Interaction Diagram - Tied Reinforced Concrete Column (Using CSA A )

Transcription

1 Interaction Diagram - Tied Reinforced Concrete Column (Uing CSA A )

2 Interaction Diagram - Tied Reinforced Concrete Column Develop an interaction diagram for the quare tied concrete column hown in the figure below about the x-axi uing CSA A proviion. Determine ix control point on the interaction diagram and compare the calculated value in the Reference and with exact value from the complete interaction diagram generated by pcolumn engineering oftware program from StructurePoint. Figure 1 Reinforced Concrete Column Cro-Section Verion: June

3 Content 1. Pure Compreion Nominal axial reitance at zero eccentricity Factored axial load reitance at zero eccentricity Maximum factored axial load reitance Bar Stre Near Tenion Face of Member Equal to Zero, ( ε = f = 0 ) c, a, and train in the reinforcement Force in the concrete and teel P r and M r Bar Stre Near Tenion Face of Member Equal to 0.5 f y, ( f = f y ) c, a, and train in the reinforcement Force in the concrete and teel P r and M r Bar Stre Near Tenion Face of Member Equal to f y, ( f = - f y ) c, a, and train in the reinforcement Force in the concrete and teel P r and M r Pure Bending c, a, and train in the reinforcement Force in the concrete and teel P r and M r Pure Tenion Strength under pure axial tenion (P rt) Correponding Moment (M rt) Column Interaction Diagram - pcolumn Software Summary and Comparion of Deign Reult Concluion & Obervation Verion: June

4 Code Deign of Concrete Structure (CSA A ) Reference Reinforced Concrete Mechanic and Deign, 1 t Canadian Edition, 2000, Jame MacGregor and Fred Michael Bratlett, Prentice Hall Canada Inc. Deign Data f c = 35 MPa f y = 400 MPa Cover = 55 mm to the center of the reinforcement Column 400 mm x 400 mm Top reinforcement = 4 No. 30 Bottom reinforcement = 4 No. 30 Solution Ue the traditional hand calculation approach to generate the interaction diagram for the concrete column ection hown above by determining the following ix control point: Point 1: Pure compreion Point 2: Bar tre near tenion face of member equal to zero, ( f = 0 ) Point 3: Bar tre near tenion face of member equal to 0.5 f y ( f = f y ) Point 4: Bar tre near tenion face of member equal to f y ( f = - f y ) Point 5: Pure bending Point 6: Pure tenion 1

5 Figure 2 Control Point 2

6 1. Pure Compreion 1.1. Nominal axial reitance at zero eccentricity P 1 f ( A A ) f A o c g t y t P kn o Where f c 0.67 CSA A (Equation 10.1) Factored axial load reitance at zero eccentricity Since thi column i a tied column with teel train in compreion: P 1 c f c( Ag At ) f y A CSA A (Equation 10.11) t ro P kn ro Where: 0.65 CSA A (8.4.2) c 0.85 CSA A (8.4.3(a)) 1.3. Maximum factored axial load reitance CSA A (Equation 10.9) P, h P 0.80P r max ro ro Pr, max kn 0.80Pro kn Pr, max 3764 kn 3

7 2. Bar Stre Near Tenion Face of Member Equal to Zero, ( ε = f = 0 ) Figure 3 Strain, Force, and Moment Arm (ε t = f = 0) Strain ε i zero in the extreme layer of tenion teel. Thi cae i conidered when calculating an interaction diagram becaue it mark the change from compreion lap plice being allowed on all longitudinal bar, to the more evere requirement of tenile lap plice. CSA A (12.15 and 16) 2.1. c, a, and train in the reinforcement c d mm Where c i depth of the neutral axi meaured from the compreion edge of the column ection. CSA A (3.2) a 1 c mm CSA A (10.1.7) Where: a = Depth of equivalent rectangular tre block CSA A (3.2) CSA A (Equation 10.2) f c CSA A (8.4.2) c 0.85 CSA A (8.4.3(a)) CSA A (10.1.3) cu F cu y 400 ( c d2) (345 55) (Compreion) > y c 345 E 200, Force in the concrete and teel C f ab CSA A (10.1.7) rc 1 c c kn f 0 kn T f A 0 kn r 1 4

8 Since > y compreion reinforcement ha yielded f f 400 MPa y The area of the reinforcement in thi layer ha been included in the area (ab) ued to compute C c. A a reult, it i neceary to ubtract α 1ϕ cf c from ϕ f before computing C r: f 1 f A 2 C kn r c c 2.3. P r and M r Pr Crc Cr Tr kN h a h h M r Crc Cr d2 Tr d M r kn.m

9 3. Bar Stre Near Tenion Face of Member Equal to 0.5 fy, ( f = fy ) Figure 4 Strain, Force, and Moment Arm (f = f y) 3.1. c, a, and train in the reinforcement Fy 400 y E 200, 000 y y tenion reinforcement ha not yielded CSA A (8.4.2) c 0.85 CSA A (8.4.3(a)) CSA A (10.1.3) cu d1 345 c cu mm cu Where c i depth of the neutral axi meaured from the compreion edge of the column ection. CSA A (3.2) a 1 c mm CSA A (10.1.7) Where: a = Depth of equivalent rectangular tre block CSA A (3.2) CSA A (Equation 10.2) f c cu ( c d2) (268 55) (Compreion) > y c Force in the concrete and teel C f ab CSA A (10.1.7) rc 1 c c kn f E , MPa 6

10 T f A kn r 1 Since > y compreion reinforcement ha yielded f f 400 MPa y The area of the reinforcement in thi layer ha been included in the area (ab) ued to compute C c. A a reult, it i neceary to ubtract α 1ϕ cf c from ϕ f before computing C r: f 1 f A 2 C kn r c c 3.3. P r and M r Pr Crc Cr Tr kn h a h h M r Crc Cr d2 Tr d M r kn.m

11 4. Bar Stre Near Tenion Face of Member Equal to fy, ( f = - fy ) Figure 5 Strain, Force, and Moment Arm (f = - f y) Thi train ditribution i called the balanced failure cae and the compreion-controlled train limit. It mark the change from compreion failure originating by cruhing of the compreion urface of the ection, to tenion failure initiated by yield of longitudinal reinforcement c, a, and train in the reinforcement Fy 400 y E 200, tenion reinforcement ha yielded y 0.65 CSA A (8.4.2) c 0.85 CSA A (8.4.3(a)) CSA A (10.1.3) cu d1 345 c cu mm cu Where c i depth of the neutral axi meaured from the compreion edge of the column ection. CSA A (3.2) a 1 c mm CSA A (10.1.7) Where: a = Depth of equivalent rectangular tre block CSA A (3.2) CSA A (Equation 10.2) f c cu ( c d2) (220 55) (Compreion) > y c 220 8

12 4.2. Force in the concrete and teel C f ab CSA A (10.1.7) rc 1 c c kn f f 400 MPa y T f A kn r 1 Since > y compreion reinforcement ha yielded f f 400 MPa y The area of the reinforcement in thi layer ha been included in the area (ab) ued to compute C c. A a reult, it i neceary to ubtract α 1ϕ cf c from ϕ f before computing C r: f 1 f A 2 C kn r c c 4.3. P r and M r Pr Crc Cr Tr kn h a h h M r Crc Cr d2 Tr d M r kn.m

13 5. Pure Bending Figure 6 Strain, Force, and Moment Arm (Pure Moment) Thi correpond to the cae where the factored axial load reitance, P r, i equal to zero. Iterative procedure i ued to determine the factored moment reitance a follow: 5.1. c, a, and train in the reinforcement Try c mm Where c i depth of the neutral axi meaured from the compreion edge of the column ection. CSA A (3.2) a 1 c mm CSA A (10.1.7) Where: CSA A (Equation 10.2) f c CSA A (10.1.3) cu Fy 400 y E 200, 000 cu ( d1 c) ( ) (Tenion) > y tenion reinforcement ha yielded c CSA A (8.4.2) c 0.85 CSA A (8.4.3(a)) cu ( c d2) ( ) (Compreion) < y c Force in the concrete and teel C f ab CSA A (10.1.7) rc 1 c c kn f f 400 MPa y 10

14 T f A kn r 1 Since < y compreion reinforcement ha not yielded f E , MPa The area of the reinforcement in thi layer ha been included in the area (ab) ued to compute C c. A a reult, it i neceary to ubtract α 1ϕ cf c from ϕ f before computing C r: f 1 f A 2 C kn r c c 5.3. P r and M r Pr Crc Cr Tr kn The aumption that c = mm i correct h a h h M r Crc Cr d2 Tr d M r kn.m

15 6. Pure Tenion The final loading cae to be conidered i concentric axial tenion. The trength under pure axial tenion i computed by auming that the ection i completely cracked through and ubjected to a uniform train greater than or equal to the yield train in tenion. The trength under uch a loading i equal to the yield trength of the reinforcement in tenion Strength under pure axial tenion (P rt) Prt f y A 1 A kn 6.2. Correponding Moment (M rt) Since the ection i ymmetrical M 0kN.m rt 12

16 7. Column Interaction Diagram - pcolumn Software pcolumn program perform the analyi of the reinforced concrete ection conforming to the proviion of the Strength Deign Method and Unified Deign Proviion with all condition of trength atifying the applicable condition of equilibrium and train compatibility. For thi column ection, we ran in invetigation mode with control point uing the CSA A In lieu of uing program hortcut, psection (Figure 9) wa ued to place the reinforcement and define the cover to illutrate handling of irregular hape and unuual bar arrangement. Figure 7 Generating pcolumn Model 13

17 Figure 8 pcolumn Model Editor (psection) 14

18 Figure 9 Column Section Interaction Diagram about the X-Axi (pcolumn) 15

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22 8. Summary and Comparion of Deign Reult Support Table 1 - Comparion of Reult Pr, kn Mr, kn.m Hand Reference * ** pcolumn Hand Reference * ** pcolumn Max compreion Allowable compreion f = f = 0.5 f y Balanced point Pure bending Max tenion * Reinforced Concrete Mechanic and Deign, 1 t Canadian Edition, Jame MacGregor and Fred Bartlett Example 11-1 ** The reference ued CSA A where the reitance factor for concrete (ϕ c) i The hand calculation and pcolumn ued CSA A where the reitance factor for concrete (ϕ c) i (Check Column Interaction Diagram Uing CSA A Example) 19

23 9. Concluion & Obervation The analyi of the reinforced concrete ection performed by pcolumn conform to the proviion of the Strength Deign Method and Unified Deign Proviion with all condition of trength atifying the applicable condition of equilibrium and train compatibility. In the calculation hown above a P-M interaction diagram wa generated with moment about the X-Axi (Uniaxial bending). Since the reinforcement in the ection i not ymmetrical, a different P-M interaction diagram i needed for the other orthogonal direction about the Y-Axi (See the following Figure for the cae where f = f y). Figure 10 Strain, Force, and Moment Arm (f = - f y Moment About x- and y-axi) 20

24 When running about the Y-Axi, we have 2 bar in 4 layer intead of 4 bar in jut 2 layer (about X-Axi) reulting in a completely different interaction diagram a hown in the following Figure. Figure 11 Comparion of Column Interaction Diagram about X-Axi and Y-Axi (pcolumn) Further difference in the interaction diagram in both direction can reult if the column cro ection geometry i irregular. In mot building deign calculation, uch a the example hown for flat plate or flat lab concrete floor ytem, all building column are ubjected to M x and M y due to lateral force and unbalanced moment from both direction of analyi. Thi require an evaluation of the column P-M interaction diagram in two direction imultaneouly (biaxial bending). StucturePoint pcolumn program can alo evaluate column ection in biaxial mode to produce the reult hown in the following Figure for the column ection in thi example. 21

25 Figure 12 Nominal & Deign Interaction Diagram in Two Direction (Biaxial) (pcolumn) 22