A THIN-WALLED RECTANGULAR BOX
BEAM UNDER TORSION:
A COMPARISON OF THE KOLLBRUNNERHAJDIN SOLUTION WITH A SOLUTION BY
DIVIDING THE BEAM INTO TWO GUIDED
VLASOV BEAMS WITH OPEN CROSSSECTION
Risto Koivula, MSc
Laboratory of Structural Engineering, Tampere University of Technology
P.O. Box 600, FIN-33101 Tampere, Finland
ABSTRACT
Due to the double symmetry of the box beam, the fields of normal stresses given rise by the axial force
or the bending moments are orthogonal with those given rise by the torsion (and distortion). Thus the
former can be independently superposed on the latter. Then for the antisymmetric warpings like
torsion and distortion the box side mid-axes can be regarded as continuous non-compressible axial
supports preventing the axial displacement only. The beam can be cut along these axes into two (or
four) guided Vlasov beams with open cross-section, which can be analysed using the formal methods
of the theory of guided beams.2, 3, 4
The idea of the theory of guided beams is the fact that for thin-walled beams, the deformations of
which are restricted by external continuous lateral or longitudinal restraints, the mutually independent
effective force and displacement quantities (warpings in terms of the generalized bending theory),
which can be determined independently by only external loads, are defined in a solving system of axes
and special points, which does not coincide with the fundamental system with the origin at the
centroid, the pole at the shear centre, and the axes parallel to the (central) principal directions. The
choice of the adequate system of axes and special points is analogous to separating correctly a group of
mutually dependent differential equations to mutually independent equations.
The main aim of this study is to create an “exact” solution of the problem in terms of the theory of
guided Vlasov beams. “Exact” means that no information is lost regards to the premises of the theory.
KEYWORDS
Thin-Walled Rectangular Box Beam, Vlasov Beam, Guided Beam, Bimoment, Warping, Torsion,
Sectorial Area, Kollbrunner-Hajdin Theory, Torsional Axis, Distortion, Distortional Axis
THE KOLLBRUNNER-HAJDIN THEORY
In the Kollbrunner-Hajdin theory of box beams with a rigid-in-plane cross-section is separated the
shear deformation, given rise by the constant shear stress flow around the tube, and the axial warping
deformation from normal stresses caused by the deplanation of the cross-section. Shear deformations
from the shear stresses given rise by warping are neglected as small compared with the former.
The displacement and force quantities are given in a right-handed system uvw, but the coordinates xyz
with the cross-sectional integrals in a left-handed system, because that system is used in the theory on
guided Vlasov beams. The shear deformation of the wall midline (profile line) is then:
γ xs ( x, s ) =
∂u x ( x, s ) ∂υ s ( x, s )
− γ xs* ( x)
+
∂x
∂s
(1)
In torsion the translation field in the profile line direction s is
υ s ( x, s ) = θ u ( x ) hC ( s ) ,
(2)
where hC(s) is the perpendicular distance of the pole C from the profile line tangent drawn at point s.
b
b
ωC(4a+4b) = 8ab
ω̂ o
s
- ω̂ o
rC(s)
a
C
x, u , θu
z
a
t
- ω̂ o
ω̂ o
y
Figure 1. The deplanation form of the rectangle box cross-section according to the
Kollbrunner-Hajdin theory
Then the axial translation field, in other words the deplanation at the profile line is:
∫ [θ ′ ( x)h
s
u x ( x, s ) =
u
C
]
(s) − γ xs∗ ds = θ u′ ( x )ωˆ ( s) ,
(3)
s0
from which is obtained
γ xs∗ =
ω C (4a + 4b)
4(a + b)
θ u′ ( x ) . Then:
(4)
sω C (4a + 4b) s
s
2ab
hC (s)ds = ω C (s) −
ωˆ (s) = ω C (s) −
s
= ∫ hC (s)ds −
∫
4(a + b)
4(a + b) S
a+b
0
ωˆ 0 = ω (b) −
2ab
2b
a−b
b = ab −
ab =
ab
a+b
a+b
a+b
(5)
(6)
Using the methods of the variational calculus, the following differential equation was obtained1, eq. III.31
ρEI ωˆ θ u( 4 ) ( x ) − GI tθ u′′( x) = mC ( x)
(7)
One more presupposition is that the moment load mC(x) is linear: in most general form in this formula
is included extra terms, including mC´´(x) and first derivative of bimoment load, bc.
The sectorial quadratic moment
4
( a − b) 2
I ωˆ = ∫ ω̂ 2 (s)dA = ta 2 b 2
a+b
3
A
(8)
The torsional moment of inertia of a closed cross-section
4tΩ 2 4t16a 2 b 2 16ta 2 b 2
=
=
It =
4(a + b)
S
a+b
(9)
The radial quadratic moment
I h = ∫ hC ( s )dA = 4tab(a + b)
2
(10)
A
The coefficient in the differential equation (7)
Ih
ρ=
=
Ih − It
4t (a + b)ab
a +b
=

2 2
16ta b
a−b
4t (a + b)ab −
a+b
2
(11)
So, for a rectangular box profile with constant wall-thickness t equation (7) takes the form
4
16ta 2 b 2
Eta 2 b 2 (a + b)θ u( 4) ( x) −
θ u′′ ( x) = muC ( x)
3
a+b
(12)
The bimoment is now obtained from equation1, eq. III.37
Bˆ C ( x ) = ρEI ωˆ θ ′′( x)
(13)
The normal stresses according to this theory are1, eq. III.42
σ x( x, s) =
Bˆ C ( x)ωˆ C ( x )
I ωˆ
(14)
THE GUIDED SYSTEM OF AXES AND SPECIAL POINTS FOR THE HALF PROFILE
The beam is divided into two guided Vlasov beams with a U cross-section along one of the two planes
of symmetry of the beam. The part beams are U profiles which are guided by non-compressible axes
along the edges of the “flanges”, in other words along the mid-lines of the box sides. These edges of
“flanges” are guided also by lateral restraints in the cutting plane to keep the distance from the original
box centre (C), around which the half box so is guided to rotate. Figure 2.
vD
θD
*
FwD
wD
MD
D
a
θD
b
b
rD(s)
ωD = ab
π/4
+
-
ωD(x)
θB
*
M uB
FwB
a
B
a-b
θC
u
N
s
C
N
z
y
D = the guided sectorial pole of the half box, in other words one distortional centre of the box
B = the centre of twist for the StVenant torsion of the half box
C = the centre of torsion of the entire box and of the laterally guided half box, the guided origin
N = the non-compressible edges
ωD(s) = the sectorial area around the distortional centre D. ωˆ (s ) = −
Figure 2.
The half-box beam as a guided beam.
a−b
ω D ( s)
a+b
In the beginning two independent degrees of freedom must be regarded for the U profile with the noncompressible axes along the edges, but without the lateral support: First, caused by the normal stresses
given rise by the bimoment warping, the cross-section tends to rotate around a sectorial pole, around
which it is possible to draw a sectorial area with zero value at the non-compressible axes. This guided
sectorial pole (D) is placed on the symmetry axis of the U cross-section at the “flange” length distant
from the web behind the web (outside the box). Deformed by the warping normal stresses only the
cross-section rotates around this guided torsional axis. Loading that beam through this axis in wdirection does not give rise to normal stresses.
Second, for cases of guided beams with two non-compressible axes, the shear deformation from the
constant shear flow between these axes must be taken into consideration as an independent degree of
freedom. Caused by the constant shear flow deformation, this axially guided along the flange edges,
but laterally non-guided U cross section tends to rotate around a St Venant torsional centre (B), which
is placed at the intersectional point of the bisectors of the angles (at equal perpendicular distant from
all three sides of the cross-section) if the wall thickness is constant. The cross-section rotates around
the point B without any tendency to deplanation and normal stresses.
The lateral guiding, fixing the cross section to rotate around the box centre axis C, constitutes a
coupling between these two for a laterally non-guided profile independent degrees of freedom. To
match the deformations from the warping torsion around D and the twisting torsion around B, we must
match the translations of point (axis) C from these two degrees freedom of the cross-section to zero.
Like the moment is separated into warping and twisting moments, also the shear force must be
separated into two components, one placed at the sectorial pole D and proportional to the StVenant
twisting moment around B, and the other placed at the StVenant shear centre B and proportional to the
warping moment (the derivative of the bimoment) around the sectorial pole D.
The guided StVenant torsional quadratic moments around B and around C, and the sectorial quadratic
moment around D are:
I tB = 2tb 2 (a + b)
I tC =
8ta 2 b 2
a+b
I ωD =
2t 2 2
a b ( a + b)
3
(15), (16), (18)
The angles of twisting torsion and warping torsion around the corresponding special axes are
θ uB ( x) =
2a
θ uC ( x )
a+b
θ uC ( x) = −
a−b
θ uC ( x)
a+b
(19), (20)
Using these we obtain the separate twisting and warping moments as functions of the angle of twist
θuC(x) around the box centre axis C (the star denotes a StVenant entity):
*
′ ( x) = GI tB
( x ) = GI tBθ uB
M uB
2a
′′ ( x )
θ uC
a+b
a−b
( 3)
′
θ uC (3) ( x)
M uD ( x) = − BD ( x) = − EI ωDθ uD ( x) = EI ωD
a+b
(21)
(22)
The shear forces are obtained based on these moments as functions of the unknown θuC(x):
FwB = − M uD ( x) /(a + b) = − EI D
*
*
FwD
( x) /(a + b) = GI tB
= M uB
a−b
θ ( 3) ( x )
2 uC
( a + b)
2a
′′ ( x)
θ uC
( a + b) 2
(23)
(24)
The two moments can be transformed according to the theory of guided beams to the box centre axis2:
*
M uC ( x ) = M uD ( x ) + 2aFwB + M uB ( x) + (a − b) FwD
( x)
= −(
2a 2
a−b 2
(3 )
( 3)
′ ( x ) = − EI ωˆ θ uC
′ ( x)
) GI tBθ uC
( x ) + GI tCθ uC
) EI ωDθ uC
( x) + (
a+b
a+b
(25)
The moment equilibrium around the box axis C gives the equation for the angle of rotation:
( 4)
′ (x) = − EI ωˆ θ uC
′′ ( x) = − muC ( x )
M uC
( x ) + GI tCθ uC
(26)
This gives the basic equation for θuC(x):
( 4)
′′ ( x ) = muC ( x)
EI ωˆθ uC
( x ) − GI tCθ uC
(27)
Iω̂ , ItC, and muC(x) are now divided by 2 compared with equation (7). Compared with the KollbrunnerHajdin equation (7) the coefficient ρ = 1, while for the Kollbrunner-Hajdin equation
a+b
ρ= 

a−b
2
(11)
The equation of the angle on twist according to both of the theories can be expressed with one formula
( 4)
θ uC
( x) − (
α
a±b
′′ ( x ) =
) 2 θ uC
3
a+b
muC ( x ) , where
2 2
2 Eta b (a ± b) 2
α=
6
1 +ν
(28)
In the sign ± the sign – belongs to the guided beam theory, and the sign + belongs to the KollbrunnerHajdin theory. ν is the Poisson koefficient.
The bimoments, torsional moments and normal stresses according to the theories are following:
The theory of guided beams
′′ ( x) = −
BD ( x) = EIωDθ uD
M uC (x ) =
σ x ( x, s) =
a −b
a+b
′′ ( x) = −
′′ ( x)
EIωDθ uC
EIωˆ θ uC
a+b
a −b
a−b
*
(3)
′ ( x)
B D′ ( x ) + M uC
( x ) = − EI ωˆ θ uC
( x ) + GI tCθ uC
a+b
BD ( x )ω D (s)
I ωD
(29)
(30)
(31)
The Kollbrunner-Hajdin theory
a+b 2
′′ ( x) = EI ωDθ uC
′′ ( x )
Bˆ C ( x ) = (
) EI ωˆ θ uC
a−b
(32)
*
( 3)
′ ( x)
M uC (x ) = − Bˆ C′ ( x) + M uC
( x) = − ρEI ωˆ θ uC
( x) + GI tCθ uC
(33)
σˆ x ( x, s ) =
Bˆ C ( x)ωˆ ( s )
I ωˆ
(14)
x
M
L
Figure 3.: A cantilever beam, loaded with a moment M at the free end.
For the cantilever beam in figure 3. with the end conditions
′ (0) = 0 ,
θ uC
θ uC (0) = 0 ,
′′ ( L) = 0
B( L) = 0 ⇒ θ uC
(34), (35), (36)
(3)
′ ( x) ≡ M
( x ) + GI tCθ uC
MuC(x) = - B´(x) + M*uC(x) = − EI ωˆ θ uC
(37)
the solution function for equation (28), when muC(x) ≡ 0, is
θ uC ( x) = Ae
α
a±b
x
+ Be
−
α
a ±b
x
+ Cx + D ,
(38)
which gives for the given end conditions

α ( x − L)

 
 a ± b  sinh
 
α
L
a ± b + tanh
θ uC ( x ) = C −

 + x  , where
α
L
α
±
a
b

 

cosh


 
a±b
C=
3M (a + b)
M
=
2 2 2
2 Eta b α
GI tC
(39)
(40)
For the guided beam solution (-), when a → b , then in equation (29) BD ( x) → 0 like
BD ( x) ≈ m M
a+b
α
e
−
α
a −b
x
(41)
and the normal stresses according to (31) σ x ( x, s) → 0 , too. The greater the side difference is, the
closer the solutions are one to another, but they never coincide.
DISCUSSION
It is seen in equation (39) that the Kollbrunner-Hajdin (+) solution gives non-adequate results when
a → b : it “introduces” normal stresses for a square box beam in torsion, which is not true, because for
a square cross-section box the sectorial area ωˆ (s) ≡ 0 according to (5) and (6). Coefficient ρ in
equation (11) is wrong. The guided beam solution gives the right result also for the square box.
The solution according to the theory of guided beams can be generalized to account the distortional
degree of freedom by coupling the already introduced warping and twisting degrees of freedom from
(18) - (22) with a spring connection. The notion of the distortional centre for the point D in this article
is introduced by professor Erkki Niemi from Lappeenranta University of Technology, independent of
the theory of guided beams, as a point loaded through which antisymmetrically in torsion, the box
beam does not distort. This notion expresses the “essence” of these four “secondary” special points of
a box beam. He has also introduced this system of axes in which all signs are positive in the formula of
normal stresses, which is useful feature for generalized warpings in the theory on guided beams.
REFERENCES
1. Kollbrunner, Curt F. , Hajdin, Nikola (1972): Dünnwandige Stäbe, Band 1, Zürich/Belgrad
ISBN-540-05643-2
2. Koivula, Risto (1998): “Transformations of Force and Displacement Quantities between Different
Systems of Axes and Special Points”, Research and Development; Proceedings of the 2nd International
Conference of Thin Walled Structures, Singapore, ISBN 0-08-043003-1
3. Koivula, Risto (2000): “Analysis of Distortion of a Thin-Walled rectangle Beam Using the Theory
of Guided Vlasov Beams” , Proceedings of the VII Finnish Mechanical Days, Tampere,
ISBN 952-15-0414-5
4. Koivula, Risto (2003): “Derivation of the Kollbrunner-Hajdin Theory of the Thin-Walled
Rectangular Box Beam under Torsion by Dividing the Beam into Guided Vlasov beams with Open
Cross-Section. Proceedings of the VIII Finnish Mechanical Days, Espoo, ISBN 951-22-6569-9