Title. Author(s)Y. IMAI; T. TSUJII; S. MOROOKA; K. NOMURA. Issue Date Doc URL. Type. Note. File Information

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1 Title CALCULATION FORULAS OF DESIGN BENDING OENTS ON TH APPLICATION OF THE SAFETY-ARGIN FRO RC STANDARD TO Autho(s)Y. IAI; T. TSUJII; S. OROOKA; K. NOURA Issue Date Doc URL Type poceedings Note The Thiteenth East Asia-Pacific Confeence on Stuc 13, 013, Sappoo, Japan. File Infomation easec13-f--5.pdf Instuctions fo use Hokkaido Univesity Collection of Scholaly and Aca

2 Calculation Fomulas of Design Bending oments on the Boundaies of Slabs Pat: Application of the Safety-agin fom RC standad to Othe Shaped Slabs Y. IAI 1*, T. TSUJII 1, S. OROOKA, and K. NOURA 3 1 Gaduate Student, Depatment of Achitectue, School of Engineeing, Tokai Univesity, Japan Pof., Depatment of Achitectue, Faculty of Engineeing, Tokai Univesity, Japan, D. Eng. 3 Gaduate Student, School of Science and Technology, Tokai Univesity, Japan,. Eng. ABSTRACT The pupose of this study is to popose calculation fomulas fo design bending moments on bounday sides of a fixed end tiangula slab and a fixed end tangential quadilateal slab subjected to unifom load. With these fomulas, you can calculate the design bending moment of these slabs moe simply. Keywods: Design bending moment, Tiangula slab, tangential quadilateal slab, Safety-magin 1. INTRODUCTION The pupose of this study is to popose calculation fomulas fo design bending moments on bounday sides of a fixed end tiangula slab and a fixed end tangential quadilateal slab subjected to unifom load. In the Pat1, the authos have assumed a cetain width in which the aveage bending moment is equal to the design bending moment given by RC standad and we poposed the equation that gives the aveage value of bending moment in the mean-width as the calculation fomula of design bending moment. In this Pat, the authos imagine the safety-magin against collapse, which RC standad fomula is thought to give to the slab. Fistly, we investigate this safety-magin in RC standad, then we apply this safety-magin to the tiangula slab and the tangential quadilateal slab and show calculation fomulas fo design bending moment of these slabs.. THE SAFETY-ARGIN OF RC STANDARD FORULA In this chapte, we calculate the safety-magin against collapse, which RC standad fomula gives a ectangula slab. This safety-magin can be calculated as the atio of the ultimate unifom load w u to the yield unifom load w y of the ectangula slab. And, by assuming that all of the yielding bending moments in any diections on an uppe o on a bottom sufaces ae the same value y, we can calculate the yield unifom load w y fom RC fomula, as the value of the bending moment equals to y. Note that, we assume that the collapse of the slab is caused only by bending so that we * Coesponding autho and Pesente: bcbm003@mail.tokai-u.jp 1

3 can get the ultimate unifom load w u by yield line theoy. In this theoy, we need the ultimate bending moment 0, and we assume that the value is constant all ove the slab, then we can wite it with the yielding bending moment y and the positive constant C as follows. 0 C y (1).1. Yield unifom load w y in RC standad fomula RC standad says that the maximum bending moment in a ectangula slab is wl x x1 () 1 1 whee w is the value of unifom load, l x is a length of the shote side, is a atio of long side to shot side. So substituting y to x1, we can have the unifom load w y that RC standad imagine as the yield unifom load as follows. 1 y 1 w y (3) l x.. Ultimate unifom load w u of the ectangula slab The ultimate unifom load of the fixed end ectangula slab subjected to unifom load calculated by yield line theoy can be efeed in some papes. Figue 1 shows the collapse mechanism of ectangula slab. Solid lines in the figue show yield lines whee positive bendings, which mean tension on the bottom and compession on the uppe suface, occu, and dashed lines show yield lines whee negative bendings occu. Angle can be calculated as shown in equation (), and the ultimate unifom load w u can be calculated as shown in equation (5). tan 1 ( ) () 8 0 w u (5) l x ( 1 3 1) Yield Line of Positive Bending Yield Line of Negative Bending Figue 1: Collapse mechanism of the ectangula slab.

4 .3. The safety-magin We can have the safety-magin in RC standad as a atio of the ultimate unifom load w u to the yield unifom load w y, and we found it depends on the atio of long side to shot side. wu C () wy (1 )( 1 3 1) 3. A PROPOSITION OF CALCULATION FORULAS FOR DESIGN BENDING OENTS We assume that the calculation fomulas fo fixed end tiangula o tangential quadilateal slabs have the same safety-magin as RC standad fomula has, and we apply the obtained safety-magin fom RC standad to each slab. Conseguently, the calculation fomulas fo design bending moments can be fomulated as the division of the ultimate unifom load by the safety-magin. Note that, we assume that y and 0 ae constant all ove these slabs Ultimate load of the tiangula slab and tangential quadilateal slab The ultimate unifom load of each slab is calculated by yield line theoy. And we need to find the collapse mechanism in which the ultimate unifom load is minimized. Figue shows the collapse mechanism of the squae slab calculated by equation (). The yield lines in positive bending come to be bisectos fom vetexes, and we ecognized that these lines coss in the cental point of the inscibed cicle. The bisectos fom vetexes of the tiangula slab and the tangential quadilateal slab coss also in the cental point of the inscibed cicle. And we can assume the collapse mechanism of each slab as shown in Figue 3. Note that, we had veified that the value of the ultimate unifom load obtained fom each collapse mechanism was the smallest value by a numeical analysis, and the mechanism is coect. Yield Line of Positive Bending Yield Line of Negative Bending Figue : Collapse mechanism of the squae slab. 3

5 Yield Line of Positive Bending Yield Line of Negative Bending Lmax Lmax Figue 3: Collapse mechanism of the tiangula slab and tangential quadilateal slab. The ultimate unifom load of each collapse mechanism is calculated by pinciple of vitual wok. In spite of the fact that thei figues ae diffeent, the ultimate unifom load can be expessed in the same fom, if you use the adios of the inscibed cicle as follows. 1C y a wu (7) Hee, is the adius. 3.. The safety-magin that apply to the tiangula slab and tangential quadilateal slab As expessed in equation (), the safety-magin in RC standad is changed by the atio. As descibed in Pat1, the atio might be 1.0. And we have the safety-magin d fo the tiangula o the tangential quadilateal slab by substituting =1 to equation () as follows. d C (8) 3.3. Design bending moments and the deceasing ate D m As descibed befoe, calculation fomulas fo design bending moments can be calculated as the division of the ultimate unifom load a w u by the safety-magin d. d w y a w d u y In this pape, we had assumed that 0 is can be witten in the fom of multiplication of C and y, but we found out that we don t need the C value to calculated the unifom load d w y as shown in equation (9). Solving equation (9) fo y, and letting d w y and y to be a design unifom load w and a design bending moment dm gives the following the calculation fomula fo design bending moment. 1 w dm (10) (9)

6 Fom hee, we estimate a deceasing atio which is given by the division of the value of the equation (10) by the maximum bending moments which occus in the slabs. Note that, calculation fomulas fo maximum bending moments on bounday sides of the tiangula slab and the tangential quadilateal slab had been poposed in pevious pape as follows t L i i 0.198( ) w, q L i i 0.190( ) w (11) Hee, t i is a maximum bending moment on the i-th bounday side of the tiangula slab, q i is a maximum bending moment on the i-th bounday side of the tangential quadilateal slab, L i is a length of the i-th bounday side. As we can undestand fom these equations, the maximum bending moment in the tiangula slab and in the tangential quadilateal slab occus on the longest bounday side in each slab. Letting the maximum bending moment t max in the tiangula slab, q max in the tangential quadilateal slab, and longest bounday side L max gives the following equations. t L max max 0.198( ) w, q L max max 0.190( ) w (1) So we have the deceasing atios t D m fo the tiangula slab and q D m fo the tangential quadilateal slab ae the atios of equation (11) to equation (1). They can be calculated as shown in equation (13). t D m dm 1 1, max L t max ( ) q D m dm 1 1 (13) max L q max ( ) Figue shows the values of deceasing atios t D m and qd m. A hoizontal axis in the figue shows L max /. A solid line in the figue expesses t D m, and a dashed line expesses q D m. Range of L max / is about 3. to 7. in the tiangula slab and about.00 to.73 in the tangential quadilateal slab. These anges come fom the assumed slab figue. The deceasing atio is aound 0% to 70% fo the tiangula slab and aound 5% to 80% fo the tangential quadilateal slab. The value of maximum bending moment can be futhe deceased, as L max / is inceased. 5

7 Deceasing Ratio Deceasing Ratio t D m q D m (Tiangle Slab) (Tangential Quadilateal Slab) L max / 7 Figue : Deceasing atio t D m and q D m.. DIFFERENCE OF CALCULATION FORULAS FOR DESIGN BENDING OENTS In this pape, the idea of the safety-magin of ectangula slabs against the collapse applied to the othe figued slabs gives the calculation fomulas fo design bending moment. In the Pat1, the idea of the mean-width gives the othe calculation fomulas. In this chapte, we descibe diffeence of these fomulas ae descibed. Figue 5 shows the deceasing atios calculated fom these fomulas. A hoizontal axis in the figue shows L max /. Solid lines in the figue show the deceasing atio obtained fom the mean-width, and dashed lines show the deceasing atio obtained fom the safety-magin. Thei tends ae simila to each othe. Its maximum diffeence is aound %, and calculation fomula fo design bending moment obtained fom the mean-width gives the safe value that obtained fom the safety-magin q D a (Tangential Quadilateal Slab, mean-width) 0.75 q D m (Tangential Quadilateal Slab, safety-magin) t D a (Tiangula Slab, mean-width) 0.0 t D m (Tiangula Slab, safety-magin) 3 L max / 5 7 Figue 5: Compaison of deceasing atio.

8 5. CONCLUSION In this pape, we poposed the calculation fomulas of design bending moment fom of the point of view of the safety-magin against the collapse. And the diffeences between this idea and that in Pat1 ae descibed. The aangement of einfocement in concete will be a futue poblem. REFERENCES Achitectual Institute of Japan (Revised 010). AIJ Standad fo Stuctual Calculation of Reinfoced Concete Stuctues, Aticle 10. Japan. in Japanese. Nomua K and oooka S (01). Calculation Fomula fo a aximum Stess and aximum Deflection of a Fixed End Quadilateal Slab Subjected to Unifom Load - In the case of tangential quadilateal, and isosceles tapezoid and ight-angled tapezoid -, Jounal of Stuctual and Constuction Engineeing in Japan. 80, pp in Japanese. Nomua K and oooka S (01). The Calculation Fomula fo a aximum Stess and a aximum Deflection of the Tiangle Slab with Consideing Effect of Suppot Condition Subjected to Unifom Load. Jounal of Stuctual Engineeing in Japan. 58B, pp in Japanese. Japan Society of Civil Enginees (00). The Stuctual echanics Handbook, 10. pp.35. in Japanese. 7

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