SHEAR EFFECT IN BEAM STIFFNESS MATRIX

By Gelu Onu 1
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INTRODUCTION

Within the limits of the elementary b e a m theory it is possible to in-

clude the shear deformation in the stiffness matrix for a beam element
by giving u p one of the restrictions of the Bernoulli's hypothesis (7).
Thus, ignoring the requirement that the radius of curvature of the elastic
axis should be contained in the cross-section plane, we will p r e s u m e that
any cross section, which is initially plane, remains plane after the b e a m
deformation.
There are several k n o w n ways of deriving the stiffness matrix which
consider the shear contribution on the basis of the preceding modified
hypothesis, for a beam element having two degrees of freedom (d.o.f.)
at each of the two ends.
However, the writer has no knowledge of a "correct" solution w h e n
the four-d.o.f. beam element is approached by m e a n s of an a s s u m e d
displacement field. The available references contain only approximate
solutions, gathered either by adding the separate effects produced by
flexure only and shear only analysis, or by introduction of separate as-
sumptions for lateral displacement and cross-section rotation (3,8).
The purpose of this technical note is to derive the "correct" stiffness
matrix and consistent geometric stiffness matrix for a four-d.o.f. b e a m
element from coupled displacement fields, including the effect of the
shear deformation. Additionally, a brief explanation of h o w the mass
matrix can be derived from the same coupled displacement fields is
included.

STIFFNESS MATRIX

The behavior of the beam element, as seen in Fig. 1, is described by

four-d.o.f., i.e., LZ1, 172, U3, a n d 114. There are two characteristic m o d e s
of flexure deformation (Fig. 2) for this element having uniform proper-
ties along its length a n d loaded only at n o d e s . The polynomials used
for the analytical representation of two deformed configurations must
be consistent with the force representation established in Fig. 1. This
implies the possibility of obtaining a b e n d i n g m o m e n t with linear vari-
ation and a constant shear along the beam for the antisymmetrical com-
ponent of the load, as well as a constant b e n d i n g m o m e n t and a zero
shear for the symmetrical component. Polynomials

m
'l-£ • <*)
Dr. Engr., Design Inst, for Air, Road and Water Transports, Ministry of Trans-
port and Communications, Bucharest, Romania.
Note.—Discussion open until February 1, 1984. To extend the closing date one
month, a written request must be filed with the ASCE Manager of Technical and
Professional Publications. The manuscript for this paper was submitted for re-
view and possible publication on July 1, 1982. This paper is part of the Journal
of Structural Engineering, Vol. 109, No. 9, September, 1983. ©ASCE, ISSN 0733-
9445/83/0009-2216/S01.00. Paper No. 18211.
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J. Struct. Eng. 1983.109:2216-2221.

 a]
Y.W1
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Y,W2

M1.U2 M2.U4
-a El GAe -1/2 1/2*
V1.U1 V2.U3

FIG. 1.—Nodal Forces and Displace- FIQ. 2.—Deformed Configurations of

ments of Beam Element Beam Element under Nodal Rotations:
(a) Antisymmetrleal Flexure Mode; (b)
Symmetrical Flexure Mode

a x
W2=-- +— (lb)
4 4a
fulfill these requirements.
The rigid body motions of the element, a translation normal to the
beam axis and a rotation, may be adequately represented by
1
W3 = - . (2a)
2

W4=-- (2b)

If Bernoulli's hypothesis is entirely accepted, the two characteristic

modes of deformation (Eqs. 1) and the rigid body motions (Eqs. 2) are
sufficient for the construction of the shape functions of the transverse
displacement field, i.e.:
-3
W4 WL_ 1 _3x jr__
N1 = W3
~ 2 4a 4a 3 '
a x x x
N2 = Wl + W2 = -- + - + —
4 4 4a 4? ;
W 4 - W 1 1 3x x3
N3 = W3 =-+ —.;
a 2 4a 4<r
a x
N 4 = Wl W2 = - + - (3)
4 4 is' 4a2
Renouncing the requirement according to which the elastic axis slope
must be equal to the cross-sectional rotation, the deformation mode
characterizing the shear must be considered. The two possible deformed
beam configurations under a constant shear (4) are schematically shown
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J. Struct. Eng. 1983.109:2216-2221.

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FIG. 3.—Displaced Shapes of Beam FIG. 4.—Additional Degree of Freedom

Element Due to Shear Loading: (a)
Freely Supported Mode; (b) Cantile-

in Fig. 3. Of the two possibilities, the characteristic deformation mode

for shearing is the freely supported mode [Fig- 3(a)]; the cantilevered
mode [Fig. 3(b)] is a combination between a rigid body motion and the
freely supported mode.
As the shear producing the displaced shape shown in Fig. 3(a) is the
consequence of an antisymmetrical flexure, it results in the characteristic
deformation mode for shearing always accompanying the antisymmetri-
cal flexure mode, and it is reasonable to admit that these two defor-
mation modes are defined by the same parameter, 175 (Fig. 4). Under
these conditions, the deformed configuration shown in Fig. 4(a) repre-
sents the deflection increase obtained when the shear deformation is
allowed, Lf5 gets a character of internal d.o.f., and

a
becomes the fifth shape function of the transverse displacement field.
Disregarding Bernoulli's hypothesis, it is permissible to append a sep-
arate term representing the strain energy due to the transverse shear to
the element strain energy (3), so that
i r (d2w\2 p r (dw\2 ir ^ 7
!i = - —-)Eldx +- \ [—)dx+-\ y2GAedx 5
The first integral leads to the conventional stiffness matrix of the beam
element. The second integral gives the geometric stiffness matrix and
accounts for the work of axial tensile force P acting through bending
displacement. The third integral is the contribution from the shear com-
ponent of the strain.
In this way, except for the transverse displacement field
w = N,„z (6a)
a second field must be considered in order to represent the rotation of
the beam cross sections due to shearing deformation:
£a^a | O

J. Struct. Eng. 1983.109:2216-2221.

 S = Nsz (6b)
Terms N,„, N s , and z axe defined as
Nw = [N1,N2,N3,N4,N5] (7a)
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N s = [0,0,0,0,N6]. (7b)
T
z = [171, [12,173,174,175] (8)
As it was assumed, the angular deformation, 7, is constant on the
cross section and, therefore, s = 7 = V/(GAe): Because the shear force,
V, is also constant along the beam, it follows that JV6 is a constant too
and its value may be carried out from a boundary condition, e.g., for x
= a, w' + s + 174 = 0, in which w' = -174 - U5/(2a) (from Eqs. 6a,
4, and 3) and s = N6 U5 (from Eqs. 6b, 7b, and 8). Thus
_1_
N6 = - (9)
"la
Substituting for w and s from Eqs. 6 into Eq. 5 gives
2
"d 2 N„/ 'd Na~
T T
z' ~dNw~ "dN t ;
17=- EI 2 dx + P dx
2 . dx . dx

GAe[Ns]J [N,] rfx z (10)

which after performing the integration becomes

1
L7 = - z T ( K + K G )z. (11)
2
K, a 5 by 5 stiffness matrix, and K G , a 5 by 5 consistent geometric
stiffness matrix, can be written as follows:
[Kn K,,l Kn K12
k = ; KG = KI2 K . (12)
Ui2 K22J 22

Kn and Kn are in fact the conventional stiffness matrix and conven-

tional geometric stiffness matrix, respectively, for the beam element (as-
suming that no shear occurs). The vectors K12 and K 12/ and the entries
K22 and K22, are
-1 ~-l
3E7 a p a
K, K„ = (13)
2? 1 lOfl 1
_ a _i - a -
3E7 GAe
K22 = —, 3 + ^ ~ K,, = - (14)
2fl 2a lOfl
Eliminating the internal d.o.f. 175, by static condensation with zero
force, a 4 by 4 reduced matrix, including the effect due to the shear
deformation, is defined by
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J. Struct. Eng. 1983.109:2216-2221.

 Kr = RT(K + K G )R (15a)
T 1 an
in which R = [14, -K^K^ ] d 14 is the 4 by 4 unity matrix. The ex-
tended form of Kr matrix may be split so that
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12 -61 -12 -6L '

EI -61 (4 + t)L2 6L (2-t)L2
K = R T KR = (15b)
(l + t)L3 -12 61 12 6L
L-6L (2~t)L2 61 (4 + t)L2J
SI
-S2 -S2 -SI
-S2
-S4 S3 S2
and KG = R T K G R = - (15c)
60L(1 + 0' -SI
S2 S2 SI
-S4 -S2
S3J S2
2
in which L = 2a; t = 12EI/(GAeL ); and A effective shear area. The
entries of the KG matrix are as follows: SI = 12(6 + Wt + 5t2); S2 =
6L; S3 = (8 + lOt + 5t2)L2; and S4 = (2 + lOf + 5t2)L2.
The stiffness matrix in Eq. 15b and the "correct" stiffness matrix in
Refs. 1, 2, 5, and 6 are identical. Also, the consistent geometric stiffness
matrix in Eq. 15c and the finite displacement matrix in Ref. 1 are the
same.

CONSISTENT MASS MATRIX

The kinetic energy of a prismatic beam element, vibrating at circular

frequency, co, including the effects of the shear deformation and rotatory
inertia, can be written (2) as
2 /•« r
(0 dw \ 1
T = —pA w2 + r2
dx +s (16a)
v dx
2
in which r = radius of gyration of the beam cross section; and p = ma-
terial density. Substituting for w and s from Eqs. 6 into Eq. 16a gives
2
T = z7 — vpA
2
{£[N,n[N ]dx w

dNw
+ r
£ dx
dNw
•
dx
+ N S dx\z
+ N,

Integrating with respect to x then gives

(16b)

T = — z'Mz (17)

in which M, a 5 by 5 mass matrix, can be written as

Mn M12 M„ Mi2
M = pA +r (18)
M] 2 M22. MI2 M22.
The first term in Eq. 18 represents the translational mass inertia, while
the second one represents the rotatory inertia. pA(Mn + r2Mu) = the
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J. Struct. Eng. 1983.109:2216-2221.

 conventional mass matrix of the beam element without shear effect. The
remainder submatrices are as follows:
-9~ -6"
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2a l a
M x2 = — ; M 12 = M.22 = • M22 =• • (19)
210 9 " 10« 6 105 5fl
_2flJ - a -
Eliminating 115, a 4 by 4 consistent mass matrix, including the effects
of shear deformation a n d rotary inertia, is defined b y
M = R'MR (20)
in which R = the same matrix as the one u s e d to obtain the 4 b y 4
stiffness matrix. If the necessary matrix multiplications are carried out
in Eq. 20, we find that the resulting consistent mass matrix is the same
as the mass matrix in Refs. 1 a n d 5.

CONCLUSIONS

A procedure for the formulation of the stiffness matrix, the consistent

geometric stiffness matrix, a n d the consistent m a s s matrix for the four-
d.o.f. beam element, including the shear effect by coupling the trans-
verse displacement field a n d the shear angular displacement field, has
been presented.

ACKNOWLEDGMENTS

The writer wishes to express his sincere thanks to Emeritus Professor

P. Mazilu from I.C. Bucharest for helpful discussions.

APPENDIX.—REFERENCES

1. Archer, J. S., "Consistent Matrix Formulation for Structural Analysis Using

Finite-Element Techniques," American Institute of Aeronautics and Astronautics
Journal, Vol. 3, No. 10, 1965, p. 1910.
2. Davis, R., Henshell, R., and Warburton, G. B., "A Timoshenko Beam Ele-
ment," Journal of Sound and Vibration, Vol. 22, No. 4, 1972, pp. 475-487.
3. Gallagher, R. H., Finite Element Analysis—Fundamentals, Prentice-Hall, Inc.,
Englewood Cliffs, N.J., 1975.
4. Mazilu, P., Statics of Structures, Vol. 2, Ed. Tehnica, Bucharest, Romania, 1959
(in Romanian).
5. Przemienieski, J. S., Theory of Matrix Structural Analysis, McGraw-Hill Book
Co., New York, N.Y., 1968.
6. Severn, R. T., "Inclusion of Shear Deflection in the Stiffness Matrix for a Beam
Element," Journal of Strain Analysis, Vol. 5, No. 4, 1970.
7. Timoshenko, S. P., Vibration Problems in Engineering, 3rd ed., Van Nostrand,
New York, N.Y., 1955.
8. Tong, P., "New Displacement Hybrid Finite Element Models for Solid Con-
tinua," International Journal for Numerical Methods in Engineering, Vol. 2, No. 1,
1970, pp. 73-83.

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J. Struct. Eng. 1983.109:2216-2221.