Generation of Biaxial Interaction Surfaces

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1 COPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA AUGUST 2002 CONCRETE FRAE DESIGN BS Technical Note This Technical Note describes how the program checks column capacity or designs reinforced concrete columns when the BS code is selected. Overview The program can be used to check column capacity or to design columns. If you define the geometry of the reinforcing bar configuration of each concrete column section, the program will check the column capacity. Alternatively, the program can calculate the amount of reinforcing required to design the column. The design procedure for the reinforced concrete columns of the structure involves the following steps: Generate axial force/biaxial moment interaction surfaces for all of the different concrete section types of the model. A typical biaxial interaction surface is shown in Figure 1. When the steel is undefined, the program generates the interaction surfaces for the range of reinforcement from 0.2 to 10 percent. Calculate the capacity ratio or the required reinforcing area for the factored axial force and biaxial (or uniaxial) bending moments obtained from each load combination at each station of the column. The target capacity ratio when calculating the required reinforcing area is taken as the Utilization Factor Limit, which is equal to 0.95 by default. The Utilization Factor Limit can be redefined in the Preferences. Design the column shear reinforcement. The following four subsections describe in detail the algorithms associated with this process. Generation of Biaxial Interaction Surfaces The column capacity interaction volume is numerically described by a series of discrete points that are generated on the three-dimensional interaction Overview Page 1 of 13

2 failure surface. In addition to axial compression and biaxial bending, the formulation allows for axial tension and biaxial bending considerations (BS , ). A typical interaction diagram is shown in Figure Figure 1: A Typical Column Interaction Surface The coordinates of these points are determined by rotating a plane of linear strain in three dimensions on the section of the column (BS ). See Figure 1. The linear strain diagram limits the maximum concrete strain, ε c, at the extremity of the section to (BS ). Generation of Biaxial Interaction Surfaces Page 2 of 13

3 The formulation is based consistently upon the basic principles of ultimate strength design and allows for any doubly symmetric rectangular, square, or circular column section (BS 3.8.4). The stress in the steel is given by the product of the steel strain and the steel modulus of elasticity, ε s E s, and is limited to the design strength of the steel, f y /γ s. The area associated with each reinforcing bar is placed at the actual location of the center of the bar and the algorithm does not assume any simplifications with respect to distributing the area of steel over the cross section of the column (such as an equivalent steel tube or cylinder). See Figure 2. The concrete compression stress block is assumed to be rectangular, with a stress value of 0.67f cu /γ c. See Figure 2. The interaction algorithm provides corrections to account for the concrete area that is displaced by the reinforcement in the compression zone. Calculate Column Capacity Ratio The column capacity ratio is calculated for each loading combination at each output station of each column. The following steps are involved in calculating the capacity ratio of a particular column for a particular loading combination at a particular location: Determine the factored moments and forces from the analysis load cases and the specified load combination factors to give N, V 2, V 3, 33, and 22. Determine the additional moments resulting from slender column effect. Compute moments caused by minimum eccentricity. Determine total design moments by adding the corresponding additional moments to the factored moments obtained from the analysis. Determine whether the point, defined by the resulting axial load and biaxial moment set, lies within the interaction volume. The following three subsections describe in detail the algorithms associated with this process. Calculate Column Capacity Ratio Page 3 of 13

c Figure 2: Idealized Stress and Strain Distribution in a Column Section Determine Factored oments and Forces Each load combination is defined with a set of load factors corresponding to the load

4 c Figure 2: Idealized Stress and Strain Distribution in a Column Section Determine Factored oments and Forces Each load combination is defined with a set of load factors corresponding to the load cases. The factored loads for a particular load combination are obtained by applying the corresponding load factors to the load cases, giving N, V 2, V 3, f33, and f22. Determine Additional oments If the column is in compression, the factored moments are magnified by adding extra moments to account for the local stability effects (BS 3.8.3). If the column axial force is tensile for a load combination, the additional moments are not considered for that load combination. Both the moments about the major and minor axes are magnified. For simplicity, the following is described for moments about either of the two axes. The program calculates the magnified moments for a particular load combination at a particular point as: where, mag = max{, f () ξ } f f +, (BS ) add Calculate Column Capacity Ratio Page 4 of 13

5 f is the factored moment for a particular load combination at a particular point. It is obtained by applying the corresponding load factors to the load cases. add is the additional moment about a particular load combination at a particular point. For both "Braced" and "Unbraced" columns, the additional moment is obtained as follows: Nau, if N > 0, column in compression, and add = (BS ) 0, if N 0, column in tension. a u is the deflection at the ultimate limit state. It is obtained as a u = β e Kh. (BS ) β e = 1 2,000 2 le. (BS ) b' l e is the effective length in the plane under consideration. It is obtained from l e = βl 0, (BS ) where β is the effective length factor, and l 0 the unsupported length corresponding to instability in the major or minor direction of the element, l 33 or l 22 in Figure 3. In calculating the value of the effective length, the β factor is taken as 1. However, the program allows the user to override this default value. Even if additional moments are considered, global P-delta analysis should be completed for all frames, especially unbraced frames. The default value of β is conservative for braced frames and for unbraced frames for which P-delta analysis is performed. It may not be conservative for unbraced frames if P-delta analysis is not performed. In that case, a value greater than 1 for β is appropriate. b' is the dimension of the column in the plane of bending considered. h is also the dimension of the column in the plane of bending considered. Calculate Column Capacity Ratio Page 5 of 13

$f () ξ is a distribution function. This is used to modify the moment at any point of a column by a certain fraction of add, as add is not added uniformly at all points of the column.$

6 Figure 3: Axes of Bending and Unsupported Length K is the correction factor to the deflection to take care of the influence of the axial force; K is conservatively taken as 1. f () ξ is a distribution function. This is used to modify the moment at any point of a column by a certain fraction of add, as add is not added uniformly at all points of the column. This function is consistent with BS Figure 3.20 for braced frames and BS Figure 3.21 for unbraced columns. The function is given for brced frames as follows (BS , BS Figure 3.20): Calculate Column Capacity Ratio Page 6 of 13

7 f () ξ 4ξ = ( 1 ξ ), ( ) ( )( 1 ξ ) 1 ( 1 ξ ) 6 ξ ( 1 ξ ) 2 + ξ 1 4 1, 2 6 ξ, 4, if both I and J ends if I end is hinged, if J end is hinged, and otherwise. are hinged, The function is given for unbraced frames as follows: f () ξ = 4ξ ( 1 ξ ), 2 ξ ( 1 ξ 2) ( 1 - ξ )( 1 + ξ ),, 2 3 ( I J ) + ( 1 I J )( 3ξ 2ξ ) ( 1)( 3ξ 2ξ ), J I, if if if J end is hinged, if if both I end is hinged, J I > I and J ends > I J, and, and are hinged, no hinge at any end, no hinge at any end. In the above expressions, ξ x L is the non-dimensional parameter to represent the location of the point being considered, ξ = x/l. is the distance of the point from the I end of the column. is the total length of the column. I is the absolute value of the end moment at the I end about the respective axis of bending. J is the absolute value of the end moment at the J end about the respective axis of bending. In addition to magnifying the factored column moments for major and minor axes bending, the minimum eccentricity requirements are also satisfied. The design moment is taken as: Calculate Column Capacity Ratio Page 7 of 13

8 = max ( mag, N e min ), (BS , BS ) where, is the design moment. mag is the magnified moment, which is obtained from the factored moment and the additional moment by the procedure described previously. e min is the minimum eccentricity, which is taken as 0.05 times the overall dimension of the column in the plane of bending considered, but not more than 20 mm (BS ): h e min = 20 mm. (BS ) 20 The minimum eccentricity is considered about one axis at a time (BS ). The sign of the moment resulting from the minimum eccentricity is taken to be the same as that of the analysis moment. It is assumed that the user performs a global P-delta analysis for both braced and unbraced frames. For P-delta analysis, it is recommended that the load combination used to obtain the axial forces in the columns be equivalent to 1.2 DL LL (White and Hajjar 1991). Determine Capacity Ratio A capacity ratio is calculated as a measure of the stress condition of the column. The capacity ratio is basically a factor that gives an indication of the stress condition of the column with respect to the capacity of the column. Before entering the interaction diagram to check the column capacity, the design forces N, 33 and 22 are obtained according to the previous subsections. The point (N, 33, 22 ) is then placed in the interaction space shown as point L in Figure 4. If the point lies within the interaction volume, the column capacity is adequate; however, if the point lies outside the interaction volume, the column is overstressed. This capacity ratio is achieved by plotting the point L and determining the location of point C. The point C is defined as the point where the line OL (if extended outwards) will intersect the failure surface. This point is determined by Calculate Column Capacity Ratio Page 8 of 13

three-dimensional linear interpolation between the points that define the failure surface. See Figure 4. The capacity ratio, CR, is given by the ratio.

9 three-dimensional linear interpolation between the points that define the failure surface. See Figure 4. The capacity ratio, CR, is given by the ratio. OL OC If OL = OC (or CR=1), the point lies on the interaction surface and the column is stressed to capacity. If OL < OC (or CR<1), the point lies within the interaction volume and the column capacity is adequate. If OL > OC (or CR>1), the point lies outside the interaction volume and the column is overstressed Figure 4: Geometric Representation of Column Capacity Ratio Calculate Column Capacity Ratio Page 9 of 13

10 The maximum of all the values of CR calculated from each load combination is reported for each check station of the column, along with the controlling N, 33, and 22 set and associated load combination number. Required Reinforcing Area If the reinforcing area is not defined, the program computes the reinforcement that will give a column capacity ratio of the Utilization Factor Limit, which is equal to 0.95 by default. This factor can be redefined in the Preferences. In designing the column rebar area, the program generates a series of interaction surfaces for eight different ratios of reinforcing steel area to column gross area. The column area is held constant and the rebar area is modified to obtain these different ratios; however, the relative size (area) of each rebar compared to the other bars is always kept constant. The smallest and the largest of the eight reinforcing ratios used are taken as 0.2 percent and 10 percent. The eight reinforcing ratios used are the maximum and the minimum ratios, plus six more ratios. The spacing between the reinforcing ratios is calculated as an increasing arithmetic series in which the space between the first two ratios is equal to one-third of the space between the last two ratios. After the eight reinforcing ratios have been determined, the program develops interaction surfaces for all eight of the ratios using the process described earlier in this Technical Note in the section entitled "Generation of Biaxial Interaction Surfaces." Next, for a given design load combination, the program generates a demand/capacity ratio associated with the each of the eight interaction surfaces. The program then uses linear interpolation between the interaction surfaces to determine the reinforcing ratio that gives a demand/capacity ratio of the Utilization Factor Limit, which is equal to 0.95 by default. The Utilization factor can be redefined in the Preferences. This process is repeated for all design load combinations and the largest required reinforcing ratio is reported. If the required reinforcement is found to be less than the minimum allowed in the code (0.4 percent), the program assigns the design reinforcement to be 0.4 percent (BS , BS Table 3.25). Required Reinforcing Area Page 10 of 13

11 If the required reinforcement is found to be more than 6 percent for both "braced" and "unbraced" frames (BS ), the program declares a failure condition. Design Column Shear Reinforcement The shear reinforcement is designed for each load combination in the major and minor directions of the column. The following steps are involved in designing the shear reinforcement for a particular column for a particular load combination resulting from shear forces in a particular direction (BS ): Calculate the design shear stress and maximum allowed shear stress from v = V, and (BS ) A cv v max = min {0.8R LW f cu, 5 Pa}, where (BS , BS ) Note A cv = b d. If v exceeds either 0.8R LW f cu or 5 N/mm 2, the section area should be increased (BS , BS ). In that case, the program reports an overstress. R LW is a strength reduction factor that applies to light-weight concrete. It is equal to 1 for normal weight concrete. The factor is specified in the concrete material properties. The program reports an overstress message when the shear stress exceed 0.8R LW or 5 Pa (BS , BS ). f cu Calculate the design concrete shear stress from (BS ) ' v c = v c N Ac Vd v c N 1 + (BS ) A c v c where, Design Column Shear Reinforcement Page 11 of 13

12 1 / 3 1 / k1k2 100 As 400 v c = R LW, (BS , Table 3.8) γ bd d m R LW is a shear strength factor that applies to light-weight concrete. It is equal to 1 for normal weight concrete. This factor is specified in the concrete material properties. k 1 is the enhancement factor for support compression and is taken conservatively as 1, (BS ) k 2 = fcu 25 1 / 3, (BS , Table 3.8) γ m = 1.25, (BS ) A s 3, (BS , Table 3.8) bd 400 1, (BS , Table 3.8) d If v Vd 1, (BS , Table 3.8) f cu 40 N/mm 2, (BS , Table 3.8) A s is the area of tensile steel and it is taken as half of the total reinforcing steel area, and d is the distance from the extreme compression fiber to the centroid of the tension steel of the outer layer. v ' v c + 0.4, provide minimum links defined by Asv 0.4b, (BS ) s 0.95f yv else if ' v c < v < v max, provide links given by Design Column Shear Reinforcement Page 12 of 13

13 A s sv v ' c (v v ) b, (BS ) 0.95f yv else if v > v max a failure condition is declared. (BS , ) f yv cannot be taken as greater than 460 Pa (BS ) in the calculation. If f yv is defined as greater than 460 Pa, the program designs shear reinforcing assuming that f yv is equal to 460 Pa. References White. D. W., and J.F., Hajjar Application of Second-Order Elastic Analysis in LRFD: Research in Practice. Engineering Journal. American Institute of Steel Construction, Inc. Vol. 28, No. 4. References Page 13 of 13

This Technical Note describes how the program checks column capacity or designs reinforced concrete columns when the ACI code is selected.

COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001 CONCRETE FRAME DESIGN ACI-318-99 Technical Note This Technical Note describes how the program checks column capacity or designs reinforced