Buckling contribution to the analysis of steel trusses

Compurrrs & Srrucrurrr Rimed in Great Britain.

Vol. 2:. No. 3. pp. 4454%.

BUCKLING

1980

0045-7949 e4 53.00 + .w C 1986 Pcrsmwn Press Ltd.

CONTRIBUTION TO THE ANALYSIS STEEL TRUSSES EL-SAYED

Structural

Engineering

Department,

(Received

Abstract-This

t

Cairo

University,

17 April 1984, in revised form 21 September

Cairo

Egypt

1984)

work deals with the stability phenomena when included in the analysis of Different rigidities are treated varying from the flexible connection to the fully research has shown that a certain gain in the weight of steel can be obtained connections. In this case, it is inevitable to treat the truss members as parts they are subjected to both normal force and bending moment effect.

AND PREVIOUS W’ORK

The problem of buckling of isolated compression members has been treated considering various methods of end restraints. Reference [I] has considered plane truss members as hinged at both ends, which leads to an upper bound on buckling lengths. When applied in design rules this decreases the allowable buckling load. Reference [2] suggested a fast approximate method for the determination of the critical load for rigidly jointed truss members. The rotations of joints not directly connected to the one under consideration are neglected, and members connected at this joint are considered to be fixed. Reference [3] gave a method of analysis where the rigidity of connections can be taken into consideration. In this method a compression member is assumed to be of negligible rigidity compared to the joint. On the other hand, a tension member creates a rigidity equal to 3EIIL at the position of the joint. If several tension members intersect at a joint, the rigidity of the connection will be 2:” (3El/L), where n is the total number of tension members intersecting at that joint. After the rigidity of the different joints had been calculated by Ref. [3], Ref. [4] then introduced charts relating the rigidities at the two ends of the member to its effective factor of buckling (k). Reference [S] has determined the critical load of a specific structure by plotting the stiffness of the frame Q against the loading parameter P. The stiffness of the frame decreases with the increase of the load parameter P and becomes zero at the critical load. Reference [6] presented a method for determining the elastic critical loading pattern of a rigidly-

f Associate

MACHALY

of Engineering,

research

semirigid steel trusses. rigid connection. This when we use semirigid of frame structure, iye.

1. INTRODUCTION

BAHAA

Faculty

OF

jointed plane truss as a whole. In this lvork joint translation is ignored. The elastic critical loads of a set of equilaterally triangulated symmetrical trusses are analysed for given stiffnesses of the upper chord, lower chord and web members.

Professor. 445

2. PURPOSE OF THE STUD1

The determination of bending stresses when taking the rigidity of connections into consideration. Different rigidities are treated. varying from the flexible connection (pure hinge) to the fully rigid connection. The determination of the effect of stability when included compared with an analysis where the effect of stability is neglected. The questions will be “Is there an increase or decrease in straining actions created in the members when taking the stability into consideration?” What will be the saving in material if the members of a plane truss are analysed as structural elements subjected to bending and axial normal force? N.B. Different authors have previously investigated the critical load applied on a given truss of determined configuration in order to prevent buckling. On the other hand. the effect of semirigid connections where stability is included in the analysis has never been investigated. 3. THE PROPOSED >IATHEMATICAL

3. I Con&uration

XlODEL

of rhe semirigid connertion

The semirigid connection was represented by a torsional spring whose stiffness is K, (Ref. [7]). The semirigid connection was assumed to be rigid as regards to horizontal and vertical deflections (14.~1. The moment is considered to be of positive sign if it acts clockwise and vice versa [Fig. I(a.b)l.

446

EL-SAYED BAHAA MACHALY

(b)

1.-._._.

(cl

Fig. 1. (a) Positive sign moment; (b) moment acting clockwise: spring. The stiffness matrix of the torsional spring [Fig. i(

is

3.2 ~~g~~i~ ratio of a coanectioa (percent o~r~gidityJ

Defining (p) as the rigidity ratio of the semirigid connection, i.e. the ratio obtained by dividing the fixed end moment into the actual end moment (see Ref. [7]), the stiffness (KS) of the semirigid connection can be formulated by the expression

+L*“. -P

L

(c) the stiffness

matrix

of the torsional

3.3 Stj~ness matrix of a hofi~5nta~ element where stability is taken into consideration (i.e. rhe eflert of axial load on the bending stiffness is incftrded in the analysis)

The different values of moments and reactions due to end rotation and translation are given by Ref. [S] according to Table 1. S and C are the stability functions given by the following expressions: For compression Ml s =

(2)

c=

where

sin

structural

element

2a cot Za)

-

tan

a -

’

a

2a-f$;;s,a, 2a _ 231 + c)

m =

E- Young’s modulus, I = intertia of cross section in the plane of bending, L= length of the member. Table

Joint translation. VI

Joint transiation, V2

2S(1

+ c) -

4a’

’

I.

Joint rotation, Q,

Joint rotation, Qz

Buckling contribution to the analysis of steel trusses

447

(al

tik81 t stt tcy

k4

l- r(f+c+ 1

“r
k”2

2s(l+c)--

m3

I

Ill12 t 2s(t tc) 3 mL2

Fig. 2.

(a) The sign convention for displacements and straining actions; (b) configuration of joint rotations and translations.

If we denote

For tension structural element -

’ = a(‘tanh c=

I -2st1 *cl ‘I ftlL2

2a coth 2a) a -a ’

- sinh 2a 2a sinh 2a - 2a cash 2a ’ ZS(1 + c)

m = 2S(l

+ c) +

4a2 ’

S,=-,

AE L.

= WI

s 4

Sz = SK,

s, =

S(1 + c)K L ’

+ c)K mL2 ’

the stiffness matrix of a horizontal structural element where the effect of axial force is taken into consideration can be written in the form

The sign convention for the displacements u, v, 0 and straining actions F,, Fy , Mare as shown in Fig.

2(a).

v, = 8, = 0.0

u2=v2=82=0.0 (0)

N.B.

(b)

Fig. 3. (a) Configuration of horizontal displacement: (b) straining actions due to horizontal displacement.

PE is the Euler buckling load, MI, M2 are the end moments, I;;., , F,.z are the vertical reactions. For the configuration ofjoint rotations 8,. O2and joint translations VI, V,, see Figs. 2(b).

EL-SAYED BAHAA MACHAL)

448 3.1 Determination due to joint structural

of end moments

rotation element

and joint with

semirigid

Equating

and end reactions

trunslation

for

(3~) in (3b) we get

a

connections

e; = ei = 0.0.

at the

ends when the effect of axial load is inclltded

3.4.1 Joint horizontal displncement WI. Consider member (l-2) shown in Fig. 3(a). It has a uniform flexural rigidity EI. If end (2) is kept fixed in position and direction as shown, while end (1) is displaced horizontally by UI, the stiffness matrix given in Section (3.3) will be reduced to the form

Ml

M; F 12 F 1

0

0 SI 0 s3 0 -K,, 0 (Sz + K,,) 0 -s, 0 -s3

F,I FYI =

I$

0

CS?

M2

0

0

Using eqn (3a) we can get the reactions shown in Fig. 3(b). 3.4.2 Joint rotation 01. Consider the member ( I2) shown in Fig. 4(a). It has a uniform flexural rigidity El. If end (2) is kept fixed in position and direction as shown, while end ( I) is rotated by 0,. the end moments M, , Mz and the vertical reactions F,, . Fy2 can be deduced according to the following procedure: The stiffness matrix given in Section (3.3) will be reduced to the form

s3

0 csr 0 -s3

- K,r

-KS,

0 0 0 From equiibrium

(C&)0;

+ (Sr

+

= 0.0,

(3b)

Ks2)e; = 0.0.

(3c)

v, = 0.0

v2=e2 = 0.0

0

(S2 + K,,) CS2 -S1 -s, cs: tS2 + K,:) 0 - Krz

’

M, = Ks,+, - K,,.e;, 0.0 = -K,,.8,

+ (Sr

(?a) + K,,).Bi

-

(3b)

C.S?.ei.

0.0 = c.s2*e; + (.s2 + K,+e;,

f-k)

Let

Det =

-K,, 0

-K,,

0

(SZ + K.,,)

CS2

cs:

(5:

.

+ K.,z)

which gives Det = K,,.S2[S2(1

L$+.y

s3

-KI

we can get

&I (3a)

+ (C&)8;

s3

0 KI

(Sz + f&2)

where K,, is the stiffness of the semirigid connection at the left end of the member. KS2 is the stiffness of the semirigid connection at the right end of the member. Ml, Mz are the moments at the newly created nodes 1 and 2 and which appeared as a result of considering the semirigid connection as a spring. Q,, Qz are the rotations at the newly created nodes 1 and 2 and which appeared as a result of considering the semirigid connection as a spring. From equilibrium.

(Sz + K,,)e;

e;

81 ’

- C’) - K.,J.

(-ld)

Similarly we get

Det(e,)

(a)

=

MI 0

-K,, (& + K,,)

0

cs2

0 CS2 IS,

.

+ K,:)

which gives Det(0,)

= M,[Si(l

- C’) + SZ(K,, + K,z)

(b)

Fig. 4. (a) Configuration ofjoint rotation 01: (b) straining actions due lo joint rotation 91.

+ K\,.K\zl.

(4e)

The rotation CI, being equal to the value Det(BI) el ==. -__

(40

Buckling contribution to the analysis of steel trusses

Substituting

‘t-+9

(4d) and (4e) in (4f) we get:

(a)

(W denoted by

a92

-2

s2Bhpc(a2-I~)82 n

n I

U? =

Ks,

(4h)

*’

p = I.O+S~(lIK,,

+ l/K,?) . - CZ)IK,,.K,2

(b)

I)]e,,

+ C(u, -

FsI = &.a,$~e,,

(4)

Fs2 = -Sp_,$.e,, Mz =

having

DeWi) =

&I

MI

-KS,

0 0

0

0 c&

S2 + KS2 = K,,.M,q&

+

K,d (4)

and

&I Det(@) =

-KS,

-KS, M, (Sz + K,,) 0

0

CS?

0

= Using

-s38a282

s38a282

lations (4h) we get [Fig. 4(b)]

(4h) in (4g) we get

M, = &$[a,

relations

2

Fig. 5. (a) Configuration of joint rotation 8:: (b) straining actions due to joint rotation 0:.

I.0

Substituting

z

t-1

Szwc)+Io

+s;(l

7

c&$~e,

.

Using (4g) and (40) we get the different reactions shown in Fig. 4(b). 3.4.3 Joint rotation (02). Consider the member (l-2) shown in Fig. 5(a). It has a uniform flexural rigidity EI. If end (1) is kept fixed in position and direction as shown, while end (2) is rotated &, the terminal moments M,, Mz and the vertical reactions Fy,, Fyz can be deduced according to the previous procedure given in paragraph (3.4.2). This gives the values Mr = &+[a2

K,,.C.SyM, . (4k)

(40)

+ Chz - 1)]ez.

M, = csz+be2,

(5)

F.v, = s,+ba2.e2,

(4d). (4j) and (4k) we get

Fyz = - s+azae2. 0; =

L(& $(l

+ Kc)

- C’) + &(K,,

+ Ksr) + K,,K,? * ‘, (4.1)

and

e; =

VII-4 Fy,

-C&KS, SitI - C’) + S2(Ks, + Ksz) + K,,K,? “,’

t

u, ‘8, = 0.0

~2~~2~82~

0.0

(a)

(4m) having F,, = S,(tq + F,.? = -s,(e;

eg, + ei),

(4n)

M2 = -Ksz.e;.

(b)

Fig. 6. (a) Configuration ofjoint translation C’,: (b) strain-

Substituting

(41) and (4m) in (4n) and using the re-

ing actions due to joint translation c’,

450

EL-SAYEDBAHAA

The expressions for the different reactions given in (5) are as shown in Fig. 5(b). 3.4.4 Joinr ~nnshfion (V,). Consider the member (l-1) shown in Fig. 6(a). It has a uniform flexural rigidity EI. The ends (I) and (2) are restrained against rotation but the end (I) is translated through a distance V,. The terminal moments M,. M2 and the vertical reactions FJ,. F,.: can be deduced according to the following procedure: The stiffness matrix given in paragraph (3.3) will be reduced to the form

MACHALY

Det(e;)

= S3.F,,[Sl(c

The translation

Det( VI) Det

(6hl

(6d) and (6e) in (6h) we get

[$(I

-

- C) - (K,,

C’) + &(K,,

-

K,:)]

+ K,$:) A K,,.K,:]

. I,‘,. FYI

S4

MI

0

M;

s3

=

F,2

0

0

-Ks,

-s,

+

Similarly

c*s2

K*,)

-s3

-s3

Mi

s,

c*s2

M2

0

0

(S2

+

.

(6j)

Ks2)

-K,2

Substituting From equilibrium

(6i)

s3

s3 (.%

(6g)

SJ

Fy, =

0:

0;

=

I

S;[ZS:(l VI

I) - K,,)l.

V, is equal to the value v

Substituting

-

(6d), (60 and (6i) in (6k) we get

we can get

Fy, = SaV,

+ S30; + S,e;,

0.0

= s3v,

+ (Sz + K,,)e;

0.0

=

@a) +

c*s+3i

8; =

s3[s2(c

S:(l

-

- C’) + S:(K,,

1) -

h’s21

+ K,:)

+ Ks,.Ks2

* V,.

(6b)

S3v, + csg; + (s2 + K,+e;.

(6k)

(6~) In the same manner

Let

e; = & s3 S3 (& + K,,)

Det =

s3

- 1) - h’,,]

- C’) + &(K,,

+ KS:) + K,,.K,:

CS2

cs2

s3

s,[&(c Sf(l

(S2

+

K,2)

. V,.

’

(61)

having which gives

M, = -K,,+;.

Det = S,[(Si(l

- C’) + S2(Kr,

.K,L)] - Sf[2&(1

+ K,?) + K,,

- C) + (K,,

- S,(O; f Oi,,

Fy2 = -&VI

+ Ksr)]

(6m)

Mr = - K,y8;.

(6d)

Using the relations of aI. a2 and p given in (4h) and using (6k) and (61) in (6m) we get

Similarly we get

Det(V,)

=

s3

&I 0

(S:! + K,,)

cs2

0

cs2

(Sz + Kz)

33

Fy,=S,[l.O-p(g) =

Det(V,)

= F,,[Si(l

-

x (z+$)].V,

’

C’) + SAK.,,

MI =

- F.v2, S3Pa,

(6n)

VI,

Mz = S~@Q V, . + K,d

Det(e;)

=

+ K,,.KJ. s3

s4

[v,

S,

0

C&

s3

0

& + K.,>

Det(Bi) = S3*F;.,(.S~(c Det(ei)

=

S4 S3 1s3

(6e)

I) - K.,?),

S3 (S2 + K.,,) C.%

,

FYI 0 0 1

1

(60

Using (6n) we get the different reactions shown in Fig. 6(b). 3.4.5 Joint trnnslntiorr ( \‘:I. Consider the member (l-2) shown in Fig. 7(a). It has a uniform flexural rigidity El. The ends (I) and (2, are restrained against rotation but the end (2) is translated through a distance V2. The terminal moment M,, hf2 and the reactions F!, , F,.! can be deduced according to the previous procedure given in paragraph (3.4.4).

Buckling contribution to the analysis of steel trusses This gives the values

F?,

=

Fy2 =

-S4

[ 1.0 -

(2)

+ SJ [ 1.0 -

(2)

x

($d+

z)]

b’:.

(7) Expressions given by (7) are the different actions shown in Fig. 7(b).

reFig. 7. (a) Configuration ofjoint translation I.:: (b) straining actions due to joint translation L.2.

3.5 The stiffness matrix of a horizontal element whose both ends are semirigid connections (effect of axial force is taken inro consideration)

Cases (a). (b). (c), (d) and (e) previously treated in paragraph (3.4) are for simplification denoted by YI, ~2. yII, ~22and ylr in the following:

Substituting (SC) in (8b) the stiffness matrix [S.tfI will be of the form tSM1

YI = P-1. y2 YII

=

P-az.

=

P[al + Cbl - 111.

712 =

1.0 -

0f

T, 0

0 7,

0

- 0TI

:J

07~

s(I+c)p

-

0 0

TJ

-T, 0

0TI

0 - T,

0 T;

-:J

0”

T,

- T3

0

Td

T;

T;

0

- Tj

-

T; Ti

3.6 Global sriJTnessmatrix (inclined axes) ~22 =

With reference

P[a2 + C(a2 - I)].

Using the terms of eqn (8) and the terms S,, S2. SS and So previously defined in paragraph (3.3). the stiffness matrix [SMI of a horizontal element whose both ends are semirigid connections will be of the form

to Fig. 8 and denoted by

c., = cos t$ =

Xh - Xj L .

___~__--r-_--_--_,__-_,-‘-_--r’_---

S II

0

I-s,10

0

f

IO

8

1

I

___,__--k__-__+_-

--,-----

_I_

I 0 I LYI? , S,YI 0 I -S.tYc ; __ ‘___ _;__ _.__~____.‘-___&_____ ’ 0 ; -s3y1 1 0 I SlYl , SZYll -J,_c

_o_

_

_I_

_o_

-

-

_

_j_

-

_~

t

_.

-

-

_j_

_ --

_

S?Y2 SZCP _oo-

-

-

(8b)

Let T, =

S,,

T; = &yt2. T* = S4Y12.

Tz = Sg,

I,

TJ = S,Y I 9

T; = S2Cp, T; = SJYZ.

the global stiffness matrix will be (SC, [SMD]

= [TJT.[SM]+[T].

(9a)

452

EL-SAYEDBAHAA

MACHALY

nodes as hinged connections. Compressive stresses did not exceed the allowable buckling stresses set by Euler and Johnson. Tensile stresses were less than the maximum stresses allowed by the specifications. Buckling lengths of all members were taken equal to the geometric lengths of the members. Slenderness ratios of both tension and compression members were within the allowable limits denoted by the specifications. The setions used in the proportioning of the members were angles with welded connections. Fig. 8. Straining actions

of an element with two semirigid

4.3 T/W rigidity rnrio

All trusses were reanalysed using the same sections that were proportioned previously (see Section 4.2). In this new analysis the rigidity ratio was varied for all joints taking the values from 0.1 to 1.0.

ends.

where ( T) and (T) T are of the form

4.4 The effect of the Normal Force

Substituting (ZJ and (TIT in (9a) the global stiffness matrix (MD)

will be of the form

5.DISCUSSIOSOFRESCLTS

4.DIFFERENTCASESUNDERSTUDY 4.1 Loading and geometrical

configuration

Six groups of trusses were studied as shown in Fig. 9. These were of spans 3, 6, 9, 12, 15 and 18 m. Diagonals were 60” inclined w.r.t. upper and lower chords. All cases were externally statically determinate, but were internally statically indeterminate using semirigid joints. Loads were concentrated at the different nodes of the upper chord. These consisted of both live and dead loads. 4.2 Proportioning

All trusses

of the truss members

have been designed

considering

Two further analyses were applied for each case of rigidity ratio for all the trusses: (a) The first analysis, termed “without,” was the classical method of analysis. Normal and shear forces, and bending moments were obtained for all members without taking the effect of the normal force for the moments induced in the members. In this case, the global stiffness matrix given in Ref. [7] was applied. (b) The second analysis, termed ‘.with,” included the effect of normal forces on the members moments, using the stability functions presented in the newly deduced global stiffness matrix shown in Section (3) of this paper.

all

This discussion will be based on the results obtained from 120computer runs. These were the runs of 10 cases of rigidity ratios for each of the six trusses, and with two types of analysis for each rigidity ratio. The following section presents a detailed discussion for the five-panelled truss of IS.0 m span. Results for this case, as well as for all other cases, have been illustrated further in graphical form. The principle results of the trusses which consist of 3, 4 and 6 panels will be presented in tabular form in Section 6. The single- and double-panelled trusses were not discussed as they do not appear

Buckling contribution

P, = 2.363 tons P3‘ 5.568 toes

to the analysis of steel trusses

453

P2 * 4726 tons P4 = 6.451 tons

Fig. 9. Six groups of trusses.

to be important with respect to common practice. Nevertheless, graphs of all cases are present for the sake of completeness. THE FIVE-PASELLED 5.1

TRUSS

Tire raJio O~momenJs for rhe two analyses sradied

Figure lO(a,b) presents the ratio of moments, for the two analyses under study at the different nodes, versus the rigidity ratio. From Fig. IO(a) it could be observed that the moment induced by considering the Normal Force effect is greater than that induced by the classical analysis for the tension members of the lower chord members (3,7, 11) as well as the diagonals (2, 6). This increase in moment may reach two times as much. It could be observed, moreover, that this moment increases with the increase in the rigidity ratio value. i.e. it increases as the joint tends to move from the semirigid to the rigid state. On the other hand, the moment in the compression members due to the Normal Force effect de-

creases by 2-30% [Fig. 10(b)]. This moment is observed to decrease with an increase in the value of the R.R. 5.2 Secondary stresses due to momenfs Figure 10(c) represents the ratio of stresses induced by the bending moments (UM)to the stresses induced by the Normal Force (uN) vs the R.R. varying from the ideal hinge case (R.R. = 0.0) to the completely rigid case (R.R. = 1.O).The dotted lines represent the analysis including the Normal Force effect, while the heavy lines represent the classical approach. This stress ratio is observed to vary from 0.0 to 0.08 for the top chord compression members. This value will fall to 0.04 for the diagonal compression members. On the other hand, this ratio is 0.14 for the bottom chord tension members, and will fall to 0.02 for the diagonal tension members. Moment stresses are observed to be greater in the analysis including the Normal Force effect, for the tension members. The opposite is true for the

454

EL-SAYED

BAHAA M.-\CHALY

/ CASE

100

. 01

02

03

04

OS

06

0.7

0.6

0.9

Cf

FIVE

PANELS.

”

RR

t.0

Fig. 10. (a) The ratio of moments for tension members. Fig. 10. (CL The ratio of stresses induced by the bending moments.

l%

09s

BI

0.6 -

:

:’ I’

LjJpi.

/’

IX-

,’ I’

9

,’ I’

! ,’

cl&CASE OF FIVE

PANELS

;

j”

0’

,’ I’

0 65

;

I!’

0.3,” <’ I’

// i

)R.R 01

02

03

04

OS

06

07

06

09

to (dl

(bl

Fig. IO. (b) The ratio of moments for compression bers.

mem-

Fig. IO. (d). Analysis including the normal force effect and the claswal analysts.

Buckling

contribution

to the analysis

1

105.. (raft

4%

compression members. This observation coincides with the result obtained in Section 5. I. This can be attributed to the fact that the Normal Force does not change with a change in the rigidity ratio value. Results in members I and 3 contradict the previous discussion. as the moment stresses in these members reached 60% and 45%. respectively. This increase in moment can be accounted for simply by considering the model which was used for the analysis. In such model, the hinged and roller supports were replaced by vertical zero length members fixed at their lower ends. This resulted in the great increase observed in the moments. This model was selected solely for the sake of the anal!.sis. A fact not to be overlooked, is that within the values of R.R. allowable for a node to be considered as a hinge, i.e. (R.R. = 0.0-0.25). the moment stress ratio never exceed 12% except for members 1,3 (where u,&rN reached 21% and 39%. respectively), and this was only due to the prementioned reasons.

.--

COMPRESS

of steel trusses

ION

a0

I ________----_______-__-I -WITHOUT

----WITH

07

191

5.3 The relation of maximum normal stresses to the allowable

0.7

RR

00

01

02

03

06

05

06

07

06

09

IO

(El

Fig. IO. (e) Ratio of average actual stress for compression

I .o

stress to the allowable members.

3.._ crau

09

0.t

0:

0.6

TENSION

MEMBERS.

-WITHOUT ___ WITH.

0.5

04

0.3

161 -___-- _---RR

_

01cl20)Ob050607C80910 III

Fig. IO. (0 Ratio of average actual stress to the allowable stress for tension members.

stresses

Maximum normal stresses were calculated for all members by summing up the stresses due to the moment and those due to the normal force effect. Figure IO(d) represents the analysis including the normal force effect (heavy lines) and the classical analysis (dashed lines). The sections used in calculating these stresses were those obtained from proportioning the truss members while considering ideal hinges at all nodes. From Fig. IO(d) it could be observed that all maximum normal stresses for the compression members did not exceed the allowable limits, except for member 5 in which there was an increase of 8% above the allowable stresses. The same applies for all tension members, where stresses did not exceed the allowable. except for member 3 which witnessed a 2% increase. A qualitative demonstration of a possible gain in the weight of the steel structure was attempted by plotting a line which represents the averages of stress ratios for all tension members, and a similar line for all compression members. For the compression members this line assumes values ranging from 0.85 to 0.92. For the tension members, these values will be 0.76 to 0.85. However, these low values for the tension members are due to the state imposed by member 6, which exercised exceedingly low stresses as observed from Fig. IO(d). Figure II represents the same qualitative demonstration for the six-panelled truss and from which the average stress ratio of the compression members will appear to vary from 0.88 to 0.91 and the corresponding values of the tension members will be 0.88 to 0.96. An optimum weight analysis must actually be conducted in order to calculate the save in weight nections.

due to the use of the semirigid

con-

EL-SAYED

BAHAA

MACHALY

ITH

%UCKlIW

,IY1.*-

_,A...

AVERA%E

TENSION

.-.-

-I0.66 f 2’

L

Cf.114

i

f

am

1

“;

asr

3

0.9:

t u ”

QQC

b” . .” b

WlTHOUT

BUCKLING

0.72

a88 aa6 0.84

0.62

0.64

a40

i

0.62

0.d

Fig. II. Qualitative

5.4 The mcrrimffm

vertical

demonstration for the six-paneled Iruss.

deflection

The

following table represents the maximum deflections values for the different trusses cakAated for the dead and live loads. This maximum deflection was found to be less then the dejection allowed by the specifications. The maximum deflection for

Number of panels Maximum deflection/span

analysis including the normal force effect demonstrates a 1% increase than that of the cfassica1 approach. It could be further observed that the maximum deflection decreases with the R.R. increase. the

6

5

Ii860

111154

4 111518

3

2

I/2786

MO79

457

Buckling contribution to the analysis of steel trusses 6. TABULAR FORM RESULTS FOR SIX-, FOUR- AND THREE-PASELLED TRUSSES

Six-panelled

Tension members

Compression members

B

A

M4IMB

Type

truss

Position

R.R. = 0.1

R.R. = 1.0

L.C.

1.05

2. I

D

1.05

2.0

T.C.

0.75

0.97

D

0.89

0.97

A

B

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = I.0

lo-? x 0.23 lo-’ x

IO--’ x 5.48 lo-’ x

lo-’ x 0.23 lo-’ x

lo-? x 4.66 lo-? x

0.88

0.96

0.88

0.92

0.78

9.78

0.19

3.1

0.42 x 10-z 0.26

8.8 x 10-z 10.3

0.43 x lo-’ 0.27

9 x lo-’ 10.3

0.87

0.91

0.87

0.89

x 10-z

x lo-’

x lo-?

x lo-?

The previous notations given in the table designate the following: T.C. = top chord members, D = diagonal. L.C. = lower chord members, A = analysis of Section (3), B = classical analysis.

Four-panelled

truss

UMIUN

UmaxlUall

A

MA~MB

B

B

A

Position

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

R.R. = 0.1

R.R. = 1.0

Tension members

L.C. D

1.0 1.11

3.8 1.66

0.01 0.01

0.03 0.17

0.0 0.01

0.01 0.09

0.85

0.9

0.85

0.9

Compression members

T.C. D

0.73 0.84

0.93 0.94

0.00 0.00

0.06 0.01

0.00 0.00

0.08 0.04

0.69

0.81

0.69

0.76

Type

Three-panelled

truss

B

A

MA/MB

B

A

Position

R.R. =O.l

R.R. = 1.0

R.R.

R.R.

R.R.

R.R.

R.R.

R.R.

R.R.

R.R.

Type

= 0.1

= 1.0

= 0.1

= 1.0

= 0.1

= 1.0

=O.l

= 1.0

Tension members

L.C. D

1.1 1.2

2.2 1.4

.005 .002

0.26 0.17

.005 .002

0.20 0.24

o ’5

0.55

0.5

0.55

Compression members

T.C. D

0.81 0.83

0.98 0.98

.015 .015

0.19 0.24

.018 ,015

0.24 0.29

0.8

0.85

0.8

0.85

EL-SAWED BAHAA

458 COSCLUSIONS 1.

The use of semirigid connections is highly recommended for the analvsis of truss members. In this members

case.

are subjected 3.

3.

4.

5. 6.

it is ineviiable

as parts of a frame to both normal

to treat

the truss

structure.

i.e. they

force

and bending

moment effect. The classical approach will involve lower values of moments for the compression members than those obtained from the analysis including the normal force effect. The tension members will on the contrary exercise an increase in moment value. For both tension and compression members, the use of the semirigid connections will induce moments which will increase with the increase of the rigidity ratio. The ratio of the moment stress to the normal force stress does not exceed 12% for practical cases where the rigidity ratio of the semirigid connection is less than 0.25. The deflection is not considered as a hindering factor in the analysis as it is safe in all cases. An optimum weight analysis has to be conducted in order to calculate the gain in weight due to the use of this approach in the analysis.

Ackno~c,IPdgments-The author wishes to express his deep gratitude to Professor Dr. M. S. Aggour for his interest through stages of this research work.

MACHALY

A lot of gratitude must be expressed towards Professor Dr. Ahmed- Aziz Kamal. Dean of the Faculty of Engineering. Cairo University. for his kind sponsorship of the hours of computer time necessarv for this research work. Also many thanks are due the computer centre staff. who facilitated this work with their cooperation. Thanks are also due Mr. Alaa Abdel-.Azim for his patience in typing the manuscript.

REFERESCES

I. F. Bleich,

Bwkling Srrengflr of .Vercll Srrltcrlcres. McGraw-Hill. New York t 1952). 2. A. Bolton, A quick approximation of the critical load of rigidly jointed trusses. Srnrcr. Engrtr. London. England (March 1955). 3. J. C. Badoux, SrabilifP des Consfrm-rions. lnstitut de la Construction metallique-Ecole Polytechnique Federale de Lausanne (1969). 4. Effective buckling lengthes of structural elements. J. Aeronaut. Sot. (September 1936). 5. W. Merchant, The failure load of rigid jointed frameworks as influenced by stability. Strrrctrrral Engnr. London, England (July 195-t). 6. Adel Helmy Salem, Buckling of rigidly-jointed plane trusses. ASCE (June 1969). , M. El-Sayed, Analysis of steel frames with semirigid connections. 19th Symp. of Statistics and Computer Sci., Cairo University, Cairo. Egypt. T/re Srubility of 8. M. R. Home and W. hlerchant. Frames. Pergamon, Elmsford. NY (1965).