Title. Author(s)NOMURA, K.; MOROOKA, S. Issue Date Doc URL. Type. Note. File Information

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1 Title CALCULATION FORMULA FOR A MAXIMUM BENDING MOMENT AND THE TRIANGULAR SLAB WITH CONSIDERING EFFECT OF SUPPO UNIFORM LOAD Autho(s)NOMURA, K.; MOROOKA, S. Issue Date Doc URL Type poceedings Note The Thiteenth East Asia-Pacific Confeence on Stuc 13, 2013, Sappoo, Japan. File Infomation easec13-a-5-6.pdf Instuctions fo use Hoaido Univesity Collection of Scholaly and Aca

2 CALCULATION FORMULA FOR A MAXIMUM BENDING MOMENT AND A MAXIMUM DEFLECTION OF THE TRIANGULAR SLAB WITH CONSIDERING EFFECT OF SUPPORT CONDITION SUBJECTED TO UNIFORM LOAD K. NOMURA 1* and S. MOROOKA 2 1 Couse of Science and Technology, School of Science and Technology, Toai Univesity, Japan 2 Depatment of Achitectue, Faculty of Engineeing, Toai Univesity, Japan ABSTRACT The pupose of this study is to popose the calculation fomula fo allowable stess fo stuctual design of any shape slab. In pevious pape, we poposed the calculation fomula fo the maximum deflection and the maximum pincipal bending moment of the fixed edge tiangula planed slabs subjected to unifom load. In this pape, we have offeed the calculation fomula about the case of edges conditions change. As a pocess, we assume vitual slab and obtain a calculation fomula solution by applying it to the calculation fomula fo the fixed suppoted all edges. And so, this pape poposes simple calculation fomulas give FEM solutions. Keywods: Tiangula slab, Degee of fixation, Unifom load, Maximum deflection, Maximum Pincipal bending moment 1. INTRODUCTION In Japan, floo slabs, that ae assumed to be isotopic plates, ae stuctually designed sepaately fom the fames. We calculate bending moments of the plate, and then we aange einfoced bas which allow the bending moments. Fo ectangula slabs, we can use the design fomula of Achitectual Institute of Japan. By this fomula, we can calculate fo the appoximately bending moments on ectangula slabs. On the othe hand, fo non-ectangula shapes, engineeing judgment would be done because the fomula by which we can calculate the bending moments does not exist. Fom an above-mentioned eason, the pupose of authos study is to popose the design fomula of any shape of slab. In pevious pape, authos pesented fomulas by which we can calculate the maximum deflection and the maximum pincipal bending moment pe unit length of the tiangula planed slabs with fixed suppoted all edges subjected to unifom load. In this pape, we have offeed the calculation fomula of the tiangula slab when the degee of fixation of the edges conditions decease. When the degee of fixation of some edges decease, values of the maximum deflection * Coesponding autho: _nomua@sta.toai-u.jp Pesente: moo@eyai.cc.u-toai.ac.jp 1

3 and the maximum bending moment except on the edges incease as against the case of suppot is fixed. In this pape, we obtain a calculation fomula solution by applying lage slab to the calculation fomula fo the fixed edge slab. Note that, the vetex angle of the tiangula slabs is fom 30 degee to 90 degee. Hee, the distibution of displacements in the fixed edge tiangula slab is shown in Fig. 1, and distibution of pincipal bending moments is shown in Fig. 2. The maximum values teated by the calculation fomula ae the deflection max (+ ma in Fig. 1) and the pincipal bending moment on i-th bounday sides M b2i ( ma in Fig. 2) and the cente pat M c1 ( ma in Fig. 2). The tageted values of these maximum values ae obtained fom finite element method (FEM) Figue 1: Contou of distibution of the displacement Figue 2: Contou of distibution of the pincipal bending moment 2. MAXIMUM VALUES BY THE DEGREE OF FIXATION DECREASES 2.1. Solution of the degee of fixation deceases How to calculate the solution when the degee of fixation deceases is the following pocesses. And the pocess was shown in Fig. 3. (1) An FEM solution is calculated by the fixed edge slab. The bending moment on the bounday side which lowes degee of fixation (we will call the side decease-side) is calculated. (2) Mae the analysis model of a pin suppot of the decease-side. The atio that degee of fixation deceases is supplied to the bending moment calculated by (1), and the bending moment which negates it is supplied to the side consideed as the pin suppot. And then, FEM solution of the slab is calculated. Note that, we will call this atio decease-atio, and it will be a pin when setting the atio as 1, will be a fixed when setting the atio as 0. (3) The solution of the degee of fixation deceases is supeposition FEM solutions of (1) and (2). 2

4 Degee of Fixation decease Unifom load Fixed edge supeposition + = Figue 3: Pocess of calculate the solution when the degee of fixation deceases Hee, we change vetex angles evey 5 degee and obtain 25 tiangula slabs, change the decease-atios 0(fixed) to 1(pin) evey 0.1 and obtain 10 inds atios and change decease-side one to all and obtain 7 inds combination of the sides. By all these pattens, we calculated 1750 inds of FEM solutions. Futhemoe, Thei FEM solutions ae calculated by midas/gen. The element is an isotopic thin plate element and is used tiangula element (DKT element) and quadangula element (DKQ element). A divided width of the FEM model is 1/30 adius of the inscibed cicle Effect on the maximum values by the degee of fixation deceases As an example of the FEM solution, distibutions of the pincipal bending moments ae shown in Fig. 4. In the shape of these figues, vetex angles ae 30, 60, 90 degees. The decease-side is the longest side, and the decease-atio is 40 % in Fig. 4(a), and is 60 % in (c). These figues wee dawn in the lage absolute value in two inds of pincipal bending moments. We will call the values M A. As fo the thic line in a figue, M A means 0. (a) (b) Figue 4: Contou of distibution of the M A (a) decease-atio is 40 %; (b) decease-atio is 60 % Compaison of these figue and Fig. 2 (fixed suppoted all edges), by decease-side, it poved that the line of M A =0 is moving in the diection of a nomal of the side. On the othe hand, the shapes of the distibution of the inne pat of M A =0 line o the oute pat esemble one of the fixed edge slab. And the tend is not concened with the shape, the decease-atio o a numbe of the decease-side. Futhemoe, as the example of change of the maximum value fom the fixed suppoted all edges by loweing the degee of fixation, atios of the maximum deflections is shown in Fig. 5. These atios wee calculated by 25 inds of above tiangle shape, 7 inds combinations of the decease-side and the decease-atios as 40 %, 60% and 80 %. The vetical axis shows the atio of the maximum deflections, the hoizontal axis shows the atio of the length of the decease-side to all sides. And 3

5 Ratio of the maximum deflection mae + shows case of the decease-atios as 40 %, mae shows as 60 % and mae shows as 80 %.This figue poved that the atio of the maximum deflection inceased by the decease-atios and atio of the length of the decease-side to all sides Decease-atio 40% 60% 80% atio of the length (decease-side/all sides) Figue 5: Ratios of the maximum deflections (case of the degee of fixation decease to case of the fixed suppoted all edges) 3. ASSUMPTION OF THE CALCULATION FORMULA As mentioned above, the distibution of the bending moment when the degee of fixation deceases esembled one of the fixed suppoted all edges. Fom an above-mentioned eason, we consideed that we can calculate the maximum values by supposing the fixed edge slab which lage to the diection of the nomal of the decease-side. Such slab was shown in Fig. 6 with the boen line, and we will call it vitual-slab. Oiginal slab Vitual-slab ~ ~ :Radius of the inscibed cicle on the vitual-slab :Radius of the inscibed cicle on the oiginal slab Figue 6: vitual-slab and its adius of the inscibed cicle And, the calculation fomula of the fixed edge tiangula slabs is fomulated in authos pevious pape. Hee, these fomulas ae shown in Eqs. (1) to (3). 4 w , t (1 )( ) (1) Et 1 max t 3 4

6 F / M M L b2i tiw, 0 198( i ). ti. (2) 2 c1 t1w, 15. t ( )( ) (3) Hee, max is the maximum deflection, w is the unifom load, is the adius of the inscibed cicle, E is the young s module, t is the thicness of the slab, 1 is the minimum vetex angle, 4 is calculated by ( 3-2 )/2, M b2i is the maximum pincipal bending moments in i-th bounday sides, L i is the length of i-th bounday sides, M c1 is the maximum pincipal bending moment in the cente pat. A maximum diffeence in max between the solution by Eq. (1) and FEM solution is aound 1%, in M b2i is aound 4% and in M c1 is aound 2% As indicated in these fomulas, a dominant vaiable of the maximum values ae the adius of the inscibed cicle. Theefoe, we suppose the calculation fomula of the adius of the inscibed cicle on the vitual-slab. And we will call this adius. And so, when the degee of fixation in the some edge deceases, the decease-atio and the atio of the length of the decease-side to all sides influence the solution. Theefoe, these two atios ae used fo the calculation fomula of. A tageted value is defined fo to fomulate the calculation fomula of. The taget value of is a solved value which gave the FEM solution ( max, M b2i, M c1 ) and the vaiable used fo the FEM analysis (w, E, t, 1, 4, L i ) to Eqs. (1) to (3). We will call the taget value F. Hee, obtained F ae shown in Fig. 7. The vetical axis shows the atio of F to the oiginal adius, the hoizontal axis shows the value which multiplied the decease-atio by the atio of the length of the decease-side to all sides. And mae shows case of the max, mae + shows M b2i, mae shows M c max M c1 M b2i Decease-atio x decease-side / all sides Figue 7: Taget values F fo Fom the distibution of Fig. 7, we conside that F is can calculate using vaiable used hoizontal axis. Whee, Slope diffes fo each maximum value, and some nonlineaity is obseved in each distibution. Theefoe, we will fomulate fo the calculation fomula of fo each maximum 5

7 value, and vaiable used hoizontal axis changes and uses a tend. was assumed such as Eqs. (4) and (5). (1 I ) (4) 1 ( BL ) n bi (5) L L L n n n i Hee, ( max :d, M b2i :b, M c1 :c) is the adius of the inscibed cicle on the vitual-slab in each maximum value, is the oiginal adius of the inscibed cicle, L 1 to L 3 ae the lengths of the sides, B is decease-atio (fixed:0, pin:1), L bi is the length decease-side, I and n ae undecided constants. These undecided constants ae found out in ode that each maximum value by the Eqs. (4) and (5) and the F may be conguous. 4. PROPOSAL OF THE CALCULATION FORMULA The esult calculated fom the FEM solution is shown. Note that, we calculated fo I by the tiple figues significant figue, and calculated fo n by a single figue. And we made these fomulas as they ae bigge than in all F. ( ) (6) d ( ) (7) c ( ) (8) b 1 ( BL ). bi L L L i 07 (9) Each maximum value is equied by calculating by Eqs. (6) to (8), by substituting it fo of Eqs. (1) to (3). Hee, these and F ae shown in Fig. 8. Fig. 8 (a) shows the case of max ( d calculated by Eq. (6)), (b) shows the case of M b2i ( b calculated by Eq. (7)), (b) shows the case of M c1 ( c calculated by Eq. (8)). The vetical axis shows the atio of F o to the oiginal adius, the hoizontal axis shows calculated by Eq. (9). And mae shows F case of the max, mae + shows M b2i, mae shows M c1, the solid lines of each figue ae the calculated values of Eq. (6), Eq. (7) o Eq. (8). These figues indicate that the calculation fomula can simply expess F. 6

8 F / o c / F / o d / F / o b / F Eq. (6) 1.3 F Eq. (7) (a) (n=0.7) (b) (n=0.7) F Eq. (8) (c) (n=0.7) 0.8 Figue 8: F and (a): Case of max (b): Case of M b2i (c): Case of M c1 5. ACCURACY OF THE SOLUTION BY THE CALCULATION FORMULA A cumulative fequency distibution of the diffeence of the FEM solution and the solution of this calculation fomula is shown in Fig. 9. The vetical axis shows the cumulative fequency, the hoizontal axis shows the diffeence of the FEM solution and the solution of this calculation fomula. Both of the axes ae pecentage notations. Note that, in M b2i, the fequency is calculated fo each bounday sides. And the side numbe 1 is the longest side, and 3 is shotest side. A maximum diffeence in M b2i is aound 10%. As fo the diffeence, 80 pecent o moe is distibuted between 5 to 10%. A maximum diffeence in max is aound 15%. As fo the diffeence, 80 pecent o moe is distibuted between 5 to 15%. A maximum diffeence in M c1 is aound 18%. As fo the diffeence, 80 pecent o moe is distibuted between 5 to 15%. 7

9 Cumulative fequency [%] max M c1 M b21 M b22 M b Diffeence of the solutions [%] 20 Figue 9: Cumulative fequency distibution of the diffeences of the FEM solutions and solutions of this calculation fomula 6. CONCLUSION In this pape, we poposed the calculation fomula of the tiangula slab when the degee of fixation of the edges conditions decease. When degee of fixation of some bounday side deceases, we consideed that we can calculate the maximum values by supposing the fixed edge slab which lage to the diection of the nomal of the decease-side. We call this fixed edge slab vitual-slab. Then, we poposed the calculation fomula of the adius of the inscibed cicle on vitual-slab. The maximum values can be calculated that the adius is applied to calculation fomula of the fixed edge tiangula slab. It is useful that we can use the fomula of fixed edge slabs togethe. And the diffeence of the calculation fomula solution and the FEM solution ae less than 20%. REFERENCES Achitectual Institute of Japan (Revised 2010). AIJ Standad fo Stuctual Calculation of Reinfoced Concete Stuctues, Aticle 10. Japan. in Japanese. Nomua K and Moooa S (2012). Calculation Fomula fo a Maximum Stess and Maximum 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. 680, pp in Japanese. Nomua K and Moooa S (2012). The Calculation Fomula fo a Maximum Stess and a Maximum 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. NOMURA K. and MOROOKA S.: Maximum Deflection and Bending Moment in a Tiangle Slab: Poceedings of IASS 2010,