Elastic Buckling of PlatesThe critical bucking stress for a simply supported plate subjected to compression in

one direction is given by

, (1)

where, t and w are the thickness and width of the plate, respectively; k is the buckling coefficient, and is dependent on the type of loading, boundary condition, and length to width ratio of the plate; Cm is the material constant given by

. (2)

For a simply supported plate of large length to width ratio (i.e., l/w greater than 4) with uniform compression along the length, a value of k = 4 can be used with very little loss in accuracy [Yu, 2000]. For a plate with 3 sides simply supported and 1 side free, under uniform compression and large l/w ratio, k = 0.425.

Elastic Buckling of Plates with HolesIn cold-formed steel structural members, holes may be provided in webs or flanges

of the member for functional requirements like piping, electrical cables, ducts and other utilities. Openings may also be required to accommodate the transverse member, which may be structural or non-structural. The presence of holes alters the stiffness and strength of the members.

Consider a simply supported plate, under uniform compression along the length, with a central hole of width wh (Figure 1). The width of the plate at the hole is (w – wh). Further, due to the presence of the hole, the boundary conditions, for the portion of the plate at the hole, change to simply supported on 3 sides and free on 1 side (S3F1).

Figure 1: Simply supported plate with circular hole under uniform longitudinal compression.

34

Sim

ply

Sup

port

ed

Sim

ply

Sup

port

ed

w

wh

Using these parameters, the buckling coefficient for a S3F1 plate can be expressed as

, (3)

where the critical buckling stress for a S3F1 plate is given by,

. (4)

Thus, by re-substitution, kw/ h can be expressed as

, (5)

and its variation with respect to k = 4 is shown in Figure 2. Substituting wh = 0 in Eq. (5) gives the buckling coefficient for a S3F1 plate of width w/2.

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1

w h /w

k

Figure 2: Buckling coefficient for plate with and without hole.

Hole SpecificationsThe hole size and shape vary depending on the manufacturers. Special shapes of holes can be requested, however, the oval hole shape shown in Figure 3 is very commonly used for webs of structural studs. These holes are centered at 610mm along the length of the stud, and centrally located on the width of the web.

Finite Element AnalysisThis study investigates the effect of holes on the critical buckling of the web plates

of studs subjected to pure compressive load along the length of the plate. The web of the studs is considered to be simply supported along the edges that intersect with the flanges. The buckling analysis is performed using the commercially available finite element software ABAQUS.

35

(a) (b)Figure 3: (a) Oval hole used in structural studs; (b) Location of the oval hole on the stud web.

Element TypeIn this study, the general purpose three-dimensional, stress/displacement, reduced

integration with hourglass control, shell element S4R (available in ABAQUS), is used to model the plates. S4R has 4 nodes (quadrilateral), with all 6 active degrees of freedom per node. S4R allows transverse shear deformation, and the transverse shear becomes very small as the shell thickness decreases.

Loading and Boundary ConditionsThe accuracy of the elastic buckling analysis using finite element method depends on the mesh size, and the accuracy of the model to simulate the actual loading and the boundary conditions. To minimize the local effect, it is necessary to use consistent nodal loading to simulate the uniformly distributed compressive load. This is achieved by applying concentrated loads which are proportional to the tributary area associated with the corresponding node. For e.g., if P is the total uniformly distributed load to be applied to a plate discretized into 4 elements, the load should be applied as shown in Figure 4.

Finite Element MeshThe coefficient of buckling k for a simply supported plate under uniform compression is 4.0. This is used to determine the fineness of the finite element mesh in the finite element analysis (FEA); the mesh size that yields the k value close to 4.0 for a simply supported plate without hole, is used to determine the buckling load and the buckling coefficient of the same plate with hole. The length of the plate is taken as 4w, where w is the plate width.

Section Properties: Plate SizesThe width of the structural stud sections given in the Steel Stud Manufacturers

Association (SSMA) catalog range from 63.5mm to 304.8mm [SSMA, 2001]. The minimum thickness ranges from 33mils to 97mils (design thickness of 0.879mm to 2.583mm).

36

w = 38.1mm

l = 1

01.6

mm

R = 19.05mm

305mm

610mm

610mm

Figure 4: Consistent distributed load proportional to the tributary area for linear S4R elements.

Analysis ResultsBuckling analyses is performed for simply supported web plates of the studs

subjected to pure compressive loading along the length. Figure 5a shows the first buckling mode for the simply supported web plate without opening of a 362S162-43 stud. This is the typical local buckling mode, where the plate forms crests and troughs of half wavelength equal to the width of the plate. Figure 5b shows the first buckling mode for the simply supported web plate of a 362S162-43 stud with central opening.

Figure 5a: First buckling mode for the web plate of 362S162-43 (w = 90.075mm).

37

P/4 P/4 P/4 P/8P/8

l = 4w

w = 92.075mm

Figure 5a: First buckling mode for the web plate of 362S162-43 (w = 90.075mm).

The buckling stress for plate without (w/o) and with (w/) hole for plates of 43mils ( t = 1.14554mm) thickness are listed in Table 1a. The data in Table 1.1 is shown in Figure 6. The k values from the FEA results for plate without hole are in good agreement with those calculated using the plate theory, i.e., Eq. (1). The k values for plate thickness of 97mils (t = 2.58318mm) show the same trend (see Figure 6b and Table 1b).

As seen from Figures 6a and 6b, the value of buckling coefficient for a plate with 1 central standard hole is different (generally smaller) from that of the plate without hole. The reduction in the k value is a function of the ratio of the width of the hole to the width of the plate; for very small holes, as expected, the k value is same as that of plate without hole. However, for larger width of the hole, values of k have a tendency to increase.

Further investigation is required to determine the factors influencing the k values for plates with holes.

38

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

w h/w

k

Plate Theory w /o Hole

Plate Theory w / HoleFEA w /o Hole

FEA w / Standard Hole (t = 1.14554)

w h = 38.1

Figure 6a: Comparison of coefficient of buckling for t = 1.14554mm (first mode).

0

2

4

6

8

10

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

w h/w

k

Plate Theory w /o HolePlate Theory w / HoleFEA w /o HoleFEA w / Standard Hole (t = 2.58318)

w h = 38.1

Figure 6b: Comparison of coefficient of buckling t = 2.58318mm (first mode).

References1. Yu, W., Cold-Formed Steel Design, 3rd Edition, John Wiley & Sons, Inc., NY, USA.2. SSMA, Product Technical Information, ICBO ER-4943P, Steel Stud Manufacturers

Association, Chicago IL, USA.

39

Table 1a: Comparison of the plate theory results with the FEA results for plate (43mils) with and without openings.

Stud

ID

w (m

m)

w/t

(t =

1.1

4554

)

wh/

w

fcr (MPa)

Plate Theory FEA

Plate Theory

w/o Hole k

Plate Theory

w/ Hole

k (w

.r.t.w

)

Ratio of k

(8)(6)

FEA

w/o Hole

k (w

.r.t.w

)

FEA w/ Standard Hole (t = 1.14554)

k (w

.r.t.w

) Ratio of k

(13)(11)

(1)(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

200P-43 50.8 44.346 0.750 373.241 4.000 2538.038 27.200 6.800 373.255 4.000 628.679 6.738 1.684

225P-43 57.15 49.889 0.667 294.906 4.000 1128.017 15.300 3.825 293.687 3.983 400.183 5.428 1.363

250S162-43 63.5 55.432 0.600 238.874 4.000 634.509 10.625 2.656 238.404 3.992 258.269 4.325 1.083

350S162-43 88.9 77.605 0.429 121.875 4.000 158.627 5.206 1.302 121.535 3.989 110.292 3.620 0.907

362S162-43 92.075 80.377 0.414 113.614 4.000 140.514 4.947 1.237 112.476 3.960 100.91 3.553 0.897

400S162-43 101.6 88.692 0.375 93.310 4.000 101.522 4.352 1.088 92.742 3.976 81.612 3.499 0.880

550S162-43 139.7 121.951 0.273 49.354 4.000 39.657 3.214 0.804 49.239 3.991 42.501 3.445 0.863

600S162-43 152.4 133.038 0.250 41.471 4.000 31.334 3.022 0.756 41.037 3.958 35.812 3.454 0.873

800S162-43 203.2 177.384 0.188 23.328 4.000 15.018 2.575 0.644 23.048 3.952 20.663 3.543 0.897

1000S162-43 254 221.729 0.150 14.930 4.000 8.782 2.353 0.588 14.748 3.951 13.626 3.651 0.924

1500P-43 381 332.594 0.100 6.635 4.000 3.482 2.099 0.525 6.555 3.952 6.536 3.940 0.997

64

65

Table 1b: Comparison of the plate theory results with the FEA results for plate (97mils) with and without openings.

Stud

ID

w (m

m)

w/t

(t =

2.5

8318

)

wh/

w

fcr (MPa)

Plate Theory FEA

Plate Theory

w/o Hole k

Plate Theory

w/ Hole

k (w

.r.t.w

)

Ratio of k

(8)(6)

FEA

w/o Hole

k (w

.r.t.w

)

FEA w/ Standard Hole (t = 1.14554)

k (w

.r.t.w

) Ratio of k

(13)(11)

(1)(2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14)

200P-97 50.8 19.666 0.750 1897.920 4.000 12905.858 27.200 6.800 1842.66 3.884 2783.803 5.867 1.511

225P-97 57.15 22.124 0.667 1499.591 4.000 5735.937 15.300 3.825 1474.151 3.932 1930.275 5.149 1.309

250S162-97 63.5 24.582 0.600 1214.669 4.000 3226.465 10.625 2.656 1215.971 4.004 1254.317 4.131 1.032

350S162-97 88.9 34.415 0.429 619.729 4.000 806.616 5.206 1.302 612.877 3.956 550.381 3.552 0.898

362S162-97 92.075 35.644 0.414 577.726 4.000 714.511 4.947 1.237 568.615 3.937 504.851 3.495 0.888

400S162-97 101.6 39.331 0.375 474.480 4.000 516.234 4.352 1.088 467.558 3.942 407.626 3.436 0.872

550S162-97 139.7 54.081 0.273 250.965 4.000 201.654 3.214 0.804 247.56 3.946 214.468 3.418 0.866

600S162-97 152.4 58.997 0.250 210.880 4.000 159.332 3.022 0.756 208.12 3.948 181.466 3.442 0.872

800S162-97 203.2 78.663 0.188 118.620 4.000 76.366 2.575 0.644 117.005 3.946 105.782 3.567 0.904

1000S162-97 254 98.328 0.150 75.917 4.000 44.657 2.353 0.588 74.897 3.946 70.238 3.701 0.938

1500P-97 381 147.493 0.100 33.741 4.000 17.704 2.099 0.525 33.38 3.957 33.289 3.946 0.997

66

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