外文翻譯---鋼筋混凝土填充框架結(jié)構(gòu)對拆除兩個相鄰的柱的響應(yīng)(節(jié)選）

1、<p>　　南 京 理 工 大 學(xué) 紫 金 學(xué) 院</p><p>　　畢業(yè)設(shè)計(論文)外文資料翻譯</p><p>　　系： 機(jī)械工程 </p><p>　　專 業(yè)： 土木工程 </p><p>　　姓

2、 名： 查富生 </p><p>　　學(xué) 號： 090105203 </p><p>　　外文出處： Computers and Geotechnics </p><p>　　附 件： 1.外文資料翻譯譯文；2.

3、外文原文。 </p><p>　　注：請將該封面與附件裝訂成冊。</p><p>　　附件1：外文資料翻譯譯文</p><p>　　鋼筋混凝土填充框架結(jié)構(gòu)對拆除兩個相鄰的柱的響應(yīng)</p><p>　　作者：Mehrdad Sasani 美國波士頓東北大學(xué)，斯奈爾400設(shè)計中心 MA02115</p><p>　　收稿日

4、期：2007年7月27日，修整后收稿日期2007年12月26日，錄用日期2008年1月24日，網(wǎng)上上傳日期2008年3月19日。</p><p><b>　　摘要：</b></p><p>　　本文是評價圣地亞哥旅館對同時拆除兩根相鄰的外柱的響應(yīng)問題，圣地亞哥旅館是個6層鋼筋混凝土填充框架結(jié)構(gòu)。結(jié)構(gòu)的分析模型應(yīng)用了有限元法和以此為基礎(chǔ)的分析模型來計算結(jié)構(gòu)的整體和局部變

5、形。分析結(jié)果跟實驗結(jié)果非常吻合。當(dāng)測量的豎向位移增加到為四分之一英寸（即6.4mm）的時候，結(jié)構(gòu)就發(fā)生連續(xù)倒塌。通過實驗分析方法評價和討論隨著柱的移除而產(chǎn)生的變形沿著結(jié)構(gòu)高度上的發(fā)展和荷載動態(tài)重分配。討論了軸向和彎曲的變形傳播的不同。結(jié)構(gòu)橫向和縱向的三維桁架在填充墻的參與下被認(rèn)為是荷載重分配的主要構(gòu)件。討論了兩種潛在的脆性破壞模型（沒有拉力加強(qiáng)的梁的脆斷和有加筋肋的梁的擠出）。分析評價了結(jié)構(gòu)對額外的重力和無填充墻時的響應(yīng)。</p&

6、gt;<p>　　Elsevier有限責(zé)任公司對此文保留所有權(quán)利。</p><p><b>　　關(guān)鍵詞：</b></p><p>　　連續(xù)倒塌，荷載重分配，對荷載抵抗能力，動態(tài)響應(yīng)，非線性分析，脆性破壞。</p><p><b>　　介紹：</b></p><p>　　作為減小由于結(jié)構(gòu)

7、的局部損壞而造成大量傷亡的可能性措施的一部分，美國總務(wù)管理局【1】和國防部【2】出臺了一系列制度來評價結(jié)構(gòu)對連續(xù)倒塌的抵抗力?！?】定義連續(xù)倒塌為，由原始單元的局部破壞在單元間的擴(kuò)展最終造成結(jié)構(gòu)的整體或不成比例的大部破壞。</p><p>　　通過Ellingwood 和Leyendecker【4】建議的方法，ASCE/SEI 7定義了兩種一般模型來減小結(jié)構(gòu)設(shè)計時連續(xù)倒塌效應(yīng)產(chǎn)生的損害，它們分為直接和間接的設(shè)計方

8、法。一般建筑規(guī)范和標(biāo)準(zhǔn)用增加結(jié)構(gòu)的整體性的間接設(shè)計方法。間接設(shè)計法也應(yīng)用于美國國防部的降低連續(xù)倒塌設(shè)計和未歸檔設(shè)備標(biāo)準(zhǔn)中。盡管間接設(shè)計法可以降低連續(xù)破壞的風(fēng)險【6，7】，對基于此法設(shè)計的結(jié)構(gòu)破壞后的表現(xiàn)的判斷是不容易實現(xiàn)的。</p><p>　　有一種基于直接設(shè)計的方法通過研究瞬間消除受載構(gòu)件，比如柱子，對結(jié)構(gòu)的影響來評價結(jié)構(gòu)的連續(xù)倒塌。美國防部和國家事務(wù)管理局的規(guī)章是要求去除一個受荷構(gòu)件，考慮其影響。這樣的規(guī)范

9、目的是評價結(jié)構(gòu)的整體性和結(jié)構(gòu)的一個單元出現(xiàn)嚴(yán)重的毀壞時的分荷能力。這種方法是研究結(jié)構(gòu)受連續(xù)倒塌的影響的程度，但是事實上初始結(jié)構(gòu)損傷的影響不止局限于某一根柱子。</p><p>　　在本論文中，應(yīng)用通過實驗證實的分析結(jié)果，評價圣地亞哥旅館抵抗連續(xù)破壞的能力，實驗中瞬間移除兩個相鄰的柱子，其中一個柱是拐角柱。為了爆除這兩個柱子，將炸藥放在預(yù)先在柱子上鉆的孔里面。柱子然后再用幾層保護(hù)材料包裹好，以避免爆炸時的沖擊波和碎

10、片影響結(jié)構(gòu)的其他部分。</p><p><b>　　建筑的特性</b></p><p>　　圣地亞哥旅館建造于1914年，在1924年又向南擴(kuò)展了一部分，此部分包括兩個分離的結(jié)構(gòu)。圖.1是從南邊看旅館的樣子。注意這張照片，旅館的第一和第三層被用黑色的布蒙了起來。這個六層的旅館是無延性的鋼筋混凝土框架結(jié)構(gòu)，其中還有由空心磚構(gòu)成的填充外墻。擴(kuò)展部分的填充墻有兩層共203m

11、m厚。第一層的樓高為6.0m，其他樓蓋高為3.2m，頂樓高度為5.13m。圖.2為其中一個擴(kuò)展部分的第二層。圖.3為對本建筑的實施計劃，即瞬間爆除一層相鄰的柱A2和A3，以評價其影響。</p><p>　　左圖：圖.1 圣地亞哥旅館的南端視角，本論文研究其中心結(jié)構(gòu) </p><p>　　右圖：圖.2 擴(kuò)展結(jié)構(gòu)的第二層（南端視角）</p><p>　　下圖：圖.3 擬

12、對旅館南擴(kuò)展部分實施的柱的移除計劃，第一層要被移除的柱用叉號標(biāo)出</p><p>　　如圖.3所示樓蓋系統(tǒng)縱向（南北向）有一個托梁。根據(jù)兩個混凝土構(gòu)件受壓的實驗結(jié)果，對一個標(biāo)準(zhǔn)的混凝土柱，受壓承載力為31MPa?；炷恋膹椥阅Ａ看蟾艦?6300MPa左右。同樣，通過橫截面12.7mm的鋼筋受拉實驗，其屈服和極限抗拉強(qiáng)度分別為427和600MPa。鋼筋的極限變形為0.17。鋼筋的彈性模量近似為200000MPa。&

13、lt;/p><p>　　這個建筑按計劃將被爆破摧毀。作為摧毀的一個步驟，第一層和第三層的填充墻被移除。移除時上面 沒有活荷載。所有的非結(jié)構(gòu)部件包括隔墻、管道設(shè)備、家具都被事先搬走了，只有梁、柱、樓板梁和在邊梁上的填充墻被留下。</p><p><b>　　傳感器布置</b></p><p>　　混凝土和鋼筋的應(yīng)變傳感器是用來測量梁和柱的應(yīng)變變化的。

14、線性電位計用來測量整體和局部變形?；炷翍?yīng)變測量儀常900mm，最大應(yīng)變?yōu)?#177;0.02.鋼筋應(yīng)變測量儀應(yīng)變極限為±0.2。應(yīng)變測量儀可以帶到幾百千赫茲。電位計用來測量建筑中梁沿一端的轉(zhuǎn)動和整體位移，這些以后將講到。電位計的分辨率為0.01mm，最大速度為1.0m/s，實驗中最大記錄速度為0.35m/s。</p><p><b>　　有限元模型</b></p>

15、<p>　　通過有限單元法，在軟件SAP2000【8】中生成一個建筑模型。梁和柱都被抽象成Bernoulli單元。T和L型梁的翼緣計算寬度為四倍的較厚板的厚度【5】。塑性鉸可以發(fā)生在任何鋼筋可能發(fā)生屈服的地方，包括單元的端點、加筋肋分離點和彎矩的屈服點。在分析中，塑性鉸的范圍是構(gòu)件高度的一半?，F(xiàn)行版本的SAP2000不能計算出單元斜裂縫的構(gòu)成。為了得出正確的構(gòu)件撓曲剛度，反復(fù)做以下步驟：首先假設(shè)建筑的所有單元都是沒有裂縫的；然

16、后，需要彎矩同構(gòu)件的出現(xiàn)裂縫的彎矩相比較。分別降低板厚和梁的慣性矩35%，使需求彎矩大于裂縫出現(xiàn)彎矩。梁外部出現(xiàn)裂縫的正負(fù)彎矩分別為58.2knM和37.9knM。需要注意的是柱子沒有裂縫出現(xiàn)。再后，再按以上方法重新分析建筑和彎矩簡圖。重復(fù)這些步驟直到所有的裂縫區(qū)域被鑒定和用模型表示出來。除了兩端區(qū)域建筑結(jié)構(gòu)里的梁上部不配筋（圖.4）。例如，梁A1-A2在距A1點305mm以后，其上部不配筋（如圖.4和5）。為了確定出可能喪失撓曲強(qiáng)度的

17、截面位置，將裂紋鉸布置在上部沒有配筋的可能的彎曲破壞點上。塑性鉸的撓曲強(qiáng)度設(shè)為于Mcr相等，當(dāng)所受的彎矩達(dá)到Mcr時，該截面即發(fā)生破壞。</p><p>　　圖.4 二層的梁A3-B3和梁A1-A2詳細(xì)配筋情況</p><p>　　樓蓋系統(tǒng)有沿縱向（南北向）的次梁。圖.6所示為一典型的樓蓋的橫截面。為了計算出次梁和板的可能的非線性響應(yīng)，用梁單元為樓蓋建立模型。次梁按T型梁計算，翼緣的計算寬

18、度為各自板厚的四倍【5】。選取軸2和軸3的縱梁和其之間的一個寬20英寸的梁間的格柵為板的計算模型。為了給出板沿橫向的計算模型，同樣用一個寬20英寸于橫梁平行的梁。在方形的板中其剪力流和梁單元的中的不一樣。所以其扭轉(zhuǎn)剛度取為整個截面剛度的一半【9】。</p><p>　　圖.5 梁的上部配筋彎曲位置（于梁A1-A2相似，在鄰近建筑靠近柱A1的地方）</p><p>　　圖.6 典型的樓蓋的

19、次梁系統(tǒng) </p><p>　　圖.7 實驗和分析的第二層柱A3的豎向位移</p><p>　　建筑的2、4、5、6層有填充墻，并在門窗等開口位置有過梁，如前面提到的第1、三層的填充墻，在爆除前已經(jīng)拆掉。填充墻是用良好的空隙磚砌成的，空心磚的凈空是其總大小的一半。填充墻的平面效應(yīng)增強(qiáng)了建筑的剛度和強(qiáng)度，并且影響建筑的對荷載反應(yīng)即變形。如果忽略墻的影響將得不到準(zhǔn)確的建筑的剛度和強(qiáng)度。&l

20、t;/p><p>　　在SAP2000中考慮了兩種填充墻的形式：一種是用平面框架模型（模型A），另一種是FEMA365【10】中建議的受壓桿件模型（模型B）。</p><p>　　4.1模型A是平面框架模型，但是，現(xiàn)行版本的SAP2000只能計算線性框架模型，不能計算裂縫的發(fā)展情況。填充墻的抗拉強(qiáng)度大概為26psi，彈性模量為644ksi【10】。由于裂縫的發(fā)展對填充墻的剛度影響很大，重復(fù)以下

21、步驟來計算裂縫的形成：</p><p>　　（1）假設(shè)填充墻是線性的而且沒有開裂，運(yùn)行非線性歷史分析。由于梁中的塑性鉸的存在，梁中彎矩大于裂縫出現(xiàn)彎矩時候，對截面慣性矩有一個折減。</p><p>　?。?）判定填充墻出現(xiàn)的依據(jù)是看其應(yīng)力于墻的抗拉強(qiáng)度大小關(guān)系。</p><p>　?。?）節(jié)點在拉應(yīng)力大于抗拉強(qiáng)度的地方分離。</p><p>

22、　　重復(fù)上面的步驟直到裂縫區(qū)域被確定。</p><p>　　4.2.模型B（受壓桿件模型）</p><p>　　如FEMA356【10】所述用受壓桿件來代替填充墻，桿件的方向根據(jù)移除柱后的結(jié)構(gòu)變形形式和開口位置確定。</p><p><b>　　4.3.柱的移除</b></p><p>　　按以下步驟模擬柱的移除。<

23、;/p><p>　　結(jié)構(gòu)是在只受永久荷載下分析的，內(nèi)力在柱端測定，將隨著柱的移除而卸荷。</p><p>　　模型的建立是在移除第一層的柱A2、A3的情況下進(jìn)行的。結(jié)構(gòu)同樣是在永久荷載下進(jìn)行靜態(tài)分析的。在此情況下，測得的柱端內(nèi)力被當(dāng)成永久外部荷載施加在結(jié)構(gòu)上。注意此分析結(jié)果跟第一步的分析是等價的。</p><p>　　第二步中大小相等方向相反的柱端力，被瞬間施加在原柱的

24、位置上，然后進(jìn)行動態(tài)分析。</p><p>　　4.4.實驗和分析結(jié)果的比較</p><p>　　結(jié)構(gòu)計算最大豎向位移在第二層的柱A3上，圖7所示為按模型A的實驗和分析的梁A3豎向位移的比較。實驗數(shù)據(jù)是用三個粘在A3兩端的傳感器記錄的。實驗和分析得到的最大位移分別是6.1mm和6.4mm，相差盡為4%。實驗和分析的位移產(chǎn)生所用時間分別為0.069S和0.066S。分析結(jié)果顯示永久位移為5.

25、3mm，比實驗結(jié)果小14%，實驗結(jié)果為6.1mm。</p><p>　　圖.8.第二層的柱A3在模型A和B下分別沿時間的豎向位移</p><p>　　圖.8.比較了第二層的柱A3分別在模型A和B下分析的沿時間的豎向位移。由圖中可以看出，按受壓桿件模型（模型B）得出的最大豎向位移為11.4mm,比用模型A得出的結(jié)果高出約80%。在圖.7.可以看出按模型A得出的結(jié)果與實驗結(jié)果是想接近的，B模型

26、得出的結(jié)構(gòu)變形過高。如果最大豎向位移偏大的話，填充墻開裂情況會更加嚴(yán)重，更偏向于受壓桿件形成，模型A和模型B得出結(jié)果差異將減小。</p><p>　　圖.9.比較了用模型A時第二層的柱A2的分析和實驗的位移值。同樣，第一次達(dá)到最大位移值的實驗和分析值非常接近，分析的永久位移值比實驗的位移值略微低些。圖.10.所示為根據(jù)模型A得出的最大豎向位移的結(jié)構(gòu)變形放大200倍后的情況。</p><p>

27、;　　圖.9.第二層的柱A2豎向位移實驗和分析結(jié)果比較</p><p>　　圖.10.按模型A，F(xiàn)EM分析的結(jié)構(gòu)變形形式（第二層的實驗得出變形形式也給出）</p><p>　　通過實測得的變形形式在圖中也用實線標(biāo)出了。在二層的梁A1-A2、A3-B3的上下端部應(yīng)力重分配復(fù)雜的地方共用了14個電位計。梁上部和對應(yīng)的下部電位計接在一起用來測量梁的扭轉(zhuǎn)變形。用上下端部電位計的差值除以電位計的距離

28、（沿梁高）。分析推算的二層梁端部變形曲線如圖中的曲線所示。由圖可以看出，分析的變形梁的變形曲線跟實驗所得結(jié)果非常吻合。</p><p>　　根據(jù)模型A分析結(jié)果表明預(yù)示鋼筋屈服的塑性鉸只有兩個，四個沒有上部配筋的截面，到達(dá)屈服極限而開裂。圖.10.給出了所有的塑性鉸及開裂位置。</p><p>　　附件2：外文原文（復(fù)印件）</p><p>　　Response of

29、 a reinforced concrete infilled-frame structure to removal of two adjacent columns</p><p>　　Mehrdad Sasani_</p><p>　　Northeastern University, 400 Snell Engineering Center, Boston, MA 02115, Unit

30、ed States</p><p>　　Received 27 June 2007; received in revised form 26 December 2007; accepted 24 January 2008</p><p>　　Available online 19 March 2008</p><p><b>　　Abstract</

31、b></p><p>　　The response of Hotel San Diego, a six-story reinforced concrete infilled-frame structure, is evaluated following the simultaneous removal of two adjacent exterior columns. Analytical models o

32、f the structure using the Finite Element Method as well as the Applied Element Method are used to calculate global and local deformations. The analytical results show good agreement with experimental data. The structure

33、resisted progressive collapse with a measured maximum vertical displacement of only one </p><p>　　c 2008 Elsevier Ltd. All rights reserved.</p><p>　　Keywords: Progressive collapse; Load redistri

34、bution; Load resistance; Dynamic response; Nonlinear analysis; Brittle failure</p><p>　　1. Introduction</p><p>　　As part of mitigation programs to reduce the likelihood of mass casualties follow

35、ing local damage in structures, the General Services Administration [1] and the Department of Defense [2] developed regulations to evaluate progressive collapse resistance of structures. ASCE/SEI 7 [3] defines progressiv

36、e collapse as the spread of an initial local failure from element to element eventually resulting in collapse of an entire structure or a disproportionately large part of it. Following the approaches</p><p>

37、　　2. Building characteristics</p><p>　　Hotel San Diego was constructed in 1914 with a south annex added in 1924. The annex included two separate buildings. Fig. 1 shows a south view of the hotel. Note that i

38、n the picture, the first and third stories of the hotel are covered with black fabric. The six story hotel had a non-ductile reinforced concrete (RC) frame structure with hollow clay tile exterior infill walls. The infil

39、ls in the annex consisted of two wythes (layers) of clay tiles with a total thickness of about 8 in (203 mm). Th</p><p>　　beams were present.</p><p>　　3. Sensors</p><p>　　Concrete a

40、nd steel strain gages were used to measure changes in strains of beams and columns. Linear potentiometers were used to measure global and local deformations. The concrete strain gages were 3.5 in (90 mm) long having a ma

41、ximum strain limit of ±0.02. The steel strain gages could measure up to a strain of ±0.20. The strain gages could operate up to a several hundred kHz sampling rate. The sampling rate used in the experiment was

42、1000 Hz. Potentiometers were used to capture rotation (integ</p><p>　　4. Finite element model</p><p>　　Using the finite element method (FEM), a model of the building was developed in the SAP2000

43、 [8] computer program. The beams and columns are modeled with Bernoulli beam elements. Beams have T or L sections with effective flange width on each side of the web equal to four times the slab thickness [5]. Plastic hi

44、nges are assigned to all possible locations where steel bar yielding can occur, including the ends of elements as well as the reinforcing bar cut-off and bend locations. The characteristics</p><p>　　The beam

45、s in the building did not have top reinforcing bars except at the end regions (see Fig. 4). For instance, no top reinforcement was provided beyond the bend in beam A1–A2, 12 inches away from the face of column A1 (see Fi

46、gs. 4 and 5). To model the potential loss of flexural strength in those sections, localized crack hinges were assigned at the critical locations where no top rebar was present. Flexural strengths of the hinges were set e

47、qual to Mcr. Such sections were assumed to lose thei</p><p>　　The floor system consisted of joists in the longitudinal direction (North–South). Fig. 6 shows the cross section of a typical floor. In order to

48、account for potential nonlinear response of slabs and joists, floors are molded by beam elements. Joists are modeled with T-sections, having effective flange width on each side of the web equal to four times the slab thi

49、ckness [5]. Given the large joist spacing between axes 2 and 3, two rectangular beam elements with 20-inch wide sections are used betwe</p><p>　　equal to one-half of that of the gross sections [9].</p>

50、<p>　　The building had infill walls on 2nd, 4th, 5th and 6th floors on the spandrel beams with some openings (i.e. windows and doors). As mentioned before and as part of the demolition procedure, the infill walls

51、in the 1st and 3rd floors were removed before the test. The infill walls were made of hollow clay tiles, which were in good condition. The net area of the clay tiles was about 1/2 of the gross area. The in-plane action o

52、f the infill walls contributes to the building stiffness and strength and</p><p>　　Using the SAP2000 computer program [8], two types of modeling for the infills are considered in this study: one uses two dim

53、ensional shell elements (Model A) and the other uses compressive struts (Model B) as suggested in FEMA356 [10] guidelines.</p><p>　　4.1. Model A (infills modeled by shell elements)</p><p>　　Infi

54、ll walls are modeled with shell elements. However, the current version of the SAP2000 computer program includes only linear shell elements and cannot account for cracking. The tensile strength of the infill walls is set

55、equal to 26 psi, with a modulus of elasticity of 644 ksi [10]. Because the formation ofcracks has a significant effect on the stiffness of the infill walls, the following iterative procedure is used to account for crack

56、formation:</p><p>　　(1) Assuming the infill walls are linear and uncracked, a nonlinear time history analysis is run. Note that plastic hinges exist in the beam elements and the segments of the beam elements

57、 where moment demand exceeds the cracking moment have a reduced moment of inertia.</p><p>　　(2) The cracking pattern in the infill wall is determined by comparing stresses in the shells developed during the

58、analysis with the tensile strength of infills.</p><p>　　(3) Nodes are separated at the locations where tensile stress exceeds tensile strength. These steps are continued until the crack regions are properly

59、modeled.</p><p>　　4.2. Model B (infills modeled by struts)</p><p>　　Infill walls are replaced with compressive struts as described in FEMA 356 [10] guidelines. Orientations of the struts are det

60、ermined from the deformed shape of the structure after column removal and the location of openings.</p><p>　　4.3. Column removal</p><p>　　Removal of the columns is simulated with the following p

61、rocedure.</p><p>　　(1) The structure is analyzed under the permanent loads and the internal forces are determined at the ends of the columns, which will be removed.</p><p>　　(2) The model is mod

62、ified by removing columns A2 and A3 on the first floor. Again the structure is statically analyzed under permanent loads. In this case, the internal forces at the ends of removed columns found in the first step are appli

63、ed externally to the structure along with permanent loads. Note that the results of this analysis are identical to those of step 1.</p><p>　　(3) The equal and opposite column end forces that were applied in

64、the second step are dynamically imposed on the ends of the removed column within one millisecond [11] to simulate the removal of the columns, and dynamic analysis is conducted.</p><p>　　4.4. Comparison of an

65、alytical and experimental results</p><p>　　The maximum calculated vertical displacement of the building occurs at joint A3 in the second floor. Fig. 7 shows the experimental and analytical (Model A) vertical

66、 displacements of this joint (the AEM results will be discussed in the next section). Experimental data is obtained using the recordings of three potentiometers attached to joint A3 on one of their ends, and to the groun

67、d on the other ends. The peak displacements obtained experimentally and analytically (Model A) are 0.242 in (6.1 mm)</p><p>　　Fig. 8 compares vertical displacement histories of joint A3 in the second floor e

68、stimated analytically based on Models A and B. As can be seen, modeling infills with struts (Model B) results in a maximum vertical displacement of joint A3 equal to about 0.45 in (11.4 mm), which is approximately 80% la

69、rger than the value obtained from Model A. Note that the results obtained from Model A are in close agreement with experimental results (see Fig. 7), while Model B significantly overestimates the def</p><p>

70、　　Fig. 9 compares the experimental and analytical (Model A) displacement of joint A2 in the second floor. Again, while the first peak vertical displacement obtained experimentally and analytically are in good agreement,

71、the analytical permanent displacement under estimates the experimental value. </p><p>　　Analytically estimated deformed shapes of the structure at the maximum vertical displacement based on Model A are shown

72、 in Fig. 10 with a magnification factor of 200. The experimentally measured deformed shape over the end regions of beams A1–A2 and A3–B3 in the second floor</p><p>　　are represented in the figure by solid li

73、nes. A total of 14 potentiometers were located at the top and bottom of the end regions of the second floor beams A1–A2 and A3–B3, which were the most critical elements in load redistribution. The beam top and correspond

74、ing bottom potentiometer recordings were used to calculate rotation between the sections where the potentiometer ends were connected. This was done by first finding the difference between the recorded deformations at the

75、 top and bottom of </p><p>　　Analytical results of Model A show that only two plastic hinges are formed indicating rebar yielding. Also, four sections that did not have negative (top) reinforcement, reached