J. Dario Aristizabal-Ochoa,Ph.D., P.E.125-Year Generation ProfessorSchool of Civil EngineeringNational UniversityMedellín, Colombia

This paper presents criteria for determining the base rotational springrestraint and the second-order analysis of a cantilevered precastconcrete column supported by an isolated reinforced concretefooting and its anchorages bearing on elastic soil on piling. Thecolumn-base-footing system is analyzed, as well as the relativeimportance of its two interfaces: the column base plate to footing andthe footing to soil. This issue is of major importance in the analysisand design of framed structures and in the determination of theirdeflections, stability, and second-order effects. This paper extends theprocedure presented in Sections 3.8.2 through 3.8.4 of the PCIDesign Handbook, Fifth Edition, and provides a practical tool forcalculating the degree of fixity at the base of precast and monolithiccolumns on isolated footings bearing directly on soil or supported bypiling. The steps for calculating the total overturning moment, overallstability, and stresses in the different materials and supporting soil, inaccordance with the ACI Building Code (ACI 318-99), are presented.The proposed model is limited to rigid footings and short-termbehavior. Further research is ongoing to determine the effects ofsecondary settlements in the soil, creep and shrinkage of theconcrete, bond slippage in the anchor bolts, and other factors.

The strength, stability, and struc-tural behavior of buildings,bridges and other structures are

highly sensitive to the overturningmoment restraint provided to the baseof columns by their connections to thefoundation, the foundation itself, andthe supporting soil or medium. Thebase moment restraint is of particularimportance in structures subjected tosevere lateral loads, such as those

caused by strong winds and earth-quakes. The behavior and strength ofcolumns at their bases affect the over-all response of buildings and bridges(including substantial increases in thelateral drifts and overturning mo-ments).

Second-order effects (lateral sway,P-∆, and “bowing” secondary mo-ments, P-δ) must be considered in theanalysis. The ACI Building Code1 and

Moment Restraint and Second-Order Analysis of a CantileveredPrecast Column Supported by an Isolated Footing

2 PCI JOURNAL

AISC LRFD Steel Manual2 providefor the proper evaluation of these ef-fects for precast cantilevered columns.

Column bases must resist overturn-ing moments in addition to axial verti-cal loads. The vertical loads cause apre-compression zone at the interfacebetween the column base plate andfooting and between the footing andsoil. When a large overturning mo-ment is applied, the pre-compressionat both interfaces on the tension sideof flexure is reduced, often to zero,creating uplift of the footing, tensionin the anchor bolts, or both simultane-ously.

On the compression side of the in-terfaces, the contact areas remain incompression but with the possibility ofexceeding the compressive strengthcapacity of the concrete or the bearingcapacity of the soil. The base plate, theanchor bolts, and the footing eachhave the ability to undergo rotationalor axial deformations, depending pri-marily on the size of the base plate,the distribution of the anchor bolts andtheir length available to deform, andthe area of the supporting soil avail-able. The behavior of the anchor boltsand base plate is also influenced byany pretension force initially appliedto the anchor bolts.3

BACKGROUNDChapter 15 of the ACI Building

Code1 contains provisions for theproper design and detailing of rein-forced concrete footings and their con-nections to columns through anchor-age and base plates. The PCI DesignHandbook,4 Fifth Edition, Sections3.8.2 to 3.8.4, presents a procedure forcalculating the overturning momentrestraint at column bases, includingthe calculation of degree of fixity andthe structural modeling of partiallyfixed columns. In this paper, the PCIprocedure is adopted and extended tofootings supported by piles. Example1 presents the application of the PCIprocedure, and Examples 2 and 3 pre-sent the application of the extendedprocedure.

Chapter 11 of the AISC LRFD SteelManual contains the design procedurepresented by Thornton5 for both heav-ily and lightly loaded column base

November-December 2002 3

plates. For moment-resisting bases insteel columns, there exist a number ofmethods that depend on the magnitudeof the axial load, its eccentricity, andthe details of the base plate and anchorbolts.2,3,6

Current research has been aimed atmodeling the behavior of the column-base-footing system and the relativeimportance of its two interfaces: thecolumn base plate to footing and thefooting to soil. Experimental studiesof steel base plates subjected to axialload and overturning moments havebeen reported by DeWolf and

Sarisley7 and Thambiratnam and Para-masivam.8

More recently, Melchers9 has pre-sented experimental results with indi-cations that include the following:

1. The footing rotation estimateshave more uncertainty than those ofthe base plate, particularly in cases offoundations on soft soils and subjectedto light vertical loads.

2. For a given lateral load, the foot-ing-to-soil interface can contributemore than one-half the total rotationfor nominally pinned bases, and con-

Fig. 1. Column supported by an isolated footing (modeling of a column-base-footingsystem): (a) Column base-plate-to-footing interface; (b) Footing-to-soil interface.

4 PCI JOURNAL

siderably more for nominally fixedbases, even in cases of very stiff sup-porting soils.

3. The behavior of the base-plate-to-footing interface depends primarily onthe base plate thickness and, to alesser extent, the bolt size.

4. The deformation of the base plateis critical to the degree of fixity at thebase of the column. Results reveal thateven for nominally fixed base connec-tions, the base can become essentiallypinned once sufficient permanent plas-tic deformation of the base plate oc-

curs at load levels of about one-half ofits ultimate moment capacity.

OBJECTIVESThe principal objective of this paper

is to present a practical method for de-termining the base rotational springrestraint and second-order analysis ofa cantilevered precast column sup-ported by an isolated reinforced con-crete footing and its anchorages bear-ing on elastic soil or on piling. Ananalysis of the column-base-footing

system and an explanation of the rela-tive importance of the two interfacesare given. The method is based onstatic equilibrium, strain compatibility,material properties, and well-estab-lished principals of soil mechanics.10,11

Three cases of an isolated footingsupported by an elastic medium arepresented. These are “two-parameter”soils,12,13 toe-bearing piles, and shaft-friction piles. The effects of the rota-tion of the footing as a rigid body, thebending of the base plate, and theelongation of the anchor bolts are alsoincluded in the proposed models.

The effects of the rotation of thefooting (which is assumed to behaveas a rigid body), the bending of thebase plate, and the elongation of theanchor bolts are included in the pro-posed models. Three numerical exam-ples demonstrate the application of theproposed models and correspondingequations in the stability, second-orderanalysis, and calculations of the totaldesign moments and deflections ofcantilever columns.

PROPOSED MODELS ANDEQUATIONS

The structural model of a rigid foot-ing supporting an eccentrically con-nected column (with properties E, A, I,and H) is shown in Fig. 1. The totalrotation of the column base θb is sim-ply the sum of the rotations of the baseplate, anchor bolts, and footing (seeFig. 1b), as expressed by Eq. (1a):

θb = θbp + θab + θf (1a)

whereθf = rotation of the footingθbp = rotation of the base plateθab = rotation caused by elongation

of the anchor boltsIf the axial load on the column is

heavy enough that there is no tensionin the anchor bolts, θbp and θab areeach zero, resulting in θb = θf andmaking the moment spring restraint atthe column base a maximum. In termsof flexibility coefficients, Eq. (1a) canbe expressed as:

θb = (γbp + γab)M + γf Mf (1b)

where

Fig. 2. Rigid footing of dimensions L × B resting on a two-parameter (ks and S) soil.

November-December 2002 5

M = overturning moment at thecolumn base

Mf = overturning moment at thefooting-to-soil interface

γbp = flexibility coefficient of thebase plate

γab = flexibility coefficient of theanchor bolts

γf = flexibility coefficient of thefooting

These three flexibility coefficientsare presented in the next two sections.

Flexibility at the Base-Plate-to-Footing Interface

The elastic structural modeling forthe moment-rotation behavior of abase plate and its anchor bolts is pre-sented in this section. The flexibilitiesof the base plate and its anchor boltscan be estimated using Eqs. (2a) and(2b), which are similar to those in-cluded in the PCI Design Handbook.Their derivations are presented in Ap-pendix A for quick reference.

whereAab = total area of anchor bolts in

tensione = eccentricity of the applied

axial load at the column base(M/P)

Ebp = modulus of elasticity of thebase plate

Eab = modulus of elasticity of theanchor bolts

g = assumed length over whichelongation of the anchor boltstakes place

w = width of the column in the di-rection of bending

Ibp = moment of inertia of the baseplate

x1 = distance from face of columnto center of the anchor bolts(positive when anchor boltsare outside the column, andnegative when they are insidethe column)

γ

γ

abab ab

bp

bp bp

g e w x

eE A w x

x x e w x

eE I w x

= + −+

=

+ + −+

[ / ( ) ]

( )

( ) [ / ( ) ]

( )

2 2 1

2

2 2 1

6

1

1

1 23

1

1

(2a)

(2b)

x2 = distance from face of columnto the base plate anchorage

The rotation of the base and the lat-eral deflection of the column cause anadditional eccentricity of the axialloads on the column, increasing thetotal base overturning moment. Notethat in Eqs. (2a) and (2b), if the eccen-tricity (e = M/P) is less than w/2 + x1

(i.e., P lies inside the center of com-pression of the bearing plate-to-boltsinterface), then θbp = θab = 0 (i.e., theeffects of base plate rotation and rota-tion caused by anchor bolt elongationdo not need to be considered).

Eqs. (2a) and (2b) indicate that thebase plate thickness, the size, distribu-tion, and anchorage of its anchorbolts, and the material characteristicsof both components play significant

roles in the strength and stiffness ofthe base-plate-to-footing interface. Ex-perimental results and further detailson the behavior and ultimate strengthof steel-bolted base plates are given byDeWolf and Sarisley,7 Thambiratnamand Paramasivam,8 and Melchers.9

Flexibility at the Footing-to-Soil Interface

Two cases for rigid footings – thoseresting directly on “two parameter”elastic soil and those resting on piling– are presented in this section.

1. Rigid footing resting directly ona “two-parameter” elastic soil. Theflexibility coefficient of a rigid footing(with dimensions L × B as shown byFig. 2) resting on a soil with elastic

Fig. 3. Plan views of a rigid footing supported on regular arrays of piles.

6 PCI JOURNAL

parameters ks and S is calculated usingEq. (3):12,13

whereks = coefficient of subgrade reactionk1 = S/ksB2

S = second “parameter,” denoted asthe surface tension reaction ofthe supporting soil.12,13

According to Vesic,14,15 the relation-ship between the soil spring constant ks

and the elastic properties of the sup-

γθ

ff

f

s

M

k LB k k

=

=

+ +

12 1

1 6 1231 1

2 (3)

glected. In this case, the flexibility ofthe footing-to-soil interface is in-creased by a factor equal to (1 + k1 +12k1

2) according to Eq. (3), and conse-quently reducing the rotational springrestraint at the base of the column.Note that Eq. (3) is valid as long as e ≤B/6 (i.e., the eccentric vertical load ap-plied at the footing-to-soil interface iswithin the kern of its area), indicatingthat the rigid footing is subjecting thesoil to compression without undergo-ing uplift.

Otherwise, for e > B/6, the nonlineareffects of that portion of the footingbeing uplifted must be taken into ac-count. Gurfinkel16 has treated thisproblem, including the combined ef-fects of biaxial bending and compres-sive axial load, using a numerical al-gorithm for three different types ofstress-strain curves for the supportingsoil.

2. Rigid footing resting directly onpiles. For this discussion, it is as-sumed that all the piles are identical –with total length Dp, cross-sectionalarea Ap, moment of inertia Ip, andmodulus of elasticity Ep – and sym-metrically distributed as shown inFigs. 3 and 4.

For a set of three piles at a distanceD between centers (see Fig. 3a):

For a set of four piles (2 × 2) at adistance D between centers (see Fig.3b):

For a set of six piles (2 × 3) at a dis-tance D between centers (see Fig. 3c):

For a set of nine piles (3 × 3) at adistance D between centers (see Fig.3d):

For a set of sixteen piles (4 × 4) at a

γ fsp pk A D

= 1

6 2 (5d)

γ fsp pk A D

= 1

4 2 (5c)

γ fsp pk A D

= 12 (5b)

γ fsp pk A D

= 22 (5a)

Fig. 4. Elevation view of a rigid footing supported on (a) toe-bearing piles and (b) friction piles.

porting soil or medium is calculatedby Eq. (4):

whereEs = Young’s modulus of support-

ing soil or mediumνs = Poisson’s ratio of supporting

soil or mediumEf = Young’s modulus of rectan-

gular footingIf = moment of inertia of rectan-

gular footingEq. (3) can be reduced to γ f =

12/ksLB3 if the surface tension S is ne-

kE E

E

L

Iss

s

s

f f

=

(4)0 65

1 2

41 12

./

−

ν

November-December 2002 7

distance D between centers (see Fig.3e):

For a set of m × n piles at a distanceD between centers (see Fig. 3f):

For bearing piles resting on rockwith a coefficient of elastic reaction kro

(see Fig. 4a):

For friction piles with their bottomends bearing on soil with a coefficientof elastic reaction kso and surroundedby soil (shaft friction) with a coeffi-cient of elastic reaction kss (see Fig 4b):

Note that Spe is the shaft friction areaof the pile (i.e., the embedment area ofeach pile in the surrounding soil = pileperimeter × embedment length =LperDe). The parameters α and λ repre-sent, respectively, the variation of theshaft friction stress along the embed-ment length of the pile and the ratio ofthe axial load bearing at the pile toe tothe total applied axial load at the pilehead.

Vesic15 suggests different values forα according to the distribution of theshaft friction force, as shown in Fig. 5,and for λ according to Eq. (7):

where

φ = angle of internal friction of thesurrounding soil

γ = specific weight of the soil atdepth De

β φ φφ

= +

sin

sin

cos

(8)

1 2

= 1 – / (7) λ βγDS Ppe p

k

D

E k

A

k S

sp

p

p so

p

ss pe

=

+[ ] + +

1

1 1λ α λ λ λ( ) ( )–

(6b)

–

kk E D

k E Dspro p p

ro p p

=+

/

/ (6a)

γ fsp pmn n k A D

=−

1

12 2( ) (5f)

γ fsp pk A D

= 1

20 2 (5e)

D = average depth (taken herein asDe/2)

Pp = total applied axial load at thetop of a pile

In addition, Vesic15 recommendsthat

and

whered = pile diameterf ′c = concrete compressive strength

of the pilesCp = empirical coefficient that de-

pends on the soil surroundingthe piles (see Table 1)

Cs = (0.93 + 0.16 )Cp

Eqs. (6a) and (6b) represent theD de /

kf

L Cssc

per s

= ′0 45.

kf d

A Csoc

p p

= ′0 45.

Fig. 5. Vesic’ssuggestedvalues for αaccording tothe distributionof the shaftfriction force.15

8 PCI JOURNAL

combined elastic modulus of the pile-rock system for toe-bearing piles andthe combined elastic modulus of thepile-surrounding soil-bearing stratumsystem for shaft-friction piles.

Note that in Eqs. (6a) and (6b), thestiffnesses of each component are con-nected in series. The values for kro, kso,and kss can be determined from in-situload testing, or suggested values can beassumed according to the correspond-ing soil characteristics.10,11,17 Also, notethat in Eqs. (7) and (8), β is a dimen-sionless factor obtained assuming thatthe friction stress along the pile shaft isconstant and that the embedded surfaceof the pile becomes a sliding or failuresurface with an average frictional forceof magnitude .

Second-Order Analysis

For a particular loading condition,the total lateral drift at the top of acantilever column can be calculated bythe well-known equation of flexibilityfor a beam-column, Eq. (9), developedby Salmon and Johnson3 and Aristiza-bal-Ochoa,18,19 which includes both theP-∆ and P-δ effects:

where

∆ = the total drift of the column atA with respect to the base B

Ma = total factored moment at topend A of the column (positiveclockwise)

Mb = total factored moment at bot-tom end B of the column(positive clockwise)

It is important to emphasize that Eq.

(9) represents the second-order equi-librium of an unbraced column underaxial compression and end moments,which can be solved directly (withouttrial-and-error procedures, as shown inExample 1 below). This is an im-provement over the method presentedin the PCI Design Handbook, whichrequires an iterative procedure. Eq. (9)is derived in Appendix B.

For a cantilever column with Ma ≠ 0and Mb = –(Mpr + Pu∆) = –Pu(e + ∆),then from Eq. (9):

where etotal = e + ∆

NUMERICAL EXAMPLES

Example 1 – Analysis of a Can-tilever Column on a Rigid FootingResting on an Elastic Soil

Given the cantilever column andfooting resting on an elastic soil ofFig. 6a, determine the overturning mo-ment restraint at B and the designloads and overturning moments, tak-ing into account the second-order ef-fects. Assume that:

σa = allowable bearing soil stress= 5000 psf (240 kPa)

ks = 200 psi/in. (0.054 N/mm3)S = 0f ′c = 5000 psi (35 MPa)E = Young’s modulus of the col-

umn = 4300 ksi (30 GPa)

Pdead = 80 kips (356 kN)Plive = 30 kips (133 kN)W (wind load) = 2 kips (8.9 kN)Check the soil bearing stress and the

slenderness ratio KH/r of the columnaccording to ACI 319-99, Section10.11.5.

etotala u

u b

e M P

P H=

– –

–

(9*)

( / )( / sin )

tan /

1 Φ ΦΦ Φ κ

Φ = PEI H

u

( / )2

M

EI H

M

EI H

H

M

a

b

b

b

( / )

sin

sin

( / )

sin

sin

Φ ΦΦ Φ

Φ Φ ΦΦ Φ

∆

cos

(9)

− +

−

= − −

2

2

κ

βγDSpe

Table 1. Cp values according to Vesic.

Soil Type Driven Piles Bored PilesSand (from dense to loose) 0.02 to 0.04 0.09 to 0.18

Clay (from hard to soft) 0.02 to 0.04 0.04 to 0.08Silt (from dense to loose) 0.03 to 0.05 0.09 to 0.12

Solution

Calculate the first-order designloads.

According to ACI Code Section 9.2,the required column strength is con-trolled by one of the following threeload cases:

Load Case 1: U = 1.4D + 1.7LLoad Case 2: U = 0.75(1.4D + 1.7L

+ 1.7W)Load Case 3: U = 0.9D + 1.3WThese three load combinations are

depicted in Fig. 6b with their respec-tive first-order eccentricities.

Check rotation between columnand footing.

Since w/2 + x1 = 20/2 + (–2) = 8 in.(> e = 6.933 in.) and e < L/6 = 12 in.,there is no tension in the anchor bolts,and the axial force resultant falls in-side the middle third of the founda-tion. Therefore, γbp = γab = 0 and γf =12/(ksLB3) = 12/(200 × 724 lb-in.) =1/(4.47897 × 108 lb-in.). Based on afirst-order analysis, the overturningmoment restraint at the base B is κb =4.47897 × 108 lb-in./radian.

Perform the second-order analysisand check final eccentricities.

In this particular problem, Ma = 0and Mb = –(WuH + Pu∆) = –Pu(e + ∆).Then, from Eq. (9):

For Load Case 2:Pu = 122.25 kips

= 0.2804 e = 4 in.

Therefore, from Eq. (10):∆ = 0.34 in.Mb = –Pu(e + ∆)

= 530.8 kip-in. etotal = e + ∆

= 4.34 in. (110 mm).

For Load Case 3: Pu = 72 kips

= 0.2152 and e = 6.93 in.

Φ =

× ×72 4300 20 12 1924 2/ [ / ( )]

Φ =

× ×122 25 4300 20 12 1924 2. / [ / ( )]

∆

Φ Φ

=

– + – 1

(10)

e

P Hu b1 1/ ( tan / )κ

November-December 2002 9

Therefore, from Eq. (10):∆ = 0.34 in.Mb = –Pu(e + ∆)

= 523.4 kip-in. etotal = e + ∆

= 7.27 in. (185 mm).For all load cases, the total eccen-

tricity etotal < (w/2 + x1) = 8 in. and L/6= 12 in. Therefore, there is no inducedtension in the anchor bolts, and theaxial force resultant falls inside themiddle third of the foundation.

Examine imaginary column forcomputer modeling.

As suggested in the PCI DesignHandbook, an imaginary beam-col-umn of flexural stiffness 3EiIi /Hi = κb

= 4.47897 × 105 kip-in./radian andaxial stiffness EiAi/Hi = ksBL = 200 ×722 lb/in.= 1036.8 kip/in. is chosen tosimulate the effects of the column-footing-soil interaction, as shown inFig. 6c. If the same material and mo-ment of inertia are chosen (i.e., Ii = I =13,333 in.4), then the span and cross-sectional area, respectively, must beHi = 384 in. or 32 ft, and Ai = 92.59 sqin. These results are similar to thosereported in the PCI Design Handbook.

Check the soil bearing stress.σa < Ptotal/BL + [W(H + hf) +

P∆]/(BL2/6)Under service conditions Ptotal =

Pdead + Plive + Weightcol + Weightfoot =80 + 30 + 6.67 + 8.1 = 124.77 kips.Then, Ptotal /BL = 3.466 ksf and [W(H+ hf) + P∆] = 2 × 17.5 + (80 + 30) ×0.34/12 = 38.12 kip-ft (assuming con-servatively that ∆ = 0.34 in.), andMtotal /(BL2/6) = 1.059 ksf. Therefore,the maximum soil stress = 3.466 +1.059 = 4.525 ksf < σa = 5000 psf(OK)

Check the slenderness ratio kH/rof the column.

With Ψa = ∞ and Ψb = (EI/H)/(κ/6)= [4300 × 204/(12 × 192)]/(4.47897 ×105/6) = 4.0, the effective length factork obtained from the alignment chart orfrom the stability equation(π /k)tan(π /k) = 1.5 is k = 3.179.Therefore:

kH/r = 3.179 × 192/5.7735 = 105.7 > 100 (too high!)

ConclusionsThe flexural fixity given to this col-

umn at its base is very low, resulting ina slenderness ratio kH/r greater than100, which, according to the ACIBuilding Code, requires a special anal-ysis that must include the effects ofshrinkage and creep, cracking, materialnonlinearity, and other considerations.This is an important oversight of thePCI Design Handbook. Therefore, tofurther reduce kH/r, it is recommendedthat the value of B be increased. In ad-dition, the fact that etotal = 7.27 in. forLoad Case 3 is very close to the limitof (w/2 + x1) = 8 in., indicating that inorder to avoid tension in the anchorbolts and a further reduction in thebase resistance, the anchor bolts shouldbe located outside of the cross section,as shown in Fig. 1a.

Note that in all three load cases,the precast cantilever column is rigidlyconnected to the footing (since γbp =γab = 0), making the rotational springrestraint at the column base a maxi-mum (with a value of κb = 4.47897 ×108 lb-in./radian). Further, it is benefi-cial to have x1 = –x2, making γbp = 0[see Eq. (2b)] for any loading condi-tion and base plate size.

Fig. 6. Cantilever column on a rigid footing resting on an elastic soil (Example 1): (a) Structural model; (b) Load combinationswith their respective first-order eccentricities; (c) Imaginary beam-column for computer analysis.

10 PCI JOURNAL

Example 2 – Cantilever ColumnSupported by a Rigid FootingResting on Toe-Bearing Piles

The rigid footing of Example 1 isbeing supported by four equallyspaced piles, as shown in Fig. 7. De-termine the moment restraint at thebase of the column. Assume that:

f ′c = concrete strength of the piles= 3000 psi (21 MPa)

Ep = Young’s modulus of the con-crete piles = 3122 ksi (21.5MPa)

Dp = 18 ft (5.5 m)d = pile diameter = 1 ft (0.305

m)D = 4 ft (1.2 m)Kro= 1.8067 kip/in.3 (0.4904

N/mm3)Assume that the loads and proper-

ties of the column are the same asthose given in Example 1.

SolutionUsing Eqs. (6a) and (5b):

= 1.6059 kip/in.3

orκb = 4.18468 × 108 lb-in./radian

ConclusionsThe flexural restraint provided by the

four toe-bearing piles shown in Fig. 7is lower than that provided by the sup-porting soil in Example 1, resulting ineven larger values for kH/r, second-order moments, and lateral deflections.To reach the same degree of fixity andthe same results of the column of Ex-ample 1, the distance D between pilesmust be increased slightly from 48 to49.7 in. Since κb is proportional to D2,it is recommended that D be increased.Assuming D = 54 in., then kH/r =98.08, which is a little improvement inthe design at no extra cost.

Example 3 – Cantilever ColumnSupported by a Rigid FootingResting on Shaft-Friction Piles

The square footing of Example 1 isbeing supported as shown in Fig. 8 byfour equally spaced friction pilesdriven 10 ft (3.1 m) deep into a densesand stratum. Determine the rotationalspring restraint at the column base.Assume that:

f ′c = concrete strength of the piles= 3000 psi (21 MPa)

Ep = Young’s modulus of the con-crete of the piles = 3122 ksi(21.5 MPa)

d = 1 ft (0.31 m)D = 4 ft (1.2 m)

k A Dfsp p

=

× × ×

×

=

=

( / ) .

. kip - in.

γ

π

1

1

48 12 4 1 6059

1

4 18468 10

2

2 2

5

kk E D

k E Dspro p p

ro p p

=+

×+

/

/

=1.8067 3122 / 216

1.8067 3122 / 216

Fig. 7. Cantilevercolumn on a rigidfooting resting ontoe-bearing piles

(Example 2).

November-December 2002 11

Dp = 10.5 ft (3.20 m)De = 10 ft (3.05 m)

γ = 143.61 lb/ft (22.56 KN/m)φ = 0.35 radiansα = 0.5Cp = 0.02Assume that the loads and properties

of the column are the same as thosegiven in Example 1.

SolutionBefore making use of Eqs. (6b) and

(5b), the following parameters must bedetermined using Vesic’s approach:15

Cp = 0.02 (for dense sand and drivenpiles; see Table 1)

Cs = (0.93 + 0.16 )Cp

= (0.93 + 0.16 ) × 0.02 = 0.02872

= 7.162 kip/in.3

= 1.2469 kip/in.3

P P Weight Weight

m ndead live col foot

=(80 30 6.67 + 8.1)

(2 2)

= 31.193 kips

( )

( )

+ + +×

+ +×

β φ φφ

=+

=+

=

sin cos

sin

(sin . )(cos . )

sin ..

1

0 35 0 35

1 0 350 2882

2

2

kf

L Cssc

per s

= ′

=×

0 45

0 45 3 0

12 0 02872

.

. ( . )

( )( . )π

kf d

A Csoc

p p

= ′

=×

0 45

0 45 3 0 12

12 4 0 022

.

. ( . )( )

( / )( . )π

10 1/

D de /

λ = 1 – /Pp

= 1 – [0.2882 × 143.61 lb/ft3 × (0.5 × 10 ft) × (π × 1 × 10 ft2)]/(31,193 lbs) = 0.7916

= 6.5285 kip/in.3

ksp

=+[ ] × + +

× × ×

1

0 7916 0 5 1 0 791610 5 12

3122

0 7916

7 1621 0 7916

12 4

12 10 12 1 2469

2

. . ( – . ). .

.( – . )

( / )

( . )

ππ

βγDSpe

Fig. 8. Cantilevercolumn on a rigidfooting resting onfriction piles(Example 3).

Finally, using Eq. (5b)

or κb = 1.701167 × 109 lb-in./radian

γ

π

fsp pk A D

=

=× × ×

=×

1

1

48 12 4 6 5285

1

1 701167 10

2

2 2

6

kip - in.

( / ) .

.

12 PCI JOURNAL

Perform second-order analysisand check of final moments and ec-centricities.

For Load Case 2: Pu = 122.25 kipsΦ = 0.2804 e = 4 in.Therefore, from Eq. (10):∆ = 0.17 in. Mb = –Pu(e + ∆) = 509.4 kip-in.etotal = e + ∆ = 4.17 in. (106 mm) For Load Case 3: Pu = 72 kipsΦ = 0.2152e = 6.93 in.Therefore, from Eq. (10):∆ = 0.17 in.Mb = –Pu(e + ∆) = 511.2 kip-in.etotal = e + ∆ = 7.10 in. (180 mm)As in Example 1, for all load cases,

the total eccentricity etotal < (w/2 + x1)= 8 in. and L/6 = 12 in. Therefore,there is no induced tension in the an-chor bolts and the axial force resultantfalls inside the middle third of thefoundation.

Check the slenderness ratio kH/rof the column.

With Ψa = ∞ and Ψb = (EI/H)/(κ/6)= [4300 × 204/(12 × 192)]/(1.701167 ×106/6) = 1.0532, the effective lengthfactor k obtained from the alignmentchart or from the stability equation(π/k)tan(π/k) = 5.6969 is k = 2.345.Therefore:

kH/r = 2.345 × 192/5.7735 = 77.98 < 100 (OK)

Conclusion

The rotational spring restraint pro-vided by the four friction piles to thecolumn shown in Fig. 8 is about 3.8and 4.1 times that provided by thesupporting soil and toe-bearing pilesin Examples 1 and 3, respectively. Asa result, the slenderness ratio kH/r, the

induced second-order moments, andlateral deflections are reduced, meet-ing stability requirements of the ACIBuilding Code.

CONCLUDING REMARKSThe analytical results discussed in

this paper indicate that the rotationalspring restraint of a column base con-nected to an isolated footing and thecantilever column’s stability dependon the following factors:

1. The physical parameters of theconnection between the column andthe footing (i.e., arrangement, size,and materials of the base plate and itsanchor bolts);

2. The geometry and size of thefooting;

3. The bearing characteristics of thesoil under the footing and its sur-roundings;

4. The type and arrangement of thefooting-soil interface and the type anddistribution of the piles; and

5. The intensities of the appliedloads and overturning moments.

Analytical results also indicate thatit is beneficial to have x1 = –x2, mak-ing γbp = 0 [see Eq. (2b)] for any load-ing condition and base plate size.However, cases in which etotal at theprecast column base is greater than(w/2 + x1) must be avoided in practice.They probably are not only unreliable,as claimed by Melchers,9 but alsocumbersome to analyze, requiring aniterative procedure to determine κb

since both γbp and γab are nonlinearfunctions of the eccentricity e [seeEqs. (2a) and (2b) with e = etotal].

In addition, the effects of the footingthickness hf on κb, which must be cal-culated according to Eq. (1b) (i.e., κb =M/θb), and on the critical axial load ofthe precast column [according to Eq.(9)] must be considered. An “exact”

second-order analysis would require amore complex model like a “step-up”column system comprising the precastcolumn and the footing itself with allthe corresponding connections.

The proposed models and corre-sponding equations are limited to rigidfootings, short term behavior, andstatic loads (i.e., linear elastic rangesof stresses, deflections, and immediatesettlements). Therefore, there is a realneed for further research to determinepractical models that include the fol-lowing:

1. The effects of the flexibility ofthe footing and piles;

2. Long-term effects (such as sec-ondary settlements in the supportingsoils, bond slippage in the anchorbolts, and creep and shrinkage in theconcrete members);

3. The actual nominal strength ofeach component and their interactionsat ultimate loading conditions (such asthe axial and shear forces and biaxialbending interactions);

4. The effects of prestressing the an-chor bolts and reinforcements; and

5. The effects of different types ofloads.

ACKNOWLEDGMENTSThe research presented in this

paper was carried out at the NationalUniversity of Colombia, School ofMines, in Medellín, Colombia. Theauthor wishes to express his apprecia-tion to DIME for their financial sup-port and encouragement and to Mr. J.Paul Smith-Pardo, graduate student atPurdue University, Indiana, while hewas at the National University ofColombia, for his cooperation and as-sistance. The author also expresses hisappreciation to the PCI JOURNAL re-viewers of this paper for their thought-ful and constructive comments.

November-December 2002 13

1. ACI Committee 318, “Building Code Requirements for Struc-tural Concrete (ACI 318-99),” American Concrete Institute,Farmington Hills, MI, 1999.

2. AISC, Manual of Steel Construction: Load & Resistance FactorDesign, Second Edition, V. II, Chapter 11, American Instituteof Steel Construction, Chicago, IL, 1993, pp. 11-54 to 11-64.

3. Salmon, C. G., and Johnson, J. E., Steel Structures: Designand Behavior, Fourth Edition, Harper-Collins, New York, NY,1996.

4. PCI Design Handbook: Precast and Prestressed Concrete,Fifth Edition, Precast/Prestressed Concrete Institute, Chicago,IL, 1999.

5. Thornton, W. A., “Design of Base Plates for Wide FlangeColumns – A Concatenation of Methods,” Engineering Jour-nal, V. 27, No. 4, Fourth Quarter, American Institute of SteelConstruction, 1990, pp. 173-174.

6. Blodgett, O. W., Design of Welded Structures, James F. Lin-coln Arc Welding Foundation, Cleveland, OH, 1966.

7. DeWolf, J. T., and Sarisley, E. F., “Column Base Plate withAxial Loads and Moments,” Journal of the StructuralDivision, V. 106, No. ST11, November 1980, American Soci-ety of Civil Engineers, pp. 2176-2184.

8. Thambiratnam, D. P., and Paramasivam, P., “Base PlatesUnder Axial Loads and Moments,” Journal of Structural Engi-neering, V. 112 , No. 5, May 1986, American Society of CivilEngineers, pp. 1166-1181.

9. Melchers, R. E., “Steel Base Plate-Footing-Soil Behavior,”Second International Workshop on Connections in Steel Struc-tures: Behavior, Strength and Design, American Institute ofSteel Construction, Chicago, IL, 1992, pp. 132-139.

10. Bowles, J. E., Foundation Analysis and Design, Fourth Edi-tion, McGraw-Hill Company, New York, NY, 1988.

11. Das, B. M., Principles of Foundation Engineering, SecondEdition, PWS-Kent Publishing Company, Boston, MA, 1990.

12. Scott, R. F., Foundation Analysis, Prentice-Hall Inc., Engle-wood Cliffs, NJ, 1981.

13. Lin, G., “Stability of Frames with Grade Beam and Soil Inter-action,” Journal of Engineering Mechanics, V. 118, No. 1,January 1992, American Society of Civil Engineers, pp. 125-140.

14. Vesic, A. S., “Bending of Beams on Isotropic Elastic Solid,”Proceedings, V. 87, No. 2, American Society of Civil Engi-neers, 1966, pp. 35-51.

15. Vesic, A. S., “Design of Pile Foundations,” National Coopera-tive Highway Research Program, Synthesis of Practice No. 42,Transportation Research Board, Washington, DC, 1977.

16. Gurfinkel, G., “Analysis of Footings Subjected to BiaxialBending,” Journal of Structural Engineering, V. 96, No. ST6,June 1970, American Society of Civil Engineers, pp. 1049-1059.

17. Fang, H-Y (Editor), Foundation Engineering Handbook, Sec-ond Edition, Van-Nostrand Reinhold, New York, NY, 1991.

18. Aristizabal-Ochoa, J. Dario, “Braced, Partially Braced, andUnbraced Columns: Complete Set of Classical Stability Equa-tions,” Structural Engineering and Mechanics, V. 4, No. 4,1996, Techno-Press, Korea, pp. 365-381.

19. Aristizabal-Ochoa, J. Dario, “Story Stability of Braced, Par-tially Braced, and Unbraced Frames: Classical Approach,”Journal of Structural Engineering, V. 123, No. 6, June 1997,American Society of Civil Engineers, pp. 799-807.

Flexibility due to the Connecting BoltsAssuming the model shown in Fig. 1, in which the left-

side anchor bolts are subject to tension with an equivalentfree length g and are still in the elastic range with a totalelongation ∆g = Tg/Eab Aab caused by the tensile force T:

From equilibrium (overturning about the right bolt in Fig.1b):

T(w + 2x1) – P(e – w/2 – x1) = 0or

Substituting T into Eq. (A1):

Flexibility due to the bearing plateAssuming the model shown in Fig. 1, which shows that

the span of the plate (x1 + x2) acting as a cantilever:

Substituting T from Eq. (A2) into Eq. (A4) and the resultinto Eq. (A3):

θ

θγ

bpbp bp

bpbp

bp bp

M x x e w x

eE I w x

M

x x e w x

eE I w x

= + + −+

= = + + −+

( ) [ / ( ) ]

( )

( ) [ / ( ) ]

( )

1 23

1

1

1 23

1

1

2 2 1

6

2 2 1

6

(2b)

δ = +T x x

E Ibp bp

( )1 23

3

(A4)

θ δbp w x

=+( )

(A3)1

θ γabab

ab abM

ge

w x

eE A w x= =

+−

+

21

21

1

or

(2a)

( )

θabab ab

M

e

e

w xg

E A w x=

+−

+2

21

1

1

( )

TM

e

e

w x=

+−

2

21

1 (A2)

θabab ab

g

w x

Tg

E A w x=

+=

+∆

( ) ( ) (A1)

1 1

REFERENCES

APPENDIX A— DERIVATION OF FLEXIBILITY EQS. (2A) AND (2B)

14 PCI JOURNAL

The reader is referred to Salmon and Johnson3 (p. 900) forfurther details on the derivation of Eq. (9). The general ap-proach to the second-order analysis of an “ideal” beam-col-umn (i.e., a perfectly straight prismatic member with per-fectly centered compressive end loads P, and with bendingend moments Ma and Mb occurring about one of the princi-pal axes of its cross section) begins with the well-knownTimoshenko equation:

whereM = Ma + Py – (Ma + Mb + P∆)x/H (B2)

Substituting (B2) into (B1):

Letting Φ2 = P/(EI/H2), the solution for Eq. (B3) is

Applying the boundary conditions of zero deflection at x = 0 and y = ∆ at x = H gives

C1 = (Ma cosΦ + Mb)/(PsinΦ)

and

C2 = Ma/P

y Cx

HC

x

H

M

P

M M P

P

x

H

a

a b

=

+

−

+ +

1 2sin cosΦ Φ

∆

+

(B4)

d y

d x

P

EIy

M

EI

M M P

EI

x

Ha a b

2

2 +

(B3) + = − + + ∆

d y

d x

M

EI

2

2 = − (B1)

APPENDIX B — DERIVATION OF EQ. (9)

Then

Differentiating once to obtain the slope:

dy

dx

M

PH

x H

M

PH

x H

H

xdy

dx H

x Hdy

dx H

M

EI H

a

b

a

b

a b

aa

= − −

−

+

= +

= +

=

+

(B6)

When = 0,

When = ,

Then, and after some algebra may be expressed as:

1

1

Φ Φ ΦΦΦ Φ

Φ∆

∆

∆

cos( / )

sin

cos( / )

sin

( / )

sin

θ

θ

θ θ

θ ΦΦ Φ ΦΦ Φ

Φ ΦΦ Φ

Φ ΦΦ Φ

Φ Φ ΦΦ Φ

∆

∆

cos

(9)

Note that from rotational compatibility at moment

equilibrium at end B:

− + −

= − + −

= − −

= − −

sin

sin ( / )

sin

sin

( / )

sin

sin ( / )

sin

sin

2 2

2 2

M

EI H

M

EI H

M

EI H

H

M

H

M

b

ba b

b

b

bb

b

θ

κ

θκ

yM

P

x H H x

H

M

P

x H x

H

x

H

a

b

= − − −

−

−

+

(B5)

sin( / )

sin

sin( / )

sin

Φ ΦΦΦ

Φ∆

November-December 2002 15

A = cross-sectional area of columnAab = total area of anchor bolts in tensionB = width of footingD = distance between centers or centroids of pilesDe = embedment length of pilesDp = total pile length

D = average depth (taken herein as De /2)d = diameter of pilesCp = empirical coefficient of soil surrounding piles

(see Table 1)Cs = empirical coefficient of soil surrounding piles

= (0.93 + 0.16 )Cp

E = Young’s modulus of the column materialEbp = modulus of elasticity of base plate Eab = modulus of elasticity of anchor boltsEs = Young’s modulus of supporting soil or mediumEf = Young’s modulus of reinforced concrete footingE = eccentricity of applied axial load at column base

= M/Pf ′c = compressive strength of concreteg = assumed length over which elongation of tension

anchor bolts takes placeH = column heighthf = footing vertical depthI = column moment of the inertiaIbp = moment of inertia of the base plateIf = footing moment of inertia = LB3/12K = effective length factor of columnkro = coefficient of elastic reaction of rock for toe-

bearing piles (Fig. 4a)ks = coefficient of subgrade reaction (according to

Vesic14)kso = coefficient of elastic reaction of bearing soil at

bottom end of friction pilekss = coefficient of elastic reaction of surrounding soil

of friction pileL = length of footingM = overturning moment at column base Mf = overturning moment at footing-soil interface

Ma = factored moment at top end A of columnMb = factored moment at bottom end B of columnm × n = rectangular array of piles [see Fig. 3(f)]P = applied axial load at column basePp = applied axial load at pile headS = surface tension reaction of supporting soilSpe = shaft friction area of pile (i.e., area of each pile

embedded in surrounding soil)= LperDe

r = radius of gyration of column cross sectionx1 = distance from face of column to center of anchor

bolts (positive if anchor bolts are outside column,and negative if they are inside)

x2 = distance from face of column to base plate an-chorage

w = width of column in the direction of bendingα = soil parameter representing variation of shaft

friction stress along embedded pile (see Fig. 5)β = coefficient of sliding friction∆ = total drift of column at top end A with respect to

its base Bνs = Poisson’s ratio of supporting soil or mediumκb = rotational stiffness of end connection at column

base Bλ = soil parameter representing the ratio of axial load

carried by pile toe (i.e., by bearing at bottom ofthe pile) to total applied axial load at pile head

γf = flexibility coefficient of footingγbp = flexibility coefficient of base plateγab = flexibility coefficient of anchor bolts

γ = specific weight of soil at depth De

φ = angle of internal friction of surrounding soil (forclay varies from 16 to 26 degrees)

Φ = stability parameter = [P/(EI/H2)]1/2

θf = rotation of footingθbp = rotation of base plateθab = rotation caused by elongation of anchor boltsΨa = alignment chart coefficients [(EI/H)c/(EI/L)g] at AΨb = alignment chart coefficients [(EI/H)c/(EI/L)g] at B

D de /

APPENDIX C — NOTATION