Derivation of Key Characteristics of Columns in Frames with Sway

JOSTEIN HELLESLAND

Mechanics Section, Department of Mathematics

University of Oslo, Oslo

NORWAY

Abstract: - The paper paper examines columns of elastic frames that are able to displace laterally under axial

load. Columns on the same level (story) of such frames may contribute to lateral support (stability) or require

lateral support (supported columns). In the former case, maximum moments will occur at column ends, but

may occur between column ends in the latter case. The column response, including the formation of moments

and shears, is often complicated, often calling for approximate analysis methods and information that presently

may not be readily available. Main attention of the paper is paid to providing results that will be helpful in

providing, in a rather simple manner, improved understanding of column mechanics, and that may be helpful also

as a supplement to full second-order analyses. Towards this goal, the major objective of the paper is threefold:

(i) to identify characteristic points, or behavioral“landmarks” in the axial load-moment solution space, through

the study of rotationally restrained elastic columns and two-column panels with sidesway; (ii) to derive simple,

novel, closed-form expressions for these, thereby providing a tool simplifying the establishment of the variation

of end and maximum moments versus axial load; (iii) to identify to what extent isolated (single) column analyses

may adequately represent horizontally interacting columns, and potential unwinding, in frames.

Key-Words: - Structural mechanics, elasticity, second-order analysis, stability, framed column behavior,

sidesway.

5HFHLYHG0DUFK5HYLVHG$XJXVW$FFHSWHG6HSWHPEHU3XEOLVKHG1RYHPEHU

1 Introduction

1.1 General

This paper examines columns of elastic frames that

are able to displace laterally under load. Columns in

such a frame, or story of a frame, may contribute to

the lateral stiffness (support), or they are may require

lateral support by the others to maintain lateral story

stability. Whether a column falls into one or the other

category depends on its relative axial load level. In

cases with no transverse load between column ends,

the maximum moment will occur at the end of a sup-

porting column, but may occur between the ends of

a supported column. Computer programs that ac-

count for geometric nonlinear (second-order) effects

are available for examining such problems. The com-

plexity of such analysis methods often obscures the

understanding of the mechanics of individual mem-

bers in a frame. In this regard, simplified methods are

often helpful as a complement.

There is an extensive literature on approximate

maximum design moment calculations for columns

of reinforced concrete or steel frames. Much of such

work has been incorporated into relevant design codes

such as, [1], [2], [3], [4]. In such provisions, columns

are either considered braced (laterally supported) or

unbraced (laterally supporting).

In contrast, easy-to-apply methods allowing pre-

diction of not only more correct maximum moments,

but also column end moments (of importance for con-

nected foundations and beams, etc.) and shears have,

apart from some isolated cases, such as, [5], seem-

ingly been almost non-existent until more recently,

[6].

Approximate methods are generally based on anal-

ysis of columns isolated from the frames of which

they are a part by incorporating appropriate end re-

straints at column ends. These are supposed to re-

flect the interaction with the surrounding structure.

Well known, and rationally justified restraint assump-

tions for regular frames, are generally adopted also for

more irregular frames. Although important, the au-

thor has not come across readily available literature

on this subject.

Based on engineering practice and research over

several years, the author has felt a need for investi-

gating this end restraint issue, and has has long rec-

ognized the potential usefulness of approaches that

would allow rather easily the establishment of the of-

ten complex moment formation in columns for any

load level. This has been the driving force behind the

present paper, which is based on a more detailed re-

port, [7].

1.2 Object and Scope

The major objective of the work is threefold. Based

on a study over the full range of axial loads of

the mechanics of the response of rotationally re-

strained elastic columns and two-column panels with

sidesway, it has been to:

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

255

Volume 18, 2023

(i) identify characteristic points, or behavioral “land-

marks”, in the axial load-moment solution space;

(ii) establish simple, novel, closed-form expressions

defining such characteristic points, and thereby

allowing easy establishment of the overall variation

of end and maximum moments versus axial load;

(iii) identify to what extent isolated (single) column

analyses may adequately represent horizontally

interacting columns, and potential unwinding, in

frames.

Apart from being of use in providing improved un-

derstanding of column mechanics, and the teaching of

the subject, such results may also be a helpful supple-

ment to full, second-order structural analyses, in the

assessment of the often complicated column response.

Towards the goals of the paper, second-order

(small rotations) theory is developed in sufficient de-

tail to allow the establishment of closed-form ex-

pressions of key characteristics, and to obtain ex-

act moment-load relationships for isolated, restrained

columns. The scope is limited to linear elastic

columns with uniform axial load and uniform sec-

tional stiffness.

2 Supporting an Supported Sway

Columns

A brief review of the overall frame sidesway mechan-

ics is in order to provide the proper context for sub-

sequent sections. Fig. 1(a) shows a laterally loaded

frame that is not braced against sidesway, i.e, it is free

to sway. In the absence of axial forces in the columns,

the lateral load (H= 4) gives rise to a first-order

sway displacement ∆0, equal at all column tops when

axial beam deformations are neglected. The lateral

load is resisted by column shears (V0) that are propor-

tional to the relative lateral stiffness of the columns.

If this stiffness is the same in all columns, first-order

shears of V0=H/4 = 1.0result in each of the

columns

Under the additional action of axial forces, caused

by vertical loads on the frame, Fig. 1(b), the displace-

ment increases to ∆ = Bs∆0, where Bsis a sway dis-

placement magnification factor that reflects the global

(story, or system) second-order effects of axial loads.

Had the individual columns, having different axial

load levels in the example, been free to sway inde-

pendently of each other, some would sway more (the

more flexible ones) and some less (the stiffer ones)

than the the overall frame. Since the columns are in-

terconnected and forced to act together, a redistribu-

tion of shears will take place from the more flexible

∆B=S∆0

2.14 0 −0.54

1234

V=2.40

αs=0.1

1 2 3

V=1

∆

0.2 1.0

111

1.2

H=4

(a) Laterally loaded sway frame −− 1st order analysis

(b) Laterally loaded sway frame −− 2nd order analysis

Figure 1: Effects of axial loads on column shears in multibay

sway frame: (a) First and (b) second-order analysis (Bs= 2.67).

ones (column 3 and 4 in the example) to the stiffer

ones (column 1 and 2).

For the vertical loads in the figure, given nondi-

mensionally in terms of the free-sway load index

αs(= N/Ncs)of each column, the redistributed

shears are those shown in Fig. 1(b) (calculated using

Eq. (31), to be presented later).

The distinction between (a) laterally “support-

ing sway columns” (“bracing columns”, “leaned-to

columns”), as illustrated by column 1 and 2 in the ex-

ample, and (b) “supported columns” (“braced”, “lean-

ing columns”), illustrated by column 4, is found to be

useful. A supported column may be visualized as one

being allowed to undergo a given sidesway and then

braced against further sway by the rest of the frame.

3 Second-order Analysis

3.1 General

Theoretical results relevant for multibay frames with

sidesway will be obtained both for panel frames and

for individual frame columns considered in isolation

from the frame. Two-column panel frames (Fig. 2)

allow horizontal interaction between columns to be

considered, whereas a simpler model consisting of a

single column (Fig. 3(a)) allow explicit expressions

to be derived for a number of column characteristics.

The latter is of considerable interest, and is pursued

below using an analysis tailor-made for this purpose.

Theoretical results for panel frames, and limitations of

single column models to represent such frames, will

be presented and discussed later.

The single column considered is shown in Fig.

3(a). It is laterally loaded (shear V), is initially

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

256

Volume 18, 2023

∆=S∆0

BLb

∆=S∆0

(a)

EI b1

EI1

b2 EI

EI b1

EI1

(b)

Figure 2: Initial and final deflection shapes of panel frames:

(a) Col. 2 is slightly stiffer than Col. 1; (b) Col. 2 is significantly

stiffer than Col. 1

∆=S∆0

N−M

−M

∆

∆0

LEI

θ1

(b)(a)

Figure 3: (a) Single column model and (b) sign convention.

straight, and has length L, uniform axial force Nand

section stiffness EI. Only in-plane bending is con-

sidered.

The rotational end restraints, reflecting the rota-

tional interaction with the rest of the frame of which

the column may be a part, can conveniently be repre-

sented by stiffness coefficients k1and k2(equal to the

moment required to give a unit rotation), respectively,

at ends 1 (top) and 2 (bottom). Nondimensionally,

they can be defined by

κj=kj

(EI/L)j= 1,2(1)

at an end j, or by other similar factors, such as the

well known Gfactors, [1], [2]. Unlike the κfactors

above, that are relative stiffness factors, the Gfactors

are relative, scaled flexibility factors. In a generalized

form they can be defined, [8], by

Gj=bo

(EI/L)

(= bo

κj

)j= 1,2(2)

where bois simply a reference (datum) factor by

which the relative restraint flexibilities are scaled.

Common datum values are bo= 6 and bo= 2 for reg-

ular unbraced and braced frames, respectively. These

values are tacitly implied in ACI, [1], and AISC, [2],

and similarly in Eurocodes, [3], [4]. For additional

discussion, see for instance, [9], [10], [11].

It should be noted, if not otherwise stated, that the

reference factor bo= 6 is used throughout this paper

in presentations in terms of Gfactors!

When the rotational restraints at a column end jis

provided by beams, the restraint stiffness at that end

can be written

kj=kb=bEIb/Lbj= 1,2(3)

Here, the summation is over all beams that are re-

straining the considered column end (joint), and bis

the bending stiffness coefficient. For beams with neg-

ligible axial forces, it is typically b= 3 for beams

pinned at the far end, b= 4 for beams fixed at the far

end and b= 2 for beams in symmetrical single cur-

vature bending. For beams in antisymmetrical double

curvature bending, b= 6.

Computations in the present paper are carried

out for the one-level cases defined in the figures

above. However, derived results are also applicable

to columns in multistory frames provided appropriate

rotational restraints are assigned to the column ends.

An approximate approach for multistory frames is to

consider the “vertical interaction” between columns

meeting at a joint by assigning a fraction of the to-

tal restraining beam stiffness sum kbat a joint to the

columns at the joint in proportion to their EI/Lval-

ues. This assumption leads to

kj=kb·(EI/L)/(EI/L)j= 1,2(4)

where EI/Lin the numerator is for the considered

column, and the summation is over all columns meet-

ing at the considered joint. This approach is generally

adopted by codes (e.g., [1], [2], [3], [4]) and is widely

covered in the literature (see for instance, [8], [10]).

It is in particular found to be acceptable for multilevel

frames with stiff beams, [5], [12]. By substituting Eq.

(4) into Eq. (2), the conventional Gfactor expression,

such as in [1] and [2], is obtained for b/bo= 1.

3.2 Basic Slope-Deflection Equations

For completeness, the major elements of the second-

order (small rotations) analysis for the single column

in Fig. 3 are developed in sufficient detail to allow

the establishment of closed form solutions in suitable

forms for use and discussion later in the paper.

The adopted sign convention is shown in Fig. 3(b).

Clockwise acting moments and rotations are defined

to be positive. Consequently, the moments shown in

Fig. 3, acting counter-clockwise, are negative (−M1

and −M2).

The member stiffness relationship, expressed by

the slope-deflection equations for a compression

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

257

Volume 18, 2023

member, can be derived from the differential equa-

tion (M=−EIv′′ ), and given by the matrix equation

below:

M1

M2=EI

LC S −(C+S)/L

S C −(C+S)/Lθ1

θ2

∆(5)

Here, the upper case Cand Sare the socalled stability

functions defined by

C=c2

c2−s2;S=s2

c2−s2(6)

where

c=1

pL21−pL

tan pL;s=1

pL2pL

sin pL −1

(7)

and

pL =LN/EI (= π√αE)(8)

In first-order theory, when the axial force is zero, C

and Stake on the familiar values of 4 and 2, respec-

tively. The formulation above was used in [10], [11].

The lower case cand sfunctions are flexibility fac-

tors; they are equal to 1/3rd of the corresponding func-

tions (ϕand ψ) in [13]. Alternative formulations are

also available ([9], [14], [15], [16]).

3.3 End Moments and Shears

From moment equilibrium at the joints (M1+k1θ1=

0,M2+k2θ2= 0), the end rotations can be expressed

by the end moments (e.g., θ1=−M1/k1). Substitu-

tion of these into Eq. (5) gives

b11 b12

b21 b22  M1

M2=−(C+S)EI∆/L2

−(C+S)EI∆/L2(9)

from which the total end moments according to

second-order theory can be solved for and expressed

M1=−(C+S) (b22 −b12)

b11b22 −b12b21 ·EI∆

L2(10)

M2=−(C+S) (b11 −b21)

b11b22 −b12b21 ·EI∆

L2(11)

Here, with the rotational stiffness of the end restraints

taken as k= 6(EI/L)/G(Eq. (2) with bo= 6),

b11 = 1+CG1

6;b12 =SG2

6;b21 =SG1

6;b22 = 1+CG2

When the determinant D=b11b22 −b12b21 ap-

proaches zero, the end moments M1and M2approach

infinity. This happens when the axial load approaches

the critical (instability) load. This limit is independent

of the lateral displacement ∆. The shear, found from

statics of the displaced column (V L =−(M1+M2+

N∆)), can in non-dimensional form be written

EI ∆/L3=−M1+M2

EI ∆/L2−(pL)2(12)

First-order results. It is convenient to present re-

sults in terms of first-order results, found by substitut-

ing C= 4 and S= 2 into Eqs. (10) and (11). The re-

sulting sway magnified first-order moments become

BsM02 =−6(G1+ 3)

2(G1+G2) + G1G2+ 3 ·EI∆

(13)

and

BsM01 =BsM02

G2+ 3

G1+ 3 (14)

and the corresponding sway magnified first-order

shear (from statics),

BsV0=−Bs(M01 +M02)/L(15)

3.4 Location and Magnitude of Maximum

Column Moment

The moment expression of the axially loaded column

subjected to the total end moments M1and M2, can be

found from the differential equation (M=−EIv′′ )

and expressed in the familiar form given by

M(x) = M2µt−cos pL

sin pL sin px +cos px(16)

where

µt=−M1/M2(17)

This end moment ratio, between total end moments

(from the second-order analysis), becomes positive

when the end moments at the two ends act in opposite

directions (“single curvature”). From the maximum

moment condition, dM (x)/dx = 0, the location xmof

the maximum moment between ends can be obtained

tan pxm=µt−cos pL

sin pL (18)

Noting that the period of Eq. (18) is π,pxmcan be

expressed by

pxm=arctan µt−cos pL

sin pL +nπ n = 0,1, ..

(19)

If, for any value of n(in practice n=0 or 1 for a re-

strained column, and n=0 for a pin-ended column),

0< pxm< pL (20)

then maximum moment is located between ends. If

no pxmvalue can be found that satisfies Eq. (20), the

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

258

Volume 18, 2023

px mpx m

π+

(<0) (<0)

y,v

Figure 4: Moment diagrams and location of maximum mo-

ments at different axial load levels.

maximum moment is located at one column end and

is then equal to the largest end moment.

Fig. 4 shows different moment diagrams that may

result. The column length is represented by pL =

π√αE. In the first case, with pxm<0, the maximum

moment is at the lower end. In the other three cases,

maximum moments occur between ends. In the third

case, two maxima occur, one on either side of the col-

umn. This may occur when end moments are equal or

nearly equal and acting in opposite directions.

By substituting Eq. (18) into Eq. (16), the maxi-

mum moment can be solved for and expressed as

Mmax =BtmaxM2(21)

where

Btmax =1

cos pxm

for µt>cos pL (22)

Btmax = 1.0for µt≤cos pL (23)

The maximum moment may become positive or

negative, as seen in Fig. 4, but in practice it is the

absolute value that is of interest. For a restrained col-

umn with N > NE(i.e., αE>1), maximum moment

will always be between ends (see also Eq. (35)).

Apart from the more general criteria set up above

for maximum moment to form between ends of the

column, the above derivation follows that given in

[10] and [11] for pin-ended members, for which αE≤

1, and consequently N≤NE.

3.5 Presentation of Moments and Shears

The analysis presented above is used to compute mo-

ments and shears in a column for different axial loads

and restraints as a function of the total relative lateral

displacement

∆ = Bs∆0(24)

between column ends, where Bsis a sway displace-

ment magnifier that represents the global (system,

story) second-order effects (“N∆” effects). This

magnifier is a function of the properties and axial

loads of all members in the frame system. Other no-

tations for Bsare, for instance, δin ACI, [1], and

B2in AISC, [2]. A number of available Bsexpres-

sions in the literature, such as, [5], [17], [18], [19],

are reviewed in [5], which also includes a general Bs

proposal that cover both lateral supporting and lateral

supported columns.

The local second-order (“Nδ”) effect in an indi-

vidual column with a specified sidesway ∆ = Bs∆0,

can be quantified by the ratios of second-order anal-

ysis results for a given axial load Nand results for

N= 0 (first-order). These ratios, or local magnifiers,

are denoted Bmax,B1,B2and Bvfor maximum mo-

ment, end moments at column end 1 and 2, and shear

force, respectively.

Thus, for frames with sway due to lateral and axial

loading only, total load effects can be expressed by

Mmax =BmaxBsM02 (25)

M1=B1BsM01 (26)

M2=B2BsM02 (27)

V=BvBsV0(28)

Since the load effects above are all explicit functions

of the lateral displacement ∆(Eqs. (10), (11), (12)),

∆will cancel out in the ratios. As a result, the local

Bcoefficients are independent of the magnitude of ∆.

The Bcoefficients depend only on the end restraints,

which is uniquely defined by the first-order moment

gradient (Eq. (14)), and on the axial load level defined

for instance by the nondimensional load parameters

αEor αs(Eq. (29)).

Mmax is expressed as a function of the first-order

moment at end 2, which per definition is taken as the

end with the larger first-order end moment (absolute

value). Note that Bmax above is different from the

Btmax in Eq. (21) (and there applied to M2obtained

from the second-order analysis).

3.6 Pseudo-critical Loads and Load Indices

In describing the response of “framed” columns, that

may be part of a larger frame, specific, individual

member axial load indices are often helpful when con-

sidering the columns in isolation, and will be used in

this paper. For an elastic, framed member of length L,

uniform axial load Nand uniform sectional bending

stiffness EI along the length, the relevant load indices

are primarily those defined below by the expressions

in Eq. (29), which are given in terms of the critical

loads defined by Eq. (30).

αcr =N

Ncr

;αs=N

Ncs

;αb=N

Ncb

;αE=N

(29)

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

259

Volume 18, 2023

where

Ncr =NE

β2;Ncs =NE

β2

;Ncb =NE

β2

;NE=π2EI

(30)

Here,

–NEis the socalled Euler load (critical load of a pin-

ended column), which is a convenient reference load

parameter in several contexts; βis the effective length

factor, taken equal to βsand βbfor the free-to-sway

and the braced case, respectively.

–Ncr is the critical load of the column at system

(frame, story) instability; Ncs and Ncb are critical

loads of the column when considered in isolation from

the rest of the frame, but with rotational restraints (to

be assumed) corresponding to the respective bend-

ing mode (sway or braced) of the real frame; Ncs

is determined for the column considered completely

free to sway, and Ncb for the column considered fully

braced. Except when the frame consists of a single

column, these are strictly pseudo-critical loads. They

can be very useful in column characterization and dis-

cussion.

As defined, the load indices are interrelated. For

instance, αE=αs/β2

sor αE=αb/β2

b. The free-

sway critical load is defined by αs= 1.0or, for in-

stance, αE= 1/β2

s, and the braced critical load by

αb= 1.0or αE= 1/β2

4 Single Column Response –

Stationary Restraints

General. Typical moment and shear response ver-

sus increasing axial load of columns with stationary

end restraints, computed with the presented second-

order analysis, are shown in Fig.,)LJ6 and )LJ7.

7KHcolumns are illustrated by the inserts in the up

SHUleft hand parts of the figures. The column top is-

LQLtially displaced laterally by an amount Bs∆0(sway-

magnified first-order displacement) and then kept

constant at this value (in a real case, by the action of

the overall frame of which the column may be con-

sidered isolated from).

The results in Fig. 5 are typical for columns with

moderately flexible end restraints, whereas Fig. 6 are

more representative for a column with stiff restraints

at one end, and Fig. 7 for columns with very stiff, and

nearly equal, rotational restraints at both ends.

The moments and shear are shown nondimension-

ally in terms of the respective Bfactors in Eqs. (25) to

(28), and the axial forces are given nondimensionally

in terms of load indices αsand αE(Eq. (29)). All re-

sults in the figure are given in terms of BsM02. There-

fore, the B1response is represented by B1·M01/M02

The curves labeled B2,secant are secant approx-

imations to the end moment curves discussed later

BmBmax

αE

B2,secant

B1,secant

M02

B1B2

1.5

1 1

111

0 −0.560.81

0.66 0.81 0.51

0.5

1.5

1.0

12345

1.5

0.5 1.0

∆

0.556 0.60

V > 0 V < 0

2=3EI/L

=2)(G

1=EI/L

=6)

(Values given relative to max. value)

Selected moment distributions

0.26

Figure 5: Moments and shear versus axial load level in column

with relative flexible end restraints (βs= 1.932,βb= 0.785,

αE= 0.268αs)

(Section 5.9). The curves labeled Bm, are maximum

moment predictions in accordance with present pro-

cedures in major design codes, and will also be dis-

cussed later (Section 7).

It is seen in the figures that both moments and

shears approach infinity (in either the positive or neg-

ative direction) as the axial load approaches the elas-

tic critical load Ncb of the fully braced column. The

corresponding load index at that instance is αb= 1, or

αE= 1/βb

2. For an elastic column, the braced criti-

cal load is independent of whether the column is fully

braced at zero or at a non-zero end displacement.

If the considered column was a part of a larger,

unbraced frame, it should be emphasized that system

instability of the frame may be reached for an axial

load (Ncr) in the column that is well below its fully

braced value Ncb. Also, lateral sway magnification

(Bs) will most often, but not always, reach large,

unacceptable values well before this load level.

Shears. In the absence of axial load, a shear of

V=BsV0is required to give a sway column a dis-

placement ∆ = Bs∆0. When an axial load is applied,

the overturning moment is increased. This tends to

further increase the sway (if it had been free to sway).

To maintain the displacement at the original specified

value, V(or Bv) must decrease, as seen in the figures.

The variation of Bvis almost linear up to and some-

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

260

Volume 18, 2023

αE

αs

=3)

(G =0)

α=1

B2,secant

M01

B1,secant

Bmax

0.5

1.0

1 2 3 4 5

1.5

0.78

1.5

0.5

0.78

k=2EI/L

Selected moment distributions

(relative to max. value)

0.65

Figure 6: Moments and shear versus axial load level in column

with stiff end restraints, G1= 3,G2= 0 (βs= 1.373,βb=

0.626)

what beyond αs= 1, and can in this range be given

by V=BvBsV0with

Bv= 1 −αs(31)

Bvis zero at αs= 1 (or αE= 1/β2

s) and becomes

negative for αs>1. This linear approximation be-

comes increasingly inaccurate (giving too small neg-

ative values) at larger αsvalues, [5]).

At αs= 1, the function of the column changes.

For αs<1(Bv>0), it is capable of resisting

external lateral loads, and is capable of supporting, or

bracing, columns with higher axial loads. For αs>1

(Bv<0), it will require lateral support, or bracing,

by the rest of the story (structure). Zero shear is the

instability criterion for a column that is free to sway.

Moment at end 2 (with the stiffer end restraint).

The largest (absolute value) of the first-order end mo-

ment will always develop at the end with the largest

rotational restraint stiffness. This can be seen in the

figures, and also in the first-order moment diagram

(for N= 0 (αE= 0)) at the bottom of Fig. 5. The

last moment diagram at the bottom is obtained for an

axial load of 0.998 αb

This end, with the largest moment, is conven-

tionally denoted end 2 and the moment M2(or

nondimensionally, B2). So also here. It decreases

continuously with increasing load level. At some

point, it becomes zero and changes direction (at about

αs= 5.1(αE= 1.37) in Fig. 5), and eventually

approaches minus infinity (at αb= 1). The responses

of the stiffer restraint cases, Fig and )LJ7, are simi

lar.

αE

αb= 1

B2,secant

B1,secant

∆

M02

αs

Bmax

1.0

0.5

1.5

1 2 3

BCm=.224

k=20EI/L

k=60EI/L

(G=0.1)

(G=0.3)

Figure 7: Moments and shear versus axial load level in column

with very stiff and almost equal end restraints (βs = 1.065, βb =

0.532, αE = 0.882αs)

Moment at end 1 (with the smaller end

restraint). The M1 response, represented by B1 ·

M01/M02 in Fig. 5 and Fig.6, is typical for cases

with relatively low restraint stiffness at end 1. The

moment at end 1 stays fairly constant, or increases

slowly, until the end moments at the two ends be-

come equal at αE = 1. From there, it increases more

sharply towards infinity at αb = 1. (αE = 1/βb

2). In

the case with stiff restraints also at end 1, such as in

Fig. 7, both end moments decrease markedly at first

with increasing axial load.

Equal or nearly equal end moments. If end

restraints at the two ends had been exactly equal, the

end moments would also be equal. With increasing

axial load, the end moments decrease continuously

to zero at αb = 1. Up to this point, the column

deflection mode would be one in antisymmetrical

double curvature bending. At αb = 1, the smallest

disturbance would cause the deflected shape to

change suddenly from double curvature bending into

the braced buckling mode. This phenomenon is often

referred to as unwinding ([20], [21]). Differences in

end restraints will always be present in practical cases.

Maximum moment. The maximum moment

(Mmax, Bmax) is initially, for zero axial load, equal

to the moment at end 2, and stays at end 2 up to an

axial load point at which it starts forming away from

end 2. Then, following an initial decrease, Bmax

starts increasing with increasing load level, and

approach plus infinity for axial loads approaching the

braced critical load.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

261

Volume 18, 2023

5 “Landmarks” in Single Column

Response

5.1 Key Characteristics

Definition of a number of key characteristics, or

”landmarks” in the moment versus axial load “map”,

are pursued here in order to facilitate a good un-

derstanding of the mechanics of laterally displaced

columns, and for enabling a quick establishment of

moment-axial load relationships. The results may be

of use both for researchers and teachers of column

mechanics, and for practicing engineers. The char-

acteristic points considered are identified by bullet

points in Fig. 5)LJDQG)LJ.

5.2 Zero Shear Force

The shear resistance of a sway columns becomes zero

at αs=αEβ2

s= 1. For αs>1, the column requires

lateral support. The effective length (buckling length)

factor βs, defining Ncs (Eq. (30)) at this stage, can

be determined exactly, or graphically from diagrams

such as the well known alignment charts (in terms

of Gfactors), or also from one of the many approxi-

mate methods available. A summary and evaluation

of such methods are given in [8]. One of several ex-

pressions, derived directly from column mechanics,

is defined by

βs=2√R1+R2−R1R2

R1+R2

(32)

where

Rj=kj

kj+c EI/L=1

1+(c/b0)Gj(33)

The cfactor is a constant, to be taken as c= 2.4in

conjunction with Eq. (32). R1and R2are degree-

of-rotational-fixity factors at ends j= 1 and j= 2,

respectively. R= 0 at a pinned end and R= 1 at a

fully fixed end.

Application to the case in Fig. 5 gives βs= 1.950

(exact 1.932), and to Fig. 7, βs= 1.076 (exact 1.065).

In general, predictions are within 0% and +1.7% of

exact results for positive Rvalues.

5.3 Moments approach Infinity

The outer axial load limit of the M−αrelationship

is defined by αb=αEβ2

b= 1, representing braced

instability. Similarly to free-sway instability, there is

a number of tools available for determining the corre-

sponding effective length factors βb(Eq. (30)). One

of several alternative formulations, [8], for this case

βb= 0.5(2 −R1)(2 −R2)(34)

where the Rfactors are identical to those defined by

Eq. (33) with c= 2.4.

This expression is accurate to within -1% and

+1.5% for positive Rvalues. Applied to the case in

Fig. 5, it gives βb= 0.785 (= exact), and applied to

Fig. 7, βb= 0.536 (exact 0.532).

For approximate system instability calculations of

multi-column frames, the method of means, [8], or

similar methods may be used.

5.4 Maximum Moment leaves Column End

The load index at which the maximum moment starts

to form away from end 2 of the column can be ex-

pressed either in terms of the column’s αEvalue or

its free-sway stability index αs, as

0.25 ≤αE≤1.0and 1.0≤αs≤β2

s(35)

These limiting values can be found from Eq. (18) as

the load at which cos pL =µtbecomes zero (xm,

measured from end 2). The minimum value is ob-

tained for µt= 0 (for a column pinned at one end)

as pL =π/2 and αE= 0.25. Similarly, the max-

imum is obtained for a column with equal end re-

straints, i.e., with µt=−1(antisymmetrical curva-

ture), as pL =πand αE= 1.0(or αs=β2

s). The

moment gradient expression, dM/dx =N θ +V,

also gives useful information as demonstrated in [7].

For instance, for a case with full fixity at an end (e.g.,

Fig. 6), θis zero. Thus, the gradient becomes nega-

tive, and the maximum moment leaves the end as V

becomes negative when αsexceeds 1.0.

5.5 Maximum Moment exceeds BsM02

The axial load level α=αmat which the max-

imum moment exceeds the larger sway-magnified

first-order end moment (BsM02), i.e. when Bmax in

the figures exceeds 1.0, is of considerable interest in

approximate analyses, and in design, as the maximum

moment may be taken conservatively equal to BsM02

below this load level.

Closed form solutions of load levels at which

Bmax = 1 could not be derived. For this reason, they

were calculated for a wide range of restraint combi-

nations using the theory presented above. Results are

presented in Fig. 8 in terms of the free-sway load in-

dex αs,m, and in Fig. 9 in terms of the braced load in-

dex αb,m. The peaks in the figures are obtained when

end restraints are equal, in which cases moments re-

main less than BsM02 (Bmax <1.0) until unwinding

take place at the braced critical load level (αb= 1). It

should be recalled that all Gfactors in the figures are

defined with b0= 6 (Eq. (2)).

Based on these results for single columns with sta-

tionary (invariant) end restraints, it can be concluded

from the figures that

Bmax <1for αs<3.5or αb<0.5(36)

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

262

Volume 18, 2023

for “practical” columns (for panels, see Eq. (45)).

The smaller αb,m values for G1>20 (k >

0.3EI/L) are of no practical significance since such

columns are essentially pin-ended. A perfect pin

(G=∞) is difficult to achieve at a connection to the

surrounding structure. For instance, the Commentary

to AISC, [2], indicates G= 10 as an upper, practical

flexibility value in effective length predictions.

αs,m G1(2)

G2(1)

= 20

0.2 110

Figure 8: Axial load level in terms of αs=N/Ncs at which

the maximum moment between ends is equal to the larger sway-

modified end moment.

αb,m G1(2) =

0.2

0.4

0.6

0.8

1.0

2(1)

Figure 9: Axial load level in terms of αb=N/Ncb at which

the maximum moment between ends is equal to the larger sway-

modified end moment.

Many structural codes accept that local second-

order effects can be neglected if they do not lead to

maximum moments in excess of 1.05 to 1.10 times

BsM02. With such criteria, the load index limits

above would be somewhat greater. For more details,

see the report, [7], on which this paper is based.

5.6 Equal End Moments, M1=M2

For M1and M2to become equal, it can be seen

from Eqs. (10) and (11) that this requires the sta-

bility functions Cand Sto become equal, and thus

also c=s(Eq. (6)). Rewriting of c=sresults in

2sin pL −pL cos pL =pL, the solution of which is

pL =π. Consequently,

M1=M2for αE= 1 (or αs=β2

s)(37)

The moments at αE= 1, found by evaluating Eq.

(11) at pL =π, become

M1=M2=−12

G1+G2+ 2 ·1.216 ·EI∆

L2(38)

or, in terms of the sway magnified first-order mo-

ments (Eqs. (13) and (14)),

B2=M2

BsM02

=4(G1+G2)+2G1G2+ 6

(G1+G2+ 2.432)(G1+ 3)

(39)

B1=M1

BsM01

=B2

G1+ 3

G2+ 3 (40)

5.7 Zero End Moment, M2= 0

In order to obtain M2= 0 (B2= 0), at the end with

the larger first-order end moment, it can be seen from

Eq. (11) that this requires that C−S=−6/G1. Thus,

1−cos pL

pL sin pL =−G1

6(41)

At M2= 0 there is, as expected, no interaction

with G2at end 2. If accurate solutions (by solving

for pL(= π√αE)by iterations) are not required, a

reasonably simple approximation that gives results

within about ±2 percent, has been found to be given

αE=4+1.1G1

1+1.1G1

(42)

For the columns in Fig)LJ and )LJ7, B2

can be se

HQWRbecome zero at about αE= 1.37, 1.67 and 3.4, re-

VSHFtively. These correspond well with the compara

WLYHvalues by Eq. (42) of 1.40, 1.67 and 3.3.

5.8 End Moments at αs= 1

The value of end moments at αs= 1, which represent

the supporting column limit, is of significant interest,

in particular for developing approximate, linearized

end moment relationships (see below). Values of B1

and B2at αs= 1 are labeled B1sand B2s, respec-

tively. It was not possible to find simple, closed-form

expressions for these in the general case with arbitrary

end restraints. Therefore, results were computed for

a wide range of end restraint combinations. They are

plotted in Fig. 10.

B2scoincides with B1sin the case with equal end

restraints (dash-dot borderline G1=G2in the fig-

ure). Results for B1s(dashed lines) and B2s(solid

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

263

Volume 18, 2023

B1s G1= 20

B2s

40 1 2 3 5

0.6

1.1

1.0

0.9

0.8

At α= 1 :

B1s =B2s

Figure 10: End moment factors at αs= 1 versus restraints (G2

of the stiffer end restraint and G1of the more flexible restraint).

lines) are located above and below the borderline, re-

spectively. Corresponding B1sand B2scurves termi-

nates therefore at the dash-dot curve.

As seen from the figure for G2= 0 (fixed end), B2s

may have values ranging between 0.79 and 0.82, and

B1sbetween about 0.82 and 1.05.

5.9 Secant Approximations of End

Moments

The results above are informative and can for instance

be used to establish useful linear approximations of

the column end moment curves. The secants through

the moment points at αs= 0 and αs= 1 are given by

B1,secant =M1

BsM01

= 1 −(1 −B1s)αs(43)

and

B2,secant =M2

BsM02

= 1 −(1 −B2s)αs(44)

where B1,s and B2,s are the moment factor values at

αs= 1 shown in Fig. 10.

Such secants provide close approximations to the

end moment curves also well beyond αs= 1, as can

be seen for the cases in Fig.,)LJ and)LJ. These

FDVHVwere computed with pairs of (B1s,B2s)-values

UHDGfrom Fig.0 of approximately (1.02,0.955),1.04,

0.79) and (0.89, 0.85), respectively. Some efforts

to derive approximate closed-form secant expressions

are presented in [6].

6 Non-stationary Restraints

6.1 Panel Columns – General

In the preceding sections, end restraint stiffnesses of

the single columns were given as constant (stationary,

invariant) values. These results are applicable also to

columns of a general frame provided the chosen col-

umn end restraints adequately represent the interac-

tion with the surrounding structure (other columns on

the same level (horizontal interaction) and columns

on other levels (vertical interaction) as discussed in

Section 3.1).

To what extent stationary restraint values can ade-

quately represent horizontal interaction is considered

in this section for panel frames with two columns (Fig.

2). More specifically, the purpose of the panel anal-

yses is, apart from studying the general mechanics of

behavior and interaction between the panel columns,

to clarify to what extent the column behavior in the

panels will affect conclusions reached earlier from the

single column studies. The author is not familiar with

available literature reporting on such aspects.

The panels are analysed using a stiffness formula-

tion of the second-order theory that incorporates the

slope-deflection equations given earlier (Section 3.2).

Axial forces in beams are neglected.

Two panels, giving deflection shapes indicated in

Fig. 2, are investigated. They are labeled Panel 1

and Panel 2, respectively. The axial column loads N1

(left column) and N2(right column) are the same in

the columns of both panels. Thus, N1=N2=N.

Member stiffnesses, expressed in terms of EI/L,

of Panel 1 are: EI1/L=EI/L,EI2/L=

1.1EI/L,EIb1/Lb= 0.333EI/Land EIb2/Lb=

1.667EI/L. End 2 of each column is taken to be at

the base (bottom), which is connected to the stiffer

beam (5 times that at the upper beam), and thus has

the larger first-order end moment.

Member stiffnesses of Panel 2 are similarly:

EI2/L= 2EI/L(twice that of Column 2 of Panel

1). Otherwise it has the same properties as Panel 1.

6.2 Isolated Column Analysis

For comparison reasons, results are also obtained for

each of the two columns of each panel by considering

them in isolation using the theory derived in Section

3.2. The stationary end restraints used in these iso-

lated column analyses (Section 3.1) are determined

by assuming a hinged support at locations of the first-

order inflection points in the panel beams. Resulting

Gvalues (Eq. (2)) are given in the figure inserts (Fig.

11, Fig. 12, Fig. 13 and Fig.14).

Alternatively, the restraint stiffness assessments

could have been based on the conventional approach

of assuming antisymmetrical beam bending stiffness

(kb= 6EIb/Lb), which is the most common ap-

proach in codes (ACI, AISC, etc.), and is equivalent

to assuming a hinged support at midspan. With the

EI/L values given above, k1=kb1= 2EI/Land

k2=kb2= 10EI/Lare obtained. Eq. (2) then gives

(G1, G2) values of (3, 0.6) for Column 1 of Panel 1,

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

264

Volume 18, 2023

Bmax

M01

1EI

G =0.59

G =2.99

11.1EI

αs1

αb= 1

Bm,t

1.5

1.0

0.5

02 3 45

m0.36

1.639

C =

2.18

(b)

(a)

αE1

Figure 11: Moments and shear versus axial load level for two

cases: (a) Column 1 (left hand) of Panel 1; (b) Column 1 in the

panel considered in isolation with approximate restraints.

(3.3, 0.66) for Column 2 of Panel 1, (3, 0.6) for Col-

umn 1 of Panel 2 and (6, 1.2) for Column 2 of Panel

2. These are not too different from the values given

in the figure inserts (Fig.11, Fig.12, Fig.13, Fig.14)

(based on as sumed support locations at first-order be-

am inflection points).

Still another alternative would be to base beam

stiffnesses on first-order theory, [5]. In the consid-

ered cases, resulting differences in computed B val-

ues according to these three alternatives will be mi-

nor, and will not alter the conclusions drawn below

on inflection-point based restraints. Additional de-

tails are given in [7].

6.3 Panel Results and Characteristics Results

for the two columns of Panel 1 are shown in Fig.11

and Fig.12 and for the two columns of Panel 2 in

Fig.13 and Fig.14. Panel column results are

shown by dashed (blue) lines and the isolated col-

umn results by solid (red) lines. The two cases are

shown by inserts in the upper left corner of the fig-

ures. The dash-dot curves labeled Bm,t, representing

present design maximum moment magnifiers, will be

discussed later.

Results are plotted versus the nominal load

indices αE

. In addition, abscissas in terms of the

free-sway index αs of the respective isolated columns

(αs = αE β 2) are added for the convenience of read-

ing and interpretation of results. Thus, zero shear of

the isolated columns will always be obtained for αs

= 1 in the figures.

System instability. System instability of the

panels will always be initiated by the most flexible

column. Since both columns are connected to the

M01

αb= 1

Bmax

G =0.67

G =3.31

21.1EI

11.1EI

αE2

1.5

1.0

0.5

12345

(b)

(a)

αs2

Bm,t

2.113

1.490

Figure 12: Moments and shear versus axial load level for two

cases: (a) Column 2 (right hand) of Panel 1; (b) Column 2 in the

panel considered in isolation with approximate restraints.

same beams, Column 1, with the highest load index

αEof the two columns, is most flexible. Panel 1,

with reasonably close load indices (αE1= 1.1αE2),

represents a more practical case than Panel 2, with

rather big difference in load indices (αE1= 2αE2).

Restraint softening and critical loads. The rota-

tional stiffnesses of the panel beams are gradually re-

duced with increasing axial column load, as the beams

“unwind” from double curvature towards more single

curvature type bending, as illustrated in Fig. 2 and in

the figure inserts. For instance, in the case of Panel 1,

Fig. 11, this beam softening implies a restraint stiff-

ness reduction to about one third of the initial (double

curvature) stiffness. As a consequence, braced panel

instability, initiated by the most flexible panel Col-

umn 1, is seen to result at a lower axial load (1.639

αE1) than that of the isolated single Column 1 (2.18

αE1).

Furthermore, a rather sudden reversal (“unwind-

ing”) of end moments is seen to take place in the stiffer

Column 2 for loads close to the panel critical load

(Fig. 12).

In the case of Panel 2, with Column 2 being sig-

nificantly stiffer than the Column 1 (αE2=αE2/2),

only a partial unwinding of beams may take place

(Fig. 2(b)). Close to panel instability, the stiff Col-

umn 2 unwinds from double to single curvature bend-

ing (Fig. 14), reflecting a braced effective length fac-

tor that is greater than 1.0. This again implies that

Column 2 contributes (through the beams) to the re-

straint of the more flexible Column 1.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

265

Volume 18, 2023

Bmax

M01

αs1

αb= 1

G =2.95

G =0.53

EI 2EI Bm,t

1.5

1.0

0.5

1345

1.805 2.218

C =

m0.366

(b)

(a)

αE1

Figure 13: Moments and shear versus axial load level for two

cases: (a) Column 1 (left hand) of Panel 2; (b) Column 1 in the

panel considered in isolation with approximate restraints.

6.4 Non-stationary vs. Stationary

Restraints

By comparing results for isolated columns and panel

columns it is possible to draw some conclusions that

should be useful in practical design work:

Shears. The axial load at which shears become

zero in the panel columns, can be predicted well by

the free-to-sway critical load of the corresponding

isolated column (Eq. (32)). The difference between

the two is found to be about ±1% for the columns

of Panel 1, and about ±3.8% for those of Panel 2 at

αs= 1.

Instability. In lieu of a full system instability

analysis, it is found that braced critical loads of panel

columns can be computed approximately from Eq.

(34) with softened (reduced) end restraints discussed

above (and as applied in Bm,t predictions later).

Alternatively, approximate yet quite accurate system

instability analyses can be carried by the “method of

means”, [8].

Moments. The moments of the panel columns

can be seen to initially follow the isolated column

moments quite closely. Consequently, the panel

columns respond initially with almost stationary end

restraints that are (by implication) nearly equal to the

restraints employed in the isolated column analyses.

Thus, Eq. (35) also provides adequate limits for

when maximum moments leave the ends of panel

columns.

Bmax

αb= 1

M01

G =1.34

G =6.11

2EI

1EI 2EI

Bmax

αs2

0.5

1.0

1.5

10.5 1.5

0.903 1.743

Bm,t

(a) (b)

αE2

Figure 14: Moments and shear versus axial load level for two

cases: (a) Column 2 (right hand) of Panel 2; (b) Column 2 in the

panel considered in isolation with approximate restraints.

Maximum moments exceed BsM02. Due to the

softening of restraints, the maximum moment branch

of the panel columns is pressed upwards at lower load

levels than in the case of the isolated columns. This

affects the limits given in Eq. (36). For the panel

columns, it is found that

Bmax <1for αs<3or αb<0.8(45)

provide reasonably conservative limits. Although

these are based on a rather limited number of panel

columns, they are believed to be reasonably represen-

tative. It should be noted that the αsand αbvalues in

Eq. (45) are computed with the critical loads Ncs and

Ncb, respectively, of the panel columns, rather than

those of the isolated single columns with stationary

end restraints.

While Ncs of the isolated and the panel column

are almost identical, Ncb of the panel column may

be lowered (by the mentioned restraint softening)

considerably below that of the single column.

Maximum moment accuracy. For the more flex-

ible Column 1 of Panel 1 (Fig. 11, EI2= 1.1EI1), it

is found that isolated column predictions of maximum

moment are at most 5% below those of the panel for

axial loads below about 85% of the panel column’s

instability load.

In the case of the most flexible Column 1 of Panel

2 (Fig. 13, EI2= 2EI1), this is the case for axial

loads below about 65% of the panel column’s instabil-

ity load. This is still a rather high load level in design

practice.

For the stiffer Column 2 of the two panels, the

isolated column predictions are conservative up to

load levels very close to the panel columns’ instabil-

ity loads (Fig.12,Fig.14). This is typical for the stiffer

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

266

Volume 18, 2023

columns in frames.

Equal end moments. The equal end mo-

ment results for the isolated single columns (Eqs.

(37) - (40)), are also found to apply to the panel

columns, except for unusual cases of panel columns

with critical loads greater then αE= 1 (e.g., Fig. 14).

Zero end moment. The applicability of Eqs.

(41) and (42) for prediction of zero end moments

(M2= 0;B2= 0) are found to be more limited for

panel columns. For the most flexible of the panel

columns (columns 1), Eqs. (41) and (42) will give

some 7-10% too large axial load values. The limits

are not applicable at all to the stiffer panel columns

(Columns 2), due to the sudden moment reversals or

unwinding that takes place in these columns close

to frame instability (brought on by the most flexible

columns).

End moments and secant approximations.

Moments at αs= 1 (B1sand B2s) in Fig. 10, and

the secant expressions in Eqs. (43) and (44) are

applicable also to the panel columns.

7 Maximum Moment in Present

Design Codes

Maximum moment approximations in present design

practice can be given by Mmax =BmBsM02, where

Bm=Cm

1−αb≥1.0; Cm= 0.6+0.4µo(≥0.4)

(46)

µ0=−BsM01

BsM02

=−M01

M02

(47)

For the sake of comparison, predictions by this ex-

pression are included in the presented moment-axial

load figures. This Bmax approximation is common in

major design codes (e.g., [1], [2], [3], [4]).

Above, αbis the critical braced load index (Eq.

(29)), Cmis a first-order moment gradient factor (that

accounts for other than uniform first-order moments),

and µ0is the ratio of first-order end moments, taken

to be positive when the member is bent in single

curvature by these moments, and negative otherwise.

In some codes, [3], Cmis limited to 0.4.

Single restrained columns. Bmpredictions by

Eq. (46), with Ncb (αb=N/Ncb) computed for end

restraints given in the respective figures for the lat-

erally displaced column analyses, are shown in Fig.

Fig.6 & Fig.7. These are based on the first-order end

moment ratios given by µ0=−(G2+ 3)/(G1+ 3)

(from Eq. (14)).

Isolated panel columns.Bmpredictions denoted

Bm,t (subscript t for tangent) for the panel columns

considered in isolation are shown in Fig1,)LJ2,

)LJ13)LJ4.7he first-order moment ratios (µ0us-

ed in the calculations are those obtained from the first-

order panel analyses. Ncb (αb=N/Ncb) are based

on assumed rotational beam restraint stiffnesses of

kb= 2EIb/Lb, (implying horizontal tangent at beam

midlength). These are 1/3rd of stiffnesses for beams

bent in antisymmetrical curvature bending, and re-

flects, in an approximate sense, the restraint soften-

ing discussed earlier, and are in accordance with most

codes of practice.

Column 2 of Panel 2 (Fig. 14) represents a case

in which the restraint softening assumption is not

very good. This is not surprising for this special case

(with Column 2 having twice the EI/Lof Column

1): Rather than receive restraint, Column 2 has been

found earlier to contribute to the restraint of Column

1. If instead, Bmin this case had been computed

with the panel critical load, which is an accepted

alternative by the codes, the rising branch would

have commenced before (moved towards the left),

giving very conservative predictions also for this case.

Conclusions. It is clear that Bm(Bm,t) given by

Eq. (46) is generally very conservative for columns

in frames with moments caused by sidesway, and in

particular for common cases of panel columns due to

restraint softening. There is room for considerable

improvements in maximum moment predictions. Pre-

sentation of some recent efforts, [6], towards this end

is outside the scope of this paper.

8 Summary and Conclusions

Theoretically derived, closed form expressions have

been presented for a number of member response

characteristics that enable quick establishment of

several typical points along the moment-axial load

curves.

These represent not only a novel contribution, but

also a tool that will be useful in providing a general

understanding of the often complicated column re-

sponse, in practical design work and as a complement

to full second-order analyses.

The appropriateness of analyzing framed columns

by single column models with stationary end re-

straints are investigated using laterally displaced sin-

gle columns isolated from laterally displaced two-

column panels.

For panel cases with smaller differences in column

stiffnesses, it is found that single column models de-

scribe the maximum moment response of the most

flexible of the panel columns quite closely for axial

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

267

Volume 18, 2023

load levels as high as 85% of the critical panel load-

ing. For the stiffer panel column, this load level is

higher.

For panel frames with large stiffness differences

between interacting columns, unwinding may occur

and reduce the load level at which the most flexible

panel column can be described well with a single col-

umn model. Even so, it is quite well for axial load

levels as high as 65% of the critical panel loading.

For the stiffer panel column, this load level is higher,

and very close to the critical panel loading.

Results are computed whereby end moments de-

scribed by secants to the axial load - moment curve

can be calculated. They are found to provide good

end moment approximations over a wide axial force

range, and are useful in particular for the assessment

of moments in adjacent restraining members, includ-

ing foundations.

Present maximum moment approximations in

structural design codes are generally found to be

very conservative for columns with moments solely

due to sidesway. There is room for considerable

improvements in such predictions, and particularly

so for columns restrained by stiff beams. Axial

load limits are established below which maximum

column moments in frames with sidesway will be

less than the sway-magnified first-order end moment

(BsM02). By taking Bm= 1, for lack of better

values, will allow more economical designs for a

wide axial force range.

Acknowledgment:

The author has long been interested in the paper

topic, both as a practicing engineer and researcher.

During a stay at the Univ. of Alberta (UA),

Edmonton, Canada, sponsored by the Research

Council of Norway and a research associateship at

UA (1981), this interest became more focused. Input

and useful discussions by the now deceased

Professor J. G. MacGregor (at UA) was greatly

appreciated. So was the running of the initial

computer analyses of the panels by S.M.A. Lai, then

a PhD student at UA.

References:

[1] ACI (American Concrete Institute). Building

code requirements for structural concrete, and

Commentary (ACI 318-M14 and ACI

318RM-14). Farmington Hills, MI, 2014.

[2] AISC (American Institute of Steel

Construction). Specifications for structural steel

buildings, and Commentary (ANSI/AISC

360-16). Chicago, IL, 2016.

[3] CEN (European Committee for Standardisation).

Design of concrete structures, part 1-1.

Eurocode 2. Brussels, 2004.

[4] CEN (European Committee for Standardisation).

Design of steel structures, part 1-1. Eurocode 3,

Brussels, 2005.

[5] Hellesland J. Extended second order

approximate analysis of frames with

sway-braced column interaction. J. of Construct.

Steel Research, Vol. 65, No. 5, 2009, pp.

1075–1086.

[6] Hellesland J. New and extended design moment

formulations for slender columns in frames with

sway. Eng. Struct., Vol. 203, No. 1 (109804),

2020.

[7] Hellesland J. Mechanics of columns in sway

frames – Derivation of key characteristics.

Research Rep. in Mechanics, Dept. of Math.,

Univ. of Oslo, Norway, No.1-Feb, 2019.

[8] Hellesland J. Evaluation of effective length

formulas and application in system instability

analysis. Eng. Struct., Vol. 45, No. 12, 2012, pp.

405–420.

[9] Chen WF, Lui EM. Stability design of steel

frames, CRC Press, Boca Raton, Fl, 1991.

[10] Galambos TV. Structural members and frames.

Prentice Hall, Inc., Englewood Cliffs, NJ, 1968.

[11] Galambos TV, Surovek AE. Structural stability

of steel: concepts and applications for structural

engineers. John Wiley & Sons, Inc., Hoboken,

NJ, 2008.

[12] Lai SM A., MacGregor JG, Hellesland J.

Geometric nonlinearities in nonsway frames. J.

Struct. Eng., ASCE, Vol. 109, No. 12, 1983, pp.

2770–2785.

[13] Timoshenko SP, Gere JM. Theory of elastic

stability, 2nd ed. McGraw-Hill Book Co, 1961.

[14] Livesley RK. The application of an electronic

digital computer to some problems of structural

analysis. The Structural Engineer, 1956,

January.

[15] Livesley RK. Matrix methods of structural

analysis, 2nd ed., Pergamon Press, New York,

NY, 1975.

[16] Bleich F. Buckling strength of metal structures.

McGraw-Hill Book Co., Inc., New York, NY,

1952.

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

268

Volume 18, 2023

[17] LeMessurier, WM. A practical method of

second order analysis. Eng. J., AISC, Vol. 14,

No. 2, 1977, pp. 49–67.

[18] Lui, EM. A novel approach for K factor

determination. Eng. J., AISC, Vol. 29, No. 4,

1992, pp. 150–150.

[19] Aristizabal-Ochoa JD. K-factor for columns in

any type of construction: Nonparadoxical

approach. J. Struct. Eng., ASCE, Vol. 120, No.

4, pp. 1272-1290.

[20] Ketter RL. Further studies of the strength of

beam-columns. J. of the Struct. Div., ASCE. Vol.

87, No. ST6, 1961, pp. 135-152.

[21] Ziemian, RD (ed.). Guide to stability design

criteria for metal structures, 6th ed., John Wiley

& Sons, Inc., Hoboken, New Jersey, USA, 2010.

Notation:

Bm= approximate maximum moment magnification factor;

Bmax = exact maximum moment magnification factor;

Bs= sway magnification factor;

Btmax = maximum moment magnification factor in second-order

analysis;

B1, B2= end moment factors, at end 1 and 2;

EI, EIb= cross-sectional stiffness of columns, and beams;

Gj= relative rotational restraint flexibility at member end j;

H= applied lateral story load (sum of column shears and

bracing force);

L, Lb= lengths of considered column and of restraining beam(s);

M0j, Mj= moment in first-order and second-order analysis, at end j;

N= axial (normal) force;

Ncr = critical load in general (=π2EI/(βL)2);

Ncb, Ncs = critical load of columns considered fully braced and

free-to-sway, respectively;

NE= Euler buckling load of a pin-ended column

(=π2EI/L2);

Rj= rotational degree of fixity at member end j;

V0, V = first-order, and total (first+second-order) shear force;

kj= rotational restraint stiffness (spring stiffness) at end j;

p=√N/EI;

αcr = member (system) stability index (=N/Ncr );

αb, αs= load index of column considered fully braced and

free-to-sway, respectively;

αE= nominal load index of a column (=N/NE);

βb, βs= effective length factor corresponding to Ncb and Ncs;

∆0,∆= first-order, and total lateral displacement;

κj= relative rotational restraint stiffness at end j

(=kj/(EI/L)).

WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICS

DOI: 10.37394/232011.2023.18.24

Jostein Hellesland

E-ISSN: 2224-3429

269

Volume 18, 2023

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflicts of Interest

The author has no conflicts of interest to declare that

are relevant to the content of this article.

Creative Commons Attribution License 4.0 (Attri-

bution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en

_US