction;}}}\\) \\( \\textsf{\\textit{\\footnotesize{(c) stress-strain diagram of concrete in terms of maximum stresses with consideration of compression softening;}}}\\) \\( \\textsf{\\textit{\\footnotesize{(d) stress-strain diagram of reinforcement in terms of stresses at cracks and average strains; (e) compression softening}}}\\) \\( \\textsf{\\textit{\\footnotesize{law; (f) bond shear stress-slip relationship for anchorage length verifications.}}}\\)</em></p>\n<p><br></p>\n<p>尽管这些假定较为简单，但已有研究表明，对于承受面内荷载的钢筋混凝土构件，类似的假定可以得到准确的预测结果（Kaufmann 1998；Kaufmann and Marti 1998），前提是所配置的钢筋能够避免在开裂时发生脆性破坏。此外，不考虑混凝土抗拉强度对极限承载力的任何贡献，这与现代设计规范的原则是一致的，因为这些规范大多基于塑性理论。</p>\n<p>然而，<strong>CSFM 并不适用于没有横向钢筋的细长构件</strong>，因为对于此类构件而言，骨料咬合、裂缝尖端残余拉应力以及销栓作用等相关机制（这些机制均直接或间接依赖于混凝土的抗拉强度）均未被考虑。虽然某些设计标准允许基于半经验规定对此类构件进行设计，但 CSFM 并非针对此类潜在脆性结构而设计的。</p>\n<h4>混凝土</h4>\n<p>CSFM 中所采用的混凝土模型基于设计规范针对截面设计所规定的单轴受压本构关系，该关系仅取决于抗压强度。CSFM 默认采用抛物线-矩形应力-应变曲线（图 2c），但设计人员也可以选择更为简化的理想弹塑性关系。当按照 ACI 规范进行验算时，只能使用抛物线-矩形应力-应变曲线。如前所述，与经典钢筋混凝土设计一样，此处忽略了混凝土的抗拉强度。</p>\n<p>开裂混凝土的有效抗压强度通过<em>k</em><em><sub>c</sub></em><sub>2</sub>折减系数根据主拉应变（ε<sub>1</sub>）自动计算得出，如图 2c 和 2e 所示。所采用的折减关系（图 2e）是对 <em>fib</em> 2010 模式规范中用于抗剪验算的建议的一种推广，该建议给出了有效混凝土强度与混凝土抗压强度之比的最大限值为 0.65，但该限值不适用于其他荷载工况。</p>\n<p><a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\"><em>IDEA StatiCa Detail</em></a>中的 CSFM 并未针对受压混凝土定义明确的应变破坏准则（即达到峰值应力后，认为其具有无限延伸的塑性段）。这一简化处理无法用于验证发生受压破坏的结构的变形能力。然而，除了图 2e 中定义的开裂混凝土折减系数（<em>k</em><em><sub>c</sub></em><sub>2</sub>）之外，通过 <em>fib</em> 2010 模式规范中定义的 <em>\\( \\eta_{fc} \\)</em> 折减系数，还考虑了随混凝土强度提高而增加的脆性，从而能够正确预测其极限承载力，具体如下：</p>\n<p>\\[f_{c,red} = k_c \\cdot f_{c} = \\eta _{fc} \\cdot k_{c2} \\cdot f_{c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>式中：</p>\n<p><em>k</em><em><sub>c </sub></em>为抗压强度的整体折减系数</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 为由于横向开裂而产生的折减系数</p>\n<p><em>f</em><em><sub>c</sub></em> 为混凝土圆柱体特征强度（用于定义<em>\\( \\eta_{fc} \\)</em>时以 MPa 为单位）。</p>\n<p>出于计算稳定性的考虑，还需对<em> k</em><em><sub>c</sub></em><sub>2</sub>系数进行折减。此折减不会影响构件的总强度。假设<em>f</em><em><sub>cd</sub></em>值为混凝土的折算强度（设计值），则<em>k</em><em><sub>c</sub></em><sub>2</sub>值按以下规则进行折减。</p>\n<p>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> < 0.11f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em>=1.0<br>0.11f</em><em><sub>cd</sub></em><em> < </em>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> < 0.37f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em> </em>在 1.0 与图 2f 所示曲线上取得的数值之间<br> 进行线性插值<em><br></em>σ<em><sub>c</sub></em><sub>2</sub><em><sub>r</sub></em><em> > 0.37f</em><em><sub>cd</sub></em><em> k</em><em><sub>c</sub></em><sub>2</sub><em> </em>直接从图 2f 中取值</p>\n<h4>钢筋</h4>\n<p>裸钢筋通常采用设计规范定义的理想化双线性应力-应变曲线（图 2d）。该曲线的定义仅需在设计阶段确定钢筋的基本特性（强度和延性等级）。用户也可以自定义应力-应变关系。</p>\n<p>通过修改裸钢筋的输入应力-应变关系来考虑拉力刚化效应，从而反映嵌入混凝土中的钢筋的平均刚度（ε<em><sub>m</sub></em>）。</p>\n<h4>粘结模型</h4>\n<p>在有限元模型中，通过采用图 2f 所示的简化刚性-理想塑性本构关系来引入钢筋与混凝土之间的粘结-滑移，其中<em>f</em><em><sub>bd</sub></em>为设计规范针对特定粘结条件所规定的极限粘结应力的设计值（折算值）。</p>\n<p>这是一个简化模型，其唯一目的是根据设计规范验证粘结相关规定（即钢筋的锚固）。当采用弯钩、弯环及类似形状的钢筋端部时，可以通过在钢筋端部定义一定的承载能力来考虑锚固长度的折减，具体内容将在后文进一步说明。 </p>"
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"value": "<p>采用标准的完全牛顿-拉夫森（Newton-Raphson，NR）算法来求解非线性有限元问题。 </p>\n<p>通常情况下，当全部荷载一次性施加时，NR 算法往往难以收敛。因此本文采用了一种常用方法：将荷载分为多个增量依次施加，并利用上一荷载增量的结果作为下一增量牛顿求解的初值。为此，在牛顿-拉夫森算法之上实现了荷载控制算法。若 NR 迭代未能收敛，则将当前荷载增量减半，并重新进行 NR 迭代。</p>\n<p>荷载控制算法的第二个目的是确定临界荷载，该临界荷载对应于特定的“终止准则”——具体包括混凝土的最大应变、粘结单元中的最大滑移、锚固单元中的最大位移，以及钢筋中的最大应变。临界荷载通过二分法求得。若模型中任意位置超过终止准则，则舍弃上一荷载增量的结果，并重新计算一个为前一增量一半大小的新增量。该过程反复进行，直至以一定的误差容限求得临界荷载为止。</p>\n<p>对于混凝土，终止准则设定为：在壳单元积分点处，受压应变达到 5%（即约比混凝土实际破坏应变大一个数量级），受拉应变达到 7%。在受拉方向，该数值的设定是为了使钢筋中的极限应变（通常约为 5%，未考虑拉力刚化效应）能够先行达到。在受压方向，该数值是在多个备选方案中选取的，既足够大以使压碎效应能够在结果中体现，又足够小以避免引发过多数值稳定性问题。</p>\n<figure data-asset-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" data-image-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c1026dcf-91ed-47ab-af2e-705ca886a9ed/Constitutive%20relationship%20of%20bond%20and%20anchorage.PNG\" data-asset-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" data-image-id=\"883637b4-6077-43ff-b6e8-ac1e86785345\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 17\\qquad Constitutive relationship of bond and anchorage elements used for anchorage length verification:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) bond shear stress slip response of a bond element; (b) force-displacement response of an anchorage element.}}}\\]</em></p>\n<p>对于钢筋，终止准则以应力形式定义。由于所模拟的是裂缝处的应力，因此受拉方向的准则对应于考虑安全系数后的钢筋抗拉强度。受压方向的准则采用相同数值。</p>\n<p>粘结单元和锚固弹簧中的终止准则为 α·δ<em>u</em><em><sub>max</sub></em>，其中 δ<em>u</em><em><sub>max</sub></em> 为规范校核中所用的最大滑移量，α = 10。</p>"
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"value": "<h3>混凝土 - 承载能力极限状态（ULS）</h3>\n<p>CSFM 中实现的混凝土模型基于 EN 1992-1-1 规定的单轴受压本构关系（用于截面设计），该关系仅取决于抗压强度。CSFM 默认采用 EN 1992-1-1 第 3.1.7 (1) 条规定的抛物线-矩形图（图 24a），但设计人员也可以选择根据 EN 1992-1-1 第 3.1.7 (2) 条规定的更简化的弹性理想塑性关系（图 24b）。与经典钢筋混凝土设计一样，忽略抗拉强度。</p>\n<figure data-asset-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" data-image-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e72b03ac-c1db-4c39-bbc2-f4d87b7522f2/Concrete%20stress-strain%20diagram%20CSFM.PNG\" data-asset-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" data-image-id=\"d99ce820-6afd-4050-a438-c0bd6d3e5e29\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 24\\qquad The stress-strain diagrams of concrete for ULS: a) parabola-rectangle diagram; b) bilinear diagram.}}}\\]</em></p>\n<p><em>IDEA StatiCa Detail</em> 中 CSFM 的实现并未针对受压混凝土考虑基于应变的显式破坏准则（即，在达到峰值应力后，考虑塑性段，其中 ε<em><sub>cu</sub></em><sub>2</sub>（ε<em><sub>cu</sub></em><sub>3</sub>）取值为 5%，而 EN 1992-1-1 假定极限应变小于 0.35%）。这一简化处理无法验证发生受压破坏的结构的变形能力。但是，除了考虑开裂混凝土的折减系数（图 25 中定义的 <em>k</em><em><sub>c</sub></em><sub>2</sub>）外，还通过 <em>fib</em> Model Code 2010 中定义的 <em>\\(\\eta_{fc}\\)</em> 折减系数考虑了混凝土强度提高时脆性增加的影响，从而正确预测了根据 EN 1992-1-1 第 3.1.3 条计算的极限承载力 <em>f</em><em><sub>cd</sub></em>，具体如下：</p>\n<p>\\[f_{cd}={\\alpha_{cc}} \\cdot \\frac{f_{ck,red}}{γ_c} = {\\alpha_{cc}} \\cdot \\frac{k_c \\cdot f_{ck}}{γ_c} = {\\alpha_{cc}} \\cdot \\frac{\\eta _{fc} \\cdot k_{c2} \\cdot f_{ck}}{γ_c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f_{ck}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>其中：</p>\n<p>α<em><sub>cc</sub></em> 为考虑抗压强度长期效应以及荷载施加方式产生不利影响的系数。该系数按 EN 1992-1-1 第 3.1.6 (1) 条确定，默认值为 1.0。</p>\n<p><em>k</em><em><sub>c</sub></em> 为抗压强度的总折减系数</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 为由于存在横向裂缝而产生的折减系数</p>\n<p><em>f</em><em><sub>ck</sub></em> 为混凝土圆柱体特征强度（在 <em>\\( \\eta_{fc} \\)</em> 的定义中以 MPa 为单位）。</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 25\\qquad The compression softening law.}}}\\]</em></p>\n<h3>混凝土 - 正常使用极限状态（SLS）</h3>\n<p>正常使用极限状态分析中对用于承载能力极限状态分析的本构模型进行了某些简化。忽略混凝土受压应力-应变曲线的塑性段，弹性段则为线性且无限延伸。不考虑压力软化规律。这些简化提高了数值稳定性和计算速度，同时只要正常使用状态下的材料应力限值明显低于其屈服点（符合 Eurocode 的要求），就不会降低解答的通用性。因此，仅当满足所有验证要求时，用于正常使用状态的简化模型才有效。</p>\n<figure data-asset-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" data-image-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bca48b51-2839-4b96-8dac-078574e47c12/Fig.%2011%20-%20Concrete%20stress-strain%20for%20serviceability%20.png\" data-asset-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" data-image-id=\"78f0e024-ae44-4ec0-b939-6ac74688ae23\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 26\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>长期效应</strong></p>\n<p>在正常使用极限状态分析中，混凝土的长期效应通过有效的无限徐变系数（\\(\\varphi\\)，默认取值为 2.5）来考虑，该系数根据 EN 1992-1-1 第 3.1.4 (3) 节及第 7.4.3 (5) 节修正混凝土的割线弹性模量（<em>E</em><em><sub>cm</sub></em>），具体如下：</p>\n<p>\\[E_{c,eff} = \\frac{E_{cm}}{1+\\varphi}\\]</p>\n<p>在考虑长期效应时，首先计算包含所有永久荷载的荷载步，并考虑徐变系数（即使用混凝土的有效弹性模量 <em>E</em><em><sub>c,eff</sub></em>），然后在不考虑徐变系数的情况下计算附加荷载（即使用 <em>E</em><em><sub>cm</sub></em>）。此外，为进行短期验证，还需进行另一次计算，其中所有荷载均在不考虑徐变系数的情况下计算。图 26 展示了长期和短期验证的两种计算方式。</p>\n<p>徐变系数由用户在材料属性中定义，应根据 EN 1992-1-1 图 3.1 进行计算。</p>\n<h3>钢筋</h3>\n<p>默认情况下，采用 EN 1992-1-1 第 3.2.7 节（图 27）中定义的裸钢筋理想化双线性应力-应变图。该图的定义仅要求在设计阶段已知钢筋的基本属性（强度和延性等级）。在已知的情况下，可以考虑钢筋（热轧、冷加工、淬火自回火等）的实际应力-应变关系。用户可以自行定义钢筋应力-应变图，但在这种情况下，无法考虑拉力刚化效应（无法计算裂缝宽度）。使用顶部水平段的应力-应变图无法验证结构耐久性。因此，需要手动验证标准延性要求。</p>\n<figure data-asset-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" data-image-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/47fb26f0-9509-403c-ac42-7d68821d59d1/Steel%20stress-strain%20diagram%20CSFM.PNG\" data-asset-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" data-image-id=\"ba3b27c3-ad63-46d8-b734-279c1a98639f\" alt=\"\"></figure>\n<p><em>\\( \\textsf{\\textit{\\footnotesize{Fig. 27 \\qquad Stress-strain diagram of reinforcement: a) bilinear diagram with an inclined top branch; b) bilinear diagram}}}\\) \\( \\textsf{\\textit{\\footnotesize{with a horizontal top branch.}}}\\)</em></p>\n<p><br></p>\n<p>拉力刚化（图 28）通过修正裸钢筋的输入应力-应变关系，自动加以考虑，从而反映嵌入混凝土中的钢筋的平均刚度（ε<em><sub>m</sub></em>）。</p>\n<figure data-asset-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" data-image-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/111ff130-8480-486a-adca-4c0068bcf66e/Tension%20stiffening%20CSFM.PNG\" data-asset-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" data-image-id=\"4a23c310-98c5-488d-a3a0-2ec9064a2f61\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 28\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>协调应力场法符合现代设计规范的要求。由于计算模型仅使用标准材料特性，设计规范中规定的分项安全系数格式可以直接应用，无需任何调整。这样，输入荷载被放大，特征材料特性则按照设计规范中规定的相应安全系数进行折减，与常规混凝土分析完全一致。系统默认采用 EN 1992-1-1 第 2.4.2.4 节规定的材料安全系数值，但用户可以在“规范与计算设置”中修改安全系数（图 29）。</p>\n<figure data-asset-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" data-image-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4cadae4a-9a8a-4f9b-935c-51395116ed4e/Material%20factors.png\" data-asset-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" data-image-id=\"7b26aa26-7ec4-4296-9296-645d3d6041b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 29\\qquad The setting of material safety factors in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>用户需要在“组合规则”中为每个非线性荷载工况组合定义荷载安全系数（图 30）。对于 <a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\">Idea StatiCa Detail</a> 中实现的所有模板，分项安全系数均已预先定义。</p>\n<figure data-asset-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" data-image-id=\"99632028-f378-4338-b74b-bef12aec3f6a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2d2607d1-29e9-4dfd-80ef-db2ba7d172bf/Combination%20factors.png\" data-asset-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" data-image-id=\"99632028-f378-4338-b74b-bef12aec3f6a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 30\\qquad The setting of load partial factors in Idea StatiCa Detail.}}}\\]</em></p>\n<p><br></p>\n<p>通过使用适当的用户自定义分项安全系数组合，用户也可以使用整体抗力系数法（Navrátil 等人，2017）配合 CSFM 进行计算，但这种方法在设计实践中几乎不被采用。一些指南建议在非线性分析中使用整体抗力系数法。然而，在简化的非线性分析（如 CSFM）中，由于其仅需要常规手算中所用的材料特性，采用分项安全系数格式仍然更为理想。</p>"
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Ø)/100</p>\n<figure data-asset-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" data-image-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7aa307dc-3cd6-4d42-8dd8-d0ff97994677/Bond%20conditions.PNG\" data-asset-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" data-image-id=\"c6ca9e31-4172-4034-a8b0-cdb2ad98d82a\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 31\\qquad EN 1992-1-1 Figure 8.2 - Description of bond conditions.}}}\\]</em></p>\n<p>在 IDEA StatiCa Detail 中，粘结条件按照图 31 c) 和 d) 予以考虑。可在软件中为每个项目单元设置浇筑方向，操作如下。</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p>这些验算均针对结构相应部位的适用限值进行（即：尽管混凝土和钢筋材料各自只有单一等级，但由于拉力刚化和压力软化效应，结构各部位最终的应力-应变曲线会有所不同）。</p>\n<p>此外还可选择模拟<strong>光圆钢筋</strong>。更多信息请参见：<a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Detail 中的光圆钢筋</a></p>\n<p><strong>总力</strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong>和极限力</strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>总力<em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em>是有限元分析的结果，可通过两种方式定义。</p>\n<p>\\[F_{tot}=A_{s}\\cdot \\sigma_{s}\\]</p>\n<p>其中<em>A</em><em><sub>s</sub></em>为钢筋截面面积，<em>σ</em><em><sub>s</sub></em>为钢筋应力。</p>\n<p>或者作为锚固力<em>F</em><em><sub>a </sub></em>与粘结力<em>F</em><em><sub>bond</sub></em><em>.</em>之和。</p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>其中<em>F</em><em><sub>a</sub></em>为锚固弹簧中的实际力，<em>F</em><em><sub>bond</sub></em>为粘结力，可通过沿钢筋长度<em>l.</em>对粘结应力<em>τ</em><em><sub>b</sub></em>进行积分得到。</p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub>为钢筋的周长。</p>\n<p>极限力<em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em>是考虑钢筋<strong>极限强度</strong>以及<strong>锚固条件</strong>（混凝土与钢筋间的粘结以及锚固弯钩、弯环等）后，钢筋单元中的最大力。</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=k\\cdot f_{yd}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot k\\cdot f_{yd}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bd}\\]</p>\n<p>其中 C<sub>s</sub>为钢筋周长，<em>l</em>为从钢筋起点到所考察点的长度。</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 32\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>其中<em>F</em><em><sub>lim,add</sub></em>为根据相邻单元之间夹角大小计算得到的附加力。<em>F</em><em><sub>lim,2</sub></em>必须始终小于<em>F</em><em><sub>u</sub></em>。</p>\n<p><br></p>\n<p>CSFM 中可用的<strong>锚固类型</strong>包括直筋（即无锚固端折减）、弯折、弯钩、弯环、焊接横筋、完全粘结以及连续钢筋。图 32 展示了纵向钢筋所有这些类型及其相应的锚固系数β，图 33 则展示了箍筋的相应情况。所采用的锚固系数取值均符合 EN 1992-1-1 第 8.4.4 节表 8.2 的规定。需要注意的是，尽管有多种可选类型，但 CSFM 仅区分三种锚固端类型：(i) 锚固长度不折减，(ii) 采用标准化锚固时锚固长度折减 30%，以及 (iii) 完全粘结。</p>\n<figure data-asset-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" data-image-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b16975dc-aeea-4e7e-bfc7-23a8f8b28c7e/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" data-image-id=\"a4b32213-4a43-4c1d-a3c3-21d42d5dfbad\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 33\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in the CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) bend; (c) hook; (d) loop; (e) welded transverse bar; (f) perfect bond; (g) continuous bar.}}}\\]</em></p>\n<p><br></p>\n<figure data-asset-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" data-image-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/86ffb525-5912-4a7f-9576-fff17481b7a1/Available%20anchorage%20types%20for%20stirrups.png\" data-asset-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" data-image-id=\"ec5159ea-3a7f-43fa-a807-a217b79d6cc9\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 33\\qquad Available anchorage types and respective anchorage coefficients for stirrups.}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Closed stirrups: (a) hook; (b) bend; (c) overlap. Open stirrups: (d) hook; (e) continuous bar.}}}\\]</em></p>\n<p>为符合 EN 1992-1-1 的要求，计算中应使用锚固弹簧，该锚固弹簧通过系数β进行修正，因此用户在定义钢筋起点和终点条件时必须使用某种可用的锚固类型。 </p>"
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"value": "<p>在设计混凝土结构时，我们会遇到两大类局部承压面积（PLA）——一类是支座，另一类是锚固区域。根据现行有效的钢筋混凝土结构设计标准 EN 1992-1-1 第 6.7 节（<em>图 34</em>），对于局部承压面积应考虑混凝土的局部压碎和横向拉力。对于面积 <em>A</em><em><sub>c0</sub></em> 上的均布荷载，混凝土的抗压承载力可根据设计分布面积 <em>A</em><em><sub>c1.</sub></em> 提高至最多三倍。</p>\n<figure data-asset-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" data-image-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/94ecb791-703a-44b7-8665-2f1526a20c1e/Partially%20loaded%20areas%20EC.PNG\" data-asset-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" data-image-id=\"d2ebd9b3-ebcd-4cb6-a090-4b0a9a1d2566\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 34\\qquad Partially loaded areas according to EN 1992-1-1.}}}\\]</em></p>\n<p>局部承压面积必须配置足够的横向钢筋，以承担该区域内产生的劈裂力。根据欧洲规范，局部承压面积横向钢筋的设计采用拉压杆法。若没有配置所需的横向钢筋，则不能考虑提高混凝土的抗压承载力。</p>\n<p><br></p>\n<p><strong>CSFM 中的局部承压面积</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 35\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>使用 CSFM，可以在考虑局部承压面积中混凝土抗压承载力提高影响的情况下，对钢筋混凝土结构进行设计和校核。由于 CSFM 是一个墙式（2D）模型，而局部承压面积是一个空间（3D）问题，因此需要找到一种能将这两类不同任务结合起来的解决方案（<em>图 35</em>）。若激活\"局部承压面积\"功能，则会根据欧洲规范（<em>图 34</em>）创建允许的锥体几何形状。所有几何碰撞均针对指定的混凝土构件几何形状以及每个 PLA 的尺寸完全在 3D 中求解。随后，将创建局部承压面积的计算模型。</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 36\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>修改材料模型被证明是一种不合适的方法，主要原因在于将属性映射到有限元网格存在问题。经确定，一种与有限元网格无关的方法是更为合适的解决方案。针对已知的压力锥体几何形状（<em>图 35</em> <em>和图 37</em>），创建了完全协调一致的虚拟压杆。这些压杆的材料属性（包括应力-应变曲线）与模型中所用的混凝土完全相同。锥体的形状决定了压杆的方向，从而使荷载从 PLA 逐渐分布到设计分布面积上。虚拟压杆的面积密度在锥体的每个部位都是可变的，它在荷载方向上增加了一个虚拟混凝土面积。在承压面积（<em>A</em><em><sub>c0</sub></em>）的水平处，根据比例 \\(\\sqrt{A_{c0} \\cdot A_{c1}} - A_{real}\\) （其中 <em>A</em><em><sub>real</sub></em> 是 2D 计算模型中假定的支座面积）增加一个虚拟混凝土面积，该面积向设计分布面积（<em>A</em><em><sub>c1</sub></em>）方向线性递减至零。这一解决方案确保了混凝土中的压应力在整个锥体体积内保持恒定。</p>\n<figure data-asset-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" data-image-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c4ff37a9-9d49-493b-946e-f048713b05cf/Partially%20loaded%20areas.PNG\" data-asset-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" data-image-id=\"47a5fe4b-0b51-4d87-a9cd-8e59e61835e4\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{c1}}{A_{c0}}} - \\frac{A_{real}}{A_{c0}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 37\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>局部承压面积的承载力根据 EN 1992-1-1 (6.7) 中规定的设计分布面积与承压面积之比进行提高。需要注意的是，这是一个设计模型，无法精确描述局部承压面积上的应力状态，其实际流动情况要复杂得多。然而，这一解决方案能够在尊重局部承压面积承载力提高的同时，实现荷载在整个模型中的正确分布。此外，它还能正确引入该区域的横向应力。</p>\n<p>在使用局部承压面积功能模拟混凝土抗压承载力提高的同时，仍需根据 EN 1992-1-1 第 6.7 (2) 节单独进行规范校核。钢筋所承担的横向拉力（劈裂力）会自动进行校核。</p>"
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0.3%）。这种简化处理无法验证以受压破坏为主的结构的变形能力。然而，除了考虑开裂混凝土的折减系数（<em>k</em><em><sub>c</sub></em><sub>2</sub>，定义见图 39）外，还通过 <em>fib</em> 2010 模式规范中定义的 <em>\\(\\eta_{fc}\\)</em> 折减系数来考虑混凝土强度提高时脆性增加的影响，从而可以正确预测强度：</p>\n<p>\\[f'_{c,lim}=\\alpha_{1}\\cdot\\phi_{c}\\cdot k_{c}\\cdot f'_{c}\\]</p>\n<p>\\[k_{c}=\\eta_{fc}\\cdot k_{c2}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>其中：</p>\n<p><em>α</em><sub>1</sub> 是 ACI 318-19 第 22.2.2.4.1 条定义的混凝土抗压强度折减系数。当使用抛物线-矩形应力-应变图时，需要用此系数折减最大压应力。这样可以使受压区的应力分布取平均值，从而得到的抗压强度小于或等于采用递减塑性段应力-应变图计算得到的抗压强度。</p>\n<p><em>Φ</em><em><sub>c </sub></em>是混凝土的强度折减系数。默认值根据 ACI 318-19 表 24.2.1 (b)(f) 设定。</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 是由于存在横向裂缝而产生的折减系数。</p>\n<p><em>f'</em><em><sub>c</sub></em> 是混凝土圆柱体强度（在定义 <em>\\( \\eta_{fc} \\)</em> 时以 MPa 为单位）。</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 39\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 是基于与 ACI 318-19 表 23.9.2 中给出的节点区系数 <em>β</em><em><sub>n</sub></em> 相同假设的折减系数，区别在于，在 CSFM 中，对每个有限单元（而非仅拉压杆模型的节点）都会检查是否存在与主压应力垂直的主拉应力。</p>\n<h3>混凝土 – 正常使用</h3>\n<p>正常使用极限状态分析对用于强度分析的本构模型进行了一定简化。混凝土受压应力-应变曲线的塑性段被忽略，弹性段则为线性且无限延伸。不考虑压力软化规律。这些简化提高了数值稳定性和计算速度，且只要正常使用状态下的最终材料应力限值明显低于其屈服点（符合 ACI 的要求），就不会降低求解的通用性。因此，仅当满足所有验证要求时，用于正常使用状态的简化模型才有效。</p>\n<figure data-asset-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" data-image-id=\"0d015331-6ce6-4a70-b087-58766f33e248\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/07b977ad-1511-48d6-b96e-12b3c67bb3b9/Concrete%20stress-strain%20for%20serviceability%20-%20ACI.png\" data-asset-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" data-image-id=\"0d015331-6ce6-4a70-b087-58766f33e248\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 40\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>长期效应</strong></p>\n<p>结构的长期性能，例如长期挠度或由持续荷载引起的裂缝宽度计算，受混凝土徐变的影响。ACI 318-19 第 24.2.4.1.3 段定义了持续荷载的时间相关系数——ξ，用以表示特定持续荷载持续时间下的徐变效应。</p>\n<p>在 Detail 软件中，通过系数 ξ 调整弹性模量 <em>E</em><em><sub>c</sub></em>，以确定结构的长期性能。调整后的弹性模量称为 <em>E</em><em><sub>c,eff</sub></em>——见图 40。</p>\n<p>假设构件的变形用应变表示，则可写为：</p>\n<p>\\[\\epsilon_{tot} = \\epsilon_{0} + \\epsilon_{creep} = \\epsilon_{0} \\cdot (1+\\xi)\\]</p>\n<p>其中：</p>\n<p><em>ε</em><em><sub>0</sub></em> 是短期应变（不考虑徐变影响），<em>ε</em><em><sub>creep</sub></em> 是由徐变引起的应变。</p>\n<p>利用胡克定律，可以写出：</p>\n<p>\\[E_{c,eff} = \\frac{f_{c}}{\\epsilon_{tot}}\\]</p>\n<p>将 \\(\\epsilon_{tot} = \\epsilon_{0} \\cdot (1+\\xi)\\) 和 \\(\\epsilon_{0} = f_{c} / E_{c}\\) 代入，可得：</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+\\xi}\\]</p>\n<p>用于确定系数 ξ 的持续荷载持续时间可以针对每个正常使用长期组合单独设定。</p>\n<figure data-asset-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" data-image-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1bb4b6d8-065d-4c52-a7e0-66ed3c01281f/Sustained%20load%20duration%20-%20ACI.png\" data-asset-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" data-image-id=\"f5a1e9f7-76c9-4bdf-9d6b-a28ade763397\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 41\\qquad Sustained load duration}}}\\]</em></p>\n<p>随后，通过有限元分析的固有特性自动考虑受压细化效应，计算出随时间变化的挠度、应力和裂缝宽度。因此，无需再乘以第 24.2.4.1.1 条中定义的系数。</p>\n<p><strong>短期效应</strong></p>\n<p>为进行短期验证，需另外进行一次计算，其中所有荷载均在不考虑持续荷载时间相关系数的情况下计算。长期和短期验证的两种计算均如图 40 所示。</p>\n<h3>钢筋</h3>\n<p>对于非预应力钢筋，采用具有明确屈服点的完全弹塑性应力-应变图，见 ACI 319-19 第 20.2.1 条。该图的定义只需已知钢筋的基本属性——强度和弹性模量。</p>\n<p>钢筋的应力-应变图也可以由用户自行定义，但在这种情况下，将无法考虑拉力刚化效应（也就无法计算裂缝宽度）。</p>\n<figure data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/77dadff9-85d4-402e-94e5-a3725f908933/Steel%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 42 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>其中：</p>\n<p><em>Φ</em><em><sub>s </sub></em>是钢筋的强度折减系数，默认值根据 ACI 318-19 表 24.2.1 设定。</p>\n<p><em>f</em><em><sub>y</sub></em> 是钢筋的屈服强度</p>\n<p><em>E</em><em><sub>s</sub></em> 是钢筋的弹性模量</p>\n<p>选取 10% 作为计算终止时的极限应变。根据 ASTM A955/A955M-20c 第 7 条，此值被视为安全值。</p>\n<p>拉力刚化（图 43）通过修改裸钢筋的输入应力-应变关系来自动考虑，以体现嵌入混凝土中的钢筋的平均刚度（ε<em><sub>m</sub></em>）。</p>\n<figure data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c045fcb6-32c6-4a92-aa15-24530fb11484/Tension%20stiffening%20CSFM%20-%20ACI.png\" data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 43\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>ACI 318-19 要求的各项验证均基于模型直接给出的结果进行评估。验证内容包括混凝土强度、钢筋强度以及锚固（粘结剪应力）。</p>\n<p><strong>混凝土</strong>受压强度评估为有限元分析得到的最大主压应力<em>f</em><em><sub>c</sub></em>（在辅助结果中也表示为 σ<sub>2</sub>）与限值<em>f'</em><em><sub>c,lim</sub></em>之比。</p>\n<p><strong>钢筋强度</strong>在拉力和压力两种情况下均按裂缝处钢筋应力<em>f</em><em><sub>s</sub></em>与规定限值<em>f</em><em><sub>y,lim</sub></em>之比进行评估。</p>\n<p><strong>粘结剪应力</strong>独立评估为有限元分析计算得到的粘结应力 τ<em><sub>b</sub></em>与粘结强度<em>f</em><em><sub>bu</sub></em>之比。</p>\n<p>然而，ACI 规范并未明确给出粘结强度，而是采用所谓的锚固长度计算方法，详见第 25.4.2 节。由于粘结强度是确定锚固长度的基本输入参数（参见 R25.4.1.1 和 ACI Committee 408 1966），粘结强度可按以下方式计算：</p>\n<p>假设将钢筋锚固到混凝土块中，锚固长度达到<em>l</em><em><sub>d</sub></em>或更长时，拔出钢筋将导致钢筋断裂而非钢筋从混凝土中拔出。该假设可用以下公式表示。</p>\n<p>\\[\\pi\\cdot d_{b} \\cdot l_{d} \\cdot f_{bu}=f_{y}\\cdot A_{s}\\]</p>\n<p>式中：</p>\n<p><em>d</em><em><sub>b</sub></em>为钢筋直径，<em>l</em><em><sub>d</sub></em>为锚固长度，<em>f</em><em><sub>bu</sub></em>为粘结强度，<em>f</em><em><sub>y</sub></em>为钢筋屈服强度，<em>A</em><em><sub>s</sub></em>为钢筋截面积。</p>\n<p>由上式可轻松推导出粘结强度的计算公式：</p>\n<p>\\[f_{bu}=\\frac{f_{y}\\cdot A_{s}}{\\pi\\cdot d_{b} \\cdot l_{d} }\\]</p>\n<p>锚固长度<em>l</em><em><sub>d</sub></em>随后按照 ACI 318-19 表 25.4.2.3 确定如下：</p>\n<p>\\[l_{d}=\\left( \\frac{f_{y}\\cdot\\psi_{t}\\cdot\\psi_{e}\\cdot\\psi_{g}}{C\\cdot\\lambda\\sqrt{f'_{c}}} \\right)\\cdot d_{b}\\]</p>\n<p>式中：</p>\n<p>对于 6 号及以下钢筋及变形钢丝，<em>C = 25</em>（公制为 2.1）；对于 7 号及以上钢筋，<em>C = 20</em>（公制为 1.7）；普通混凝土 λ = 1.0；<em>ψ</em><em><sub>t</sub></em>、<em>ψ</em><em><sub>e</sub></em><sub>、</sub> <em>ψ</em><em><sub>g</sub></em>按照 ACI 318-19 表 25.4.2.3 确定。 </p>\n<p>仅支持无涂层或镀锌钢筋，因此<em>ψ</em><em><sub>e</sub></em><em> = 1.0</em>。<em>ψ</em><em><sub>g</sub></em>根据钢筋等级自动确定，<em>ψ</em><em><sub>t</sub></em>则根据钢筋在模型中的位置以及可在软件中为每个项目设置的浇筑方向自动推导，具体如下。</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 46\\qquad Direction of concreting}}}\\]</em></p>\n<p>这些验证均针对结构相应部分的适用限值进行（也就是说，尽管混凝土和钢筋材料各自只有单一等级，但由于拉力刚化和压力软化效应，结构不同部分最终的应力-应变曲线会有所不同）。</p>\n<p>还可以选择建立<strong>光圆钢筋</strong>模型。详情请参见：<a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Detail 中的光圆钢筋</a></p>\n<p><strong>总力 </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> 和限值力 </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>总力<em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em>是有限元分析的结果，可通过两种方式定义。</p>\n<p>\\[F_{tot}=A_{s} \\cdot f_{s}\\]</p>\n<p>式中<em>A</em><em><sub>s</sub></em>为钢筋截面积，<em>f</em><em><sub>s</sub></em>为钢筋中的应力。</p>\n<p>或表示为锚固力 <em>F</em><em><sub>a </sub></em>与粘结力 <em>F</em><em><sub>bond</sub></em>之和<em>。</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>式中<em>F</em><em><sub>a</sub></em>为锚固弹簧中的实际力，<em>F</em><em><sub>bond</sub></em>为沿钢筋长度<em>l</em>对粘结应力<em>τ</em><em><sub>b</sub></em>积分得到的粘结力。</p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub>为钢筋周长。</p>\n<p>限值力<em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em>是考虑钢筋<strong>强度</strong>以及<strong>锚固条件</strong>（混凝土与钢筋之间的粘结、锚固弯钩、弯环等）后钢筋单元中的最大力。</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>式中 C<sub>s</sub>为钢筋周长，<em>l</em>为从钢筋起点到目标点的长度。</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 47\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>式中<em>F</em><em><sub>lim,add</sub></em>为根据相邻单元之间夹角大小计算得到的附加力。<em>F</em><em><sub>lim,2</sub></em>必须始终小于<em>F</em><em><sub>u</sub></em>。</p>\n<p><br></p>\n<p>CSFM 中可用的<strong>锚固类型</strong>包括直钢筋（即无锚固端折减）、90 度弯钩、180 度弯钩、完全粘结以及连续钢筋。图 48 展示了纵向钢筋所有这些类型及其对应的锚固系数 β。所采用锚固系数的取值是通过比较 ACI 318-19 第 25.4.3.1 节的公式与 ACI 318-19 第 25.4.2.3 节的公式得出的。需要注意的是，尽管可选类型各不相同，CSFM 仅区分三种锚固端类型：(i) 锚固长度不折减，(ii) 标准化锚固情况下锚固长度折减 30%，(iii) 完全粘结。</p>\n<figure data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b76bc446-995d-4d16-8ef9-4aa26671edda/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 48\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>箍筋的锚固系数始终为 β = 1.0。</p>\n<p>为符合 ACI 规范要求，计算中应使用锚固弹簧，该锚固弹簧会按 β 系数进行修正，因此用户在定义钢筋起始和终止条件时必须使用其中一种可用的锚固类型。 </p>"
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"value": "<p>正常使用极限状态评估针对应力限制、裂缝宽度和挠度限值进行。混凝土和钢筋单元的应力校核方式与 ACI 318-19 规定的承载能力校核方式类似。</p>\n<h3>应力限制</h3>\n<p>对于 U 级和 T 级预应力构件，应验证使用荷载下允许的混凝土压应力。根据表 R24.5.2.1，对于假定已开裂的混凝土，不需要进行应力限制校核。用户需要在设计构件设置中设定预应力构件的等级。</p>\n<figure data-asset-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" data-image-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/5dff4f86-fd02-432a-812c-cf520aabe92b/Prestressed%20member%20class.png\" data-asset-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" data-image-id=\"aebd4701-afaa-4f1f-b7f6-e531c65ed403\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 53\\qquad Prestressed flexural member class selection}}}\\]</em></p>\n<p>ACI 318-19 第 24.5.4.1 条规定，承受瞬时荷载的构件的允许压应力为 <em>0.6f</em><em><sub>c</sub></em><em>'。</em>设定 <em>0.45f</em><em><sub>c</sub></em><em>'</em> 的压应力限值是为了降低预应力混凝土构件因反复荷载而破坏的概率。该限值同时也被认为可以合理防止过度徐变变形。在较高应力水平下，随着施加应力的增加，徐变应变往往增长得更快。</p>\n<p>混凝土压应力通过以下比值来评估：由有限元正常使用极限状态分析得到的最大主压应力 <em>f</em><em><sub>c</sub></em> <em>= σ</em><em><sub>c</sub></em><sub>2</sub><em><sub> </sub></em>与根据表 24.5.4.1 设定的限值之比。</p>\n<figure data-asset-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" data-image-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/26aa9ff8-a409-41a2-b69b-b28fc2841ec0/Concrete%20compressive%20stress%20limits%20at%20service%20loads%20-%20ACI.png\" data-asset-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" data-image-id=\"5f5abc59-7c83-43de-9aa6-045ba160e215\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 54\\qquad Concrete compressive stress limits at service loads}}}\\]</em></p>\n<p>在软件中，<em>预应力加持续荷载</em>被视为长期组合，<em>预应力加总荷载</em>被视为短期组合。</p>\n<h3>挠度</h3>\n<p>根据所选组合类型（长期或短期），评估长期挠度或短期挠度。最大允许挠度值应由用户确定，并应符合 ACI 138-19 第 24.2 条的规定。 </p>\n<figure data-asset-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" data-image-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/35ae9de1-6a34-4952-a6e7-ffc528e1e5aa/Deflection%20limit%20value%20selection.png\" data-asset-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" data-image-id=\"977137a7-f1f0-4e67-8f44-06634328b1a4\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 55\\qquad Maximum allowable deflection value}}}\\]</em></p>\n<p>在软件中，可以分别显示恒荷载 <em>Δ</em><em><sub>DL</sub></em> 和活荷载 <em>Δ</em><em><sub>LL</sub></em> 产生的挠度，以及总挠度 <em>Δ</em><em><sub>Tot</sub></em><sub> </sub>（恒荷载+活荷载），同时显示变形形状。</p>\n<p>无法对被截断端部的挠度进行校核。</p>\n<h3>裂缝宽度</h3>\n<p><br></p>\n<p>针对正常使用极限状态的短期或长期组合计算裂缝宽度和裂缝方向。由于 ACI 未直接规定裂缝宽度限值，用户须自行指定裂缝宽度限值 <em>w</em><em><sub>lim</sub></em>。</p>\n<p>校核结果表示如下：</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>其中：</p>\n<p><em>w</em> 由有限元分析计算得到的短期或长期裂缝宽度，</p>\n<p><em>w</em><em><sub>lim</sub></em> 用户定义的裂缝宽度限值。</p>\n<p>软件中使用的裂缝宽度计算方法（本文档中也有更详细的说明）符合 ACI 224R-01 的规定。因此，可以使用 ACI 224R-01 表 4.1 来确定裂缝宽度的限值。</p>\n<figure data-asset-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" data-image-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4af498a4-6b3b-4043-be8f-f10522f5b188/Reasonable%20crack%20widths%20-%20ACI.png\" data-asset-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" data-image-id=\"00675749-f338-4b86-80b7-14648ef6e0b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 56\\qquad Reasonable crack widths for reinforced concrete under service load}}}\\]</em></p>\n<p>裂缝宽度的计算有两种方式（稳定开裂和非稳定开裂）。在一般情况下（稳定开裂），通过对钢筋一维单元上的应变进行积分来计算裂缝宽度。裂缝方向则根据距给定钢筋一维有限单元中心最近的三个二维混凝土单元积分点计算得出。虽然这种计算裂缝方向的方法并不对应裂缝的实际位置，但它仍能提供具有代表性的数值，从而得到可与规范要求的钢筋位置处裂缝宽度值进行比较的裂缝宽度结果。</p>"
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2010 中定义的 <em>\\(\\eta_{fc}\\)</em> 折减系数考虑混凝土强度提高导致的脆性增加，从而能够合理预测强度，其定义如下：</p>\n<p>\\[f'_{c,lim}=\\alpha_{1}\\cdot\\phi_{c}\\cdot k_{c}\\cdot f'_{c}\\]</p>\n<p>\\[k_{c}=\\eta_{fc}\\cdot k_{c2}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>其中：</p>\n<p><em>α</em><sub>1</sub> 是 AASHTO LRFD (2024) 第 5.6.2.2 条中定义的混凝土抗压强度折减系数。当使用抛物线-矩形应力-应变图时，需要用此系数折减最大压应力。这样可对受压区的应力分布取平均，使所得的抗压强度小于或等于使用带下降塑性段应力-应变图计算得到的抗压强度<em>。</em></p>\n<p><em>Φ</em><em><sub>c </sub></em>是混凝土的抗力系数。默认值根据 AASHTO LRFD (2024) 第 5.5.4.2 条设置。</p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 是由于存在横向裂缝而产生的折减系数。</p>\n<p><em>f'</em><em><sub>c</sub></em> 是混凝土圆柱体强度（用于 <em>\\( \\eta_{fc} \\)</em> 定义时单位为 MPa）。</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 58\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>k</em><em><sub>c</sub></em><sub>2</sub> 是基于与 AASHTO LRFD (2024) 5.8.2.5.3a 条及表 5.8.2.5.3a-1 中给出的混凝土效率系数 <em>ν</em> 相同假设的折减系数，不同之处在于，在 CSFM 中，对每个有限单元（而不仅仅是拉压杆模型的节点）都要检查是否存在垂直于主压应力方向的主拉应力。</p>\n<h3>混凝土 – 正常使用</h3>\n<p>正常使用分析对用于强度分析的本构模型进行了一定的简化。受压混凝土应力-应变曲线的塑性段被忽略，弹性段则为线性且无限延伸。压力软化规律不予考虑。这些简化提高了数值稳定性和计算速度，且只要正常使用状态下的结果材料应力限值明显低于其屈服点（与 AASHTO LRFD 正常使用极限状态方法一致），就不会降低解的通用性。因此，用于正常使用验算的简化模型仅在满足所有验证要求时才有效。</p>\n<figure data-asset-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" data-image-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/be2c4149-9e8e-4595-b5a5-7e9fa87c20f3/Concrete%20stress-strain%20for%20serviceability%20-%20AASHTO.png\" data-asset-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" data-image-id=\"fdcc5f99-090b-4af6-af2f-efa12840c367\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 59\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>长期效应</strong></p>\n<p>当在顶部功能区中选择长期效应时，长期本构关系（Fig. 59 中的红色曲线）用于裂缝宽度计算、总挠度计算，以及预应力构件的应力限值验证。在 IDEA StatiCa Detail 软件中，长期效应验证采用有效弹性模量，如 AASHTO LRFD (2024) C5.12.5.3.6-1 所述。</p>\n<p>\\[E_{eff} = \\frac{E_{c}}{1+\\psi}\\]</p>\n<p>其中：<br><em>E</em><em><sub>c</sub></em> 是 AASHTO LRFD (2024) 第 5.4.2.4 条中定义的弹性模量<br><em>ψ</em> 是 AASHTO LRFD (2024) 第 5.4.2.3.2 条中定义的徐变系数</p>\n<p>徐变系数由用户在材料属性中定义。</p>\n<p><strong>短期效应</strong></p>\n<p>为进行短期验证，将执行另一次计算，其中所有荷载均在不考虑徐变系数的情况下计算。长期和短期验证的两种计算均如 Fig. 59 所示。</p>\n<h3>钢筋</h3>\n<p>非预应力钢筋采用具有明确屈服点的完全弹塑性应力-应变图，见 AASHTO LRFD (2024) 第 5.4.3 条。此图的定义仅需要已知钢筋的基本属性——强度和弹性模量。</p>\n<p>用户也可以自定义钢筋的应力-应变图，但在这种情况下，无法考虑拉力刚化效应（也无法计算裂缝宽度）。 </p>\n<figure data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/77dadff9-85d4-402e-94e5-a3725f908933/Steel%20stress-strain%20diagram%20CSFM%20-%20ACI.png\" data-asset-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" data-image-id=\"2d9c6401-28af-4bfe-bc92-1d6f830f7c93\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 60 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>其中：</p>\n<p><em>Φ</em><em><sub>s </sub></em>是钢筋的抗力系数。默认值根据 AASHTO LRFD (2024) 第 5.5.4.2 条设置。</p>\n<p><em>f</em><em><sub>y</sub></em> 是钢筋的屈服强度</p>\n<p><em>E</em><em><sub>s</sub></em> 钢筋的弹性模量</p>\n<p>计算终止时的极限应变取为 10%。根据 ASTM A955/A955M-20c 第 7 条，此值被认为是安全的。</p>\n<p>拉力刚化（Fig. 61） 通过修改裸钢筋的输入应力-应变关系而自动考虑，以反映嵌入混凝土中钢筋的平均刚度（ε<em><sub>m</sub></em>）。</p>\n<figure data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/c045fcb6-32c6-4a92-aa15-24530fb11484/Tension%20stiffening%20CSFM%20-%20ACI.png\" data-asset-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" data-image-id=\"c9add949-2ad5-4922-8e6c-0d75fb47cb70\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 61\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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a) b) ‘good’ bond conditions for all bars; c) d) unhatched zone – ‘good’ bond conditions, hatched zone – ‘poor’ bond conditions}}}\\]</em></p>\n<p>可以在软件中设置混凝土浇筑方向。</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 65\\qquad Direction of concreting}}}\\]</em></p>\n<p>5.11.2.1.2条中确定的所有其他系数均等于1.0，因为软件仅支持常重混凝土，且仅支持未涂层钢筋。</p>\n<p>受压钢筋的粘结剪应力和粘结强度的计算方式与受拉钢筋类似，但使用的是AASHTO LRFD (2014) 第5.11.2.2条中的方程。</p>\n<p>此外还可以选择模拟<strong>光圆钢筋</strong>。更多信息请参见：<a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Detail中的光圆钢筋</a></p>\n<p><br></p>\n<p><strong>总力 </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> 和限值力 </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>总力<em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em>是有限元分析的结果，可以通过两种方式定义。</p>\n<p>\\[F_{tot}=A_{b} \\cdot f_{s}\\]</p>\n<p>式中<em>A</em><em><sub>b</sub></em>为钢筋截面面积，<em>f</em><em><sub>s</sub></em>为钢筋中的应力。</p>\n<p>或者作为锚固力<em>F</em><em><sub>a </sub></em>与粘结力<em>F</em><em><sub>bond</sub></em><em>之和。</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>式中<em>F</em><em><sub>a</sub></em>为锚固弹簧中的实际力，<em>F</em><em><sub>bond</sub></em>为通过沿钢筋长度<em>l</em>对粘结应力<em>τ</em><em><sub>b</sub></em>进行积分得到的粘结力。</p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub>为钢筋的周长。</p>\n<p>限值力<em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em>是考虑钢筋的<strong>强度</strong>以及<strong>锚固条件</strong>（混凝土与钢筋之间的粘结以及锚固弯钩、弯环等）后，钢筋单元中的最大力。</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{b}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{b}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>式中C<sub>s</sub>为钢筋的周长，<em>l</em>为从钢筋起点到目标点的长度。</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1a6bbdca-e56b-47e1-a85f-00d4317689a8/Flim.png\" data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 66\\qquad Definition of the limit force Flim}}}\\]</em></p>\n<p><br></p>\n<p>\\[F_{lim,2}=F_{lim,1}+F_{lim,add}\\]</p>\n<p>式中<em>F</em><em><sub>lim,add</sub></em>为根据相邻单元间夹角大小计算得到的附加力。<em>F</em><em><sub>lim,2</sub></em>必须始终小于<em>F</em><em><sub>u</sub></em>。</p>\n<p><br></p>\n<p>CSFM（协调应力场法）中可用的<strong>锚固类型</strong>包括直钢筋（即无锚固端折减）、90度弯钩、180度弯钩、完全粘结和连续钢筋。图67展示了纵向钢筋的所有这些类型及其相应的锚固系数β。所采用的锚固系数值是通过比较AASHTO LRFD (2014) 5.11.2.1节的方程和AASHTO LRFD (2014) 5.11.2.4.1节的方程得出的。需要注意的是，尽管存在多种可选方案，CSFM（协调应力场法）实际上区分三种锚固端类型：（i）锚固长度不折减，（ii）标准化锚固情况下锚固长度折减30%，以及（iii）完全粘结。</p>\n<figure data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b76bc446-995d-4d16-8ef9-4aa26671edda/Available%20anchorage%20types%20for%20longitudinal%20rebars.png\" data-asset-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" data-image-id=\"85c164c0-d864-4723-8c34-a84a426100b2\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 67\\qquad Available anchorage types and respective anchorage coefficients for longitudinal reinforcing bars in CSFM:}}}\\]</em></p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{(a) straight bar; (b) 90-degree hook; (c) 180-degree hook; (d) perfect bond; (e) continuous bar}}}\\]</em></p>\n<p>箍筋（适用于梁单元）的锚固系数始终为 β = 1.0。</p>\n<p>为符合AASHTO要求，计算中应使用锚固弹簧。锚固弹簧由β系数进行修正，因此用户在定义钢筋起始和末端条件时必须使用可用的锚固类型之一。 </p>"
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"value": "<p>在设计混凝土结构时，我们会遇到两大类局部受压区（PLA）——第一类包括<strong>支座</strong>，另一类则是<strong>锚固区</strong>。 </p>\n<p>根据现行有效的钢筋混凝土结构设计规范，对于<strong>支座</strong>，应考虑混凝土的局部压碎和横向拉力。对于面积 <em>A</em><em><sub>1</sub></em> 上的均布荷载，混凝土的抗压承载力可根据设计分布面积 <em>A</em><em><sub>2</sub></em> 提高至多两倍。参见 AASHTO LRFD（2024）第 5.6.5 条。<br></p>\n<figure data-asset-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" data-image-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e777285d-ac56-4c61-801e-d2edd9ae0318/PLA%20AASHTO.png\" data-asset-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" data-image-id=\"635e6258-ee56-41b1-8137-b791039b6b3b\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 68\\qquad Partially loaded areas for bearings according to AASHTO LRFD (2024) Article 5.6.5}}}\\]</em></p>\n<p>对于后张<strong>锚固区</strong>，应遵循 AASHTO LRFD（2024）第 5.8.4.4 条。</p>\n<p>局部受压区必须配置足够的横向钢筋，以传递该区域内产生的劈裂力。如果没有所需的横向钢筋，则不能考虑提高混凝土的抗压承载力。</p>\n<p><br></p>\n<p><strong>CSFM 中的局部受压区</strong></p>\n<figure data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/3dcea2b1-7700-46f3-a938-4c08204d52e8/Fictitious%20struts.PNG\" data-asset-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" data-image-id=\"77fdebe4-afac-4ee7-aee5-716984d4e6d3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 69\\qquad Fictitious struts with concrete finite element mesh.}}}\\]</em></p>\n<p>使用 CSFM（协调应力场法），可以在设计和校核钢筋混凝土结构的同时，考虑局部受压区混凝土抗压承载力提高的影响。由于 CSFM 是一种墙（2D）模型，而局部受压区是一个空间（3D）问题，因此需要找到一种解决方案，将这两种不同类型的问题结合起来（<em>图 69</em>）。如果激活了\"局部受压区\"功能，将根据 ACI 规范创建允许的锥体几何形状（<em>图 68</em>）。针对给定的混凝土构件几何形状和每个 PLA 的尺寸，所有几何冲突均完全在 3D 中求解。随后，创建局部受压区的计算模型。</p>\n<figure data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6ae87bd2-682b-4b92-ab1f-4b12e9d3a0df/Cone%20geometry.png\" data-asset-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" data-image-id=\"05c2e193-bc14-42b5-bc07-da8610febda8\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 70\\qquad Allowable cone geometries.}}}\\]</em></p>\n<p>对材料模型进行修改被证明是一种不合适的方法，主要原因是将属性映射到有限单元网格上存在困难。研究确定，一种独立于有限单元网格的方法是更合适的解决方案。针对已知的受压锥体几何形状（<em>图 70</em><em>和图 71</em>），创建了完全协调的虚拟压杆。这些压杆的材料属性与模型中使用的混凝土完全相同，包括应力-应变曲线。锥体的形状决定了压杆的方向，从而将荷载从 PLA 逐渐分布到设计分布面积。虚拟压杆的面积密度在锥体的每个部分都是可变的，它会沿荷载方向增加一个虚拟混凝土面积。在受荷面积（<em>A</em><em><sub>1</sub></em>）处，根据比例 \\(\\sqrt{A_{1} \\cdot A_{2}} - A_{real}\\) （其中 <em>A</em><em><sub>real</sub></em> 是 2D 计算模型中假定的支座面积）增加一个虚拟混凝土面积，该面积朝设计分布面积（<em>A</em><em><sub>2</sub></em>）方向线性递减至零。这一解决方案确保了混凝土的压应力在整个锥体体积内保持恒定。</p>\n<figure data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1dae350c-2f3a-445d-930f-f383e991dcca/Partially%20loaded%20areas%20-%20ACI.png\" data-asset-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" data-image-id=\"aff079fa-74f7-4575-a46b-8e589950238a\" alt=\"\"></figure>\n<p>\\[\\rho \\left( {\\beta ,z} \\right) = \\left( {\\sqrt {\\frac{A_{2}}{A_{1}}} - \\frac{A_{real}}{A_{1}}} \\right)\\,\\cdot\\,\\left( {1 - \\frac{z}{h}} \\right)\\,\\cdot\\,\\frac{1}{{\\cos \\beta }}\\]</p>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 71\\qquad Fictitious struts in the computational model}}}\\]</em></p>\n<p>局部受压区的承载力根据 AASHTO LRFD（2024）第 5.6.5 条规定的设计分布面积与受荷面积之比提高。 需要注意的是，这是一种设计模型，无法精确描述局部受压区上实际流动情况要复杂得多的应力状态。然而，该解决方案能够在尊重局部受压区提高的承载能力的同时，将荷载正确地分布到整个模型中。此外，它还能在该区域正确引入横向应力，从而正确设计抵抗劈裂力的钢筋。</p>\n<p>AASHTO LRFD（2024）第 5.8.4.4 条列出了 <em>0.85f</em><em><sub>c</sub></em><em>'</em> 的允许<strong>承压</strong>应力。密度受到限制，以使公式 5.6.5-3 中给出的最大双倍承载力不被超出。 </p>\n<p>对于<strong>锚固区</strong>，PLA 在软件中的使用方式与支座相同。因此，必须手动检查第 5.8.4.4 和 5.8.4.5 条中定义的局部区和整体区的压应力。因此，PLA 仅用于避免超出局部区的应变准则，从而防止计算过早终止。另一方面，抵抗整体区（第 5.8.4.5 条定义）中崩裂、平面内剥落和边缘拉应力的钢筋，可以在软件中直接且便捷地进行校核。</p>"
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"value": "<p>正常使用极限状态评估针对应力限制、裂缝宽度和挠度限值进行。混凝土和钢筋单元的应力按照 AASHTO LRFD 进行校核，方式与承载能力极限状态所规定的方法类似。</p>\n<h3>应力限制</h3>\n<p>混凝土压应力仅针对预应力构件（当模型中存在 Prestressing 荷载工况时）进行评估，评估方式为：从有限元正常使用分析得到的最大主压应力 <em>f</em><em><sub>c</sub></em> <em>= σ</em><em><sub>c</sub></em><sub>2</sub><em><sub> </sub></em>与限值的比值，该限值依据 AASHTO LRFD 表 5.9.2.3.2a-1 设定。</p>\n<figure data-asset-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" data-image-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aacf656a-eb2c-4aae-a8b2-6b1c16cdc864/Compressive%20Stress%20Limits%20in%20Prestressed%20Concrete%20at%20Service%20Limit%20State%20-%20AASHTO.png\" data-asset-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" data-image-id=\"0946a8a5-4fdf-4626-ad28-c49499d4d6eb\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 72\\qquad Concrete compressive stress limits at service loads}}}\\]</em></p>\n<p>在软件中，<em>Prestress plus permanent load</em>（预应力加永久荷载）被视为 Sustain（持久）荷载，而<em>Prestress, permanent, and transient load</em>（预应力、永久荷载和瞬时荷载）被视为 Total（总）荷载。</p>\n<figure data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/bf637c0b-a972-4836-a119-45f59df9ed58/Combination%20types%20-%20AASHTO.png\" data-asset-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" data-image-id=\"ac528856-0620-4e95-9877-ea4415ba38b5\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 73\\qquad Serviceability combination types}}}\\]</em></p>\n<p>此外，始终可以使用考虑或不考虑徐变系数的材料模型，对短期效应和长期效应分别进行分析——参见\"材料模型 (AASHTO)\"一节。</p>\n<figure data-asset-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" data-image-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/74bf96e4-46c2-4bb5-9b9f-2ab9ebf8f58b/Stress%20limitation%20model%20type%20-%20AASHTO.png\" data-asset-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" data-image-id=\"ceefbba0-847a-4d2d-8c6c-5430e5e9c43d\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 74\\qquad Serviceability material models}}}\\]</em></p>\n<h3>挠度</h3>\n<p>对于每个启用了挠度评估的组合，均计算瞬时挠度和总挠度。 </p>\n<ul>\n <li>对于瞬时挠度，采用 AASHTO LRFD (2024) 第 5.4.2.4 条规定的弹性模量 <em>E</em><em><sub>c</sub></em>。 </li>\n <li>对于总挠度，采用 AASHTO LRFD (2024) 第 C5.12.5.3.6 条规定的有效弹性模量 <em>E</em><em><sub>c,eff</sub></em>。 </li>\n</ul>\n<p>参见本文档中的\"材料模型 (AASHTO) - 混凝土 – 正常使用极限状态\"一章。</p>\n<p>挠度校核功能本身在顶部功能区中启用。用户根据所分析构件的类型，按照 AASHTO LRFD (2024) 第 2.5.2.6.2 条设置挠度限值。</p>\n<figure data-asset-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" data-image-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b95113f5-c76d-4a13-8962-99fd504ce2f2/Deflection%20check%20AASHTO.png\" data-asset-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" data-image-id=\"ddf1f284-82ac-44ca-a815-6e64b4471afe\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 75\\qquad Maximum allowable deflection value}}}\\]</em></p>\n<p>无法对修剪端处的挠度进行校核。</p>\n<h3>裂缝宽度</h3>\n<p>裂缝宽度和方向仅针对启用了裂缝宽度评估的组合，按长期效应计算（采用 AASHTO LRFD (2024) 第 C5.12.5.3.6 条规定的 <em>E</em><em><sub>c,eff</sub></em>）。基于用户指定限值的校核如下：</p>\n<p>\\[\\frac{w}{w_{lim}}\\]</p>\n<p>式中：</p>\n<p><em>w</em> 由有限元分析计算得到的裂缝宽度，</p>\n<p><em>w</em><em><sub>lim</sub></em> 用户定义的裂缝宽度限值。</p>\n<p>限值 <em>w</em><em><sub>lim</sub></em> 应根据构件类型和暴露等级，依照 AASHTO LRFD (2024) 第 5.6.7 条及其条文说明确定。 </p>\n<p>裂缝宽度有两种计算方式（稳定裂缝和非稳定裂缝）。在一般情况下（稳定裂缝），裂缝宽度通过对钢筋一维单元上的应变进行积分来计算。裂缝方向则根据距给定钢筋一维有限单元中心最近的三个二维混凝土单元积分点计算得出。虽然这种计算裂缝方向的方法并不对应裂缝的真实位置，但它仍能提供具有代表性的数值，从而得到可与规范要求的、在钢筋位置处的裂缝宽度值进行比较的裂缝宽度结果。</p>"
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"value": "<h3>混凝土 - 强度</h3>\n<p>CSFM 中用于强度计算的混凝土模型基于抛物线-塑性应力-应变曲线。与经典钢筋混凝土设计一样，不考虑抗拉强度。</p>\n<figure data-asset-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" data-image-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/dc47139c-3c53-4397-bfa6-71fe09d5c24b/Concrete%20stress-strain%20diagram%20CSFM%20-%20AUS.png\" data-asset-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" data-image-id=\"1ce5c049-0015-4d84-8bd2-9bacc8e4b5b4\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 76\\qquad The stress-strain diagram of concrete for Strength analysis}}}\\]</em></p>\n<p><em>IDEA StatiCa Detail</em> 中 CSFM 的实现未考虑受压混凝土应变方面的明确破坏准则（即达到峰值应力后，考虑最大值为 5% 的 ε<em><sub>c</sub></em><sub>0</sub> 塑性分支，而 AS 3600 第 8.3.1 条假定极限应变小于 0.3%）。这一简化不允许验证受压破坏结构的变形能力。然而，除了考虑开裂混凝土因子（在图 77 中定义的 <em>k</em><em><sub>c</sub></em><sub>2</sub>）外，还通过 <em>fib</em> Model Code 2010 中定义的 <em>\\(\\eta_{fc}\\)</em> 折减系数考虑了随混凝土强度提高而增加的脆性，从而能够正确预测强度，如下所示：</p>\n<p>\\[f'_{c,lim}=\\alpha_{2}\\cdot\\phi_{s}\\cdot \\beta \\cdot \\eta_{fc}\\cdot f'_{c}\\]</p>\n<p>\\[{\\eta _{fc}} = {\\left( {\\frac{{30}}{{{f'_{c}}}}} \\right)^{\\frac{1}{3}}} \\le 1\\]</p>\n<p>其中：</p>\n<p><em>α</em><sub>2</sub> 是 AS 3600 第 8.3.1 条中定义的混凝土抗压强度折减系数<br>在使用抛物线-矩形应力-应变图时，有必要通过该系数降低最大压应力。这样可以对受压区的应力分布进行平均化处理，使得所得的抗压强度小于或等于使用具有递减塑性分支的应力-应变图计算得出的<em>抗压强度</em>。第 8.1.3 章为矩形应力块定义了类似的方法。</p>\n<p><em>Φ</em><em><sub>s </sub></em>是混凝土的应力折减系数。默认值根据 AS 3600 表 2.2.3 设置。</p>\n<p><em>β</em> 是由于横向裂缝存在而产生的折减系数（本文中也称为 <em>k</em><em><sub>c</sub></em><sub>2</sub>）</p>\n<p><em>f'</em><em><sub>c</sub></em> 是混凝土圆柱体强度（在 <em>\\( \\eta_{fc} \\)</em> 的定义中以 MPa 为单位）。</p>\n<figure data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/085222c7-055a-4870-9bcb-8f18bd65620f/Compression%20softening%20CSFM.PNG\" data-asset-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" data-image-id=\"b9d5ff6a-d0b5-43f3-a686-dddbe6675ac1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 77\\qquad The compression softening law.}}}\\]</em></p>\n<p><em>β</em> 是基于与第 2.2.3 章中定义的有效抗压强度系数相同原理的折减系数。可以在 AS3600:2018 Sup 1:2022 第 C2.2.3 条中找到确定该系数所依据的文献（包括 AS3600 标准的相关背景）。</p>\n<h3>混凝土 – 正常使用极限状态</h3>\n<p>正常使用极限状态分析对用于强度分析的本构模型进行了某些简化。受压混凝土应力-应变曲线的塑性分支被忽略，而弹性分支为线性且无限延伸。不考虑压力软化规律。这些简化提高了数值稳定性和计算速度，并且只要正常使用极限状态下的最终材料应力限值明显低于其屈服点（符合 AS3600 的要求），就不会降低解的通用性。因此，用于正常使用极限状态的简化模型仅在满足所有验证要求时才有效。</p>\n<figure data-asset-id=\"1a187098-8984-42f2-b203-d261cab0f727\" data-image-id=\"1a187098-8984-42f2-b203-d261cab0f727\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/5b3dc17b-2a5b-4258-8495-b5d436e4885b/Concrete%20stress-strain%20for%20serviceability%20-%20AUS.png\" data-asset-id=\"1a187098-8984-42f2-b203-d261cab0f727\" data-image-id=\"1a187098-8984-42f2-b203-d261cab0f727\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 78\\qquad Concrete stress-strain diagrams implemented for serviceability analysis: short- and long-term verifications.}}}\\]</em></p>\n<p><br></p>\n<p><strong>长期效应</strong></p>\n<p>在正常使用极限状态分析中，混凝土的长期效应通过根据 AS 3600 第 3.1.8 条的设计徐变系数（<em>φ</em><em><sub>cc</sub></em>，默认取值为 2.5）来考虑，该系数按如下方式修正混凝土的割线弹性模量（<em>E</em><em><sub>c</sub></em>）：</p>\n<p>\\[E_{c,eff} = \\frac{E_{c}}{1+\\varphi_{cc}}\\]</p>\n<p>荷载增量按以下顺序依次计算：预应力 - 恒载 - 活载，如图 78 所示，每个增量使用相应的有效弹性模量。徐变系数由用户在材料属性中定义，应根据 AS 3600 第 3.1.8.3 条计算。</p>\n<figure data-asset-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" data-image-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f9c75c70-4a16-4077-963e-7ccbed22202a/Desgn%20creep%20factor%20-%20AUS.png\" data-asset-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" data-image-id=\"7c1e2af1-4d0f-46da-8cf0-d5bee4931cf3\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 79\\qquad Definition of the design creep factor}}}\\]</em></p>\n<p><strong>短期效应</strong></p>\n<p>为了进行短期验证，需要进行另一次计算，其中所有荷载均在不考虑持续荷载的时间相关系数的情况下计算。图 78 中展示了长期和短期验证的两种计算。</p>\n<h3>钢筋</h3>\n<p>对于非预应力钢筋，采用具有明确屈服点的完全弹塑性应力-应变图，见 AS 3600 第 3.2 节。该图的定义仅需要了解钢筋的基本属性——强度和弹性模量。</p>\n<p>钢筋的应力-应变图也可以由用户自行定义，但在这种情况下，无法考虑拉力刚化效应（无法计算裂缝宽度）。 </p>\n<figure data-asset-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" data-image-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4e33b934-9d0f-4ba7-9764-4f31801c752b/Steel%20stress-strain%20diagram%20CSFM%20-%20AUS.png\" data-asset-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" data-image-id=\"b5b99d46-a4ed-4625-853e-cdc4c4ede122\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 80 \\qquad Stress-strain diagram of reinforcement}}}\\]</em></p>\n<p>其中：</p>\n<p><em>Φ</em><em><sub>s </sub></em>是钢筋的强度折减系数。默认值根据 AS 3600 表 2.2.3 设置。</p>\n<p><em>f</em><em><sub>y</sub></em> 是钢筋的屈服强度</p>\n<p><em>E</em><em><sub>s</sub></em> 是钢筋的弹性模量</p>\n<p>拉力刚化（图 81）通过修正裸钢筋的输入应力-应变关系自动考虑，以捕捉埋置于混凝土中的钢筋的平均刚度（ε<em><sub>m</sub></em>）。</p>\n<figure data-asset-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" data-image-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b27b5ab6-24ea-410b-901a-fccbd7e4005f/Tension%20stiffening%20CSFM%20-%20AUS.png\" data-asset-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" data-image-id=\"c9465d3e-05e3-4514-a218-3a96876ed503\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 81\\qquad Scheme of tension stiffening.}}}\\]</em></p>"
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"value": "<p>AS 3600 要求的各项验证基于模型直接输出的结果进行评估。验证内容包括混凝土强度、钢筋强度及锚固（粘结剪应力）。</p>\n<p><strong>混凝土强度</strong>（受压）以有限元分析所得最大主压应力 <em>f</em><em><sub>c</sub></em>（辅助结果中亦记为 σ<sub>2</sub>）与限值 <em>f'</em><em><sub>c,lim</sub></em> 之比进行评估。</p>\n<p><strong>钢筋强度</strong>在受拉和受压状态下均以裂缝处钢筋应力 <em>f</em><em><sub>s</sub></em> 与规定限值 <em>f</em><em><sub>sy,lim</sub></em> 之比进行评估。</p>\n<p><strong>粘结剪应力</strong>单独以有限元分析计算所得粘结应力 τ<em><sub>b</sub></em> 与极限设计粘结应力 <em>f</em><em><sub>bu</sub></em> 之比进行评估。</p>\n<p>极限设计粘结应力 <em>f</em><em><sub>bu</sub></em> 的确定采用 AS3600:2018 Sup 1:2022 中公式 C13.1.2.2：</p>\n<p>\\[f_{bu}=\\frac{k_{2}}{k_{1} \\cdot k_{3}} \\cdot (0.5 \\cdot \\sqrt{f'_{c}})\\]</p>\n<p>其中 <em>f'</em><em><sub>c</sub></em><em> ≤ 65 MPa</em>（公式中单位为 MPa），<em>k</em> 系数按 AS 3600 第 13.1.2.2 条确定如下：</p>\n<p><em>k</em><em><sub>3</sub></em><em> = 0.7</em> （所有钢筋的保守取值）<br><em>k</em><em><sub>2</sub></em><em> = (132 - d</em><em><sub>b</sub></em><em>) / 100</em> （<em>d</em><em><sub>b</sub></em> 为钢筋直径，单位毫米）<br>= 1.3（适用于水平钢筋，其下方浇筑混凝土厚度超过 300 mm），否则取 1.0</p>\n<p><em>k</em><em><sub>1</sub></em> 由模型中钢筋的位置及浇筑方向自动推导，浇筑方向可在软件中针对每个项目条目进行设置，如下所示。</p>\n<figure data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e00845bc-3d60-4315-a8b3-67d4a52666a4/Direction%20of%20concreting.png\" data-asset-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" data-image-id=\"8a2ed21c-590e-4061-8c46-c5cc4c60ade1\" alt=\"\"></figure>\n<p><em>\\[ \\textsf{\\textit{\\footnotesize{Fig. 84\\qquad Direction of concreting}}}\\]</em></p>\n<p>基本锚固长度 <em>L</em><em><sub>sy,tb</sub></em> 按 AS 3600 公式 13.1.2.2 计算如下：</p>\n<p>\\[L_{sy,tb}=\\frac{0.5\\cdot k_{1}\\cdot k_{3}\\cdot f_{sy}\\cdot d_{b}}{k_{2}\\cdot \\sqrt{f'_{c}}}\\ge 29 \\cdot k_{1}\\cdot d_{b}\\]</p>\n<p>由公式可见，基本锚固长度 <em>L</em><em><sub>sy,tb</sub></em> 存在下限，因此软件中极限设计粘结应力 <em>f</em><em><sub>bu</sub></em> 也须相应加以限制，即：</p>\n<p>\\[f_{bu}\\le \\frac{f_{sy}}{116 \\cdot k_{1}} \\]</p>\n<p>其中 <em>f</em><em><sub>sy</sub></em> 单位为 MPa。</p>\n<p><em>f</em><em><sub>bu</sub></em> 限值的推导过程如下：</p>\n<p>\\[f_{bu}= \\frac{f_{sy}\\cdot A_{s}}{ \\pi \\cdot d_{b} \\cdot L_{sy,tb}}=\\frac{f_{sy}\\cdot \\pi \\cdot d_{b}^{2}}{4 \\cdot \\pi \\cdot d_{b} \\cdot 29 \\cdot k{1} \\cdot d_{b}} =\\frac{f_{sy}}{116 \\cdot k_{1}} \\]</p>\n<p>软件中还提供<strong>光圆钢筋</strong>的建模选项。更多信息请参阅：<a data-item-id=\"182f8ba8-899b-44fc-a1c7-59d562ef8c6c\" href=\"\">Detail 中的光圆钢筋</a></p>\n<p><br></p>\n<p><strong>总力 </strong><em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em><strong> 与限制力 </strong><em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em></p>\n<p>总力 <em><strong>F</strong></em><em><strong><sub>tot</sub></strong></em> 为有限元分析的结果，可通过以下两种方式定义。</p>\n<p>\\[F_{tot}=A_{s} \\cdot f_{s}\\]</p>\n<p>其中 <em>A</em><em><sub>s</sub></em> 为钢筋截面面积，<em>f</em><em><sub>s</sub></em> 为钢筋中的应力。</p>\n<p>或表示为锚固力 <em>F</em><em><sub>a </sub></em>与粘结力 <em>F</em><em><sub>bond</sub></em> 之和<em>。</em></p>\n<p>\\[F_{tot}=F_{a}+F_{bond}\\]</p>\n<p>其中 <em>F</em><em><sub>a</sub></em> 为锚固弹簧中的实际力，<em>F</em><em><sub>bond</sub></em> 为粘结力，可通过沿钢筋长度 <em>l</em> 对粘结应力 <em>τ</em><em><sub>b</sub></em> 积分求得。</p>\n<p>\\[F_{bond}=C_{s} \\cdot \\int_{0}^{l}\\tau_{b}\\left( x \\right)dx\\]</p>\n<p>C<sub>s</sub> 为钢筋的周长。</p>\n<p>限制力 <em><strong>F</strong></em><em><strong><sub>lim</sub></strong></em> 是综合考虑钢筋<strong>强度</strong>及<strong>锚固条件</strong>（混凝土与钢筋之间的粘结以及锚固弯钩、环等）后，钢筋单元所能承受的最大力。</p>\n<p>\\[F_{lim}=min\\left( F_{lim,bond}+F_{au},F_{u} \\right)\\]</p>\n<p>\\[F_{u}=f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{au}=\\beta\\cdot f_{y,lim}\\cdot A_{s}\\]</p>\n<p>\\[F_{lim,bond}=C_{s}\\cdot l \\cdot f_{bu}\\]</p>\n<p>其中 C<sub>s</sub> 为钢筋周长，<em>l</em> 为从钢筋起点到计算截面的长度。</p>\n<figure data-asset-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\" data-image-id=\"d3675eaf-0adb-4512-9366-58e4bdf171b1\"><img 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"value": "<p>协调应力场法（CSFM）是一种基于二维平面应力的计算方法，其中混凝土使用二维有限元建模，钢筋则通过约束连接一维钢筋单元。模型中还可以添加特殊类型的一维单元来表示粘结预应力钢筋，这些钢筋可以建模为先张法或后张法。</p>\n<p>预应力钢筋的建模方式与普通钢筋类似，均使用传递轴力的线性单元。每个预应力钢筋单元均由其截面积和材料属性来表征。这些属性由所采用规范（EN 1992-1-1、ACI 318-19 等）规定的特征材料曲线给出。</p>\n<p><strong>EUROCODE</strong></p>\n<p>预应力钢筋的应力-应变图：a) EN 1992-1-1 中定义的应力-应变图；b) 先张法钢筋的初始应变</p>\n<figure data-asset-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" data-image-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/aa25e678-c691-4887-9f8f-b5ae0c4a4fb2/prestressing%20model_Detail_01.png\" data-asset-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" data-image-id=\"7d9fac4b-fa97-49d3-a624-ddfab1bf7dee\" alt=\"\"></figure>\n<p><strong>ACI</strong></p>\n<p>预应力钢筋的应力-应变图：a) 应力-应变图；b) 先张法钢筋的初始应变</p>\n<figure data-asset-id=\"7b26f280-9951-4255-98c4-90f558de030f\" data-image-id=\"7b26f280-9951-4255-98c4-90f558de030f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1c112ef0-c06a-4141-9d09-1e3cfa42d079/prestressing%20model_Detail__ACI.png\" data-asset-id=\"7b26f280-9951-4255-98c4-90f558de030f\" data-image-id=\"7b26f280-9951-4255-98c4-90f558de030f\" alt=\"\"></figure>\n<p><br></p>\n<p>钢筋单元通过粘结模型与混凝土模型的二维单元连接，其方式与经典混凝土钢筋相同。 </p>\n<ul>\n <li>阅读<a data-item-id=\"85424e98-41cd-4bdd-a978-e4b540a10be5\" href=\"\">有限单元类型</a></li>\n</ul>\n<p>粘结模型单元允许预应力钢筋与混凝土之间发生相对变形，并具有相应的非线性特性。这可以正确地模拟钢筋与混凝土的粘结，以及先张法钢筋的锚固模型。后张法钢筋的端部构造（例如锚板）通过在预应力钢筋末端设置具有相应锚具刚度的单元来建模，而端部预应力则作为面荷载施加到混凝土模型中，作用区域为锚板尺寸大小的面积。该模型无法正确描述锚固区附近局部的三轴应力状态，因此该区域必须单独考虑。 </p>\n<p>由于假定预应力钢筋附近的混凝土处于受压状态，因此在预应力钢筋中不考虑混凝土相互作用引起的钢筋拉力刚化效应。</p>\n<h2>先张法钢筋</h2>\n<p>先张法钢筋在构件浇筑之前施加预应力，预应力钢筋几乎总是沿直线布置，因此不会产生摩擦预应力损失。当混凝土达到所需强度后，钢筋从锚固台座上放张，从而激活预应力钢筋，并将力从钢筋传递到混凝土。这种效应在物理上等效于钢筋的过冷，可通过类似于温度荷载的初始应变来建模。这就得到了上图 b) 所示的预应力钢筋应力-应变图。计算模型会自动计算结构对所施加预应力的变形响应，因此可直接通过单元的弹性应变确定预应力损失。</p>\n<p>由于预应力力已知，因此预应力应力 <em>σ</em><em><sub>pmo</sub></em> 也已知，钢筋的材料曲线用于表示应力与变形的关系，可写为：</p>\n<p><em>\\[{{σ}_{p}}=~{{f}}({{ε}}-{{ε}_{0}})\\]</em></p>\n<p>假设钢筋中的预应力低于屈服强度（即满足 EN 1992-1-1 第 5.10.3 节所定义的条件），初始应变也可按以下公式计算：</p>\n<p><em>\\[{{ε}_{0}}=\\frac{{{σ}_{pm0}}}{{{E}_{p}}}\\]</em></p>\n<p><em>ε</em><em><sub>0</sub></em> - 预应力引起的初始应变<br><em>σ</em><em><sub>pm0</sub></em> - 放张前的应力<br><em>E</em><em><sub>p</sub></em> - 预应力钢筋的弹性模量</p>\n<p>先张法钢筋的特殊之处在于其端部的锚固是通过多种不同机理实现的——钢筋与混凝土在分子层面的粘附作用、钢筋表面与混凝土界面产生的摩擦力、螺旋钢筋对混凝土的机械挤压作用，以及预应力钢筋直径增大所产生的楔形机理（即霍伊尔效应，Hoyer effect）。上述效应在 CSFM 计算模型中通过修改先张法钢筋端部区域锚固模型的属性来体现。</p>\n<p>先张法钢筋与混凝土的相互作用：a) 螺旋钢筋挤压混凝土；b) 霍伊尔效应</p>\n<figure data-asset-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" data-image-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/035bbeed-4b37-4477-b848-8ee98b174f72/prestressing%20model_Detail_02.png\" data-asset-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" data-image-id=\"cd6cee68-68e6-44b3-921a-4ccf8cd4df35\" alt=\"\"></figure>\n<h2>后张法钢筋</h2>\n<p>后张法钢筋在结构浇筑完成之后施加预应力。张拉设备直接支承在结构上，因此消除了预应力引起的结构弹性应变所导致的损失。达到所需预应力力后，钢筋被锚固，随后对钢束管道进行灌浆，从而实现钢筋与结构的粘结。因此，在对后张法钢筋建模时，计算被划分为若干个荷载步——张拉预应力、施加其他永久荷载以及施加可变荷载。</p>\n<p>带有附加一维预应力钢筋单元的混凝土有限元网格：</p>\n<figure data-asset-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" data-image-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/a028db63-b458-44e7-945b-bedabb1a6785/prestressing%20model_Detail_03.png\" data-asset-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" data-image-id=\"3b267c80-ee0e-457f-af00-f74c91a48d7d\" alt=\"\"></figure>\n<h4>“张拉预应力”荷载步</h4>\n<p>在对钢筋施加预应力时，钢筋的刚度不计入结构的整体刚度中。在该荷载步中，模型不考虑线性单元的刚度，钢筋单元被替换为与预应力应力和钢筋截面积相对应的等效荷载，如上图所示。当预应力达到全值且该荷载步收敛后，读取各线性单元的变形，并根据该变形确定预应力钢筋各线性单元的初始应变 <em>ε</em><em><sub>0</sub></em>。</p>\n<p>预应力应力可沿钢筋长度手动定义，也可根据钢筋几何形状自动计算。若选择自动计算损失，则会考虑锚固过程中的摩擦损失（根据 EN 1992-1-1 第 5.10.5.2 节，或 ACI 318-19 第 20.3.2 节）以及钢筋滑移（锚具楔块的压入）。由于所有预应力钢筋均在一个荷载步中施加，因此不考虑分批张拉引起的损失。</p>\n<h4>预应力钢筋参与工作后的后续荷载步</h4>\n<p>在后续荷载步（施加其他永久荷载和可变荷载）中，采用与先张法钢筋相同的处理方法。此时考虑预应力钢筋的全部刚度，考虑钢筋与周围混凝土的粘结作用，并通过初始应变 <em>ε</em><em><sub>0</sub></em> 修正预应力钢筋的应力-应变图。该应变对每个单元均不同，来源于前一荷载步“张拉预应力”的计算结果。由于考虑了钢筋与混凝土的粘结作用，模型能够正确考虑外部荷载引起的结构弹性变形对预应力的改变。</p>"
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data-codename=\"theoretical_background_detail___service_limit_stat\"></object>\n<h1><br></h1>\n<h2>7 依据澳大利亚标准 AS 3600（2018）的结构验算</h2>\n<p>采用 CSFM 对结构进行评估时，分为两种不同的分析：一种针对正常使用荷载组合，另一种针对强度荷载组合。正常使用分析假设构件在系数荷载下的性能是满足要求的，且在正常使用荷载水平下材料不会达到屈服条件。这种方法允许在正常使用分析中采用简化的本构模型（混凝土应力-应变图采用线性段），以提高数值稳定性和计算速度。</p>\n<p>CSFM 是一种满足第 6.1.1 节和第 6.1.2 节一般规则的结构分析方法，并在第 6.1.3 节中被定义为 (f) 非线性应力分析——进一步说明见第 6.6 节。 </p>\n<p>CSFM 的分析考虑了第 6.6.3 节所定义的所有相关非线性和非弹性效应（收缩除外）。 </p>\n<p>为满足第 6.6.4 节和第 6.6.5 节的要求——更多内容可参见 AS3600:2018 Sup 1:2022 第 C6.6 节——该方法已在多所高校进行了验证与确认。相关验证与确认结果的详细文章可参见以下链接。</p>\n<ul>\n <li><a href=\"https://www.ideastatica.com/support-center-verifications?label=detail\">验证：Detail 2D</a></li>\n</ul>\n<p>由于 IDEA StatiCa Detail 是一款实用的设计软件，计算中采用 28 天龄期的折减特征圆柱体抗压强度 <em>f'</em><em><sub>c</sub></em>，具体内容将在下一章中说明。</p>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n83eff71e_cb6d_0152_d7f9_4621fe4095f0\"></object>\n<object type=\"application/kenticocloud\" 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data-codename=\"theoretical_background_detail___serviceability_ver_77b5f2c\"></object>\n<h1><br></h1>\n<h2>8 预应力 - 模型说明</h2>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"n1ecc3d2d_b87a_01f1_a493_9161e2eae77d\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"prestressing_in_detail___model_description__body_\"></object>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"component\" data-codename=\"aec1d9ba_c119_0122_d551_4e09e764c5f9\"></object>\n<h1><br></h1>\n<h2>参考文献</h2>\n<p>ACI Committee 318. 2019. <em>Building Code Requirements for Structural Concrete (ACI 318-19) and Commentary</em>. Farmington Hills, MI: American Concrete Institute.</p>\n<p><br></p>\n<p>Alvarez, Manuel. 1998. <em>Einfluss des Verbundverhaltens auf das Verformungsvermögen von Stahlbeton</em>. IBK Bericht 236. Basel: Institut für Baustatik und Konstruktion, ETH Zurich, Birkhäuser Verlag.</p>\n<p><br></p>\n<p>Beeby, A. W. 1979. “The Prediction of Crack Widths in Hardened Concrete.” <em>The Structural Engineer</em> 57A (1): 9–17.</p>\n<p><br></p>\n<p>Broms, Bengt B. 1965. “Crack Width and Crack Spacing In Reinforced Concrete Members.” <em>ACI Journal Proceedings</em> 62 (10): 1237–56. https://doi.org/10.14359/7742.</p>\n<p><br></p>\n<p>Burns, C.. 2012. “Serviceability Analysis of Reinforced Concrete Members Based on the Tension Chord Model.” IBK Report Nr. 342, Zurich, Switzerland: ETH Zurich.</p>\n<p><br></p>\n<p>Crisfield, M. A. 1997. <em>Non-Linear Finite Element Analysis of Solids and Structures</em>. Wiley.</p>\n<p><br></p>\n<p>European Committee for Standardization (CEN). 2015. <em>1 Eurocode 2: Design of concrete structures - Part 1-1: General rules and rules for buildings</em>. Brussels: CEN, 2005.</p>\n<p><br></p>\n<p>Fernández Ruiz, M., and A. Muttoni. 2007. “On Development of Suitable Stress Fields for Structural Concrete.” <em>ACI Structural Journal</em> 104 (4): 495–502.</p>\n<p><br></p>\n<p>Kaufmann, W., J. Mata-Falcón, M. Weber, T. Galkovski, D. Thong Tran, J. Kabelac, M. Konecny, J. Navratil, M. Cihal, and P. Komarkova. 2020. “<em>Compatible Stress Field Design Of Structural Concrete</em>. Berlin, Germany.”AZ Druck und Datentechnik GmbH, ISBN 978-3-906916-95-8.</p>\n<p><br></p>\n<p>Kaufmann, W., and P. Marti. 1998. “Structural Concrete: Cracked Membrane Model.” <em>Journal of Structural Engineering</em> 124 (12): 1467–75. https://doi.org/10.1061/(ASCE)0733-9445(1998)124:12(1467).</p>\n<p><br></p>\n<p>Kaufmann, W.. 1998. “Strength and Deformations of Structural Concrete Subjected to In-Plane Shear and Normal Forces.” Doctoral dissertation, Basel: Institut für Baustatik und Konstruktion, ETH Zürich. https://doi.org/10.1007/978-3-0348-7612-4.</p>\n<p><br></p>\n<p>Konečný, M., J. Kabeláč, and J. Navrátil. 2017. <em>Use of Topology Optimization in Concrete Reinforcement Design</em>. 24. Czech Concrete Days (2017). ČBS ČSSI. https://resources.ideastatica.com/Content/06_Detail/Verification/Articles/Topology_optimization_US.pdf.</p>\n<p><br></p>\n<p>Marti, P. 1985. “Truss Models in Detailing.” <em>Concrete International</em> 7 (12): 66–73.</p>\n<p><br></p>\n<p>Marti, P. 2013. <em>Theory of Structures: Fundamentals, Framed Structures, Plates and Shells</em>. First edition. Berlin, Germany: Wiley Ernst & Sohn.</p>\n<p>http://sfx.ethz.ch/sfx_locater?sid=ALEPH:EBI01&genre=book&isbn=9783433029916.</p>\n<p><br></p>\n<p>Marti, P., M.Alvarez, W. Kaufmann, and V. Sigrist. 1998. “Tension Chord Model for Structural Concrete.” <em>Structural Engineering International</em> 8 (4): 287–298.</p>\n<p>https://doi.org/10.2749/101686698780488875.</p>\n<p><br></p>\n<p>Mata-Falcón, J. 2015. “Serviceability and Ultimate Behaviour of Dapped-End Beams (In Spanish: Estudio Del Comportamiento En Servicio y Rotura de Los Apoyos a Media Madera).” PhD thesis, Valencia: Universitat Politècnica de València.</p>\n<p><br></p>\n<p>Meier, H. 1983. “Berücksichtigung Des Wirklichkeitsnahen Werkstoffverhaltens Beim Standsicherheitsnachweis Turmartiger Stahlbetonbauwerke.” Institut für Massivbau, Universität Stuttgart.</p>\n<p><br></p>\n<p>Navrátil, J., P. Ševčík, L. Michalčík, P. Foltyn, and J. Kabeláč. 2017. <em>A Solution for Walls and Details of Concrete Structures</em>. 24. Czech Concrete Days.</p>\n<p><br></p>\n<p>Schlaich, J., K. Schäfer, and M. Jennewein. 1987a. “Toward a Consistent Design of Structural Concrete.” <em>PCI Journal</em> 32 (3): 74–150.</p>\n<p><br></p>\n<p>Standards Australia. 2018. <em>Concrete Structures (AS 3600:2018)</em>. Sydney, NSW: Standards Australia.</p>\n<p><br></p>\n<p>Standards Australia. 2022. <em>Concrete Structures – Commentary (Supplement 1 to AS 3600:2018)</em>. Sydney, NSW: Standards Australia.</p>\n<p><br></p>\n<p>Vecchio, F.J., and M.P. Collins. 1986. “The Modified Compression Field Theory for Reinforced Concrete Elements Subjected to Shear.” <em>ACI Journal</em> 83 (2): 219–31. </p>"
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"value": "<p>在实际工程中，<strong>拉压杆法（S&T）</strong>和<strong>应力场法</strong>是设计钢筋混凝土及预应力混凝土结构 D 区的标准方法。<strong>协调应力场法（CSFM）</strong>是在这些经典理论的基础上发展而来的，具有高度自动化的特点，并且符合设计规范的要求。尽管方法本身简单，但它能够非常真实地反映混凝土结构在承载能力极限状态（ULS）和正常使用极限状态（SLS）下的响应。CSFM 已在<a data-item-id=\"b4790cf9-a605-45b3-b41b-e36909ad4291\" href=\"\">IDEA StatiCa Detail</a>中实现。 </p>\n<figure data-asset-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" data-image-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7fd8d041-20d1-40a8-9a71-eb9cdce27155/7.png\" data-asset-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" data-image-id=\"a7b3dcf1-10ed-4b44-99e3-f59b4bd2a7fe\" alt=\"\"></figure>\n<p><em>图 1 a) 带开洞的墙 b) 剪力墙 c) 带企口端及开洞的梁 d) 桥墩 e) 桥梁横隔板 </em></p>\n<p>混凝土结构截面的标准设计方法适用于符合伯努利-纳维平截面假定（B 区）的部分。不适用该假定的部位称为<strong>不连续区（D 区）</strong>。这些区域包括存在集中荷载或截面突变的结构部位，例如企口端（图 1c）、深梁、带开洞的墙（图 1a、1b），以及牛腿或承台。在桥梁工程领域，则包括墩帽（图 1d）、横隔板（图 1e）、转向块等。</p>\n<h2>1. 拉压杆法</h2>\n<p>定义拉压杆模型时的基本假定是忽略混凝土的抗拉强度。一个简单的桁架模型由受压和受拉单元组成，用于表征承载能力极限状态下的行为。总体而言，这并非复杂问题，对于有经验的工程师来说，定义一个基本的拉压杆模型（图 2a）应该不成问题。然而，即使是这一基本任务，按照设计规范对模型进行正确评估也可能是一项繁琐、手工且反复迭代的过程。</p>\n<figure data-asset-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" data-image-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7aecb55f-dac0-47db-a25c-48082e5e70c4/Untitled%20design.png\" data-asset-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" data-image-id=\"59f28d4a-b793-4501-a11a-6ae6245cab70\" alt=\"\"></figure>\n<p><em>图 2 a) 拉压杆模型方案 1 b) 拉压杆模型方案 2 c) 拉压杆模型方案 </em></p>\n<p>必须对拉杆、节点区以及压杆中的横向拉应变进行校核。若模型未通过校核，则须调整拉压杆几何形状，或选择不同的拉压杆模型（图 2b、2c）。这常常导致结构工程师只确定一次拉压杆模型的几何形状，而仅对钢筋进行校核。这可能带来重大误差。模型的选择始终取决于经验。对于更复杂的结构详图，选择能充分匹配结构实际行为的拉压杆模型，可能不像上述案例那样容易。此外，拉压杆法仅是一种用于承载能力极限状态设计的方法。它无法用于<strong>正常使用极限状态（变形、开裂）</strong>的设计，而这些恰恰是关键性指标，尤其对于重要结构而言，它们直接影响结构的使用寿命。</p>\n<h2>2. 协调应力场法 - CSFM</h2>\n<p>CSFM 是一种现代<strong>非线性方法</strong>，用于分析 D 区以及行为可简化为平面应力（即二维模型）的构件。 然而，它仍然基于规范中一个基本且安全的假定：<strong>混凝土不承受拉力</strong>，所有拉力必须由钢筋承担。协调应力场法（CSFM）是拉压杆法与应力场法的发展演变，消除了上述方法的主要缺点：模型选择的不确定性、难以实现自动化，以及无法评估正常使用极限状态。</p>\n<figure data-asset-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" data-image-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/9e760312-8401-44c8-aae1-020b514876fb/2.png\" data-asset-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" data-image-id=\"6552ad81-c0fa-4071-9b95-00d09eb9fea4\" alt=\"\"></figure>\n<p><em>图 3 a) 平面应变 b) 主应力 c) CSFM</em></p>\n<p>CSFM 的原理可以通过钢筋混凝土结构基本平面单元的<strong>平面应力</strong>状态来说明。图 3a 展示了平面应力状态下的基本二维单元，这在所有弹性力学与材料力学教科书中都能见到。这是结构中某一点的应力状态，例如可通过有限单元法（FEM）的线弹性分析获得。该单元承受水平法向应力 σ<sub>x</sub>、竖向法向应力 σ<sub>z</sub>，以及剪应力 τ<sub>xz</sub>。由这些应力可以确定所谓的<strong>主应力</strong>及其由角度 θ 定义的方向（图 3b）。此时该单元承受主拉应力 σ<sub>1</sub> 和主压应力 σ<sub>2</sub>。</p>\n<p>那么，经 CSFM 分析后，同一单元的应变将呈现怎样的状态？应变如图 3c 所示。受压混凝土沿主压应力 σ<sub>2</sub> 方向出现，并生成应力为 σ<sub>c2</sub> 的应力场。如前所述，基本假定是混凝土不承受拉力。因此，横向主拉应力 σ<sub>1</sub> 不会由混凝土承担，裂缝将垂直于该方向形成。因此应力 σ<sub>c1r</sub> 必须为零。为避免二维单元发生破坏，所有拉应力必须由钢筋（图 3c 中蓝色标示）承担，钢筋必须作为计算模型的一部分。 </p>\n<p>如果使用 CSFM 对整个待求解二维区域进行<strong>连续</strong>应力分析，其结果是混凝土中形成连续的受压场，同时钢筋中产生拉应力与压应力。图 4 给出了 CSFM 应力场的简化图形表示。除混凝土与钢筋的承载比外，图中还给出了沿各区域计算所得应力 σ<sub>c2</sub> 方向的变化。</p>\n<figure data-asset-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" data-image-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/dc56c80f-67f3-481d-a2c8-26dacc258bb2/csfm%20explained%20%281%29.png\" data-asset-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" data-image-id=\"9739b6d6-2cbc-4745-a590-4a85f7e1862f\" alt=\"\"></figure>\n<p><em>图 4 IDEA StatiCa Detail 的整体计算结果 </em></p>\n<p>使用<strong>CSFM 对详图或结构进行分析基于有限单元法</strong>。混凝土采用二维墙单元建模，钢筋则采用一维构件单元建模（图 7）。由于这是一个非线性问题，分析并非一步完成。计算过程中荷载以增量方式施加，非线性方程组的求解则采用<strong>牛顿-拉夫逊法</strong>。 </p>\n<p>虚拟的弥散裂缝（ε<sub>1</sub> 为平均值）\"形成\"于垂直主应力方向，该方向可能在非线性计算过程中随每一级荷载增量下单元\"逐步开裂\"而发生变化。综上所述，本方法采用的是一种虚拟的无应力旋转裂缝模型。 </p>\n<p>基于 CSFM 的有限单元法求解结果是一个协调的应力场（即模型中混凝土不会分解为若干独立作用的压杆），以及在整个待求解二维区域内连续分布的应变状态。这是相较于经典拉压杆法的一大优势，并使计算模型能够实现自动化和精细化处理，具体内容将在以下各段中说明。</p>\n<figure data-asset-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" data-image-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/40e41bf7-4a81-4245-bd1b-843963c104e8/3.png\" data-asset-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" data-image-id=\"c5bf3113-2223-4ddc-bbab-2131db37ac0c\" alt=\"\"></figure>\n<p><em>图 5 混凝土软化原理</em></p>\n<p>CSFM 简单的公式表述使得可以直接采用设计规范中标准的单轴抛物线-矩形受压应力-应变曲线来描述混凝土。众所周知，当混凝土受到横向裂缝损伤时，其抗压强度会降低（图 5）。这种所谓的<strong>压力软化</strong>效应通过自动考虑混凝土的有效抗压强度纳入本方法。 </p>\n<p>根据横向拉应变 ε<sub>1</sub> 的水平，确定折减系数 k<sub>c</sub>，并对混凝土的应力-应变曲线进行相应调整（图 5）。由于整个结构中的应变场是已知的，因此可根据局部横向拉应变 ε<sub>1</sub> 的水平，在各个截面自动计算混凝土的有效抗压强度。</p>\n<figure data-asset-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" data-image-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/51628156-5014-4501-8f28-9f3b93783840/1.png\" data-asset-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" data-image-id=\"6c73faf0-64d4-41ce-b816-520ccadff05a\" alt=\"\"></figure>\n<p><em>图 6 拉力刚化原理</em></p>\n<p>此外，CSFM 还考虑了裂缝间受拉混凝土对钢筋的<strong>刚化效应</strong>，即所谓的拉力刚化。在计算模型中，采用平均钢筋应变 ε<sub>m</sub>。据此对钢筋的应力-应变曲线进行相应修正（图 6）。这使得能够真实反映因开裂而受损的钢筋混凝土结构的刚度。然而，混凝土的抗拉强度仍不计入极限承载力。裂缝处钢筋的最大应力 σ<sub>sr</sub> 是设计中的关键控制指标（图 6）。</p>\n<p>CSFM 使用<strong>设计规范</strong>中定义的常规单轴材料模型（应力-应变曲线）。随后采用标准方法——分项安全系数法——对承载能力极限状态进行评估。该方法简单实用，适合工程实践，并且符合设计规范的要求。 </p>\n<p>尽管这是一种非线性有限元分析，结构工程师却无需在计算中输入额外的材料属性和混凝土特性参数——这些参数在设计阶段可能根本无法获得，而基于断裂力学的非线性有限元分析则必须使用这些参数。如前所述，除了承载能力极限状态外，CSFM 分析的一大优势还在于能够评估<strong>正常使用极限状态：挠度、应力限值，尤其是裂缝宽度</strong>。</p>\n<figure data-asset-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" data-image-id=\"6c090b06-f906-4e6e-9016-d73de172f321\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/8fd3f66a-63b9-4aba-bb10-9e12889ed7d4/Finite%20element%20model.png\" data-asset-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" data-image-id=\"6c090b06-f906-4e6e-9016-d73de172f321\" alt=\"\"></figure>\n<p><em>图 7 IDEA StatiCa Detail 中有限元模型表示示例</em></p>\n<p>（图 7）CSFM 中的有限元模型由以下几种类型的有限单元组成：</p>\n<ul>\n <li>用于钢筋的具有轴向刚度的一维单元</li>\n <li>用于混凝土的二维等参单元</li>\n <li>用于带端部处理的钢筋锚固模型的端部弹簧</li>\n <li>用于模拟钢筋与混凝土间粘结作用的特殊二维单元</li>\n <li>粘结单元与混凝土之间的刚性约束及插值约束（多点约束，MPC）</li>\n</ul>\n<p>如果所设计的钢筋能够防止构件发生脆性破坏，则已证明尽管公式表述简单，CSFM 仍能对结构的响应和极限承载力给出非常良好的预测。换言之，该方法并不适用于例如无横向抗剪钢筋、可能表现出脆性行为的梁的设计。该方法的<a data-item-id=\"1e879886-9e36-49e1-acb1-e6001361531f\" href=\"\">验证</a>，包括相关试验，见文献 [1]。关于该方法更详细的说明超出了本文的范围，可参见<a data-item-id=\"0000c94c-b603-48c4-8d31-bc56d7c95886\" href=\"\">理论背景</a>。</p>\n<p>显然，CSFM 的原理具有一般性，因此其应用并不限于 D 区，还可用于对整个构件（例如预制梁）进行建模，只要该构件可简化为平面二维模型。该方法及其在软件（IDEA StatiCa Detail）中的实现也已得到扩展，现已支持指定<strong>预应力及后张钢筋</strong>。</p>\n<h2>3. 墩帽设计示例</h2>\n<p>图 8 中的桥墩墩帽设计展示了 CSFM 的实际应用。该墩为一座三跨连续梁桥（跨径为 30.0 m、42.0 m、30.0 m）的第二个桥墩。该钢筋混凝土桥墩墩顶采用 C40/50 混凝土，其厚度（沿桥梁纵向）为 2.0 m。</p>\n<figure data-asset-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" data-image-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/7bd9ebbf-5d30-448a-93e2-c89641d05677/4.png\" data-asset-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" data-image-id=\"9d541a10-b879-4d35-a6f8-4e85fa9843a6\" alt=\"\"></figure>\n<p><em>图 8 墩帽：a) 设计汇总；b) 承载能力极限状态下混凝土的压应力； c) 承载能力极限状态下钢筋的拉应力；d) 正常使用极限状态下的裂缝宽度</em></p>\n<p>在墩帽顶部，首先设计了一根横梁，配置 B500 钢筋 20xϕ28+20xϕ25——即顶部四层钢筋。图 8a 展示了承载能力极限状态下的设计汇总，显示了混凝土中的压应力、压应力方向以及钢筋中的应力。图 8b 和图 8c 则进一步给出了混凝土和钢筋中更详细的应力分布。横向钢筋的应力略低于屈服强度，混凝土的应力（及相应应变）在承载能力极限状态下也令人满意。然而，裂缝宽度计算结果（图 8d）表明，该设计不满足<strong>正常使用极限状态</strong>的要求：w<sub>max</sub> = 0.36 mm > w<sub>lim</sub> = 0.3 mm。<strong>为满足裂缝宽度限值要求，需将</strong>横梁的钢筋增加至 20xϕ32+20xϕ28。若 w<sub>lim</sub> = 0.2 mm（例如靠近产生盐雾的道路旁的桥墩，环境作用等级为 XF2），则横梁钢筋甚至需增加至 24xϕ32+24xϕ28。</p>\n<h2>结论</h2>\n<p>CSFM 适合工程实践，因为它采用了<strong>设计规范中定义的简单材料模型</strong>。除承载能力极限状态外，该方法还可用于正常使用极限状态的设计，而这在以往采用拉压杆模型时是难以想象的。通过在<strong>IDEA StatiCa Detail</strong>中实现该方法，工程师能够真实地捕捉结构的响应，从而高效、安全地设计和评估不连续区及更大规模的构件组合。</p>\n<p>CSFM 的开发主要得益于苏黎世联邦理工学院（ETH）结构工程系主任 Walter Kaufmann 教授的工作。他及其团队也<a data-item-id=\"0dd36e25-63b2-4d63-a33e-6043644fda4f\" href=\"\">对该方法及其软件实现进行了验证</a>。</p>\n<h2>参考文献</h2>\n<p>[1] KAUFMANN, Walter, et al.: Compatible stress field design of structural concrete, ETH Zurich, 2020, ISBN 978-3-906916-95-8,</p>\n<p>[2] KAUFMANN, W., MARTI, P.: Structural Concrete: Cracked Membrane Model. Journal of Structural Engineering 124 (12): 1467-75, 1998 https://doi.org/10.1061/(ASCE)0733-9445(1998)124:12(1467)</p>\n<p>[3] KRAUS, M., M. WEBER, W. KAUFMANN, W, BOBEK, L.: Numerical analysis of experimentally tested frame corners with opening moments using the Compatible Stress Field Method (CSFM). In: Computational Modelling of Concrete and Concrete Structures, pp. 694-03. CRC Press, 2022 <a href=\"https://doi.org/10.1201/9781003316404\">https://doi.org/10.1201/9781003316404</a></p>\n<h2>作者</h2>\n<p>Ing. Pavel Kaláb, Ph.D.</p>\n<p>IDEA StatiCa s.r.o.</p>\n<p><br></p>"
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"value": "<h2>引言 </h2>\n<p>本文讨论了使用<a data-item-id=\"eaab962d-ba44-4ee0-8fa7-45193c9f52b5\" href=\"\">CSFM</a>分析含少量箍筋梁中剪切破坏的方法。为此，本文分析了 Huber (2016)、Piyamahant (2002) 以及 Vecchio 和 Shim (2004) 对简支钢筋混凝土梁进行的一系列试验。这些试验涵盖了大量参数，包括不同的尺寸、剪跨比以及抗剪和纵向钢筋的用量。本节描述了使用 CSFM 对来自这些试验计划的 17 个试验进行的分析，探讨了 CSFM 正确模拟各种截然不同破坏模式的能力，这些破坏模式从箍筋断裂和未断裂的剪切破坏，到弯曲破坏以及剪弯混合破坏不等。 </p>\n<p>试验设置 图 6.17 展示了所分析试验的几何形状、试验设置和配筋布置。表 6.10 给出了抗剪钢筋（直径（<em>Ø</em><em><sub>t</sub></em>）、间距（<em>s</em><em><sub>t</sub></em>）和几何配筋率（<em>ρ</em><em><sub>t,geo</sub></em>））、纵向受弯钢筋（数量（<em>n</em><em><sub>l</sub></em>）和直径（<em>Ø</em><em><sub>l</sub></em>））以及几何参数（有效高度（<em>d</em>）、剪跨比（<em>a/d</em>）和梁宽（<em>b</em>））的信息。Huber (2016) 进行的试验 R1000m60 和 R500m351 采用单肢弯钩箍筋，而所有其他试验均采用双肢封闭箍筋。在 Piyamahant (2002) 所分析的试验中，几何形状和受弯钢筋保持不变，而在其他两项研究中则有所变化。 </p>\n<figure data-asset-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" data-image-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b04ab1b6-8164-42a4-8153-b0bca529109d/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.17.png\" data-asset-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" data-image-id=\"7a127e09-323a-47c2-bb68-f6d7305df916\" alt=\"\"></figure>\n<figure data-asset-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" data-image-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f016576c-b58f-4934-8a91-92a8f02ada8b/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%2010.png\" data-asset-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" data-image-id=\"ae1b1eb6-ac3f-45d5-908f-58e08887a726\" alt=\"\"></figure>\n<figure data-asset-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" data-image-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/1c39493d-2611-4b20-8595-b15bd109181c/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%2010-2.png\" data-asset-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" data-image-id=\"460dd1dc-d307-4f49-b7d8-f46b081238f6\" alt=\"\"></figure>\n<object type=\"application/kenticocloud\" data-type=\"item\" data-rel=\"link\" data-codename=\"four_point_bending_tests_on_t_beams__definition_of\"></object>\n<h2>材料属性</h2>\n<p>表 6.11 汇总了用于 CSFM 分析的抗剪钢筋、受弯钢筋和混凝土的材料属性。用于 <a data-item-id=\"2ebdaf9c-827f-4fd6-9f82-28bc96970a64\" href=\"\">CSFM</a> 分析所需的大部分材料属性均可从相应的试验报告中获得。表 6.11 中标明了必须假定的数值。 </p>\n<figure data-asset-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" data-image-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/d40a91ae-9e7a-4cbe-9980-8bc295693359/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11.png\" data-asset-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" data-image-id=\"8493aef1-7d61-4eca-8e25-aeda94512c10\" alt=\"\"></figure>\n<figure data-asset-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" data-image-id=\"b38d80f8-07f0-4992-b707-36da923011d3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/037428f1-0d3f-4404-9321-db2a2a0f0c8c/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11-2.png\" data-asset-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" data-image-id=\"b38d80f8-07f0-4992-b707-36da923011d3\" alt=\"\"></figure>\n<figure data-asset-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" data-image-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6388402c-7243-4c35-9936-241e9bbc86e5/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.11-3.png\" data-asset-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" data-image-id=\"78bf381f-664d-41ad-a30e-cb1dc0204a8e\" alt=\"\"></figure>\n<h2>使用 CSFM 建模</h2>\n<p>根据试验设置，在 CSFM 中对几何形状、钢筋、<a data-item-id=\"50ed723b-9b87-4870-a69f-e05b5a8a8150\" href=\"\">支座和加载</a>条件进行了建模。图 6.18 以 Vecchio 和 Shim (2004) 的试验 A3 为例，展示了建模过程。</p>\n<figure data-asset-id=\"be95afab-6efa-4ca2-af2d-187495214492\" data-image-id=\"be95afab-6efa-4ca2-af2d-187495214492\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ee8f6a35-2573-4491-a59a-c75249bae2ec/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.18.png\" data-asset-id=\"be95afab-6efa-4ca2-af2d-187495214492\" data-image-id=\"be95afab-6efa-4ca2-af2d-187495214492\" alt=\"\"></figure>\n<p>针对每个试验，使用以下参数进行了四次数值计算：</p>\n<ul>\n <li>网格尺寸，在梁高方向上从 5（这些特定示例的默认值）经 10 变化到 20 个有限单元。由于默认网格已经非常粗糙，本研究仅分析了更细的网格，除 M0 外均采用 10 个单元的网格。 </li>\n <li> 是否考虑拉力刚化效应。默认情况下，CSFM 会考虑拉力刚化。 </li>\n <li>是否考虑箍筋中可能存在的非稳定裂缝。若考虑（默认设置），则采用拉拔模型（POM）定义箍筋中的拉力刚化（所有梁的几何配筋率均低于（<em>ρ</em><em><sub>cr</sub></em>），因此从不使用拉力弦模型）。若不考虑，则通过 TCM 来计入拉力刚化。</li>\n</ul>\n<p><em>\\[ρ_{\\text{cr}} = \\frac{f_{\\text{ct}}}{f_{\\text{y}} - (n-1)f_{\\text{ct}}}\\]</em></p>\n<p>其中<em>：</em></p>\n<ul>\n <li>\\(f_y\\)<em> - </em>钢筋屈服强度</li>\n <li>\\(f_{ct}\\) - 混凝土抗拉强度</li>\n <li>\\(n = \\frac{E_s}{E_c}\\) - 弹性模量比</li>\n</ul>\n<p>表 6.12 给出了每次数值计算所用的参数。M0 对应于采用 CSFM 默认设置的模型。</p>\n<figure data-asset-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" data-image-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/654a5eaa-78a1-4ba4-aa24-c24a2bb360bd/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.12.png\" data-asset-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" data-image-id=\"f066bc6c-98fe-48c4-b90d-1eb356184726\" alt=\"\"></figure>\n<h2>与试验结果的比较</h2>\n<p>本节将 <a data-item-id=\"8b9399db-b927-491a-a50f-c66ad97560af\" href=\"\">CSFM</a> 给出的极限荷载和破坏模式与试验结果进行了比较。为了同时验证 CSFM 在正常使用性能和变形能力方面的适用性，还将模型给出的荷载-变形响应与所选梁的试验结果进行了比较。</p>\n<h4>破坏模式与极限荷载</h4>\n<p>表 6.13 汇总了试验中测得的极限剪力（<em>V</em><em><sub>u,exp</sub></em>）、CSFM 预测的极限剪力（<em>V</em><em><sub>u,calc</sub></em>），以及各自的破坏模式。该表还给出了每个数值模型中实测极限荷载与计算极限荷载比值的平均值和变异系数（CoV）。在所有分析中（除 M3 忽略了拉力刚化外），CSFM 均预测箍筋发生剪切破坏。这与 Huber (2016) 和 Piyamahant (2002) 试验中观察到的破坏机制吻合较好，但与 Vecchio 和 Shim (2004) 中观察到的破坏机制不符。未能准确捕捉破坏模式，在此情况下导致极限荷载的估计略偏保守。总体而言，默认参数给出了良好的强度估计，但略偏不安全一侧（平均偏差为 6%）。</p>\n<p>图 6.19 通过试验极限剪力与计算极限剪力之比（<em>V</em><em><sub>u,exp</sub></em><em>/V</em><em><sub>u,calc</sub></em>），展示了 CSFM 强度预测对所分析各数值参数的敏感性。极限荷载对所选有限单元尺寸有明显的敏感性（见图 6.19 a）。最粗网格与最细网格（M0 和 M2）之间的最大差异达到 36%（Piyamahant (2002) 的试验 4），平均差异约为 15%。采用默认参数（模型 M0 中梁高方向上 5 个有限单元）的预测结果略高估了试验强度（约 5%）。当将网格细化为梁高方向上 10 个或 20 个有限单元（分别对应模型 M1 和 M2）时，可获得极佳的强度预测结果，且略偏于极限荷载的安全一侧。改变有限单元网格尺寸时，未观察到破坏模式发生变化。即使采用默认网格尺寸，结果也相当令人满意，考虑到若干试验表现出脆性剪切破坏，而这类破坏用设计方法预测颇具挑战性。</p>\n<p>拉力刚化的考虑方式对强度预测有着高度相关的影响，如图 6.19 b-c 所示。通过 POM（CSFM 中的默认设置）考虑箍筋中的拉力刚化，平均而言与试验结果具有极佳的一致性（见图 6.19 b）。然而，忽略拉力刚化会导致极限荷载平均高估约 22%（见表 6.12）。忽略拉力刚化时，破坏模式转变为弯曲破坏（见表 6.12），与观察到的剪切破坏模式不符。结果对所采用的压力软化关系也非常敏感。如图 6.19 c 所示，在箍筋中使用拉力弦模型（模型 M4）而非拉拔模型（模型 M1），所得结果略优于忽略拉力刚化的情形（模型 M3），但仍严重高估极限荷载约 15%（见表 6.12）。因此可以得出结论：在这些算例中，采用拉拔模型对于正确模拟承载性能至关重要。 </p>\n<figure data-asset-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" data-image-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/4c6d49ee-719b-4472-9a67-06b062b2df81/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13.png\" data-asset-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" data-image-id=\"31352232-bbe5-4fb4-b54e-b0809010c318\" alt=\"\"></figure>\n<figure data-asset-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" data-image-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/f3c49504-d203-4f82-a25e-0ab44da27aaa/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13-2.png\" data-asset-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" data-image-id=\"2fcb4c3b-de08-410f-93c2-989b65091df0\" alt=\"\"></figure>\n<figure data-asset-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" data-image-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/6fcfdbf1-9666-447a-988f-40df2e112587/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20table%206.13-3.png\" data-asset-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" data-image-id=\"48a1d13e-083a-4eb5-927c-4812e5f4e37d\" alt=\"\"></figure>\n<figure data-asset-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" data-image-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/23242854-167d-40ac-8182-edd05e68f870/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19.png\" data-asset-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" data-image-id=\"82f061b1-5fb4-4ea7-8342-375c6db59907\" alt=\"\"></figure>\n<figure data-asset-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" data-image-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/a1037f6e-2499-4ade-bb10-681d8ea41e54/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19-2.png\" data-asset-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" data-image-id=\"3ccbbd8a-d639-4133-be1c-6e4e9c0abade\" alt=\"\"></figure>\n<figure data-asset-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" data-image-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/2391dda1-eba7-4458-8142-a7ff352da592/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.19-3.png\" data-asset-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" data-image-id=\"52e40f52-8d8f-4d36-a4c6-7885d25979c3\" alt=\"\"></figure>\n<p>图 6.20 展示了连续应力场结果（主压应力（<em>σ</em><em><sub>c</sub></em>）和钢筋应力（<em>σ</em><em><sub>sr</sub></em>）处的应力），图中标出了针对 Vecchio 和 Shim（2004）试件 A1 和 A3 预测的剪切破坏部位。这些结果是采用数值参数 M1 计算得到的（默认参数，但网格尺寸为默认值的一半）。从应力场可以看出，由弯曲引起的受压区压应力已进入塑性段（99.5%）。然而，由于所采用的混凝土压碎判据，箍筋断裂发生在混凝土压碎之前。 </p>\n<figure data-asset-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" data-image-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/86e2648e-f52e-4162-8ad8-a9615fc572c6/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.20.png\" data-asset-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" data-image-id=\"72d9b512-35a8-4c59-bf70-2fa1c8913dba\" alt=\"\"></figure>\n<figure data-asset-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" data-image-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/65718f07-1206-42f2-990a-97e7e18d0547/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.20-2.png\" data-asset-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" data-image-id=\"5faec16e-18c3-4336-aaaf-92b23a315aed\" alt=\"\"></figure>\n<h2>荷载-变形响应</h2>\n<p>图 6.21 将采用 M1（受弯钢筋考虑 TCM、箍筋考虑 POM）和 M3（忽略任何拉力刚化效应）数值参数计算得到的荷载-变形响应，与试验 R500m352、T1、A1 和 A3 实测的荷载-变形响应进行了比较。荷载 <em>V</em> 对应所施加的剪力，<em>u</em> 对应跨中挠度（见图 6.20a）。</p>\n<figure data-asset-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" data-image-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/b2a3cfdf-9892-4c29-b6b0-a26d1f47bb22/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21.png\" data-asset-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" data-image-id=\"4f5fd5df-0fbe-493e-aab6-f3613ea8a079\" alt=\"\"></figure>\n<figure data-asset-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" data-image-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/e90b12fb-ba6f-4ce8-97c3-ebca7c6212f6/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21-2.png\" data-asset-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" data-image-id=\"43e6f9e6-0867-450f-86f0-447652d4cba9\" alt=\"\"></figure>\n<figure data-asset-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" data-image-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\"><img src=\"https://assets-us-01.kc-usercontent.com:443/66e7a155-be94-0096-73e6-c55dfc7e5788/ba5f9efa-97c3-461f-b6fc-5e3fdd4f0ff9/Shear%20tests%20in%20beams%20with%20low%20amounts%20of%20stirrups%20fig%206.21-3.png\" data-asset-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" data-image-id=\"005c3655-78cd-4a12-b72d-adb1628d298d\" alt=\"\"></figure>\n<p>考虑拉力刚化效应后，在整个加载历程中均能较好地预测试验挠度，但峰值荷载处的挠度略被低估。特别是在 Vecchio 和 Shim（2004）的试验 A3 中，由于数值分析预测箍筋首先发生断裂，因此无法正确捕捉试验中因受弯钢筋屈服而出现的平台段。忽略拉力刚化效应会导致极限荷载和变形高估。对于不考虑拉力刚化的分析所得出的这些结论，在采用 M4 参数（箍筋和受弯钢筋均采用 TCM）时同样成立。</p>\n<h2>结论</h2>\n<p>针对含少量箍筋的简支梁试验，比较 CSFM 结果与观测到的试验行为，可得出以下结论： </p>\n<ul>\n <li>CSFM 对极限荷载给出了良好的估计，采用默认数值参数时略被高估（平均高估 5%）。由剪切和受弯区混凝土压碎共同导致的破坏模式较难捕捉；CSFM 预测的是箍筋断裂导致的破坏，这使得强度预测结果偏于保守。 </li>\n <li>极限荷载预测结果对有限单元网格尺寸的变化有一定的敏感性。细化默认有限单元网格后可获得最佳预测结果。因此，在进行最终校核时，始终建议研究有限单元尺寸对结果的影响。 </li>\n <li>忽略拉力刚化会导致极限荷载和变形能力被显著高估。即使通过拉力弦模型对箍筋中的拉力刚化进行建模，预测的极限荷载仍明显偏于不安全一侧。对于配筋量较低的箍筋，通过�Fault拔模型考虑非稳定裂缝的影响可获得最佳结果。这也是 CSFM 中默认采用的拉力刚化模型。 </li>\n</ul>\n<object 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