Lecture_25 - Chapter Seven Design of Reinforced Concrete...

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Unformatted text preview: Chapter Seven _____________________________________________________________________ Design of Reinforced Concrete Columns 7.1 Columns in Buildings 7.1.1 Types of columns • Columns in a building carry loads from the beams and slabs down to the foundations. They are primarily compression members. • In general, columns also have to resist bending moments due to the continuity of the structure. M N N M N Fig. 7.1-1 Loading and internal forces in columns • Two common types of columns (i) braced column – where the lateral loading is resisted by walls or some other form of bracing. A braced column is designed to resist vertical loads only (such as dead and imposed loads). (ii) unbraced column – where the lateral loading is resisted by the bending action of columns. An unbraced column is designed to resist both vertical and horizontal (wind/earthquake) loads. 246 wind shear walls (b) (a) wind (c) Fig. 7.1-2 (a) Plan; (b) braced frame; (c) unbraced frame 7.1.2 Reduction of imposed floor loads • Critical loading arrangement For a braced column: the critical loading arrangement is usually that which causes the largest moment in + largest axial force. Fig. 7.1-3 shows the critical arrangement of the ultimate loads for the design of (1) the centre column at the 1st-floor level, and (2) the left-hand column at all floor level. Fig. 7.1-3 A critical loading arrangement 247 • Reduction of total imposed floor loads Table 7.1 Reduction of total imposed floor loads on columns, walls and foundations. British code difference Hong Kong Building Regulations (CAP. 123) 7.2 Column Classification and Failure Modes 7.2.1 Classification Table 7.2 Short and slender columns (BS 8110: Part 1: 1997, clause where lex – effective height with respect to major axis bending ley – effective height with respect to minor axis bending 248 7.2.2 Effective height • The effective height of a column is specified as l e = β lo (7.1) where lo – the clear distance between the column end restraints β − a coefficient which depends on the degree of end restraints y x lox x loy y to T.O.S. column cross-section Fig. 7.2-1 Clear distances lox and loy between the column end restraints with respect to major and minor axis bending 249 Table 7.3 Values of b for braced and unbraced columns 7.2.3 Failure modes • A short column usually fails by crushing, but a slender column is liable to fail by buckling. o The crushing load Nuz of a truly axially loaded column Nuz = 0.45fcuAc + 0.95fyAsc where Ac – concrete area and Asc – longitudinal steel area. o The additional moment Neadd of a slender column causes further lateral deflection. If the applied load N exceeds a critical value Ncritical, the column buckles. Eüler formula for the critical load of a pin-ended strut: N critical = π 2 EI l2 250 Fig. 7.2-2 Column failure modes l h 7.3 Reinforcement Details 7.3.1 Longitudinal steel • A minimum of four bars, which are not less than size 12, is required in a rectangular column and six bars in a circular column. • Asc/Ac ≤ 6%, to avoid congestion and hence unsatisfactory compaction of concrete (IStructE: ρmax = 4% is preferable.) • Asc/Ac ≥ 0.4%, to resist bending moment which may exist: emin N N M Mmin = N emin where emin = 0.05 h ( > 20 mm) M N N 251 • Splices of reinforcement column Fig. 7.3-1 Details of splices in column reinforcement 7.3.2 Links (stirrups) (b) (a) Fig. 7.3-1 Links in column • Minimum size = 1 × size of the largest compression steel, but not 4 less than 6 mm. • Maximum spacing = 12 × size of the smallest compression bars. (I StructE recommends that link spacing be not greater than the smallest cross-sectional dimension of the column). 252 • The links should be arranged so that every corner bar and alternate bar or group in an outer layer of longitudinal steel is supported by a link passing round the bar and having an included angle not greater than 135°. • No bar not restrained by a link is to be further than 150 mm from a restrained bar (Fig. 7.3-1a). 7.4 Design of Short Columns 7.4.1 Braced axially-loaded columns This type of columns can be considered to occur when they support a symmetrical and very rigid structure. (1) Typical modes of failure premature buckling (a) (b) Fig. 7.4-1 (a) Short axially-loaded column; (b) reinforcement buckling after ultimate strength is reached To prevent premature buckling of longitudinal reinforcement: o Links should be sufficiently closely spaced. o Link size should be adequate. 253 (2) Ultimate load capacity N = 0.45fcuAc + 0.95fyAsc (7.2) Ac – the net area of the concrete Asc – the area of the longitudinal reinforcement To allow for eccentricity of the loading (Mmin = N emin, where emin = 0.05h or 20 mm, whichever is the lesser), BS 8110 limits the capacity for axial load to about 90% of that from Eq. (7.2); thus N = 0.4 f cu Ac + 0.8 f y Asc (7.3) For a rectangular column and to allow for the area of concrete displaced by the longitudinal steel, Eq. (7.3) becomes N = 0.4 f cu bh + Asc (0.8 f y − 0.4 f cu ) (7.4) Example 7.4-1: Axially loaded short column A short braced axially loaded column 300 mm square in section is reinforced with four 25 mm diameter bars. Find the ultimate axial load that the column can carry and the pitch and diameter of the links required. The materials are grade 30 concrete and grade 460 reinforcement. Steel area Asc = 1963 mm2 Concrete area Ac = 3002 – 1963 = 88037 mm2 N= 0.4 × 30 × 88037 0.8 × 1963 × 460 + = 1056 + 722 = 1778 kN 10 3 10 3 254 The links are not to be less than 6 mm in diameter or one-quarter of the diameter of the longitudinal bars. The spacing is not to be greater than 12 times the diameter of the longitudinal bars. Provide 8 mm diameter links at 300 mm centres ([email protected]). The column section is shown in the following figure. From the code the cover for mild exposure is 25 mm. T8-300 Example 7.4-2: Axially loaded short column A short braced column has to carry a design axial load of 1366 kN. The column size is 250 mm × 250 mm. Determine the steel area required for the longitudinal reinforcement and select suitable bars. The materials are grade 30 concrete and grade 460 reinforcement. Substitute in the expression for the ultimate load, 1366 ×103 = 0.4 × 30 × (250 2 − Asc ) + 0.8 × 460 Asc Asc = 1730 mm 2 Provide four 25 mm diameter bars to give a steel area of 1963 mm2. Check: 100 As 100 × 1963 = = 3.14 . This is satisfactory. bd 250 2 255 7.4.2 Braced columns supporting an approximately symmetrical arrangement of beams • The moments of these columns are small due primarily to unsymmetrical arrangement of the live load. • The ultimate load capacity N = 0.35 f cu Ac + 0.70 f y Asc (7.5) 7.4.3 Columns resisting moments and axial forces In practice, the area of longitudinal steel for these columns is determined using design charts in BS 8110: Part 3 or in the IStructE design manual. (a) (b) Fig. 7.4-2 (a) Column section; (b) BS 8110 column design charts 256 • Text Example 9.3 (pp.228-232): Column design using design charts. • Example 7.4-3: Short column subjected to axial load and moment using design chart A short braced column is subjected to a design axial load of 1480 kN and a design moment of 54 kNm. The column section is 300 mm × 300 mm. Determine the area of steel required. The materials are grade 30 concrete and grade 460 reinforcement. Assume 25 mm diameter bars for the main reinforcement and 8 mm diameter links. The cover on the links is 25 mm. d = 300 – 25 – 8 –12.5 = 255 mm d/h = 255/300 = 0.85 Use the chart in BS 8110 shown below, where d/h = 0.85. N 1480 × 10 3 M 54 × 10 6 = = 16.4 ; = = 2. 0 bh 300 2 bh 2 300 3 100Asc / bh = 2.0. Then Asc = 2.0 × 3002/100 = 1800 mm2 Provide four 25 mm diameter bars to give an area of 1963 mm2. 257 ...
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