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Plate 1
Date : 1971
Subject : Compressive Buckling of Perforated Plate Elements
Title : The first Speciality Conference on Cold-Formed Structures
Author : Pennington Vann, W
Structure: Plate
Material : Steel
Width : 50T
Diameter of Hole:
0.3B
Height of the 10T
Stiffener :
BoundCond: Simply supported along all four edges
Loading : Compressed in one direction by imposing uniform in-plane displacements
of magnitude eB/2 along the top and bottom edges.
Thus the average in-plane strain in the direction of the loading is e and
The unloaded edges are free to move in or out and to
deform in the plane of the plate
This paper concerned with the strength of a light-gage steel member in axial compression when one
or more of the plate elements of which the member is composed has circular holes along its centreline.
1. Sample analytical results are given for the elastic buckling of a plate with a single hole. This information is intended to
provide some insight into the effect of the hole and the expected influence of a stiffening lip around the edge os the ho
2. Some experimental results are given for the elasti buckling of analogous light-gage elements.
3. Finally, results are presented for the ultimate strength of several light-gage wall studs which are perforated all along th
Hole Diameter: 1", 2", 3" and 4"
Four circular holes of successively larger diameter were cut in the web in the same location,
and after each hole was cut several buckling tests were conducted.
Also, before each larger hole was cut, a half inch width of the plate around the edge of the
previous smaller hole was bent out at approximately 45 degree to form a
stiffening lip in the shape of a cone,
and the buckling load for the stiffening hole was determined.
Loading : Specimen was bent about the weak axis with the web on the compression side.
Two symmetrically placed loads were used, so that the central segment of the web
was placed in uniform compression.
These central segment was approximately five times the flat width of the web.
Reinforcing bars were welded to the flanges of the section to prevent
them from yielding prematurley in tension.
s is
The buckling quantity,cr the non-dimensional average stress at buckling
and 's' is the average edge stress, 'T' plate thickness s
scr = 2
æT ö
Eç ÷
è Bø
Table 1: EFFECT OF HOLE AND STIFFENER ON AVERAGE BUCKLING STRESS
(Simply Supported Square Plate, Circular Hole with D/B=0.3)
Description Critical Avg. Ratio to Scr
Stress, Scr for no hole for unstiffened hole
(case no.1)
Plate with no hole 3.72 1.00
Plate with unstiffened hole 3.42 0.92
Plate with stiffened hole 5.25 1.41
B = Plate width
D = Diameter of central circular hole
Scr = non-dimensional average stress at buckling
Table 2: Measured Ultimate Loads on Wall Studs
(B/T =57.3; Yield Stress= 37.3; D/B=0.33)
Case No Description Ultimate Load Ratio to Pu for
Ratio to Pu
Pu, ksi unstiffened hole
for no hole
(Case No. 1) (Case No.2)
1.0 Stud with no 29.9 1.0 0.9
Hole
2.0 Stud with un- 31.7 1.1 1.0
stiffened hole
3.0 Stud with 31.5 1.1 1.0
stiffened hole
Conclusions:
1. The theoretical and experimental results presented indicte that unless a central unflanged hole is fairly large, it will have
a very small effect on the elastic buckling of a plate, and that a flanged hole can be expected to make the elastic
buckling load greater than the corresponding unpierced plate.
2. Tentative results concerning the ulitmate strength of pierced plates indicate that a small unflanged hole has
essentially no effect on the ultimate strength,and a stiffened hole may or may not affect the ultimate strength.
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Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
m in-plane displacements
n of the loading is e and
in axial compression when one
circular holes along its centreline.
te with a single hole. This information is intended to
influence of a stiffening lip around the edge os the hole.
ogous light-gage elements.
light-gage wall studs which are perforated all along their length.
eb in the same location,
round the edge of the
e web on the compression side.
hat the central segment of the web
times the flat width of the web.
the section to prevent
Ratio to Scr
for unstiffened hole
(case no.2)
1.09
1.00
1.54
less a central unflanged hole is fairly large, it will have
ed hole can be expected to make the elastic
es indicate that a small unflanged hole has
ay or may not affect the ultimate strength.
ported and edges not free to pull in
ported and unloaded edges free to pull in
pported and one edge free
mped and two unloaded edges simly supported
d two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
PLATE 2
Year : 1971
Title : Buckling Behaviour and Post-Buckling Strength of Perforated Stiffened Compression Elements
Source : The first Speciality Conference on Cold-Formed Structures
Author : Yu,W.W. and Davies, C.S
Structure: Beam
Material : Steel
This paper deals only with the buckling load and post-buckling strength of
stiffened compression elements having circular and square holes.
In order to verify the effect of holes on the buckling load and post-buckling strength of the perforated stiffened
elements, 28 short-column tests and 8 beam tests have been conducted to cover the following parameters:
(a) Shape of holes: circular andsquare holes
(b) Overall width-to-thickness ratio: 36.6 to 73.8
(c) Hole opening to overall width ratio (d/w or h/w): 0 to 0.722
(d) Yield point of steel: 34.4 to 59.3ksi
Column - Short columns (approx. 20 in. long)
- test specimen composed of two c-shaped channels(6-1/2 x 2-1/2 in., nominal size)
- central perforations, either circular or square
Beam - Test specimens were track sections
- Circular holes ranging from 1 to4 inches in diameter
Conclusions:
1. The presence of holes may reduce the buckling load of the stiffened elements
2. The reduction of buckling load of the stiffened compression elements is more pronounced for square
holes than for circular holes due to the differencein stress concentration and the shape of holes.
3. Test data indicated that for stiffened compression elements with circular holes, the
uniform stress approach may be used to predict the buckling load.
4. Winter's effective width equation for solid plates can be modified fordetermination of
the effective width of perforated stiffened elements
5. Even though the buckling load for the perforated stiffened elements is affected by the square holes than circular
holes, the post-buckling strength of the elements with square and circular holes are found
to be nearly the same if the diameter of a circular hole is the same as the width of a square hole.
ession Elements
erforated stiffened
are holes than circular
Plate 3
Date : 1975
Subject : Buckling and Post-Buckling Behaviour of Plates with Holes
Title : Aeronatical Quarterly
Author : Ridchie, D. and Rhodes, J
Structure: Plate
Material : Mild Steel
In view of the general lack of information on the post-buckling behaviour of perforated
plates and on the buckling of rectangular perforated plates, an experimental and theoretical
investigation was undertaken on suquare plates and rectangular plates with aspect ratio of 2:1.
The theoretical investigation took the form of the development of a method of post-buckling
analysis and comparison of the results produced by it to the experimentally observed behaviour.
The boundary conditions chosen to study were
1. Uniformly compressed loaded edges
2. Simply-supported edges
3. Stress-free unloaded edges
Table 1: Effects of various loading and boundary conditions studied by various investigators
Author Loaded edge Unloaded edge
Rotational Inplane Rotational
Restraint Restraint Restraint
Levy [T] None Constant stress None
Kumai [E][T] None Constant stress None
Schlack [E][T] None Constant displacement None
Yoshiki [E][T] None Constant stress None
Kawai and None Constant stress None
Ohtsubo [T] None Constant displacement None
None Constant displacement None
Kumai [E][T] Clamped Constant stress Clamped
[E] denotes experimental study
[T] denotes theoretical study
The theoretical analysis employs an approximate approach using a combination of Rayleigh-Ritz method
and finite element methods.
Experimental Investigation:
YModulus : 208000
PsRatio : 0.3
YldStress 297.0
Plate Length: 254 mm and 508 mm
Plate Width : 254 mm
Thickness : 1.57 mm
Hole Diameter: 0.1 to 0.7 times of the plate width
Loading Mechanism : The unloaded vertical edges of the plate are supported in the frame by knife edges and
the loaded edges are supported in rollers resting on needle bearings contained in the blocks
through which the laoding is transmitted.
Conclusions:
1. The post-buckling analysis described in this paper shown to predict accurately the buckling loads and modes of square and
rectangular plates with holes under the conditions of uniform edge displacement anduniformedge stress.
2. Agreement between experiment and theory is shown to be good at buckling.
3. In the post-buckling range it is found that the theoretical analysis is reasonably accurate for small holes,
but loses accuracy when dealing with large holes.
4. The post buckling analysis for the uniform displacement case was found to predict accuartely the out-of-plane deflection
magnitudes and to show the trends of stress distribution after buckling, although the accuracy of the stress predictions af
5. The simplified collapse analysis gives results which agree with those obtained experimentally for the plates tested,
but the authors would hesitate to recommend its direct application to deal with a wide range without further experimentat
Comparision between the experimental results and theoretical relationship between buckling load and hole size in
square and rectngular simply supported plates loaded by uniform displacements is illustrated in Fig.8 and 9
of the above article.
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
imental and theoretical
s with aspect ratio of 2:1.
method of post-buckling
entally observed behaviour.
ed by various investigators
Inplane
Restraint
None
None
Edges held straight
None
None
Edges held straight
None
None
mbination of Rayleigh-Ritz method
pported in the frame by knife edges and
g on needle bearings contained in the blocks
accurately the buckling loads and modes of square and
splacement anduniformedge stress.
s is reasonably accurate for small holes,
ound to predict accuartely the out-of-plane deflection
ing, although the accuracy of the stress predictions after buckling is limited.
se obtained experimentally for the plates tested,
o deal with a wide range without further experimentation.
onship between buckling load and hole size in
lacements is illustrated in Fig.8 and 9
ported and edges not free to pull in
ported and unloaded edges free to pull in
pported and one edge free
mped and two unloaded edges simly supported
d two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 4
Date : 1982
Subject : Elastic Buckling of Perforated Square Plates for Various Loading and Edge Condititons
Title : Proc., International Confernece on finite element methods
Author : Shanmugam, N.E and Narayanan, R
Structure: Plate
Material : Steel
Openings : centrally placed perforations of Square or Circular shapes
Openings having a maximum diameter or side equal to half the length of the plate
BoundCond: 1. Simply supported edges
2. Clamped edges
Loading : Bi-axial in-plane compressive stresses or
to uniform shear stresses along the edges
Studies using a Finite Element Formulation on the elastic buckling of perforated steel plates subjected to bi-axial
in-plane compressive stresses or to uniform shear stresses along the edges are reported.
Only square plates containing centrally placed perforations of square or circular shapes are considered.
Two boundary conditions, viz. Simply supported edges and clamped edges are investigated.
Buckling coefficients and patterns of buckling are reported for the cases considered. Approximate formulae for
evaluating the buckling coeeficients conservatively are suggested.
The studies are confined to cutouts having a maximum diameter or side, equal to hal the length of the plate.
Based on studies using a Finite Element Formulation, the following approximate relationships
for buckling coefficients are suggested:
Loading Simply supported edge clamped edge
Biaxial compression k=1.9-1.1[d/a] k=5.1-4.5[d/a]+14[d/a]^2
Under Shear ks=9-16[b/a] ks=14-21[b/a]
a - side of the square plate
d - diameter of the circular cutout or
diagonal of the square cutout
Conclusions:
1. The value of the buckling coefficient for a simply supported plate containing a square perforation and subjected to
biaxial in-plane compressive stresses decreases with the increase in the size of the perforation. [similar for circular cutout
2. When the palteis clamped at the edges and subjected to bi-axial compressive stresses, the value of the buckling
coefficient drops only marginally for very small cutouts; thereafter, it increases with the increase in the size of the cutout
3. The values of the buckling coefficients in shear decrease with an increasein the size of the cutout, when the plate
is clamped or simply supported.
4. Aprroxiamte formulae have been suggested for the computation of buckling stresses conservatively.
ates for Various Loading and Edge Condititons
side equal to half the length of the plate
perforated steel plates subjected to bi-axial
edges are reported.
or circular shapes are considered.
dges are investigated.
cases considered. Approximate formulae for
ide, equal to hal the length of the plate.
proximate relationships
clamped edge
k=5.1-4.5[d/a]+14[d/a]^2
ks=14-21[b/a]
containing a square perforation and subjected to
the size of the perforation. [similar for circular cutout]
ompressive stresses, the value of the buckling
it increases with the increase in the size of the cutout.
ncreasein the size of the cutout, when the plate
f buckling stresses conservatively.
Plate 5
Date : 1983
Subject : Elastic Buckling of Flat Panels Containing Circular and Square Holes
Title : Proc., International Conference on Instability and Plastic Collapse of Structures
Author : Sabir, A.B. and Chow, F.Y
Structure: Plate
Loading: Inplane loadings considered are
Uniaxail, Biaxial orshear distributed uniformly alonf the straight edges of the plates
Boundary Conditions: 1. Simply supported
2. Clamped
Material : Steel
The finite element method of analysis is employed to determine the elastic critical buckling
loads of flat square panels containing circular and square holes.
FEM
1. The elements used for calculating the inplane stressesprior to buckling were first developed for generalplane elasticity
problems, teir convergence performane were extensively tested and shown to be superior to other existing elements.
2. They are based on generalised strian rather than displacement assumptions and satisfy the requirement of zero strain du
3. Furthermore these elements were designed to include an additional degree offreedom, namely the inplane rotation.
4. In this way they are made suitable for caseswhere the inplane rotation as well as other degrees of freedom may be
restrained and also when the plate is combined with other structural components having this rotation as an essential extern
5. The elastic critical buckling loads are obtained for plates with centrally located square and circular holes when
subjected to the above mentioned laoding cases and having several types of boundary conditions at the straight edges.
Variations of buckling coefficient for different boundary conditions(simply supported and clamped) and different loading
conditions for square plates with square and circular holes are presented in Figure 3 to Figure 7 of this paper[for Referenc
Conclusions:
The results presented in this paper show
1. The use of the strain based inplane elements requires less number of elements into which the plate is
to be divided into in order that sufficiently accurate converged results are obtained.
2. The ability of these elements toproduce the classical critical loads for plates with no holes in the limit as the size of the
This is not the case with inplane elements based on prescribed displacement functions unless an inordicatley large number o
3. Any unusual rapidly changing configuration can be dealt with since no deterioration
of results is encountered as the aspect ratio of the elements varied.
4. The use of long rectangular elements makes it possible to assemble the overall structural matrices in a simple systematic
way and enables the use of the less efficient triangular elements only in regions where they are absolutely needed.
Circular and Square Holes
bility and Plastic Collapse of Structures
ormly alonf the straight edges of the plates
elastic critical buckling
kling were first developed for generalplane elasticity
shown to be superior to other existing elements.
umptions and satisfy the requirement of zero strain due to rigid body displacements.
l degree offreedom, namely the inplane rotation.
ation as well as other degrees of freedom may be
components having this rotation as an essential external degree of freedom.
trally located square and circular holes when
ypes of boundary conditions at the straight edges.
simply supported and clamped) and different loading
nted in Figure 3 to Figure 7 of this paper[for Reference].
of elements into which the plate is
for plates with no holes in the limit as the size of the hole is reduced.
acement functions unless an inordicatley large number of elements are used.
e the overall structural matrices in a simple systematic
y in regions where they are absolutely needed.
Plate 6
Date : 1983
Subject : Buckling and Elasto-Plastic Collapse of Perforated Plates
Title : Proc., International Confernece on Instability and Plastic Collapse of Structures
Author : Azizian, Z.G and Roberts, T.M
Structure: Plate
Material : Steel
This paper describes the buckling and geometrically non-linear elasto-plastic analysis of perforated plates by the finite elem
Triangular elements are used to model the plates and a number of solution refinements are discussed.
The elasto plastic stress strain relationships are based on Ilyushin's approximate area yield function.
Solutions are presented for axially compressed square plates with central square and circular holes
K values for perforated plates, obtained from the present analysis shows general agreement
with existing results: Penningtan-Wann and Shanmugam and Narayanan.
Conclusions:
1. The buckling load of a uniaxially compressed plate with a centrally placed hole is almost independent
of the hole size up to half the width of the palte and may even increase for larger hole sizes.
2. The ultimate laod of a uniaxially compressed plate with a centrally placed hole is influenced significantly by the size
of the hole. The reduction in the ulitmate laod is most pronounced for low b/t values.
|::
bility and Plastic Collapse of Structures
-plastic analysis of perforated plates by the finite element method.
ution refinements are discussed.
approximate area yield function.
ntral square and circular holes
hows general agreement
placed hole is almost independent
ase for larger hole sizes.
placed hole is influenced significantly by the size
Plate 7
Date : 1984
Subject : Ultimate Capacity of Uniaxially Compressed Perforated Plates
Title : Thin-Walled Structures, (Vol.2)
Author : Narayanan, R and Chow, F.Y
Structure: Plate
Boundary Condition: Simply Supported
1. An approximate method of predicting the ultimate load carrying capacity and the psot-buckling behaviour of
perforated plates typically used in engineering structures is presented.
2. The theory is based on a mechanism solution, used in conjuction with an elastic loading path derived from energy methods
3. Experiments have shown that a good approximation of the loading and unloading paths for simply supported plates
containing square and circular openings has been obtained by the suggested theoretical treatment.
4. Curves suitable for the use of designers have been suggested which can be used directly
for practical plates containing centrally placed holes.
In the experimental investiagations, all the plates tested were square and were simply supported at all edges.
Some 23 plates were tested, which included parametric variations is:
1. The size of the plate in relation to its thickness
2. The shape of the cutout (circular or square)
3. The size of the cutout in relation to the size of the plate.
4. The eccentricity of the cutout
Table 1 : DETAILS OF TEST SPECIMENS CONTAINING CENTRAL CUTOUTS
Group Specimen No.
Circular Hole a t a/t = b/t d or a' d/a or
a'/a
(mm) (mm) (mm)
CIR2a 125.0 1.615 77.40 25.0 0.2
CIR2b 125.0 1.615 77.40 25.0 0.2
1 CIR3a 125.0 1.615 77.40 37.5 0.3
CIR4a 125.0 1.615 77.40 50.0 0.4
CIR4b 125.0 1.615 77.40 50.0 0.4
CIR5a 125.0 1.615 77.40 62.5 0.5
CIR6 86.0 2.032 42.30 25.0 0.291
CIR7 86.0 1.615 53.23 25.0 0.291
2 CIR8 86.0 0.972 88.48 25.0 0.291
CIR9 86.0 0.693 124.10 25.0 0.291
CIR10 86.0 2.032 42.30 40.0 0.465
CIR11 86.0 1.615 53.25 40.0 0.465
CIR12 86.0 0.972 88.48 40.0 0.465
Square Hole
SQ2 125.0 1.615 77.40 25.0 0.2
SQ3 125.0 1.615 77.40 37.5 0.3
3 SQ4 125.0 1.615 77.40 50.0 0.4
SQ5 125.0 1.615 77.40 62.5 0.5
Table 2 : DETAILS OF TEST SPECIMENS CONTAINING ECCENTRICALLY LOCATED CUTOUTS
Specimen No. a t a/t = b/t d or a' d/a or
a'/a
Circular (mm) (mm) (mm)
UEC 1 125.0 0.972 128.60 37.5 0.3
UEC 2 125.0 0.972 128.60 62.5 0.5
Square
UES 1 125.0 0.972 128.60 37.5 0.3
UES 2 125.0 0.972 128.60 62.5 0.5
UES 3 125.0 0.972 128.60 37.5 0.3
UES 4 125.0 0.972 128.60 62.5 0.5
The theoretical analysis developed in this paper is approximate; the ultimate laod is estimated from the point of
intersection of a theoretical elastic loading curve with the unloading line obtained from the rigid plastic theory.
Table 3: EXPERIMENTAL RESULTS FOR UNIAXIALLY LOADED PLATES HAVING CENTRAL HOLES
Group Specimen No. a/t d/a or Observed Values
a'/a Pcr Avg. Ku Avg. Failure
Load
(kN) (kN)
1 2 3 4 5 6
Circular
PL 77.40 0.0 25.064 4.013 39.32
CIR2a 77.40 0.2 22.504 3.604 37.46
CIR2b 77.40 0.2 23.228 3.720 38.70
1 CIR3a 77.40 0.3 21.311 3.413 33.94
CIR4a 77.40 0.4 19.706 3.156 29.57
CIR4b 77.40 0.4 18.358 2.940 28.39
CIR5a 77.40 0.5 19.482 3.120 27.35
CIR6 42.30 0.291 - - 42.17
CIR7 53.23 0.291 - - 26.18
2 CIR8 88.48 0.291 6.341 3.205 12.35
CIR9 124.10 0.291 2.320 3.235 7.33
CIR10 42.30 0.465 - - 33.64
CIR11 53.25 0.465 - - 22.14
CIR12 88.48 0.465 5.926 2.995 10.89
Square Hole
SQ2 77.40 0.2 22.600 3.620 33.48
SQ3 77.40 0.3 20.290 3.250 28.85
3 SQ4 77.40 0.4 18.230 2.920 25.52
SQ5 77.40 0.5 19.170 3.070 21.86
Curves suitable for the use of designers have been proposed and can be employed to
determine the ultimate capacity of approximately square plates containing centrally placed holes.
The approximate post-bucled stress distribution at the loaded adge of the perforated plate may be written as
æd -do ö p2 éæ a4 ö 2 py
ù
s xh = s cr ç
h
(
÷ + E d 2 - d o2) êç 3 + 4 ÷ - 4 sin
ç ú
è d ø 16 a 2 ëè b ÷ø bû
A simplified form of Von Mises criterion is used in the analysis of collapse mechanism and is given by
2 2
M æ P ö æ S ö
+ç ÷ +ç ÷ =1
M p ç Pp
è
÷ çS ÷
ø è pø
Conclusions:
1. An approximate method of evaluating the ultimate capacity of simply supported perforated plates under
uniaxial compression has been suggested using simple elastic and plastic concepts.
2. The method avoids tedious calculations which would become necessary when 'large deflection theory' or nonlinear finite e
3. Good approximation of the loading and unloading paths are simply supported perforated paltes
under compression has been obtained by the theoretical treatment.
4. Since the analysis is based on small deflection theory, the method is not valid
for wide plates with a/t values in excess of 80 or so.
ed Perforated Plates
pacity and the psot-buckling behaviour of
th an elastic loading path derived from energy methods.
and unloading paths for simply supported plates
gested theoretical treatment.
h can be used directly
e and were simply supported at all edges.
Imperfec Imperfec/t YldStress
(mm) N/mm2
0.229 0.142 323.3
0.097 0.060 323.3
0.136 0.084 323.3
0.304 0.188 323.3
0.127 0.079 323.3
0.279 0.173 323.3
0.254 0.143 334.7
0.229 0.142 323.3
0.102 0.105 317.6
0.051 0.074 322.8
0.102 0.050 334.7
0.279 0.173 323.3
0.152 0.156 317.6
0.097 0.060 323.3
0.141 0.087 323.3
0.113 0.070 323.3
0.209 0.129 323.3
TRICALLY LOCATED CUTOUTS
e e/a Imperfec Imperfec/t YldStress
(mm) (mm) N/mm2
12.5 0.1 0.254 0.261 317.6
12.5 0.1 0.102 0.105 317.6
12.5 0.1 0.127 0.131 317.6
12.5 0.1 0.229 0.236 317.6
25.0 0.2 0.078 0.080 317.6
25.0 0.2 0.132 0.136 317.6
ultimate laod is estimated from the point of
line obtained from the rigid plastic theory.
PLATES HAVING CENTRAL HOLES
Failure Load/ Predicted (8)/(7)
Squash Load Strengths
Pxh/Psq
7 8
0.603 0.610 1.012
0.574 0.560 0.976
0.593 0.560 0.944
0.520 0.510 0.981
0.453 0.470 1.038
0.435 0.470 1.080
0.419 0.420 1.002
0.721 0.700 0.971
0.583 0.615 1.055
0.465 0.480 1.032
0.381 0.410 1.076
0.575 0.560 0.974
0.493 0.510 1.034
0.410 0.410 1.000
0.525 0.525 1.024
0.460 0.460 1.041
0.400 0.400 1.023
0.340 0.340 1.015
Mean 1.015
Standard Deviation 0.037
aining centrally placed holes.
of the perforated plate may be written as
lapse mechanism and is given by
ply supported perforated plates under
sary when 'large deflection theory' or nonlinear finite element analysis is used.
upported perforated paltes
Plate 8
Date : 1984
Subject : Strength of Perforated Plates Subjected to In-plane Loading
Title : Thin-Walled Structures, (Vol.2)
Author : Roberts, T.M. and Azizian, Z.G
Structure: Plate
Material : Steel
This paper describes the buckling and geometrically non-linear elasto-plastic
analysis of perforated plates by the finite element method.
The elasto plastic stress strain relationships are based on Ilyushin's approximate area yield function.
Solutions are presented for square plates with central square and circular holes subjected to
uniaxial compression, biaxial compression and pure shear.
K values for simply supported perforated square plates subjected to uniaxial compression,
biaxial compression and pure shear are shown in Fig. 2 of the above article.
Conclusions:
1. For simply supported plates subjected to uniaxial and biaxialcompression the buckling load
is almost independent of the size of the hole for 'd/b'from '0'to '0.5'.
2. For simply supported plates subjected to pure shear the buckling laod decreases continuously with increasing size of the
3. The ulitmate or collapse load ofall the plates studied decreases with increasing size of the hole.
For uniaxial or biaxial compression the reduction in the ultimate load is most significant forlow b/t values.
4. For pure shear there is a significant reduction in the ultiamte laod over the entire laod over the entire practical range of
5. The finite element solutions agree reasonably well with existing empirical and approximate solutions which have been veri
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
d to In-plane Loading
approximate area yield function.
cular holes subjected to
uniaxial compression,
ession the buckling load
aod decreases continuously with increasing size of the hole.
th increasing size of the hole.
is most significant forlow b/t values.
over the entire laod over the entire practical range of b/t.
mpirical and approximate solutions which have been verified by experiment.
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 9
Date : 1984
Subject : Buckling of Plates Containing Openings
Title : Proc., Seventh International Speciality Conference on Cold-Formed Steel Structures
Author : Narayanan, R and Chow, F.Y
Structure: Plate
Material : Steel
The buckling behaviour of simply supported square plates containing circular or square holes
and subjected to uniaxial or biaxial compression or to shear loading is investigated.
Parameters: 1. Size of the hole
2. eccentricity of hole's location with respect to the centre of the plate.
Boundary Conditions: 1. Simply supported
2. Clamped
Opening: 1. Square
2. Circular
The results of the parametric studies using an appropriate
finite element formaulation are presented in the above article.
Buckling Co-efficients: 1. Figures 3 and 4 - Plates compressed uniaxially under clamped and simply supported condition
containing eccentrically located square cutouts.
2. Figures 5 and 6 - Plates compressed uniaxially under clamped and simply supported condition
containing eccentrically located circular cutouts.
3. Figures 7 and 8 - Plates compressed biaxially under clamped and simply supported condition
containing eccentrically located square cutouts.
4. Figures 9 and 10 - Plates compressed biaxially under clamped and simply supported condition
containing eccentrically located circular cutouts.
5. Figures 11 and 12 - Plates under shear loading for a clamped and simply supported condition
square holes located in the tension diagonal.
6. Not shown - Plates under shear loading for a clamped and simply supported condition
circular holes located in the tension diagonal.
7. Figures 13 and 14 - Plates under shear loading for a clamped and simply supported condition
square holes located in the compression diagonal.
8. Not shown - Plates under shear loading for a clamped and simply supported condition
circular holes located in the compression diagonal.
Experimental Investigation:
Series 1 23 tests on square plates containing centrally placed and eccetrically placed
square or circular opeings and subjected to uniaxial compression.
Series 2 23 tests on square plates containing centrally placed and eccetrically placed
square or circular opeings and subjected to biaxial compression.
Series 3 38 tests on plates containing centrally placed holes or eccentrically placed
openings and subjected to shear
The test results were compared with the corresponding values
obtained by using the finite element formulation.
Conclusions:
1. It was found that small diameter holes did not influence the buckling coefficients, irrespective of their location.
2. When the holes were larger than 0.2a, and these values reduced with the increase in eccentricity of the hole location.
3. However for plates under uniform shear loading, two opposing results were obtained for holes located in the compression
tension diagonals; in the former case, the eccentricity reduced the buckling coefficient and in the latter, increased it.
|::
Plate 21
Date : [***-98]
Subject : Effects of Openings of the buckling of Cylindrical Shells Subjected to Axial Compression
Title : Thin-Walled Structures, Vol. 31, pp. 187-202
Author : J. F. Jullien and A. Limam
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
onference on Cold-Formed Steel Structures
circular or square holes
spect to the centre of the plate.
niaxially under clamped and simply supported condition
cated square cutouts.
uniaxially under clamped and simply supported condition
cated circular cutouts.
biaxially under clamped and simply supported condition
cated square cutouts.
biaxially under clamped and simply supported condition
cated circular cutouts.
loading for a clamped and simply supported condition
oading for a clamped and simply supported condition
loading for a clamped and simply supported condition
e compression diagonal.
oading for a clamped and simply supported condition
he compression diagonal.
rally placed and eccetrically placed
to uniaxial compression.
rally placed and eccetrically placed
to biaxial compression.
aced holes or eccentrically placed
ng coefficients, irrespective of their location.
ith the increase in eccentricity of the hole location.
lts were obtained for holes located in the compression and
buckling coefficient and in the latter, increased it.
drical Shells Subjected to Axial Compression
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 10
Date : 1984
Subject : Strength of Biaxially Compressed Perforated Plates
Title : Proc., Seventh International Speciality Conference on Cold-Formed Steel Structures
Author : Narayanan, R. And Chow, F.Y
Structure: Plate
Material : Steel
An approximate method of predicting the post-buckling behaviour and the ultimate carrying capacity
of perforated plates under biaxial compression.Design curves using the theory outlined
in this paper are given in Fig.10 of the above article.
The ultimate load is approximately estimated from the point of intersection of the
elastic loading curve and the plastic unloading curve.
A series os biaxial compression tests were carried out on perforated plates containing
centrally located circular and square openings and compared with the theoretical results.
Table 1: Details of Specimens with Centrally Located Holes
Group Specimen No.
Circular Hole a t a/t d or a' d/a or
a'/a
(mm) (mm) (mm)
BL5 125.0 1.615 77.4 0.0 0
BC2 125.0 1.615 77.4 25.0 0.2
1 BC3 125.0 1.615 77.4 37.5 0.3
BC4 125.0 1.615 77.4 50.0 0.4
BC5 125.0 1.615 77.4 62.5 0.5
BC6 86.0 2.032 42.3 25.0 0.291
BC7 86.0 1.615 55.3 25.0 0.291
2 BC8 86.0 0.972 88.5 25.0 0.291
BC9 86.0 2.032 42.3 40.0 0.465
BC10 86.0 1.615 55.3 40.0 0.465
BC11 86.0 0.972 88.5 40.0 0.465
Square Hole
BSQ2 125.0 1.615 77.4 25.0 0.2
BSQ3 125.0 1.615 77.4 37.5 0.3
3 BSQ4 125.0 1.615 77.4 50.0 0.4
BSQ5 125.0 1.615 77.4 62.0 0.5
Table 2: Comparison of Test Results with Theoretically Predicted Strengths
Circular Hole a/t d/a or Experimental Measured Values
a'/a Pcr mean Kb mean Failure
Load
(kN) (kN)
1 2 3 4 5 6
BL5 77.4 0 12.235 1.960 27.71
BC2 77.4 0.2 11.050 1.770 23.45
1 BC3 77.4 0.3 10.050 1.610 18.90
BC4 77.4 0.4 9.805 1.570 15.95
BC5 77.4 0.5 8.990 1.440 14.35
BC6 42.3 0.291 - - 27.10
BC7 55.3 0.291 - - 16.65
1 BC8 88.5 0.291 3.500 1.630 7.00
BC9 42.3 0.465 - - 20.90
BC10 55.3 0.465 - - 15.60
BC11 88.5 0.465 3.195 1.496 5.65
Square Hole
BSQ2 77.4 0.2 10.930 1.751 19.60
BSQ3 77.4 0.3 10.180 1.630 16.60
1 BSQ4 77.4 0.4 9.395 1.504 15.00
BSQ5 77.4 0.5 10.085 1.615 12.10
Conclusions:
1. A method of evaluating the ulitmate capacity of simply supported rectangular perforated plates under biaxial compressio
is suggested in this paper. The method is based on the elastic loading behaviour and the plastic unloading characteristic of
2. Tests carried out on a specially fabricated rig shows that the observed collapse loads are close to the predicted
ultimate capacity of the plates; predictions obtained are slightly unconservative.
3. For the range of hole sizes considered, the loss in strength due to the introduction of the opening is rapid when the plate
slenderness (a/t) is smaller than 50; for higher values of a/t, the loss in strength is gradually reduced.
4. Design curves which can be used to assess the ultimate strength of square plates with square and circular openings
have been proposed and these can be used directly; similar curves can be readily obtained for rectangular plates using the m
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
onference on Cold-Formed Steel Structures
d the ultimate carrying capacity
Imperfec Imperfec/t YldStress
(mm) N/mm2
0.171 0.106 323.3
0.195 0.121 323.3
0.062 0.038 323.3
0.235 0.146 323.3
0.104 0.064 323.3
0.053 0.026 334.7
0.155 0.096 323.3
0.078 0.080 317.6
0.085 0.042 334.7
0.148 0.092 323.3
0.069 0.071 317.6
0.134 0.083 323.3
0.158 0.098 323.3
0.097 0.060 323.3
0.103 0.064 323.3
Failure Load/ Predicted (8)/(7)
Squash Load Strengths
Pxh/Psq
7 8
0.425 0.385 0.906
0.360 0.320 0.889
0.290 0.298 1.028
0.245 0.272 1.110
0.220 0.250 1.136
0.480 0.470 0.979
0.371 0.390 1.051
0.259 0.275 1.062
0.370 0.385 1.041
0.348 0.325 0.934
0.209 0.238 1.139
0.301 0.310 1.030
0.255 0.275 1.078
0.230 0.240 1.043
0.186 0.210 1.129
rectangular perforated plates under biaxial compression
g behaviour and the plastic unloading characteristic of the plate.
erved collapse loads are close to the predicted
the introduction of the opening is rapid when the plate
s in strength is gradually reduced.
f square plates with square and circular openings
n be readily obtained for rectangular plates using the method outlined in the paper.
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 11
Date : 1984
Subject : Elastic Buckling of Perforated Plates under Shear
Title : Thin-Walled Structures, (Vol.2)
Author : Narayanan,R and Der-Avanessian, N.G
Structure: Plate
Boundary Conditions: Supported Edge
Simply
Clamped Edge
Opening: Square hole
Circular hole
Parameters: 1. Aspect Ratio of the plate ( b/h = 1.0 and 1.5 )
2. The dimensions of the cut out
3. The location of the hole.
4. The edge support conditions
Loading: Shear loading
Studies on the elastic critical stresses of perforated plates in shear were carried out using
the finite element method of analysis.
The cases considered are:
1. Square and Rectangular plates with central circular cut-outs.
2. Square Plates with Centrally placed rectangular cut-outs.
3. Square Plates with eccentrically placed rectangular cut-outs.
4. Square Plates with reinforced circular cut-outs.
5. Square Plates with reinforced rectangular cut-outs.
Buckling coefficients for all the cases are avialable in Fig. 1 to Fig 16 of the above aritcle.
Approximate formula for the use of designers have been suggested for the practical caese
where the hole diameter is generally not greater than half the width of the plate.
Table 1: Convergence study for Shear Buckling Coefficients, using the Finite Element Method
Number of elements Simply supported clamped edges
used ko % error ko % error
Square plate
Exact value 9.34 - 14.71 -
32 8.28 -11.35 - -
72 8.75 -6.31 12.50 -15.00
128 8.97 -3.96 13.24 -10.00
200 9.08 -2.78 13.90 -5.50
512 9.15 -2.03 14.27 -3.00
Rectangular plate
Exact value 7.12 - 11.50 -
108 6.05 -15.03 9.60 -16.50
224 6.65 -6.60 10.50 -8.70
300 6.86 -3.65 11.04 -4.00
588 6.99 -1.82 11.24 -2.20
Conclusions:
1. It was found that for rectangular paltes containing central circular sut-outs there is a near-linear relationship between
the shear buckling coefficient and the diameter/diagonal of palte ratio. The value of 'k' increases as the diameter of the h
2. For square plates containing rectangular openings an approximately linear relationship was also found between 'k/ko' and
area of cut-out/total area of the plate ratio. The value of k decreases with an increase in the size of the rectangular hole.
3. For the case of square plate with an eccentric circular hole, the value of k diminishes compared with a centrally placed
circular cut-out when the centre of the hole is moved along the compression diagonal, there is significant increase in the va
4. The load corresponding to the elastic critical stress of a plate with a central circular hole is improved considerably due t
presence of reinforcement rings around the hole. It was found that in all cases the critical load equivalent
to an unperforated plate was achieved by employing a relatively small size of reinforcement ring.
5. For the case of a square plate containing a square opening, reinforced with strips above and below the cut-out, it was
found that the laod corresponding to elastic critical stresses of an equivalent unperforated plate can be achieved if the
total length of each reinforcement strip is at least 1.5 times the width of the cut-out and the thickness is at least three
times the thickness of the plate for all practical widths of reinforcement. The variation of k with the width of the reinfor
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
there is a near-linear relationship between
alue of 'k' increases as the diameter of the hole increases.
lationship was also found between 'k/ko' and the
n increase in the size of the rectangular hole.
iminishes compared with a centrally placed
agonal, there is significant increase in the value of k.
l circular hole is improved considerably due to the
s the critical load equivalent
trips above and below the cut-out, it was
unperforated plate can be achieved if the
cut-out and the thickness is at least three
e variation of k with the width of the reinforcement strip is very small.
d edges not free to pull in
d unloaded edges free to pull in
two unloaded edges simly supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 12
Date : 1985
Subject : Ultimate Capacity of Plates Containing Holes under Linearly Varying Edge Displacements
Title : Computers and Structures, Vol. 21, No. 4
Author : Narayanan, R. and Chan, S.L
Structure: Plate
Material : Steel
An approximate method is suggested for evaluating the ultimate strength of plates containing circular
holes having simply supported boundary conditions (with edges free to pull in or
edges kept straight) when subjected to linearly varying edge displacements.
The theoretical treatment outlined in this paper is based on the Energy approach originally
proposed by Horne and Narayanan for unperforated plates under uniform loading.
TABLE 1. TEST RESULTS FOR PLATES UNDER A TRAPEZOIDAL EDGE LOAD
No Ratio of two extreme axb t b/t d/b e/b
edge strains
1 * 85.6x85.8 1.93 43.88 0.189 0.128
2 2 86.0x85.9 1.93 44.56 0.194 0.125
3 3 86.0x85.9 1.93 44.51 0.191 0.126
4 * 86.0x85.7 1.93 44.40 0.187 0.223
5 2 85.8x85.9 1.93 44.51 0.184 0.225
6 3 86.0x85.7 1.93 44.40 0.184 0.218
7 * 85.9x85.6 1.93 44.35 0.278 0.127
8 2 85.7x85.5 1.93 44.30 0.276 0.130
9 3 86.0x85.8 1.93 44.46 0.280 0.128
10 * 85.8x85.5 1.02 83.82 0.187 0.223
11 2 85.8x85.6 1.02 83.92 0.189 0.220
12 3 85.7x85.7 1.02 84.02 0.191 0.208
13 * 85.5x85.7 1.02 84.02 0.188 0.123
14 2 85.7x85.5 1.02 83.82 0.191 0.120
15 3 85.7x85.5 1.02 84.02 0.189 0.124
16 * 85.7x85.5 1.02 83.82 0.282 0.115
17 2 85.7x85.8 1.02 84.12 0.280 0.114
18 3 85.7x85.6 1.02 83.92 0.282 0.114
Specimens marked * were subjected to a triangular edge displacement
TABLE 2: COMPARISION WITH NARAYANAN AND CHOW's EXPERIMENTAL RESULTS
Specimen No b/t diameter/b Measured Measured Meas.Load/
YldStress Load Squash Load
(N/mm2) (kN)
CIR2a 77.40 0.200 323.3 37.46 0.574
CIR2b 77.40 0.200 323.3 38.70 0.593
CIR3a 77.40 0.300 323.3 33.94 0.520
CIR4a 77.40 0.400 323.3 29.57 0.453
CIR4b 77.40 0.400 323.3 28.39 0.435
CIR5a 77.40 0.500 323.3 27.35 0.419
CIR6 42.30 0.291 334.7 42.17 0.721
CIR7 53.25 0.291 323.3 26.18 0.583
CIR8 88.48 0.291 317.6 12.35 0.465
CIR9 124.10 0.291 322.8 7.33 0.381
CIR10 42.30 0.465 334.7 33.64 0.575
CIR11 53.25 0.465 323.3 22.14 0.493
CIR12 88.48 0.465 317.6 10.89 0.410
Standrad Deviation
Conclusions:
1. A rapid method of evaluating the ulitmate strengths has been presented for approximately square plates containing
circular openings and having simply supported boundary conditions (with edges free to pull in or edges kept straight) when s
2. The theoretical treatment is approximate, but adequate.
3. It has been shown that for many practical plates (e.g. b/t<50), there is very little difference in the
ulitmate capacity, whether the edges are free to pull in or kept straight.
4. Holes located away form the centre do not cause any significant difference in the ulitmate capacity.
5. Plates subjected to increasing ratio of edge strains showed decreasing strength.
6.
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
oles under Linearly Varying Edge Displacements
ngth of plates containing circular
rgy approach originally
Measured Measured Meas.Load/ Pre.Load/ Pre.Load/
YldStress Load Squash Load Squash Load Meas.Load
(N/mm2) (kN)
220.4 17.2 0.471 0.456 0.947
220.4 24.0 0.657 0.625 0.951
220.4 21.2 0.580 0.568 0.979
220.4 17.3 0.475 0.454 0.956
220.4 23.9 0.654 0.625 0.960
220.4 19.7 0.540 0.572 1.059
220.4 15.9 0.437 0.430 0.984
220.4 21.0 0.577 0.587 1.017
220.4 19.4 0.532 0.538 1.011
223.6 6.2 0.318 0.347 1.091
223.6 8.2 0.420 0.454 1.081
223.6 8.1 0.414 0.421 1.017
223.6 6.3 0.322 0.345 1.071
223.6 8.7 0.446 0.459 1.029
223.6 7.4 0.379 0.423 1.116
223.6 6.0 0.308 0.338 1.097
223.6 7.9 0.404 0.448 1.109
223.6 7.3 0.374 0.410 1.096
Mean 1.032
Standrad Deviation 0.060
XPERIMENTAL RESULTS
Pre.Load/ Pre.Load/
Squash Load Meas.Load
0.528 0.920
0.528 0.890
0.518 0.996
0.488 1.077
0.488 1.122
0.454 1.084
0.674 0.935
0.626 1.074
0.490 1.054
0.423 1.110
0.545 0.948
0.529 1.073
0.455 1.110
Mean 1.030
Standrad Deviation 0.081
ented for approximately square plates containing
ith edges free to pull in or edges kept straight) when subjected to uniformly varying edge displacements.
re is very little difference in the
fference in the ulitmate capacity.
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 13
Date : 1985
Subject : Experiments on Perforated Plates Subjected to Shear
Title : Journal of Strain Analysis, Vol 20, No. 1
Author : Narayanan, R and Chow, F.Y
Structure: Plate
Openings: Square and Circular
Experiments on the buckling and ultimate capacity of thin walled steel web plates containg circular and square cut-outs are
Some 64 tests were carried out on plates containing either one or two holes.
The parameters varied included the size of the hole and the eccentricity of the location of the hole.
Four series of tests were performed on plates containing opeings of various sizes.
Actual sizes of panels: 215.9 x 215.9 mm
Average Thickness : 0.976 mm
Approximate formulae are suggested in the paper for computing the shear buckling coefficents
for plates with central and eccentric holes, and are suitable for use by designers.
A method of predicting accurately, the ultimate capacity of the web is presented.
Table 1. Experimental results for square shear plates having centrally placed holes
Measured Values
Type of cut-out d/a Pcr Pult Predicted values
a'/a for Clamped edges
(kN) (kN)
(1) (2) (3) (4) (5)
All panels with a 0 14.08 57.00 14.71
central circular hole 0.1 12.10 53.60 12.90
0.2 10.00 51.43 11.70
0.3 9.06 45.00 10.40
0.4 8.05 - 9.15
0.5 7.60 37.97 8.10
0.6 6.35 - 7.15
All panels with a 0.1 13.02 Not tested 13.75
central square hole 0.2 11.44 Not tested 11.70
0.3 9.00 Not tested 9.65
0.4 8.50 Not tested 8.15
0.5 6.78 Not tested 6.80
0.6 4.70 Not tested 5.70
Table 2. Experimental results for square shear panels having eccentrically located circular holes
Measured Values
Location of hole d/a e/a Pcr Pult Predicted values
for Clamped edges
(kN) (kN)
(1) (2) (3) (4) (5) (6)
Hole located in 0.1 0.1 12.75 13.56
tension diagonal 0.2 12.05 13.30
0.3 12.80 55.60 13.29
0.4 11.80 13.31
0.2 0.1 10.29 45.00 11.20
0.2 9.40 10.97
0.3 10.00 10.89
0.3 0.1 9.30 46.80 9.27
0.2 7.80 7.93
0.3 6.85 7.67
0.4 0.1 6.26 6.40
0.2 5.08 5.69
0.5 0.1 4.76 5.00
0.2 4.47 4.20
Hole located in 0.1 0.1 13.25 13.70
compression diagonal 0.2 12.60 14.15
0.3 13.36 14.47
0.4 13.16 14.69
0.2 0.1 12.89 50.80 13.00
0.2 14.57 50.70 15.79
0.3 0.1 10.08 46.40 11.95
0.2 14.12 15.83
0.4 0.1 9.32 45.60 9.91
0.2 13.08 45.80 14.38
0.5 0.1 8.42 8.82
Table 3. Experimental results for square shear panels having eccentrically located square holes
Measured Values
Location of hole d/a e/a Pcr Pult Predicted values
for Clamped edges
(kN) (kN)
(1) (2) (3) (4) (5) (6)
Hole located in 0.1 0.1 13.63 - 13.82
tension diagonal 0.2 12.39 49.00 13.97
0.3 13.40 50.60 13.47
0.4 11.44 - 13.52
0.2 0.1 10.99 42.00 10.08
0.2 8.58 42.60 9.86
0.3 7.36 42.60 9.76
0.3 0.1 8.36 36.40 7.60
0.2 7.51 36.60 6.93
0.3 5.81 38.80 6.38
0.4 0.1 5.12 29.80 5.56
0.2 4.39 31.00 4.82
0.5 0.1 4.36 26.20 4.41
0.2 3.61 26.00 3.66
Hole located in 0.1 0.1 13.89 - 14.80
compression diagonal 0.2 12.97 52.20 15.35
0.3 14.08 54.60 15.23
0.4 13.21 - 15.35
0.2 0.1 12.83 41.50 13.35
0.2 13.45 43.80 16.15
0.3 14.92 46.80 18.60
0.3 0.1 10.58 33.60 11.53
0.2 13.31 35.60 17.05
0.4 0.1 9.70 - 9.72
0.2 15.17 32.80 18.30
0.5 0.1 8.83 - 8.67
Table 4. Experimental results for square shear panels having two eccentrically located holes
Measured Values
Location of hole d/a Pcr Pult Predicted values
for Clamped edges
(kN) (kN)
(1) (2) (3) (4) (5)
Hole located in 0.10 9.34 56.00 8.04
tension diagonal 0.19 7.42 43.80 5.89
0.29 4.75 41.80 3.60
0.38 3.84 39.40 2.37
Hole located in 0.10 11.51 53.40 10.30
compression diagonal 0.19 9.93 42.80 7.70
0.29 5.70 36.20 5.90
0.38 5.28 30.80 3.60
Eccentricity, e=53.975 mm
Conclusions:
1. For plates containing central cut-outs (both square and circualr) the buckling coefficient drops as the diameter of the ho
An approximate equation suitable for design use is suggested.
2. For small holes where 'd/a' or 'a'/a'<0.1, the buckling coefficient is virtually insensitive to the location of holes,
whether located eccentrically in the compression diagonal or in the tension diagonal.
3. For hole sizes larger than 0.1a, the value of buckling coefficient reduces as the centre of hole is shifted along the tensio
diagonal towards the corner. However, when the centre of the hole ismoved along the compression diagonal towards
the corner of the plate, there is a considerable increase in the value of buckling coefficient. for design purposes,
an approxiamte equation has been suggested for paltes containing a cut-out located in tension diagonal.
4. For plates with two holes located at the two ends of a diagonal, the buckling coefficient reduced with the increase in
the hole sizes. A plate with two holes located in the compression diagonal gives a higher value of
buckling resistance than that of a similar plate with holes located in the tension diagonal.
5. A method of accurately predicting the ultimate capacity of perforated webs is presented. For plates with a single hole,
the ultimate strengths are insensitive to the location of the holes, but reduce with the increase of the hole sizes.
6. For plates with two holes, the ultimate strengths are higher when the holes are located in the tension diagonal,
compared with webs with similar holes located in the compression diagonal. However, these conclusions are only restricted
to webs with very stiff members where a full tension field would develop across the entire web in the post critical stage.
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
el web plates containg circular and square cut-outs are reported.
city of the location of the hole.
shear buckling coefficents
Buckling Coefficients Ultimate Strengths
Predicted values for Observed (5)/(7) (6)/(7) Predicted (10)/(4)
Simply Supported edges Values Values
(kN)
(6) (7) (8) (9) (10) (11)
9.34 13.95 1.054 0.670 52.99 0.930
8.45 11.99 0.929 0.705 52.02 0.982
7.30 9.91 1.181 0.737 50.10 0.974
6.05 8.97 1.159 0.674 48.42 1.076
4.90 7.97 1.148 0.615 44.27 -
3.85 7.53 1.076 0.511 40.72 1.073
2.90 6.29 1.137 0.461 36.88 -
8.50 12.90 1.066 0.659 50.47 -
6.75 11.33 1.035 0.596 47.63 -
5.15 8.91 1.083 0.578 43.36 -
3.90 8.42 0.968 0.463 38.26 -
2.90 6.72 1.012 0.432 32.63 -
2.10 4.66 1.223 0.451 26.81 -
Mean 1.082 0.581 1.007
Standard 0.085 0.106 0.064
Deviation
entrically located circular holes
Buckling Coefficients Ultimate Strengths
Predicted values Predicted values for Observed (6)/(8) Predicted (10)/(5)
for Clamped edges Simply Supported edges Values Values
(kN)
(7) (8) (9) (10) (11)
8.47 12.63 1.074 51.16
8.48 11.94 1.114 51.33
8.38 12.68 1.048 51.89 0.933
8.51 11.69 1.138 51.33
7.30 10.20 1.098 49.59 1.102
6.94 9.31 1.178 49.70
6.96 9.91 1.099 49.74
5.26 9.21 1.007 47.57 1.016
5.14 7.72 1.027 47.60
5.11 6.79 1.130 47.67 1.036
3.83 6.20 1.032 44.38
3.70 5.03 1.131 44.48
2.82 4.71 1.062 40.74
2.58 4.43 0.948 40.75
9.76 13.12 1.044 51.07
9.86 12.48 1.134 50.09
9.90 13.23 1.094 50.57
9.68 13.04 1.127 50.43
8.01 12.77 1.018 48.71 0.959
9.80 14.43 1.094 47.27 0.932
6.66 9.98 1.197 46.28 0.997
8.32 13.98 1.132 45.23
6.29 9.23 1.074 43.95 0.964
8.65 12.96 1.110 42.78 0.934
4.50 8.34 1.058 40.66
Mean 1.087 0.986
Standard 0.056 0.058
Deviation
entrically located square holes
Buckling Coefficients Ultimate Strengths
Predicted values Predicted values for Observed (6)/(8) Predicted (10)/(5)
for Clamped edges Simply Supported edges Values Values
(kN)
(7) (8) (9) (10) (11)
8.62 13.50 1.024 50.46 -
8.64 12.27 1.139 50.38 0.970
8.72 13.27 1.015 50.21 0.990
9.41 11.33 1.193 49.61 -
6.30 10.88 0.927 47.47 1.130
6.29 8.50 1.160 47.54 1.110
6.22 7.29 1.339 47.58 1.110
4.59 8.28 0.918 43.33 1.190
4.52 7.44 0.931 43.42 1.180
4.42 5.75 1.110 43.49 1.120
3.25 5.07 1.097 38.26 1.280
3.07 4.35 1.108 38.21 1.230
2.29 4.32 1.021 32.65 1.240
2.08 3.58 1.022 32.16 1.230
9.98 13.76 1.076 49.30 -
9.82 12.85 1.195 48.69 0.930
9.60 13.94 1.093 49.04 0.890
9.78 13.09 1.173 48.96 -
8.35 12.71 1.050 46.61 1.120
11.10 13.32 1.212 44.23 1.010
13.20 14.78 1.258 43.17 0.920
7.28 10.48 1.100 42.90 1.200
10.10 13.18 1.294 41.53 1.100
5.23 9.61 1.011 38.39 -
9.47 15.03 1.218 37.95 1.100
3.88 8.75 0.991 33.44 -
Mean 1.103 1.1
Standard 0.112 0.11
Deviation
eccentrically located holes
Buckling Coefficients Ultimate Strengths
Predicted values for Observed (5)/(7) Predicted (9)/(4)
Simply Supported edges Values Values
(kN)
(6) (7) (8) (9) (10)
7.40 9.25 0.969 52.39 0.936
3.50 7.35 0.801 50.91 1.162
1.58 4.70 0.766 48.26 1.155
0.92 3.80 1.241 44.93 1.140
10.30 11.40 1.018 51.02 0.955
7.70 9.84 0.919 45.20 1.056
5.90 6.29 1.094 37.54 1.037
3.60 5.23 1.008 29.92 0.971
e buckling coefficient drops as the diameter of the hole increases.
s virtually insensitive to the location of holes,
duces as the centre of hole is shifted along the tension
smoved along the compression diagonal towards
of buckling coefficient. for design purposes,
ut-out located in tension diagonal.
e buckling coefficient reduced with the increase in
gonal gives a higher value of
the tension diagonal.
ated webs is presented. For plates with a single hole,
ut reduce with the increase of the hole sizes.
the holes are located in the tension diagonal,
gonal. However, these conclusions are only restricted
velop across the entire web in the post critical stage.
Mean 0.977 1.052
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 14
Date : 1985
Subject : Ultimate Strength of Axially Compressed Stiffened Plates Containing Openings
Title : Proc., International Conference on Metal Structures
Author : Shanmugam, N.E., Paramasivam, P and Lee, S.L
Structure: Plate
Material : Steel
An approximate method using the effective width concept is proposed to predict the ultimate
strength of axially compressed stiffened panels containing openings.
The method allows for the loss in stiffness of plates by making use of the 'effective width' concept.
Systematic experiments that have been carried out on small-scale steel models of perofrated stiffened plates are reported
TABLE 1 : DETAILS OF TEST SPECIMENS
Specimen b t b/t d or h d/b or
(mm) (mm) (mm) h/b
CIR 1 99.8 0.95 105 0 0.0
CIR 2 100.3 0.95 106 20 0.2
CIR 3 99.8 0.95 105 40 0.4
CIR 4 99.9 0.95 105 60 0.6
CIR 5 99.2 0.95 104 70 0.7
CIR 6 59.9 0.95 63 24 0.4
CIR 7 60 0.95 63 36 0.6
CIR 8 60 0.95 63 42 0.7
CIR 9 80 0.95 84 32 0.4
CIR 10 80 0.95 84 48 0.6
CIR 11 80.1 0.95 84 56 0.7
SQR 1 100.2 0.95 105 20 0.2
SQR 2 100.1 0.95 105 40 0.4
SQR 3 99.8 0.95 105 60 0.6
SQR 4 99.9 0.95 105 70 0.7
Table 2: Summary of Results
Specimen b/t d/b or Psq Expt. Theoret.
h/b Failure Failure
(kN) Load (kN) Load (kN)
(1) (2) (3) (4) (5) (6)
CIR 1 105 0.0 38.70 24.72 26.26
CIR 2 106 0.2 34.42 23.71 25.60
CIR 3 105 0.4 29.25 23.30 24.04
CIR 4 105 0.6 25.48 22.16 23.44
CIR 5 104 0.7 22.19 21.42 21.37
CIR 6 63 0.4 23.72 20.51 22.62
CIR 7 63 0.6 20.96 20.24 20.66
CIR 8 63 0.7 19.73 17.27 19.16
CIR 9 84 0.4 26.31 20.56 23.37
CIR 10 84 0.6 22.85 19.75 21.94
CIR 11 84 0.7 21.09 18.43 20.70
SQR 1 105 0.2 33.95 22.72 24.90
SQR 2 105 0.4 28.81 21.55 23.14
SQR 3 105 0.6 25.38 21.58 23.17
SQR 4 105 0.7 22.43 19.18 21.49
The effective width method, suggested by Yu and Davies, though approximate, is efficient
and economical for design office use where repetitive analyses are required at the preliminary design stage.
Steps involved in computing the collapse load:
Step 1. Assume the maximum edge stress to reach 90% of yield stress
Step 2. Determine the effectivewidth of the flange plate between the stiffeners by Yu and Davies formula
E æ d öé t E æ d öù
be = 1.9t ç1 - 0.226 ÷ ê1 - 0.415 ç1 - 0.0379 ÷ú
se è b øë b-d se è b øû
Step 3. The effective column cross section will now be of T shape consisting of the stiffeners, effective flange width and
overhanging portions of the flange palte, if any. Calcualte the radius of gyrations about the centroidal axis.
Step 4. Calcualte themean stress at collapse of the effective column by making use of the Perry-Robertson formula
sa 1 é s ù ì1 é s eu ù s eu ü
2
ï ï
= ê1 + (1 + h ) eu ú - í ê1 + (1 + h ) ú - ý
sy 2êë sy úû ï4 ê
ë sy ú û sy ï
î þ
Step 5. Calcualte the collapse load as the product of the mean stress and the area of the effective column cross section.
Conclusions:
1. The method, though approximate, is efficient and economical for design office use where
repetitive analyses are required at the preliminary design state.
2. The results clearly show that the introduction of openings has a significant
effect on the ultimate capacity of the stiffened panels.
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
d Stiffened Plates Containing Openings
d to predict the ultimate
f the 'effective width' concept.
eel models of perofrated stiffened plates are reported.
Dw Tw
(mm) (mm)
15.9 1.6
16.0 1.6
15.7 1.55
16.1 1.6
15.9 1.5
16.5 1.45
16.4 1.45
16.5 1.45
16.4 1.45
16.5 1.45
16.6 1.45
15.4 1.6
15.0 1.55
16.0 1.6
15.5 1.55
(5)/(4) (6)/(4) (7)/(8)
(7)
0.64 0.68 0.94
0.69 0.74 0.93
0.80 0.82 0.98
0.87 0.92 0.95
0.97 0.96 1.01
0.86 0.95 0.91
0.97 0.99 0.98
0.88 0.97 0.91
0.78 0.89 0.88
0.86 0.96 0.90
0.87 0.98 0.89
0.67 0.73 0.92
0.75 0.80 0.94
0.85 0.91 0.93
0.86 0.96 0.90
proximate, is efficient
equired at the preliminary design stage.
he stiffeners by Yu and Davies formula
nsisting of the stiffeners, effective flange width and
of gyrations about the centroidal axis.
by making use of the Perry-Robertson formula
and the area of the effective column cross section.
esign office use where
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 15
Date : 1986
Subject : Stiffened Flanges Containing Openings
Title : Journal of Structural Engineering, Vol. 112, No. 10
Author : Shanmugam, N.E., Paramasivam, P and Lee, S.L
Structure: Plate
Material : Steel
An approximate method using the effective width concept is proposed to predict the ultimate
strength of axially compressed stiffened panels containing openings.
The method allows for the loss in stiffness of plates by making use of the 'effective width' concept.
Systematic experiments that have been carried out on small-scale steel models of perofrated stiffened plates are reported
TABLE 1 : DETAILS OF TEST SPECIMENS
Specimen b t b/t d or h d/b or
(mm) (mm) (mm) h/b
CIR 1 99.8 0.95 105 0 0.0
CIR 2 100.3 0.95 106 20 0.2
CIR 3 99.8 0.95 105 40 0.4
CIR 4 99.9 0.95 105 60 0.6
CIR 5 99.2 0.95 104 70 0.7
CIR 6 59.9 0.95 63 24 0.4
CIR 7 60 0.95 63 36 0.6
CIR 8 60 0.95 63 42 0.7
CIR 9 80 0.95 84 32 0.4
CIR 10 80 0.95 84 48 0.6
CIR 11 80.1 0.95 84 56 0.7
SQR 1 100.2 0.95 105 20 0.2
SQR 2 100.1 0.95 105 40 0.4
SQR 3 99.8 0.95 105 60 0.6
SQR 4 99.9 0.95 105 70 0.7
Table 2: Summary of Results
Specimen b/t d/b or Psq Expt. Theoret.
h/b Failure Failure
(kN) Load (kN) Load (kN)
(1) (2) (3) (4) (5) (6)
CIR 1 105 0.0 38.70 24.72 26.26
CIR 2 106 0.2 34.42 23.71 25.60
CIR 3 105 0.4 29.25 23.30 24.04
CIR 4 105 0.6 25.48 22.16 23.44
CIR 5 104 0.7 22.19 21.42 21.37
CIR 6 63 0.4 23.72 20.51 22.62
CIR 7 63 0.6 20.96 20.24 20.66
CIR 8 63 0.7 19.73 17.27 19.16
CIR 9 84 0.4 26.31 20.56 23.37
CIR 10 84 0.6 22.85 19.75 21.94
CIR 11 84 0.7 21.09 18.43 20.70
SQR 1 105 0.2 33.95 22.72 24.90
SQR 2 105 0.4 28.81 21.55 23.14
SQR 3 105 0.6 25.38 21.58 23.17
SQR 4 105 0.7 22.43 19.18 21.49
The effective width method, suggested by Yu and Davies, though approximate, is efficient
and economical for design office use where repetitive analyses are required at the preliminary design stage.
Steps involved in computing the collapse load:
Step 1. Assume the maximum edge stress to reach 90% of yield stress
Step 2. Determine the effectivewidth of the flange plate between the stiffeners by Yu and Davies formula
E æ d öé t E æ d öù
be = 1.9t ç1 - 0.226 ÷ ê1 - 0.415 ç1 - 0.0379 ÷ú
se è b øë b-d se è b øû
Step 3. The effective column cross section will now be of T shape consisting of the stiffeners, effective flange width and
overhanging portions of the flange palte, if any. Calcualte the radius of gyrations about the centroidal axis.
Step 4. Calcualte themean stress at collapse of the effective column by making use of the Perry-Robertson formula
sa 1 é s ù ì1 é s eu ù s eu ü
2
ï ï
= ê1 + (1 + h ) eu ú - í ê1 + (1 + h ) ú - ý
sy 2êë sy úû ï4 ê
ë sy ú û sy ï
î þ
Step 5. Calcualte the collapse load as the product of the mean stress and the area of the effective column cross section.
Conclusions:
1. The method, though approximate, is efficient and economical for design office use where
repetitive analyses are required at the preliminary design state.
2. The results clearly show that the introduction of openings has a significant
effect on the ultimate capacity of the stiffened panels.
|::
Plate 19
Date : [***-99]
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
The accuracy of the analytical method by using ABAQUS was established by analysing
a number of perforated plates tested by Narayanan and Chow.
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
Design Equation : Pult/Psqu = C1( d/b) ^2+ C2 (d/b) + C3 ; 0 < d/b < 0.7 ; S=(b/t)/100 ; 20 < b/t <70
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
|::
d to predict the ultimate
f the 'effective width' concept.
eel models of perofrated stiffened plates are reported.
Dw Tw
(mm) (mm)
15.9 1.6
16.0 1.6
15.7 1.55
16.1 1.6
15.9 1.5
16.5 1.45
16.4 1.45
16.5 1.45
16.4 1.45
16.5 1.45
16.6 1.45
15.4 1.6
15.0 1.55
16.0 1.6
15.5 1.55
(5)/(4) (6)/(4) (7)/(8)
(7)
0.64 0.68 0.94
0.69 0.74 0.93
0.80 0.82 0.98
0.87 0.92 0.95
0.97 0.96 1.01
0.86 0.95 0.91
0.97 0.99 0.98
0.88 0.97 0.91
0.78 0.89 0.88
0.86 0.96 0.90
0.87 0.98 0.89
0.67 0.73 0.92
0.75 0.80 0.94
0.85 0.91 0.93
0.86 0.96 0.90
proximate, is efficient
equired at the preliminary design stage.
he stiffeners by Yu and Davies formula
nsisting of the stiffeners, effective flange width and
of gyrations about the centroidal axis.
by making use of the Perry-Robertson formula
and the area of the effective column cross section.
esign office use where
ply supported and edges not free to pull in
ply supported and unloaded edges free to pull in
mply supported and one edge free
es clamped and two unloaded edges simly supported
ped and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
7 ; S=(b/t)/100 ; 20 < b/t <70
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
Plate 16
Date : 1987
Subject : Simplified Procedures for the Strength Assessment of Axially Compressed Plates with or wit
Title : Proc., International Conference on Steel and Aluminum Structures
Author : Narayanan, R
Structure: Plate
Material : Steel
Cases : 1. Uniaxially Compressed Perforated Plates
2. Biaxially Compressed Perforated Plates
3. Perforated Plates under Linearly Varying Edge Displacements
An effective width formulation based on the energy method is presented for the rapid
computation of the strength of plates having various boundary conditions and subjected to
uniaxial or biaxial compression, or uniformly varying edge displacement. The method has
been extended satidfactorily to plates containing openings; comparisions with available
test results demonstrate the adequacy of the method.
Design Curves for each cases are avilable in Figure 11 , 14 and 15 respectively in the
above article.
A summary is provided in the table;
fuller details of the tests are available in the refernce cited.
Table 1: Summary of tests
Details of Tests Summary of tests
and refernce Number of Mean value for Standard
Tests Predicted/Measured LoadDeviation
Uniaxially Compressed 23 1.010 0.080
Plates
Biaxially Compressed 16 0.980 0.055
Plates
Plates subjected to 22 0.998 0.061
uniformly varying
displacements
Uniaxially compressed 18 1.015 0.037
plates containing
openings
Biaxially compressed 15 1.046 0.074
plates containing
openings
Plates with openings 18 1.032 0.060
subjected to uniformly
carying edge displacements
Conclusions:
Simplified effective width methods, based on Energy approach, have been formulated for the rapid assessment of the
ultiamte capacity of plates with or without openings subjected to uniaxial or biaxial compression.
sessment of Axially Compressed Plates with or without Openings
d Aluminum Structures
Edge Displacements
formulated for the rapid assessment of the
or biaxial compression.
Plate 17
Date : 1994
Subject : Strength of Stiffened Plates with Openings
Title : Proc., Twelfth Internatinal Specialty Conference on Cold-Formed Steel Structures
Author : Mahendran, M., Shanmugam, N.E and Richard Liew, J.Y
Structure: Plate
Material : Steel
The load-deflection and ultimate strength behaviour of longitudinally stiffened plates with openings
was studied usin a second-order elastic post-buckling analysis and a rigid-plastic analysis.
The following procedure was used to calculate the ultimate load of perforated stiffened plates.
Step 1 From the intersection point of elastic and rigid-plastic curves
in Figs 2 and 3, obtain (Avg. Sress/ Yield Stress) for the plate
component (perforated or unperforated) between the stocky stiffeners
Step 2 Calculate the effective width Be of the plate component using the
relationship, Be= B (Avg. Stress/Yield Stress) where B is the plate width.
Step 3 The effective column cross section thus becomes a number of T-Sections consisting of
the stiffeners, the plate of width Be and the overhanging parts of the plate. For the effective
cross section calculate the total area Ae, the radius of gyration, Rex= Sqrt(Iex/Ae)
about the centroidal axis and the slederness=Length/Rex.
Step 4 For the calculated slenderness, determine the compressive strength
sc
using the Perry strut formula ( BS5950, 1990) s eulers yield
sc =
f + (f 2 - s eulers yield )
0.5
Step 5 Calculate the ultimate load which is the product of compressive
strength and the total effective area of the stiffened plate.
In order to determine the accuracy of the simplified procedure, the predicted ultimate loads
were compared with available perforated stiffened plate tests results by Liew and Shanmugam .
TABLE 1: BUCKLING COEFFICEINT FOR SIMPLY SUPPORTED PLATES WITH SQUARE OPENINGS
Size of Opening b'/b 0 0.1 0.2 0.3 0.4
Buckling Coefficient 4.00 3.75 3.40 3.15 3.05
The above Procedure was used for the tested panels and the results are given in the table Below.
Table 2: EXPERIMENTAL INVESTIGATION
b t b' b/t b'/b ds
(mm) (mm) (mm)
1 100 0.95 20 105 0.20 15.4
2 100 0.95 40 105 0.40 15.0
3 100 0.95 60 105 0.60 16.0
4 100 0.95 70 105 0.70 15.5
5 135 2.1 62 65 0.45 50.0
Conclusions:
1. The ultimate strength behaviourof longitudinally stiffened plates with openings under axialcompression
was studied using second-order elastic and rigid-plastic analyses and laboratory experiemnts.
2. The ultimate strengths of stiffened paltes were obtained from the point of intersection of the elastic and
rigid-plastic curves and the perry strut formula and compared with experimental results.
3. Theoretical predictions compared reasonably well andthus validatedthe simple method usedin this investigation.
4. Effects of the size of opening, the initial geometrical imperfections and the plate slenderness ratio on
the strength of perforated stiffened plates were also studied.
ference on Cold-Formed Steel Structures
stiffened plates with openings
igid-plastic analysis.
erforated stiffened plates.
d rigid-plastic curves
Stress) for the plate
between the stocky stiffeners
plate component using the
ress) where B is the plate width.
becomes a number of T-Sections consisting of
the overhanging parts of the plate. For the effective
, the radius of gyration, Rex= Sqrt(Iex/Ae)
e the compressive strength
sc
s eulers yield
sc =
f + (f 2 - s eulers yield )
0.5
roduct of compressive
the stiffened plate.
predicted ultimate loads
ts by Liew and Shanmugam .
LATES WITH SQUARE OPENINGS
0.5 0.6 0.7
3.03 3.01 3.00
are given in the table Below.
ts Psq Pexp Pthe Pexp/Psq Pthe/Psq Pexp/Psq
(mm) (kN) (kN) (kN)
1.60 33.95 22.72 25.37 0.67 0.75 0.90
1.55 28.81 21.55 22.82 0.75 0.79 0.94
1.60 25.38 21.58 22.19 0.85 0.87 0.97
1.55 22.43 19.18 20.39 0.86 0.91 0.94
5.00 582.80 378.00 391.90 0.65 0.67 0.97
with openings under axialcompression
laboratory experiemnts.
e point of intersection of the elastic and
dthe simple method usedin this investigation.
ns and the plate slenderness ratio on
Plate 18
Date : 1996
Subject : Design of Stiffened Plates with Openings
Title : Journal of The Institution of Engineers, Singapore, Vol.36, No.2, pp.15-21
Author : Mahendran, M., Shanmugam, N.E and Richard Liew, J.Y
Structure: Plate
Material : Steel
The load -deflection and ultimate strength behaviour of longitudinally stiffened plates with openings
was studied using a second-order elastic post-buckling analysis and a rigid-plastic analysis.
The ultimate strength was predicted from the intersection point of elastic and rigid-plastic curves and the Perry strut form
Comparison with experimental results shows that satisfactory prediction of ulitmate strength can be obtained by this simpl
Effects of the size of opening, the initial geometrical imperfections and the plate slenderness ratio on the strength of perf
The following procedure was used to calculate the ultimate load of perforated stiffened plates.
Step 1 From the intersection point of elastic and rigid-plastic curves
in Figs 2 and 3, obtain (Avg. Sress/ Yield Stress) for the plate
component (perforated or unperforated) between the stocky stiffeners
Step 2 Calculate the effective width Be of the plate component using the
relationship, Be= B (Avg. Stress/Yield Stress) where B is the plate width.
If any of the plate components between the stiffeners does not have an opening,
the appropriate Be should be used in the effective cross section based
on the corresponding intersection point.
Step 3 The effective column cross section thus becomes a number of T-Sections consisting of
the stiffeners, the plate of width Be and the overhanging parts of the plate. For the effective
cross section calculate the total area Ae, the radius of gyration, Rex= Sqrt(Iex/Ae)
about the centroidal axis and the slederness=Length/Rex.
Step 4 For the calculated slenderness, determine the compressive strength
using the Perry strut formula ( BS5950, 1990)
s eulers yield
sc =
f + (f 2 - s eulers yield )
0.5
Step 5 Calculate the ultimate load which is the product of compressive
strength and the total effective area of the stiffened plate.
In order to determine the accuracy of the simplified procedure, the predicted ultimate loads
were compared with available perforated stiffened plate tests results by Liew and Shanmugam .
TABLE 1: BUCKLING COEFFICEINT FOR SIMPLY SUPPORTED PLATES WITH SQUARE OPENINGS
Size of Opening b'/b 0.0 0.10 0.20 0.3 0.4
Buckling Coefficient 4.0 3.75 3.40 3.15 3.05
The above Procedure was used for the tested panels and the results are given in the table Below.
Table 2(a) Summary of results for the specimens tested by Liew (1986)
Opening b'/b Psq kN Pexp Kn Pthe kN Pexp/Psq
1 No 0.00 587 397 427 0.68
2 One in middle panel 0.45 547 400 413 0.73
3 One in an edge panel 0.45 547 402 413 0.74
4 One in each of edge 0.45 508 400 398 0.79
panels
5 One in each three 0.45 468 378 384 0.81
panels
b=135mm; t=2.1mm; b/t=65; ds=50mm; ts=5mm; L=780mm; Ys=310MPa; E=200000
Table 2(b) Summary of results for the specimens tested by Shanmugam et al. (1986)
b' b'/b ds ts Psq kN Pexp Kn
1 20 0.2 15.4 1.60 33.95 22.72
2 40 0.4 15.0 1.55 28.81 21.55
3 60 0.6 16.0 1.60 25.38 21.58
4 70 0.7 15.5 1.55 22.43 19.18
b=100mm; t=0.95mm;b/t=105; L=130mm; Ys=235MPa
b'= size of square opening
ts=thickness of stiffener
Conclusions:
1. The ultimate strength behaviourof longitudinally stiffened plates with openings under axialcompression
was studied using second-order elastic and rigid-plastic analyses and laboratory experiemnts.
2. The ultimate strengths of stiffened paltes were obtained from the point of intersection of the elastic and
rigid-plastic curves and the perry strut formula and compared with experimental results.
3. Theoretical predictions compared reasonably well andthus validatedthe simple method usedin this investigation.
4. Effects of the size of opening, the initial geometrical imperfections and the plate slenderness ratio on
the strength of perforated stiffened plates were also studied.
Singapore, Vol.36, No.2, pp.15-21
y stiffened plates with openings
rigid-plastic analysis.
elastic and rigid-plastic curves and the Perry strut formula.
tion of ulitmate strength can be obtained by this simple method.
and the plate slenderness ratio on the strength of perforated plates were also studied.
erforated stiffened plates.
d rigid-plastic curves
Stress) for the plate
between the stocky stiffeners
plate component using the
ress) where B is the plate width.
the stiffeners does not have an opening,
effective cross section based
becomes a number of T-Sections consisting of
the overhanging parts of the plate. For the effective
, the radius of gyration, Rex= Sqrt(Iex/Ae)
sc
e the compressive strength
s eulers yield
sc =
f + (f 2 - s eulers yield )
0.5
roduct of compressive
the stiffened plate.
predicted ultimate loads
ts by Liew and Shanmugam .
LATES WITH SQUARE OPENINGS
0.5 0.6 0.7
3.03 3.01 3
are given in the table Below.
Pthe/Psq Pexp/Psq
0.73 0.93
0.76 0.97
0.76 0.97
0.78 1.00
0.82 0.98
mugam et al. (1986)
Pthe kN Pexp/Psq Pthe/Psq Pexp/Psq
25.37 0.67 0.75 0.90
22.82 0.75 0.79 0.94
22.19 0.85 0.87 0.97
20.39 0.86 0.91 0.94
with openings under axialcompression
laboratory experiemnts.
e point of intersection of the elastic and
dthe simple method usedin this investigation.
ns and the plate slenderness ratio on
Plate 19
Date : 1999
Subject : Design Formula for Axially Compressed Perforated Plates
Title : Thin-Walled Structures, Vol. 34, pp. 1-20
Author : N.E. Shanmugam, V. Thevendran and Y.H. Tan
Structure: Plate
Material : Steel
YModulus : 200000
PsRatio : 0.3 (assumed)
Section : Plate
Loading : Uniaxial or biaxial Compression
1. A design formula to predict the ultimate laod capacity of perforated paltes with different boundary
conditions and subjected to compressive load is proposed.
2. This was achieved by using the finite element package ABAQUS to carry out an elasto-plastic finite
element analysis on plates with different combinations of boundary conditions, viz. Simply supported or
clamped or the combination of the conditions along with the free edge condition.
3. Two different types of loading conditions, uniaxial or biaxial compression, were investigated.
4. Square or circualr shapes were considered for centrally placed openings and their sizes were varied tocover those that a
5. Plates of varying slenderness were studied.
6. Best fit regression analysis was employed in developingthe design formula to predict the ulitmate load of perforated plat
7. A simplified for the ultimate load in terms of opening area ratio and plate slenderness was formulated.
8. The proposed formulae was verified using ABAQUS results to check the accuracy in predicting the strength of perforat
Table 1: Details of the plate specimens tested by Narayanan and Chow
Specimen No B t b/t D D/B
PL1a 125 1.615 77.4 0 0
CIR2a 125 1.615 77.4 25 0.2
CIR3a 125 1.615 77.4 37.5 0.3
CIR4a 125 1.615 77.4 50 0.4
CIR5a 125 1.615 77.4 62.5 0.5
SQ2 125 1.615 77.4 25 0.2
SQ3 125 1.615 77.4 37.5 0.3
SQ4 125 1.615 77.4 50 0.4
SQ5 125 1.615 77.4 62.5 0.5
Table 2: Comparison of ABAQUS and experimental results
Specimen No D/B Pabaqus Pexp Pabaqus/Pexp
PL1a 0 40.45 39.32 1.03
CIR2a 0.2 36.49 37.46 0.97
CIR3a 0.3 34.97 33.94 1.03
CIR4a 0.4 30.93 29.57 1.05
CIR5a 0.5 29.41 27.35 1.08
SQ2 0.2 32.46 33.48 0.97
SQ3 0.3 27.49 28.85 0.95
SQ4 0.4 26.18 25.52 1.03
SQ5 0.5 23.46 21.86 1.07
Table 3: Comparison of proposed formula and Experimental results
Specimen b/t d/b Pdes/Psq Pexp/Psq Pdes/Pexp
CIR2a 77.4 0.200 0.610 0.574 1.06
CIR3a 77.4 0.300 0.561 0.520 1.08
CIR4a 77.4 0.400 0.501 0.453 1.11
CIR5a 77.4 0.500 0.430 0.419 1.03
CIR6 42.3 0.291 0.702 0.721 0.97
CIR10 42.3 0.465 0.549 0.575 0.95
CIR11 53.3 0.465 0.519 0.493 1.05
CIR12 88.5 0.465 0.428 0.410 1.04
d refers to size of opening
b refers to plate width
Plates were analysed using the finite element method(FEM), and extensive studies were
carried out covering parameters such as plate slenderness, opening size, boundary conditions
and the nature of loading.
A design formula to determine the ultimate load carrying capacity was established
based on a best-fit regression analysis using the results from the finite element analyses.
Design Equation : Pult/Psqu = C1( Ah/Ap) + C2 (Ah/Ap)^2 + C3
Ah = Area of the opening
Ap = Surface area of the square plate
C1,C2,C3 = Co-efficients in terms of plate slenderness , S=0.01 (b/t)
Boundary conditions: 1. Uniaxially Compressed - Four edges simply supported and edges not free to pull in
2. Uniaxially Compressed - Four edges simply supported and unloaded edges free to pull in
3. Biaxially Compressed - Four edges simply supported
4. Uniaxially Compressed - Three edges simply supported and one edge free
5. Biaxially Compressed - Four edges clamped
6. Uniaxially Compressed - Two loaded edges clamped and two unloaded edges simly supported
7. Biaxially Compressed - Two edges clamped and two edges simply supported
8. Uniaxially Compressed - Four edges clamped
Co-efficients value for each case:
2
æA ö æA ö
Design Equation :Pult = C1 ç h ÷ + C 2 ç h ÷ + C3
çA ÷ çA ÷
Psqu è pø è pø
Boundary conditions: C1 C2
1 -0.1S^2-0.39S-0.061 0.6S^2+0.61S-1.0671
2 -0.4S^2-0.51S+0.1125 1.59S-1.3522
3 -10S^3-14S^2+5.8S+0.66 0.1S^2+1.44S-1.14
4 10S^3-12S^2+3.74S-0.28 -0.6S^2+2.13S-1.2913
5 -2S^2+2.2S+0.94 3S^2-2.2S-0.062
0 -0.89
6 -4S^2+1.58S-0.1286 3S^2-1.52S-0.8674
0 -0.96
7 3.0S^2-3.25S+0.404 -2.0S^2+3.58S-1.51
2
Pult ædö ædö b
Design Equation : ; : :
d
= C1 ç ÷ + C 2 ç ÷ + C3 0< < 0 .7 S = (b t ) / 100 20 < < 70
Psqu èbø èbø b t
8 -5S^3+5S^2-1.29S+0.0891 2S^2-1.37S-0.8465
0 -0.97
Conclusions:
1. It is found that the proposed equation gives a slightly conservative prediction with an error of
less than 10% for most cases and this is acceptable design.
2. The ultimate carrying capacity of perforated plates was found to be affected by various parameters studied.
3. The increase in hole size and slenderness ratio reuslts in a significant loss in the ultimate strength ofperforated plates.
4. The strength of perforated plates with simply supported edges is lower as compared to that of plates with clamped edge
5. It was noted that keeping the unloaded edges straight enhances the ultimate load of plates. The
plates with circualr holes generally have higher ultimate loads compared to the square perforated plates.
|::
ltes with different boundary
ry out an elasto-plastic finite
ions, viz. Simply supported or
on, were investigated.
s and their sizes were varied tocover those that are used in practice.
ula to predict the ulitmate load of perforated plates for various cases.
te slenderness was formulated.
e accuracy in predicting the strength of perforated paltes.
Imperfection Imperfection/t Yield stress
0.229 0.141 323.3
0.229 0.142 323.3
0.136 0.084 323.3
0.304 0.188 323.3
0.279 0.173 323.3
0.097 0.060 323.3
0.141 0.087 323.3
0.113 0.070 323.3
0.209 0.129 323.3
y supported and edges not free to pull in
y supported and unloaded edges free to pull in
ply supported and one edge free
s clamped and two unloaded edges simly supported
ed and two edges simply supported
C3 Note
-0.59S+1.1
-0.7S+1.1633
0.6S^2-1.57S+1.2
-1.14S+1.1
-S^2+0.48S+0.76 0 < d/b < 0.4
-0.89 0.4< d/b < 0.7
-S^2+0.49S+0.9585 0 < d/b < 0.4
0.973 0.4< d/b < 0.7
-0.1S^2-0.95S+1.14 0 < d/b < 0.7
-0.6S^2+0.4S+0.9549 0 < d/b < 0.4
0.981 0.4< d/b < 0.7
diction with an error of
fected by various parameters studied.
oss in the ultimate strength ofperforated plates.
r as compared to that of plates with clamped edges.
timate load of plates. The
o the square perforated plates.
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