MATH: Reinforced Concrete Design 02

Chapter 8
Long Columns
SLENDERNESS EFFECTS IN COLUMNS
The slenderness of columns depends on its unsupported length and the geometry of its section. As the slenderness increases, the tendency that it will buckle also increases.
To visualize the effect of slenderness, let us imagine a stick (say wire or broomstick) with the same cross-section area but with varying length, being compresses until it breaks as shown in figure 8.1
Obviously, Figure 8.1 (a) and Figure 8.1 (b) would have larger value in P until it fails by crushing. This is the situation of a short column. However, Figure 8.1 (c) will fail by lateral buckling due to its length (which increases the slenderness) and the value of P would be lesser than that of (a) and (b).
According to Section 5.10.10.1 of NSCP, design of compression members shall be based in forces and moments determined from analysis of the structure. Such analysis should take into aaccount influences of axial loads and variable moment of inertia on member stiffness and fixed-end moments, effects of deflections on moments and forces, and the effects of duration of loads. In lieu of this procedure, the slenderness effects in compression members may be evaluated in accordance with the approximate procedure presented in Sec. 5.10.11
APPROXIMATE EVALUATION OF SLENDERNESS EFFECTS (SECTION 5.10.11)
Unsupported Length of Compression Members
Unsupported length la of a compression member should be taken as the clear distance between floor slabs, beams, or other members capable of providing lateral support for that compression member. Where column capitals or haunches are present, unsupported length should be measured to the lower extremety of capital or haunch in the place considered.
Effective Length Factors (5.10.11.2.1 & 5.10.11.2.2)
For compression members braced against sideway, effective length factor k should be taken as 1.0, unless analysis shows that a lower value is justified. For compression members not braced against sideway, effective length factor k should be determined with the due consideration of effects of cracking and reinforcement on relative stiffness, and should be greater than 1.0.
Radius of Gyration
Radius of gyration r may be taken equal to 0.30 times the overall dimension in the direction stability is being considered for rectangular compression members, and 0.25 times the diameter for circular compression members. For other shapes, r may be computed for the gross concrete section.
For rectangular compression members:
Eq. 8 – 1 r=0.3h
For circular compression members:
Eq. 8 - 2 r=0.25D
CONSIDERATION OF SLENDERNESS EFFECTS
According to Section 5.10.11.4.1 of the Code, for compression members braced against sidesway, effects of slenderness may be neglected when
klur100, an analysis as defined in Sec.5.10.10.1 (see Page 237) should be made.
Braced and Unbraced Frames
As a guide in judging whether a frame is braced or unbraced, the Commentary on ACI 318-83 indicates that a frame may be considered braced if the bracing elements, such as shear walls, shear trusses, or other means resisting lateral movement if a storey, have a total stiffness at least six times the sum of the stiffnesses of all the columns resisting lateral movement in that storey.
Alignment Charts
The ACI Committee 441 has proposed that k should be obtained from the Jackson and Moreland alignment chart as shown in Figure 8.2. To use this chart, a parameter ΨA for end A of column AB, and a similar parameter ΨB must be computed for end B. The parameter Ψ at one end of the column equals the sum of the stiffnesses ( EI/L) of the column meeting at that joint (including the column in question), divided by the sum of all the stiffnesses of the beam meeting at that joint. Once ΨAand ΨB are known k is obtained by placing a straightedge between ΨAand ΨB . The point where the straightedge crossed the middle monograph is k.
Eq. 8 – 3 Ψ= EILof columns EILof beams


For columns for which the slenderness ratio lies between 22 and 100, and therefore the slenderness effect on load-carrying capacity must be taken into account, either an elastic analysis can be performed to evaluate the effects of lateral deflections and other effects secondary stresses, or an approximate method based on MOMENT MAGNIFICATION may be used.
MOMENT MAGNIFIER METHOD
The effect in slenderness in long columns may be approximately accounted for in design empirically increasing the factored design moment. According to Section5.10.11.5.1 of the Code, compressing member should be designed using the factored axial load Pu from a conventional frame analysis and a magnified factored moment Mc defined by:
Eq. 8 – 4 Mc=δbM2b+ δsM2s
Where
M1b = value if smaller factored end moment on a compression member due to the loads that result in no appreciable sideways, calculated by conventional elastic frame analysis, positive if member is bent in single curvature, negative if bent in double curvature.
M2b = value of larger factored end moment on compression member due to loads that result in appreciable sideway, calculated by conventional elastic frame analysis. According to Section5.10.11.5.4, for use in Eq. 8 – 4
M2b Pu15+0.03h
Where h is the column dimension in the direction of bending.
M2s = value of larger factored end moment on compression member due to loads that result in appreciable sidesway (such as wind, earthquake, and other lateral loads), calculated elastic frame analysis. According to Section 5.10.11.5.1,
M2s Pu15+0.03h
Where h is the column dimension n the direction of bending.
δb = moment magnification factor that takes into account the effect of member curvature in a frame braced against sidesway and is equal to
Eq. 8 – 5 δb=Cm1-PuϕPc 1.0
δb = moment magnification factor that takes into account the effect of member curvature in a frame braced against sidesway and is equal to
Eq. 8 – 5 δb=Cm1-PuϕPc 1.0
δs = moment magnification factor that takes into account the lateral drift of a column caused by lateral and gravity loading the frame not being braced against sidesway and is equal to
Eq. 8 – 6 δs=11- Puϕ Pc 1.0
And
Eq. 8 – 7 Pc=π2EIklu2










For slabs with beams between supports, the slab portion of colmn strips shall be proportioned to resist that portion of column strip moments not resisted by beams.
Factored Moments in Beams (Section 5.13.6.5)
Beams between supports shall be proportioned to resist 85 percent of column strip moments if (α1L2/L1) is equal to or greater than 1.0. For value of (α1L2/L1) between 1.0 and zero, proportion of column strip moments resisted by beams shall be obtained by linear interpolation between 85 and zero percent.
Factored Moments in Middle Strips
That portion of negative and positive factored moments not resisted by column strips shall be proportionately assigned to corresponding half middle strips. Each middle strip shall be proportioned to resist the sum of the moments assigned to its two half middle strips. A middle strip adjacent to and parallel with an edge supported by a wall shall be proportioned to resist twice the moment assigned to the half middle strip corresponding to the first row of interior supports.
Modification of Factored Moments
Negative and positive factored moments may be modified by 10 percent provided the total static moment for a panel in the direction considered is not less than that required by Eq. 10 – 6.
Factored Shear in Slab System with Beams
Beans with (α1L2/L1) equal to or greater than 1.0 shall be proportioned to resist shear caused by factored loads on tributary areas bounded by 45deg lines drawn from the corners of the panels and the center-lines of the adjacent panels parallel to the long sides. Beams with (α1L2/L1) less than 1.0 ,ay be proportioned to resist shear obtained by linear interpolation, assuming beams carry no load at α=0.
















Problem 8.1
A square column having an unsupported length of 6m is used in a frame braced against sidesway (k = 1.0). The column is bent in single curvature and subjected to factored end moments of 80 kN-m at the top and 60 kN-m at the bottom. Find its minimum dimension such that slenderness may not be considered in the analysis.
Solution
M1b= smaller factored end moment= +60kN-m
M2b=larger factored end moment=80kN-m
So that slendernedd may not be considered.
klur2), see Pag337
Thus, hmin=133mm











COMBINED FOOTINGS
Combined footings support more than one column. One situation where these footings may be used is when the columns are so close together so that isolated or individual footing would run into each other. Another situation is when the column is very near the property line, such that an isolated footing would extend across the line. A trapezoidal footing or strap (t) footing may also be used is the two adjacent column are very near the property line.
In any of these shapes, it is very important to let the centroid of the footing coincide with the centroid of the combined column loads. In this manner, the bearing pressure underneath the footing coincide with the centroid of the combined column loads. In this manner, the bearing pressure underneath the footing would be uniform and it prevents uneven settlement.


















STRAP OR CANTILEVER FOOTING
When space is restricted for a single column footing, the soil pressure under the footing can be made uniform by combining it with the adjacent column or columns using a rectangular or trapezoidal shape. As the distance between such columns increases, the cost of such shapes rises rapidly, For column spacing more than 4.5m, a strap footing may be more economical. It consists of a separate footing under each column connected by a beam or strap to distribute the column loads.
The footings are sized to produce the same constant pressure under its base. This is attained when the centroid of their areas coincide with resultant of the column loads.



















Usually, the strap is raised above the bottom of the footing so as not to bear on the soil. The strap should be designed as a rectangular beam spanning between the columns. The loads on it include its own weight (when it does not rest on soil), the upward pressure from the footings, and the column loads. The width of strap is usually selected equal to that of the largest column plus 100 mm to 200 mm so that column forms can be supported on top of the strap. Its depth is determined by the maximum bending moment.
The main reinforcement in the strap is placed near the top and stirrups are normally needed near the columns. Longitudinal placement steel is also set near the bottom, plus reinforcement to guard against settlement stresses.
The footing under the exterior column may be designed as a wall footing. The interior footing should be designed as a single-column footing; however its critical section for punching shear should be computed on a section parallel to the strap at a distance d/2 from its face.