COMBINED COMPRESSION AND BENDING: COLUMNS

Transcription

1 CHAPTER REINFORCED CONCRETE Reinforced Concrete Design A Fundamental Approach - Fifth Edition Fifth Edition COMBINED COMPRESSION AND BENDING: COLUMNS A. J. Clark School of Engineering Department of Civil and Environmental Engineering 9a SPRING 2004 B Dr. Ibrahim. Assakkaf ENCE 454 Design of Concrete Structures Department of Civil and Environmental Engineering Universit of Marland, College Park CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 1 Axial Compression Columns are defined as members that carr loads in compression. Usuall the carr bending moments as well, about one or both axes of the cross section. The bending action ma produce tensile forces over a part of the cross section. Despite of the tensile forces or stresses that ma be produced, columns are Generall referred to as compression members because the compression forces or stresses dominate their behavior.

2 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 2 Axial Compression (cont d) In addition to the most common tpe of compression members (vertical elements in structures), compression members include: Arch ribs Rigid frame members inclined or otherwise Compression elements in trusses shells CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 3

3 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 4 Reinforced Concrete Columns CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 5 Ohio River Bridge. Tpical cantilever and suspended span bridge, showing the truss geometr in the end span and cantilevered portion of the t main span. (Madison, Indiana)

4 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 6 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 7

5 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 8 Failure of Columns Failure of columns could occur as a result of material failure b Initial ielding of the steel at the tension face; Initial crushing of the concrete at the compression face; or Loss of lateral stabilit (buckling) If a column fails due to initial material failure, it is then considered short or non-slender column. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 9 Failure of Columns (cont d) As the length of the column increases, the probabilit that failure will occur b buckling also increases. The slenderness ratio kl u /r is a measure of the tpe of column. According to ACI, if kl u 22 r then the column is considered a short column kl u = effective length k =factor that depends on end condition of column and condition of bracing l u = unsupported length of column r = radius of gration = I / A

6 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 10 Buckling and Axial Compression Buckling is a mode of failure generall resulting from structural instabilit due to compressive action on the structural member or element involved. Examples Overloaded metal building columns. Compressive members in bridges. Roof trusses. Hull of submarine CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 11 Buckling and Axial Compression Figure 1a

7 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 12 Buckling and Axial Compression Figure 1b CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 13 Buckling and Axial Compression Definition Buckling can be defined as the sudden large deformation of structure due to a slight increase of an existing load under which the structure had exhibited little, if an, deformation before the load was increased.

8 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 14 Buckling and Axial Compression Figure 2. Buckling Failure of Reinforced Concrete Columns CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 15 Critical Buckling Load, P cr The critical buckling load (Euler Buckling) for a long column is given b 2 π EI = (1) P cr 2 L where E = modulus of elasticit of the material I = moment of inertia of the cross section L = length of column

9 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 16 Columns Ba The spacing of columns in plan establishes what is called a Ba. For example, if the columns are 20 ft on center in one direction and 25 ft in the other direction, the ba size is 20 ft 25 ft. Larger ba sizes increase the user s flexibilit in space planning. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 17 Columns Ba Ba Size : 20 ft 25 ft 20 ft 25 ft

10 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 18 Column load transfer from beams and slabs 1) Tributar area method: Half distance to adjacent columns Load on column = area ด floor load Floor load = DL + LL DL = slab thickness ด conc. unit wt. x Example: x = 16.0 ft, = 13.0 ft, LL = 62.4 lb/ft 2, slab thickness = 4.0 in. Floor load = 4.0 (150)/ = lb/ft 2 Load on column = (16.0)(13.0)(112.4) = 23.4 kips =10,800 kg CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 19 Column load transfer from beams and slabs 2) Beams reaction method: Collect loads from adjacent beam ends B1 B2 B4 R B1 R B1 R B2 R B2 B1 C1 B2 B3

11 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 20 Load summation on column section for design ROOF 2nd FLOOR Design section Load on 2nd floor column = Roof floor + Column wt. 1st FLOOR Design section Ground level Load on 1st floor column = load on 2nd floor column + 2nd floor + Column wt. Footing Design section Load on pier column = load on 1st floor column + 1st floor + Column wt. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 21 Tpes of Columns Tpes of Reinforced Concrete Columns Members reinforced with longitudinal bars and lateral ties. Members reinforced with longitudinal bars and continuous spirals. Composite compression members reinforced longitudinall with structural steel shapes, pipe, or tubing, with or without additional longitudinal bars, and various tpes of lateral reinforcement.

12 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 22 Tpes of Columns Tpes of Reinforced Concrete Columns Tie Spiral Longitudinal steel s = pitch Tied column Spirall reinforced column CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 23 Tpes of Columns Tpes of Reinforced Concrete Columns Composite columns

13 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 24 Tpes of Columns Tpes of Columns in Terms of Their Strengths 1. Short or Non-Slender Columns A column is said to be short when its length is such that lateral buckling need not be considered. Most of concrete columns fall into this categor 2. Slender Columns When the length of the column is such that buckling need to be considered, the column is referred to as slender column. It is recognized that as the length increases, the usable strength of a given cross section is decreased because of buckling problem CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 25 Tpes of Columns Tpes of Columns in Terms of the Position of the Load on the Cross Section 1. Concentricall Loaded Columns Concentricall loaded columns (see Figure 3)carr no moment. In practice, however, all columns have to be designed for some unforeseen eccentricit. 2. Eccentricit Loaded Columns Eccentricit loaded columns are subjected to moment in addition to the axial force. The moment can be converted to a load P and eccentricit e (see Figure 4)

14 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 26 Tpes of Columns Tpes of Columns in Terms of the Position of the Load on the Cross Section P Figure 3. Concentricall Loaded Column CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 27 Tpes of Columns P P P P M x e M M x e x x x x e x Figure 4a. Axial load plus uniaxial moment Figure 4b. Axial load plus biaxial moment

15 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 28 Strength of Short or Slender Columns If a compression member is loaded parallel to its axis b a load P without eccentricit, the load P theoreticall induces a uniform compressive stress over the cross-sectional area. If the compressive load is applied a small distance e awa from the longitudinal axis, however, there is a tendenc for the column to bend due to the moment M = Pe. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 29 Strength of Short or Slender Columns Concentric Axial Loading in a Plane of Smmetr When the line of action of the axial load P passes through the centriod of the cross section, it can be assumed that the distribution of normal stress is uniform throughout the section, i.e, f=p/a. Such a loading is said to be centric, as shown in Figure 3.

16 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 30 Strength of Short or Slender Columns Eccentric Axial Loading in a Plane of Smmetr When the line of action of the concentrated load P does not pass through the centroid of the cross section, the distribution of normal stress is no longer uniform. Such loading is said to eccentric, as shown in Figure 4. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 31 Strength of Short or Slender Columns Eccentric Axial Loading in a Plane of Smmetr The stress due to eccentric loading on a beam cross section is given b P M f ± M x = ± (2) A I x I x

17 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 32 Strength of Short or Slender Columns Example 1 The T-section shown in the figure is used as a short post to support a compressive load P of 150 kips. The load is applied on centerline of the stem at a distance e = 2 in. from the centroid of the cross section. Determine the normal stresses at points A and B on a transverse plane C-C near the base of the post. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 33 Strength of Short or Slender Columns Example 1 (cont d) P z e 6 in 2 in x 2 in A B 6 in x C C Section C-C

18 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 34 Strength of Short or Slender Columns Example 1 (cont d) Computing the cross-sectional properties: x Area = x C 3 = 6 in 2 in 2 A = 2[ 6 2] = 24 in ( 6 2) + ( 6 + 1)( 6 2) 24 = 5 in.from point A 2 in A x C = 5 in B 6 in I 2 = ( 5) 6( 3) 4( 1) = 136 in N.A. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 35 Strength of Short or Slender Columns Example 1 (cont d) Equivalent force sstem: P = 150 kip acts through centroid M = Pe = ( 150)( 2) 12 = 3,600 kip in Computations of normal stresses: f f A B = = P A P A M + I M I ( 5) x = () 3 x = x f = ± ± I x = P A M 4.78 ksi (T) = ksi (C) M I x

19 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 36 Concentricall Loaded Columns Background The concrete column that is loaded with a compressive axial load P at zero eccentricit is probabl nonexistent, and even the axial/small eccentricit combination is relativel rare. Nevertheless, the case of columns that are loaded with compressive axial loads at small eccentricit e is considered first. In this case we define the situation in which the induced small moments are of little significance. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 37 Concentricall Loaded Columns Notations for Columns Loaded with Small Eccentricities A g = gross area of the column section (in 2 ) A st = total area of longitudinal reinforcement (in 2 ) P 0 = nominal or theoretical axial load at zero eccentricit P n = nominal or theoretical axial load at given eccentricit P u = factored applied axial load at given eccentricit ρ g = ratio of total longitudinal reinforcement area to cross-sectional area of column: A st ρ g or ρt = (3) Ag

20 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 38 Concentricall Loaded Columns Strength of Short Axiall Loaded Columns P 0 f Steel Stress Section A-A A A f c Concrete Strain CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 39 Concentricall Loaded Columns Strength of Short Axiall Loaded Columns P 0 [ ΣF = 0 ] ( Ag Ast ) f Ast P = f + 0 c f f f c F s = A st f F c = (A g - A st ) f c From experiment (e.g., ACI): c ( Ag Ast ) f Ast P = f + 0 where A g = Gross area of column section A st = Longitudinal steel area

21 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 40 Concentricall Loaded Columns Column Failure b Axial Load P u P u Axial load Initial failure Tied column Heav spiral ACI spiral Light spiral 0 Axial deformation CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 41 Concentricall Loaded Columns ACI Code Requirements for Column Strength φpn P u (4) φp Spirall reinforced column: n ( max ) = 0.85φ [ 0.85 f c ( Ag Ast ) + f Ast ], φ = Tied column: φp n ( max ) = 0.80φ [ 0.85 f c ( Ag Ast ) + f Ast ], φ = (5) (6)

22 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 42 Concentricall Loaded Columns Table 1. Resistance or Strength Reduction Factors Structural Element Factor φ Beam or slab; bending or flexure 0.90 Columns with ties 0.65 Columns with spirals 0.70 Columns carring ver small axial load (refer to Chapter 9 for more details) or Beam: shear and torsion 0.75 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 43 Concentricall Loaded Columns ACI Code Requirements for Column Strength (cont d) Normall, for design purposes, (A g A st ) can be assumed to be equal to A g without great loss in accurac. Accordingl, Eqs. 5 and 6, respectivel, give A ρt = A st g Pn Ag = 0.68 f c + 0.8ρt f Pn Ag = 0.78 f ρ f c t (7a) (7b)

23 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 44 Concentricall Loaded Columns ACI Code Requirements for Column Strength (cont d) For first trial section, with appreciable eccentricit, the designer can tr the following equations for assuming gross section area A g : A g P 0.45 c t n ( f + f ρ ) for tied columns (8a) A g P 0.55 c t n ( f + f ρ ) for spirall reinforced columns (8b) CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 45 Concentricall Loaded Columns Example 2 A non-slender (short column) column is subjected to axial load onl. It has the geometr shown and is reinforced with three No. 9 bars on each of the two faces parallel to the x axis of bending. Calculate the maximum nominal axial load strength P n(max). Assume that f = 60,000 psi and f = 4000 psi. c

24 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 46 Concentricall Loaded Columns Example 2 (cont d) 2.5 in in./in f c 3 No in. C = A f s f ( A A ) c g st 3 No. 9 A s f 12 in. 2.5 in. Strain Stress CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 47 Concentricall Loaded Columns Example 2 (cont d) A s = A s 2 = 3 in Therefore, A st = 6 in 2. Eq. 6 gives P n ( max ) = 0.80[ 0.85( 4000) [( ) ] + 60,000( 6) ] = 924,480 lb If A g A st is taken to equal A g, then Eq. 6 results in P n ( max ) = 0.80[ 0.85( 4000)( 12 20) + 60,000( 6) ] = 940,800 lb

25 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 48 Concentricall Loaded Columns Example 3 (cont d) A 20-in.-diameter, non-slender, spirall reinforced circular column is smmetricall reinforced with six No. 8 bars as shown. Calculate the strength P n(max) of this column if subjected to axial load onl. Use f c = 4000 psi and f = 60,000 psi. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 49 Concentricall Loaded Columns Example in./in f c 6 No. 8 A st f 3 Ast 3 f + C c A st f 3 Strain Stress

26 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 50 Concentricall Loaded Columns Example 3 (cont d) A st = 4.74 in π Ag = 4 Therefore, Eq. 5 gives P n 2 (6 No.8 bars) 2 2 ( 20) = 314 in ( max ) = 0.85[ 0.85( 4000)( [ ) ] + 60,000( 4.74) ] = 1,135,501lb If A g A st is taken to equal A g, then Eq. 5 results in P n ( max ) = 0.85[ 0.85( 4000)( 314) + 60,000( 4.74) ] = 1,149,200 lb CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 51 Concentricall Loaded Columns ACI Code Limits on percentage of reinforcement A st 0.01 ρ g = 0.08 (9) Ag Lower limit: Upper limit: To prevent failure mode of plain concrete To maintain proper clearances between bars

27 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 52 Code Requirements Concerning Column Details Minimum Number of Bars The minimum number of longitudinal bars is four within rectangular or circular ties Three within triangular ties Six for bars enclosed b spirals Clear distance between Bars The clear distance between longitudinal bars must not be less than 1.5 times the nominal bar diameter nor 1 ½ in. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 53 Code Requirements Concerning Column Details Clear distance between Bars (cont d) Table 2 (Table 9 Handout) ma be used to determine the maximum number of bars allowed in one row around the peripher of circular or square columns. Cover Cover shall be 1 ½ in. minimum over primar reinforcement, ties or spirals.

28 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 54 Code Requirements Concerning Column Details Table 2. Preferred Maximum Number of Column Bars in One Row Table 1 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 55 Code Requirements Concerning Column Details Tie Requirements According to Section of ACI Code, the minimum is No. 3 for longitudinal bars No. 10 and smaller Otherwise, minimum tie size is No. 4 (see Table 2 for a suggested tie size) The center-to-center spacing of ties must not exceed the smaller of 16 longitudinal bar diameter, 48 tie-bar diameter, or the least column dimension.

29 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 56 Code Requirements Concerning Column Details Spiral Requirements According to Section of ACI Code, the minimum spiral size is 3/8 in. in diameter for cast-in-place construction (5/8 is usuall maximum). Clear space between spirals must not exceed 3 in. or be less than 1 in. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 57 Code Requirements Concerning Column Details Spiral Requirements (cont d) The spiral steel ratio ρ s must not be less than the value given b where A f A c f g c ρ s( ) = min (10) volume of spiral steel in one turn ρs = volume of column core in height ( s) s = center-to-center spacing of spiral (in.), also called pitch A g = gross cross-sectional area of the column (in 2 ) A c = cross-sectional area of the core (in 2 ) (out-to-out of spiral) f = spiral steel ield point (psi) 60,000 psi = compressive strength of concrete (psi)

30 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 58 Code Requirements Concerning Column Details Spiral Requirements (cont d) An Approximate Formula for Spiral Steel Ratio A formula in terms of the phsical properties of the column cross section can be derived from the definition of ρ s. In reference to Fig. 5, the overall core diameter (out-to-out of spiral) is denoted as D c, and the spiral diameter (center-to-center) as D s. The cross-sectional area of the spiral bar or wire is given the smbol A sp. CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 59 Code Requirements Concerning Column Details Spiral Requirements (cont d) D c Spiral D s Figure 5. Definition of D c and D s

31 CHAPTER 9a. COMBINED COMPRESSION AND BENDING: COLUMNS Slide No. 60 Code Requirements Concerning Column Details Spiral Requirements (cont d) From the definition of ρ s, an expression ma written as follows: AspπDs actual ρ s = 2 (11) πd / 4 s ( )() c If the small difference between D c and D s is neglected, then in terms of D c, the actual spiral steel ratio is given b 4Asp actual ρ s = (12) D s c