104 MARCH 2003 /

BY SAM ESKILDSEN

Spread footings are typically designed to resistthree different types of loads: gravity forces,moments, and uplift forces. Some footing designs aremore complex, involving net bearing forces, upliftforces, moments, surcharge loads, and passiveresistance from cantilevering slabs-on-ground. Thisarticle presents a procedure that allows designers toaccount for all of these design loads in their analysisand view the results in an interaction diagram.

As with concrete and concrete masonry unit (CMU)columns, an interaction diagram representing theenvelope of allowable combinations of axial load andmoment that can be resisted by a rectangular footingcan be created. The procedure presented in thisarticle only considers moment about one axis of thefooting. While developing the procedure, I madeseveral simplifying assumptions, and they are: Plane sections remain plane; The soil cannot resist tension. However, upward

forces generated by applied moments are assumedto be resisted by gravity loads: the footing weight,the soil weight, any permanent surcharge present,and any passive resistances (for example, cantile-vering slabs); and

G46

Spreadsheets help reduce design timefor footings with complex loading

The relationship between the stress and strain of thesoil is linear when the applied loads are less thanthe ultimate capacity of the soil.Referring to Fig. 1 and 2, consider four distinct

stress distribution ranges of the soil.In Range 1, all soil beneath the footing is in

compression. Due to the presence of a moment, the soilon the left experiences more stress than the soil on theright. Eventually with increasing moment, the soilstress at the extreme right will be zero. This is the startof Range 2, which assumes the footing self-weight, soilabove the footing, permanent surcharges, and passiveresistance counteract any upward forces or uplift.

In the context of this discussion, surcharge refers tothe footing weight, weight of soil above the footing,and gravity loads other than the axial load applied tothe footing from the structure. Such an appliedsurcharge could be a slab-on-ground that lies over thefooting. Typically, a calculation is done to determinehow far the slab-on-ground over the footing willcantilever beyond the limits of the footing whensupporting its own weight. The procedure outline inthis article allows inclusion of this type of resistancein the surcharge called passive resistance.

/ MARCH 2003 105

When the upward stress at the extreme right (Fig. 1)equals the surcharge value, Range 3 begins. Soil to theright of the neutral axis is capable of resisting stressequal to the applied surcharge. No stress beyond thesurcharge is considered. In Range 3, the surcharge isconsidered activated, providing resistance tofooting uplift (that is, the soil does not supply resis-tance through tension). Range 4 is similar to Range 3except the footing is not resisting a net uplift.

G46Figure 1 presents Ranges 1 and 2 where

l = footing length;bp = allowable soil gross bearing pressure less the

active portion of the surcharge (footingweight, soil weight, gravity surcharges, loadfrom slab directly above footing);

reduced = incremental bearing pressure, less than bp; andup = uplift stress (less than the maximum) that can

be resisted by the footing weight, soil weight,and surcharges (a value less than sc, which isthe value of the total applied surcharge).

The axial load and moment supported by a footingstressed in Ranges 1 and 2 can be expressed (replaceup with reduced for use in Range 1)

AP average= or ( )AP upbp 21

+=

where A is the footing area. Note that P is the axialload applied to the footing not including thesurcharge, soil weight, and footing weight

WM upbp2

=

where W is the footing width.Points along the interaction diagram for Ranges 1

and 2 are obtained by incrementally changing reduced(in Range 1) and up (in Range 2) from bp to the totalapplied surcharge sc.

For Range 2, as the soil stress at the extreme leftshifts from bp to sc, the neutral axis location alsomoves to the left. One can determine the location ofthe neutral axis at the point when the soil stress at theextreme right reaches sc

( )scbpbp

+=

Data points for the interaction curve in Ranges 3and 4 can be obtained by keeping the soil stress at theleft equal to

bp and incrementally shifting the location

of the neutral axis l to the left.From Fig. 1 (for Range 3) and geometry we can write

bp

scla

= allb =

Range 4

Range 3

Range 2

Range 1

300

Axial Load

400

200

0

600

500

200 Moment 0 100 300 400 500 Fig. 1: An example of a footing interaction diagram, showing the four different soilstress ranges for different axial force and moment load combinations

106 MARCH 2003 /

l

2

l

2

l

ll

bp

Stress at End of Range

Compressive

Stress at

Footing

Bottom

(upward

resistance)

2

l

l ll

l

2

l

bp

sc

a b

Range 1

Range 2

Range 3

Range 4 See Range 3 Figure.

reduced

up

Compressive Stress at

Footing Top

(downward resistance)

Stress at End of Range

Stress at End of Range

Stress at Start of Range

Stress at Start of Range

Stress at Start of Range

Stress at End of Range

Stress at Start of Range

2

l

2

l

bp

KEY:

Fig. 2: The four distinct stress distribution ranges of the soil under a footing

/ MARCH 2003 107

Dividing the total downward resistance into atriangular region at a and a rectangular region at b

sca aF 21

= scb bF = thus bat FFF +=

with a total compression force of:

lWF bpcomp 21

=

we can add forces to arrive at the resistance to axial load P

For convenience in obtaining an expression for themoment, we can locate the resultant of the twodownward forces by adding moments at the left edgeof the footing

Thus the moment that can be resisted by the footingwith a neutral axis location dl is

Two important considerations in footing design arethe factors of safety against uplift and overturning. Afactor of safety of 1.5 against uplift is easily obtainedusing the previous procedure and limiting sc to sc /1.5.

Unfortunately, the previous procedure does not lenditself to the determination of an ultimate overturningmoment. Traditionally, the factor of safety againstoverturning is computed

Such a computation assumes that at the ultimatecondition the soil beneath the footing will continue todeform plastically after reaching its ultimate stress.Applying this same assumption, the ultimate overturningmoment at any given axial load can be computed fromFig. 3. Given an axial load, l can be located

and the allowable ultimate moment computed

G46Interaction diagrams for footings may be obtained

by entering the preceding equations into a spread-sheet. The equations given here assume the footingweight, soil weight, and surcharge may be used toresist upward forces when evaluating the capacity of afooting. A factor of safety against uplift can be

Fig. 3: The ultimate overturning moment at any given axial load can be computed assuming thesoil beneath the footing will continue to deform plastically after reaching its ultimate stress

MPlSF

2=

scbp

scWlP

l

85.+

+=

( )lllWM bp 85.2185. =( ) ( )( )lllWllsc 85.2185. +

WllFllFMtcomp ]23

2[ + =

lF

blFalF

t

ba + +=

232

( )WFFFP bacomp +=

108 MARCH 2003 /

provided by reducing the value of sc used in Ranges2 to 4 by 1.5. Overturning stability can be achieved bycomputing the ultimate moment as given in the previousstability section and reducing it appropriately, typicallyby 1.5. If a spreadsheet is used to generate values of theinteraction curve for Range 1 through 4, the computedallowable axial load from the interaction diagram canbe substituted into the ultimate moment equations.The resulting moment can then be compared to themoment obtained from Ranges 1 through 4 using theequations from the interaction section. The lesser valuefrom these four ranges can then be plotted.

!!"Why not just include the surcharge in the axial load

and determine the footing capacity based on thestandard triangular pressure distribution? Simply put, itis not efficient to handle passive surcharges withtraditional methods. Passive surcharge does subtractfrom the available bearing pressure. If the designer is

not intent on considering passive resistance, then thismethod is not an absolute necessity. When a spread-sheet includes the preceding calculations, however, amethod of this form has other advantages. Such amethod is able to handle challenging and unusualcases. When input into a spreadsheet, the software candraw the interaction diagram and plot the loads as theyrelate to the interaction curve. Figure 4 is a screenimage of a spreadsheet that uses the methods dis-cussed previously. The designer has a better picture ofthe overall situation and can make rapid changes to theinput without performing lengthy hand calculations.

Though the method is complex, only one sitting isrequired to input it into a spreadsheet. The flowchartshown as Fig. 5 summarizes the procedure for inputinto a computer program or spreadsheet. Results fromthe spreadsheet are graphical and give the engineer amuch better feel for a design with complex loading.

Received and reviewed under Institute publication policies.

Fig. 4: Screen image of a spreadsheet that uses the procedure outlined in the article

/ MARCH 2003 109

ACI member Sam Eskildsen is a projectengineer with LBYD Inc. in Birmingham,AL. He is a graduate of Auburn Universityand a member of ACI Committee 355,Anchorage to Concrete.

Start

Vreduced = Vbp

Compute P and M:

AP reducedbp VV 2

1

WMreducedbp

2

VV

Incrementally reduce Vreduced

Is Vreduced in compression?

No- Start

Range 2

Yes

Compute P and M:

AP upbp VV 2

1

WMupbp

2

VV

Incrementally reduce Vup

Is Vup less than VSC

No- Start

Range 3 and 4

Yes

Compute

scbp

bp

VVV

G

Compute P and M:

WFFFP bacomp WllFllFM

tcomp]

232[

w E

Incrementally reduce G

Is G =0 No

Yes

Stop

Disclaimer: The information about the computer software

reported in this article is solely that of the author(s).

Publication here does not represent endorsement of the

software, nor of author(s) claims about it, by this magazine

or by the American Concrete Institute; nor have any of the

American Concrete Institute staff tested or used the

software mentioned. This article is provided as information

only to our readers, and it is urged that users of the

software cited exercise proper and sufficient technical and

other necessary knowledge when testing and applying the

mentioned software. For any additional information about

the software, it is recommended that the author of the

article be contacted directly.

Fig. 5: Flowchart summarizing the procedure for input into a computer program or spreadsheet