Wang 2016

8/18/2019 Wang 2016

1/12

8/18/2019 Wang 2016

2/12

investigate the failure mode and shear buckling strength of the web-

post. An empirical formula which predicted the ultimate vertical shear

buckling strength of web-posts was formulated for the particular web

opening shapes. Lawson, et al. [14] presented simplied equations for

web-post buckling based on strut model, which was calibrated against

results of nite element analyses. Under vertical shear force, the web-

post buckled inan “S ” shape. Hence, it was discordant to check its stabil-

ity using the column buckling curves. And the benecial effect of thein-

clined tension zone was not included in the design.

As shown in Fig. 1(b), it was much reasonable to check the stability

of the web-post based on the plate shear buckling theory. Redwood and

Demirdjian [15] investigated buckling behavior of the web-post in the

CSB with hexagonal openings through studying the upper part of the

web-post under horizontal shear force, as shown in Fig. 2. The horizon-

tal shear buckling strength, V h,cr, of the free body was calculated by

V h;cr ¼ k Eetw

h0=twð Þ2

ð1Þ

where k was the shear buckling coef cient of the upper part of theweb-

post under horizontal shear force. E was the Young's modulus of steel. e

was the width of web-post. t w was the web-post thickness. h0 was the

height of web opening. From the force equilibrium of the free bodyshown in Fig. 2, the vertical shear buckling strength of the web-post

was calculated by

V cr ¼ h−2yi

s V h;cr ð2Þ

where h was the section height of CSB. yi was the distance from the

ange to the centroid of Tee-section. s was the distance between two

adjacent web openings. The buckling strength of the web-post could

be easily determined if k was provided. However, Redwood and

Demirdjian [15] only presented curves to calculate the shear buckling

coef cient k of the web-post with limited geometric parameters.

Substitute Eq. (1) into Eq. (2), the vertical shear buckling strength of

the web-post can be obtained by

V cr ¼ k h−2yi

s

Eetw

h0=twð Þ2

ð3Þ

In this paper, after obtaining the vertical shear buckling strength of

the web-post through FEM simulation, the shear buckling coef cient k

was calculated through reverse analysis. And then an equation was

Fig. 1. Strut inthe web-post. (a) Inclined compressionstrut. (b) Compressionand tension

eld.

Fig. 2. Free body in the CSB with polygonal web openings.

Fig. 3. Conguration of studied CSB.

Fig. 4. FEM of the CSB with hexagonal web openings . (a) Mesh of the CSB with

hexagonal web openings. (b) Load and boundary conditions of the CSB with hexagonal

web openings.

174 P. Wang et al. / Journal of Constructional Steel Research 121 (2016) 173–184

8/18/2019 Wang 2016

3/12

proposed to calculate k based on the parameter study results. Studies in

this paper followed four steps:

(1) A Finite Element Model (FEM) was proposed and veried by

available test results [10,15,19,20] on CSBs failed in web-post

buckling;

(2) Shear buckling strengths calculated by hand calculated method

proposed by Redwood and Demirdjian [15] were compared

with the FEM simulation results to show its precision;

(3) Parameters affected the shear buckling coef cient k of the web-

post were studied; and then an equation to calculate k was

proposed;

(4) The vertical shear buckling strength of the web-post obtained

using the proposed buckling coef cient k was veried by FEM

and test results [15,19].

2. Finite element model and verication

2.1. Model description and constitutive model of FEM

The CSB was made of Q345 steel with yield strength of 345 MPa and

Poisson's ratio of 0.3. The material behavior provided by ABAQUS [16]

using the PLASTIC option allows a nonlinear stress–strain curve to be

Fig. 5. Comparison of load-middle span deection curves and buckling deformations. (a) Dimension of test beams [10]. (b) Load-middle span deection curves. (c) Bucklingdeformations.

Table 1

Tested CSBs with hexagonal openings [15,19].

Specimens h

(mm)

bf (mm)

t w(mm)

t f (mm)

e

(mm)

h0(mm)

b

(mm)

n L

(mm)

f yw(MPa)

f yf (MPa)

10-5a [15] 380.5 66.9 3.56 4.59 77.8 266.2 76.2 4 1220 352.9 345.6

10-5b [15] 380.5 66.9 3.56 4.59 77.8 266.2 76.2 4 1220 352.9 345.6

10-6 [15] 380.5 66.9 3.56 4.59 77.8 266.2 76.2 6 1828 352.9 345.6

10-7 [15] 380.5 66.9 3.56 4.59 77.8 266.2 76.2 8 2438 352.9 345.6

10-1 [19] 370.59 69.09 3.58 4.39 58.17 245.87 69.85 12 3048 357.1 342

10-3 [19] 376.43 70.61 3.61 4.45 57.91 260.53 127 8 3048 357.1 342

12-1 [19] 476.25 78.49 4.69 5.33 73.41 352.81 101.6 8 3048 311.6 307

12-3 [19] 449.58 78.23 4.62 5.35 71.37 302.51 149.35 6 3048 311.6 307

175P. Wang et al. / Journal of Constructional Steel Research 121 (2016) 173–184

8/18/2019 Wang 2016

4/12

used. The rst part of the nonlinear curve represents the elastic part up

to the proportional limit stress with Young's modulus of 205 GPa. Since

the non-linear buckling analysis involved large inelastic strains, the

nominal stress–strain curves were converted to true stress and

logarithmic plastic true strain curves. The steel constitutive model in

the proposed FEM used Von Mises yield surfaces with associated plastic

ow, which allowed for isotropic yield. The Von Mises yield surface wasdened by giving the value of the uniaxial yield stress as a function of

uniaxial equivalent plastic strain [17,18].

The studied CSB had four openings along the beam, as shown in

Fig. 3. The vertical concentrated force was applied at the middle span

of the beam. The width and thickness of the two anges were

180 mm and 20 mm, respectively. α was the incline angle of web

Fig. 6. CSB with hexagonal openings [15,19].

Table 2

Shear buckling strengths of CSBs with hexagonal openings obtained from FEM and tests.

Specimens V cr,TEST(kN)

V cr,FEM(kN)

V cr,TEST/ V cr,FEM

10-5a [15] 46.35 45.75 1.05

10-5b [15] 50.45 45.75 1.10

10-6 [15] 47.4 45.75 1.03

10-7 [15] 42.2 41.75 1.01

10-1 [19] 39.55 38.75 1.02

10-3 [19] 36.92 40.64 0.91

12-1 [19] 57.33 58.43 0.98

12-3 [19] 58.22 61.21 0.95

Fig. 7. Comparison of web-post buckling deformations. (a) Test results [19]. (b) FEM

results.

Fig. 8. CSB with cellular openings [20].

Table 3

Test specimen NPI_240 and NPI_280 [20].

Specimens h

(mm)

bf (mm)

t w(mm)

t f (mm)

e

(mm)

h0(mm)

L

(mm)

f yw(MPa)

f yf (MPa)

NPI_240 355.6 106 8.7 13.1 94 251 2846 390 390

NPI_280 406.9 119 10.1 15.2 163 271 2820 290 290

Table 4

Shear buckling strengths of CSBs with circular openings obtained from FEM and tests.

Specimens V cr,TEST [20]

(kN)

Mean value

(kN)

V cr,FEM(kN)

V cr,TEST/ V cr,FEM

NPI_240 Test3 142.05 142.58 152.62 0.93

NPI_240 Test4 143.1

NPI_280 Test1 188.8 187.89 176.81 1.06

NPI_280 Test2 192.1

NPI_280 Test3 186.2

NPI_280 Test4 184.45

Table 5

Comparison of deection obtained from FEM and tests [20].

Specimens wTEST(mm)

Mean value

(kN)

wFEM(mm)

wTEST/ wFEM

NPI_240 Test3 13.892 14.019 15.91 0.88

NPI_240 Test4 14.146

NPI_280 Test1 20.181 20.143 20.036 1.01

NPI_280 Test2 20.694

NPI_280 Test3 19.805

NPI_280 Test4 19.891

Fig. 9. Shearbuckling coef cient forweb-postproposedby Redwoodand Demirdjian [15].


8/18/2019 Wang 2016

5/12

Fig. 10. Values of k obtained from FEM and proposed by Redwood and Demirdjian [15]. (a) Effects of t w on k, (b) effects of h0/ h on k, and (c) effects of e/ t w on k.

Table 6

Dimensions of web-post to study effects of t w on k.

Group No. e/ t w h0/ e h0/ h t w (mm) e (mm) h0(mm) h (mm)

1

20 3, 4, 5, 6, 7, 8 0.74

6.0 120 360, 480, 600, 720, 840, 960, 480, 640, 800, 960, 1120, 1280

2 8.0 160 480, 640, 800, 960, 1120, 1280 640, 853, 1066, 1280, 1493, 1706

3 10.0 200 600, 800, 1000, 1200, 1400, 1600 800, 1066, 1333, 1600, 1866, 2133

Table 7

Dimensions of web-post to study effects of h0/ h on k.

Group no. e/ t w h0/ e h0/ h t w (mm) e (mm) h0 (mm) h (mm)

120 3, 4, 5, 6, 7, 8

0.746.0

1 20 3 60 , 48 0, 6 00, 7 20, 84 0, 9 60 48 0, 6 40 , 80 0, 96 0, 1 12 0, 12 80

4 0.51 120 360, 480, 600, 720, 840, 960 720, 960, 1200, 1440, 1680, 1920

Table 8

Dimensions of web-post to study effects of e/ t w on k.

Group no. e/ t w h0/ e h0/ h t w (mm) e (mm) h0 (mm) h (mm)

1 203, 4, 5, 6, 7, 8 0.74 6.0

12 0 3 60 , 48 0, 60 0, 7 20 , 84 0, 96 0, 4 80 , 64 0, 8 00 , 9 60 , 11 20 , 1 28 0

5 15 90 270, 360, 450, 540, 630, 720 480, 600, 720, 840, 960


8/18/2019 Wang 2016

6/12

opening edge. hf was the web height of the Tee-section above the web

opening. l was the half spanof the CSB. t f was the thicknessof theange.

h and bf were the section height and ange width of the CSB.

2.2. Boundary conditions

For the symmetry of the model, only half of the beam was modeled,

asshown in Fig. 4(a). Displacements in y (Uy) and z (Uz)direction of the

bottom ange at the left end of the beam were restrained to simulate

the pin support. The Uz at the top ange was restrained to simulate

the lateral brace. At the right end of the beam, the displacement in x

(Ux) direction and the rotation around y-axis (URy) of theweb were re-

strained. The Uz was also restrained to prevent the beam from failing at

thelateral torsionalbuckling. Forthe top andbottomanges,the Ux and

rotation around z-axis (URz) were restrained. The vertical shear force

was applied at the bottom ange, as shown in Fig. 4(b).

The beam was meshed using the shell element S4R in ABAQUS, a 4-

node quadrilateral shell element with reduced integration and a large-strain formulation, with mesh size of 10 × 10 mm. Theeigenvalue buck-

ling analysis was employed to obtain the elastic buckling strength and

the buckling modes of the web-post.

2.3. Model veri cation

Test results on CSBs with cellular openings by Tsavdaridis and

D'Mello [10], on CSBs with hexagonal openings by Redwood and

Demirdjian [15] and by Zaarour and Redwood [19], and on CSBs with

cellular openings by Erdal and Saka [20] were used to validate the pro-

posed FEM. The beams selected all failed in web-post buckling.

CSBs with cellular web openings tested by Tsavdaridis and D'Mello

[10] were made of UB457 × 152 × 52 and had steel grade of S355, asshown in Fig. 5(a). The yield strength equaled to 375.3 MPa and

359.7 MPa for the web and the anges, respectively; and the ultimate

stresses of the web and the anges were 492.7 MPa and 480.9 MPa, re-

spectively. Thesimply supported beam was loadedat mid-span through

a hydraulic jack. The load-middle span deection curves and failure

modes obtained from FEM agreed well with those from tests, as

shown in Fig. 5(b). The shear buckling strengths of tested specimen

A1 and B1 obtained from test were 133.3 kN and 101.5 kN, respectively,

which were half of the applied concentrated load. The V cr,TEST/ V cr,FEMwere only 0.99 and 1.05 for the two tested beams, respectively. V cr,TESTand V cr,FEM were the shear buckling strengths obtained from test and

FEM, respectively. Failure modes and buckling deformations obtained

from FEM and test all buckled in “S ” shape, as shown in Fig. 5(c),

which proved the validity of the FEM.Test results on CSBs with hexagonal openings carried out by Red-

wood and Demirdjian [15] and Zaarour and Redwood [19] were com-

pared with the FEM results. The tested CSBs were simply supported

and were loaded by a concentrated force at middle span. Vertical stiff-

eners were applied at the two supports and the loading point at the

middle span. Dimensions of tested beams and yield strengths of the

web ( f yw) and the anges ( f yf ) were listed in Table 1. h was the section

height of the CSB. h0 was the height of web opening. bf and t f were the

width and thickness of the ange, respectively. t w was the thickness of

the web. e was the width of web-post. b was the horizontal length of

Table 9

Parameters of CSBs for studying effects of e/ t w.

Group no. Incline angle

α

e/ t w h0/ t w hf / t w t w (mm) e (mm) h0 (mm) hf (mm)

Group I 60° 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60 80 15 8.0 80, 120, 160, 200, 240, 280, 320, 360, 400, 440, 480 640 120

Group II 10.0 100, 150, 200, 250, 300, 350, 400, 450, 500, 550, 600 800 150

Group III 12.0 120, 180, 240, 300, 360, 420, 480, 540, 600, 660, 720 960 180

Table 10

Parameters of CSBs for studying effects of h0/ t w.

Group no. Incline angel

α


G ro up I V 60 ° 15 40 , 50 , 60 , 70 , 80 , 90 , 10 0, 1 10, 1 20,

130, 140

15 3.0 45 120, 150, 180, 210, 240, 270, 300, 330, 360, 390, 420 45

Group V 6.0 90 240, 300, 360, 420, 480, 540, 600, 660, 720, 780, 840 90

Group VI 10.0 150 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400 150

Table 11

Parameters of CSBs for studying effects of hf / t w.

Group no. Incline angle

α


Group VII 60° 15 80 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23 8.0 120 640 24, 40, 56, 72, 88, 104, 120, 136, 152, 168, 184

Group VIII 10.0 150 800 30, 50, 70, 90, 110, 130, 150, 170, 190, 210, 230

Group IX 12.0 180 960 36, 60, 84, 108, 132, 156, 180, 204, 228, 252, 276

Table 12

Parameters of CSBs for studying effects of t w.

Group

no.

Degrees

α


Group X 60° 15 80 15 3.0, 4.0, 5.0 ,6.0,

7.0,8.0, 9.0, 10.0, 11.0

45, 60, 75, 90, 105, 120, 135, 150, 165 240, 320, 400, 480, 560,

640, 720, 800, 880

45, 60, 75, 90, 105, 120,

135, 150, 165Group XI 20 60, 80, 100, 120, 140, 160, 180, 200, 220

Group XII 30 90, 120, 150, 180, 210, 240, 270, 300, 330


8/18/2019 Wang 2016

7/12

Table 13

Parameters of CSBs for studying effects of α .

Group no. Incline angel

α

Tangent value e/ t w h0/ t w hf / t w t w (mm) e (mm) h0 (mm) hf (mm)

Group XIII 85°, 80°, 75°, 70°, 65°, 60°, 55°, 50° 11.43, 5.67, 3.73, 2.14, 1.73, 1.43, 1.20 15 80 15 6.0 90 480 90

Group XIV 8.0 120 640 120

Group XV 10.0 150 800 150

Fig.11. Effectsof web-post dimensionson vertical shearbucklingstrengthof the web-post. (a)Effectsof e/ t w onshearbucklingstrength, (b)effects of h0/ t w on shearbuckling strength,

(c) effects of hf / t w on shear buckling strength, (d) effects of t w on shear buckling strength, and (e) effects of α on shear buckling strength.


8/18/2019 Wang 2016

8/12

the opening edge. L wasthelength of the beam, as shown in Fig. 6. n was

the number of openings of the CSB.

Shear buckling strengths obtained from FEM and tests [15,19] were

listed in Table 2. It could be seen that the shear buckling strengths pre-

dicted by FEM agreed well with those measured from tests [15,19]. The

V cr,TEST/ V cr,FEM variedfrom0.91and 1.05 of theeight beams with a mean

value of 0.99 and a standard deviation of 0.0572.

For beam 10-5a and 10-5b [15], shear buckling deformations of the

web-post obtained from FEM and test were shown in Fig. 7. The FEMand test results both demonstrated that the web-post buckled in “S ”

shape.

Erdal and Saka [20] carried out experiment on cellular steel beams.

Test results of specimen NPI_240 and NPI_280 which failed in web-post

buckling were used to verify the FEM. The cellular steel beams were sup-

portedby round rollers and loaded by a concentrate force at middle span.

Lateral supports were installed at beam ends to prevent the lateral tor-

sional buckling, as shown in Fig. 8. Dimensions of NPI_240 and NPI_280

were listed in Table 3. Shear buckling strengths and vertical deections

obtained from FEM and tests were listed in Tables 4 and 5, respectively.

Results obtained from the FEM with mesh size of 10 mm agreed well

with those measured from tests. The V cr,TEST/ V cr,FEM varied from 0.93

and 1.06 of the six beams with a mean value of 0.99 and a standard devi-

ation of 0.0667. The wTEST/ wFEM varied from 0.88 and 1.01 of the six

beams. wTEST was the deection of the cellular steel beam obtained

from the tests, and wFEM was the deection obtained from FEM.

3. Precision of k proposed by Redwood and Demirdjian [15]

If V cr had been obtained through FEM simulation, the buckling coef-

cient k could be obtained through reverse analysis by

k ¼ V cr h0=twð Þ

2

Eetw

s

h−2yið4Þ

where V cr was the vertical shear buckling strength of the web-post,which could be obtained through FEM eigenvalue analysis.

Thehorizontal shear buckling strength of theweb-post could not ex-

ceed its shear yield strength

V h; p ¼ 0:58etw f y ð5Þ

Substitute Eqs. (2) and (5) into Eq. (4). The corresponding shear

yielding coef cient kp

k p ¼ 0:58 f yE

h0tw

2ð6Þ

Redwood and Demirdjian [15] proposed the shear buckling coef -

cient k as a function of the ratio of web opening height to web-post

width h0/ e, the ratio of web opening height to section height h0/ h, and

the ratio of web-post width to web thickness e/ t w. For h0/ h equaled to

0.51 or 0.74, k could be obtained through Fig. 9 [15]. dg in Fig. 9 [15] rep-

resented section heightof theCSB, which wasdenoted as h in this paper.

Comparison of shear buckling coef cients of web-posts obtained

using the nite element method and method proposed by Redwood

and Demirdjian [15] was shown in Fig. 10(a), (b) and (c). The yield

strength of steel, f y, was 345 MPa. The incline angle of opening edge

was 60°.Effects of web thickness on the shear buckling coef cients of the

web-post were shown in Fig. 10(a). Dimensions of the web-post were

listed in Table 6. Three t w were studied, which were 6.0 mm, 8.0 mm

and 10.0 mm. For theweb-post with thesame e/ t w, h0/ e and h0/ h, k pro-

posed by Redwood and Demirdjian [15] were the same. However, FEM

results showed that k changed with the changes of t w. So it was not

enough to formulate the buckling coef cient by only using thethree di-

mensionless parameters e/ t w, h0/ e and h0/ h. And the k calculated by

Fig. 12. Effects of e/ t w on k.

Fig. 13. Effects of h0/ t w on k.

Fig. 14. Effects of hf / t w on k.


8/18/2019 Wang 2016

9/12

method proposed by Redwood and Demirdjian [15] was much lower

than that obtained through FEM.

Effects of h0

/ h on the shear buckling coef cients of the web-post

were shown in Fig. 10(b). Dimensions of the web-post studied were

listed in Table 7. Two h0/ h were studied, which were 0.74 and 0.51. Ac-

cording to the curves obtained by Redwood and Demirdjian [15], with

theincrease of h0/ e, the slope ofthe k line with different h0/ h was almost

the same, as shown in Fig. 10(b).However, FEMresults showedthat the

slope of the k line decreased with the increase in h0/ h. k obtained by

Redwood and Demirdjian [15] was still smaller than FEM results.

Effects of e/ t w on k were shown in Fig. 10(c). Dimensions of the web-

post studied were listed in Table 8. Three e/ t w were studied, while t wwas kept constant. k decreased with increase in e/ t w, if it wascalculated

by method presented by Redwood and Demirdjian [15]. FEM results

showed that k nearly kept constant with the increase in e/ t w, if the

web-post had the same t w. k obtained by Redwood and Demirdjian

[15] was smaller than FEM results.

4. Shear buckling strength of web-post

Parameters that affected the buckling behaviors of the web-post

were studied rst to nd out how and to what degrees each parameter

inuenced the V cr. If the vertical shear buckling strength V cr had been

obtained through FEM simulation, then the shear buckling coef cient

k can be calculated by Eq. (4). Then a practical method to calculate k

was proposed based on curve tting of FEM simulation results. And

the vertical shear buckling strength of the web-post was calculated by

Eq. (3) at last.

4.1. Parameters affected vertical shear buckling strength of web-post

CSBs listed in Tables 9–13 were studied to show effects of web-post

dimensions on vertical shear buckling strength of a CSB with hexagonal

webopenings. Analysis results were shown in Fig. 11. Dimensions of the

web-post were represented by dimensionless factors, which were the

ratio of web-post width to web thickness e/ t w, the ratio of web opening

height to web thickness h0/ t w, the ratio of web height of Tee-section

above web opening to web thickness hf / t w, the web thickness t w (in

mm), and the incline angle of web opening edge α .

As shown in Fig. 11(a), (b) and (c), with the increase in e/ t w, h0/ t w

and hf / t w, the shear buckling strength decreased. h0/ t w had great effecton the shear buckling capacity. And with the increase in t w and α , the

shear buckling strength increased, as shown in Fig. 11(d) and (e).

4.2. Effects of geometry parameters on the shear buckling coef cient k

After V cr was obtained through FEM simulation, the buckling coef -

cient k can be obtained by Eq. (4). Parameters that affected k were

investigated through FEM.

4.2.1. Effects of e/t w on k

Three groups of CSBs were studied to show effects of e/ t w on k. Pa-

rameters of the studied CSBs were listed in Table 9. As shown in

Fig. 12, k decreased non-linearly with the increase in e/ t w and it could

be expressed by an exponent function with negative index.

Fig. 16. Effects of α on k.

Fig. 17. Distribution of k obtained from proposed method and FEM.

Table 14

Statistics of (kCAL − kFEM)/ kFEM.

Statistics project Value

Mean 0.004653

Standard error 0.002179

Median 0.000391

Standard deviation 0.027217

Sample variance 0.000741

Minimum −0.043890

Maximum 0.1183996

Sum 0.6978705

Count 150

Fig. 15. Effects of t w on k.


8/18/2019 Wang 2016

10/12

4.2.2. Effects of h0 /t w on k

Totally 33 CSBs listed in Table 10 were studied to show ef-

fects of h0/ t w on k. As shown in Fig. 13, k increased linearly

with the increase in h0/ t w and it could be formulated by a linear

function.

4.2.3. Effects of h f /t w on k

33 CSBs were studied to illustrate effects of h f / t w on k . Dimen-

sions of CSBs studied were listed in Table 11. As shown in Fig. 14,

k decreased non-linearly with the increase in hf / t w and it could

be formulated by an exponential function.

Fig. 18.Comparisonsof V cr obtained by proposed methodand FEM. (a)Vertical shear buckling strengthof web-post with different e/ t w, (b)vertical shear buckling strength ofweb-post

with different h0/ t w, (c)vertical shear bucklingstrengthof web-post with different hf / t w, (d)vertical shear buckling strength of web-post with different t w, and(e) vertical shear buckling

strength of web-post with different.


8/18/2019 Wang 2016

11/12

4.2.4. Effects of t w on k

As listed in Table 12, 27 CSBs were studied to show effects of t w on k.

As shown in Fig. 15, k decreased linearly with the increase in t w. It could

also be expressed by a linear function.

4.2.5. Effects of α on k

As listed in Table 13, 24 CSBs were studied to show effects of α on k.

α was represented by its tangent value tan(α ).As shown in Fig. 16, k in-

creased linearly with the increase in tan α . It could also be expressed by

a linear function.

4.3. Practical equations for calculating shear buckling coef cient

Parameter studies showed that k decreased non-linearly with the in-

crease in e/ t w and hf / t w, asshownin Figs. 12 and 14. It increased linearly

with the increase in α and h0/ t w, as shown in Figs. 16 and 13. It de-

creased linearly with t w (in mm), as shown in Fig. 15. As results of nu-

merical regression analysis, k could be calculated by

k ¼ k1 k2 k3 k4 k5≤k p ð7Þ

where

k1 ¼ 0:782 þ 19:712 e

tw

−1:011

ð8Þ

k2 ¼ 0:0325 þ 0:00292h0tw

ð9Þ

k3 ¼ 0:275 þ 0:239exp −0:093h f =twð Þ ð10Þ

k4 ¼ 16:998−0:312tw ð11Þ

k5 ¼ 2:037 þ 0:0360tanα ð12Þ

Comparisons of k obtained from FEM and Eq. (7) were shown in

Figs. 12–16. k obtained by Eq. (7) agreed well with FEM results. Fig. 17

showed differences between k obtained by Eq. (7) and FEM. The differ-

ences were mainly within ±5%, which showed that Eq. (7) could give

accurate predictions on shear buckling coef cient. Here, kCAL was ob-tained by Eq. (7) and kFEM was obtained from FEM analyses. Statistics

of (kCAL − kFEM)/ kFEM were listed in Table 14. The mean value was

only 0.004653 and the standard error was only 0.002179.

4.4. Veri cation

The vertical shear buckling strength of the web-post in a CSB was

predicted by Eq. (3), where k was calculated by Eq. (7). Comparisons

of V cr obtained by Eq. (3) and FEM analysis were shown in Fig. 18. It

could be seen that, for CSBs listed in Tables 9–13, V cr obtained byEq. (3) agreed well with FEM analyses.

Test results on CSBs with hexagonal openings [15,19] were used to

validate the proposed method. Calculated parameters of CSBs were

listed in Table 15. Buckling coef cients k1 ~ k5 and shear buckling

strengths of CSBs obtained from the proposed method and test were

listed in Table 16. The average of (V cr,CAL − V cr,TEST)/ V cr,TEST was 19.7%.

Web-posts in specimen 10-5a, 10-5b,10-6 and 10-7 buckled in theelas-

tic state;and those in specimen 10-1, 10-3, 12-1 and12-3 failed in shear

yielding. For the proposed method was based on the elastic buckling of

the web-post, it overestimated the shear buckling strength when the

web-post buckled in the elastic–plastic state, such as those with thick

web thickness. Through introducing a safety factor of 1.2, the precision

of the proposed method was increased. The average of ( V cr,CAL /1.2 −

V cr,TEST)/ V cr,TEST was reduced to −0.22%, as listed in Table 16.

5. Conclusions

Elastic buckling behaviors of the web-post in the CSB under vertical

shear were investigated using nite element method. Through treating

the upper part of the web-post as a free body under horizontal shear

force, whose shear buckling strength can be calculated by the thin-

plate shear buckling theory, the vertical shear buckling strength of the

web-post was obtained after providing the shear buckling coef cient k.

The shear buckling coef cient k wasnot only affected by thedimen-

sionless parameters of the web-post, such as e/ t w, h0/ t w, and hf / t w, but

also the thickness of the web-post t w and incline angle of the opening

edge α . k was calculated through reverse analysis after the vertical

shear buckling strength of the CSB was obtained through FEM simula-tion. e/ t w, h0/ t w, and h f / t w, t w and α all had great inuences on the

shear buckling coef cient. A simplied method was proposed to calcu-

late k. The vertical shear buckling strength of the web-post employing

the proposed k agreed well with FEM results.

For the proposed method was based on the elastic buckling of the

web-post, it overestimated the shear buckling strength when the

web-post buckled in the elastic–plastic state, such as those with thick

web thickness. The average of (V cr,CAL − V cr,TEST)/ V cr,TEST was 19.7%.

Through introducing a safety factor of 1.2, the precision of the proposed

method was increased. The average of (V cr,CAL /1.2 − V cr,TEST)/ V cr,TESTwas reduced to −0.22%. The simplied method for calculating shear

buckling coef cient k in elastic–plastic state needed to be investigated

later.

The proposed equation for calculating the shear buckling strength of web-posts was derived from the CSBs with hexagonal web openings.

Table 16

Shear buckling strength of CSBs obtained by proposed method and tests [15,19].

Specimens k1 k2 k3 k4 k5 kp k V cr P TEST (kN) (V cr,CAL − V cr,TEST)/ V cr,TEST (V cr,CAL /1.2 − V cr,TEST)/ V cr,TEST

10-5a [15] 1.65 0.25 0.34 15.89 2.10 5.58 4.64 54.23 46.35 17.0% −2.5%

10-5b [15] 1.65 0.25 0.34 15.89 2.10 5.58 4.64 54.23 50.45 7.5% −10.4%

10-6 [15] 1.65 0.25 0.34 15.89 2.10 5.58 4.64 54.23 47.4 14.4% −4.7%

10-7 [15] 1.65 0.25 0.34 15.89 2.10 5.58 4.64 54.23 42.2 28.5% 7.1%

10-1 [19] 1.96 0.23 0.33 15.88 2.10 4.77 4.99 57.42 39.55 45.2% 21.0%

10-3 [19] 1.97 0.24 0.34 15.87 2.07 5.26 5.30 40.97 36.92 11.0% −7.5%

12-1 [19] 2.00 0.25 0.35 15.53 2.10 4.99 5.82 79.47 57.33 38.6% 15.5%

12-3 [19] 2.02 0.22 0.34 15.56 2.07 3.78 4.89 55.71 58.22 −4.3% −20.3%

Average 21.0% 0.8%

Table 15

Calculated parameters of CSBs with hexagonal web openings.

Specimens e/ t w h0/ t w hf hf / t w t w α tanα yi s (mm)

10-5a [15] 21.85 74.78 52.56 14.76 3.56 60.00 1.73 13.11 308.00

10-5b [15] 21.85 74.78 52.56 14.76 3.56 60.00 1.73 13.11 308.00

10-6 [15] 21.85 74.78 52.56 14.76 3.56 60.00 1.73 13.11 308.00

10-7 [15] 21.85 74.78 52.56 14.76 3.56 60.00 1.73 13.11 308.00

10-1 [19] 16.25 68.68 57.97 16.19 3.58 60.00 1.73 14.86 256.04

10-3 [19] 16.04 72.17 53.50 14.82 3.61 45.00 1.00 13.26 369.82

12-1 [19] 15.65 75.23 56.39 12.02 4.69 60.00 1.73 14.62 350.0212-3 [19] 15.45 65.48 68.19 14.76 4.62 45.00 1.00 18.46 441.44


8/18/2019 Wang 2016

12/12

Whether the CSBs with other web opening shapes could use the pro-

posed equation needed to be investigated later.

Acknowledgments

Theauthors wish to acknowledge thesupport from theFundamental

Research Funds of Shandong University (No. 2015JC 046), the Natural

Science Foundation of Shandong Province (ZR2015EM041) and the Nat-

ural Science Foundation of China (51578322) for the work reported inthis paper.

References

[1] K.F. Chung, T.C.H. Liu, A.C.H. Ko, Investigation on Vierendeel mechanism in steelbeams with circular web openings, J. Constr. Steel Res. 57 (5) (2001) 467–490.

[2] T.C.H. Liu, K.F. Chung, Steel beams with large web openings of various shapes andsizes: nite element investigation, J. Constr. Steel Res. 59 (9) (2003) 1159–1176.

[3] P.J. Wang, Q.J. Ma, X.D. Wang, Investigationon Vierendeel mechanism failureof cas-tellated steel beams with llet corner web openings, Eng. Struct. 74 (1) (2014)44–51.

[4] P. Panedpojaman, T. Thepchatri, S. Limkatanyu, Novel simplied equations forVierendeel design of beams with (elongated) circular openings, J. Constr. SteelRes. 112 (1) (2015) 10–21.

[5] E. Ehab, Interaction of buckling modes in castellated steel beams, J. Constr. Steel Res.67 (5) (2011) 814–825.

[6] P. Panedpojaman, T. Thepchatri, S. Limkatanyu, Novel design equations for shear

strength of local web-post buckling in cellular beams, Thin-Walled Struct. 76 (1)(2014) 92–104.

[7] A.R. Zainal Abidin, B.A. Izzuddin, Meshless local buckling analysis of steel beamswith irregular web openings, Eng. Struct. 50 (special issue IASS-IACM 2012: Model-ling and Computations) (2013) 197–206.

[8] S. Durif, A. Bouchaïr, O. Vassart, Experimental tests and numerical modeling of cel-lular beams with sinusoidal openings, J. Constr. Steel Res. 82 (1) (2013) 72–87.

[9] S. Durif, A. Bouchaïr, O. Vassart, Experimental and numerical investigation on web-post member from cellular beams with sinusoidal openings, Eng. Struct. 59 (2014)587–598.

[10] K.D. Tsavdaridis, C. D'Mello, Web buckling study of the behaviour and strength of perforated steel beams with different novel web opening shapes, J. Constr. SteelRes. 67 (10) (2011) 1605–1620.

[11] K.D. Tsavdaridis, C. D'Mello, Optimisation of novel elliptically-based web openingshapes of perforated steel beams, J. Constr. Steel Res. 76 (1) (2012) 39–53.

[12] British Standard Institution, BS5950-1:2000, Structural Use of Steelworks in Build-

ing, BSI, 2000.[13] P.J. Wang, X.D. Wang, N. Ma, Vertical shear buckling capacity of web-posts in castel-lated steel beams with llet corner hexagonal web openings, Eng. Struct. 75 (1)(2014) 315–326.

[14] R.M. Lawson, J. Lim, S.J. Hicks, W.I. Simms, Design of composite asymmetric cellularbeams and beams with large web openings, J. Constr. Steel Res. 62 (6) (2006)614–629.

[15] R.G. Redwood, S. Demirdjian, Castellated beam web buckling in shear, J. Struct. Eng.ASCE 124 (8) (1998) 1202–1207.

[16] ABAQUS Standard User's Manual, Hibbitt, Karlsson and Sorensen, Inc., USA, 2008vols. 1, 2 and 3. Version 6.8-1.

[17] R.D. Krieg, D.B. Krieg, Accuracies of numerical solution methods for the elastic-per-fectly plastic model, ASME J. Press. Vessel. Technol. 99 (4) (1977) 510–515.

[18] H.L. Schreyer, R.F. Kulak, J.M. Kramer, Accurate numerical solutions for elastic–plas-tic models, ASME J. Press. Vessel. Technol. 101 (3) (1979) 226–234.

[19] W. Zaarour, R.G.Redwood,Web bucklingin thin webbedcastellated beams,J. Struct.Div. Proc. ASCE 122 (8) (1996) 860–866.

[20] F. Erdal, M.P. Saka, Ultimate load carrying capacity of optimally designed steel cellu-lar beams, J. Constr. Steel Res. 80 (1) (2013) 355–368.

http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0060http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0060http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0080http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0080http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0100http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0095http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0090http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0085http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0080http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0080http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0075http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0070http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0065http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0060http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0060http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0055http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0050http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0045http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0040http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0035http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0030http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0025http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0020http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0015http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0010http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005http://refhub.elsevier.com/S0143-974X(16)30036-0/rf0005

Documents

Wang 2016