STEEL COMMITTEE OF CAUFORNIA

TECHNICAL INFORMATION & PRODUCT SERVICE

NOVEMBER 1990

Design of Small Base Plates for Wide Flange Columns*W. A. THORNTON

The 9th Edition• of the AISC Manual of Steel Construc-tion uses the Murray-Stockwell2 method for analysis ofsmall base plates, i.e., plates that are only slightly larger thanthe column depth d and width bf. It combines this methodwith the cantilever method of the 8th3 and earlier editionsfor large base plates. The Murray-Stockwell method assumesa bearing pressure of Ft,, the maximum permitted, over anH-shaped contact area under the column cross-sectionbetween the plate and the concrete. The cantilever method,on the other hand, assumes a uniform bearing pressure, fp< Fp, over the entire base plate surface of area BxN(Fig. 1). Thus, the two methods assume very different bear-ing pressure distributions and are difficult to combine intoa single method.

A solution to this dilemma is to return to the 8th Editionassumption of uniform pressure between the base plate andthe concrete. This assumption is conservative with respectto the base plate thickness determination because the truepressure distribution will be less near the plate edges andmore under the column cross-section, which cross-sectionalso provides support for the plate at its top surface. Sincethe plate is assumed more heavily loaded distant from its

A. Thornton, PhD, PE, is chief engineer, Cives Steel Com-pany, Roswefi, GA, and is chairman of AISC Committee onManual, Textbooks, and Codes.

supports than it will be, a plate thickness determined underthis load will be thicker than it needs to be.

To supplement the cantilever method for large base plates,which is actually a yield line method, it is consistent againto use yield line theory applied to the portion of the baseplate contained within the column depth and width. Hap-pily, exact solutions to this problem are available in the liter-ature.4 Consider Fig. 2, which shows a plate supported onthree edges and free on the fourth. The dimensions of theplate are taken as the column depth d and the half columnwidth bfi2, rather than the more correct d - 2tf and (bf -t,.)/2. This is done for simplicity and is conservative. If thethree supported edges are taken as completely fixed, i.e.,no displacement and no rotation about an axis parallel to eachedge, the required base plate thickness with a factor of safetyof 2 is

tp = o.t,j (1)

whereft, = uniform pressure between base plate and concrete

= P/BxN, ksiF.,. -- yield stress of base plate, ksi

G ,, f 3 G - l•--6-G- +I'•

where r/ = d/bf

Reproduced from AISC Engineering Journal, Volume 27, No. 3, 3rd Quarter 1990

dIOr

/

m

.95Or

m ,

NUnsuppo•ed

Edge

b

SuppoSedEdge

Fig. I. Column base plate geometry and symbols (from AISC'). Fig,. 2. Small base plate geometry and support conditions.

The expression for et given in Eq. 2 can be approximated by

et = 4 • (3)

with an error of -2.97 % (unconservative) to +6.00% (con-servative) in the range of ?7 from • to 3. Then, Eq. I becomeswith Eq. 3

1where • has been replaced by • with an error of 2%.

Combining Eq. 4 with the cantilever method for large baseplates, let

n'= (5)

and

I = max(m,n,n) (6)

where m and n are defined in Fig. 1. Then the required platethickness is

tr = 2 t J (7)

If the base plate is small with N • d, it may be unconser-vative to assume complete fixity of the base plate to the col-umn flanges. If the plate of Fig. 2 is completely fixed to thecolumn web along the side of length d but simply supported,i.e., no displacement but rotation unrestrained, along thesides of length bf/2, the required base plate thickness witha factor of safety of 2 is given by Eq. 1, with

•,2,•: + l,/--7•--lJ (8)

This expression for et can be approximated bY

et = '/2,J-• (9)

with an error of -0% (unconservative) and +t7.7% (con-servative) in the range of ,/from g to 3. In the more com-mon range of g _< ,/ < 2, the error is only +8.00% (con-servative). Using Eq. 9 in Eq. 1,

= 2(U•q•j) J (10)t,

Combining Eq. 10 with the cantilever method for large baseplates, let

n ' = i,• (11)

I = max(m,n,n') (12)

2

/Z_tp = 2 1 • (13)

The formulation for the two models just discussed can beseen to be exactly the same except for n'. Let the first for-mulation, for which n' = be referred to as Model 1and the second, with n' = 'Ax/db/be referred to as Model2. It will be instructive to see how these two models com-pare with a method suggested by Ahmed and Krepss andthe method of the AISC 8th Edition Manual. To this end,consider Table 1. The nine examples of this table show thatboth Models 1 and 2 produce plate thicknesses !ess than orequal to the method of the AISC 8th Edition. The methodof Ahmed and Kreps produces plate thicknesses betweenModels 1 and 2 for small base plates of square columns, buttends to produce plates too thick for nonsquare columns(T/ > 1), as seen from Examples 7, 8 and 9. In the case ofExamples 8 and 9, it produces plates thicker than the 8thEdition method.

Considering the results shown in Table 1, and recognizingthat Model 2 is clearly conservative while still producingplates thinner or at most as thick as the method of the AISC8th Edition Manual, it is recommended that Model 2, i.e.,

n' = I,•Xf'•f

I = max (m,n,n')

t, = 2l•,.

be used to replace the current AISC 9th Edition Manual baseplate design method for axial load.

The equivalent Load and Resistance Factor Design (LRFD)equation for base plate thickness is:

I,,[ 2Put, = •l 0.9F, BN (14)

whereP,, = total factored column load

NOTATION

The symbols used in this paper follow the usage of the AISCManual, 8th or 9th Edition.

REFERENCES

1. American Institute of Steel Construction, Manual of SteelConstruction, 9th Edition, 1989, pages 3-106 through3-110.

2. Murray, T. M., "Design of Lightly Loaded Column BasePlates;' AISC Engineering J., Volume 20, No. 4, 4thQuarter, 1983, pp. 143-152.

3. American Institute of Steel Construction, Manual of SteelConstruction. 8th Edition, 1980, pp. 3-99 through 3-102.

4. Park, R. and Gamble, W. L., Reinforced Concrete Slabs,Wiley, 1980, pp. 329-331.

5. Ahmed, S. and Kreps, R. R., "Inconsistencies in Col-umn Base Plate Design in the New AISC ASD (July 1989)Manual, AISC Engineering J., 3rd Quarter, 1990, pp.106-107.

6. DeWolf, J. T., and Ricker, D. T., Column Base Plates,AISC Steel Design Guide Series, No, 1, 1990, pp. 13-15.

7. Fling, R. S., "Design of Steel Bearing Plates/' AISCEngineering J., Volume 7, No. 2, 2nd Quarter, April 1970,pp. 37-40.

T a b l e 1.E x a m p l e s To C o m p a r e M e t h o d s (Fy = 36 ksi for al l c a s e s )

Data n'/tp(in.lin.)

Col. P d bt N B fp m n Mod. Mod. Ahmed & AISCExample Source Sect. (kips) (in.) (in.) (in.) (in.) (ksi) (in.) (in.) I 2 Kreps 8th Ed.

1. AISC Des. Guidea W 1 0 x l 0 0 200 11.10 10.34 11.5 11 1.58 .48 1.36 2.14 2.68 2.33 3.92.90 1.12 .98 1.64

2. Ahmed ,• Krepsb W12x106 331 12.89 12.22 14 13 1.82 .88 1.61 2.51 3.14 2.71 4.771.13 1.41 1.22 2.15

3. -- W12x106 300 12.89 12.22 14 13 1.65 .88 1.61 2.51 3.14 2.71 4.771.07 1.34 1.16 2.04

4. -- W12x106 300 12.89 12.22 16 16 1.17 1.88 3.11 2.51 3.14 2.71 4.771.12 1.13 1.12 1.72

5. AISC 8th Ed. W 1 0 x l 0 0 525 11.10 10.34 19 17 1.63 4.23 4.36 2.14 2.68 2.33 3.921.86 1.86 1.86 1.86

6. AISC8thEd. W12x106 600 12.89 12.22 18 16 2.08 2.88 3.11 2.51 3.14 2.71 4.771.50 1.51 1.50 2.29

7. Flingc 1 4 x 8 W F -- 14 8 -- -- .75 -- -- 2.12 2.65 2.94 3.68.61 .77 .85 1.06

8. -- W 2 4 x 6 8 · 450 23.73 8.965 24 9 2.08 -- -- 2.92 3.65 4.98 4.231.41 1.76 2.40 2.04

9. -- W36x160 1000 36.01 12.00 38 14 1.88 1.90 2.20 4.16 5.20 7.56 5.631.90 2.38 3.46 2.57

a. See Ref. 6b. See Ref. 5c. See Ref. 7, Fling gets tp = 0.711 in. for this example

I

3

Inconsistencies in Column Base Plate Designin the New AISC ASD Manual*SALAHUDDIN AHMED and ROBERT R. KREPS

The new AISC steel design manual (ninth edition)' sug-gests a new procedure for computing the thicknesses of col-umn base plates to rectify problems associated with the some-what conservative design approach adapted in its earlierversion. However, a close scrutiny of the suggested methodreveals that the new approach is sometimes overly conser-vative and even inconsistent.

Referring to Fig. 1 (pg. 3-106 of the AISC Manual),

P = Total column load, kipsAj = B x N = Area of plate, in.2

A, = Full cross-sectional area of concrete support, in.'Fh = Allowable bending stress in base plate, ksiv = Allowable bearing stress in support, ksi

fp = actual bearing pressure, ksif,' = Compressive strength of concrete, ksitp = Thickness of plate, in.

$alahuddin Ahmed, Ph.D., is structural engineer, LeonhardtKreps LeFevre, Toledo, Ohio.Robert R. Kreps, P.E., is principal, Leonhardt Kreps LeFevre,Toledo, Ohio.

Referring to page 3-108 of the Manual, the following proce-dure is followed to compute base plate size:

For a given P, f! and A2, minimum area of base plate iscomputed and reasonable values of B and N are selected.

Based on the column dimensions and selected B and N, quan-tities m and n are computed and the larger of the two con-trols. In the next step, the value of L (Fig. 2) is computedfrom the following expression, Fv = P/(2 + d + b -

2L)/L, which is quadratic in L. Solving for L,

L = [(d + b) _+ x/((d + /i)2 _ 4P/Fp)]/4

The Manual is silent as to which of the two solutions shouldbe used in further computation. However, a careful studyof the equation reveals that the smaller of the two L values

should be used. The required base plate thickness is thencomputed based on the larger of m and n calculated and thevalue of L, as described in the Manual.

The quantity L is computed based on an area with a pres-sure of Ft, and not fy. Thus it is not clear why fi, is used inthe expression t = Lx/(3fv/Fh) (pg. 3-107 of the Manual).

dllri

b

i i i l.8Ob.

4, '•'

N

, r D

r I

e L - - , T -

2L

I, I / l , - . , ×z [

Figure I Figure 2

· - Reproduced from AISC Engineering Journal, Volume 27, No. 3, 3rd Quarter 1990

4

Example

Let

P = 331 kipsColumn W12x106 (d = 12.89, b = 12.22)

f,'. = 3 ksiF,. = '36 ksiPier: 34 in. x 34 in.

1. A• = (1/(34 x 34))(331/(.35 x 3))" = 86

A• = 331/(.7 x 3) -- 158 Controls2. A = .5[.95 x 12.89 - .8 x 12.22] = 1.235

N = • + 1.235 = 13.8, use 14 in.

B = 158/14 = 11.3, use 13 in.A• = 14 x 13 = 182 in.-'

3. fv = 331/182 = 1.82 ksi4. m = [14 - 0.95 x 12.89]/2 = 0.88 in.

n = [13 - 0.80 x 12.22]/2 = 1.61 in. Controls

Fv = .35 x 3 x (34 x 34/182) = 2.65 ksi > 2.1 ksi,

use 2.1 ksiL =

= (25.11 - x/-•.036)/4 = 6.23 in.

tv = 1.61 x x/(1.82/(.25x36)) = 0.72 in.tt, = 6.23 x x/(3xl.82/27) = 2.80 in. Controls

It may be noted that the thickness of 2.80 in. is greaterthan what would be obtained according to the eighth edition

AISC Manual.Now if one repeated the same calculations with a load of

332 kips, L would become imaginary and as per the manualwould be ignored. As a result the required thickness wouldbe 0.72 in., less than that required for a lighter load.

Therefore the authors feel that the new way of computingL is basically inconsistent and likely to result in too thicka base plate when L controls and too thin a base plate whenL is imaginary and thus ignored.

SUGGESTED METHOD OF ANALYSIS

Let us assume that the pressure under the base plate is uni-form and is equal to P/Al. Let us also assume that the plate

is essentially fixed at the web and flanges of the column.Thus what we have here is a plate with one long and twoshort edges fixed and the fourth edge free with a uniform

load. One can go back to various moment coefficients avail-

able in the literature to compute maximum moment in theplate. Considering the width to length ratios of usual col-umn sections, the authors suggest a moment coefficient of

0.022 so that the maximum moment in the plate is 0.022 ×fp × d2 kip-in./in., where d is the depth of the column.

Sr•qa. = 0.022 x fp x d2/Fht = V(6S,e4a.) = V(O. 132fpd2/Fb) (1)

Therefore, to compute the base plate thickness,

a) Compute m and n as discussed in the Manual and select

a thickness based on the larger of the two.

b) Use the larger of the two thicknesses obtained in step

(a) and by Eq. 1.Applying this method to the example above,

a) Compute thickness to be 0.72 in. for the larger of mand n.

b) Use Eq. I for t = 4(0.132 × 1.82 × 12.892127)= 1.22 in. Controls

REFERENCES

1. American Institute of Steel Construction, Inc., Manualof Steel Construction, Allowable Stress Design, ninth edi-tion, Chicago, IL, July 1989.

2. Winter, Urquhart, O'Rourke, Niison, Design of ConcreteStructures, seventh edition, McGraw-Hill Book Company,New York.

This publication expresses the opinion of the author, and care has been taken to insurethat all data and information furnished are as accurate as possible. The author andpublisher cannot assume or accept any responsibility or liability for errors in the dataor information and 'in the use of such information.

The information contained herein is not intended to represent official attitudes, recom-mendations or policies of the Structural Steel Educational Council. The Council is notresponsible for any statements made or opinions expressed by contributors to thispublication.

5

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