Universit
¨
at Karlsruhe (TH)
Institut f
¨
ur Baustatik
Shear correction factors
in Timoshenko’s beam theory
for arbitrary shaped cross–sections
F. Gruttmann, W. Wagner
Mitteilung 6(2001)
BAUSTATIK
Universit
¨
at Karlsruhe (TH)
Institut f
¨
ur Baustatik
Shear correction factors
in Timoshenko’s beam theory
for arbitrary shaped cross–sections
F. Gruttmann, W. Wagner
Mitteilung 6(2001)
BAUSTATIK
c
Prof. Dr.–Ing. W. Wagner Telefon: (0721) 608–2280
Institut f
¨
ur Baustatik Telefax: (0721) 608–6015
Universit
¨
at Karlsruhe E–mail: bs@.uni-karlsruhe.de
Postfach 6980 Internet: http://www.bs.uni-karlsruhe.de
76128 Karlsruhe
Shear correction factors in Timoshenko’s
beam theory for arbitrary shaped
cross–sections
F. Gruttmann
Institut f¨ur Statik
Technische Universit¨at Darmstadt
Alexanderstraße 7
64283 Darmstadt
Germany
W. Wagner
Institut f¨ur Baustatik
Universit¨at Karlsruhe (TH)
Kaiserstraße 12
76131 Karlsruhe
Germany
Abstract In this paper shear correction factors for arbitrary shaped beam cross–
sections are calculated. Based on the equations of linear elasticity and further assump-
tions for the stress field the boundary value problem and a variational formulation are
developed. The shear stresses are obtained from derivatives of the warping function.
The developed element formulation can easily be implemented in a standard finite ele-
ment program. Continuity conditions which occur for multiple connected domains are
automatically fulfilled.
1 Introduction
The problem of torsional and flexural shearing stresses in prismatic beams has been stud-
ied in several papers. Here, publications in [1, 2, 3] are mentioned among others. Further-
more the text books of e.g. Timoshenko and Goodier [4] or Sokolnikoff [5] give detailed
representations of the topics. A finite element formulation has been discussed by Mason
and Herrmann [6]. Based on assumptions for the displacement field and exploiting the
principle of minimum potential energy triangular finite elements are developed. Zeller [7]
evaluates warping of beam cross–sections subjected to torsion and bending.
In the present paper shear correction factors for arbitrary shaped cross-sections using
the finite element method are evaluated. The considered rod is subjected to torsionless
bending. Different definitions on this term have been introduced in the literature, see
Timoshenko and Goodier [4]. Here we follow the approach of Trefftz [3], where uncoupling
of the strain energy for torsion and bending is assumed. The essential features and novel
aspects of the present formulation are summarized as follows.
All basic equations are formulated with respect to an arbitrary cartesian coordinate system
which is not restricted to principal axes. Thus the origin of this system is not necessarily a
special point like the centroid. This relieves the input of the finite element data. Based on
the equilibrium and compatibility equations of elasticity and further assumptions for the
stress field the weak form of the boundary value problem is derived. The shear stresses are
obtained from derivatives of a warping function. The essential advantage compared with
stress functions introduced by other authors, like Schwalbe [2], Weber [1]orTrefftz[3]
1
is the fact that the present formulation is also applicable to multiple connected domains
without fulfilment of further constraints. Within the approach of [2, ?, 3] the continuity
conditions yield additional constraints for cross sections with holes. In contrast to a
previous paper [8] the present formulation leads to homogeneous Neumann boundary
conditions. This simplifies the finite element implementation and reduces the amount of
input data in a significant way. Within our approach shear correction factors are defined
comparing the strain energies of the average shear stresses with those obtained from the
equilibrium. Other definitions are discussed in the paper. The computed quantities are
necessary to determine the shear stiffness of beams with arbitrary cross–sections.
2 Torsionless bending of a prismatic beam
We consider a rod with arbitrary reference axis x and section coordinates y and z.The
parallel system ¯y = y − y
S
and ¯z = z − z
S
intersects at the centroid. According to
Fig. 1 the domain is denoted by Ω and the boundary by ∂Ω. The tangent vector t with
associated coordinate s and the outward normal vector n =[n
y
,n
z
]
T
form a right–handed
system. In the following the vector of shear stresses τ =[τ
xy
,τ
xz
]
T
due to bending is
derived from the theory of linear elasticity. For this purpose we summarize some basic
equations of elasticity.
y
z
n
n
s
Ω
t
t
y
z
S
∂Ω
y
z
S
S
n
t
Fig. 1: Cross–section of a prismatic beam
The equilibrium equations neglecting body forces read
σ
x
,
x
+τ
xy
,
y
+τ
xz
,
z
=0
σ
y
,
y
+τ
yz
,
z
+τ
xy
,
x
=0
σ
z
,
z
+τ
xz
,
x
+τ
yz
,
y
=0,
(1)
where commas denote partial differentiation. Furthermore, the compatibility conditions
in terms of stresses have to be satisfied
(1 + ν)∆σ
x
+ s,
xx
=0 (1+ν)∆τ
yz
+ s,
yz
=0
(1 + ν)∆σ
y
+ s,
yy
=0 (1+ν)∆τ
xy
+ s,
xy
=0
(1 + ν)∆σ
z
+ s,
zz
=0 (1+ν)∆τ
xz
+ s,
xz
=0.
(2)
2
Here, ∆ denotes the Laplace operator, ν is Poisson’s ratio, and s = σ
x
+ σ
y
+ σ
z
, respec-
tively.
We proceed with assumptions for the stress field. The shape of the normal stresses σ
x
is
given according to the elementary beam theory, thus linear with respect to ¯y and ¯z.The
stresses σ
y
,σ
z
and τ
yz
are neglected. The transverse shear stresses follow from derivatives
of the warping function ϕ(y, z). Thus, it holds
σ
x
= a
y
(x)¯y + a
z
(x)¯z
σ
y
= σ
z
= τ
yz
=0
τ
xy
= ϕ,
y
−f
1
τ
xz
= ϕ,
z
−f
2
(3)
where we assume that a
y
and a
z
are linear functions of x. Furthermore the functions
f
1
(z)=−
ν
2(1 + ν)
a
y
(z − z
0
)
2
f
2
(y)=−
ν
2(1 + ν)
a
z
(y − y
0
)
2
. (4)
are specified, where ()
denotes the derivative with respect to x. Using a definition for
torsionless bending the constants y
0
and z
0
are derived in the appendix. As is shown in
this section considering the functions f
1
(z)andf
2
(y) one obtains a differential equation
by which the equilibrium and compatibility equations can be advantageously combined.
The rod is stress free along the cylindrical surface which yields the boundary condition
τ
xy
n
y
+ τ
xz
n
z
=0. (5)
Next, the derivative of the normal stresses σ
x
,
x
:= f
0
(y, z)reads
f
0
(y, z)=a
y
¯y + a
z
¯z. (6)
The unknown constants a
y
und a
z
are determined with
Q
y
=
(Ω)
τ
xy
dAQ
z
=
(Ω)
τ
xz
dA. (7)
The integral of the shear stresses τ
xy
considering (1)
1
and applying integration by parts
yields
(Ω)
τ
xy
dA =
(Ω)
[τ
xy
+¯y (τ
xy
,
y
+τ
xz
,
z
+f
0
)]dA
=
(Ω)
[(¯yτ
xy
),
y
+(¯yτ
xz
),
z
]dA +
(Ω)
¯yf
0
dA
=
(∂Ω)
¯y (τ
xy
n
y
+ τ
xy
n
z
)ds +
(Ω)
¯yf
0
dA.
(8)
The boundary integral vanishes considering (5). Thus, inserting eq. (6)weobtain
(Ω)
τ
xy
dA =
(Ω)
¯y(a
y
¯y + a
z
¯z)dA. (9)
3
