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Stress concentration factors around a penetration in a strip or slab of finite dimensions
NUCLEAR ENGINEERING AND DESIGN 10 (1969) 356-360. NORTH-HOLLAND PUBLISHING COMPANY, AMSTERDAM
STRESS CONCENTRATION
FACTORS
AROUND
A PENETRATION
IN A STRIP OR SLAB OF FINITE DIMENSIONS J.M.O'CONNELL Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, England Received 29 April 1969
Experimental and theoretical examination of the stress concentration effect of openings in pressure vessels are often carried out on rectangular slabs of finite width. This report examines the errors involved in using a penetration in a finite width slab to simulate a penetration in a much larger area. Such errors are particularly critical in the examination of openings in concrete vessels due to the susceptibility of the material to cracking at relatively low values of tensile stress.
1. Introduction The current series of concrete pressure vessels contain several examples o f large openings. For example the gas circulators are m o u n t e d in openings through the walls. A knowledge of the deformations of these openings is important in the design of the blower mountings. In some proposed designs, the boilers are m o u n t e d in vertical cylindrical openings in the vessel wall. Here again, a knowledge o f the stress situation on the ligament is essential to a safe vessel design. Strains and deformations in the vicinity of a penetration in the vessel wall could be determined experimentally by building a number of scale models o f the vessel. However, to make any meaningful measurements would require models o f reasonable size. The cost of such models and equipment favours the use of an alternative solution. A flat slab model is often used. If a flat slab containing a circular penetration is subjected to a uniaxial tensile or compressive stress field, the solution for tension or compression in two perpendicular directions can be easily obtained by superposition. So if the stress field in an unpierced vessel is known, the local stresses around a penetration in the vessel can be calculated by superposing
one or more uniaxial stress patterns on the unpierced stress field. The fiat slab containing a circular penetration is an attractive alternative to the full scale model vessel, in that a large number of slabs, containing a variety of reinforcing configurations, could be manufactured at a fraction of the cost of scale models of the vessel. Ideally, it would be desirable to choose a slab width to penetration diameter ratio such that stresses around the penetration periphery approach the semiinfinite condition. However, it is usually necessary to compromise between the cost o f the rig and the acceptable deviation between the finite and semiinfinite results. The curves given in this article show how the deviation between the finite and semi-infinite solutions varies with the ratio o f penetration diameter to slab width. The magnitude of the stress ratios plotted is of particular significance when brittle materials such as concrete are being considered, because the slab may fail in tension before sufficient strain has occurred to allow accurate measurements to be made. The ratio of penetration diameter to slab width is also important in theoretical calculations. In using analytical tools such as finite elements or finite dif-
STRESS CONCENTRATION FACTORS AROUND A PENETRATION ference methods, the accuracy of the result is related to the mesh size. Therefore, it is tempting to use the available elements and nodes to produce a fine mesh on a small 'slab', rather than a coarse mesh on a large 'slab'. The use of too small a 'slab' may give erroneous results.
357
=O°
"-.~x~
o I-
3o"
r
Z
2. Stresses in plates o f finite width
o Z
2.1. Infinite plate The influence of a penetration of circular shape on the stress distribution in an elastic flat plate which is infinitely large in both the 'OX' and 'OY' directions is well understood. It has been shown experimentally, Coker and Filon (1959) and theoretically, Timoshenko and Goodier (1933), that the maximum stress concentration factors at the periphery of an unreinforced penetration in a plate subjected to a uniaxial stress field are 3 and - 1. These results can be used in conjunction with the principle of superposition and the unpenetrated vessel stresses to determine the stress distribution in the vicinity of a penetration in a structure whose width is several times the diameter of the penetration.
Z
8
-!
g
- a ~c $,4.y&l 1 lll|-I
o
+i
+2
HOOP STRESS CONCENTRATION F A ~ .
2.2. Penetration & a strip of finite width but infinite
length Howland [3], analysed the stress distribution in the vicinity of an unreinforced penetration in a strip of finite width but infinite length, subjected to a uniaxial stress field. The hoop S.C.F. around the contour of the penetration is plotted against 'X', that is, the ratio of penetration diameter (d) to strip width (D) in fig. 1. Maximum hoop compressive and tensile stress concentration factors at the periphery of the penetration are plotted against 'X' in fig. 2. From the figure it will be observed that these stresses increase with increasing values of '~'. 2.3. Penetration in a square slab or finite width At present no analytical solution is available for the stress distribution in the vicinity of a penetration in a finite square slab. However, it is possible to use the finite element method, Clough [4], Campbell [5], to get a very close approximation to the answer to the problem. A finite width square slab containing a circular penetration subject to a uniaxial stress field
Fig. 1. Distribution of hoop stress concentration factor around the periphery of a circular penetration in a strip of finite width plotted against the hole diameter to strip width ratio (X). is shown in fig. 3. By symmetry, to analyse this configuration, it is only necessary to consider one quadrant of the slab and the triangular mesh which is superimposed on the slab is shown in fig. 4. To obtain increased accuracy, the size of the triangle is reduced in the region of increasing stress gradient, that is, as the periphery of the penetration is approached. To examine the variation of stress concentration factor with change in penetration diameter (d) to slab width (D) ratios, a series of meshes of the type shown in fig. 4 were generated. These meshes covered (d/D) ratios, ranging from 0.5 to 0.1, the latter approximating very closely to the infinite condition. The associated geometries are read into the computer as part of the data necessary to solve the problem.
NoM
358
J.M.O'CONNEL L
+5"0 Lu >
+4'0
K, = STRESS CONCENTRATION
FACTOR DEFINED
K z = STRESS CONCENTRATION FIBRE DEFINED BY
FACTOR AT
K5 = STRESS
FACTOR DEFINED
BY
f~ = - K,P D~
f,° - K,~p
EXTREME
Dt
CONCENTRATI, ON
WHERE J, AND :~3 ARE THE MAXIMUM COMPRESSIVE
BY
= _ K,,,P
/
AND
O~ TENSILE STRESSES AT THE PENETRATION PERIPHERY, U
n. 0 U
2
+3.0 P +Z-C
z
_o I-z uJ u z 0 u
÷i,O
Kz
o'.,
0
i
o12
0-3 d/ooR
0"5
0"4
,X
i.-- -I-O in
K3
Z U,J
-2-O
Fig. 2. Maximum compressive and tensile hoop stress concentration factors plotted against penetration diameter to strip width ratio. Uniaxial stress field. K, = STRESS CONCENTRATION
+5"0
+4.0
FACTOR
KZ= MAXIMUM EXTREME
STRESS FIBRE
K 3 = MAXIMUM DEFINED
STRESS CONCENTRATION BY
DEFINED
BY
CONCENTRATION FACTOR AT DEFINED BY
~(, =
+ K , r~°M
XI~ =
"J"K~6-NOIVt
Kt
/
FACTOR
WHERE J, AND~3 ARE THE MAXIMUM COIvIPRESSIVE" AND TENSILE STRESSES AT THE PENETRATION PERIPHERY.
Z
+3.0
D c~ pO +2'C u
*1.0
i-
Z 0 P
~
z w
o
0"2
0"3 d
/D
~
j
-i
I
i
0'1
o
0"4
~"-~...0'5 Xz
-FO;
w
ff m
- 2 .O
-3'0 K3
Fig. 3. Maximum compressive and tensile stress concentration factors in a finite width square slab plotted against penetration diameter to slab width ratio (uniaxial stress field).
STRESS C O N C E N T R A T I O N F A C T O R S A R O U N D A P E N E T R A T I O N
359
The hoop S.C.F. at points 1 and 3 on the penetration periphery and point 2 on the unloaded edge are plotted against ';~' in fig. 3 for the unreinforced case. From the figure, it will be seen that any attempt at experimental investigation of the uniaxial case in brittle material is difficult as cracking will occur in the tensile zone before appreciable compression has been achieved elsewhere. By invoking the principle of superposition the hoop S.C.F. around the periphery of a penetration in a finite width square slab loaded biaxially, was calculated and the results are shown in fig. 5. From the figure it will be observed that the finite square slab subject to equi-biaxial compression is far less sensitive to finite width effects thanthe same slab subjected to uniaxial loading. Timoshenko [2] states that for a square slab with an unreinforced penetration, and width to diameter ratio of four, the deviations between finite and infinite results is 5%. Whilst this statement holds true for biaxial loading, the deviation associated with uniaxial loading is 35% tensile and 10% compressive.
Fig. 4. Finite element m e s h for duct.
O
K, = STRESS CONCENTRATION
FACTOR DEFINED BY ~, = ~. K,~',~o~
KI~--STRESS
FACTOR
111
N
ill
#
I.
CONCENTRATION
DEFINED BY
X3 = ,I-Ks.~'-NOM
I'll + 3-0
T I I l GNOM
K, K~
=( ~2'C r~ o IU
+I.0
Z
o
or,
IZ W
i
0'.2
0-3
o!4
-6~.s
d
%
~ -I'0
8 w -2.C
-3-C Fig. 5. M a x i m u m compressive hoop stress concentration factor at the periphery o f a penetration in a finite square slab subjected to an equi-biaxial stress field.
360
J.M.O'CONNELL
3. Conclusions 1. For strips of finite width the hoop stress concentration factor associated with the uniaxial stress field increases rapidly with increasing radius of penetration. In the unreinforced case for example, for '~' = 0.25, the deviation between the finite and semiinfinite solutions for tensile stress is 20%. This increase in tensile stress is significant in materials such as concrete which are weak in tension. 2. For square slabs of finite width, the hoop S.C.F. associated with the uniaxial stress field increases more rapidly with increasing radius of penetration than does that associated with strips of finite width. For '~,' = 0.25, the deviation in the unreinforced case between the maximum tensile stress associated with the finite and infinite slab respectively, is 35%. 3. In designing experiments involving penetrated strips or slabs made of brittle materials, the maximum tensile S.C.F. needs to be considered, because in arriving at compressive strains from which meaningful measurements can be made, the associated tensile S.C.F. may lead to the propagation of cracks in the
strip or slab. Such a situation would render the principle of superposition invalid. Such cracking can often be eliminated by applying an initial biaxial compression to the slab. Uniaxial loading may then be superposed, and the resulting stress pattern obtained by observing the changes in strain which occur.
Acknowledgement This paper is published by permission of the Central Electricity Generating Board.
References [1] E.G.Coker and L.N.Filon, Photoelasticity (Cambridge University Press, 1957). [2] Timoshenko and Goodier, Theory of Elasticity (McGrawHill Book Company, New York, 1951). [3] Howland, Phil. Trans. of the Royal Society, A, 229 (1929). [4] R.W.Clough, The Finite Element Method of Structural Mechanics (Wiley, New York, 1965). [ 5 ] J.S.Campbell, C.E.G.B. Report RD/B/N.834 (1968).
STRESS CONCENTRATION
FACTORS
AROUND
A PENETRATION
IN A STRIP OR SLAB OF FINITE DIMENSIONS J.M.O'CONNELL Central Electricity Generating Board, Berkeley Nuclear Laboratories, Berkeley, Gloucestershire, England Received 29 April 1969
Experimental and theoretical examination of the stress concentration effect of openings in pressure vessels are often carried out on rectangular slabs of finite width. This report examines the errors involved in using a penetration in a finite width slab to simulate a penetration in a much larger area. Such errors are particularly critical in the examination of openings in concrete vessels due to the susceptibility of the material to cracking at relatively low values of tensile stress.
1. Introduction The current series of concrete pressure vessels contain several examples o f large openings. For example the gas circulators are m o u n t e d in openings through the walls. A knowledge of the deformations of these openings is important in the design of the blower mountings. In some proposed designs, the boilers are m o u n t e d in vertical cylindrical openings in the vessel wall. Here again, a knowledge o f the stress situation on the ligament is essential to a safe vessel design. Strains and deformations in the vicinity of a penetration in the vessel wall could be determined experimentally by building a number of scale models o f the vessel. However, to make any meaningful measurements would require models o f reasonable size. The cost of such models and equipment favours the use of an alternative solution. A flat slab model is often used. If a flat slab containing a circular penetration is subjected to a uniaxial tensile or compressive stress field, the solution for tension or compression in two perpendicular directions can be easily obtained by superposition. So if the stress field in an unpierced vessel is known, the local stresses around a penetration in the vessel can be calculated by superposing
one or more uniaxial stress patterns on the unpierced stress field. The fiat slab containing a circular penetration is an attractive alternative to the full scale model vessel, in that a large number of slabs, containing a variety of reinforcing configurations, could be manufactured at a fraction of the cost of scale models of the vessel. Ideally, it would be desirable to choose a slab width to penetration diameter ratio such that stresses around the penetration periphery approach the semiinfinite condition. However, it is usually necessary to compromise between the cost o f the rig and the acceptable deviation between the finite and semiinfinite results. The curves given in this article show how the deviation between the finite and semi-infinite solutions varies with the ratio o f penetration diameter to slab width. The magnitude of the stress ratios plotted is of particular significance when brittle materials such as concrete are being considered, because the slab may fail in tension before sufficient strain has occurred to allow accurate measurements to be made. The ratio of penetration diameter to slab width is also important in theoretical calculations. In using analytical tools such as finite elements or finite dif-
STRESS CONCENTRATION FACTORS AROUND A PENETRATION ference methods, the accuracy of the result is related to the mesh size. Therefore, it is tempting to use the available elements and nodes to produce a fine mesh on a small 'slab', rather than a coarse mesh on a large 'slab'. The use of too small a 'slab' may give erroneous results.
357
=O°
"-.~x~
o I-
3o"
r
Z
2. Stresses in plates o f finite width
o Z
2.1. Infinite plate The influence of a penetration of circular shape on the stress distribution in an elastic flat plate which is infinitely large in both the 'OX' and 'OY' directions is well understood. It has been shown experimentally, Coker and Filon (1959) and theoretically, Timoshenko and Goodier (1933), that the maximum stress concentration factors at the periphery of an unreinforced penetration in a plate subjected to a uniaxial stress field are 3 and - 1. These results can be used in conjunction with the principle of superposition and the unpenetrated vessel stresses to determine the stress distribution in the vicinity of a penetration in a structure whose width is several times the diameter of the penetration.
Z
8
-!
g
- a ~c $,4.y&l 1 lll|-I
o
+i
+2
HOOP STRESS CONCENTRATION F A ~ .
2.2. Penetration & a strip of finite width but infinite
length Howland [3], analysed the stress distribution in the vicinity of an unreinforced penetration in a strip of finite width but infinite length, subjected to a uniaxial stress field. The hoop S.C.F. around the contour of the penetration is plotted against 'X', that is, the ratio of penetration diameter (d) to strip width (D) in fig. 1. Maximum hoop compressive and tensile stress concentration factors at the periphery of the penetration are plotted against 'X' in fig. 2. From the figure it will be observed that these stresses increase with increasing values of '~'. 2.3. Penetration in a square slab or finite width At present no analytical solution is available for the stress distribution in the vicinity of a penetration in a finite square slab. However, it is possible to use the finite element method, Clough [4], Campbell [5], to get a very close approximation to the answer to the problem. A finite width square slab containing a circular penetration subject to a uniaxial stress field
Fig. 1. Distribution of hoop stress concentration factor around the periphery of a circular penetration in a strip of finite width plotted against the hole diameter to strip width ratio (X). is shown in fig. 3. By symmetry, to analyse this configuration, it is only necessary to consider one quadrant of the slab and the triangular mesh which is superimposed on the slab is shown in fig. 4. To obtain increased accuracy, the size of the triangle is reduced in the region of increasing stress gradient, that is, as the periphery of the penetration is approached. To examine the variation of stress concentration factor with change in penetration diameter (d) to slab width (D) ratios, a series of meshes of the type shown in fig. 4 were generated. These meshes covered (d/D) ratios, ranging from 0.5 to 0.1, the latter approximating very closely to the infinite condition. The associated geometries are read into the computer as part of the data necessary to solve the problem.
NoM
358
J.M.O'CONNEL L
+5"0 Lu >
+4'0
K, = STRESS CONCENTRATION
FACTOR DEFINED
K z = STRESS CONCENTRATION FIBRE DEFINED BY
FACTOR AT
K5 = STRESS
FACTOR DEFINED
BY
f~ = - K,P D~
f,° - K,~p
EXTREME
Dt
CONCENTRATI, ON
WHERE J, AND :~3 ARE THE MAXIMUM COMPRESSIVE
BY
= _ K,,,P
/
AND
O~ TENSILE STRESSES AT THE PENETRATION PERIPHERY, U
n. 0 U
2
+3.0 P +Z-C
z
_o I-z uJ u z 0 u
÷i,O
Kz
o'.,
0
i
o12
0-3 d/ooR
0"5
0"4
,X
i.-- -I-O in
K3
Z U,J
-2-O
Fig. 2. Maximum compressive and tensile hoop stress concentration factors plotted against penetration diameter to strip width ratio. Uniaxial stress field. K, = STRESS CONCENTRATION
+5"0
+4.0
FACTOR
KZ= MAXIMUM EXTREME
STRESS FIBRE
K 3 = MAXIMUM DEFINED
STRESS CONCENTRATION BY
DEFINED
BY
CONCENTRATION FACTOR AT DEFINED BY
~(, =
+ K , r~°M
XI~ =
"J"K~6-NOIVt
Kt
/
FACTOR
WHERE J, AND~3 ARE THE MAXIMUM COIvIPRESSIVE" AND TENSILE STRESSES AT THE PENETRATION PERIPHERY.
Z
+3.0
D c~ pO +2'C u
*1.0
i-
Z 0 P
~
z w
o
0"2
0"3 d
/D
~
j
-i
I
i
0'1
o
0"4
~"-~...0'5 Xz
-FO;
w
ff m
- 2 .O
-3'0 K3
Fig. 3. Maximum compressive and tensile stress concentration factors in a finite width square slab plotted against penetration diameter to slab width ratio (uniaxial stress field).
STRESS C O N C E N T R A T I O N F A C T O R S A R O U N D A P E N E T R A T I O N
359
The hoop S.C.F. at points 1 and 3 on the penetration periphery and point 2 on the unloaded edge are plotted against ';~' in fig. 3 for the unreinforced case. From the figure, it will be seen that any attempt at experimental investigation of the uniaxial case in brittle material is difficult as cracking will occur in the tensile zone before appreciable compression has been achieved elsewhere. By invoking the principle of superposition the hoop S.C.F. around the periphery of a penetration in a finite width square slab loaded biaxially, was calculated and the results are shown in fig. 5. From the figure it will be observed that the finite square slab subject to equi-biaxial compression is far less sensitive to finite width effects thanthe same slab subjected to uniaxial loading. Timoshenko [2] states that for a square slab with an unreinforced penetration, and width to diameter ratio of four, the deviations between finite and infinite results is 5%. Whilst this statement holds true for biaxial loading, the deviation associated with uniaxial loading is 35% tensile and 10% compressive.
Fig. 4. Finite element m e s h for duct.
O
K, = STRESS CONCENTRATION
FACTOR DEFINED BY ~, = ~. K,~',~o~
KI~--STRESS
FACTOR
111
N
ill
#
I.
CONCENTRATION
DEFINED BY
X3 = ,I-Ks.~'-NOM
I'll + 3-0
T I I l GNOM
K, K~
=( ~2'C r~ o IU
+I.0
Z
o
or,
IZ W
i
0'.2
0-3
o!4
-6~.s
d
%
~ -I'0
8 w -2.C
-3-C Fig. 5. M a x i m u m compressive hoop stress concentration factor at the periphery o f a penetration in a finite square slab subjected to an equi-biaxial stress field.
360
J.M.O'CONNELL
3. Conclusions 1. For strips of finite width the hoop stress concentration factor associated with the uniaxial stress field increases rapidly with increasing radius of penetration. In the unreinforced case for example, for '~' = 0.25, the deviation between the finite and semiinfinite solutions for tensile stress is 20%. This increase in tensile stress is significant in materials such as concrete which are weak in tension. 2. For square slabs of finite width, the hoop S.C.F. associated with the uniaxial stress field increases more rapidly with increasing radius of penetration than does that associated with strips of finite width. For '~,' = 0.25, the deviation in the unreinforced case between the maximum tensile stress associated with the finite and infinite slab respectively, is 35%. 3. In designing experiments involving penetrated strips or slabs made of brittle materials, the maximum tensile S.C.F. needs to be considered, because in arriving at compressive strains from which meaningful measurements can be made, the associated tensile S.C.F. may lead to the propagation of cracks in the
strip or slab. Such a situation would render the principle of superposition invalid. Such cracking can often be eliminated by applying an initial biaxial compression to the slab. Uniaxial loading may then be superposed, and the resulting stress pattern obtained by observing the changes in strain which occur.
Acknowledgement This paper is published by permission of the Central Electricity Generating Board.
References [1] E.G.Coker and L.N.Filon, Photoelasticity (Cambridge University Press, 1957). [2] Timoshenko and Goodier, Theory of Elasticity (McGrawHill Book Company, New York, 1951). [3] Howland, Phil. Trans. of the Royal Society, A, 229 (1929). [4] R.W.Clough, The Finite Element Method of Structural Mechanics (Wiley, New York, 1965). [ 5 ] J.S.Campbell, C.E.G.B. Report RD/B/N.834 (1968).