Design of Compression membersAxially Loaded columns by S.PraveenKumar Assistant Professor Department of Civil Engineering PSG College of Technology Coimbatore
[email protected],
[email protected]
Introduction ► A column is an important components of R.C. Structures. ► A column, in general, may be defined as a member carrying direct axial
load which causes compressive stresses of such magnitude that these stresses largely control its design. ► A column or strut is a compression member, the effective length of which exceeds three times the least lateral dimension. ► When a member carrying mainly axial load is vertical, it is termed as column ,while if it is inclined or horizontal, it is termed as a strut. ► Columns may be of various shape such as circular, rectangular, square, hexagonal etc.
2
Classification of columns Based on Type of Reinforcement a) Tied Columns-where the main longitudinal bars are enclosed within closely spaced lateral ties( all cross sectional shapes) b) Spiral columns-where the main longitudinal bars are enclosed within closely spaced and continuously wound spiral reinforcement (Circular, square, octagonal sections) c) Composite Columns-where the reinforcement is in the form of structural steel sections or pipes, with or without longitudinal bars
3
Based on Type of Loading a) Columns with axial loading (applied concentrically) b) Columns with uniaxial eccentric loading c) Columns with biaxial eccentric loading
4
occurrence of ‘pure’ axial compression in a column (due to concentric loads) is relatively rare. ► Generally, flexure accompanies axial compression — due to ‘rigid frame’ action, lateral loading and/or actual(or even, unintended/accidental) eccentricities in loading. ► The combination of axial compression (P) with bending moment (M) at any column section is statically equivalent to a system consisting of the load P applied with an eccentricity e = M/P with respect to the longitudinal centroidal axis of the column section. ► In a more general loading situation, bending moments (Mx and My) are applied simultaneously on the axially loaded column in two perpendicular directions — about the major axis (XX) and minor axis (YY) of the column section. This results in biaxial eccentricities ex= Mx /P and ey = My /P, as shown in [Fig.(c)]. ► The
5
► Columns
in reinforced concrete framed buildings, in general, fall into the third category, viz. columns with biaxial eccentricities. ► The biaxial eccentricities are particularly significant in the case of the columns located in the building corners. ► In the case of columns located in the interior of symmetrical, simple buildings, these eccentricities under gravity loads are generally of a low order (in comparison with the lateral dimensions of the column), and hence are sometimes neglected in design calculations. ► In such cases, the columns are assumed to fall in the first category, viz. columns with axial loading. ► The Code, however, ensures that the design of such columns is sufficiently conservative to enable them to be capable of resisting nominal eccentricities in loading 6
Based on Slenderness Ratio Columns (i.e., compression members) may be classified into the following two types, depending on whether slenderness effects are considered insignificant or significant: 1. Short columns 2. Slender (or long) columns. ‘Slenderness’ is a geometrical property of a compression member which is related to the ratio of its ‘effective length’ to its lateral dimension. This ratio, called slenderness ratio, also provides a measure of the vulnerability to failure of the column by elastic instability (buckling) — in the plane in which the slenderness ratio is computed.. Columns with low slenderness ratios, i.e., relatively short and stocky columns, invariably fail under ultimate loads with the material (concrete, steel) reaching its ultimate strength, and not by buckling. 7
On the other hand, columns with very high slenderness ratios are in danger of buckling (accompanied with large lateral deflection) under relatively low compressive loads, and thereby failing suddenly. Braced columns & unbraced column In most of the cases, columns are also subjected to horizontal loads like wind, earthquake etc. If lateral supports are provided at the ends of the column, the lateral loads are borne entirely by the lateral supports. Such columns are known as braced columns.(When relative transverse displacement between the upper and lower ends of a column is prevented, the frame is said to be braced (against sideway)). Other columns, where the lateral loads have to be resisted by them, in addition to axial loads and end moments, are considered as unbraced columns. (When relative transverse displacement between the upper and lower ends of a column is not prevented, the frame is said to be unbraced
(against sideway).
8
In such cases, the effective length ratio k varies between 0.5 and 1.0
9
In such cases, the effective length ratio k
varies between 1.0 and infinity
10
Reinforcement in column ► Concrete is strong in compression. ► However, longitudinal steel rods are
always provided to assist in
carrying the direct loads. ► A minimum area of longitudinal steel is provided in the column, whether it is required from load point of view or not. ► This is done to resist tensile stresses caused by some eccentricity of the vertical loads. ► There is also an upper limit of amount of reinforcement in RC columns, because higher percentage of steel may cause difficulties in placing and compacting the concrete. ► Longitudinal reinforcing bars are “tied” laterally by “ties” or “stirrups” at suitable interval so that the bars do not buckle 11
Codal Provisions(IS-456-2000)
12
Page No:41 & 42 –IS 456-2000
13
Page No:94–IS 456-2000
Page No:42–IS 456-2000
14
15
Clause26.53.1- Page No:48–IS 456-2000
16
Functions of longitudinal reinforcement ► To share the vertical compressive load, thereby reducing the overall
size of the column. ► To resist tensile stresses caused in the column due to (i) eccentric load (ii) Moment (iii) Transverse load. ► To prevent sudden brittle failure of the column. ► To impart certain ductility to the column. ► To reduce the effects of creep and shrinkage due to sustained loading.
17
18
Clause 26.53.3.2Page No:49–IS 456-2000
19
Clause 26.5.3.2 Page No:49–IS 456-2000
20
Clause 26.5.3.2 Page No:49–IS 456-2000
Functions of Transverse reinforcement ► To prevent longitudinal buckling of longitudinal reinforcement. ► To resist diagonal tension caused due to transverse shear due to
moment/transverse load. ► To hold the longitudinal reinforcement in position at the time of concreting. ► To confine the concrete, thereby preventing its longitudinal splitting. ► To impart ductility to the column. ► To prevent sudden brittle failure of the columns. 21
Clause 26.5.3.2 Page No:49–IS 456-2000
Cover to reinforcement For a longitudinal reinforcing bar in a column, the nominal cover shall not be less than 40mm, nor less than the diameter of such bar. In the case of columns of minimum dimension of 200mm or under, whose reinforcing bars does not exceed 12mm, a cover of 25mm may be used. Clause 26.4.2.1 Page No:49–IS 456-2000
22
SP 34- 1987 Page No:88
23
Assumptions in Limit State of Collapse -Compression
24
Clause 38.1 Page No:69–IS 456-2000
Clause 38.1 Page No:69–IS 456-2000
25
Clause 38.1 Page No:69–IS 456-2000
26
27
28
29
Detailing of columns
30
a) Termination of column bars inside slab b) Fixed end joint in a column
31
c) Typical detail of beam column junction at external column
32
33
Design of Compression membersUniaxial Bending
Based on Type of Loading a) Columns with axial loading (applied concentrically) b) Columns with uniaxial eccentric loading c) Columns with biaxial eccentric loading
2
occurrence of ‘pure’ axial compression in a column (due to concentric loads) is relatively rare. ► Generally, flexure accompanies axial compression — due to ‘rigid frame’ action, lateral loading and/or actual(or even, unintended/accidental) eccentricities in loading. ► The combination of axial compression (P) with bending moment (M) at any column section is statically equivalent to a system consisting of the load P applied with an eccentricity e = M/P with respect to the longitudinal centroidal axis of the column section. ► In a more general loading situation, bending moments (Mx and My) are applied simultaneously on the axially loaded column in two perpendicular directions — about the major axis (XX) and minor axis (YY) of the column section. This results in biaxial eccentricities ex= Mx /P and ey = My /P, as shown in [Fig.(c)]. ► The
3
► Columns
in reinforced concrete framed buildings, in general, fall into the third category, viz. columns with biaxial eccentricities. ► The biaxial eccentricities are particularly significant in the case of the columns located in the building corners. ► In the case of columns located in the interior of symmetrical, simple buildings, these eccentricities under gravity loads are generally of a low order (in comparison with the lateral dimensions of the column), and hence are sometimes neglected in design calculations. ► In such cases, the columns are assumed to fall in the first category, viz. columns with axial loading. ► The Code, however, ensures that the design of such columns is sufficiently conservative to enable them to be capable of resisting nominal eccentricities in loading 4
Column under axial compression and Uni-axial Bending ► Let
us now take a case of a column which is subjected to combined action of axial load (Pu) and Uni-axial Bending moment (Mu). ► This case of loading can be reduced to a single resultant load Pu acting at an eccentricity e such that e= Mu / Pu . ► The behavior of such column depends upon the relative magnitudes of Mu and Pu , or indirectly on the value of eccentricity e. ► For a column subjected to load Pu at an eccentricity e, the location of neutral axis (NA) will depend upon the value of eccentricity e. ► Depending upon the value of eccentricity and the resulting position (Xu) of NA., We will consider the following cases. 5
Case I : Concentric loading: Zero Eccentricity or nominal eccentricity (Xu =∞) Case II : Moderate eccentricity (Xu > D) Case III : Moderate eccentricity (Xu = D) Case IV : Moderate eccentricity (Xu < D)
Case I (e=0 and e
6
Case II (Neutral axis outside the section)
7
Case III (Neutral axis along the edge)
8
Case IV (Neutral Axis lying within the section)
9
Modes of Failure in Eccentric Compression ► The mode of failure depends upon the relative magnitudes of
eccentricity e. (e = Mu / Pu ) Eccentricity e = Mu / Pu e = Mu / Pu e = Mu / Pu
Range Small Large In between two
Behavior Compression Flexural Combination
Failure Compression Tension Balanced
10
Column Interaction Diagram ► A column subjected to varying magnitudes of P and M will act with its
neutral axis at varying points.
11
Method of Design of Eccentrically loaded short column The design of eccentrically loaded short column can be done by two methods I) Design of column using equations II) Design of column using Interaction charts
12
Design of column using equations
13
14
Design of column using Interaction charts
15
16
Design of Compression membersBiaxial Bending
Introduction ►A
column with axial load and biaxial bending is commonly found in structures because of two major reasons: Axial load may have natural eccentricities, though small, with respect to both the axes. Corner columns of a building may be subjected to bending moments in both the directions along with axial load Examples 1) External façade columns under combined vertical and horizontal load 2) Beams supporting helical or free-standing stairs or oscillating and rotary machinery are subjected to biaxial bending with or without axial load of either compressive or tensile stress. 2
Biaxial Eccentricities ► Every
column should be treated as being subjected to axial compression along with biaxial bending by considering possible eccentricities of the axial load with respect to both the major axis(xx-axis) as well as minor axis (yy-axis). ► These eccentricities, designated as ex and ey with respect of x and y axes, may be atleast emin though in majority of cases of biaxial bending, these may be much more then emin.
3
4
Method Suggested by IS 456-2000 ► The
method set out in clause 39.6 of the code is based on an assumed failure surface that extends the axial load-moment diagram (Pu-Mu) for single axis bending in three dimensions. Such an approach is also known as Breslar’s Load contour method. ► According to the code, the left hand side of the equation
5
Shall not exceed 1. Thus we have The code further relates αn to the ratio of Pu/Puz as under: Pu/Puz Between 0.2 and 0.8 For intermediate values, linear interpolation may be done from figure. Load Puz is given by Load Puz may be evaluated from chart 63 of ISI Handbook(SP-16-2000)
6
Design of Column Step-1-Assume the cross-section of the column and the area of reinforcement along with its distribution, based on moment Mu given by equation where a may vary between 1.10 to 1.20lower of a for higher axial loading (Pu/Puz) Step-2- Compute Puz either using Equation or chart. Find ratio of Pu/Puz. Step-3- Determine Uniaxial Moment Capacities Mux1 and Muy1 combined with axial load Pu , using Appropriate Interaction curves(Design charts) for case of column subjected to axial load (Pu ) and Uniaxial Moment.
7
Step-4-Compute the values of Mux/Mux1 and Muy/Muy1 from chart 64 of SP-16, Find the permissible value of Mux/Mux1 corresponding top the above values of Muy/Muy1 and Pu/Puz .If actual value of Mux/Mux1 is more than the above value found from chart 64 of SP 16, the assumed section is unsafe and needs revision. Even if the assumed value is over safe, it needs revision for the sake of economy.
8
Simplified method as per BS8110
9
Design of Slender Columns
Introduction ►A
Compression member may be considered as slender or long when the slenderness ratio lex/D and ley/b are more than 12. ► Thus, if lex/D > 12, the column is considered to be slender for bending about x-x axis, while if ley/b > 12, the column is considered to be slender for bending about y-y axis. ► When a short column is loaded even with an axial load, the lateral deflection is either zero or very small. ► Similarly when a slender column is loaded even with axial load, the lateral deflection ∆, measured from the original centre line along its length, becomes appreciable.
2
Methods of Design of Slender Columns The design of a slender column can be carried out by following
simplified methods 1) The Strength Reduction Coefficient method 2) The Additional moment Method 3) The Moment Magnification Method The reduction coefficient method, given by IS 456-2000 is recommended for working stress design for service load and is based on allowable stresses in steel and concrete. The additional moment method is recommended by Indian and British codes. The ACI Code recommends the use of moment magnification method. 3
4
Page no:71,72-IS 456-2000
Determination of Total Moment
5
6
Bending of columns in frames
(a) Braced (b) unbraced
7
Procedure for Design of Slender Column Step-1- Determine the Effective Length and Slenderness Ratio in each direction Step-2- (a) Determine Initial Moment (Mui) from given primary end moments Mu1 and Mu2 in each direction. (b) Calculate emin and Mu,min in each direction. (c) Compare moments computed in steps (a) and (b) above and take the greater one of the two as initial moment Mui ,in each direction. Step-3- (a) Compute additional moment (Ma) in each direction, using equation
8
(b) Compute total moment (Mut ) in each direction from using equation without considering reduction factor (ka)
(c) Make Preliminary design for Pu and Mut and find area of steel. Thus p is known. Step-4- (a) Obtain Puz. Also obtain Pb in each direction, for reinforcement ration p determined above. (b) Determine the value of ka in each direction. (c) Determine the Modified design value of moment in each direction Mut = Mui + ka Ma 9
Step-5- Redesign the column for Pu and Mut . If the column is slender about both the axes, design the column for biaxial bending, for (Pu , Muxt) about x-axis and (Pu , Muyt) about y-axis. Note-When external moments are absent, bending moment due to minimum eccentricity should be added to additional moment about the corresponding axes.
10
