Numerical Investigation of Shear Buckling and Post-buckling of Thin Steel

Plates with FRP Strengthening

Mohamad Alipour1, Alireza Rahai

2, Devin K. Harris

1

1Department of Civil and Environmental Engineering, University of Virginia,

Charlottesville, VA, USA

2Department of Civil and Environmental Engineering, Amirkabir University of

Technology (Tehran Polytechnic), Tehran, Iran

ABSTRACT

The behavior of thin steel plates under shear loading is governed by early diagonal

buckling and subsequent formation of an inclined tension field. This behavior

describes the load carrying mechanism in deep steel plate girder webs and steel plate

shear walls. Traditionally, steel stiffeners are used on the thin plate in some

applications to mitigate out-of-plane buckling and encourage shear yielding, which is

preferable and a more stable and energy-dissipating load carrying mechanism.

However, the cost and practical difficulties associated with welding these stiffeners,

especially on very thin plates and for in-service rehabilitation purposes, are

considered as major drawbacks.

In this paper, the behavior of thin steel plates strengthened with FRP strips was

numerically investigated. FRP wraps were hypothesized to act as an elastic support

for the thin steel plate in the early loading stages and as an auxiliary load transfer path

in the inelastic tension field action. Numerical investigations were carried out using

the finite element method and were divided into two phases; buckling and post-

buckling. Elastic eigenvalue analysis for buckling and full inelastic analysis with

geometrical and material nonlinearity for the post-buckling phase were carried out.

FRP fracture and damage were incorporated in the models. Optimum angle of FRP

fibers was studied both for buckling mitigation and post-buckling behavior

enhancement. A number of FRP strengthening configurations together with a range of

common FRP materials were also employed in the analyses. It was concluded that

bonding FRP patches on the steel plate can effectively encourage behavior

enhancements, especially given appropriate configuration and material properties.

KEYWORDS:

FRP Composites, Shear Buckling, Post Buckling, Steel Plate Shear Wall, Finite

Element Method (FEM)

INTRODUCTION

Steel and Composite Plate Shear Walls

Steel plate shear wall (abbreviated hereafter as SPSW) is a lateral load resistant

system composed of a very thin steel plate connected along its four sides to two

columns and two successive beams. SPSW systems are internal structural

components used to resist lateral loads exerted by wind or earthquake. In this

configuration, the plate is loaded in shear and will buckle similar to a slender plate-

girder web. A post-buckling phenomenon will form an inclined tension field which

accounts for the system shear strength as schematically depicted in Figure 1. The

inclined tension strips formed along the dominant principal tension brings to mind the

possibility of using an orthotropic material to strengthen this mechanism.

Figure 1. Inclined tension field in SPSW

Although the system has been proven to have acceptable behavior, a number of issues

emerge. Plate instabilities restrain energy dissipation of the system, which is

demonstrated as pinching of the hysteresis curves. A thick or stiffened plate will yield

in shear which is a more ductile behavior and increases energy dissipation. On the

other hand, buckling-induced out-of-plane deformations of the thin plate may cause

non-structural damage. The inclined tension field also exerts strong inward forces on

beams and columns compared with a thick plate, which yields in shear before

buckling happens.

In a series of cyclic tests on SPSWs, Hatami et al. (2012) and Hatami and Rahai

(2008) tested three one-story SPSW specimens with a 3-mm thick steel plate. In one

of the models, the steel infill plate was made composite by attaching a 0.176-mm

thick CFRP layer using epoxy resin. Comparing the acquired data, the researchers

observed less damage in the retrofitted specimen along with rupture of some bolts

connecting the plate to the boundary members. They also found the FRP layer

responsible for a 37% increase in energy dissipation and 50% in lateral stiffness. It

was stated however that FRP bonding issues had decreased the ductility of the wall by

8%.

Alipour (2010) carried out a numerical investigation on buckling and post-buckling

behavior of FRP-composited steel shear walls. The buckling behavior of a steel infill

plate under pure shear was studied using elastic eigen-value analysis and its post-

buckling behavior was simulated using plasticity for steel and progressive damage for

FRP materials. Both the buckling and post-buckling analyses demonstrated the ability

of the FRP layer to act as an elastic support for the thin steel plate (Alipour, 2010).

Rahai and Alipour (2011) also investigated the fiber direction and geometry for FRP-

composited steel shear walls using the finite element method. It was found that the

optimum direction for fibers is the direction of tension field in the plate as can be

calculated using an energy-based formula (Rahai and Alipour, 2011). Tabrizi and

Rahai studied FRP strips as edge reinforcement for perforated SPSWs and reported

promising results (Tabrizi and Rahai 2011). The current paper builds on the

experiences from these numerical analyses and seeks to investigate the buckling and

post-buckling behavior of the SPSW system with different configurations of FRP

strengthening.

METHOD OF ANALYSIS

In this investigation, a step-by-step numerical analysis was used to describe the global

behavior of a FRP-strengthened SPSW. First, a SPSW was designed according to

AISC provisions. Numerical simulation of the model is then carried out using the

finite element method. Two sets of parametric analyses were then carried out on the

model. First, the shear buckling behavior of the system is studied using linear elastic

eigen-value analyses. Then, a series of nonlinear analyses incorporating steel

plasticity and fracture in FRP materials were conducted.

Model Properties

The steel plate shear wall model of this study was designed using the guidelines

outlined in the AISC seismic provisions for structural steel (AISC 2005). According

to these provisions, the design of a SPSW is considered acceptable when the infill

plate undergoes considerable yielding prior to yielding in the boundary members.

Moreover, with an increase in story drift, yielding must occur in the ends of the

beams to ensure a safe load path for the gravity loads (plastic hinge formation). For

sizing the SPSW sections, a force equal to yield stress of the infill plate was applied

on the beams and columns at the angle of tension field and the boundary frame

members were designed accordingly. Reduced beam section (RBS) connections were

used to reduce the beam plastic moment on the column ends and ensure plastic hinge

formation at the beam ends. The designed SPSW details together with its yield pattern

at 2.5% drift are illustrated in Figure 2. The yield pattern confirms the design being

compliant with AISC provisions.

(a) (b)

Figure 2. SPSW model: (a) designed model details, (b) Yield pattern at 2.5% drift

Finite Element Modeling

Modeling and analysis of the SPSW were carried out using the ABAQUS finite

element package (Abaqus 2008). Models were simulated using 2-dimensional planar

parts and the four-node reduced integration S4R shell elements were used in the

discretization. First, eigen-value buckling analysis was performed on the models to

acquire the buckling shapes of the model and the scaled first buckling mode shape

was introduced to the models to account for the initial imperfections of the plate.

Geometric and material nonlinearity effects were also accounted for.

ASTM A36 and ASTM A572 Gr.50 steel was used for the infill plate and the

boundary frame members respectively. Elastic perfectly plastic stress-strain curves

with Fy=248 MPa for the infill plate and Fy=345 MPa for the frame, and E=209 GPa

and ν=0.3 for the elastic range were used.

Behavior of FRP materials was modeled through elastic behavior, damage initiation

and damage evolution models incorporated in ABAQUS (Abaqus 2008). Orthotropic

elastic behavior was defined through the introduction of elastic coefficients (E11, E22,

G12, ν12) in a local coordinate system with the main 1-direction along the fibers.

Initiation of damage at a specific point is examined through the calculation of Hashin

failure criteria (Hashin and Rotem 1973, Hashin 1980) and once a criterion is met in a

longitudinal, transverse or shear mode, corresponding stiffness was reduced

according to a linear softening rule. The softening rule governing damage evolution

in each failure mode was based on the energy dissipated during fracture (Abaqus

2008).

Four representative FRP materials were selected from the literature: An ordinary

strength and stiffness carbon fiber reinforced plastic (CFRP), a high modulus

composite (HM-CFRP), a high-strength CFRP (HS-CFRP), and finally a glass fiber

reinforced plastic material (GFRP) (Zhao and Zhang 2007, Accord and Earls 2006,

Jones 1999, Buyukozturk et al. 2003). Table 1 shows the mechanical properties of the

FRP materials used in the analysis. In this table, XT and X

C denote the tensile and

compressive strengths in fiber direction, YT and Y

C are for the matrix and S is the

shear strength.

Table 1. Mechanical properties of the materials used

Material

E11 E22 G12 V12 XT X

C Y

T Y

C S

(GPa) (GPa) (GPa) - (MPa) (MPa) (MPa) (MPa) (MPa)

CFRP 146.8 11.4 6.1 0.3 1,730 1,379 66.5 268.2 58.7

HM-CFRP 450 11.4 6.1 0.3 1,540 1,232 66.5 268.2 58.7

HS-CFRP 210 11.4 6.1 0.3 3,200 2,560 66.5 268.2 58.7

GFRP 20.3 11.4 6.1 0.3 855 684 66.5 268.2 58.7

Validation of the Modeling Approach

To verify the modeling of a SPSW and the nonlinearity and post-buckling phenomena

in its behavior, a benchmark case study of a four-story shear wall tested at the

University of Alberta was selected and simulated (Driver et al 1997). Using the same

procedures as those adopted in current paper, shear force of the first story was plotted

against first story lateral displacement and a comparison of the simulated pushover

curve with the laboratory hysteresis curves is shown in Figure 3. The good agreement

achieved demonstrates the ability of the adopted procedures to simulate key features

of SPSW behavior.

(a) (b)

Figure 3. Benchmark case study simulation: (a) FE model, (b) agreement of results

(Driver et al 1997)

To investigate the ability of the finite element procedures used in this study to predict

the interaction of metal plasticity and FRP fracture and damage behaviors, a fiber

metal laminate (FML) specimen found in the literature was modeled and analyzed

(Lapczyk and Hurtado 2007). The FLM consisted of three thin aluminum layers

bonded with two GFRP layers under tension with a central hole as depicted in Figure

4. Lapczyk and Hurtado modeled the FLM using solid elements for the aluminum and

GFRP layers and cohesive elements for the adhesive films, while we used the same

modeling assumptions outlined in the previous section. Results available in the work

by Lapczyk and Hurtado and the simulations of this study demonstrate good

agreement.

(a) (b)

Figure 4. FRP/metal interaction study (a) test setup (b) agreement of results (Lapczyk

and Hurtado 2007)

ANALYSIS RESULTS

This section presents the results of the numerical investigations in two separate

sections. First, the performance in buckling phase studied using eigen-value buckling

analysis is presented. Later, the post-buckling behavior is discussed through inelastic

analyses incorporating steel plasticity and fracture and damage in FRP composites. In

each one of these sections, two strengthening configurations of full-wrap FRP and

diagonal strips are separately studied.

Performance in Buckling Phase

Parametric Study on Full Wrap configuration.

In the first configuration, the 3mm-thick steel plate is fully covered with two 1.5 mm

CFRP layers at (θ/-θ) from horizontal. The angle θ is then gradually incremented and

the shear buckling force per unit edge length is calculated for each angle and plotted

in Figure 5. The optimal fiber angle is 45° as shown by Figure 5 which is in

agreement with the fact that pure shear corresponds to diagonal compression and

tension at (45/-45).

Figure 5. Shear buckling force for different fiber angles

For a steel infill with (+45/-45) layers of full wrap FRP, the effect of increasing FRP

thickness is illustrated in Figure 6. The order of strengthening effects between the

four materials is justifiable considering the order of their moduli of elasticity. In the

same geometry and orientation, the highest buckling prevention is gained by the

material with the highest modulus of elasticity (HM-CFRP).

Figure 6. Buckling force for different FRP materials and thicknesses (full-wrap)

Diagonal Strips Configuration

In the second configuration, two diagonal FRP strips were attached to the steel plate

with widths of 150, 300 and 450 mm (Figure 7). The shear buckling critical stress

was calculated for each plate strengthened with the three FRP widths and different

thicknesses and shown in Figure 8.

050

100150200250300350400450

0 2 4 6

Cri

tica

l S

hea

r buck

ling F

orc

e

Per

Unit

Len

gth

of

Edge

(N/m

m)

FRP full wrap thickness (mm)

GFRP

CFRP

HM-CFRP

HS-CFRP

(a) (b) (c)

Figure 7. Configuration of diagonal FRP strips (a) w=150mm (b) w=300mm (c)

w=450mm

(a) (b)

(c) (d)

Figure 8. Buckling stress comparison for diagonal strip configuration with different

widths and thickness (a) GFRP, (b) HS-CFRP, (c) CFRP, (d) HM-CFRP

Using Figure 8, considering a cross-sectional area for the FRP strip, it is obvious from

the evaluation of buckling stress that a narrower and thicker strip provides better

buckling prevention than a thinner strip with a greater width. Figure 9 provides a

representative comparison between the buckling mode of a plate strengthened with a

150x4.5 mm strip and that of a plate with 450x1.5mm strip. For this comparison, the

bending stiffness of the thicker strip (as estimated by bh3/12) is 9 times that of the

thinner strip and therefore the diagonal of the plate tends to resist out-of-plane

bending better and force the plate to buckle in its second mode shape, thereby

increasing its shear buckling stress.

(a)

(b)

Figure 9. First shear buckling mode shapes for the plate with (a) 450x1.5 mm FRP

strips, and (b) 150x4.5mm strips

Performance in the Post-Buckling Phase

Retrofit Using FRP materials

Similar to the buckling phase studies, the steel infill plate of the SPSW was

strengthened with FRP materials in two patterns. First, the plate was fully covered

with FRP layers with the fibers oriented at (θ/-θ) as shown in Figure 10 (a). The

second configuration was diagonal FRP strips with varying widths and the fibers were

oriented along the length of the strip as previously depicted in Figure 7.

For each model, the load displacement pushover curve was extracted for each fiber

angle (θ). The curve was then idealized as a bilinear curve with equivalent initial

slope, yield point and ultimate strength by equating the enclosed area under the real

and the idealized curves as depicted in Figure 10 (b). Using the idealized curve,

quantitative measurement of the behavior of the composite SPSW was done through

strength parameters (Fu and Fy), lateral stiffness (K) and area enclosed by the load-

displacement curve (A).

Figure 10. FRP retrofit assumptions (a) Full wrap FRP bonding pattern (b) Bilinear

idealization of load-displacement curves

Optimum Fiber Angle in Full Wrap.

Figure 11 provides an illustration of the influence of fiber angle on strength and

stiffness behavior of the full wrap SPSW. The analyses show that for a SPSW with

two 1.5mm

-thick layers of CFRP, ultimate force and enclosed area are maximized at

37.5°, while the stiffness of the system is maximized at 45° and yield force at 42.5°.

Figure 11. Influence of fiber angle on strength and stiffness parameters of the full

wrap model (a) Ultimate strength, (b) area under the curve (c) lateral stiffness

Comparison of different FRP materials in Full Wrap

Table 2 provides a summary of the contributions to the behavior of the system by

different FRP materials. The models pertaining to this table are retrofitted using two

1.5 mm thick FRP layers oriented at an angle of 45°. This table shows considerable

strength and stiffness gains together with minor reductions in ductility. The highest

strength is obtained in the model with HS-CFRP and the highest lateral stiffness

occurs in the model with HM-CFRP. The model retrofitted using ordinary CFRP

shows considerable increases of 42.7% and 32.8% in strength and stiffness

respectively. The GFRP materials used seem to be relatively weak for strengthening a

SPSW.

Table 2. Comparison of different FRP materials for 1.5mm

full-wrap fibers at 45°

Material Fu (kN) A (kN.mm) K(kN/mm) Ductility

No FRP 3,396 214,507 280 7.6

CFRP 4,846 284,933 330 7.2

Increase (%) 42.7 32.8 18.0 -4.5

HS-CFRP 5,197 301,980 348 7.3

Increase (%) 53.0 40.8 24.7 -3.0

GFRP 3,949 236,973 290 7.5

Increase (%) 16.3 10.5 3.6 -0.8

HM-CFRP 3,890 262,226 384 8.0

Increase (%) 14.6 22.2 37.4 6.2

Figure 12 shows the general load-displacement curves of SPSWs retrofitted using

different FRP materials. The curves show a similar ascending trend except for the

model with HM-CFRP which demonstrates premature softening.

Figure 12. Load-displacement curves for SPSWs strengthened with different FRPs

To investigate the reason for this different behavior of the HM-CFRP system in

Figure 12, the Von Mises stress in the infill plate and the tensile damage variable in

the diagonal FRP layer at 2.5% story drift were evaluated (Figure 13). This figure

shows that the FRP layer is ruptured along the perimeter and the steel plate undergoes

strain hardening. This is to be expected considering the fact that the HM-CFRP

material has the smallest fracture strain capacity (0.34%) compared with the other

three materials which have 1.18%, 1.52% and 4.54% fracture strains for CFRP, HS-

CFRP and GFRP materials, respectively. It can be inferred that if the fracture strain of

the FRP material is small, severe damage and rupture of the fibers might occur, which

can produce a detrimental effect on the strength behavior of the SPSW.

0

2000

4000

6000

0 10 20 30 40 50 60 70

Sto

ry S

hea

r (k

N)

Story Lateral Displacement (mm)

HS-CFRP HM-CFRP CFRP

GFRP No FRP

(a) (b)

Figure 13. Behavior of the full wrap system at 2.5% drift (a) Von Mises equivalent

stress distribution in the steel infill plate, (b) tensile damage in the FRP wrap

Effect of the width of a diagonal FRP strip configuration.

An important question regarding the use of FRP materials to retrofit a SPSW is about

the spread of the FRP layer over the steel infill plate. Table 3 compares the

parameters of a SPSW retrofitted using 300mm

-wide diagonal strips with those of a

fully wrapped SPSW. It is notable that although the amount of FRP material used in

the full-wrap model is 6 times that of the diagonal strip model, the resulting increase

in strength is much smaller (e.g. 2.5 times increase in strength). This implies that

concentrating the FRP material in the diagonal region rather than spreading it on the

whole surface of the plate produces a better result. The reason is that the diagonal

region on the infill plate experiences the largest strains as the inward forces of the

plate gives the frame members an hourglass shape while the ends of columns are

forced away by the support provided by the ends of the upper and lower beams.

Table 3. Comparison of the model with 300mm

strip, with a full wrap model

(thickness=1.5mm

)

SPSW System Fu (kN) A (kN.mm) K (kN/mm)

No FRP 3,391 214,108 280

SPSW with full-wrap HM-CFRP 3,890 262,226 384

Increase (%) 14.7 22.5 37.2

SPSW with 300mm strip of HM-CFRP 3590.3 227397.8 305.2

Increase (%) 5.9 6.2 9.0

Full-wrap/strip ratio of increase in Fu, K or A 2.5 3.6 4.1

Table 4 provides a comparison between the parameters of two SPSWs with 150 mm

and 300 mm diagonal strips. The thickness of both CFRP strips is 6 mm and thus the

amount of material used in the 300-mm wide strip is twice that of the 150 mm-wide

strip. It can be seen that although the amount of FRP material used is doubled in the

300-mm model, the ratio of strength or stiffness increase never exceeds 1.8, which

implies that using a narrower strip of FRP results in a better retrofit which is again a

result of the strain concentration on the diagonal of the plate.

Table 4. Influence of FRP strip width on strength and stiffness parameters (strip

thickness=6 mm)

SPSW System Fu (kN) A (kN.mm) K (kN/mm)

No FRP 3,391 214,108 280

SPSW with 300mm CFRP strip 5,062 286,094 326

Increase (%) 49.3 33.6 16.4

SPSW with 150mm CFRP strip 4,336 256,670 307

Increase (%) 27.9 19.9 9.8

w300/w150 ratio of increase in K, A, Fu 1.8 1.7 1.7

CONCLUSIONS

Models of steel plate shear walls were designed and simulated using the finite

element method. FRP materials were attached to the steel infill plate in full wrap and

diagonal strip patterns and numerical analysis and comparisons of results of these

models were carried out in the two phases of buckling and post-buckling. The results

of the investigations reported in this paper showed that:

The optimum fiber angle for buckling prevention is 45º. Larger values of

elastic modulus resulted in greater improvement in shear buckling capacity.

In the diagonal strip configuration, for the same FRP cross section, the

thickest and narrowest strip results in the greatest buckling resistance.

The optimum orientation of fibers is in the direction of the inclined tension

field in the steel plate.

Both of the proposed retrofit patterns show significant improvements in terms

of shear strength and lateral stiffness.

Concentrating FRP on the diagonal region rather than its full spread on plate

yields improved results due to the strain concentration on the diagonal.

REFERENCES

Alipour, M., (2010). “Evaluation of the effect of FRP layer characteristics on SPSW

behavior.” MSc dissertation, Department of civil engineering., Amirkabir

University of Technology (Tehran Polytechnic), Tehran, Iran.

AISC (American institute of steel construction). (2005) ANSI/AISC 341-05: Seismic

Provisions for Structural Steel Buildings, AISC, Chicago, U.S.

Accord, N.B., Earls, C.J. (2006). “Use of Fiber-Reinforced Polymer Composite

Elements to Enhance Structural Steel Member Ductility”, Journal of Composites

for Construction, ASCE, Vol. 10, No. 4, pp. 337-344.

ABAQUS (2008). ABAQUS/Standard Analysis User’s manual, Version 6.8-1.

Hibbitt, Karlsson, Sorenson Inc, (HKS).

Buyukozturk, O., Gunes, O., Karaca, E. (2003). "Progress on understanding

debonding problems in reinforced concrete and steel members strengthened using

FRP composites", Construction and Building Materials, pp. 9-19.

Driver, R.G., Kulak, G.L., Kennedy, D.J.L., Elwi, A.E. (1997). “Seismic Behavior of

Steel Plate Shear Walls”. Report No. 215. Edmonton, Canada.

Hashin Z., Rotem A. (1973). “A fatigue failure criterion for fiber-reinforced

materials”. Journal of Composite Materials; 7:448.

Hashin Z. (1980). “Failure criteria for unidirectional fiber composites.” Journal of

Applied Mechanics; 47:329–34.

Hatami, F., Rahai A.R. (2008). “An investigation of FRP composite steel shear walls

(CSSW) under cyclic loading on laboratory”, 14th

world conf. on Earthquake

engineering. China, 2008.

Hatami, F., Ghamari, A., Rahai, A. (2012). “Investigating the properties of steel shear

walls reinforced with Carbon Fiber Polymers” (CFRP). Journal of Constructional

Steel Research, 70, 36-42.

Jones, R.M. (1999). “Mechanics of Composite Materials”, second ed., Taylor and

Francis, pp. 56-74.

Lapczyk, I., Hurtado, J.A. (2007). “Progressive damage modeling in fiber reinforced

materials.” Journal of Composites, Part A, 38: 2333–2341.

Rahai, A. R., Alipoura, M. (2011). “Behavior and characteristics of innovative

composite plate shear walls.” Procedia Engineering, 14, 3205-3212.

Tabrizi, M. A., Rahai, A. (2011). “Perforated Steel Shear Walls with FRP

Reinforcement of Opening Edges.” Australian Journal of Basic and Applied

Sciences, 5(10), 672-684.

Zhao X.L., Zhang, L. (2007). “State of the art review on FRP strengthened steel

structures.” Journal of Engineering Structures 29 (8), 2007, pp. 1808–1823.

The images used in this paper are created by the authors unless otherwise indicated.