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On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections
Computers and Structures 80 (2002) 691–695 www.elsevier.com/locate/compstruc
Technical note
On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections A. Arpaci *, E. Bozdag Faculty of Mechanical Engineering, Istanbul Technical University, Gumussuyu, 80191 Istanbul, Turkey Received 13 March 2001; accepted 9 January 2002
Abstract This work relates to the analysis of triply coupled vibrations of thin-walled beams having nonsymmetrical open cross-sections. The governing differential equations for coupled bending and torsional vibrations are derived and solved exactly. A recent study on the same subject is criticized and discussed in theoretical and numerical aspects. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Vibration; Coupled vibration; Beam
1. Introduction Beams of thin-walled open cross-sections are widely used in structural design. In general, the centroid and the shear center do not coincide; hence the flexural and the torsional vibrations are coupled. The case that the cross-section has a single axis of symmetry and consequently the flexural vibrations in one direction are coupled with the torsional vibrations is extensively studied by many researchers using Euler–Bernoulli theory. But the number of studies dealing with coupled flexural–flexural–torsional vibrations is rather limited. Yaman [1] investigates the triply coupled vibrations of open-section channels by wave analysis method. The coupled wave numbers, various frequency response curves and the mode shapes are presented for undamped and structurally damped channels. Recently, Tanaka and Bercin [2] have extended the approach of Bishop et al. [3] used previously to investigate the double coupling, which owes much to the work of Dokumacı [4], to triply coupled vibrations of thin-walled beams.
*
Corresponding author. Tel.: +90-212-293-1300x2493; fax: +90-212-245-0795. E-mail address: [email protected] (A. Arpaci).
The mentioned authors [2] have derived the governing differential equations of motion and obtained solutions for various boundary conditions using Mathematica. The present work is motivated by the fact that the governing differential equations of motion derived by them exhibit a confusion of co-ordinate system. It is well known that the product of inertia terms must be included in the formulations of bending deflections unless the principal centroidal axes are used. In the theory developed by the mentioned authors although the flexural displacement and the offsets of the shear center are determined with respect to the axes which are perpendicular and parallel to the web of unsymmetrical channel, the equations are formulated as if the principal axes are used. The purpose of the present study is to correct the theory and to give accurate results.
2. Equations of motion In Fig. 1 is shown a typical cross-section of no axial symmetry where x and y axes are taken through the centroid and parallel to the sides. X and Y are the axes passing through the shear center S and parallel to the centroidal axes. The equations for bending and nonuniform torsion under static loads are
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 2 5 - 1
692
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
Fig. 1. Co-ordinate system.
d4 u d4 v þ EI ¼ qX xy dz4 dz4 4 4 dv du EIx 4 þ EIxy 4 ¼ qY dz dz d4 u d2 u EIw 4 GJ 2 ¼ mt dz dz
kx ¼ Ix =Ixy ;
EIy
ky ¼ Iy =Ixy ;
ax ¼ qAx2 =EIxy ;
ð1Þ
kw ¼ GJ =EIw ;
ay ¼ qAx2 =EIy ;
2
aw ¼ qAx2 e1 =EIw
2
bx ¼ qAx e1 =EIxy ;
k0 ¼ qI0 x2 =EIw
by ¼ qAx e2 =EIy ;
bw ¼ qAx2 e2 =EIw
D ¼ d=dz ð4Þ
Thus the governing differential equations are: where u and v are deflections of the shear center in X and Y directions respectively, u is angle of rotation of crosssection, EIx and EIy are flexural rigidities, EIxy is coupling stiffness, EIw is warping stiffness, GJ is torsional constant, qX and qY are intensities of distributed forces and mt is intensity of distributed torque along shear center axis. According to the d’Alembert’s principle the static loads are replaced by inertia forces and the equations of motion are thus obtained as follows,
EIy
4
ð5Þ
2
bw U aw V þ ðD kw D k0 ÞU ¼ 0 Setting the determinant of the above system equal to zero yields an ordinary differential equation of 12th order, for each function, as follows a1 D12 þ a2 D10 þ a3 D8 þ a4 D6 þ a5 D4 þ a6 D2 þ a7 F ¼ 0 ð6Þ
a1 ¼ kx ky ; a2 ¼ kw ky kw kx a3 ¼ ax ðay þ k0 Þkx þ k0 ky a4 ¼ ax kw þ ay kw kx ; a5 ¼ ax ay aw bx aw by þ ax k0
o4 u o2 u o2 v o2 u o2 u GJ 2 þ qAe1 2 qAe2 2 þ qI0 2 ¼ 0 4 oz oz ot ot ot ð2Þ
where q is mass density, A is cross-sectional area, e1 and e2 are co-ordinates of centroid G, I0 is polar moment of inertia about shear center S and t is time. For harmonic vibrations the displacements and the torsional rotation may be expressed in the form uðz; tÞ ¼ UðzÞ sin xt; vðz; tÞ ¼ V ðzÞ sin xt;
D4 U þ ðkx D4 ax ÞV bx U ¼ 0
where
o4 u o4 v o2 u o2 u þ EIxy 4 þ qA 2 qAe2 2 ¼ 0 4 oz oz ot ot
o4 v o4 u o2 v o2 u EIx 4 þ EIxy 4 þ qA 2 þ qAe1 2 ¼ 0 oz oz ot ot EIw
ðD4 ay ÞU þ ky D4 V þ by U ¼ 0
þ ðay k0 bw by Þkx bw bx ky a6 ¼ ax ay kw ;
a7 ¼ aw ay bx þ ax bw by ax ay k0 ð7Þ
and F denotes U, V or U. Taking a solution of the form Cerz for the amplitudes and introducing the variable s ¼ r2 , the following characteristic equation can be obtained a1 s6 þ a2 s5 þ a3 s4 þ a4 s3 þ a5 s2 þ a6 s þ a7 ¼ 0
ð8Þ
The general solutions of U, V and U are then, uðz; tÞ ¼ UðzÞ sin xt
ð3Þ
U¼
12 X i¼1
Let the following shorthand notations are defined as
Ai eri z ;
V ¼
12 X i¼1
Bi eri z
and
U¼
12 X
Ci eri z
i¼1
ð9Þ
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
where Ai , Bi and Ci are three sets of constants. They are not all independent, however. By substituting above expressions into Eq. (5) only 12 integration constants are obtained.
693
them, it has not been possible to carry out the calculations for the same cross-section. Another example is provided in order to exhibit the good agreement between the results of the present theory and that of [1], the latter being obtained by the wave propagation approach.
3. Boundary conditions Clamped end: The well-known boundary conditions for a clamped end are that the translations, rotations and slopes are zero. Hence, U ¼ 0; V ¼ 0; U ¼ 0; U 0 ¼ 0; V 0 ¼ 0; U0 ¼ 0 Notice that the condition U0 ¼ 0 is due to the warping effect considered in the present theory, implied by the restricted longitudinal displacement of the cross-section which originally was plane [5], and is not considered when Saint–Venant theory is used. Hinged end: A hinged end implies restraint against translations and rotation but not against warping; that is, the end of the beam does not rotate but is free to warp. This means the longitudinal stress is zero, or, in other words, U00 must be zero [5]. It is well known that the bending moments are also zero. Hence the boundary conditions are U ¼ 0; V ¼ 0; U ¼ 0; U 00 ¼ 0; V 00 ¼ 0; U00 ¼ 0 Free end: For a free end, the bending moments, the torsional moment and the shear forces are equal to zero. As no restraint against warping is implied the longitudinal stress must be zero. The boundary conditions are then U 00 ¼ 0; V 00 ¼ 0; GJ U0 EIw U000 ¼ 0; U 000 ¼ 0; V 000 ¼ 0; U00 ¼ 0
4.1. Example 1 A thin-walled uniform beam with unsymmetric channel section (Fig. 1) is considered. The geometric and material properties of the beam are listed in Table 1. The relative error due to co-ordinate confusion is shown in Fig. 2 for two kinds of boundary conditions. 4.2. Example 2 A thin-walled uniform beam with a Z cross-section is considered as the second example. Table 1 contains the geometric and material properties. The relative error is shown in Fig. 3 for two kinds of boundary conditions. 4.3. Example 3 Yaman [1] considers a simply supported thin-walled beam with unsymmetric channel section as shown in Fig. 1 and having the following geometric and material properties:
Table 1 Geometric and material properties of beams studied in Examples 1 and 2 Properties
Beam cross-section Example 1
Example 2
4. Numerical evaluation Application of the boundary conditions to Eq. (9) at z ¼ 0 and z ¼ ‘ will yield 12 simultaneous linear homogeneous equations. The frequency equation which can be numerically solved to give the values of x is obtained by setting the determinant of the system equal to zero. The relative error is defined as ðx x0 Þ=x, where x is the frequency obtained by the present theory and x0 is the frequency obtained in Ref. [2]. The natural frequency analysis is performed for two kinds of cross-section; and two kinds of boundary conditions are considered for either of them. The numeric results are obtained by using the present theory and the theory employed by Tanaka and Bercin [2]. Because the mentioned authors do not give the value of product of inertia for the cross-section considered by
EIx (N m2 ) EIy (N m2 ) EIxy (N m2 ) EIw (N m4 ) GJ (Nm2 ) I0 (m4 ) q (kg/m3 ) A (m2 ) e1 (m) e2 (m) ‘ (m)
1.093Eþ08 1.600Eþ03 1.874Eþ03 9.746E02 2.034Eþ02 1.052E07 7.860Eþ03 3.050E04 7.669E03 8.988E03 9.800E01
1.509Eþ04 6.234Eþ03 7.252Eþ03 9.934E01 9.720Eþ01 2.720E05 7.860Eþ03 2.975E01 4.596E03 8.363E03 8.000E01
694
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
Fig. 2. Relative error versus mode number for various end conditions (Example 1).
Fig. 3. Relative error versus mode number for various end conditions (Example 2).
Table 2 Natural frequencies (Hz) of beam studied as Example 3 Mode number [1] Present study
1
2
3
4
5
6
7
45.49 47.00
69.91 73.39
101.73 102.36
149.82 145.34
154.79 154.53
207.43 206.70
259.89 258.76
EIx ¼ 355:6 ðN m2 Þ;
EIy ¼ 1568 ðN m2 Þ;
2
4
EIxy ¼ 297:5 ðN m Þ;
EIw ¼ 0:4977 ðN m Þ; I0 ¼ 4:6 108 ðm4 Þ;
2
GJ ¼ 1:352 ðN m Þ; 3
q ¼ 2700 ðkg=m Þ;
wave propagation theory employed in [1]. The agreement between them seems to be good.
A ¼ 9:68 105 ðm2 Þ;
e1 ¼ 9:09 103 ðmÞ; ‘ ¼ 1:00 ðmÞ:
5. Conclusions
e2 ¼ 10:43 103 ðmÞ;
Table 2 shows the natural frequencies of above referenced beam obtained by the present theory and by the
Figs. 2 and 3 show that considerable errors will arise if the coupling stiffness EIxy is not considered in nonprincipal co-ordinate system. This is more profound when the product of inertia is of a great value as in Z cross-section. The relative errors are very small at some
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
modes. This can be attributed to the fact that the flexural coupling is not important at torsion-dominated modes.
References [1] Yaman Y. Vibration of open-section channels: a coupled flexural and torsional wave analysis. Journal of Sound and Vibration 1997;204(1):131–58.
695
[2] Tanaka M, Bercin AN. Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Computers and Structures 1999;71:1–8. [3] Bishop RED, Cannon SM, Miao S. On coupled bending and torsional vibration of uniform beams. Journal of Sound and Vibration 1989;131(3):457–64. [4] Dokumacı E. An exact solution for coupled bending and torsion vibrations of uniform beams having single crosssectional symmetry. Journal of Sound and Vibration 1987;119(3):443–9. [5] Gere JM. Torsional vibrations of beams of thin-walled open section. Journal of Applied Mechanics 1954;21:381–7.
Technical note
On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections A. Arpaci *, E. Bozdag Faculty of Mechanical Engineering, Istanbul Technical University, Gumussuyu, 80191 Istanbul, Turkey Received 13 March 2001; accepted 9 January 2002
Abstract This work relates to the analysis of triply coupled vibrations of thin-walled beams having nonsymmetrical open cross-sections. The governing differential equations for coupled bending and torsional vibrations are derived and solved exactly. A recent study on the same subject is criticized and discussed in theoretical and numerical aspects. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Vibration; Coupled vibration; Beam
1. Introduction Beams of thin-walled open cross-sections are widely used in structural design. In general, the centroid and the shear center do not coincide; hence the flexural and the torsional vibrations are coupled. The case that the cross-section has a single axis of symmetry and consequently the flexural vibrations in one direction are coupled with the torsional vibrations is extensively studied by many researchers using Euler–Bernoulli theory. But the number of studies dealing with coupled flexural–flexural–torsional vibrations is rather limited. Yaman [1] investigates the triply coupled vibrations of open-section channels by wave analysis method. The coupled wave numbers, various frequency response curves and the mode shapes are presented for undamped and structurally damped channels. Recently, Tanaka and Bercin [2] have extended the approach of Bishop et al. [3] used previously to investigate the double coupling, which owes much to the work of Dokumacı [4], to triply coupled vibrations of thin-walled beams.
*
Corresponding author. Tel.: +90-212-293-1300x2493; fax: +90-212-245-0795. E-mail address: [email protected] (A. Arpaci).
The mentioned authors [2] have derived the governing differential equations of motion and obtained solutions for various boundary conditions using Mathematica. The present work is motivated by the fact that the governing differential equations of motion derived by them exhibit a confusion of co-ordinate system. It is well known that the product of inertia terms must be included in the formulations of bending deflections unless the principal centroidal axes are used. In the theory developed by the mentioned authors although the flexural displacement and the offsets of the shear center are determined with respect to the axes which are perpendicular and parallel to the web of unsymmetrical channel, the equations are formulated as if the principal axes are used. The purpose of the present study is to correct the theory and to give accurate results.
2. Equations of motion In Fig. 1 is shown a typical cross-section of no axial symmetry where x and y axes are taken through the centroid and parallel to the sides. X and Y are the axes passing through the shear center S and parallel to the centroidal axes. The equations for bending and nonuniform torsion under static loads are
0045-7949/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 0 2 5 - 1
692
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
Fig. 1. Co-ordinate system.
d4 u d4 v þ EI ¼ qX xy dz4 dz4 4 4 dv du EIx 4 þ EIxy 4 ¼ qY dz dz d4 u d2 u EIw 4 GJ 2 ¼ mt dz dz
kx ¼ Ix =Ixy ;
EIy
ky ¼ Iy =Ixy ;
ax ¼ qAx2 =EIxy ;
ð1Þ
kw ¼ GJ =EIw ;
ay ¼ qAx2 =EIy ;
2
aw ¼ qAx2 e1 =EIw
2
bx ¼ qAx e1 =EIxy ;
k0 ¼ qI0 x2 =EIw
by ¼ qAx e2 =EIy ;
bw ¼ qAx2 e2 =EIw
D ¼ d=dz ð4Þ
Thus the governing differential equations are: where u and v are deflections of the shear center in X and Y directions respectively, u is angle of rotation of crosssection, EIx and EIy are flexural rigidities, EIxy is coupling stiffness, EIw is warping stiffness, GJ is torsional constant, qX and qY are intensities of distributed forces and mt is intensity of distributed torque along shear center axis. According to the d’Alembert’s principle the static loads are replaced by inertia forces and the equations of motion are thus obtained as follows,
EIy
4
ð5Þ
2
bw U aw V þ ðD kw D k0 ÞU ¼ 0 Setting the determinant of the above system equal to zero yields an ordinary differential equation of 12th order, for each function, as follows a1 D12 þ a2 D10 þ a3 D8 þ a4 D6 þ a5 D4 þ a6 D2 þ a7 F ¼ 0 ð6Þ
a1 ¼ kx ky ; a2 ¼ kw ky kw kx a3 ¼ ax ðay þ k0 Þkx þ k0 ky a4 ¼ ax kw þ ay kw kx ; a5 ¼ ax ay aw bx aw by þ ax k0
o4 u o2 u o2 v o2 u o2 u GJ 2 þ qAe1 2 qAe2 2 þ qI0 2 ¼ 0 4 oz oz ot ot ot ð2Þ
where q is mass density, A is cross-sectional area, e1 and e2 are co-ordinates of centroid G, I0 is polar moment of inertia about shear center S and t is time. For harmonic vibrations the displacements and the torsional rotation may be expressed in the form uðz; tÞ ¼ UðzÞ sin xt; vðz; tÞ ¼ V ðzÞ sin xt;
D4 U þ ðkx D4 ax ÞV bx U ¼ 0
where
o4 u o4 v o2 u o2 u þ EIxy 4 þ qA 2 qAe2 2 ¼ 0 4 oz oz ot ot
o4 v o4 u o2 v o2 u EIx 4 þ EIxy 4 þ qA 2 þ qAe1 2 ¼ 0 oz oz ot ot EIw
ðD4 ay ÞU þ ky D4 V þ by U ¼ 0
þ ðay k0 bw by Þkx bw bx ky a6 ¼ ax ay kw ;
a7 ¼ aw ay bx þ ax bw by ax ay k0 ð7Þ
and F denotes U, V or U. Taking a solution of the form Cerz for the amplitudes and introducing the variable s ¼ r2 , the following characteristic equation can be obtained a1 s6 þ a2 s5 þ a3 s4 þ a4 s3 þ a5 s2 þ a6 s þ a7 ¼ 0
ð8Þ
The general solutions of U, V and U are then, uðz; tÞ ¼ UðzÞ sin xt
ð3Þ
U¼
12 X i¼1
Let the following shorthand notations are defined as
Ai eri z ;
V ¼
12 X i¼1
Bi eri z
and
U¼
12 X
Ci eri z
i¼1
ð9Þ
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
where Ai , Bi and Ci are three sets of constants. They are not all independent, however. By substituting above expressions into Eq. (5) only 12 integration constants are obtained.
693
them, it has not been possible to carry out the calculations for the same cross-section. Another example is provided in order to exhibit the good agreement between the results of the present theory and that of [1], the latter being obtained by the wave propagation approach.
3. Boundary conditions Clamped end: The well-known boundary conditions for a clamped end are that the translations, rotations and slopes are zero. Hence, U ¼ 0; V ¼ 0; U ¼ 0; U 0 ¼ 0; V 0 ¼ 0; U0 ¼ 0 Notice that the condition U0 ¼ 0 is due to the warping effect considered in the present theory, implied by the restricted longitudinal displacement of the cross-section which originally was plane [5], and is not considered when Saint–Venant theory is used. Hinged end: A hinged end implies restraint against translations and rotation but not against warping; that is, the end of the beam does not rotate but is free to warp. This means the longitudinal stress is zero, or, in other words, U00 must be zero [5]. It is well known that the bending moments are also zero. Hence the boundary conditions are U ¼ 0; V ¼ 0; U ¼ 0; U 00 ¼ 0; V 00 ¼ 0; U00 ¼ 0 Free end: For a free end, the bending moments, the torsional moment and the shear forces are equal to zero. As no restraint against warping is implied the longitudinal stress must be zero. The boundary conditions are then U 00 ¼ 0; V 00 ¼ 0; GJ U0 EIw U000 ¼ 0; U 000 ¼ 0; V 000 ¼ 0; U00 ¼ 0
4.1. Example 1 A thin-walled uniform beam with unsymmetric channel section (Fig. 1) is considered. The geometric and material properties of the beam are listed in Table 1. The relative error due to co-ordinate confusion is shown in Fig. 2 for two kinds of boundary conditions. 4.2. Example 2 A thin-walled uniform beam with a Z cross-section is considered as the second example. Table 1 contains the geometric and material properties. The relative error is shown in Fig. 3 for two kinds of boundary conditions. 4.3. Example 3 Yaman [1] considers a simply supported thin-walled beam with unsymmetric channel section as shown in Fig. 1 and having the following geometric and material properties:
Table 1 Geometric and material properties of beams studied in Examples 1 and 2 Properties
Beam cross-section Example 1
Example 2
4. Numerical evaluation Application of the boundary conditions to Eq. (9) at z ¼ 0 and z ¼ ‘ will yield 12 simultaneous linear homogeneous equations. The frequency equation which can be numerically solved to give the values of x is obtained by setting the determinant of the system equal to zero. The relative error is defined as ðx x0 Þ=x, where x is the frequency obtained by the present theory and x0 is the frequency obtained in Ref. [2]. The natural frequency analysis is performed for two kinds of cross-section; and two kinds of boundary conditions are considered for either of them. The numeric results are obtained by using the present theory and the theory employed by Tanaka and Bercin [2]. Because the mentioned authors do not give the value of product of inertia for the cross-section considered by
EIx (N m2 ) EIy (N m2 ) EIxy (N m2 ) EIw (N m4 ) GJ (Nm2 ) I0 (m4 ) q (kg/m3 ) A (m2 ) e1 (m) e2 (m) ‘ (m)
1.093Eþ08 1.600Eþ03 1.874Eþ03 9.746E02 2.034Eþ02 1.052E07 7.860Eþ03 3.050E04 7.669E03 8.988E03 9.800E01
1.509Eþ04 6.234Eþ03 7.252Eþ03 9.934E01 9.720Eþ01 2.720E05 7.860Eþ03 2.975E01 4.596E03 8.363E03 8.000E01
694
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
Fig. 2. Relative error versus mode number for various end conditions (Example 1).
Fig. 3. Relative error versus mode number for various end conditions (Example 2).
Table 2 Natural frequencies (Hz) of beam studied as Example 3 Mode number [1] Present study
1
2
3
4
5
6
7
45.49 47.00
69.91 73.39
101.73 102.36
149.82 145.34
154.79 154.53
207.43 206.70
259.89 258.76
EIx ¼ 355:6 ðN m2 Þ;
EIy ¼ 1568 ðN m2 Þ;
2
4
EIxy ¼ 297:5 ðN m Þ;
EIw ¼ 0:4977 ðN m Þ; I0 ¼ 4:6 108 ðm4 Þ;
2
GJ ¼ 1:352 ðN m Þ; 3
q ¼ 2700 ðkg=m Þ;
wave propagation theory employed in [1]. The agreement between them seems to be good.
A ¼ 9:68 105 ðm2 Þ;
e1 ¼ 9:09 103 ðmÞ; ‘ ¼ 1:00 ðmÞ:
5. Conclusions
e2 ¼ 10:43 103 ðmÞ;
Table 2 shows the natural frequencies of above referenced beam obtained by the present theory and by the
Figs. 2 and 3 show that considerable errors will arise if the coupling stiffness EIxy is not considered in nonprincipal co-ordinate system. This is more profound when the product of inertia is of a great value as in Z cross-section. The relative errors are very small at some
A. Arpaci, E. Bozdag / Computers and Structures 80 (2002) 691–695
modes. This can be attributed to the fact that the flexural coupling is not important at torsion-dominated modes.
References [1] Yaman Y. Vibration of open-section channels: a coupled flexural and torsional wave analysis. Journal of Sound and Vibration 1997;204(1):131–58.
695
[2] Tanaka M, Bercin AN. Free vibration solution for uniform beams of nonsymmetrical cross section using Mathematica. Computers and Structures 1999;71:1–8. [3] Bishop RED, Cannon SM, Miao S. On coupled bending and torsional vibration of uniform beams. Journal of Sound and Vibration 1989;131(3):457–64. [4] Dokumacı E. An exact solution for coupled bending and torsion vibrations of uniform beams having single crosssectional symmetry. Journal of Sound and Vibration 1987;119(3):443–9. [5] Gere JM. Torsional vibrations of beams of thin-walled open section. Journal of Applied Mechanics 1954;21:381–7.