Materials Engineering 1995, Vol. 6, n. 1-2 pp. 197-210

STRUCIURAL DESIGN EQUATION FOR FRP COLUMNS

Ever Barbero

Assistant ProfessorMechanical and Aerospace Engineering

Member ASME, SAMPE, ASCE, ASTM - Constructed Facilities CenterWest Virginia University - Morgantown, WV 26506-6101

(304) 293-3111 ext. 337

ABSTRACT: A design equation for fiber reinforced plastic columns is presented in this paperbased on the interaction between local (flange) and global (Euler) buckling observed duringtesting of the FRP columns included in this investigation. An existing interaction equation isadapted to account for the modes of failure observed in columns made of fiber reinforcedcomposite materials. Experimental data generated during this investigation is presented andused to validate the interaction equation and to obtain the interaction constant. A slendernessratio is proposed and used to present a plot of buckling for all sections and column lengths(short, long, and intermediate). An'expression for the optimum column length to be used inthe experimental determination of the interaction constant is proposed.

1. INTRODUCTION

Pultruded FRP beams and columns are being used for civilengineering structural applications[l.2.3] because of their superiorproperties compared to conventional materials (steel, concrete, wood,aluminum, etc.), such as light weight and high corrosion resistance.FRP materials can be cost competitive with conventional ones when

Paper presented at the International Seminar ,«The application of fiber reinforced plastics(FRP) in civil structural engineering». October 22-23 1993, Bologna - Italy

197

mass production methods like pultrusion are used to reduce their cost.In the pultrusion process, fibers impregnated with a polymer resin arepulled through a heated die that provides the shape of the cross-sectionto the final product. Pultrusion is a continuous process for manufac­turing of prismatic sections of virtually any shape[4].

Pultruded structural members have open or closed thin-walledcross-sections. For long composite columns, global (Euler) buckling isexpected to occur before any other instability failure. The bucklingequation has to account for the anisotropic nature of the material, andtheoretical predictions correlate well with experimental data for longcolumns(5J. Local buckling occurs first for short columns leading eitherto large deflections and finally to overall buckling, or to materialdegradation due to large deflections (crippling). Because of the largeelongation of failure allowed by both the fibers and the resin, the FRPmaterial remains linearly elastic for large deflections and strains unlikeconventional materials that yield (steel) or crack (concrete) formoderate strains~ Therefore, buckling is the governing failure for thistY1?_~C?f ,cross-sections and the critical bucklin~ load is directly relatedwith the load carrying capacity of the member.

The experimental data for short and long column buckling(5.6.7,&Jsuggest the existence of an intermediate-column region where thecritical loads are lower than the predictions of both local and globalbuckling theories. Since fiber reinforced plastics (FRPs) remainlinearly elastic for large values of strain, buckling in the intermediaterange occurs due to mode interaction(9) between the local (flange) andglobal (Euler) buckling modes rather than between local buckling andyield as in the case of steel columns. In this paper, a phenomenologicaldesign equation is presented to represent the buckling envelope forpultruded composite I-beams. The formulation accounts for the in­teraction between local (flange) and global (Euler) buckling modes., Anadjustable interaction parameter is used to also account for variousmaterial imperfections and interaction levels apparent in pultrudedmembers. An interaction constant, determined through a series of testsdescribed in this paper, is used to account for the buckling mode in­teraction behavior observed experimentally.

2. INTERACTION; EQUATION

Column design equations for wood members are usually viewed asempirical equations for the compressive strength as a function of slen-

198

derness[lO). In steel columns, interaction takes place between globalbuckling and yield. In timber columns, column failure is described bythe interaction between crushing and buckling. The following equationis widely used

r + a == 1 + cra (1)

qactual failure load

local buckling load

actual strength=---------local buckling strength

(2)

actual failure load actual strengtha =--------=---------Euler buckling load ,.. Euler buckling l.strength

to model the interaction(ll]. The parameter c can be viewed as con­trolling the amount of interaction between the two failure modes(Fig. 1). The parameter c accounts for the effects of nonlinear com­pression and inhomogeneity of the material and may also be used toinelude the-eff-ectofcrookedness as-long--asthe-columnte-stspec-imensare randomly selected from an appropriate sample of columns.

Note that by the definition of r and a, linear interaction is definedas r + s = 1 which is a conservative approximation for design pur­poses. Equation (1) models linear interaction if c = 0 and the nonin­teraction case if c == 1. Interaction between local and global bucklingresults in an intermediate value of c, which is found from experiments.

A universal slenderness ratio for pultruded I-beam columns canbe defined as

~Ll=kL --

D1C2

where k is the effective length coefficient, L is the column length, D isthe bending stiffness and PL is the theoretical local buckling load. Theeffective length coefficient k is given by analytical formulation forcolumns with various end conditions found in many strength ofmaterials textbooks and design manuals. Since the global bucklingload is PB = 1C

2 D / L:l, and using Eq. (2) we obtain PB == PL / A2 whichsubstituted into Eq. (1) gives

C 12 tf - (1 + 12) q + 1 == 0 (3)

199

,0.2 0.4 0.6 0.8

s = P / PEuler

1

oo

0.8

~(J C'.2

Q... 0.6"- C'

~Q... '0

II 0.4C'

0-~

~

0.2

Figure 1 - Various values for the interaction constant c.

The root of this equation is

q = Pa- = 1 +1/11_4/( 1 + 1/1

2)2 1P

L2c V 2 c - cl1

(4)

which represents actual values of q determined experimentally anddescribed by the interaction parameter c, with 1 given by Eq. (2).Therefore, Eq. (4) can be used as a design equation over the entirerange of column slenderness, short, intermediate, and long. The ben­ding stiffness D of a FRP column is computed from the informationprovided by the manufacturer for each section following the

200

1 1/2 oz OC

~:oz OC

Roving 22 (61 yield)oz OC

Roving 22 (61 yield)1/2 OCoz

00

~ ~

32 :2.....~ .~

~ ~

~ ~

lO to'--" '--"

~ ~

tn If'")

0) 0)E .5.;: >~ ~ /: V2 oz OC

oz OC00 2 oz OC

Roving 22 (61 yield)

~1 1/2 DCozRoving 22 (61 yield)

Figure 2· Creative Pultrusions schematic for the 152 mm x 152 mm x 6.4 mm (6" x 6" x 1/4")WF I-beam.

methodology developed by Barbero(12,13) for pultruded compositebeams. This information includes the type of fibers and matrixmaterial, the location and orientation of the fibers and the fiber con­tent in the cross-section (Fig. 2). Table (1) shows the flange and webbending stiffness components for a 102 x 102 x 6.4 mm and 152 x 152x 6.4 mm pultruded H-beams, respectively, for bending about theweak axis. The local buckling loads were obtained experimentally(1).The. maximum interaction between local and global buckling occurswhen q = s. Using A = 1 into Eq. (2), the column length for whichmaximum interaction occurs is

201

Table 1. Weak axis bending stiffness components oftbe flanges and .web for eacb of tbeR-beam sections tested (note: DI , = D. = 0 for eacb section).

102 x 102 x 6.4 mm 152 x 152 x 6.4 mm

Stiffness Flanges Web Flanges Web

Component (MN-em) (kN-em) (MN-em) (kN-em)

D ll 13.176 43.720 30.295 46.217

D22 4.958 20.747 11.628 21.052

D 12 1.878 8.181 4.435 8.267

D66 1.652 6.603 3.852 6.736

(5)

Table (2) shows the value ofL· along with the local buckling loadand the weak axis bending stiffness D for each H-beam section con-

202

Table 2. Maximum InteraetioD lengtb L* for eacb pultnlded H-beam section.

IL*

I

PJocaJ

ID

I

(em) (kN) (MN-em2)

102 x 102 x 6.4 105.9 223.25 253.71

152 x 152 x 6.4 221.5 175.12 872.31

sidered in this paper, which corresponds to a length in which maximuminteraction between local and global buckling is expected. Therefore,experimental data to obtain the interaction constant should begathered at this length.

Using the universal slenderness ratio and the ratio of q = Pcr / PL

the theoretical buckling envelopes for all H-beam sections can becollapsed into one universal curve (shaded area in Fig. 3).

3 EXPERIMENTAL RESULTS

The intermediate column tests were performed on the followingpultruded H-beams: 152 x 152 x 6.4 mm and 102 x 102 x 6.4 mm. Alarger number of tests were done on the 152 x 152 x 6.4 mm H-beambecause it has a wider range of lengths where interaction occurs. Thetesting procedure for the intermediate column lengths consisted ofloading the specimen with controlled axial displacement. The load, theaxial displacement, and the transverse deflection at the midspan weremeasured by a load cell and two LVDTs. The data was fIltered fromnoise and recorded using a data acquisition system and a movingaverage technique(14]. The test was continued until the ultimate load ofthe column was achieved, recorded with a peak indicator on theMaterials Testing System (MTS) control console. The data reductionconsisted of obtaining a maximum load the column was capable ofsupporting recorded by the peak indicator on the MTS control con­sole.

Since the span of the intermediate column range had not been

203

432

A

1o

o

1.2

~ Buckling Envelope

0.9

Ci0.2

CL

0.6 c = 0.84"'-

L.0

CL

0.3

Figure 3 - Universal buckling envelope accounting for local-global buckling interaction.

previously established, the exact column lengths in the intermediaterange were unknown. Hence, the testing lengths were taken betweenthe longest local buckling test available and the shortest globalbuckling test previously performed(s.lsJ• All tests were conducted withthe weak axis of the section in the pinned-pinned configuration. Alarger number of samples correspond to lengths close to the predictedL*, only one test was done while at lengths sufficiently far from L*.The length, theoretical local buckling load, theoretical global bucklingload and maximum experimental load for each length tested are shownin Table (4) for a 152 x 152 x 6.4 mm section. A significant decrease incritical load is obtained around the value. of L *; which indicates in­teraction between the local and global buckling modes. This interac-

204

Table 3. Experimental Intermediate test data OD the 151 x 151 x 6,4 mm H-beam.

Length Local Global Experimental

(em) (kN) (kN) (kN)

144.8 175.33 418.00 174.34

175.3 175.33 280.23 148.99

175.3 175.33 280.23 157.23

175.3 175.33 280.23 151.15

175.3 175.33 280.23 156.45

205.7 175.33 203.38 136~83

205.7 175.33 203.38 133.06

205.7 175.33 203.38 138.18

236.2 175.33 152.64 116.67

236.2 175.33 152.64 115.40

236.2 175.33 152.64 120.10

266.7 175.33 121.03 99.23

297.2 175.33 97.48 78.19

327.7 175.33 80.18 67.13

Table 4. Experlmeotallotermedlate test data OD the 101 x 101 x 6,4 mm H-beam.

Length Local Global Experimental

(em) (kN) (kN) (leN)

114.3 223.52 191.68 171.08

114.3 223.52 191.68 161.98

147.3 223.52 115.39 98.53

147.3 223.52 115.39 95.29

177.8 223.52 79.21 71.58

tion was apparent during the test in which local flange buckling oc­curred in combination with lateral deflection.

205

1 //

//

0.8 ESJ //

0

a1 C = 0.84 0

0 00 /

a.. 0.6 /./

"- /

a../

* /0.4 y/ 0

II// /

C) y//

152 152 x 6.40'.2 / 0 x/ 0 102 x 102 x 6.4

//

00 0.2 0.4 0.6 0.8 1

s P / PEulerFigure 4 - Nondimensional q VI. S plot for the combined interaction test data (c = 0.84).

The length, theoretical local buckling load, theoretical globalbuckling load, and maximum experimental load for each columntested are shown in Table (5) for a 102 x 102 x 6.4 mm section. Adecrease in the experimental load, when compared to theory, is ex­perienced indicating interaction between the local and global bucklingmodes. It was apparent during the test that both local buckling of theflaIlges and global buckling occurred simultaneously. These obser­vations can be made because the test is not as catastrophic as a resultof controlling the axial displacement with the hydraulic servo­controlled MTS.

206

54321o

o

1.2c 152 x 152 x 6.4

- - -~ -, 0 102 x 102 x 6.4,0.9

,,,,«1 d, c = 0.840 0,0

\a..0.6

,Euler Data\

"-\'\,

a.. \\

0.3

Figure j • Nondimensional q vs. A. plot with intermediate and global buckling test data (c =0.84).

4. DETERMINATION OF INTERACfION CONSTANT

The experimental data is plotted in the interaction diagram shownin Fig. (4). From this figure, symmetry appears to exist between the ex­perimental failure loads. Thus, the use of Eq. (1) as an interactionequation appears to be valid. Using the data in Tables (3) and (4), theinteraction constant c was determined for the pultruded H-beam sec­tions. For each sample, the value of the interaction constant c wasfound by solving Eq. (1) for c

q + s -1c =

qs

207

The interaction constant c for each type of cross-section was in­ferred from the experimental data by averaging the value of c of allsamples, resulting in c = 0.84. For the 152 x 1526.4 mrn section, an in­teraction constant of c = 0.85 was obtained. Similarly, for the 102 x102 x 6.4 mm, an interaction constant of c = 0.83 was obtained. Witha value of the interaction constant c = 0.84, the interaction equationfits quite well with the experimental data obtained from- the inter­mediate test.

Using the interaction constant c to approximate the interactionbetween local and global buckling, an overall buckling envelope isshown by solid line in Fig. (5). Thus, it is proposed to use Eq. (4) as adesign equation for the buckling of pultruded composite H-beams.The buckling envelope shown by a solid line in Fig. (5) is valid for anypultruded H-beam section (similar to the sections used in this in­vestigation) provided the values of the bending stiffness D, the localbuckling load PL and the interaction constant c are known. Also shownin the figure is additional experimental data obtained by Barbero andTomblin(5] in the global column range. As seen in Fig. (5), the designequation closely approximates the value of the·failure load obtained inexperiments.

Interaction is very pronounced for columns with-lengths closer toL *, as seen in the experiments. In this region, the column failure oc­curs and rapidly inducing large deformations and material failure(fiber breakage and delaminations). The importance of the interactioncurve is evident from a design viewpoint. If the interaction value werenot used, approximately a 25070 difference between the theoreticallypredicted local buckling loads and the actual column failure loadwould be experienced at L*. Furthermore, the proposed designequation is simpler to use than an Euler equation for long columns anda local buckling equation for short columns.

7. CONCLUSIONS

The proposed design equation predicts well the actual criticalloads of FRP columns of the type used in this investigation. The valueof the interaction parameter c = 0.84 provides a good estimate of theinteraction between local and global buckling for the columns tested.The buckling envelope for pultruded composite H-beams shows goodcorrelation with the experimental data developed in this investigation.The interaction curve and design envelope presented- provides a basis

208

for the design and use of pultruded structural members in engineeringapplications. Interaction of local and global buckling modesdominates the intermediate range of lengths of FRP columns used inthis investigation. The definition of universal slenderness 1 andbuckling strength q proposed leads to a simple d~sign equation to beused for the buckling of FRP columns. Since the proposed designequation is based on a phenomenological approach, the interactionconstant must be determined experimentally for each new sectionfollowing the- procedure presented in this work. The validity of theproposed design equation for other types of sections not included inthis investigation should be verified by additional experimentation.

8. ACKNOWLEDGEMENTS

The financial support and materials provided by CreativePultrusions, Inc., is appreciated. The financial support of WestVirginia Department of Highways (Grant RPT699) and National­Science Foundation (Grant 8802265) is aknowledged.

REFERENCES

[ 1] GangaRao, H.V.S. and Barbero, E.l. (1991). Structural Application 01 Composites inConstruction, Encyclopedia on Composites, 6, 173-187, Ed. S. Lee, VCH Pub.

[ 2] Barbero, E.l. «Construction: Applications and Design», Lubin's Handbook of Com­posites, 2nd Ed., Stan Peters Ed., to appear.

[ 3] Barbero, E.l. and GangaRao, H.V.S. «Structural Applications of Composites in In­frastructure»), SAMPE Journal, Part I, 27(6), 9-16, 1991, Part II, 28(1), 9-16, 1992.

[ 4] Creative Pultrusions Design Guide (1988), Creative PultrusioDS, Inc., Pleasantville In­dustrial Park, Alum Bank, PA.

I 5] Barbero, E.l. and Tomblin, J.S. (1992), «Euler Buckling of Thin-Walled CompositeColumns,» Thin-Walled Structures, 17, 237-258, 1993.

[ 6] Raftoyiannis, I.G. (1991), Buckling 01 Pultruded Composite Columns, MSCE Thesis,Department of Qvil Engineering, West Virginia University, WV.

[ 7] Tomblin, J.S. and Barbero, E.l. (1992), «Local Buckling Experiments on FRP Colum­- ns,») Journal oj Thin- Walled Structures, In Press.

[ 8] Yuan, R.L., Hashen, Z., Green, A., and Bisarnsin, T. (1991), «Fiber-Reinforced PlasticComposite Columns», Advanced Composite Materials in Civil Engineering Structures,Ed. S.L. Iyer, ASCE, lan. 31 - Feb. 1, Las Vegas, 205-211.

[ 9] Raftoyiannis, I.G., 1994, Buckling Mode Interaction in FRP Columns, Ph.D. Disser­tation, West Virginia University, January.

[10] Zahn, 1.J. (1990), «Interaction of Rupture and Buckling in Wood Members,» ASCEJournal 01 Structural Engineering.

(11] Ylinen, A. (1956), <<A Method for Determining the Buckling Stress and the RequiredCross-Sectional Area for Centrally Loaded Straight Columns in Elastic and InelasticRange,») Publications 01 the International Association lor Bridge and StructuralEngineering, Zurich, Vol. 16.

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[12] Barbero, E.J. (1991), «Pultruded Structural Shapes - From the Constituents to the Struc­tural Behavior,» SAMPE Journal, 27 (1), 25-30.

[13] Barbero, E.J., Lopez-Anido, R., and Davalos, J.F., «On the mechanics of thin-walledlaminated composite beams,» JoumiJI 01 Composite Materia/s, 27(8), 806-829, 1993.

(14] Labtech Notebook (1990), (version 6), Laboratory Technologies Corporation,Wilmington, MA.

[lS] Tomblin, J.S. (1991), «A Universal Design Equation for Pultruded Composite Colum­ns,» MSME Thesis, Department of Mechanical and Aerospace Engineering, WestVirginia University, WV.

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