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First-order generalised beam theory
J. Construct. Steel Research 31 (1994) 187-220
© 1994 Elsevier Science Limited Printed in Malta. All rights reserved 0143-974X/94/$7.00 ELSEVIER
First-Order Generalised Beam Theory J. M. Davies & P. L e a c h Department of Civil Engineering and Construction, University of Salford, Salford, UK, M5 4WT (Received 5 January 1994)
ABSTRACT This paper introduces the basic principles of Generalised Beam Theory and shows how this theory may be used to analyse cold-formed sections in which distortion of the cross-section is significant. The calculation procedure is illustrated by a detailed numerical example.
NOTATION A a, b, c B,C,D bi E F-L F-Q F-O G I i,j,k J K k L M m N
Area of cross-section Ordinates of warping function Section properties for individual modes Width of bith plate element Young's modulus In-plane movement of a face at its mid-point Movement normal to a face at its mid-point Rotation of the chord line of a face Shear modulus Second m o m e n t of area Mode number (usually in the form of a forward superscript) St Venant torsional constant Stiffness matrix Foundation constant Span of member Bending m o m e n t Uniformly applied torque or transverse bending m o m e n t Axial load 187
188
P q S s t
V 13, W
W x, y z
,fl,2 F v O" T
4, O~
J. M. Davies, P. Leach
Point load Uniformly distributed load for individual mode Shear force Distance around member Thickness Generalised displacement function Displacement components in GBT Stress resultant Horizontal and vertical axes Distance along member Non-dimensional coefficients defined in text Warping constant Poisson's ratio Direct stress Shear stress Rotation Sectorial coordinate
Primes indicate differentiation with respect to the distance z along the member. ,~ indicates a unit value of a quantity, e.g. a warping function. A forward superscript indicates the mode number.
INTRODUCTION Generalised Beam Theory (GBT) seeks, at the same time, both to unify and to extend conventional theories for the analysis of prismatic thinwalled structural members. The analytical treatment of the four fundamental modes of deformation, namely extension, bending about the two principal axes and torsion, is united within a consistent notation. These four modes may be referred to as the 'rigid-body' modes because they do not involve any distortion of the cross-section. The notation is then extended to include higher-order modes of deformation which involve cross-sectional distortion. This allows elegant and economical solutions to a wide range of complex problems and provides a natural transition from beam theory to folded plate theory. Generalised Beam Theory can also be further extended to include second-order effects (local and global buckling), and this is considered in a companion paper. Evidently, Generalised Beam Theory is particularly applicable to the analysis of cold-formed steel sections and, in this paper, it will be intro-
First-order Generalised Beam Theory
189
duced from this standpoint. At the same time, it should be appreciated that it also has much wider application. The development of the theory has been pioneered by Professor R. Schardt and his colleagues at the University of Darmstadt in Germany, a work that has extended over more than 20 years. Unfortunately, little has been written during this period and almost nothing in English. The definitive reference at this present time is a recent book by Schardt.1 This paper attempts a somewhat different and more concise presentation of the fundamentals of the theory. By doing so it hopes to introduce the English-speaking world to a major step forward in structural mechanics.
MODES OF DEFORMATION AND WARPING FUNCTIONS A unifying feature of the theory is the concept of 'warping functions' whereby each mode k of deformation is associated with a distribution of axial strain kt~. Thus, the first mode of deformation is a uniform distribution of axial strain over the cross-section. For this mode, the 'warping function' 1~= _ 1 for all points s of the cross-section. The second and third modes are bending modes and the associated warping functions are linear distributions of strain about the two principal axes. The fourth mode is torsion and here the term warping has its conventional meaning as the warping function is the sectorial coordinate which reflects the distribution of axial strain due to a bi-moment. It is important to realise at the outset that all these warping functions are orthogonal. Practically, this means that in any first-order analysis they can be considered quite independently and their effects combined by simple superposition. Mathematically, the orthogonality condition is expressed as
f
i~kl]dA=0
for i # k
(1)
A
The above concepts are illustrated in Fig. 1. Figure l(a) shows a crosssection with five nodes. According to GBT, the five nodes can 'warp' independently and the warping functions are linear between the nodes. Therefore, the warping functions each have five degrees of freedom and the section has five orthogonal deformation modes together with their associated warping functions. Four of these are the rigid-body modes
190
J. M. Davies, P. Leach
J
/
®® (a) Cross - section with five nodes
1. Axial 8tress
2. Bending
'L_I--
\
3. Bending
4. Torsion Dleplaced 8hape
Wa
n
(b) Rigid body modes (k-1... 4)
Displaced Shape
WarpingFunction
(c) Cross - section distortion mode (k-S)
Fig. 1. Typical displaced shapes and warping functions.
shown in Fig. l(b) and the fifth is the mode of cross-sectional distortion shown in Fig. l(c). In simple applications, the number of modes is equal to the number of nodes (or fold lines) in the cross-section. More modes of distortion can be introduced by inserting intermediate nodes between the primary nodes. Each mode k has associated with it: • • • •
a warping function as already described; a corresponding pattern of cross-sectional displacements; distortional bending stresses (for modes 5 and above); section properties kc, kD and kB.
First-order Generalised Beam Theory
191
For the rigid-body modes 1 to 4 the section properties are familiar, thus: 1C =
2C= 3C = 4C = 4D =
cross-sectional area; second m o m e n t of area about first principal axis; second m o m e n t of area about second principal axis; warping constant; St Venant torsional constant;
1D -- IB = 0
2D =
2B=0 3D = 3B = 0
4B = 0
For the higher-order modes which involve cross-sectional distortion, all three section properties are in general non-zero. The analogy between conventional theory and GBT for section properties is summarised in Table 1. It can be seen that C represents the stiffness (in the relevant mode) with regard to direct stress, D represents the stiffness with regard to the shear stresses arising from twisting, and B represents the stiffness with regard to transverse bending stress. B therefore exists only for the distortional modes 5 and above. Schardt 1 traces the history of structural mechanics and shows how early investigators found the first four orthogonal modes of deformation of prismatic elements but made no attempt to unify them. Later work on prismatic folded plate structures moved away from these classical beginnings and a totally independent theory was created. This work has now evolved into the finite strip method with its implicit assumption that all problems can be solved numerically and there is little need to return to classical structural mechanics. Wlassow 2 came closest to GBT and Schardt believes that, but for his early death, it would have been only a matter of time before his research into prismatic structures evolved into a similar formulation. This paper and its companion paper on second-order theory demonstrate, among other things, that there is still some life in classical structural mechanics and that this is not just a matter for academics but has practical usefulness. Indeed, ultimately, GBT may offer the most practical way to deal with the difficult problem of the distortional buckling of cold-formed TABLE 1 Section Properties in GBT and Conventional Notations
Conventional theory
Deformation mode
Generalised beam theory
1
A
~C
2 3 4
I~ Iyr F
J
2C 3C 4C
4D
k
?
?
kC
gD
?
kB
192
d. M. Davies, P. Leach
sections. However, before we can consider this, there is much basic ground to be covered. GBT NOTATION The notation used for GBT in this paper follows that developed by Schardt I and is summarised at the start of the paper. In general, terms will also be defined as they are introduced, but two points are worthy of particular note: • a forward superscript is used to denote the mode number; • ~ over a symbol denotes a unit value of a quantity, e.g. a warping function or a related quantity derived from it.
THE FUNDAMENTAL
BEAM EQUATION
The basic equation of GBT is g k c k v " ' _ _ G k O k V " d-kBkV:kq
(2)
In this equation, E G
= Young's modulus; = shear modulus; kc, kD, kB = section properties applicable to mode k; kv =generalised deformation in mode k; kq = distributed load applicable to mode k; and primes indicate differentiation with respect to z which lies along the length of the member. That eqn (2) includes the basic modes of bending and torsion is easy to see. In conventional notation, the differential equations for bending about the two principal axes and torsion are d4y EIx~ d z 4 = qy d4x Elyy ~ = q:,
d4q~
(3) d2~
E F -d~z4 -- G J -d~z~ = m
First-order Generalised Beam Theory
193
These are evidently special cases of eqn (2). The full eqn (2) applies to modes 5 and above.
EQUIVALENCE
OF GBT AND CONVENTIONAL NOTATIONS
The full equivalence between the conventional and GBT notations is shown in Table 2. One of the most powerful features of GBT is that it extends naturally from the familiar rigid-body modes into the higher-order modes using a unified notation. This notation also includes a unified treatment of stresses and stress resultants so that, in GBT terms,
kw = - E kc kv t, ~_ f a k~l d A
(4)
is a generalisation of
Mx= - E l x x dz 2 =
try dA (bending moment)
(5) B = -- E F d2rp ~z 2 =
f ago dA (bi-moment)
where F is the warping constant and go is the sectorial coordinate.
SOLUTION
OF PROBLEMS
UTILISING GBT
The use of GBT to obtain the stresses and deformations of a member under specified loading and support conditions involves two quite distinct steps. In the first step, only the cross-section is considered in order to obtain for each mode k the warping functions k/~ and the associated section properties RC, kD and kB, together with other relevant information concerning the deformations and stresses induced by this mode. In the second step, this basic information is used to obtain solutions of eqn (2) for particular cases. The analysis implicit in the first step can be complex and, in general, it is carried out using standard computer programs. It is the authors' experience that GBT is best introduced without going into the details of the general form of the first step, but rather concentrating on the general principles and applications. This allows the power and versatility
A = [" dA
Area
Differential equation
EAu"= - N
N ¢r= - A
Normal stress
o Ixxt(s)
EIxxy"=qr
r l(s)
Mzy lxx
Mx = - EIxxy"
N = EAu'
Compatibility equation ¢rx=
A
Shear stress
Iyr= f x2 dA A
xw'
x
w
Bendin9 2
F = ,.I A
/ (/)2 dA
~¢p'
c5
q~
Warpin9 torsion
o Iyyt(s)
EIyrx"=q~
~2(s) =
Myx ~ry= - lrr
My= - Elvrx"
A
Ft(s)
-Bo~ F
EFq)''-GJq)"=m
~(s)- ~
or,-
B = - EF~p"
A
N=fadA Mx=faydA Mr=faxda B=fecoda
lxx = f y2 dA A
A
Stress resultant
Torsional constant
Second moment of area
yv'
u
Warping
A
y
1
Unit warping
v
u
Deformation
Bendin9 1
Extension
Quantity
TABLE 2 Equivalence of Conventional Theory and Generalised Beam Theory
o kCt(s)
k W k6 kC
EkCkV"--kGkDV"+kBkV=kq
k,(s) =
k~ =
kW= -- E kC ~V"
A
kw=fak~da
A
i C = fkfi2 dA
ku = k~ kv,
k~
%
GBT-includin9 higher modes
First-order Generalised Beam Theory
195
of the method to be appreciated before the more complex considerations become dominant. This is possible because, for simple problems, the first step can be carried out manually, an operation which has much to commend it as it provides a valuable insight into the principles involved. Furthermore, explicit expressions have been derived 1 for the properties of channel, z- and hat sections, including the higher-order modes, so that GBT can be readily used for a wide range of practical problems with the minimal introduction given in this paper.
SECTION PROPERTIES AND WARPING FUNCTIONS Table 3 shows the full results obtained by applying the first part of GBT to the cross-section shown in Fig. 2. The various terms are defined for each mode k as follows:
kc, kD kB are section properties; k/~
=
kF-L kF-Q kF-O
= = =
k/,~
=
ks
= =
kr7' k~
warping function defined at each node of the cross-section and assumed to be linear between the nodes (the remaining quantities are associated with unit value of this warping function); in-plane movement of a face at its mid-point; movement normal to the face at its mid-point; rotation of chord line of face; transverse bending moment at each node (distortional modes only); shear force in each face (as a function of kw'); nodal displacements in horizontal and vertical directions.
It should be noted that all the quantities tabulated for modes 1 to 4 are already obtainable from the standard procedures of structural mechanics which are given in basic texts on the subject. Those tabulated for mode 5 can also be derived manually using the fact that the warping function 5t~ must be orthogonal to ~ , 2t~, 3t~, 4~. The manual evaluation of all the quantities in Table 3 will be illustrated in detail later. Once the section properties have been evaluated, solutions to eqn (2) for different load and boundary conditions can be found. After obtaining a solution to eqn (2) for a particular case, multiplying the tabulated quantities by the relevant value of kv allows a complete pattern of transverse stresses and displacements to be obtained. The section shown in Fig. 2 is an open section that is unrestrained in space and with nodes only at the fold lines. Various other features may be
-
Mode k=3: 1 2 3 4 5 C = 145"8
6'3640 4'2426 -6"3640 4"2426 6'3640
Mode k=2: 1 - 8.4853 2 - 10-6066 3 0.00013 4 10"6066 5 8-4853 C = 502"2
1.0000 1.0000 1-0000 1.0000
- 1.0000
1
~
2 3 4 5 C = 10.8
M o d e k = 1:
Node
0.0000 0.0000 0.0000
0-0000
F- Q
0"7071 -0-7071 -0-7071 0'7071
D = 0-00
- 0-7071 -0.7071 0"7071 0-7071
D =0-00
0"7071 0'7071 --0'7071 --0'7071
0"7071 -0.7071 -0.7071 0'7071
D =0-00
0.0000 0.0000 0.0000
0.0000
F-L
0-0000 0"0000 0'0000 0"0000
0-0000 0"0000 0-0000 0-0000
0.0000 0.0000 0.0000 0-0000
F- 0
- 1"0000
B=O-O0
0-00013
1"0000 1'0000 1-0000 1-0000
-
1'0000
0"0000 0"0000 0-0000 0"0000 0'0000
0.0000 0.0000 0.0000 0.0000 0.0000
~
0-0000 0"0000 0"00(30 0"0000
B=O'O0
1"0000
1"0000
1"00(30 1'0000
0.0000 0'0000 0.0000 0.0000 0.0000 B=O.O0
~
TABLE 3 Cross-Sectional Values for Section s h o w n in Fig. 2
0"0000
0-0000 0"0000 0"0000 0-0000
0"01300 0"0000 0-0000 0-0(030 0"0000
0.0000 0.0000 0.0000 0"0000 0.0000
rfi
0'0524 0"6545 - 0-6545 - 0.0524
- 0"0247 -0"7318 - 0"7318 - 0.0247
0.0000 0-0000 0.0000 0.0000
S/W'
e'~
t'.,,
e:
C = 0"720
Mode k=5: 1 2 3 4 5
C =939.0
- 1'0000 0'3333 -0-2000 0"3333 - 1-0000
Mode k=4: 1 38.7097 2 - 7.8629 3 0-0000 4 7.8629 5 -38-7097
D =0'000401
-0'4444 0-0356 -0"0356 0'4444
0"1076 0"2044 0"2044 0"1076
-2.0242 - 8-0242 8.0242 2-0242
D =0.324
15.5242 - 0.5242 -0.5242 15.5242
-0-4412 -0"3394 0"0000 0-3394 0-4412
13.4692 11-3479 0.7413 11-3479 13.4692
B =0.02127
--0"0480 -0"0320 0'0320 0'0480
B =0.00
1.0000 1.0000 1.0000 1.00~
0"1873 0'2891 -0"0503 0"2891 0"1873
-8-4853 - 10.6066 0.0000 10.6066 8-4853
0.0000 0"0000 -0"3323 0-0000 0.0000
0.0000 0.00130 0.0000 0.0000 0.0000
- 1"0417 1'0417 - 1"0417 1.0417
0-0331 0.0331 0.0331 0.0331
,..q
~t
r,~
J. M. Davies, P. Leach
198
A z,w b z - 15
(~)i
b 2 • 15
0.741
1 6.364
•/
.....
.
.
.
.
2.121
b
•
"3
Fig. 2. Cross-section for example.
incorporated in the derivation of the section properties and warping functions, namely: The section may be open or closed. The section may include branches, though at the present time this is not so well documented. Intermediate nodes may be added in order to increase the number of distortional modes. The free movement of the cross-section may be restrained either rigidly or elastically. This latter facility has many applications, some of which are illustrated in Fig. 3. Bearing in mind that these are cross-sections of members which may have arbitrary loading and support conditions in the longitudinal direction, the wide range of problems to which GBT is applicable becomes apparent.
SOLUTION
OF THE FUNDAMENTAL
BEAM EQUATION
Equation (2) may be solved in a number of different ways depending on the nature of the relevant modes of deformation and the loading and boundary conditions. As the unique features of GBT are revealed only when cross-section distortion is present, attention here is concentrated on the general case. Equation (2) is identical in form to the differential equation for the displacement of an axially loaded beam on an elastic foundation. In order to apply beam on elastic foundation solutions to GBT problems, the
First-order Generalised Beam Theory
°l ®
| ®
®
C~Cy-y-y-,Cy-v~Cv--y~ \
Cy-~Cy~C~ \
\
Ic) Symmetry condition
. . . . . . . . . .
®
®
(b) Elastic lateral and torsional restraint of purlin
(a) Lateral restraint of purlin from sheeting
//
199
\
(d) antisymmetry condition
.
"........
®
,¢ ®
:
,
',
,
®
(e) Periodic construction
(f) Rigid restraint
Fig. 3. Examples of restrained cross-sections.
following substitutions must be made: second m o m e n t of area axial load (tension positive) foundation constant uniformly distributed load
I :~ k c N - G kD k -- kB q = kq
and, when the solution has been completed, the stress resultant M --kw
It follows that the solutions for many relatively simple loading and support conditions are well documented. 3'4 Indeed, Ref. 3 is a whole book devoted to the topic and explicit solutions such as those given therein will provide an appropriate method for a wide range of practical problems. However, many other practical problems do not have simple loading and boundary conditions and for these it is necessary to resort to numerical methods of solution. Two such methods have been widely used in practice, namely the finite difference method and the finite element method.
200
J. M. Davies, P. Leach
FINITE DIFFERENCE
SOLUTION
O F E Q N (2)
Schardt and his associates at Darmstadt have exclusively used the finite difference method in their solutions of the general case of the fundamental beam equation. The method is most easily understood when it has a readily visualised physical form and, for this reason, it is usually described with reference to a beam on an elastic foundation. The solution system for this problem is given in Appendix A. As with many finite difference applications, the treatment of the relevant boundary conditions is non-trivial and the treatment for the cases usually encountered is also given in Appendix A. FINITE ELEMENT
SOLUTIONS OF GBT PROBLEMS
To the best of the authors' knowledge, they were the first to utilise the finite element method in the context of GBT. The advantage of this approach is that eqn (2) falls into the class of equations for which the finite element solution is exact. This means that, for many first-order problems, the computational requirements become almost trivial. The detailed derivation of the relevant element and consistent load vector has been given by the first author 5 for the beam on elastic foundation problem shown in Fig. 4, for which the governing differential equation is (6)
E l y " ' - Ny" + ky = q
The element stiffness equations have the form
ry,l[]r,l M'/= K
Y'I
(7)
Py2
Yl:~ PY~I N ~ '~M1 I ' ~
M
.~Y2
" ' ' ' 6PY2 elaatic foundation
L Fig. 4. Beam on elastic foundation element.
First-order Generalised Beam Theory
201
where the terms of the 4 x 4 stiffness matrix [K] are given in Appendix B. The consistent load vector for a uniformly distributed load has the form
MF
(8)
PF --MF
where PF and MF are also given in Appendix B. It may be noted that with this formulation, nodal loads and boundary conditions can be applied directly in the usual way.
MANUAL CALCULATION
USING GBT
As an illustration of manual calculation using GBT, the problem shown in Fig. 5 will be solved. A beam with the cross-section shown in Fig. 2 is simply supported over a span of 120cm and subject to a vertical uniformly distributed load acting along the centre of the cross-section. It is immediately obvious that, in order to obtain a sensible solution to this problem, it is essential to take into account distortion of the crosssection. As has already been explained, the warping functions for modes 1 to 4, the rigid-body modes, may be obtained using elementary structural mechanics. For any section, mode 1 is always a unit axial strain and the warping function lt~= - 1 across the entire section. Mode 2 represents bending about the vertical axis through the centroid. The associated warping function is a linear distribution of strain in the
li
/f
/ / / / / /
/ /
Fig. 5. Example using GBT.
J. M. Davies, P. Leach
202
horizontal direction, the strain at any point being proportional to the horizontal distance from the z-axis. Similarly, mode 3 represents bending about the horizontal axis through the centroid and the associated warping function is the vertical distance of the various points of the cross-section from this axis. Finally, mode 4 represents a unit twist about the shear centre and the warping function 4~ is the sectorial coordinate. The calculation of sectorial coordinates is not trivial but computational algorithms have been published 6 and they have been tabulated for some common sections] Once the warping functions for modes 1 to 4 have been identified, the other associated quantities in Table 3 for these modes follow directly. If they are not obvious by inspection, they can be calculated using the general procedures which will now be explained with reference to mode 5. The warping function for mode 5 can be determined from the condition that it must be orthogonal to the other modes, i.e.
f
ill
k/~ dA
= 0
(9)
for i ¢- k
A
For the symmetrical section shown in Fig. 2 subject to a vertical load, it is evident that the mode required must be symmetrical. A symmetric mode will automatically be orthogonal to the antisymmetric modes 2 and 4 so that it is only necessary to ensure orthogonality with modes 1 and 3. If the warping function is assumed to have ordinates {a, b, c, b, a}, then
f
it7 5fidA
gives
A
3×0"3×(-1)×
--
+15×0"3x(-1)×
i.e. - 0-45a -- 2"7b - 2.25c = 0
and
f
ail St~dA=0
gives
A
3xO-3x
[6"3640a 3 4"2426b
6"3640b6 4"2426a ]
=0
First-order Generalised Beam Theory
+15x0.3
203
I4'2426b 3 6"3640c 4"2426c- 6"3640b1 t 6 =0
i.e. 2.5456a+3"8183b-6.3640c=O Solving these two simultaneous equations gives b = -0.3333a c=0.2000a The absolute magnitudes of the warping ordinates for modes 5 and above are indeterminate and it is usual to normalise them to make the largest numerically equal to plus or minus unity. It then follows that sO1 ~u2 =0"3333 5~3 =0"2000 5~4=0"3333 s~s = - - 1 =
-
1
These values of fi give rise to slopes Of~/Os which cause in-plane displacements of each element of the section in the s-direction. Thus 5F_L 1 _ ~ 1 - f i 2 _ - 1 - 0 . 3 3 3 3 bl 3
5F_L2_ fi2-tt3 b2
_
-0.4444
0"3333+0"2_0.0356 15
and 5 F - L 3 and 5 F - L 4 follow by symmetry: 5F-L3 = -0.0356 5F-L4 = 0-4444
These in-plane displacements define various movements of the plate elements which can be determined by simple geometry. Neglecting for the present the implied transverse bending of the plates by considering the connection at node 3 to be a hinge, the movements shown by the full line in Fig. 6 can be calculated. It is easy to show that these give rise to the
204
.J. M. Davies, P. Leach
IW V
: ,
0.~.0356 (~:~'-~,,-0.4444 0.4444j'~ ~(~) Fig. 6. Cross-sectionalmovementsin mode5.
Fig. 7. Bendingmoments for mode 5.
following movements:
5F-Q=0.0836 5F-0=-0.0320 5f=_0.4073 5~= 0-2044 0.2044 0.0836
--0.0320 0.0320 0-0320
-0.3394 0 0.3394 0.4073
0.2212 0.2891 -0-0503 0-2891 0.2212
It is now necessary to introduce continuity at node 3 by rotating plates 2 and 3 at this node without moving nodes 2 and 4, whose positions are determined by geometry. Nodes 1 and 5 are not constrained during this operation. This introduces the bending moment distribution shown in Fig. 7, where 3
Et 3
5ff/3 b2 1 2 [ 1 - v 2] 5F-02 3 x 21 000 × 0"33 x --0"0320 15 × 10.92 = 0.3323 kN/cm It also introduces additional rotations at nodes 2 and 4 which are numerically equal to ½(F-02)=0.0160 with corresponding changes in F-Q and F-O for plates 1 and 4 and ~ and v~ for nodes 1 and 5. This brings the values calculated above into agreement with Table 3. The final displaced shape of the cross-section is then shown by the chain-dotted line in Fig. 6.
First-order GeneralisedBeam Theory
205
The section properties can now be calculated using generalisations of the familiar formulae:
5C=fS~2dA
[compare I = f y 2 d A , etc.]
A
=2x3x0-3[(- 1)2--1 X 0.33333+0"33332] 0.3[0.33332-0.2×0.33333+0"22] + 2 x 15 ×
= 0"720 cm 4 1
5
(10)
n
D = ~ r=l -3-1 x 2
F-O2b, t 3 [compare J = ½ E b t 3] ×
0"33[0"04802 X 3+0"03202 x 15]
= 0"000401 cm 2
(11)
Note: The formula used above has the advantage of familiarity and gives a good approximation to RD. The general theory also incorporates a more precise expression: 5 B = f 1211 - v 2] rfi2 ds
J
Et a
S
_
1 F2 × 0"33232 - - 3x 51"923 /
151
= 0"02127 kN/cm 2
(12)
Finally, for completeness, the unit shear flows can now be calculated for
kW'= 1 using
a(s)t(s) f S
%(s)t(st=~1d
ds
0
compare
zt = -- 1
f y dA
(13) for unit shear force]
J. M. Davies, P. Leach
206
The shear force in each face then follows: r+l
%= f kz~(s)t(s)ds
(14)
r
Thus, for face 1 sfi(s) = 0"4444s - 1
.'. 5%(s) . t(s)= ~ 2 [O'4444s2 5S, = 00@2 [0"44~4sa
q 1. sJ = 0-4167/cm
for
s= 3
1 .s213 -
2- _]o- - 1"0417
and for face 2 5fi(s) = - 0.0356s + 0.3333 ÷0"3333s] 5zs(s) • t(s) = - 0.4167 + 00_~732[ -0"0356s2 2 =0
for s = 15 0.3333s2"~] 1'
2
}Ao
= 1.0417 The full pattern of shear flow for this mode is shown in Fig. 8. This completes the derivation of the full set of section properties for mode 5 given in Table 3 and concludes the first phase of the calculations. Having determined the section properties, the next step is to solve the fundamental beam equation for each of the modes which is to be included in the solution. It is immediately apparent that the analyst can exercise choice here and can investigate the significance of individual modes. Furthermore, the boundary conditions do not have to be the same for the different modes. These are considerable advantages of GBT which are not available in other methods. We note here that as the structure and its loading are symmetrical, the deformation must be symmetrical. It follows that only modes 3 (bending
First-order Generalised Beam Theory
207
0
0.4167
0
Fig. 8. Shear forces and shear flow for mode 5.
about the horizontal principal axis) and 5 (distortion of the cross-section) play any role. The section will be considered to be simply supported with respect to both these modes. The load terms in the fundamental beam equation represent the virtual work of the loads acting on the modal displacements. They can be calculated in two alternative ways, both of which lead to the same result: (a) By considering the virtual work of the horizontal and vertical components of the nodal loads acting on the modal displacements k~ and k~, i.e. n+l
kq= Z qY,rk~rq-qz, rk~,r
(15)
r=l
so that here, noting that the applied load is in the negative direction and that both 3/~3 and 5k 3 are also negative, 3q = 1"0 5q = 0"0503 (b) By considering equilibrium of the shears ks where the governing equation is
as
*q= ~ *q,kSr
(16)
r=l
For mode 3, the general beam equation (2) reduces to E
3C 3V't" =
3q,
i.e.
shown in Fig. 9,
E1 V"" = q
208
J. M. Davies, P. Leach
~L1"0
. / N . O.70~ ~707
1
qr
m\
/,
41" -0.0524 0.0524 0.05035 Sr
3S r
Fig. 9. Resolution of load between modes. which is the usual equation of the Engineers' theory of bending neglecting distortion of the cross-section. The solution is therefore standard and at mid-span:
I i.e. M - ----ff--[ qL21 3W(L/2)=3qL2 8 1"0 x 1202 8 and
- 180 kN cm
3V(L/2)_ 53qL4 I 384E 3C
5qL4] i.e. w = 384EI
5 x - l ' O x 1204 =0"08818 cm 384 x 21 000 x 145-8 The longitudinal stresses then follow from
k~= kc
i.e. a =
(17)
First-order Generalised Beam Theory
209
i.e. for mode 3, node 1, aal-
-- 180 x 6"364 145-8 -7"857 kN/cm 2 (tension positive)
The complete pattern of mid-span stresses for mode 3 is given in Table 4(a). The deformations at mid-span are also given in Table 4. Because mode 3 is a rigid-body mode, the calculation of 3v and 3w at the nodes is trivial. For mode 5, the general beam equation (2) is complete, i.e. E sc SV"'--G SD Sv"+SB SV=Sq
The analogous beam on elastic foundation equation is E I y " - Ny" + ky = q TABLE 4 Resultant Stresses and Deflections a t M i d - S p a n (a) L o n g i t u d i n a l s t r e s s e s a t m i d - s p a n
Node:
1
aa sa
7'86 - 26'40
Ztr
-- 18'54
2 5"24 8"800 14"04
(b)
Node: 3v
1 0
3
4
5
- 7'86 - 5"28
5'24 8"80
7"86 - 26' 40
- 13"14
14"04
- 18-54
4
5
Horizontal deflections a t m i d - s p a n 2 0
3 0
0
0
5v
- 0"909
- 0-700
0
0"700
0"909
Ev
- 0"909
- 0"700
0
0"700
0"909
(c) V e r t i c a l d e f l e c t i o n s a t m i d - s p a n ( u p w a r d s p o s i t i v e )
Node:
1
2
3
4
5
Sw
Sw
-- 0.088 0-386
-- 0.088 0-596
-- 0.088 -- 0" 104
- 0.088 0-596
- 0.088 0.386
Zw
0-298
0-508
- 0' 192
0"508
0"298
J. M. Davies, P. Leach
210
with the following substitution: Span Young's modulus Second moment of area (Tensile) axial force Foundation modulus Distributed load
L E I N k q
-- 120 cm = 21 000 kN/cm 2 =SC = G SD = SB = Sq
= 0"72 cm 4 = 8077 × 0.000401 = 3-239 kN = 0-02127 kN/cm 2 = 0-0503 kN/cm
This can be solved in one of three ways: (a) Using an explicit solution (as given, for example, by Hetenyi3): q f y= ~
1
1
+cos
2atfl(cosh ~L + cos ilL)
[2~fl(cosh ~x cos fix'
cosh ~ x ' ) + ( ~ z - fl2)(sinh ~x sin fix' + s i n fix sinh atx')]t ./
(18) where x = distance along the beam, x ' = L - x
=
22 + 4E----]
fl =
4EI
and
F.I
Substituting into this e q u a t i o n gives, at mid-span,
y(L/2) = sV(L/2) = 2-061 cm and, either explicitly by differentiation, or numerically,
EIy"(L/2) = M ( L / 2 ) - 5W(L/2)= 19.01 k N cm (b) Using finite differences. If the span is divided into 6 slices, the finite difference p r o b l e m expressed in matrix form is s h o w n in Table 5. It is necessary to solve 14 simultaneous equations. These are simple to prog r a m b u t the use of a c o m p u t e r is, of course, m a n d a t o r y . The solution gives values of SV a n d SWat each node. At mid-span, the values are
y(L/2) - sV(L/2) = 2.060 cm M(L/2) =- sW(L/2) = 19.03 k N cm (c) Using finite elements. If only the stresses a n d deflections at m i d - s p a n are required, the simple two-element model s h o w n in Fig. 10 will suffice
b= -N+
12
kAx 2
12EI Ax 2
= -3"2389÷
Ax = 120/6 = 20 cm; a
b+ 2N c b
--a
b+ 2N 0
2a -a
2a
0
a
m
ma
-a
Y3
Y2
Y2
2a -a
5
0 b+2N
-a
-a
b c b+2N
0
~76
-a 2a
Y6
2Ax 0
- 2Ax 0
R1
- 2 2 -1
- 2 - 10 -1
M2
-
-1 2 -1
-1 10 -1
Ma
12
0"02127 x 202
kAx 2 +2.5299 kN; c = 2 N + - 7 ~ . = 13.568 k N 1.2
12 x 21000 x 0"720 202 = 453"6 kN; N = 3"2389 k N (axial force)
-a
a
Y5
2a
Y4
TABLE
-1 2 -1
-1 -10 -1
M4
Finite Difference Solution for M o d e 5 of Example
-1 2 -1
-1 -10 -1
M5
-1 2 -2
-1 -10 -2
M6
0 2Ax
0 -- 2Ax
R7
qAx 2 qAx 2 qAx 2 qAx 2 qAx 2 qAx 2 qAx 2
0 0 0
bO
.~.
212
J. M. Davies, P. Leach
I I I I 111
I I 111111IIIIII1
Fig. 10. Model for finite element analysis.
and give exact answers. This model has only four degrees of freedom and therefore requires the solution of just four simultaneous equations. The solution is y ( L / 2 ) - 5 V ( L / 2 ) = 2.061 cm M ( L / 2 ) - s W ( L / 2 ) = 19.01 k N cm
The longitudinal stresses for this mode then follow from eqn (14) as for mode 3, e.g. 5trl-
5W stY1 19-01 × ( - 1) 5C 0-72 -
2 6 . 4 0 k N c m 2, etc.
The complete pattern of stresses is given in Table 4(a). The deflections at different points of the cross-section can also be easily calculated. For instance, at section 1, 5W 1 = 5V 5w 1 =
2"061 × 0"1873 = 0"3860 cm
The remaining deflections at mid-span can be calculated similarly and these are also given in Table 4. The discerning reader will immediately notice the significance of the distortion terms. Quite clearly, a conventional analysis based on the Engineers' theory of bending would give the mode 3 terms only and hence, quite misleading results. It should be noted at this point that the transverse bending m o m e n t distribution due to distortion of the cross-section is also available. Thus, at mid-span, 5m 3 = s V ( L / 2 ) s~ 3 = 2.061 x (-0.3323) = -0-6849 k N cm/cm
It follows that GBT can provide the complete pattern of quantities that are of interest to the designer/analyst and it can do this in a particularly concise and useful form. The authors believe that for this, and for m a n y other reasons, it deserves to be more widely known.
First-order Generalised Beam Theory
213
CONCLUSIONS Generalised Beam Theory is important for two reasons: • The purely theoretical reason that it represents a major step forward in our understanding of structural principles. • The very practical reason that it provides a powerful and economical method of analysis for a wide range of problems. This paper has described the principles of the theory (largely without formal proof) and shown how it may be applied to bending problems in which cross-sectional distortion is important. Subsequent papers will generalise the derivation of section properties and present a computer program whereby they may be calculated. They will also show how the method may be applied to other practical problems. In a companion paper, the authors show how second-order effects may be included in the analysis and apply it to a range of buckling problems. The investigation of buckling problems using Generalised Beam Theory is particularly important because, by separating and then combining the various buckling modes, such problems as the interaction between local and global buckling are amenable to a particularly elegant treatment.
ACKNOWLEDGEMENT The authors wish to pay tribute to Professor R. Schardt for sharing his ideas so readily with the authors in many fruitful discussions. They also acknowledge that a number of the ideas presented in this paper find their inspiration in Chapter 1 of Ref. 1.
REFERENCES 1. Schardt, R., Verallgemeinerte Technische Biegetheorie [Generalised Beam Theory]. Springer Verlag, Berlin, Heidelberg, 1989. 2. Wlassow, W. S., Allgemeine Schalentheorie und ihre Anwendung in der Technik. Akademie Verlag, Berlin, 1958. 3. Hetenyi, M., Beams on Elastic Foundations. University of Michigan Press, Ann Arbor, MI, 1946. 4. Schardt, R. & Okur, H., Hilfswerte f/ir die L6sung der Differential-gleichung [Aids for the solution of the differential equation] ayiV(x)-by"(x)+cy(x)= p(x). Stahlbau, 1 (1971) 6-17. 5. Davies, J. M., An exact finite element for beam on elastic foundation problems. J. Struct. Mech., 14(4) (1986) 489-99.
214
J. M. Davies, P. Leach
6. Roberts, T. M., Section properties of thin-walled bars of open cross-section. Struct. Engr, 4313(3) (1985) 63-7. 7. Davies, J. M., Torsion of light gauge steel members. In Design of Cold-Formed Steel Members, ed. J. Rhodes. Elsevier Science Publishers, London, 1991, pp. 229q54.
A P P E N D I X A: F I N I T E D I F F E R E N C E S O L U T I O N OF FUNDAMENTAL BEAM EQUATION The solution is presented in terms of the beam on elastic foundation which has the following differential equations:
Ei d4Y_~z d2y dx 4 .. dx2 +ky=q
(A1)
M
E1 d 2 y dx 2
(A2)
where E1 =flexural rigidity of the beam; N = axial force (tensile positive); k = modulus of elastic foundation (kN/mm); q = uniformly distributed load; M = bending moment (hogging positive). The notation is further illustrated in Fig. A1. Equations (A1) and (A2) are more conveniently expressed as d2y +
-M-N~x
(A3)
2 ky=q P
I cl''' X |
N ,I-
...-
lY "
Jq'
"
'
....
I
" "
'
"
•
J
/ / / / / / / / / / / / / /
"
. '
. "
/ /
. •
. "
/
. ,
/
.
/
.'' "
/
N -.~
. . '.,(. ", " . i .
"
Di stLri:(~ t e d
.-
~ ; ,'..' •
"
I el'°'
I
; , •
'
I
M • . . . .
'
Point Load
//
'
•
"
/
/
•
• .
"
•
/ /
Beam
Foundation of '
/
Fig. AI. Notation for finite difference solution.
Modulus 'k'
First-order Generalised Beam Theory
215
M+EZ3=0 dx2
644)
Using this form of the equations, the finite difference operations are expressed in terms of the two variables y and M:
for point i
(for A3)
-Uyi-,+2Uyi-_y,+,-Mi_,-lOMi-Mi+,=PiAX
(A5)
(for A4)
byi-,+Cyi+byi+,-M~-~+2Mi_Mi+~=[Pi+P,i]Ax
(A6)
where
12EZ a=-jQb=
-Iv+%
d= -2N+3
c=2N+
kAx2
e=2N+
Equations (A5) and (A6) can be conveniently cient table:
Yi-1
---a b
Yi
Yi+l
2a c
-a
Mi-1
b
-1 -1
Mi
Mi+l
-10 2
-1 -1
5kAx2 6 2kAx2 3
expressed
in terms of a coeffi-
Right-hand
side
AxPi Ax(Pi + Fqi)
Boundary conditions As with all finite difference solutions, the boundary conditions non-standard coefficients into the equations. The following tables cover the cases most usually encountered. 1. Clamped end
introduce coefficient
J. M. Davies, P. Leach
216
Instead of Yl = 0 , the support force R is introduced as an additional unknown.
M3
Right-hand side
-2 -10
0 -1
-2API -AP2
-2 2
0 - 1
2Ax(P1 +Fql) Ax(P 2 q- fq2 )
M1
Mz
M3
Right-hand side
- 10 -1
-2 -10
0 -1
-2AxP1
2 - 1
-2
0 - 1
2Ax(P1 +Fql)
R
Y2
Y3
M1
-2Ax 0
-2a 2a
0 -a
-10 -1
2Ax 0
2a c
0 b
M2
2 - 1
2. Sliding clamp
Yl
Y2
Y3
2a -a
-2a 2a
-a
c b
2b c
0 0 b
2
AxP2 Ax(P2 + Fq2)
3. Pinned
(~ Ill
Instead of M t = 0 and Yl =0, the fictitious deflection force R are introduced, respectively, as unknowns.
.~2 and the support
217
First-order Generalised Beam Theory
)72
Y2
--a 0
-a 2a
Y3
R
0
-2Ax 0
-a
(b + 2 N ) ( b + 2 N ) 0 c
0 b
M2
Ma
-2 - 10
0 - 1
-2AxP1 -AxP2
-2
0 - 1
2Ax(PI+Fq~)
2Ax 0
2
Right-hand side
Ax(P2 + Fq2)
4. Free
I n s t e a d of M~ = 0 , the fictitious deflection )72 is i n t r o d u c e d as u n k n o w n .
Yl
Y2
2a -a
Y3
(-a+N)
0
2a e b
-a
d c
0 b
)72
M2
(--a-N)
M3
2
0
0
- 10
- 1
0 0
-2
0 - 1
2
Right-hand side -2AxPI -AxP2 2Ax(PI +Fql)
Ax(P 2 + Fq2)
5. Internal support
Yj- 1 2a --a
c b 0
Rj 0 -Ax 0
0 Ax 0
Yi+ ~
Mr- 1
Mr
Mr+ 1
Right-hand side
0 -a 2a
-10 - 1 0
-1 - 10 -1
0 - 1 -10
--AxP~-l -AxPj -AxPj+I
0 b c
2 - 1 0
--1 2 -1
0 - I 2
Ax(Pj_,+F~d_,) Ax(Pfq,j) Ax(Pr+l+Fq,r+l)
218
J. M. Davies, P. Leach
A P P E N D I X B: F I N I T E E L E M E N T S O L U T I O N O F FUNDAMENTAL BEAM EQUATION The solution is presented in terms of the beam on elastic foundation which has the following differential equation: EI d4Y_ Iv d2y dx 4 -, d x 2 + k y = q where El=flexural rigidity of the beam; N = axial force (tensile positive); k ---modulus of elastic foundation (kN/mm); q = uniformly distributed load. The subject finite element has been shown previously as Fig. 4. Basic case: - 2x/~EI <~N <~2x/~EI P=B[A]-~d
F :l where
P=
I A=
Py2
Y2
.M2.]
y~
sin flL
oOS ,
k-flc°shctL
where ~ =
•/
/~2
/ fl = . / 2 2
V
N q 4EI N 4EI
0
0
sin ~tL
sin flL
acoshctL
flcosflL
sinh ~L
l
-acosflL.J
First-order Generalised Beam Theory
B~I F B=/8 1
B12
BI3
B14q
B22
B23
B24]
B31
B32
B33
B341
I_Ba
B42 B43
219
B44_]
w h e r e B11 = -- El" $1 [f13 --
3~2fl]COSi l L - Nfl
cos flL
n 1 2 = - - El" S 2 [(z 3 _ 3aft 2 ] COS flL + Net cos flL n13 = El" S 1 [ fl 3 _ 3a 2fl] cosh o~L + N ~ cosh ~L n14 = El" S2[~ 3 -- 3ctfl2]cosh o~L-Nct cosh ~ L n21 = - E I
B22 =
-
[~2 _ f l 2 ] s i n flL
El" Sl(2afl)sin flL
B23 = EI" S2(2afl)sinh a L B24 = - - E l [~2 _ f12] sin h otL n31 = -- El" S i [ fl 3 _ 3a 2fl ] cosh a L - Nfl cosh aL B32 = - El" $2 [ct 3 _ 3aft 2 ] cosh a L + N a cosh ctL B33 = E1 " Sl [fl 3 - 3a2 fl ]cos i l L + N i l cos flL B34 = El" $2 [ a a - - 3aft2 ] cos i l L - Net cos flL B41 = El" S2(2afl)sinh a L B42 = - E1 [ct 2 _ fl 2 ] sinh ctL B43 = B44 =
-
-
E1 [~2 _ fl 2 ] sin flL E l . Sl(2ctfl)sin flL
a n d w h e r e $1 = S2 = 1-0 for the basic case.
Construction of consistent load vector for a uniformly distributed load of q per unit length
P=
MF -- M F
ctflq(ct2 + f12) (cosh a L - cos ilL) w h e r e Pv = 224(fl sinh a L + a sin ilL) q(ct2 + f l 2 )
M v - 4 2 4 ( f l sinh ~ L + c t sin ilL)
(fl sinh o~L-o~ sin ilL)
220
J. M. Davies, P. Leach
Case of N > 2 x / / ~ In the equations given above: (a) ~ is unchanged and
=
-
.
(b) Change sin flL to sinh flL and cos tL to cosh ilL. (c) Change (~2--fl2) to (0t2"Jt-fl2), (~2"Jt-fl2) to (a2--f12), (a3--3a2) to (eta + 3~fl 2) and (f13-3a2fl) to (f13 + 3ct2fl). (d) Change $1 from 1.0 to - 1 . 0 .
Case of N<2x/kEI In the equations given above:
(a) 0 t = ; / ~ 2 -- 4E--I N and i =
/--~. 2 - ~N (tension positive).
(b) Change sinh aL to sin aL and cosh ~L to cos aL. (c) Change (a2_f12) to ( _ a 2 _ f 1 2 ) and (a2+f12) to ( _ a 2 + f 1 2 ) . (d) Change $2 from 1-0 to -1-0. Special case of zero coefficients
If the axial load is N = 0 , standard beam on elastic foundation theory suffices. For this case, ~ = fl = 2 and the foregoing derivations simplify but remain valid. For the special case k = 0 , i.e. no elastic foundation, the above derivation breaks down. However, there are well-known equations for the resulting axially loaded bending element that are given in m a n y standard textbooks. It is therefore necessary to treat k = 0 as a special case and to use the available explicit stiffness matrix based on stability functions. As an alternative to treating k = 0 as a special case, solutions that are sufficiently accurate for all practical purposes can be obtained by inserting a very small value for k in the general solution.
© 1994 Elsevier Science Limited Printed in Malta. All rights reserved 0143-974X/94/$7.00 ELSEVIER
First-Order Generalised Beam Theory J. M. Davies & P. L e a c h Department of Civil Engineering and Construction, University of Salford, Salford, UK, M5 4WT (Received 5 January 1994)
ABSTRACT This paper introduces the basic principles of Generalised Beam Theory and shows how this theory may be used to analyse cold-formed sections in which distortion of the cross-section is significant. The calculation procedure is illustrated by a detailed numerical example.
NOTATION A a, b, c B,C,D bi E F-L F-Q F-O G I i,j,k J K k L M m N
Area of cross-section Ordinates of warping function Section properties for individual modes Width of bith plate element Young's modulus In-plane movement of a face at its mid-point Movement normal to a face at its mid-point Rotation of the chord line of a face Shear modulus Second m o m e n t of area Mode number (usually in the form of a forward superscript) St Venant torsional constant Stiffness matrix Foundation constant Span of member Bending m o m e n t Uniformly applied torque or transverse bending m o m e n t Axial load 187
188
P q S s t
V 13, W
W x, y z
,fl,2 F v O" T
4, O~
J. M. Davies, P. Leach
Point load Uniformly distributed load for individual mode Shear force Distance around member Thickness Generalised displacement function Displacement components in GBT Stress resultant Horizontal and vertical axes Distance along member Non-dimensional coefficients defined in text Warping constant Poisson's ratio Direct stress Shear stress Rotation Sectorial coordinate
Primes indicate differentiation with respect to the distance z along the member. ,~ indicates a unit value of a quantity, e.g. a warping function. A forward superscript indicates the mode number.
INTRODUCTION Generalised Beam Theory (GBT) seeks, at the same time, both to unify and to extend conventional theories for the analysis of prismatic thinwalled structural members. The analytical treatment of the four fundamental modes of deformation, namely extension, bending about the two principal axes and torsion, is united within a consistent notation. These four modes may be referred to as the 'rigid-body' modes because they do not involve any distortion of the cross-section. The notation is then extended to include higher-order modes of deformation which involve cross-sectional distortion. This allows elegant and economical solutions to a wide range of complex problems and provides a natural transition from beam theory to folded plate theory. Generalised Beam Theory can also be further extended to include second-order effects (local and global buckling), and this is considered in a companion paper. Evidently, Generalised Beam Theory is particularly applicable to the analysis of cold-formed steel sections and, in this paper, it will be intro-
First-order Generalised Beam Theory
189
duced from this standpoint. At the same time, it should be appreciated that it also has much wider application. The development of the theory has been pioneered by Professor R. Schardt and his colleagues at the University of Darmstadt in Germany, a work that has extended over more than 20 years. Unfortunately, little has been written during this period and almost nothing in English. The definitive reference at this present time is a recent book by Schardt.1 This paper attempts a somewhat different and more concise presentation of the fundamentals of the theory. By doing so it hopes to introduce the English-speaking world to a major step forward in structural mechanics.
MODES OF DEFORMATION AND WARPING FUNCTIONS A unifying feature of the theory is the concept of 'warping functions' whereby each mode k of deformation is associated with a distribution of axial strain kt~. Thus, the first mode of deformation is a uniform distribution of axial strain over the cross-section. For this mode, the 'warping function' 1~= _ 1 for all points s of the cross-section. The second and third modes are bending modes and the associated warping functions are linear distributions of strain about the two principal axes. The fourth mode is torsion and here the term warping has its conventional meaning as the warping function is the sectorial coordinate which reflects the distribution of axial strain due to a bi-moment. It is important to realise at the outset that all these warping functions are orthogonal. Practically, this means that in any first-order analysis they can be considered quite independently and their effects combined by simple superposition. Mathematically, the orthogonality condition is expressed as
f
i~kl]dA=0
for i # k
(1)
A
The above concepts are illustrated in Fig. 1. Figure l(a) shows a crosssection with five nodes. According to GBT, the five nodes can 'warp' independently and the warping functions are linear between the nodes. Therefore, the warping functions each have five degrees of freedom and the section has five orthogonal deformation modes together with their associated warping functions. Four of these are the rigid-body modes
190
J. M. Davies, P. Leach
J
/
®® (a) Cross - section with five nodes
1. Axial 8tress
2. Bending
'L_I--
\
3. Bending
4. Torsion Dleplaced 8hape
Wa
n
(b) Rigid body modes (k-1... 4)
Displaced Shape
WarpingFunction
(c) Cross - section distortion mode (k-S)
Fig. 1. Typical displaced shapes and warping functions.
shown in Fig. l(b) and the fifth is the mode of cross-sectional distortion shown in Fig. l(c). In simple applications, the number of modes is equal to the number of nodes (or fold lines) in the cross-section. More modes of distortion can be introduced by inserting intermediate nodes between the primary nodes. Each mode k has associated with it: • • • •
a warping function as already described; a corresponding pattern of cross-sectional displacements; distortional bending stresses (for modes 5 and above); section properties kc, kD and kB.
First-order Generalised Beam Theory
191
For the rigid-body modes 1 to 4 the section properties are familiar, thus: 1C =
2C= 3C = 4C = 4D =
cross-sectional area; second m o m e n t of area about first principal axis; second m o m e n t of area about second principal axis; warping constant; St Venant torsional constant;
1D -- IB = 0
2D =
2B=0 3D = 3B = 0
4B = 0
For the higher-order modes which involve cross-sectional distortion, all three section properties are in general non-zero. The analogy between conventional theory and GBT for section properties is summarised in Table 1. It can be seen that C represents the stiffness (in the relevant mode) with regard to direct stress, D represents the stiffness with regard to the shear stresses arising from twisting, and B represents the stiffness with regard to transverse bending stress. B therefore exists only for the distortional modes 5 and above. Schardt 1 traces the history of structural mechanics and shows how early investigators found the first four orthogonal modes of deformation of prismatic elements but made no attempt to unify them. Later work on prismatic folded plate structures moved away from these classical beginnings and a totally independent theory was created. This work has now evolved into the finite strip method with its implicit assumption that all problems can be solved numerically and there is little need to return to classical structural mechanics. Wlassow 2 came closest to GBT and Schardt believes that, but for his early death, it would have been only a matter of time before his research into prismatic structures evolved into a similar formulation. This paper and its companion paper on second-order theory demonstrate, among other things, that there is still some life in classical structural mechanics and that this is not just a matter for academics but has practical usefulness. Indeed, ultimately, GBT may offer the most practical way to deal with the difficult problem of the distortional buckling of cold-formed TABLE 1 Section Properties in GBT and Conventional Notations
Conventional theory
Deformation mode
Generalised beam theory
1
A
~C
2 3 4
I~ Iyr F
J
2C 3C 4C
4D
k
?
?
kC
gD
?
kB
192
d. M. Davies, P. Leach
sections. However, before we can consider this, there is much basic ground to be covered. GBT NOTATION The notation used for GBT in this paper follows that developed by Schardt I and is summarised at the start of the paper. In general, terms will also be defined as they are introduced, but two points are worthy of particular note: • a forward superscript is used to denote the mode number; • ~ over a symbol denotes a unit value of a quantity, e.g. a warping function or a related quantity derived from it.
THE FUNDAMENTAL
BEAM EQUATION
The basic equation of GBT is g k c k v " ' _ _ G k O k V " d-kBkV:kq
(2)
In this equation, E G
= Young's modulus; = shear modulus; kc, kD, kB = section properties applicable to mode k; kv =generalised deformation in mode k; kq = distributed load applicable to mode k; and primes indicate differentiation with respect to z which lies along the length of the member. That eqn (2) includes the basic modes of bending and torsion is easy to see. In conventional notation, the differential equations for bending about the two principal axes and torsion are d4y EIx~ d z 4 = qy d4x Elyy ~ = q:,
d4q~
(3) d2~
E F -d~z4 -- G J -d~z~ = m
First-order Generalised Beam Theory
193
These are evidently special cases of eqn (2). The full eqn (2) applies to modes 5 and above.
EQUIVALENCE
OF GBT AND CONVENTIONAL NOTATIONS
The full equivalence between the conventional and GBT notations is shown in Table 2. One of the most powerful features of GBT is that it extends naturally from the familiar rigid-body modes into the higher-order modes using a unified notation. This notation also includes a unified treatment of stresses and stress resultants so that, in GBT terms,
kw = - E kc kv t, ~_ f a k~l d A
(4)
is a generalisation of
Mx= - E l x x dz 2 =
try dA (bending moment)
(5) B = -- E F d2rp ~z 2 =
f ago dA (bi-moment)
where F is the warping constant and go is the sectorial coordinate.
SOLUTION
OF PROBLEMS
UTILISING GBT
The use of GBT to obtain the stresses and deformations of a member under specified loading and support conditions involves two quite distinct steps. In the first step, only the cross-section is considered in order to obtain for each mode k the warping functions k/~ and the associated section properties RC, kD and kB, together with other relevant information concerning the deformations and stresses induced by this mode. In the second step, this basic information is used to obtain solutions of eqn (2) for particular cases. The analysis implicit in the first step can be complex and, in general, it is carried out using standard computer programs. It is the authors' experience that GBT is best introduced without going into the details of the general form of the first step, but rather concentrating on the general principles and applications. This allows the power and versatility
A = [" dA
Area
Differential equation
EAu"= - N
N ¢r= - A
Normal stress
o Ixxt(s)
EIxxy"=qr
r l(s)
Mzy lxx
Mx = - EIxxy"
N = EAu'
Compatibility equation ¢rx=
A
Shear stress
Iyr= f x2 dA A
xw'
x
w
Bendin9 2
F = ,.I A
/ (/)2 dA
~¢p'
c5
q~
Warpin9 torsion
o Iyyt(s)
EIyrx"=q~
~2(s) =
Myx ~ry= - lrr
My= - Elvrx"
A
Ft(s)
-Bo~ F
EFq)''-GJq)"=m
~(s)- ~
or,-
B = - EF~p"
A
N=fadA Mx=faydA Mr=faxda B=fecoda
lxx = f y2 dA A
A
Stress resultant
Torsional constant
Second moment of area
yv'
u
Warping
A
y
1
Unit warping
v
u
Deformation
Bendin9 1
Extension
Quantity
TABLE 2 Equivalence of Conventional Theory and Generalised Beam Theory
o kCt(s)
k W k6 kC
EkCkV"--kGkDV"+kBkV=kq
k,(s) =
k~ =
kW= -- E kC ~V"
A
kw=fak~da
A
i C = fkfi2 dA
ku = k~ kv,
k~
%
GBT-includin9 higher modes
First-order Generalised Beam Theory
195
of the method to be appreciated before the more complex considerations become dominant. This is possible because, for simple problems, the first step can be carried out manually, an operation which has much to commend it as it provides a valuable insight into the principles involved. Furthermore, explicit expressions have been derived 1 for the properties of channel, z- and hat sections, including the higher-order modes, so that GBT can be readily used for a wide range of practical problems with the minimal introduction given in this paper.
SECTION PROPERTIES AND WARPING FUNCTIONS Table 3 shows the full results obtained by applying the first part of GBT to the cross-section shown in Fig. 2. The various terms are defined for each mode k as follows:
kc, kD kB are section properties; k/~
=
kF-L kF-Q kF-O
= = =
k/,~
=
ks
= =
kr7' k~
warping function defined at each node of the cross-section and assumed to be linear between the nodes (the remaining quantities are associated with unit value of this warping function); in-plane movement of a face at its mid-point; movement normal to the face at its mid-point; rotation of chord line of face; transverse bending moment at each node (distortional modes only); shear force in each face (as a function of kw'); nodal displacements in horizontal and vertical directions.
It should be noted that all the quantities tabulated for modes 1 to 4 are already obtainable from the standard procedures of structural mechanics which are given in basic texts on the subject. Those tabulated for mode 5 can also be derived manually using the fact that the warping function 5t~ must be orthogonal to ~ , 2t~, 3t~, 4~. The manual evaluation of all the quantities in Table 3 will be illustrated in detail later. Once the section properties have been evaluated, solutions to eqn (2) for different load and boundary conditions can be found. After obtaining a solution to eqn (2) for a particular case, multiplying the tabulated quantities by the relevant value of kv allows a complete pattern of transverse stresses and displacements to be obtained. The section shown in Fig. 2 is an open section that is unrestrained in space and with nodes only at the fold lines. Various other features may be
-
Mode k=3: 1 2 3 4 5 C = 145"8
6'3640 4'2426 -6"3640 4"2426 6'3640
Mode k=2: 1 - 8.4853 2 - 10-6066 3 0.00013 4 10"6066 5 8-4853 C = 502"2
1.0000 1.0000 1-0000 1.0000
- 1.0000
1
~
2 3 4 5 C = 10.8
M o d e k = 1:
Node
0.0000 0.0000 0.0000
0-0000
F- Q
0"7071 -0-7071 -0-7071 0'7071
D = 0-00
- 0-7071 -0.7071 0"7071 0-7071
D =0-00
0"7071 0'7071 --0'7071 --0'7071
0"7071 -0.7071 -0.7071 0'7071
D =0-00
0.0000 0.0000 0.0000
0.0000
F-L
0-0000 0"0000 0'0000 0"0000
0-0000 0"0000 0-0000 0-0000
0.0000 0.0000 0.0000 0-0000
F- 0
- 1"0000
B=O-O0
0-00013
1"0000 1'0000 1-0000 1-0000
-
1'0000
0"0000 0"0000 0-0000 0"0000 0'0000
0.0000 0.0000 0.0000 0.0000 0.0000
~
0-0000 0"0000 0"00(30 0"0000
B=O'O0
1"0000
1"0000
1"00(30 1'0000
0.0000 0'0000 0.0000 0.0000 0.0000 B=O.O0
~
TABLE 3 Cross-Sectional Values for Section s h o w n in Fig. 2
0"0000
0-0000 0"0000 0"0000 0-0000
0"01300 0"0000 0-0000 0-0(030 0"0000
0.0000 0.0000 0.0000 0"0000 0.0000
rfi
0'0524 0"6545 - 0-6545 - 0.0524
- 0"0247 -0"7318 - 0"7318 - 0.0247
0.0000 0-0000 0.0000 0.0000
S/W'
e'~
t'.,,
e:
C = 0"720
Mode k=5: 1 2 3 4 5
C =939.0
- 1'0000 0'3333 -0-2000 0"3333 - 1-0000
Mode k=4: 1 38.7097 2 - 7.8629 3 0-0000 4 7.8629 5 -38-7097
D =0'000401
-0'4444 0-0356 -0"0356 0'4444
0"1076 0"2044 0"2044 0"1076
-2.0242 - 8-0242 8.0242 2-0242
D =0.324
15.5242 - 0.5242 -0.5242 15.5242
-0-4412 -0"3394 0"0000 0-3394 0-4412
13.4692 11-3479 0.7413 11-3479 13.4692
B =0.02127
--0"0480 -0"0320 0'0320 0'0480
B =0.00
1.0000 1.0000 1.0000 1.00~
0"1873 0'2891 -0"0503 0"2891 0"1873
-8-4853 - 10.6066 0.0000 10.6066 8-4853
0.0000 0"0000 -0"3323 0-0000 0.0000
0.0000 0.00130 0.0000 0.0000 0.0000
- 1"0417 1'0417 - 1"0417 1.0417
0-0331 0.0331 0.0331 0.0331
,..q
~t
r,~
J. M. Davies, P. Leach
198
A z,w b z - 15
(~)i
b 2 • 15
0.741
1 6.364
•/
.....
.
.
.
.
2.121
b
•
"3
Fig. 2. Cross-section for example.
incorporated in the derivation of the section properties and warping functions, namely: The section may be open or closed. The section may include branches, though at the present time this is not so well documented. Intermediate nodes may be added in order to increase the number of distortional modes. The free movement of the cross-section may be restrained either rigidly or elastically. This latter facility has many applications, some of which are illustrated in Fig. 3. Bearing in mind that these are cross-sections of members which may have arbitrary loading and support conditions in the longitudinal direction, the wide range of problems to which GBT is applicable becomes apparent.
SOLUTION
OF THE FUNDAMENTAL
BEAM EQUATION
Equation (2) may be solved in a number of different ways depending on the nature of the relevant modes of deformation and the loading and boundary conditions. As the unique features of GBT are revealed only when cross-section distortion is present, attention here is concentrated on the general case. Equation (2) is identical in form to the differential equation for the displacement of an axially loaded beam on an elastic foundation. In order to apply beam on elastic foundation solutions to GBT problems, the
First-order Generalised Beam Theory
°l ®
| ®
®
C~Cy-y-y-,Cy-v~Cv--y~ \
Cy-~Cy~C~ \
\
Ic) Symmetry condition
. . . . . . . . . .
®
®
(b) Elastic lateral and torsional restraint of purlin
(a) Lateral restraint of purlin from sheeting
//
199
\
(d) antisymmetry condition
.
"........
®
,¢ ®
:
,
',
,
®
(e) Periodic construction
(f) Rigid restraint
Fig. 3. Examples of restrained cross-sections.
following substitutions must be made: second m o m e n t of area axial load (tension positive) foundation constant uniformly distributed load
I :~ k c N - G kD k -- kB q = kq
and, when the solution has been completed, the stress resultant M --kw
It follows that the solutions for many relatively simple loading and support conditions are well documented. 3'4 Indeed, Ref. 3 is a whole book devoted to the topic and explicit solutions such as those given therein will provide an appropriate method for a wide range of practical problems. However, many other practical problems do not have simple loading and boundary conditions and for these it is necessary to resort to numerical methods of solution. Two such methods have been widely used in practice, namely the finite difference method and the finite element method.
200
J. M. Davies, P. Leach
FINITE DIFFERENCE
SOLUTION
O F E Q N (2)
Schardt and his associates at Darmstadt have exclusively used the finite difference method in their solutions of the general case of the fundamental beam equation. The method is most easily understood when it has a readily visualised physical form and, for this reason, it is usually described with reference to a beam on an elastic foundation. The solution system for this problem is given in Appendix A. As with many finite difference applications, the treatment of the relevant boundary conditions is non-trivial and the treatment for the cases usually encountered is also given in Appendix A. FINITE ELEMENT
SOLUTIONS OF GBT PROBLEMS
To the best of the authors' knowledge, they were the first to utilise the finite element method in the context of GBT. The advantage of this approach is that eqn (2) falls into the class of equations for which the finite element solution is exact. This means that, for many first-order problems, the computational requirements become almost trivial. The detailed derivation of the relevant element and consistent load vector has been given by the first author 5 for the beam on elastic foundation problem shown in Fig. 4, for which the governing differential equation is (6)
E l y " ' - Ny" + ky = q
The element stiffness equations have the form
ry,l[]r,l M'/= K
Y'I
(7)
Py2
Yl:~ PY~I N ~ '~M1 I ' ~
M
.~Y2
" ' ' ' 6PY2 elaatic foundation
L Fig. 4. Beam on elastic foundation element.
First-order Generalised Beam Theory
201
where the terms of the 4 x 4 stiffness matrix [K] are given in Appendix B. The consistent load vector for a uniformly distributed load has the form
MF
(8)
PF --MF
where PF and MF are also given in Appendix B. It may be noted that with this formulation, nodal loads and boundary conditions can be applied directly in the usual way.
MANUAL CALCULATION
USING GBT
As an illustration of manual calculation using GBT, the problem shown in Fig. 5 will be solved. A beam with the cross-section shown in Fig. 2 is simply supported over a span of 120cm and subject to a vertical uniformly distributed load acting along the centre of the cross-section. It is immediately obvious that, in order to obtain a sensible solution to this problem, it is essential to take into account distortion of the crosssection. As has already been explained, the warping functions for modes 1 to 4, the rigid-body modes, may be obtained using elementary structural mechanics. For any section, mode 1 is always a unit axial strain and the warping function lt~= - 1 across the entire section. Mode 2 represents bending about the vertical axis through the centroid. The associated warping function is a linear distribution of strain in the
li
/f
/ / / / / /
/ /
Fig. 5. Example using GBT.
J. M. Davies, P. Leach
202
horizontal direction, the strain at any point being proportional to the horizontal distance from the z-axis. Similarly, mode 3 represents bending about the horizontal axis through the centroid and the associated warping function is the vertical distance of the various points of the cross-section from this axis. Finally, mode 4 represents a unit twist about the shear centre and the warping function 4~ is the sectorial coordinate. The calculation of sectorial coordinates is not trivial but computational algorithms have been published 6 and they have been tabulated for some common sections] Once the warping functions for modes 1 to 4 have been identified, the other associated quantities in Table 3 for these modes follow directly. If they are not obvious by inspection, they can be calculated using the general procedures which will now be explained with reference to mode 5. The warping function for mode 5 can be determined from the condition that it must be orthogonal to the other modes, i.e.
f
ill
k/~ dA
= 0
(9)
for i ¢- k
A
For the symmetrical section shown in Fig. 2 subject to a vertical load, it is evident that the mode required must be symmetrical. A symmetric mode will automatically be orthogonal to the antisymmetric modes 2 and 4 so that it is only necessary to ensure orthogonality with modes 1 and 3. If the warping function is assumed to have ordinates {a, b, c, b, a}, then
f
it7 5fidA
gives
A
3×0"3×(-1)×
--
+15×0"3x(-1)×
i.e. - 0-45a -- 2"7b - 2.25c = 0
and
f
ail St~dA=0
gives
A
3xO-3x
[6"3640a 3 4"2426b
6"3640b6 4"2426a ]
=0
First-order Generalised Beam Theory
+15x0.3
203
I4'2426b 3 6"3640c 4"2426c- 6"3640b1 t 6 =0
i.e. 2.5456a+3"8183b-6.3640c=O Solving these two simultaneous equations gives b = -0.3333a c=0.2000a The absolute magnitudes of the warping ordinates for modes 5 and above are indeterminate and it is usual to normalise them to make the largest numerically equal to plus or minus unity. It then follows that sO1 ~u2 =0"3333 5~3 =0"2000 5~4=0"3333 s~s = - - 1 =
-
1
These values of fi give rise to slopes Of~/Os which cause in-plane displacements of each element of the section in the s-direction. Thus 5F_L 1 _ ~ 1 - f i 2 _ - 1 - 0 . 3 3 3 3 bl 3
5F_L2_ fi2-tt3 b2
_
-0.4444
0"3333+0"2_0.0356 15
and 5 F - L 3 and 5 F - L 4 follow by symmetry: 5F-L3 = -0.0356 5F-L4 = 0-4444
These in-plane displacements define various movements of the plate elements which can be determined by simple geometry. Neglecting for the present the implied transverse bending of the plates by considering the connection at node 3 to be a hinge, the movements shown by the full line in Fig. 6 can be calculated. It is easy to show that these give rise to the
204
.J. M. Davies, P. Leach
IW V
: ,
0.~.0356 (~:~'-~,,-0.4444 0.4444j'~ ~(~) Fig. 6. Cross-sectionalmovementsin mode5.
Fig. 7. Bendingmoments for mode 5.
following movements:
5F-Q=0.0836 5F-0=-0.0320 5f=_0.4073 5~= 0-2044 0.2044 0.0836
--0.0320 0.0320 0-0320
-0.3394 0 0.3394 0.4073
0.2212 0.2891 -0-0503 0-2891 0.2212
It is now necessary to introduce continuity at node 3 by rotating plates 2 and 3 at this node without moving nodes 2 and 4, whose positions are determined by geometry. Nodes 1 and 5 are not constrained during this operation. This introduces the bending moment distribution shown in Fig. 7, where 3
Et 3
5ff/3 b2 1 2 [ 1 - v 2] 5F-02 3 x 21 000 × 0"33 x --0"0320 15 × 10.92 = 0.3323 kN/cm It also introduces additional rotations at nodes 2 and 4 which are numerically equal to ½(F-02)=0.0160 with corresponding changes in F-Q and F-O for plates 1 and 4 and ~ and v~ for nodes 1 and 5. This brings the values calculated above into agreement with Table 3. The final displaced shape of the cross-section is then shown by the chain-dotted line in Fig. 6.
First-order GeneralisedBeam Theory
205
The section properties can now be calculated using generalisations of the familiar formulae:
5C=fS~2dA
[compare I = f y 2 d A , etc.]
A
=2x3x0-3[(- 1)2--1 X 0.33333+0"33332] 0.3[0.33332-0.2×0.33333+0"22] + 2 x 15 ×
= 0"720 cm 4 1
5
(10)
n
D = ~ r=l -3-1 x 2
F-O2b, t 3 [compare J = ½ E b t 3] ×
0"33[0"04802 X 3+0"03202 x 15]
= 0"000401 cm 2
(11)
Note: The formula used above has the advantage of familiarity and gives a good approximation to RD. The general theory also incorporates a more precise expression: 5 B = f 1211 - v 2] rfi2 ds
J
Et a
S
_
1 F2 × 0"33232 - - 3x 51"923 /
151
= 0"02127 kN/cm 2
(12)
Finally, for completeness, the unit shear flows can now be calculated for
kW'= 1 using
a(s)t(s) f S
%(s)t(st=~1d
ds
0
compare
zt = -- 1
f y dA
(13) for unit shear force]
J. M. Davies, P. Leach
206
The shear force in each face then follows: r+l
%= f kz~(s)t(s)ds
(14)
r
Thus, for face 1 sfi(s) = 0"4444s - 1
.'. 5%(s) . t(s)= ~ 2 [O'4444s2 5S, = 00@2 [0"44~4sa
q 1. sJ = 0-4167/cm
for
s= 3
1 .s213 -
2- _]o- - 1"0417
and for face 2 5fi(s) = - 0.0356s + 0.3333 ÷0"3333s] 5zs(s) • t(s) = - 0.4167 + 00_~732[ -0"0356s2 2 =0
for s = 15 0.3333s2"~] 1'
2
}Ao
= 1.0417 The full pattern of shear flow for this mode is shown in Fig. 8. This completes the derivation of the full set of section properties for mode 5 given in Table 3 and concludes the first phase of the calculations. Having determined the section properties, the next step is to solve the fundamental beam equation for each of the modes which is to be included in the solution. It is immediately apparent that the analyst can exercise choice here and can investigate the significance of individual modes. Furthermore, the boundary conditions do not have to be the same for the different modes. These are considerable advantages of GBT which are not available in other methods. We note here that as the structure and its loading are symmetrical, the deformation must be symmetrical. It follows that only modes 3 (bending
First-order Generalised Beam Theory
207
0
0.4167
0
Fig. 8. Shear forces and shear flow for mode 5.
about the horizontal principal axis) and 5 (distortion of the cross-section) play any role. The section will be considered to be simply supported with respect to both these modes. The load terms in the fundamental beam equation represent the virtual work of the loads acting on the modal displacements. They can be calculated in two alternative ways, both of which lead to the same result: (a) By considering the virtual work of the horizontal and vertical components of the nodal loads acting on the modal displacements k~ and k~, i.e. n+l
kq= Z qY,rk~rq-qz, rk~,r
(15)
r=l
so that here, noting that the applied load is in the negative direction and that both 3/~3 and 5k 3 are also negative, 3q = 1"0 5q = 0"0503 (b) By considering equilibrium of the shears ks where the governing equation is
as
*q= ~ *q,kSr
(16)
r=l
For mode 3, the general beam equation (2) reduces to E
3C 3V't" =
3q,
i.e.
shown in Fig. 9,
E1 V"" = q
208
J. M. Davies, P. Leach
~L1"0
. / N . O.70~ ~707
1
qr
m\
/,
41" -0.0524 0.0524 0.05035 Sr
3S r
Fig. 9. Resolution of load between modes. which is the usual equation of the Engineers' theory of bending neglecting distortion of the cross-section. The solution is therefore standard and at mid-span:
I i.e. M - ----ff--[ qL21 3W(L/2)=3qL2 8 1"0 x 1202 8 and
- 180 kN cm
3V(L/2)_ 53qL4 I 384E 3C
5qL4] i.e. w = 384EI
5 x - l ' O x 1204 =0"08818 cm 384 x 21 000 x 145-8 The longitudinal stresses then follow from
k~= kc
i.e. a =
(17)
First-order Generalised Beam Theory
209
i.e. for mode 3, node 1, aal-
-- 180 x 6"364 145-8 -7"857 kN/cm 2 (tension positive)
The complete pattern of mid-span stresses for mode 3 is given in Table 4(a). The deformations at mid-span are also given in Table 4. Because mode 3 is a rigid-body mode, the calculation of 3v and 3w at the nodes is trivial. For mode 5, the general beam equation (2) is complete, i.e. E sc SV"'--G SD Sv"+SB SV=Sq
The analogous beam on elastic foundation equation is E I y " - Ny" + ky = q TABLE 4 Resultant Stresses and Deflections a t M i d - S p a n (a) L o n g i t u d i n a l s t r e s s e s a t m i d - s p a n
Node:
1
aa sa
7'86 - 26'40
Ztr
-- 18'54
2 5"24 8"800 14"04
(b)
Node: 3v
1 0
3
4
5
- 7'86 - 5"28
5'24 8"80
7"86 - 26' 40
- 13"14
14"04
- 18-54
4
5
Horizontal deflections a t m i d - s p a n 2 0
3 0
0
0
5v
- 0"909
- 0-700
0
0"700
0"909
Ev
- 0"909
- 0"700
0
0"700
0"909
(c) V e r t i c a l d e f l e c t i o n s a t m i d - s p a n ( u p w a r d s p o s i t i v e )
Node:
1
2
3
4
5
Sw
Sw
-- 0.088 0-386
-- 0.088 0-596
-- 0.088 -- 0" 104
- 0.088 0-596
- 0.088 0.386
Zw
0-298
0-508
- 0' 192
0"508
0"298
J. M. Davies, P. Leach
210
with the following substitution: Span Young's modulus Second moment of area (Tensile) axial force Foundation modulus Distributed load
L E I N k q
-- 120 cm = 21 000 kN/cm 2 =SC = G SD = SB = Sq
= 0"72 cm 4 = 8077 × 0.000401 = 3-239 kN = 0-02127 kN/cm 2 = 0-0503 kN/cm
This can be solved in one of three ways: (a) Using an explicit solution (as given, for example, by Hetenyi3): q f y= ~
1
1
+cos
2atfl(cosh ~L + cos ilL)
[2~fl(cosh ~x cos fix'
cosh ~ x ' ) + ( ~ z - fl2)(sinh ~x sin fix' + s i n fix sinh atx')]t ./
(18) where x = distance along the beam, x ' = L - x
=
22 + 4E----]
fl =
4EI
and
F.I
Substituting into this e q u a t i o n gives, at mid-span,
y(L/2) = sV(L/2) = 2-061 cm and, either explicitly by differentiation, or numerically,
EIy"(L/2) = M ( L / 2 ) - 5W(L/2)= 19.01 k N cm (b) Using finite differences. If the span is divided into 6 slices, the finite difference p r o b l e m expressed in matrix form is s h o w n in Table 5. It is necessary to solve 14 simultaneous equations. These are simple to prog r a m b u t the use of a c o m p u t e r is, of course, m a n d a t o r y . The solution gives values of SV a n d SWat each node. At mid-span, the values are
y(L/2) - sV(L/2) = 2.060 cm M(L/2) =- sW(L/2) = 19.03 k N cm (c) Using finite elements. If only the stresses a n d deflections at m i d - s p a n are required, the simple two-element model s h o w n in Fig. 10 will suffice
b= -N+
12
kAx 2
12EI Ax 2
= -3"2389÷
Ax = 120/6 = 20 cm; a
b+ 2N c b
--a
b+ 2N 0
2a -a
2a
0
a
m
ma
-a
Y3
Y2
Y2
2a -a
5
0 b+2N
-a
-a
b c b+2N
0
~76
-a 2a
Y6
2Ax 0
- 2Ax 0
R1
- 2 2 -1
- 2 - 10 -1
M2
-
-1 2 -1
-1 10 -1
Ma
12
0"02127 x 202
kAx 2 +2.5299 kN; c = 2 N + - 7 ~ . = 13.568 k N 1.2
12 x 21000 x 0"720 202 = 453"6 kN; N = 3"2389 k N (axial force)
-a
a
Y5
2a
Y4
TABLE
-1 2 -1
-1 -10 -1
M4
Finite Difference Solution for M o d e 5 of Example
-1 2 -1
-1 -10 -1
M5
-1 2 -2
-1 -10 -2
M6
0 2Ax
0 -- 2Ax
R7
qAx 2 qAx 2 qAx 2 qAx 2 qAx 2 qAx 2 qAx 2
0 0 0
bO
.~.
212
J. M. Davies, P. Leach
I I I I 111
I I 111111IIIIII1
Fig. 10. Model for finite element analysis.
and give exact answers. This model has only four degrees of freedom and therefore requires the solution of just four simultaneous equations. The solution is y ( L / 2 ) - 5 V ( L / 2 ) = 2.061 cm M ( L / 2 ) - s W ( L / 2 ) = 19.01 k N cm
The longitudinal stresses for this mode then follow from eqn (14) as for mode 3, e.g. 5trl-
5W stY1 19-01 × ( - 1) 5C 0-72 -
2 6 . 4 0 k N c m 2, etc.
The complete pattern of stresses is given in Table 4(a). The deflections at different points of the cross-section can also be easily calculated. For instance, at section 1, 5W 1 = 5V 5w 1 =
2"061 × 0"1873 = 0"3860 cm
The remaining deflections at mid-span can be calculated similarly and these are also given in Table 4. The discerning reader will immediately notice the significance of the distortion terms. Quite clearly, a conventional analysis based on the Engineers' theory of bending would give the mode 3 terms only and hence, quite misleading results. It should be noted at this point that the transverse bending m o m e n t distribution due to distortion of the cross-section is also available. Thus, at mid-span, 5m 3 = s V ( L / 2 ) s~ 3 = 2.061 x (-0.3323) = -0-6849 k N cm/cm
It follows that GBT can provide the complete pattern of quantities that are of interest to the designer/analyst and it can do this in a particularly concise and useful form. The authors believe that for this, and for m a n y other reasons, it deserves to be more widely known.
First-order Generalised Beam Theory
213
CONCLUSIONS Generalised Beam Theory is important for two reasons: • The purely theoretical reason that it represents a major step forward in our understanding of structural principles. • The very practical reason that it provides a powerful and economical method of analysis for a wide range of problems. This paper has described the principles of the theory (largely without formal proof) and shown how it may be applied to bending problems in which cross-sectional distortion is important. Subsequent papers will generalise the derivation of section properties and present a computer program whereby they may be calculated. They will also show how the method may be applied to other practical problems. In a companion paper, the authors show how second-order effects may be included in the analysis and apply it to a range of buckling problems. The investigation of buckling problems using Generalised Beam Theory is particularly important because, by separating and then combining the various buckling modes, such problems as the interaction between local and global buckling are amenable to a particularly elegant treatment.
ACKNOWLEDGEMENT The authors wish to pay tribute to Professor R. Schardt for sharing his ideas so readily with the authors in many fruitful discussions. They also acknowledge that a number of the ideas presented in this paper find their inspiration in Chapter 1 of Ref. 1.
REFERENCES 1. Schardt, R., Verallgemeinerte Technische Biegetheorie [Generalised Beam Theory]. Springer Verlag, Berlin, Heidelberg, 1989. 2. Wlassow, W. S., Allgemeine Schalentheorie und ihre Anwendung in der Technik. Akademie Verlag, Berlin, 1958. 3. Hetenyi, M., Beams on Elastic Foundations. University of Michigan Press, Ann Arbor, MI, 1946. 4. Schardt, R. & Okur, H., Hilfswerte f/ir die L6sung der Differential-gleichung [Aids for the solution of the differential equation] ayiV(x)-by"(x)+cy(x)= p(x). Stahlbau, 1 (1971) 6-17. 5. Davies, J. M., An exact finite element for beam on elastic foundation problems. J. Struct. Mech., 14(4) (1986) 489-99.
214
J. M. Davies, P. Leach
6. Roberts, T. M., Section properties of thin-walled bars of open cross-section. Struct. Engr, 4313(3) (1985) 63-7. 7. Davies, J. M., Torsion of light gauge steel members. In Design of Cold-Formed Steel Members, ed. J. Rhodes. Elsevier Science Publishers, London, 1991, pp. 229q54.
A P P E N D I X A: F I N I T E D I F F E R E N C E S O L U T I O N OF FUNDAMENTAL BEAM EQUATION The solution is presented in terms of the beam on elastic foundation which has the following differential equations:
Ei d4Y_~z d2y dx 4 .. dx2 +ky=q
(A1)
M
E1 d 2 y dx 2
(A2)
where E1 =flexural rigidity of the beam; N = axial force (tensile positive); k = modulus of elastic foundation (kN/mm); q = uniformly distributed load; M = bending moment (hogging positive). The notation is further illustrated in Fig. A1. Equations (A1) and (A2) are more conveniently expressed as d2y +
-M-N~x
(A3)
2 ky=q P
I cl''' X |
N ,I-
...-
lY "
Jq'
"
'
....
I
" "
'
"
•
J
/ / / / / / / / / / / / / /
"
. '
. "
/ /
. •
. "
/
. ,
/
.
/
.'' "
/
N -.~
. . '.,(. ", " . i .
"
Di stLri:(~ t e d
.-
~ ; ,'..' •
"
I el'°'
I
; , •
'
I
M • . . . .
'
Point Load
//
'
•
"
/
/
•
• .
"
•
/ /
Beam
Foundation of '
/
Fig. AI. Notation for finite difference solution.
Modulus 'k'
First-order Generalised Beam Theory
215
M+EZ3=0 dx2
644)
Using this form of the equations, the finite difference operations are expressed in terms of the two variables y and M:
for point i
(for A3)
-Uyi-,+2Uyi-_y,+,-Mi_,-lOMi-Mi+,=PiAX
(A5)
(for A4)
byi-,+Cyi+byi+,-M~-~+2Mi_Mi+~=[Pi+P,i]Ax
(A6)
where
12EZ a=-jQb=
-Iv+%
d= -2N+3
c=2N+
kAx2
e=2N+
Equations (A5) and (A6) can be conveniently cient table:
Yi-1
---a b
Yi
Yi+l
2a c
-a
Mi-1
b
-1 -1
Mi
Mi+l
-10 2
-1 -1
5kAx2 6 2kAx2 3
expressed
in terms of a coeffi-
Right-hand
side
AxPi Ax(Pi + Fqi)
Boundary conditions As with all finite difference solutions, the boundary conditions non-standard coefficients into the equations. The following tables cover the cases most usually encountered. 1. Clamped end
introduce coefficient
J. M. Davies, P. Leach
216
Instead of Yl = 0 , the support force R is introduced as an additional unknown.
M3
Right-hand side
-2 -10
0 -1
-2API -AP2
-2 2
0 - 1
2Ax(P1 +Fql) Ax(P 2 q- fq2 )
M1
Mz
M3
Right-hand side
- 10 -1
-2 -10
0 -1
-2AxP1
2 - 1
-2
0 - 1
2Ax(P1 +Fql)
R
Y2
Y3
M1
-2Ax 0
-2a 2a
0 -a
-10 -1
2Ax 0
2a c
0 b
M2
2 - 1
2. Sliding clamp
Yl
Y2
Y3
2a -a
-2a 2a
-a
c b
2b c
0 0 b
2
AxP2 Ax(P2 + Fq2)
3. Pinned
(~ Ill
Instead of M t = 0 and Yl =0, the fictitious deflection force R are introduced, respectively, as unknowns.
.~2 and the support
217
First-order Generalised Beam Theory
)72
Y2
--a 0
-a 2a
Y3
R
0
-2Ax 0
-a
(b + 2 N ) ( b + 2 N ) 0 c
0 b
M2
Ma
-2 - 10
0 - 1
-2AxP1 -AxP2
-2
0 - 1
2Ax(PI+Fq~)
2Ax 0
2
Right-hand side
Ax(P2 + Fq2)
4. Free
I n s t e a d of M~ = 0 , the fictitious deflection )72 is i n t r o d u c e d as u n k n o w n .
Yl
Y2
2a -a
Y3
(-a+N)
0
2a e b
-a
d c
0 b
)72
M2
(--a-N)
M3
2
0
0
- 10
- 1
0 0
-2
0 - 1
2
Right-hand side -2AxPI -AxP2 2Ax(PI +Fql)
Ax(P 2 + Fq2)
5. Internal support
Yj- 1 2a --a
c b 0
Rj 0 -Ax 0
0 Ax 0
Yi+ ~
Mr- 1
Mr
Mr+ 1
Right-hand side
0 -a 2a
-10 - 1 0
-1 - 10 -1
0 - 1 -10
--AxP~-l -AxPj -AxPj+I
0 b c
2 - 1 0
--1 2 -1
0 - I 2
Ax(Pj_,+F~d_,) Ax(Pfq,j) Ax(Pr+l+Fq,r+l)
218
J. M. Davies, P. Leach
A P P E N D I X B: F I N I T E E L E M E N T S O L U T I O N O F FUNDAMENTAL BEAM EQUATION The solution is presented in terms of the beam on elastic foundation which has the following differential equation: EI d4Y_ Iv d2y dx 4 -, d x 2 + k y = q where El=flexural rigidity of the beam; N = axial force (tensile positive); k ---modulus of elastic foundation (kN/mm); q = uniformly distributed load. The subject finite element has been shown previously as Fig. 4. Basic case: - 2x/~EI <~N <~2x/~EI P=B[A]-~d
F :l where
P=
I A=
Py2
Y2
.M2.]
y~
sin flL
oOS ,
k-flc°shctL
where ~ =
•/
/~2
/ fl = . / 2 2
V
N q 4EI N 4EI
0
0
sin ~tL
sin flL
acoshctL
flcosflL
sinh ~L
l
-acosflL.J
First-order Generalised Beam Theory
B~I F B=/8 1
B12
BI3
B14q
B22
B23
B24]
B31
B32
B33
B341
I_Ba
B42 B43
219
B44_]
w h e r e B11 = -- El" $1 [f13 --
3~2fl]COSi l L - Nfl
cos flL
n 1 2 = - - El" S 2 [(z 3 _ 3aft 2 ] COS flL + Net cos flL n13 = El" S 1 [ fl 3 _ 3a 2fl] cosh o~L + N ~ cosh ~L n14 = El" S2[~ 3 -- 3ctfl2]cosh o~L-Nct cosh ~ L n21 = - E I
B22 =
-
[~2 _ f l 2 ] s i n flL
El" Sl(2afl)sin flL
B23 = EI" S2(2afl)sinh a L B24 = - - E l [~2 _ f12] sin h otL n31 = -- El" S i [ fl 3 _ 3a 2fl ] cosh a L - Nfl cosh aL B32 = - El" $2 [ct 3 _ 3aft 2 ] cosh a L + N a cosh ctL B33 = E1 " Sl [fl 3 - 3a2 fl ]cos i l L + N i l cos flL B34 = El" $2 [ a a - - 3aft2 ] cos i l L - Net cos flL B41 = El" S2(2afl)sinh a L B42 = - E1 [ct 2 _ fl 2 ] sinh ctL B43 = B44 =
-
-
E1 [~2 _ fl 2 ] sin flL E l . Sl(2ctfl)sin flL
a n d w h e r e $1 = S2 = 1-0 for the basic case.
Construction of consistent load vector for a uniformly distributed load of q per unit length
P=
MF -- M F
ctflq(ct2 + f12) (cosh a L - cos ilL) w h e r e Pv = 224(fl sinh a L + a sin ilL) q(ct2 + f l 2 )
M v - 4 2 4 ( f l sinh ~ L + c t sin ilL)
(fl sinh o~L-o~ sin ilL)
220
J. M. Davies, P. Leach
Case of N > 2 x / / ~ In the equations given above: (a) ~ is unchanged and
=
-
.
(b) Change sin flL to sinh flL and cos tL to cosh ilL. (c) Change (~2--fl2) to (0t2"Jt-fl2), (~2"Jt-fl2) to (a2--f12), (a3--3a2) to (eta + 3~fl 2) and (f13-3a2fl) to (f13 + 3ct2fl). (d) Change $1 from 1.0 to - 1 . 0 .
Case of N<2x/kEI In the equations given above:
(a) 0 t = ; / ~ 2 -- 4E--I N and i =
/--~. 2 - ~N (tension positive).
(b) Change sinh aL to sin aL and cosh ~L to cos aL. (c) Change (a2_f12) to ( _ a 2 _ f 1 2 ) and (a2+f12) to ( _ a 2 + f 1 2 ) . (d) Change $2 from 1-0 to -1-0. Special case of zero coefficients
If the axial load is N = 0 , standard beam on elastic foundation theory suffices. For this case, ~ = fl = 2 and the foregoing derivations simplify but remain valid. For the special case k = 0 , i.e. no elastic foundation, the above derivation breaks down. However, there are well-known equations for the resulting axially loaded bending element that are given in m a n y standard textbooks. It is therefore necessary to treat k = 0 as a special case and to use the available explicit stiffness matrix based on stability functions. As an alternative to treating k = 0 as a special case, solutions that are sufficiently accurate for all practical purposes can be obtained by inserting a very small value for k in the general solution.