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E. M. Lui
Department of Civil and Environmental Engineering
Syracuse University
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New Directions in Civil Engineering
SERIES EDITOR: W. F. CHEN Purdue University
Contents
Preface, vii
Notation, ix
1 Analysis of Beam-Columns,
2 Design of Beam-Columns, 39
3 Second-Order Rigid Frame Analysis, 145
4 Simplified Second-Order Rigid Frame Analysis, 203
5 Behavior and Modeling of Semi-Rigid Connections, 235
6 Analysis of Semi-Rigid Frames, 275
7 Design of Semi-Rigid Frames, 343
Index, 373
v
Preface
vii
Notation
ix
X NOTATION
Material parameters
E modulus of elasticity of steel (29 000 ksi)
G shear modulus of elasticity of steel ( 11 200 ksi)
Fy, ay yield stress
v Poisson's ratio
Connection parameters
C, D curve-fitting constants
K standardization parameter
M connection moment
Mu connection ultimate moment capacity
n shape parameter
Rk connection stiffness
Rk R~IEI = nondimensional connection stiffness
1.1 Introduction
Beam-columns are structural members subjected to combined axial forces and
bending moments. The bending moments that are present in a beam-column consist
of two types: primary bending moments which arise from moments applied or
induced at the ends of the member and/or moments from transverse loadings on the
member; secondary bending moments which arise as a result of the axial force acting
through the lateral displacement of the member. The so-called P-r5 moments are
moments caused by the axial force acting through the lateral displacement of the
member relative to its chord (Fig. 1.1 a), whereas the P-Ll moments are moments
caused by the axial force acting through the relative lateral displacement of the two
ends of the member (Fig. 1.1 b).
Secondary moments generally produce detrimental effects to slender compres-
sion members and so they must be accounted for in design. The nature and the
manner in which these secondary moments are incorporated in the analysis and
design of structural members in frames will be addressed in this and the following
chapters.
Beam-columns can be considered as the basic element of a structural frame.
Beams and columns are special cases of beam-columns. In the case of beams, the
p
; ~ -=::::;::;------
--.............::=----__..1=----r- p
Deflected configuration
(a) P - o Effect
1
2 CHAPTER 1
Deflected configuration
effect of axial force on the primary bending moments is negligible, but in the case
of beam-columns this effect may be such as to add significant additional moments
to the primary moments. At the present time, the design methodology for frames is
based essentially on the behavior of individual members that comprise the frame
rather than on the entire frame itself, i.e. the design is performed on a member level.
Thus the usual procedure for a frame design is first to carry out an elastic frame
analysis and to determine the axial force and bending moments on each member.
The design of each member is then carried out one at a time by using an ultimate
strength interaction equation that expresses a safe combination of axial force and
bending moments that the member can sustain. A detailed discussion of these
beam-column design interaction equations is given in Chapter 2. At the present
time, it suffices to say that the design of a structural frame is merely a selection of
members that comprise the frame. Realistically, a frame should be designed based
on an interactive system behavior rather than on a collection of individual
behaviors of beam-columns. Nevertheless, this will require a considerable change in
the concept and philosophy of structural design which is currently not feasible;
however, with the present rapid infusion of computing into structural engineering,
such a challenge may be achieved in the next decade.
Although a structural frame is designed at the member level, the member is by
no means treated as a totally isolated element. It is easily conceivable that an
isolated member behaves rather differently from a framed member because of the
interaction effect that always exists among adjacent members of a frame. To account
for this interaction approximately, the concept of effective length has been widely
ANALYSIS OF BEAM-COLUMNS 3
Inelastic lateral
torsional buckling
Elastic lateral
torsional buckling
Another form of failure which may occur in the member is local buckling. Local
buckling is the buckling of component elements of the cross-section. An element
with a high width to thickness ratio is very susceptible to local buckling. Like lateral
torsional buckling, local buckling may occur in the elastic or inelastic range. The
effect of local buckling is to reduce the load-carrying capacity of the cross-section.
Local buckling is accounted for in design by the use of a reduced width for the
buckled element.
The analysis of beam-columns is an inherent complicated problem. To trace a
load-deflection curve, like that shown in Fig. 1.2, one must resort to some type of
approximate or numerical technique. This is because the differential equations
governing the inelastic behavior of a beam-column are highly nonlinear even for the
simplest loading case (Chen and Atsuta, 1976, 1977). Although a number of
methods are available for the analysis of beam-columns, they all involve some form
of simplifying assumptions to make the problem tractable. In Sections 1.5 and 1.6,
two such methods are described. The first method was developed by Newmark
(1943) for the analysis of the in-plane bending response of beam-columns. The
second method was deve1oped by Cranston ( 1983) for the ana1ysis of the biaxia1
bending behavior of beam-columns.
P= LazdA (1.4.1)
M = LyazdA (1.4.2)
X ---f---,
(a) (b)
In the above equations, az is the normal stress and A is the area of the cross-section.
Depending on the yield pattern of the cross-section, three possible stress
distributions can be identified. They are shown in Fig. 1.3b and are referred to as
elastic, primary plastic, and secondary plastic stress distributions. The resulting
nondimensional moment-curvature-thrust relationships that correspond to these
stress distributions are as follows.
For 0 ... t/1 ... (1 - p)
m = t/1 (1.4.3a)
For (1 - p) ... cp ... 11(1 - p)
2(1-p)3/2
m = 3(1 - p)- ·Jt/1 (1.4.3b)
For t/1 ?:: 11( 1 - p)
1.6
t/> = .!!?.._
"'y
Fig. 1.4 M-ff>-P curves for rectangular cross-sections
1 The strain distribution across the cross-section is linear (i.e. plane sections before
bending remain plane after bending).
2 The axial force acts through the centroid of the cross-section.
3 Shear deformation is neglected.
4 The shape of the cross-section remains unchanged throughout the course of the
loading.
Suppose an initial load vector {P0 , M 0 }T that corresponds to a known initial
deformation vector {e0 , <1> 0 }T is known, a new load vector {P1, MdT that corresponds
to a new deformation vector {e 1 , <l>dT can be written as
>
Fig. 1.5 Arbitrary cross-section subjected to an axial force and a bending moment
ANALYSIS OF BEAM-COLUMNS 9
and
(1.4.9)
In the above equations, P, M, e and ~ are the axial thrust, bending moment, axial
strain and curvature, respectively.
A relationship between the incremental load vector {M, L\M}T and the
incremental deformation vector {.1e, .1~}T can be formulated as follows. Knowing
dP= aP de+ aP d~
ae ~ (1.4.10)
aM aM (1.4.11)
ae
dM=-de+-d~
~
(1.4.12)
(1.4.13)
or
(1.4.14)
The elements of the section incremental stiffness matrix S can be obtained using
finite differences. For example
aP M P2-P1 (1.4.15)
ae~ .1e=~
If central difference is used, then P 1 is evaluated from a strain (e- ~.1e) and P 2 is
evaluated from a strain (e + ~.1e), where e is the current state of strain and .1e is an
assumed axial strain increment. In a numerical implementation, P 1 and P 2 are
1Q CHAPTER 1
(1.4.16)
(1.4.17)
where y is the distance measured from the centroidal axis of the cross-section to the
centroid of the elemental area.
A similar approach can be applied to obtain values for the off-diagonal terms
aPia <I> and aM/ae. Once S is obtained, the following procedure can be used to
trace the M-<1>-P relationship of a general cross-section. The complete M-<1>-P
curve is traced in a step-by-step manner. The basic premise is that once an initial
load vector {P0 , M 0 }T that corresponds to an initial deformation vector {e0 , <1> 0}T
is known, a subsequent load vector {P1, MdT for a deformation vector {e 1, <l>dT
can be obtained by following the iterative steps outlined.
1 Assume an incremental load vector {AP, dM} T.
2 Calculate the section incremental stiffness matrix S as described above.
3 Evaluate the incremental deformation vector using Eq. (1.4.14).
4 Calculate the deformation vector {e 1 , <I> 1}T using Eq. (1.4.9).
5 Calculate the load vector {P 1 , M 1}T that corresponds to the deformation vector
calculated in step 4 by numerically integrating Eqs. ( 1.4.1) and ( 1.4.2) in
conjunction with a known stress-strain relationship.
6 Calculate {P1, M 1}T using Eq. (1.4.8).
7 Compare the load vector obtained in step 5 with that obtained in step 6. If the
difference between them is within an acceptable tolerance, the solution is said to have
converged. Otherwise, the difference between these two load vectors {AP, dM}T is
used as the new incremental load vector. Steps 2 through 7 are repeated until
convergence is achieved.
ANALYSIS OF BEAM-COLUMNS 11
By using the above procedure, M-cf>-P curves can be generated which can
then be utilized in a beam-column analysis. To improve efficiency, it is advan-
tageous to express these curves in the form of mathematical expressions (Chen,
1971) analogous to the analytical expressions for the M-cf>-P relationship of a
rectangular section. Recall that for a rectangular section, three stress distributions
can be identified (Fig. 1.3b) which correspond to three regimes: elastic (when no
fiber has yielded), primary plastic (when the fibers in the compression zone of the
cross-section have yielded), and secondary plastic (when fibers in both the
compression and tension zones of the cross-section have yielded). These regimes
are shown schematically for an M-cf>-P curve in Fig. 1.6. The general mathe-
matical expressions for each regime can be written as follows.
For 0 :o;;:; r/> :o;;:; r/> 1
m=ar/> (1.4.18)
For r/> 1 :o;;:; r/> :o;;:; r/> 2
(1.4.19)
(1.4.20)
mpc --------------------------------
Secondary
plastic
m,
J p = constant j
<1>,
Fig. 1.6 Three regimes of moment-curvature curve for a constant axial force
12 CHAPTER I
In the above equations, a, b, c and fare curve-fitting constants, m and r/> are the
nondimensional moment and curvature as defined in Eq. (1.4.4), and mpc is the
limit moment as shown in Fig. 1.6.
The constants a, b, c and fare determined in terms of m 1 , m 2 , mpco r/> 1 and r/> 2
from the continuity conditions between adjacent regimes of the moment-curvature-
thrust curve. The continuity conditions are as follows.
At the primary yield point
(1.4.21)
( 1.4.22)
The solution of Eqs. ( 1.4.21) and ( 1.4.22) yields the following expressions for a, b,
c and/
m,
a=- (1.4.23)
rP!
m2..Jr/>z- m,.../rjJ, (1.4.24)
b=
.../rf>z- .../rf>t
mz-m, (1.4.25)
c-- 1/...jrjJ, - 11.../r/>2
According to a study by Chen (1971 ), the values of m" m 2, mpc• r/> 1 and r/> 2 are
practically independent of the size of the section. Therefore, for a given cross-
sectional shape, only one set of expressions is needed to describe its m-rf>-p
relationship. Approximate expressions for m 1 , m 2 , mpc• r/> 1 and r/> 2 for wide flange
cross-sections are summarized in Table 1.1. Similar expressions for square and
circular tubes can be found in the books by Chen and Atsuta (1976) and Chen and
Han (1985). Figures 1.7, 1.8 and 1.9 show a comparison of the m-rf>-p curves
obtained analytically or numerically with the curve obtained using Eqs. ( 1.4.18) to
(1.4.20). Good approximations are observed. For fabricated cylindrical tubes
commonly used in offshore structures, additional studies on the cross-section
behavior, including the effects of hydrostatic pressure, cross-sectional imperfections,
residual stresses, local buckling and cyclic loading, have been reported recently by
Chen and Sohal ( 1988).
ANALYSIS OF BEAM-COLUMNS 13
Table 1.1 Approximate m-<J>-p expressions for wide flange sections (Chen and Atsuta, 1976)
Strong axis bending, no residual stress
For all p
m1 = 1-p
</>1 = 1-p
1.2
p=P/P. = 0
y 0.1
0.2
1.0
0.8 0.4
M
m=-
My
0.6
0.6
0.4
0.8
----Actual
-Approx. -_I-wsx31
0 23 4 5
l/J = cJ>
Cl>y
Fig. 1.7 Comparison of actual and approximate m-¢-p curves (I-section without residual stresses bent
about the strong axis)
1.4.2, this numerical approach requires the discretization of the cross-section into
small elemental areas. The cumulative effect of the entire cross-section is
obtained by summing the effects of all the elements that comprise the cross-
section.
Figure 1.10 shows a cross-section subjected to an axial force P and moments Mx
and My about the x- and y-axis, respectively. By assuming that plane sections
remain plane, the normal strain of element i can be expressed as a linear function
of x andy as
(1.4.27)
where e0 is the strain at the centroid of the cross-section, <l>x is the curvature with
respect to the x-axis, <l>y is the curvature with respect to the y-axis, and er is the
residual strain. Because of the nonlinear nature of the problem, it is convenient to
express the strain in Eq. (1.4.27) in incremental form as
(1.4.28)
ANALYSIS OF BEAM-COLUMNS 15
1.2
0.2
1.0
0.8 0.4
m=~
My
0.6
0.6
0.4
0.8
l
0.30y
----Actual
- --W8x31
--Approx.
0 23 4 5
tP=cp
cpy
Fig. 1.8 Comparison of actual and approximate m-rp-p curves (!-section with residual stresses bent about
the strong axis)
Note that L1er = 0 since the residual strain is a constant for a given element. The
corresponding stress increment is
( 1.4.29)
where Eeff(i) is the effective modulus of the i-th element which can be taken as the
slope of the known a-e curve for the case of a uniaxial state of stress or the slope
of the effective stress-strain (Chen and Han, 1988) for the case of a biaxial state of
stress that exists in the element.
The axial force and bending moment increments are related to the stress
increments by
(1.4.32)
16 CHAPTER 1
p=P/Py= 0
0.4
0.6
0.8
Actual
Approx.
3 4
Fig. 1.9 Comparison of actual and approximate m-rf>-p curves (tubular section without residual stresses,
MIMP = (1t/4)(MIM,))
In the above equations, the subscript i refers to the i-th element of the cross-section
and the summation is carried out over the entire cross-section. By substitution of
Eq. (1.4.29) into Eqs. (1.4.30) to (1.4.32), we have, after rearrangement
( 1.4.33)
where
sll = IEetf(i)Ai
Fig. 1.10 General cross-section subjected to an axial force and biaxial bending moments
(1.4.34)
In the above equation, M' is the incremental generalized stress vector, M is the
incremental generalized strain vector, and S is the section tangent stiffness matrix.
Once S which corresponds to a given state of stress (or strain) is evaluated, it is a
simple matter to find the path of generalized strain X for a given path of generalized
stress F through a step-by-step incremental calculation in conjunction with an
iteration procedure. The procedure is described below.
18 CHAPTER I
1 For given incremental generalized stress vector M, evaluate the section tangent
stiffness matrix S based on the state of stresses at the beginning of the increment.
2 Calculate the incremental generalized strain vector Musing Eq. (1.4.36).
3 Obtain total generalized strains by adding the incremental strains M to the
current state of generalized strains.
4 Calculate the total generalized stresses using the known stress-strain relationship
and the following relationships
(1.4.37)
(1.4.38)
(1.4.39)
5 Alternatively, calculate the total generalized stress vector by adding the incremen-
tal generalized stresses M in step 1 to the current state of generalized stresses.
6 Compare the total generalized stresses calculated in step 4 and step 5. If the
discrepancy is negligible, the solution is said to have converged. Otherwise, their
difference is used as the incremental generalized stress and steps 1 to 6 are repeated
until convergence.
Once the M-4>-P relationship is known, a beam-column analysis can be carried
out. In this chapter, two rather efficient and powerful numerical methods for
beam-column analysis are presented. They are the Newmark method and the
Cranston method.
Ms
~-~~~~-·-··· ..._.;
... _.. ~~p - X
vk
(1.5.1)
where M 1 is the primary moment due to in-span loads, end moments, and reactions.
Pvk is the secondary moment due to the P-t5 effect.
4 Using the known moment-curvature-thrust relationship for the cross-section,
compute the curvature <l>k at each station.
5 Evaluate a new set of deflections at the stations by the conjugate beam method.
This involves the following.
(a) Assume a curvature distribution between the stations. A linear or quadratic
<I> distribution as shown in Fig. 1.12 can be used as an approximation.
(b) Calculate the equivalent nodal loads using the formulas given in the figure.
The formulas in Figs. 1.12a and 1.12c should be used if one of the stations is
an end station or if there is an abrupt change in curvature due to a sudden
change in M or EI at one of the stations.
(c) Calculate the shear and moment of the conjugate beam (which are
equivalent to the slope and deflection of the real beam) using the calculated
nodal loads as conjugate beam loads.
6 Compare the deflections calculated in step 5 with the assumed deflections in
step 1. If the discrepancy is negligible, a solution is said to have been obtained.
Otherwise, use the calculated deflections as the new set of assumed deflections and
repeat steps 3 to 5 until convergence.
The above procedure must be repeated for every increment of applied load
in order to trace the load-deflection response of the member. The peak point
20 CHAPTER I
--~--- ..................
....... .,
I
ri-1 ...__ _ _ _ _ _ _ _ _ _ _ _ _ _ - .JI ri+1
(c) (d)
An illustrative example
Using the Newmark method and the moment-curvature-thrust relationship given
below, determine the deflection at midspan of an initially crooked beam-column
shown in Fig. l.l3a for the following two loading cases: (a) P = 0. 5Py and M = 0.4My
(Fig. l.l3b) and (b)P=0.5Py and M=0.8My (Fig. 1.13c). Assume ay=0.001E.
Solution
For PIPy = 0.5, m = M!My and if>= <1>/<l>y, the moment-curvature-thrust relationship
is given by
if>= m, m,;;; 0.5
p-E-~--------'-------- l+p
r-- --·· . .-___,
-x
! L
Y Deflected shape
~
Cross section
0}=0.1L
+ .,_._liii=:::::'=:;~t~/~~~===='
0.4My V0 = (0.001L) sin (nx)/L 0.4My
0.5Py ;:;;;;;:;::::::o-
.jii,..;r.j-- 0.5Py
0.5Py +0 O.BMy
~--J_--- ~--~
2 3 ------'
O.BMy
4~ 0.5Py
1
cf> = .j(2.25 -2m) ' 1.0,;;; m < 1.125
The basic quantities needed in the following calculations for the rectangular
beam-column shown in Fig. 1.13a are
Py = Aay = bhay
bh 2 h 0.1L PyL
My =Say =(lay =()PY =-6-Py = 60
(~ r<l>y 8~0 =
In the following calculations (Tables 1.2 and 1.3), the assumed additional
deflections vk were taken as the first-order deflections of the member. The
equivalent nodal loads Rk were calculated using the equations in Fig. 1.12c for
stations 0 and 4 with r; _ 1 = - r; + 1 , and using the equation in Fig. 1.12b for
stations 1, 2 and 3.
Table 1.2 Case (a). For P = 0.5P, and M = 0.4JI, with n = 4 (Fig. 1.13b)
Station Common
factor
0 2 3 4
Primary moment
M, 0.4 0.4 0.4 0.4 0.4 My
Initial imperfection
Vok 0 0.0007 0.001 0.0007 0 L
Cycle I calculations
Assumed additional
deflections vk 0 0.00075 0.001 0.00075 0 L
P-J moment
P(vok + vk) 0 0.000725 0.001 0.000725 0 PyL
Change common factor 0 0.0435 0.060 0.0435 0 M,
Total moment
M, + P(v 0k + vk) 0.4 0.4435 0.460 0.4435 0.4 M,
M-cJ>-P relationship
q,k 0.4 0.4435 0.460 0.4435 0.4 cJ>y
Calculated additional
deflections v~) 0 0.00084 0.00112 0.00084 0 L
Cycle 2 calculations
Assumed additional
deflections v k 0 0.00084 0.00112 0.00084 0 L
P-J moment
P(vok + vk) 0 0.00077 0.00106 0.00077 0 PyL
Change common factor 0 0.0462 0.0636 0.0462 0 My
Total moment
M 1 + P(v 0k + vk) 0.4 0.4462 0.4636 0.4462 0 M,
M-cJ>-P relationship
q,k 0.4 0.4462 0.4636 0.4462 0.4 cJ>y
24 CHAPTER I
Calculated additional
deflections v~> 0 0.00084 0.00112 0.00084 0 L
Since vf> = vk at the second cycle, the solution has converged. The total deflection at midspan is
Vm;dspan = Voz + Vz = O.OOIL + 0.00112L = 0.00212£
given first. A more detailed discussion follows. A full description can be found in
the 1983 report by Cranston.
1 Divide the member into n segments by (n + 1) stations. Denote the length of the
k-th segment as lk.
2 Discretize the cross-section into small elemental areas. For the i-th element,
denote its area as A;, normal strain as ez; and normal stress as azi·
3 Define a stress-strain relationship for the material.
4 Assume a set of displacements for the stations. For the end stations, the end
slopes are also assumed.
5 Perform a cross-section analysis (see Section 1.6.1).
6 Adjust the end slopes and applied loads until equilibrium and compatibility are
satisfied at the ends and at a control station.
7 Calculate the deflections at other stations.
8 Compare the calculated deflections with the assumed deflections. If the two sets
of deflections agree within appropriate limits, a valid solution is said to have been
obtained.
Figure 1.14 shows the beam-column under investigation. The beam-column is
held by restraining systems at ends A and B (stations 0 and n). The member is
prevented from sway movement, but is allowed to rotate at the ends. The member
may possess initial deformations u0 k and v0 k (k = 1 to n - 1) in the x and y
directions, respectively. The loadings consist of an axial load AP acting at the
centroid of the cross-section and end moments llixA• AMyA• AMxs and AMyB acting
as shown. A. is an analysis load factor. The lateral displacements of the member
under loads are denoted by uk and vk (k = 1 to n- 1) measured from line AB to the
centroid of the cross-section in the x and y directions, respectively. The end slopes
are denoted by exA• eyA• exB and eyB· All quantities are taken to be positive as shown
in the figure. Curvatures are positive when the slope is decreasing in the direction
ANALYSIS OF BEAM-COLUMNS 25
Table 1.3 Case (b). For P = 0.5Py and M = 0.8My with n = 4 (Fig. 1.13c)
Station Common
factor
0 2 3 4
Primary moment
Ml 0.8 0.8 0.8 0.8 0.8 My
Initial imperfection
Vok 0 0.0007 0.001 0.0007 0 L
Cycle I calculations
Assumed additional
deflections vk 0 0.0015 0.002 0.0015 0 L
P-6 moment
P(vok + vk) 0 0.0011 0.0015 0.0011 0 PyL
Change common factor 0 0.066 0.090 0.066 0 My
Total moment
M1 + P(v 0k + vk) 0.8 0.866 0.890 0.866 0.8 My
M-<JJ-P relationship
q,k 1.020 1.244 1.344 1.244 1.020 q,y
Calculated additional
deflections v~l 0 0.00235 0.00317 0.00235 0 L
Cycle 2 calculations
Assumed additional
deflections vk 0 0.00235 0.00317 0.00235 0 L
P-6 moment
P(vok + vk) 0 0.00153 0.00209 0.00153 0 PyL
Change common factor 0 0.0918 0.125 0.0918 0 My
Total moment
M 1 + P(v 0k + v~) 0.8 0.892 0.925 0.892 0.8 My
M-<JJ-P relationship
q,k 1.020 1.353 1.512 1.353 1.020 q,y
26 CHAPTER I
Station Common
factor
0 2 3 4
Calculated additional
deflections v~l 0 0.00257 0.00348 0.00257 0 L
Cycle 3 calculations
Assumed additional
deflections vk 0 0.00257 0.00348 0.00257 0 L
P-o moment
P(vok + vk) 0 0.00164 0.00224 0.00164 0 PyL
Change common factor 0 0.0984 0.134 0.0984 0 My
Total moment
M, + P(v 0k + vk) 0.8 0.898 0.934 0.898 0.8 My
M-!1>-P relationship
!l>k 1.020 1.380 1.561 1.380 1.020 !l>y
Calculated additional
deflections v~l 0 0.00263 0.00357 0.00263 0 L
Cycle 4 calculations
Assumed additional
deflections vk 0 0.00263 0.00357 0.00263 0 L
P-o moment
P(vok + vk) 0 0.00167 0.00229 0.00167 0 PyL
Change common factor 0 0.100 0.137 0.100 0 My
Total moment
M1 + P(v 0k + vk) 0.8 0.900 0.937 0.900 0.8 My
M-!1>-P relationship
!l>k 1.020 1.389 1.577 1.389 1.020 !l>y
ANALYSIS OF BEAM-COLUMNS 27
Station Common
----------------------factor
0 2 3 4
Conjugate beam method
Rk 0.657 1.359 1.514 1.359 0.657 (L/4)ct>y
(Jk 2.116 0.757 -0.757 -2.116 (L/4)ct>y
vk 0 2.116 2.873 2.116 0 (L/4fct>y
Calculated additional
deflections v~ 4 ) 0 0.00265 0.00359 0.00265 0 L
Since vi4)"" vk at the fourth cycle, the solution is considered to have converged. The total deflection at
midspan is
Vm;dspan = Vo2 + v2 = 0.001L + 0.00359L = 0.00459L
More cycles are required to obtain a converged solution for the case of M = 0.8My because the
beam-column is stressed into the primary plastic range, whereas for the case of M = 0.4MY the
beam-column is fully elastic.
(1.6.1)
Equation ( 1.6.1) is identical to Eq. ( 1.4.27) except for the sign of the term <l>yXi· This
is due to the difference in sign convention used. By assuming values for e0 , <l>x and
<l>y, ezi can be calculated, from which azi and the tangent stiffness Eti can be
obtained by using the known stress-strain behavior of the material, allowing for
unloading if necessary (Fig. 1.15). The cross-sectional force and bending moments
at the k-th station can be calculated using the following equations
(1.6.2)
(1.6.3)
28 CHAPTER I
A.P
(1.6.4)
where lxi is the second moment of the element about the element centroidal axis
parallel to the x-axis and ly; is the second moment of the element about the element
centroidal axis parallel to the y-axis. The inclusion of the terms involving I xi and ly;
in calculating the bending moment accounts for the fact that the normal stress azi
is not constant over the element. If the cross-section is fully elastic, Eqs. (1.6.3) and
(1.6.4) will give the exact bending moments. If the cross-section becomes inelastic,
the use of the terms improves the accuracy of the calculation and enables the
ANALYSIS OF BEAM-COLUMNS 29
Loading
Strain
where
MrxA = AMxA-MRxA• MryA = AMyA-MRyA
MrxB = AMxB- MRxB• MryB = AMyB- MRyB
are the resultant moments at the A-th and B-th ends of the member about the x- and
y-axis, respectively. MR is the end restraint moment delivered by the restraining
systems at the ends of the member. They are derived from the MR-e relationships
30 CHAPTER I
(1.6.8)
( 1.6.9)
aP aP aP
aeo act> X act>y
{:}· aMx
aeo
aMy
aeo
aMx aMx
a«l>x act>y
aMy aMy
a«l>x a«l>y
r~} ~<l>x
~<l>y
(1.6.11)
(1.6.14b)
(1.6.14c)
ANALYSIS OF BEAM-COLUMNS 31
c5<I>x produces a stress change in element i equal to Y; Eti c5<I>x and a bending
moment change equal to lx; Et; c5<I>x about the x-axis. Thus
(1.6.15b)
: -:::=.LA;x;Eti ( 1.6.16a)
y
(1.6.16b)
The deflections are calculated starting at end A. Intermediate values for slopes
and deflections, denoted by (}' and v' respectively, are calculated by assuming that
the end slope at A equals the assumed value (OxA)a. v8 calculated on this basis will
not equal zero as expected and corrections to the intermediate values for slopes and
deflections are made afterwards.
The intermediate values for slopes and deflections are calculated by assuming
that the curvature varies linearly within each segment as
o;A =( OxA)a + 110xa (1.6.18a)
vA.=O (1.6.18b)
o;k = (}~k-1) + [(<l>x(k-1) + <l>xk)h]12 + f1(}xk fork= 1 ton (1.6.18c)
vk = vk-1 + (}~k-1A + [lk 2(2<1>x(k- I)+ <l>xk)]l6 fork= 1 ton (1.6.18d)
0;B =- 0;n (1.6.18e)
vB = v~ (1.6.18£)
The corrected values for the end slopes and deflections are given by
( OxA}c = ( OxA}a - VB IL ( 1.6.19a)
( Oxa)c = ( Oxa)a + VB IL (1.6.19b)
(1.6.19c)
The above procedure is also applied to calculate the rotation about the y-axis
and deflection in the x-axis.
Normally, the calculated values for the end slopes and the deflection at the
control point do not agree with the assumed values in the first iteration. As a result,
adjustments to the assumed values must be made. This is carried out as follows.
Define
ANALYSIS OF BEAM-COLUMNS 33
(1.6.20a)
/::;.(JxA ={ (JxA)c- {(JxA)a (1.6.20b)
f1(JxB =( Oxa)c- ( Oxs)a (1.6.20c)
f1(JyA = { OyA)c -{ OyA)a (1.6.20d)
118ya =( Oya)c -( Oya)a (1.6.20e)
where (C)c is the calculated deflection at the control station in the control direction
and (C)a is the assumed deflection at the control station in the control direction. It
should be noted that the control deflection can be chosen in either direction. Also,
both the control station and direction can be changed during the analysis. The
modifications to the assumed quantities are obtained by solving the following
matrix
a(I1C) a(I1C) a(I1C) a(I1C) a(!1C)
a;:;:- iJ((JxA)a CJ( Oxs)a iJ((JyA)a iJ(Oys)a
a(f1(JxA) a(f1(JxA) a(f1(JxA) a(f1(JxA) a(f1(JxA) 11Aa
11C
----ax;- a((JxA)a a(Oxa)a a(OyA)a a(Oya)a (MJxA)a
f1(JxA
a(f18xa) a(f18xa) a(f18xa)a(f18xa) a(f18xa)
(118xa)a
f1(JxB ----ax;- a(OxA)a a(Oxs)a a( OyA)a a( Oya)a
(f1(JyA)a
f1(JyA
a(f18yA) a(f10yA) a(f10yA) CJ(f1(JyA) a(f1(JyA)
f1(JyB (!18ya)a
~ a(OxA)a a( Oxs)a a( OyA)a a( Oya)a
a(f18ya) a(f18ya) a(f18ya) CJ(f18ya) a(f18ya)
~ a(OxA)a a(Oxa)a a( OyA)a a( Oya)a
(1.6.21)
Symbolically, Eq. (1.6.21) can be written as
!1U =A 11Ua (1.6.22)
from which
11Ua =A- 1 11U (1.6.23)
The terms in matrix A are obtained by considering the effects of unit changes in each
of the assumed quantities in tum on the calculated deflected shape, assuming the
section incremental stiffness matrix S in Eq. ( 1.6.12) to be constant. This process is
described below.
(1.6.24b)
(1.6.24c)
From Eq. ( 1.6.13), the change in axial strain and curvatures can be evaluated
(1.6.25)
Equation (1.6.25) is applied to all stations and the curvatures are then
integrated numerically to obtain new slopes and deflections. Using these values and
by differentiating Eq. ( 1.6.20), we obtain
( 1.6.26a)
(1.6.26b)
(1.6.26d)
(1.6.26e)
(1.6.27b)
(1.6.27c)
where oMRxA/oOxAis the slope of the MRxA-OxA relationship of the end restraint
evaluated at OxA = ( OxA>a·
Curvatures are calculated and integrated as for the case of unit change in Aa and
the second column of matrix A is obtained from
ANALYSIS OF BEAM-COLUMNS 35
a(~C) a(C)c
a(e xA)a = a(e xA)a = (C)c I(llx.>•• I ( 1.6.28a)
(1.6.28b)
( 1.6.28c)
(1.6.28d)
(1.6.28e)
The procedure follows that for (OxA)a as described above and will not be repeated
here.
Once all elements of the A matrix in Eq. (1.6.21) are calculated, Eq. (1.6.23) can be
used to calculate the modifications ~Ua. These modifications are added to the
assumed values to obtain new assumed values and the procedure is repeated until
convergence.
When convergence is achieved in stage 1, stage 2 can commence. In this stage,
the calculated deflections at other stations are compared with the assumed
deflections. If disagreement exists, the assumed deflections are replaced by the
calculated deflections. This procedure is repeated until convergence is achieved.
When a solution satisfying both stages 1 and 2 exists, a valid solution is said to have
been obtained.
deflection for the current cycle. A realistic set of assumed values qi for a given
variable can be calculated using the following parabolic extrapolation equation
( 1.6.29)
where qi _ I> qi _ 2 and qi _ 3 are the solutions of the variable obtained in cycles (i- 1),
(i- 2) and (i- 3), respectively, and
(1.6.30a)
(1.6.30b)
qi
,...,:
---------------------------------------------------------""
_, '
qi-1
/
~
:'
:
'''
''
'
'''
''
q i-3 ----------
~-2 p
References
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Engineering Division, ASCE, 97, ST2; 529-544.
Chen, W.F. and Atsuta, T. (1976) Theory of Beam-Columns, Vol. 1: In-Plane Behavior and Design,
McGraw-Hill, New York, NY, 513 pp.
Chen, W.F. and Atsuta, T. (1977) Theory of Beam-Columns, Vol. 2: Space Behavior and Design,
McGraw-Hill, New York, NY, 732 pp.
38 CHAPTER I
Chen, W.F. and Han, D.J. ( 1985) Tubular Members in Offshore Structures, Pitman, Marshfield, MA,
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ACT Structural Journal, 86, 2; 163-167.
Galambos, T.V. (1968) Structural Members and Frames, Prentice-Hall, Englewood Cliffs, NJ, 373 pp.
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References