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Cracked columns under compression fixed ends
CRACKED
COLUMNS UNDER FIXED ENDS
COMPRESSION
H. W. LIU and CHORNG-SHIN CHU Metallurgical Research Laboratories, Department of Chemical Engineering and Metallurgy, Syracuse University
HAROLD LIEROWITZ The George Washington University Abstract-The deflection, the load carrying capacity, the stress intensity factor, and the fracture load of a slender column with fixed ends have been calculated. The cofumn has a single-edge crack at the mid-section and it has an initial deflection. A crack reduces the flexurai rigidity of a column. It increases the lateral deflection and decreases the load carrying capacity of a column. The lateral deflection at the cracked section increases the bending moment there; thereby, it increases the stress intensity factor and decreases the fracture load.
1. INTRODUCTION LIEBOWITZ
ef ai.[l] studied experimentally the stability and fracture of cracked columns under compression. Liebowitz and Claus [2] have subsequently anaiyzed the fracture of notched columns utilizing the stress intensity factor and Neuber’s stress concentration expressions. Okamura et a1.[3] have calculated the deflection, the load carrying capacity, the stress intensity factor and the fracture load of a slender and single-edge-cracked column with hinged ends under an eccentric compression. The calculations agree very well with the measurements by Liebowitz ef al. A similar approach to this problem using stress intensity expressions was also made by Cotterell[4]. For a column with fixed ends, the Euler load is the load carrying capacity. if the column is perfectly smooth and straight. But a column may contain geometric imperfections such as crookedness, notches and cracks. In this study, a single-edge-cracked column with fixed ends and an initial deflection was analyzed. The deflection. the load carrying capacity, the stress intensity factor and the fracture load were calculated. The crack reduces the flexural rigidity of the column; therefore, it increases the deflection, decreases the load carrying capacity, and causes a normal mode crack tip stress field. The analysis follows closely that in [3]. The essential elements of the analysis will be reviewed here, but the details will refer to the earlier work [3].
2. A CRACKED COLUMN UNDER A COMPRESSION-
FIXED ENDS
Figure 1 shows a single-edge-cracked column with fixed ends under compression P. The crack is at the mid-section of the column. Its length is “a”. The column has a rectangular cross section with width W and thickness B. The column has an initial deflection y, = So sin%. The maximum initial deflection, S, is at the mid-section. When IS,,is zero, the column is ZiY
220
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
Fig. 1. A cracked column under compression-fixed
ends.
perfectly straight. In this section, the deflection and the load carrying capacity will be calculated. The bending moment, M, at any cross section of the column is
(2)
M=P(Y+Yo)+Mo
where y is the lateral deflection of the column caused by the load P, and M0 is the moment at the fixed ends to maintain zero slope there. The differential equation for beam deflection is d”y Eldx2
= -P(y+y,+p
MO .
To assist in the analysis, the column is divided into three sections, the center section of length, I:“. which contains the crack, and two outer sections each of length 1. The moment and the slope of the column are continuous at the intersection of the center and the outer sections, and at x = 0 y=o (4) dy z=o. Let A2= P/El,(3)is then (5) The general solution of (5) is y=Asinhx+BcosAx+
(2hl)Z 6 sin%--j;. MCI 73 -(2A1)2 ’
(6)
‘71
Cracked columns
The end conditions of y = 0 and ~~1~~ = 0 at ,Y= 0 lead to (7) and A=-
Therefore,
2nAl %-2- (2h$O
(8)
the deflection is 2X/60 2hlsinz-rsinhx (2A1)2(
y = 712-
>
+B(COsh.~-II)
At the intersection of the outer and the center sections, the moment and the slope are continuous, The moment and the slope, M and 19,of the outer section are related to its lateral deflection. The moment at both ends of the center section and the relative rotation between these two ends, M* and B*, are related to each other by the flexural compliance of the cracked center section, which is increased by the crack. The effect of the reduced flexural rigidity is not realized until the crack is opened, The crack remains closed until the tensile bending stress on the cracked side of the column exceeds the compression. Therefore, the bending moment has to exceed PWI6,before the reduced flexural rigidity is realized. Consequently, M* and 0* are related to each other by
where k is the rotational spring constant of the cracked center section. In Fig. 2, the variation of the compliance, l/k, is shown schematically as a function of the moment. The solid line ABCD is the one assumed for this study. The continuity conditions of the moment and the slope at the intersection between the outer and the center sections and the fact that 0* = 20 = 2 dylllx lead to HA2 B cos Al -
+h’ 27rAl 2So sin Al+ 7T2- (2hl) 7? - (2hl)Z6”
1
M= (PW/6)(Y,/Y,)
BENDING
MOMENT
AT CRACKED
SECTION
Fig. 2. Compliance of a cracked column as influenced by bending moment at cracked section.
222
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
The constant B can be evaluated from the above equation
[
zSO cos Al-&U sin Al+? B=
1
(12)
P6”.
(13)
sin Al+ oh1 cos Al
where p = E1/2kl. The moment M, at the fixed end is related to B by (7), hence cos Al--PAlsin Al+? I
M,=
sin Al+ Bhl cos Al
With the constant B known, the deflection can be calculated from (9)
2X1 2Alsin>T-rr rr2 - (2A1)2
+
sin Ax . 1
(14)
The total deflection, yT = y + y,,, is given by Y~_~hl~-_2~~~hl~~[coshl-Ph~sin~~+~~ (cos Ax-
sin Ali- /3Al cos Al
s,-
+
rr2- ;?A1)?
I)
rrsin?-2Alsin
Ax . I
(13
If one assumes 1” = 0,t the total deflection at the mid-section &, is given by 6 2=
63
~~-;;hl)~
[z-sin Al+2hf(cos
Al- 1) +pAI(n-2Al +/3hl$-(cos 0
sin Al)]
Al- 1) /(sin Al+phlcos I
Al).
(16)
The effect of the reduced flexural rigidity is contained in the parameter /X A sample calculation is shown in Fig. 3. For this column, p = 0.013. The deflection increases with load and the quantity r&/W. As /3 approaches zero, the solution approaches that of an untracked column. For both /3 and a0 approach zero, (16) gives the Euler load. The load for infinite deflection is given by the solution to the equation ,6X1cot Al = -1.
The solution gives the load carrying capacity,
P,,, of the cracked
(17) column. The ratio
tThis assumption of I* = 0 is equivalent to a mechanical model of a column of length 21, with a rotational spring at the mid-section. The spring constant is k[3].
Cracked columns EULER LOAD I
TOTAL
Fig. 3.
LATERAL
DEFLECTION-&
(INCH)
Deflectionof a cracked column.
is plotted in Fig. 4. P, is the Euler load. For an untracked column, 0 = 0, and PC,.= P,. P,.* is independent of initial deflection. If an ideal crack exists in a coiumn, the crack is closed by the compressive stress. The crack surfaces bear a compressive load, and the effect of the crack is not realized until the bending moment at the cracked section exceeds PW/6. Therefore, the load carrying capacity of the cracked column is higher than Pep In this case, if the initial detlection, a,, is zero, the load carrying capacity of the column is the Euler load. If a thin slot or a sharp notch is used to simulate a crack, the flexural rigidity of the column is reduced and the load carrying capacity is given by P,,, provided the column is not fractured by the mode-1 crack tip stress field. The rotation of the center section, 8*, consists of two parts: one part is the rotation of a smooth column of length 1*, and the other part is caused by the increased compliance due to the crack. When the lateral deflection was calculated with (16), l* was assumed zero, so that the rotation of the smooth center section has been accounted for. Hence, only the additional rotation caused by the crack in the center section needs to be included. Therefore, it is the difference between the compliances of a cracked and an untracked center section that contributes to the decrease of k. Let C and Co be PJP,
F.F.M.
VOL.3.
NO. 3-f
‘24
H. W. LIU, CHORNG-SHIN
0
5
CHU and H. LlEBOWlTZ
IO
I5
20
P=EI/ZkP Fig. 4. Load carrying capacity of a cracked column.
a crack. Hence k = I /(C - C,). When the crack length approaches zero, C approaches C,, k to x, and p to zero. The compliance, C, can be calculated from the stress intensity factor, K,,, for beam bending. Okamura et al. [ 31 have calculated the spring constant k as compliances
of the center
section
with and without
k=
EBW”
(18)
where E is Young’s modulus and v is Poisson’s ratio for a plane strain case and it is zero for a plane stress case, and the function F (a/ W) is
- 103.36 lnFig.S,thefunctionF(a/W)isplottedvs. (u/W). With the spring constant known, the parameter, ,B, can be calculated. Then the deflection of the column can be calculated with (14), (15) and (16), and (17) can be written as (20)
hlcothl=-
3(1 -v&F~$)
Cracked
Fig. 5. The function
P,, = BUCKLING 0
0. I CRACK
LOAD 0.2 LENGTH
Fig. 6. Load carrying
225
columns
F(n/W).
OF AN UNCRACKED 0.8 TO COLUMN
capacity,
COLUMN
0.4 THICKNESS
0.5 0.6 RATIO-a/W
P,., of a cracked
0.7
column
In Fig. 6, the ratio P,,/P, is plotted vs. the ratio (a/W), with the quantity ( I - 9) W/l as parameter. The maximum load that can be sustained by a cracked column could be the Euler load, the load carrying capacity as given by (20), or the fracture load. All of these three quantities of a set of cracked columns are plotted in Fig. 7. The cross-sectional area
226
H. W. LIU, CHORNG-SHIN
---
FRACTURE
CHU and H. LIEBOWITZ
LOAD,
EO. (251,
EULER
-v 5 m J IOzi 0 J CL I
-_
8, ~0.1
CJ -_
‘21;
m I_ --.
_ _
I
0
10.25”
\ 0
-
&7
--
fs 9
-o---_
EULER
--_ -_
-y
21:
LOAD 22.25”
I z ;i a z
LOAD I
- L..
I I IO-
.~.__ -
TT-
-__
!
t -----+ EULER
LOAD I = 16.25”
-__
---%,-_-;\-.2f
!
I
II
0
I
I
0.1
0.2 NOTCH
I
0.3 0.4 DEPTH - o (INCH)
(
j
Fig. 7. Load carrying capacity and fracture load of single edge notched columns-707.5Th5
I1
Al.
of the columns is $ in. X $in. The lengths are 10.25 in., 16.25 in., 22.25 in., and 28.25 in. The columns were made of 7075-T65 11 aluminum with single edge notches at the midsection. The elastic modulus of the material is 10.4 X lo6psi. The measured failure loads were lower than the load carrying capacity calculated with (20), as indicated by the dash-dot curves. The failure of these columns could be caused by fracture. The bending moment causes a tensile mode crack stress field. When the net tensile stress intensity exceeds the fracture toughness of the material, fracture takes place. The stress intensity factor and the fracture of cracked columns will be discussed in the next section. Both the calculated curves in Figs. 6 and 7, and the experimental data indicate that the effect of a crack on the load carrying capacity decreases with the ratio W/l. 3. STRESS INTENSITY FACTOR OF CRACKED COLUMNS The bending moment at the cracked section causes a positive tensile mode crack tip stress field, and the positive mode-1 stress field is reduced by the compression. The stress intensity factors, K, and K,, of a cracked beam under pure bending and of a single edge notched plate under tension were calculated by Gross and Srawley [51 K,B W2 = Y,= 6Ma112
1.99-2.47
(“) w
+ 12*97 (yz w
-23 . 17(“)3 w
+24 *80 (a)l w
(21)
227
Cracked columns
and
= 199-0.41
Zif. + 18 70 if_ ‘-38 48 -!?- 3+S3 8.5 a (wj ’ iw) * iw> . (wi_
(22)
The bending moment at the cracked section is
pa0 7T2-
Z!TZ
(71sin Al- 2hl) + phl$ 7;zhr)” (sin Al+@XIcos hi)
cos XI O
(23)
.
Substituting into (2 1) and rearranging the equation, one obtains
X
Tr’
-(7r sin AI--2A1)~+/3Af -72Af)2
cos Al . I
CW
The net stress intensity factor of the column is the difference between those of bending and compression K= I&-Kty,
The functions ~ffjW~“~Y~ and Y,/Y, are plotted vs. (~~IW) in Figs. 8 and 9. In Fig. 10, a sample calculation for both K and K, is plotted vs. (All2 with &‘W as parameter. The dashed lines are for K’,* and the solid lines are for K. At low load levels, and at low values of So/W, the difference between K, and K is considerable. This indicates that the compressive stress intensity factor is significant. As the load or 6, increases, the bending stress intensity factor becomes dominant, and the compressive stress intensity factor becomes negligible. In Fig. 7, the calculated fracture loads of a set of columns made of 7075-T65 1 I aluminum were plotted as the dashed curves. The previous study by Okamura et rrf.[3] indicated that the fracture toughness of the material was between 30-37 KSIV???. The fracture loads were calculated based on a fracture toughness value of 35 KSIV’%. For these calculations, an initial deflection of O-1 in. was assumed. The calculated fracture loads agree with the measured data reasonably well. The maximum deviation is less than 25 per cent. When the bending moment at the mid-section is below PW/6, the crack is closed. Therefore, the crack does not have any effect on the compliance of the column. As the moment increases beyond P W/6, the crack begins to open. In this study, the compliance is assumed to increase to the full value of a cracked column, when the moment is larger
228
H. W. LIU, CHORNG-SHIN
0
0.1
02
CHU and H. LIEBOWITZ
03
0.4
o/w Fig. 8. (u/W)“~I;(U/W) as a function of o/W.
(a/W)
Fig. 9. The ratio Y,,/Y,, as a function of (a/W).
0.5
0.6
Cracked
columns
o/W=O.l8 E= 10.4
x IO6 PSI
-K
Yill
0
I
I I
I
8
2 $5
Fig. 10. Stress intensity
P&I
factor
of a cracked
column.
than PW/6. But the crack is not fully open until K, > K,,. When this condition written in terms of the moment at the midsection, it becomes
is
(26) The actual variation of the compliance with moment in the region PW/6 -( M : (P W/6) (Y,,/Y,,,) is not known, and it is shown schematically by the dashed line in Fig. 2. SUMMARY 1. A crack reduces the flexural rigidity of a column. The rotational spring constant of the cracked section was calculated from Srawley and Gross’s stress intensity factor of a single-edge-cracked beam under bending. 2. The deflection, the load carrying capacity, the stress intensity factor, and the fracture load of a cracked column were calculated. The column has a single-edge-crack at the mid-section with both ends fixed. It has an initial deflection.
230
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
3. A crack increases the lateral deflection and decreases the load carrying capacity of a column. 4. The increased lateral deflection increases bending moment at the cracked section. Hence, it increases the stress intensity factor of a cracked column. Acknowledgements-The investigation was conducted at the Metallurgical Laboratories of Syracuse University and the Catholic University of America. The support of ONR and Naval Ship Systems Command, Nonr 666(21). made the investigation possible. The authors are indebted to Dr. Hiroyuki Okamura of Tokyo University for his valuable discussion, to Mr. W. Gavigan for his assistance in numerical calculation, and to Mmes. Helen Turner and Barbara Howden for the preparation of the manuscript.
REFERENCES [l] H. Liebowitz, H. Vanderveldt and D. W. Harris, Carrying capacity of notched column. Int. J. Solids Structures 3,4,489-500 (1967). [2] H. Liebowitz and W. D. Claus, Jr., Failure of notched columns. Engng Fracture Mech. April (1968). .3] Hiroyuki Okamura, H. W. Liu, C. Chu and H. Liebowitz, A cracked column under compression. Engng Fracture Mech. I, 3 (1969). 141 B. Cotterell, On the carrying capacity of notched columns. Presented at the ASME meeting in New York City, December (1968). [5] B. Gross and J. E. Srawley, Stress-intensity factors for single-edge-notch specimens on bending or combined bending and tension by boundary collocation of a stress function. NASA TN D-2603 (1965). (Received 17 February 1969)
COLUMNS UNDER FIXED ENDS
COMPRESSION
H. W. LIU and CHORNG-SHIN CHU Metallurgical Research Laboratories, Department of Chemical Engineering and Metallurgy, Syracuse University
HAROLD LIEROWITZ The George Washington University Abstract-The deflection, the load carrying capacity, the stress intensity factor, and the fracture load of a slender column with fixed ends have been calculated. The cofumn has a single-edge crack at the mid-section and it has an initial deflection. A crack reduces the flexurai rigidity of a column. It increases the lateral deflection and decreases the load carrying capacity of a column. The lateral deflection at the cracked section increases the bending moment there; thereby, it increases the stress intensity factor and decreases the fracture load.
1. INTRODUCTION LIEBOWITZ
ef ai.[l] studied experimentally the stability and fracture of cracked columns under compression. Liebowitz and Claus [2] have subsequently anaiyzed the fracture of notched columns utilizing the stress intensity factor and Neuber’s stress concentration expressions. Okamura et a1.[3] have calculated the deflection, the load carrying capacity, the stress intensity factor and the fracture load of a slender and single-edge-cracked column with hinged ends under an eccentric compression. The calculations agree very well with the measurements by Liebowitz ef al. A similar approach to this problem using stress intensity expressions was also made by Cotterell[4]. For a column with fixed ends, the Euler load is the load carrying capacity. if the column is perfectly smooth and straight. But a column may contain geometric imperfections such as crookedness, notches and cracks. In this study, a single-edge-cracked column with fixed ends and an initial deflection was analyzed. The deflection. the load carrying capacity, the stress intensity factor and the fracture load were calculated. The crack reduces the flexural rigidity of the column; therefore, it increases the deflection, decreases the load carrying capacity, and causes a normal mode crack tip stress field. The analysis follows closely that in [3]. The essential elements of the analysis will be reviewed here, but the details will refer to the earlier work [3].
2. A CRACKED COLUMN UNDER A COMPRESSION-
FIXED ENDS
Figure 1 shows a single-edge-cracked column with fixed ends under compression P. The crack is at the mid-section of the column. Its length is “a”. The column has a rectangular cross section with width W and thickness B. The column has an initial deflection y, = So sin%. The maximum initial deflection, S, is at the mid-section. When IS,,is zero, the column is ZiY
220
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
Fig. 1. A cracked column under compression-fixed
ends.
perfectly straight. In this section, the deflection and the load carrying capacity will be calculated. The bending moment, M, at any cross section of the column is
(2)
M=P(Y+Yo)+Mo
where y is the lateral deflection of the column caused by the load P, and M0 is the moment at the fixed ends to maintain zero slope there. The differential equation for beam deflection is d”y Eldx2
= -P(y+y,+p
MO .
To assist in the analysis, the column is divided into three sections, the center section of length, I:“. which contains the crack, and two outer sections each of length 1. The moment and the slope of the column are continuous at the intersection of the center and the outer sections, and at x = 0 y=o (4) dy z=o. Let A2= P/El,(3)is then (5) The general solution of (5) is y=Asinhx+BcosAx+
(2hl)Z 6 sin%--j;. MCI 73 -(2A1)2 ’
(6)
‘71
Cracked columns
The end conditions of y = 0 and ~~1~~ = 0 at ,Y= 0 lead to (7) and A=-
Therefore,
2nAl %-2- (2h$O
(8)
the deflection is 2X/60 2hlsinz-rsinhx (2A1)2(
y = 712-
>
+B(COsh.~-II)
At the intersection of the outer and the center sections, the moment and the slope are continuous, The moment and the slope, M and 19,of the outer section are related to its lateral deflection. The moment at both ends of the center section and the relative rotation between these two ends, M* and B*, are related to each other by the flexural compliance of the cracked center section, which is increased by the crack. The effect of the reduced flexural rigidity is not realized until the crack is opened, The crack remains closed until the tensile bending stress on the cracked side of the column exceeds the compression. Therefore, the bending moment has to exceed PWI6,before the reduced flexural rigidity is realized. Consequently, M* and 0* are related to each other by
where k is the rotational spring constant of the cracked center section. In Fig. 2, the variation of the compliance, l/k, is shown schematically as a function of the moment. The solid line ABCD is the one assumed for this study. The continuity conditions of the moment and the slope at the intersection between the outer and the center sections and the fact that 0* = 20 = 2 dylllx lead to HA2 B cos Al -
+h’ 27rAl 2So sin Al+ 7T2- (2hl) 7? - (2hl)Z6”
1
M= (PW/6)(Y,/Y,)
BENDING
MOMENT
AT CRACKED
SECTION
Fig. 2. Compliance of a cracked column as influenced by bending moment at cracked section.
222
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
The constant B can be evaluated from the above equation
[
zSO cos Al-&U sin Al+? B=
1
(12)
P6”.
(13)
sin Al+ oh1 cos Al
where p = E1/2kl. The moment M, at the fixed end is related to B by (7), hence cos Al--PAlsin Al+? I
M,=
sin Al+ Bhl cos Al
With the constant B known, the deflection can be calculated from (9)
2X1 2Alsin>T-rr rr2 - (2A1)2
+
sin Ax . 1
(14)
The total deflection, yT = y + y,,, is given by Y~_~hl~-_2~~~hl~~[coshl-Ph~sin~~+~~ (cos Ax-
sin Ali- /3Al cos Al
s,-
+
rr2- ;?A1)?
I)
rrsin?-2Alsin
Ax . I
(13
If one assumes 1” = 0,t the total deflection at the mid-section &, is given by 6 2=
63
~~-;;hl)~
[z-sin Al+2hf(cos
Al- 1) +pAI(n-2Al +/3hl$-(cos 0
sin Al)]
Al- 1) /(sin Al+phlcos I
Al).
(16)
The effect of the reduced flexural rigidity is contained in the parameter /X A sample calculation is shown in Fig. 3. For this column, p = 0.013. The deflection increases with load and the quantity r&/W. As /3 approaches zero, the solution approaches that of an untracked column. For both /3 and a0 approach zero, (16) gives the Euler load. The load for infinite deflection is given by the solution to the equation ,6X1cot Al = -1.
The solution gives the load carrying capacity,
P,,, of the cracked
(17) column. The ratio
tThis assumption of I* = 0 is equivalent to a mechanical model of a column of length 21, with a rotational spring at the mid-section. The spring constant is k[3].
Cracked columns EULER LOAD I
TOTAL
Fig. 3.
LATERAL
DEFLECTION-&
(INCH)
Deflectionof a cracked column.
is plotted in Fig. 4. P, is the Euler load. For an untracked column, 0 = 0, and PC,.= P,. P,.* is independent of initial deflection. If an ideal crack exists in a coiumn, the crack is closed by the compressive stress. The crack surfaces bear a compressive load, and the effect of the crack is not realized until the bending moment at the cracked section exceeds PW/6. Therefore, the load carrying capacity of the cracked column is higher than Pep In this case, if the initial detlection, a,, is zero, the load carrying capacity of the column is the Euler load. If a thin slot or a sharp notch is used to simulate a crack, the flexural rigidity of the column is reduced and the load carrying capacity is given by P,,, provided the column is not fractured by the mode-1 crack tip stress field. The rotation of the center section, 8*, consists of two parts: one part is the rotation of a smooth column of length 1*, and the other part is caused by the increased compliance due to the crack. When the lateral deflection was calculated with (16), l* was assumed zero, so that the rotation of the smooth center section has been accounted for. Hence, only the additional rotation caused by the crack in the center section needs to be included. Therefore, it is the difference between the compliances of a cracked and an untracked center section that contributes to the decrease of k. Let C and Co be PJP,
F.F.M.
VOL.3.
NO. 3-f
‘24
H. W. LIU, CHORNG-SHIN
0
5
CHU and H. LlEBOWlTZ
IO
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20
P=EI/ZkP Fig. 4. Load carrying capacity of a cracked column.
a crack. Hence k = I /(C - C,). When the crack length approaches zero, C approaches C,, k to x, and p to zero. The compliance, C, can be calculated from the stress intensity factor, K,,, for beam bending. Okamura et al. [ 31 have calculated the spring constant k as compliances
of the center
section
with and without
k=
EBW”
(18)
where E is Young’s modulus and v is Poisson’s ratio for a plane strain case and it is zero for a plane stress case, and the function F (a/ W) is
- 103.36 lnFig.S,thefunctionF(a/W)isplottedvs. (u/W). With the spring constant known, the parameter, ,B, can be calculated. Then the deflection of the column can be calculated with (14), (15) and (16), and (17) can be written as (20)
hlcothl=-
3(1 -v&F~$)
Cracked
Fig. 5. The function
P,, = BUCKLING 0
0. I CRACK
LOAD 0.2 LENGTH
Fig. 6. Load carrying
225
columns
F(n/W).
OF AN UNCRACKED 0.8 TO COLUMN
capacity,
COLUMN
0.4 THICKNESS
0.5 0.6 RATIO-a/W
P,., of a cracked
0.7
column
In Fig. 6, the ratio P,,/P, is plotted vs. the ratio (a/W), with the quantity ( I - 9) W/l as parameter. The maximum load that can be sustained by a cracked column could be the Euler load, the load carrying capacity as given by (20), or the fracture load. All of these three quantities of a set of cracked columns are plotted in Fig. 7. The cross-sectional area
226
H. W. LIU, CHORNG-SHIN
---
FRACTURE
CHU and H. LIEBOWITZ
LOAD,
EO. (251,
EULER
-v 5 m J IOzi 0 J CL I
-_
8, ~0.1
CJ -_
‘21;
m I_ --.
_ _
I
0
10.25”
\ 0
-
&7
--
fs 9
-o---_
EULER
--_ -_
-y
21:
LOAD 22.25”
I z ;i a z
LOAD I
- L..
I I IO-
.~.__ -
TT-
-__
!
t -----+ EULER
LOAD I = 16.25”
-__
---%,-_-;\-.2f
!
I
II
0
I
I
0.1
0.2 NOTCH
I
0.3 0.4 DEPTH - o (INCH)
(
j
Fig. 7. Load carrying capacity and fracture load of single edge notched columns-707.5Th5
I1
Al.
of the columns is $ in. X $in. The lengths are 10.25 in., 16.25 in., 22.25 in., and 28.25 in. The columns were made of 7075-T65 11 aluminum with single edge notches at the midsection. The elastic modulus of the material is 10.4 X lo6psi. The measured failure loads were lower than the load carrying capacity calculated with (20), as indicated by the dash-dot curves. The failure of these columns could be caused by fracture. The bending moment causes a tensile mode crack stress field. When the net tensile stress intensity exceeds the fracture toughness of the material, fracture takes place. The stress intensity factor and the fracture of cracked columns will be discussed in the next section. Both the calculated curves in Figs. 6 and 7, and the experimental data indicate that the effect of a crack on the load carrying capacity decreases with the ratio W/l. 3. STRESS INTENSITY FACTOR OF CRACKED COLUMNS The bending moment at the cracked section causes a positive tensile mode crack tip stress field, and the positive mode-1 stress field is reduced by the compression. The stress intensity factors, K, and K,, of a cracked beam under pure bending and of a single edge notched plate under tension were calculated by Gross and Srawley [51 K,B W2 = Y,= 6Ma112
1.99-2.47
(“) w
+ 12*97 (yz w
-23 . 17(“)3 w
+24 *80 (a)l w
(21)
227
Cracked columns
and
= 199-0.41
Zif. + 18 70 if_ ‘-38 48 -!?- 3+S3 8.5 a (wj ’ iw) * iw> . (wi_
(22)
The bending moment at the cracked section is
pa0 7T2-
Z!TZ
(71sin Al- 2hl) + phl$ 7;zhr)” (sin Al+@XIcos hi)
cos XI O
(23)
.
Substituting into (2 1) and rearranging the equation, one obtains
X
Tr’
-(7r sin AI--2A1)~+/3Af -72Af)2
cos Al . I
CW
The net stress intensity factor of the column is the difference between those of bending and compression K= I&-Kty,
The functions ~ffjW~“~Y~ and Y,/Y, are plotted vs. (~~IW) in Figs. 8 and 9. In Fig. 10, a sample calculation for both K and K, is plotted vs. (All2 with &‘W as parameter. The dashed lines are for K’,* and the solid lines are for K. At low load levels, and at low values of So/W, the difference between K, and K is considerable. This indicates that the compressive stress intensity factor is significant. As the load or 6, increases, the bending stress intensity factor becomes dominant, and the compressive stress intensity factor becomes negligible. In Fig. 7, the calculated fracture loads of a set of columns made of 7075-T65 1 I aluminum were plotted as the dashed curves. The previous study by Okamura et rrf.[3] indicated that the fracture toughness of the material was between 30-37 KSIV???. The fracture loads were calculated based on a fracture toughness value of 35 KSIV’%. For these calculations, an initial deflection of O-1 in. was assumed. The calculated fracture loads agree with the measured data reasonably well. The maximum deviation is less than 25 per cent. When the bending moment at the mid-section is below PW/6, the crack is closed. Therefore, the crack does not have any effect on the compliance of the column. As the moment increases beyond P W/6, the crack begins to open. In this study, the compliance is assumed to increase to the full value of a cracked column, when the moment is larger
228
H. W. LIU, CHORNG-SHIN
0
0.1
02
CHU and H. LIEBOWITZ
03
0.4
o/w Fig. 8. (u/W)“~I;(U/W) as a function of o/W.
(a/W)
Fig. 9. The ratio Y,,/Y,, as a function of (a/W).
0.5
0.6
Cracked
columns
o/W=O.l8 E= 10.4
x IO6 PSI
-K
Yill
0
I
I I
I
8
2 $5
Fig. 10. Stress intensity
P&I
factor
of a cracked
column.
than PW/6. But the crack is not fully open until K, > K,,. When this condition written in terms of the moment at the midsection, it becomes
is
(26) The actual variation of the compliance with moment in the region PW/6 -( M : (P W/6) (Y,,/Y,,,) is not known, and it is shown schematically by the dashed line in Fig. 2. SUMMARY 1. A crack reduces the flexural rigidity of a column. The rotational spring constant of the cracked section was calculated from Srawley and Gross’s stress intensity factor of a single-edge-cracked beam under bending. 2. The deflection, the load carrying capacity, the stress intensity factor, and the fracture load of a cracked column were calculated. The column has a single-edge-crack at the mid-section with both ends fixed. It has an initial deflection.
230
H. W. LIU, CHORNG-SHIN
CHU and H. LIEBOWITZ
3. A crack increases the lateral deflection and decreases the load carrying capacity of a column. 4. The increased lateral deflection increases bending moment at the cracked section. Hence, it increases the stress intensity factor of a cracked column. Acknowledgements-The investigation was conducted at the Metallurgical Laboratories of Syracuse University and the Catholic University of America. The support of ONR and Naval Ship Systems Command, Nonr 666(21). made the investigation possible. The authors are indebted to Dr. Hiroyuki Okamura of Tokyo University for his valuable discussion, to Mr. W. Gavigan for his assistance in numerical calculation, and to Mmes. Helen Turner and Barbara Howden for the preparation of the manuscript.
REFERENCES [l] H. Liebowitz, H. Vanderveldt and D. W. Harris, Carrying capacity of notched column. Int. J. Solids Structures 3,4,489-500 (1967). [2] H. Liebowitz and W. D. Claus, Jr., Failure of notched columns. Engng Fracture Mech. April (1968). .3] Hiroyuki Okamura, H. W. Liu, C. Chu and H. Liebowitz, A cracked column under compression. Engng Fracture Mech. I, 3 (1969). 141 B. Cotterell, On the carrying capacity of notched columns. Presented at the ASME meeting in New York City, December (1968). [5] B. Gross and J. E. Srawley, Stress-intensity factors for single-edge-notch specimens on bending or combined bending and tension by boundary collocation of a stress function. NASA TN D-2603 (1965). (Received 17 February 1969)