Ch. 10 Design of Short Columns Subject to Axial Load and Bending

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1 Ch. 10 Design o Short Columns Subjet to Axial Load and Bending Axial Loading and Bending Development o Interation Diagram Column Design Using P-M Interation Diagram Shear in Columns Biaxial Bending Examples

2 Axial Load and Bending Equivalent

3 Development o Interation Diagrams

4 b u = M P h Y X As3, d3 As2, d2 As4, d4 s2 s3 s4 a s3 s2 s4 P F s4 C F s3 F s2 P n As1, d1 s1 Setion Strain Stress Equilibrium C F si 0.85 ab A si si where h a h M n C Fsi di P-M interation urve an be onstruted by plotting (M n,p n ) or various s1 or. s1 si y F s1

5 Example 10.2: olumn setion 6 #29 bars 356 mm 64 mm 482 mm 64 mm 610 mm

6 Example 10.2: Strain Proile s a mm s y 64 mm 180 mm 302 mm 64 mm 244 mm =366 mm 610 mm

7 Example 10.2: Fore Equilibrium s MPa T s A s s kN kn 2598 kn 801 kn 92 mm C 0.85 ab kn C s A 801kN 155 mm s s P n M 149 mm 64 mm 241 mm 241 mm 64 mm C n C C h 2 s T s 305 mm 305 mm kN a F 2 si h d 2 si kn m 3

8 Example 10.2: Tension Failure New Strain Proile s > y ( y ) 64 mm 610 mm

9 Example 10.2: P-M Interation Curve 6595 kn 2828 kn kn-m 2245 kn 759 kn-m 403 kn-m 1602 kn

10 P-M Interation Diagram

11 P-M Interation Depending on Steel

12 P-M Curve with Strength Redution Fator (M n, P n ) (M n, P n )

13 st y st g n st y st g n A A A P A A A P P-M Curve or Design

14 Modiiation o Strength Redution Fator

15 Column Design Using P-M Interation Diagrams Column design harts are nothing but P-M interation urves arranged in one o various ways The primary purpose o design harts is to make olumn design quik and easy without onstruting the P-M urve or every speii olumn It is important to keep the bar arrangement as lose as possible between the hart and reality

18 Estimation o Column Size Least dimension o retangular setion 200 mm Minimum diameter o irular setion 300 mm For tied olumns For spiral olumns A g Pu y A g Pu y

19 Example 10.3 The short 1420 in ( mm) tied olumn is to be used to support P D =125 k (556 kn), P L =140 k (623 kn), M D =75 k-t (102 kn-m) and M L =90 k-t (122 kn-m). I =4 ksi (27.6 MPa) and y =60 ksi (414 MPa), selet reinoring bars to be plaed in its end aes only using appropriate ACI olumn interation diagrams. (64 mm) Sine olumn design harts in SI units are not available, US ustomary units are used. (356 mm) (381) M u 1.2M D (508 mm) M u M n (64 mm) P u P e 1.2P n M P 1.6P M kips Pu kips ( t y assumed ) 0.65 n n D L L k t 360 k t 7.51in

20 Example 10.3 The short 1420 in ( mm) tied olumn is to be used to support P D =125 k (556 kn), P L =140 k (623 kn), M D =75 k-t (102 kn-m) and M L =90 k-t (122 kn-m). I =4 ksi (27.6 MPa) and y =60 ksi (414 MPa), selet reinoring bars to be plaed in its end aes only using appropriate ACI olumn interation diagrams. (356 mm) (64 mm) Pn Ag Pn e Ag h ( 0.7, Graph A3) ( 0.8, Graph A4) ( 0.75) (381) (508 mm) Ast bh mm 2 Use 6 #29( Ast, provided 3870 mm ) s Sine t y (64 mm) 0.65 O.K. 2

23 Example 10.4 Design a short olumn or P u =600 kips (2670 kn), M u =125 k-t (170 kn-m), =4 ksi (27.6 MPa), y =60 ksi (414 MPa). Plae bars uniormly around all our aes. Determination o olumn setion, assuming 0.02 A g 0.45 Use 8#32( A u in (256in / assumed Pn A M A g n g Pu A 0.04 ( 0.6, ( 0.7, ) , Graph A6) , Graph A7) ( 0.69) A bh mm st P g M u h A h g y st, provided mm s s y y 2 ) 256in mm 406 mm mm mm 406

26 Example 10.5 Selet reinoring bars or the short round spiral olumn i =4 ksi (27.6 MPa), y =60 ksi (414 MPa), P u =500 kips (2225 kn), M u =200 k-t (271 kn-m) mm 2 64 mm 381 mm 64 mm 508 mm Pu A M A ( 0.75 A u st g g Use 6#29( A Sine h ( 0.7, ( 0.8, A s g y by interpolation) mm st, provided 3870 mm 1.0,, 0.7 t s s y y y 0.45, Graph A11) 0.55, Graph A12) 2 ) 2

29 Example 10.7 Using the appropriate interation urves, determine the value o P n or the short tied olumn i e x =10 in (254 mm). Assume =4 ksi (26.7 MPa) and y =60 ksi (414 MPa). (76 mm) 3#32(2445 mm 2 ) 3#32(2445 mm 2 ) (305 mm) From Fig.10.15, Pn e A h g 0.24 is read P n 0.24 (356) (508) e (76 mm) A g h e h kips

30 0.0316

31 Shear in Columns mm., units in positive in ompression and all load, atored axial where tension, with axial ii) when aounting or lexure noted above or 8 4 (approximate) ompression, with axial ii) 1 where (aounting or lexure) (approximate) 6 1 load, without axial i) N N d b A N. V M d h N M M d b A N V M d V d b. d b M d V ρ V d b V u w g u u u u m w g u u u w w u u w w

32 Biaxial Bending

33 1 P P P P P ni where ni nx ny o 1 P Biaxial Bending Bresler Formula nx 1 P ny 1 P at an eentriity e at an eentriity e o the nominal axial load apaity o the setion when the load is plaed at a given eentriity along both axes the nominal axial load apaity o the setion when the load i splaed x the nominal axial load apaity o the setion when the load is plaed y the nominal axial load apaity o the setion when the load is plaed with a zero eentriity. It is usually taken as 0.85 A g A st y

34 Example 10.8 Determine the design apaity P ni o the short tied olumn subjeted to biaxial bending. Use =4 ksi (27.6 MPa), y =60 ksi (414 MPa), e x =16 in (406 mm) and e y =8 in (203 mm). (64 mm) Bending about X axis (8 #29) (64 mm) 254 (381 mm) (508) (635) (64 mm) e h 25 Pn e (Graph A8) A h P nx g kips

35 e/h=

36 Example 10.8 Determine the design apaity P ni o the short tied olumn subjeted to biaxial bending. Use =4 ksi (27.6 MPa), y =60 ksi (414 MPa), e x =16 in (406 mm) and e y =8 in (203 mm). (64 mm) (8 #29) (64 mm) 254 (381 mm) (508) (635) (64 mm) Bending about Y axis e h 15 Pn e (Graph A6 and A7) Ag h Pny 452.3kips 8

37 e/h=

38 e/h=

39 (8 #29) (64 mm) Example 10.8 Determine the design apaity P ni o the short tied olumn subjeted to biaxial bending. Use =4 ksi (27.6 MPa), y =60 ksi (414 MPa), e x =16 in (406 mm) and e y =8 in (203 mm). (64 mm) 254 (381 mm) (508) (635) (64 mm) Axial loading apaity or M P 0.85 o Ag y Ast kips Using Bresler's ormula Pni Pnx Pny Po P 253.5kips ni 0

Purpose of reinforcement P/2 P/2 P/2 P/2

Purpose of reinforcement P/2 P/2 P/2 P/2 Department o Civil Engineering Purpose o reinorement Consider a simpl supported beam: P/2 P/2 3 1 2 P/2 P/2 3 2 1 1 Purpose o Reinorement Steel reinorement is primaril use beause o the nature o onrete