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Influence of the flexural rigidity of diaphragms on the stability of thin-walled box columns subjected to uniform compression
Thin-Walled Structures 8 (1989) 63-71
Influence of the Flexural Rigidity of Diaphragms on the Stability of Thin-Walled Box Columns Subjected to Uniform Compression
M. Kr61ak Institute of Applied Mechanics, Technical University of-L6d~, Poland (Received 8 August 1988: revision received 7 October 1988; accepted 20 October 1988)
ABSTRACT The influence of the diaphragm flexural rigidity on the critical load of a thinwalled column loaded by uniform compression is examined. The problem has been solved approximately by applying an energy method. The results of numerical calculations are presented in graphical form.
1 INTRODUCTION Thin-walled beams or columns are often stiffened by diaphragms or transverse ribs in order to retain the shape of the structure. In theoretical calculations diaphragms have usually been considered as elements that are very flexible in bending and rigid in tension or compression. Such an assumption has proved a considerable simplification. The results of model tests examining stability and the post-buckling state of girders t have shown that there is elastic interaction between the diaphragms and the girder walls with substantial bending stresses in the region adjacent to their junction. Theoretical analysis of the behaviour of lightly stiffened box girder diaphragms by means of a nonlinear finiteelement method has been carried out in Refs 2-5. Test results for two large-scale box girders with diaphragms have been given by Einarsson and Dowling. 6"7 Simple design rules for lightly stiffened box girder 63
Thin-Walled Structures 0263-8231/89/$03.50© 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain.
64
M Krolak
diaphragms have been proposed in Refs 8,9 and in Ref. 10 the ultimate strength of such elements is examined. The aim of the present paper is to analyse the influence of the flexural rigidity, of diaphragms on the critical load of a uniformly compressed column.
2 F O R M U L A T I O N OF T H E P R O B L E M The problem concerns the local elastic stability of a thin-walled column with rectangular cross-section and diaphragms perpendicular to its axis. In this problem the diaphragm flexural rigidity and the interaction between the walls and the diaphragms have been taken into account. It has been assumed that the distance I between the diaphragms is small, so that after local buckling no additional nodal lines will appear between diaphragms. The basic dimensions of the column cross-section are shown in Fig. 1.
3 SOLUTION OF T H E P R O B L E M The problem has been solved in an approximate way by means of an energy method. The section of the column considered, with coordinate system corresponding to a particular wall, is presented in Fig. 2. Taking the double symmetry of the column cross-section into account, the corresponding plate elements of the section are denoted by indices 1 and 2. The deflection functions of walls 1 and 2 are assumed to be in the form: wi = t2[~'i,X(Z 3 - 2Z 2 -t- 1)r/;(r/~ - 21/,-2+ 1)
+ ~';2X(X - 1)r/i(r/i - l) + ~',3Z (X3 - 2Z 2 + 1)r/i(r/i +~';4Z(X- 1 ) r / , ( q ~ - 2 q ~ + 1)1
i = 1,2
I
L
P i
1.
~' Fig.
1.
.J
--
1)
(1)
Stability of thin-walled box columns
65
• Xr:Xz=X
z~,,-, j____ ,2 /I
//
/pi
,? .
F"
_r,
tS
Fig. 2.
where t2 is the thickness of the wall 2; ~'0 parameters of the deflection functions;z = and 1/,. = dimensionless coordinates. The deflection functions for the walls, assumed to be in the form of polynomials, have been chosen in order to represent the buckling forms of the column section, which would be appropriate for different ratios of the thicknesses of wall 1 to wall 2 (t* -and for different ratios of the flexural rigidity of wall 2 to that ofthe diaphragm (y = From the condition of the equality of the slopes between neighbouring walls and the equality of the bending moments along each junction line, the following relations involving the parameters (0 have been obtained:
x/l;
yi/bi
t~/t2)
D2/Dd).
My
b,
¢,, = ~ ¢ 2 , - ~
b,(
¢,2 = - D , \ g )
D2 b , )
1 + D--7b7 ¢23 ¢22
(2)
V, fb,):
¢,3 = -O, I ~ ] ¢23
b,
b'(102 ~, D--~02 b--'~)b'
where D~ and Dz are the flexural rigidities of walls 1 and 2 respectively. The conditions of the equality of the bending moments on the junction lines between the walls and the diaphragms have been satisfied by
66
M Krolak
/, (t,,%,,% o Y - ~ 4 - - ~
|.1
/
Fig. 3.
applying to the edges of the diaphragms bending moments calculated for walls 1 and 2 atx = 0. Taking into account the antisymmetric form ofthe buckling of the column walls against the diaphragms (Fig. 3), it is necessary, to load the diaphragm edges by double values of the bending moments (M,I),. = 0.y and (Mr2)x = 0.y:. After the expansion of the functions I/(17 - 1) and r/(rl s - 217: + 1) into Fourier series, the expressions for the bending moments M.,.i may be written in the following form: tt [,- [
8
(M.,,)-,=,,.y, = -2D, T [ g , : [ + ~',4
~
Z
nrrrli)
n3
-
" . = t.3..~
(M,.,).,.= o.y, = - 2 D : ~
sin
~ . =Z,.3.5
2'- -
n:
E
(3)
m3
m = 1.3,5
(96 sin rnrrr/., ) ] + ~'24 ~.,--~1.3.5 ~
mS
The problem of the flexure of the diaphragm loaded by edge bending moments has been solved using the finite Fourier transformation method. The deflection function of the diaphragm is assumed in the form
Stabili O' of thin-walled box columns
4 ~ ]~ w* (n, m)sin ct.y, sin e,.Y._ bib2 . ,.
Wd
67
(4)
where nrr tin
-
bl '
mzr ~m
-
b2
while w* (n, m) is the Fourier transform of the function wd 0'~ ,Y2). Taking the double symmetry of the diaphragm load into account, the transform w*(n, m) can be expressed in the form ~ w*(n, m) = 2 a.A,,, + [3.,B. Da(a~ + fl~,)2
when n, m are odd
w*(n.m) = 0
when n, m are even
(5)
is the flexural rigidity of the diaphragm, while A,. and B. are the coefficients of expansion for the Fourier series of the edge moment expressions. The defection function of the diaphragm being expressed by the parameters 52j of the defection function, wE is required to satisfy the following conditions for the slopes between the walls and the diaphragm:
D d
=
ex
),< : o.y, : l,,/_"
dx
],<: o..v.. = <./2
=
-
key2/y..:o.,.,
:~,1,_
dy, i.,., :o..,:
= <./_,
~,
(6)
The following relations have been obtained from the conditions: 5_3 = alq,_: + a25,_4
(7)
521 = a3522 -1- a4524 in which a t. . . . . a. are constants dependent on b i, b 2,1,D i, D> Dp. Owing to relations (2) and (7) the defection functions of the walls (1) and diaphragm (4) can be written as a function of only two independent parameters, 522 and (_,4. From the condition of minimum total potential energy of the column section (including the bending energy of walls and diaphragms and the work ofexternal loading) expressed in terms of the parameters 522 and 5_.4, a set of two homogeneous algebraic equations, depending on these parameters, can be obtained. Equating to zero the characteristic determinant of this set, the value of the critical load is obtained. The
68
M. Krolak
derived formulae have been p r o g r a m m e d on a personal computer. Calculations have been carried out using n o n - d i m e n s i o n a l quantities obtained using the following notation:
b, t* 1 "
bl
a-l,~P cr
tl , 6t:
Dj , D,
y-
D, , Dj
pcrt, b~ /7202
4 ANALYSIS O F C A L C U L A T I O N RESULTS Let us first analyse the influence of the flexural rigidity of the diaphragms on the stability o f thin-walled c o l u m n s having square cross-section. The geometrical d i m e n s i o n s of the c o l u m n s analysed, are described by the following parameters: a =/3 = 1.0 a n d a =/3 = 1-25, r = 0-001 + 8. t* = tl/t2 = 1"0 (6 = 1-0). The calculation results of the dimensionless critical loads in terms of the p a r a m e t e r characterizing the flexural rigidity of the d i a p h r a g m s are presented in Fig. 4. As can be seen from the graphs, the d i a p h r a g m s with thickness equal to that of the c o l u m n walls (y = 1) cause an increase in the critical load of 35--40% in c o m p a r i s o n to very flexible diaphragms. Figure 5 presents the graphs of the dimensionless critical load p*, in terms of the distance ! between the d i a p h r a g m s for three different rigidities o f the d i a p h r a g m s g = 0.001, 7 = 1.0 a n d V = 1000. It can be seen in Fig. 5 that in the range considered the critical load for the c o l u m n
t t • ~-U:1,0 t,
8,O
~,0
~o
2,0
Fig. 4.
a0
Stability o f thin-walled box columns
69
: 0001
~:0.1 -- 1ooo
I it" - "--u-=70 ' -
tS
m"
oy
I.o
Fig.
1,3 5.
in the case of extremely flexible diaphragms (y = 1000) depends on their spacing, i, to a small degree, while for rigid diaphragms (• = 0-001) the critical load quickly increases with a decrease in the diaphragm spacing. In the case of the column having ), = 0.001 the increase in the critical load for a = 1.3, in comparison to the case when a = 0.7, amounts to about 70%. From Fig. 6 the influence of the flexural rigidity of the diaphragms on the critical load can be analysed for different thicknesses t~ with the coefficients a and/3 equal to 2.5 and 1.25 respectively. The graphs given in Fig. 6 show that for the type of columns considered the dimensionless critical loadp*r increases with the flexural rigidity ofthe diaphragms and the thickness t~ of wall 1 (thickness tz of wall 2 is regarded as a constant quantity). The almost horizontal part of the curve for~" = 0.001 indicates buckling of wall 2 which for t* > 1.5 is practically clamped at all edges.
5 CONCLUSIONS The results from the analysis performed indicate that it is advisable to take into account the flexural rigidity of the diaphragms during calculations of the local stability of thin-walled columns with densely spaced diaphragms. This allows for a more exact determination at the critical load of the structure under consideration. The critical loads of uniformly compressed thin-walled columns, with densely spaced diaphragms whose thicknesses are equal to that of the walls, are fairly significantly greater than those of the columns with perfectly flexible
70
M. Kr6lak
3ol
Joe =125 /5' --25 1
20
a': l,O
10
I I I
0,5
~,0
15
t*
20
Fig. 6.
diaphragms. The program devised for this problem can also be used for the calculation of the critical loads of thin-walled rectangular prisms subjected to uniform compression.
REFERENCES 1. Leyko, J., Kr61ak, M., Jakubowski, S., Kelm, A. & Kowal-Michalska, K., Badania do~wiadczalne zachowania sie w stanie zakrytycznym du~ych modeli cienko~ciennych d~wigar6w skrzynkowych. Archiwum Budowy Maszyn, XXXI (1984), Z.1-2. 2. Crisfield, M. A., A note on the elastic analysis of box girder diaphragms using finite elements. Department of the Environment TRRL Report SR 118UC, Transport and Road Research Laboratory, Crowthorne, 1974. 3. Crisfield, M.A. & Puthli, R. S., A finite element method applied to the collapse of stiffened box-girder diaphragms. In Steel Plated Structures, eds P. J. Dowling, J. E. Harding & P. A. Frieze, Crosby Lockwood Staples, 1977. 4. Puthli, R.S. & Crisfield, M.A., The strength of stiffened box-girder
Stability of thin-walled box columns
5. 6. 7. 8. 9. 10. 11.
71
diaphragms. TRRL Supplementary Report 353, Department ofthe Environmerit/Department of Transport, 1977. Crisfield, M. A., Theoretical and experimental behaviour of lightly stiffened box-girder diaphragms. TRRL Laboratory Report 961, Department of the Environment/Department of Transport, 1980. Einarsson, B. & Dowling, P.J., Tests on simply stiffened rectangular diaphragmsmmodel 1. CESLIC Report BG54, Dept Civil Engineering, Imperial College, London, 1979. Einarsson, B. & Dowling, P.J., Tests on simply stiffened rectangular diaphragms--model 2, CELSIC Report BG57, Dept Civil Engineering, Imperial College, London, 1979. Crisfield, M. A. & Puthli, R. S.,-A design approach for lightly stiffened boxgirder diaphragms. TRRL Supplementary Report 393, Department of the Environment, 1978. Einarsson, B., Dowling, P.J. & Slatford, J. E., Design of steel box girder diaphragms. CELSIC Report BG59, Dept Civil Engineering, Imperial College, London, 1980. Herzog, M. A. M., Ultimate strength of diaphragm in steel box girders. J. Inst. Engng. (India), Civ. Eng. Div., 64 (1984) 371-3. Nowacki, W., D'wigary Powierzchniowe, PWN, Warszawa, 1979.
Influence of the Flexural Rigidity of Diaphragms on the Stability of Thin-Walled Box Columns Subjected to Uniform Compression
M. Kr61ak Institute of Applied Mechanics, Technical University of-L6d~, Poland (Received 8 August 1988: revision received 7 October 1988; accepted 20 October 1988)
ABSTRACT The influence of the diaphragm flexural rigidity on the critical load of a thinwalled column loaded by uniform compression is examined. The problem has been solved approximately by applying an energy method. The results of numerical calculations are presented in graphical form.
1 INTRODUCTION Thin-walled beams or columns are often stiffened by diaphragms or transverse ribs in order to retain the shape of the structure. In theoretical calculations diaphragms have usually been considered as elements that are very flexible in bending and rigid in tension or compression. Such an assumption has proved a considerable simplification. The results of model tests examining stability and the post-buckling state of girders t have shown that there is elastic interaction between the diaphragms and the girder walls with substantial bending stresses in the region adjacent to their junction. Theoretical analysis of the behaviour of lightly stiffened box girder diaphragms by means of a nonlinear finiteelement method has been carried out in Refs 2-5. Test results for two large-scale box girders with diaphragms have been given by Einarsson and Dowling. 6"7 Simple design rules for lightly stiffened box girder 63
Thin-Walled Structures 0263-8231/89/$03.50© 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain.
64
M Krolak
diaphragms have been proposed in Refs 8,9 and in Ref. 10 the ultimate strength of such elements is examined. The aim of the present paper is to analyse the influence of the flexural rigidity, of diaphragms on the critical load of a uniformly compressed column.
2 F O R M U L A T I O N OF T H E P R O B L E M The problem concerns the local elastic stability of a thin-walled column with rectangular cross-section and diaphragms perpendicular to its axis. In this problem the diaphragm flexural rigidity and the interaction between the walls and the diaphragms have been taken into account. It has been assumed that the distance I between the diaphragms is small, so that after local buckling no additional nodal lines will appear between diaphragms. The basic dimensions of the column cross-section are shown in Fig. 1.
3 SOLUTION OF T H E P R O B L E M The problem has been solved in an approximate way by means of an energy method. The section of the column considered, with coordinate system corresponding to a particular wall, is presented in Fig. 2. Taking the double symmetry of the column cross-section into account, the corresponding plate elements of the section are denoted by indices 1 and 2. The deflection functions of walls 1 and 2 are assumed to be in the form: wi = t2[~'i,X(Z 3 - 2Z 2 -t- 1)r/;(r/~ - 21/,-2+ 1)
+ ~';2X(X - 1)r/i(r/i - l) + ~',3Z (X3 - 2Z 2 + 1)r/i(r/i +~';4Z(X- 1 ) r / , ( q ~ - 2 q ~ + 1)1
i = 1,2
I
L
P i
1.
~' Fig.
1.
.J
--
1)
(1)
Stability of thin-walled box columns
65
• Xr:Xz=X
z~,,-, j____ ,2 /I
//
/pi
,? .
F"
_r,
tS
Fig. 2.
where t2 is the thickness of the wall 2; ~'0 parameters of the deflection functions;z = and 1/,. = dimensionless coordinates. The deflection functions for the walls, assumed to be in the form of polynomials, have been chosen in order to represent the buckling forms of the column section, which would be appropriate for different ratios of the thicknesses of wall 1 to wall 2 (t* -and for different ratios of the flexural rigidity of wall 2 to that ofthe diaphragm (y = From the condition of the equality of the slopes between neighbouring walls and the equality of the bending moments along each junction line, the following relations involving the parameters (0 have been obtained:
x/l;
yi/bi
t~/t2)
D2/Dd).
My
b,
¢,, = ~ ¢ 2 , - ~
b,(
¢,2 = - D , \ g )
D2 b , )
1 + D--7b7 ¢23 ¢22
(2)
V, fb,):
¢,3 = -O, I ~ ] ¢23
b,
b'(102 ~, D--~02 b--'~)b'
where D~ and Dz are the flexural rigidities of walls 1 and 2 respectively. The conditions of the equality of the bending moments on the junction lines between the walls and the diaphragms have been satisfied by
66
M Krolak
/, (t,,%,,% o Y - ~ 4 - - ~
|.1
/
Fig. 3.
applying to the edges of the diaphragms bending moments calculated for walls 1 and 2 atx = 0. Taking into account the antisymmetric form ofthe buckling of the column walls against the diaphragms (Fig. 3), it is necessary, to load the diaphragm edges by double values of the bending moments (M,I),. = 0.y and (Mr2)x = 0.y:. After the expansion of the functions I/(17 - 1) and r/(rl s - 217: + 1) into Fourier series, the expressions for the bending moments M.,.i may be written in the following form: tt [,- [
8
(M.,,)-,=,,.y, = -2D, T [ g , : [ + ~',4
~
Z
nrrrli)
n3
-
" . = t.3..~
(M,.,).,.= o.y, = - 2 D : ~
sin
~ . =Z,.3.5
2'- -
n:
E
(3)
m3
m = 1.3,5
(96 sin rnrrr/., ) ] + ~'24 ~.,--~1.3.5 ~
mS
The problem of the flexure of the diaphragm loaded by edge bending moments has been solved using the finite Fourier transformation method. The deflection function of the diaphragm is assumed in the form
Stabili O' of thin-walled box columns
4 ~ ]~ w* (n, m)sin ct.y, sin e,.Y._ bib2 . ,.
Wd
67
(4)
where nrr tin
-
bl '
mzr ~m
-
b2
while w* (n, m) is the Fourier transform of the function wd 0'~ ,Y2). Taking the double symmetry of the diaphragm load into account, the transform w*(n, m) can be expressed in the form ~ w*(n, m) = 2 a.A,,, + [3.,B. Da(a~ + fl~,)2
when n, m are odd
w*(n.m) = 0
when n, m are even
(5)
is the flexural rigidity of the diaphragm, while A,. and B. are the coefficients of expansion for the Fourier series of the edge moment expressions. The defection function of the diaphragm being expressed by the parameters 52j of the defection function, wE is required to satisfy the following conditions for the slopes between the walls and the diaphragm:
D d
=
ex
),< : o.y, : l,,/_"
dx
],<: o..v.. = <./2
=
-
key2/y..:o.,.,
:~,1,_
dy, i.,., :o..,:
= <./_,
~,
(6)
The following relations have been obtained from the conditions: 5_3 = alq,_: + a25,_4
(7)
521 = a3522 -1- a4524 in which a t. . . . . a. are constants dependent on b i, b 2,1,D i, D> Dp. Owing to relations (2) and (7) the defection functions of the walls (1) and diaphragm (4) can be written as a function of only two independent parameters, 522 and (_,4. From the condition of minimum total potential energy of the column section (including the bending energy of walls and diaphragms and the work ofexternal loading) expressed in terms of the parameters 522 and 5_.4, a set of two homogeneous algebraic equations, depending on these parameters, can be obtained. Equating to zero the characteristic determinant of this set, the value of the critical load is obtained. The
68
M. Krolak
derived formulae have been p r o g r a m m e d on a personal computer. Calculations have been carried out using n o n - d i m e n s i o n a l quantities obtained using the following notation:
b, t* 1 "
bl
a-l,~P cr
tl , 6t:
Dj , D,
y-
D, , Dj
pcrt, b~ /7202
4 ANALYSIS O F C A L C U L A T I O N RESULTS Let us first analyse the influence of the flexural rigidity of the diaphragms on the stability o f thin-walled c o l u m n s having square cross-section. The geometrical d i m e n s i o n s of the c o l u m n s analysed, are described by the following parameters: a =/3 = 1.0 a n d a =/3 = 1-25, r = 0-001 + 8. t* = tl/t2 = 1"0 (6 = 1-0). The calculation results of the dimensionless critical loads in terms of the p a r a m e t e r characterizing the flexural rigidity of the d i a p h r a g m s are presented in Fig. 4. As can be seen from the graphs, the d i a p h r a g m s with thickness equal to that of the c o l u m n walls (y = 1) cause an increase in the critical load of 35--40% in c o m p a r i s o n to very flexible diaphragms. Figure 5 presents the graphs of the dimensionless critical load p*, in terms of the distance ! between the d i a p h r a g m s for three different rigidities o f the d i a p h r a g m s g = 0.001, 7 = 1.0 a n d V = 1000. It can be seen in Fig. 5 that in the range considered the critical load for the c o l u m n
t t • ~-U:1,0 t,
8,O
~,0
~o
2,0
Fig. 4.
a0
Stability o f thin-walled box columns
69
: 0001
~:0.1 -- 1ooo
I it" - "--u-=70 ' -
tS
m"
oy
I.o
Fig.
1,3 5.
in the case of extremely flexible diaphragms (y = 1000) depends on their spacing, i, to a small degree, while for rigid diaphragms (• = 0-001) the critical load quickly increases with a decrease in the diaphragm spacing. In the case of the column having ), = 0.001 the increase in the critical load for a = 1.3, in comparison to the case when a = 0.7, amounts to about 70%. From Fig. 6 the influence of the flexural rigidity of the diaphragms on the critical load can be analysed for different thicknesses t~ with the coefficients a and/3 equal to 2.5 and 1.25 respectively. The graphs given in Fig. 6 show that for the type of columns considered the dimensionless critical loadp*r increases with the flexural rigidity ofthe diaphragms and the thickness t~ of wall 1 (thickness tz of wall 2 is regarded as a constant quantity). The almost horizontal part of the curve for~" = 0.001 indicates buckling of wall 2 which for t* > 1.5 is practically clamped at all edges.
5 CONCLUSIONS The results from the analysis performed indicate that it is advisable to take into account the flexural rigidity of the diaphragms during calculations of the local stability of thin-walled columns with densely spaced diaphragms. This allows for a more exact determination at the critical load of the structure under consideration. The critical loads of uniformly compressed thin-walled columns, with densely spaced diaphragms whose thicknesses are equal to that of the walls, are fairly significantly greater than those of the columns with perfectly flexible
70
M. Kr6lak
3ol
Joe =125 /5' --25 1
20
a': l,O
10
I I I
0,5
~,0
15
t*
20
Fig. 6.
diaphragms. The program devised for this problem can also be used for the calculation of the critical loads of thin-walled rectangular prisms subjected to uniform compression.
REFERENCES 1. Leyko, J., Kr61ak, M., Jakubowski, S., Kelm, A. & Kowal-Michalska, K., Badania do~wiadczalne zachowania sie w stanie zakrytycznym du~ych modeli cienko~ciennych d~wigar6w skrzynkowych. Archiwum Budowy Maszyn, XXXI (1984), Z.1-2. 2. Crisfield, M. A., A note on the elastic analysis of box girder diaphragms using finite elements. Department of the Environment TRRL Report SR 118UC, Transport and Road Research Laboratory, Crowthorne, 1974. 3. Crisfield, M.A. & Puthli, R. S., A finite element method applied to the collapse of stiffened box-girder diaphragms. In Steel Plated Structures, eds P. J. Dowling, J. E. Harding & P. A. Frieze, Crosby Lockwood Staples, 1977. 4. Puthli, R.S. & Crisfield, M.A., The strength of stiffened box-girder
Stability of thin-walled box columns
5. 6. 7. 8. 9. 10. 11.
71
diaphragms. TRRL Supplementary Report 353, Department ofthe Environmerit/Department of Transport, 1977. Crisfield, M. A., Theoretical and experimental behaviour of lightly stiffened box-girder diaphragms. TRRL Laboratory Report 961, Department of the Environment/Department of Transport, 1980. Einarsson, B. & Dowling, P.J., Tests on simply stiffened rectangular diaphragmsmmodel 1. CESLIC Report BG54, Dept Civil Engineering, Imperial College, London, 1979. Einarsson, B. & Dowling, P.J., Tests on simply stiffened rectangular diaphragms--model 2, CELSIC Report BG57, Dept Civil Engineering, Imperial College, London, 1979. Crisfield, M. A. & Puthli, R. S.,-A design approach for lightly stiffened boxgirder diaphragms. TRRL Supplementary Report 393, Department of the Environment, 1978. Einarsson, B., Dowling, P.J. & Slatford, J. E., Design of steel box girder diaphragms. CELSIC Report BG59, Dept Civil Engineering, Imperial College, London, 1980. Herzog, M. A. M., Ultimate strength of diaphragm in steel box girders. J. Inst. Engng. (India), Civ. Eng. Div., 64 (1984) 371-3. Nowacki, W., D'wigary Powierzchniowe, PWN, Warszawa, 1979.