the elastic lateral stability of truss_1960

7/28/2019 The Elastic Lateral Stability of Truss_1960

http://slidepdf.com/reader/full/the-elastic-lateral-stability-of-truss1960 1/9

May, 1960 147

The Elastic Lateral Stability of Trusses

by M. R. Horne, Sc.D., A.M.I.C.E.

SynopsisThe article gives an approximate nalysis, using

the energymethod, or the elastic stabi lityout oftheirplane of trusseswithoutstanding compressionchords, the tension hord being held in position.Previous discussions of this problem have suallyconsidered it in erms of thestabil ity of the com-pression chord as a member subjected to variouselasticestraints. In the present treatment,hetruss is considered as a whole, account being takenof the flexural and torsional rigidities of all the mem-bers, and of part ial estra int of the tension chordagainstwisting. The analysis isnterpretednrelation to a number of common arrangements of

vertical and diagonal web members. The analysisis derived for a truss in which the compression chordis of uniform section, carrying a uniform thrust,bu t he use of some of the results asapproximatecriteria of stability for non-uniform conditions isalso discussed.

Introduction

Thisarticledealswith he ateralstability of anelastic plane russ with parallel chords, such as th atshown in Fig. 1 a). The nalysis pplies to nyarrangement of vertical and diagonal web members,the only estriction being that no account is takenof intersectionswithin the web. The russ s ub-jected to any combination of uniform bending moment

and overall axial load. The compression chord ABin Fig. l (a )) sustains an axial load P and is laterallysupportedonlyby he web members of the russ,except a t A and B, where it is assumed to be restrainedagainst deflection out of the plane ABCD. The chordCD may carry any axiaload Q (tensileor compressive),and is laterally supported at the panel points. Allow-ancesmaden the analysis or lastic estraintsagainstwisting of th e chord CD about ts longi-tudinalaxis, hese estraints being applied at panelpoints. Although this simplified form of loadingis assumed, some of the results may be applied approxi-mately to a truss more realistically loaded as shownin Fig. l (b ) provided the vertical loads are applied

to he lower chord only, and he web members donot sustain compressive axial loads which are a highproportion of theiraxial loading capacities as pin-ended struts.The analysis s of particular nterestinrelation to trusses composed of tubular members,but is not estricted to such cases. The ointsareall assumed to be rigid, and he members prismaticbetween panelpoints.For the sake of analysis it isalso assumed th at he chords re ach of uniformsection, while each set of web members (for example,the verticals) is of uniform section throughout hetruss.

When theaxial load P in the compression chordreaches a criticalvalue, the russ buckles laterallyas shown in Fig. 2(a).The buckling mode maybe

in a single half-wave, or in a series of almostequalhalf-waves. The compression chord is acted upon bya series of torques,moments and shear forces fromthe web members.

JI

(b)0 LAT ER AL RESTRAINTS.

TRUSS UNDER (a) UNIFORM THRUST AN D TENSION

( b ) VARYING BENDING MOMENT.

FIG. 1

' In most treatments of this problem, the tabilityof the compression chord is investigated by assumingthat he estra ints from the web members may berepresented by the action of springs lying n a hori-

zontalplane, as shown in Fig. 3(a). Bleichl suggestsmethods for solving this idealisation of the problem,allowing for changing axial load and cross-sectionof the compression chord and variations in the spacingand stiffness of the springs. For he simpler caseof a uniform axial load in a chord of uniform section,withequal spacing and stiffness of the springs, anearlyapproximate solution by Engesser2 gives goodresults. If the springsn Fig. 3(a) require a forcek to produce unit deflection, andhe springs are

TRUSS IN THE BUCKLED STATE.

FIG. 2.



148 The Structural Engineer

I T A B L E 1 IMEMBER FLEXUUAL

RIGIDITY.IGIDITY

TORSIONAL

I ITENSION BOOM

COMPRESSION BOOM

VERTICALS

DIAGONALS~ ~ ~ ~ ~ ~ _ _ _ ~

B = ( B ~ + 3~)i n 3+c = (c , + c 5 ) sm3+A = {FLEXURAL STIFFNESS OF TPANSVERSL

MEMBER3 A T EACH LOWER PANELPOINT. }

spaced at intervals S , Engesser replaces the springsby a ontinuous lastic medium which resistsunit

kdisplacement by a force of - per unit ength of the

chord, as shown inFig. 3(b). It is then ound tha tthe chord buckles in half-waves of length I where

S

. .

in which E I is the flexural rigidity of the chord forbending out of t he plane of the russ.The criticalvalue of the axial load P is

Engesser showed experimentally that equation (2) is

remarkablyaccurate provided -> 1 -8 and his has

been confirmed theoretically by Bleichl. Equation(2) has been used extensively for bridge trusses in

which the web members in Fig. (1) arealmost com-pletely irection fixed a t theireet.The Engessersolution assumes that t t lc chord is infinitely long andthat the ends are fixed against lateral dcflcction withsufficient rigidityor the buckling load not o bcreduced by buckling a t the ends.The effect of com-pletely free ends has been discussed by Zimmermann3,whileChwalla4 considers ends of finite rigidity.Thecase of chords of finite length in a continuous elasticmedium, but with ends fixed against lateral deflection,has been discussed by Timoshenko5, who deals notonlywith uniform axial loads in the chord, but alsowit h axial loads varying parabolically.

Reference to Fig. 2 shows that he compressionchord in a truss is not only restrained against lateral

deflection (in direction OX), but tha t the chord mem-bers also provide restraint against the twisting of thechord aboutts longitudinal xisparallel to 02) .The rotation of the chord at panel points about axes

IS

parallel to OY is also partially restrained] this restraintbeing due to twisting n the vertical chord membersand to bending and twisting in the diagonal members.The buckling of a uniformly compressed, uniformchord member withequally spaced lateral and rota-tional estraintshas been discussed by Budianskyet ale. Forhe compression chord in Fig. 2, thistreatment would allow for restraints at panelpointsagainst deflections parallel to OX and for restraintagainst rotation about axes parallel to OY, but wouldignore the effects associated with the twisting of thechord about ts own longitudinal xis. The effectof bending momentn the vertical members, andtheir nteractionwith he wisting of the chordhasbeen discussed by Hrennikoff‘who, however, ignoressome of the other restra ints .An exhaustive s tudy of the problem of buckling of

chords npony russes has recently been conductedby Holt8. It will e appreciated that a completesolution is extremely involved, and Holt’s trea tmentis too complicated forpractical use. In a review oftheoretical and experimental work on the bucklingof top chords in pony russes, Handag comes to heconclusion th at th e simple formula of Engesser gives

results as good as any for the collapse loads of actualtrusses, and that the complications of most treatmentsarenot worthwhile. This sperhapsnot urprisingin view of Bleich’s findings inrelation to Engesser’sformula. The trusses considered byanda arecomposed of open section members, so that effectsassociated with he resistance of the chord and wcbmembers to twistingare of no mportancc providedlocal torsional buckling does not occur. Whenclosed sections are used, and buckling rigidities becomeof the same order as flexural rigidities, the Engcsserformula can no longer be expected to give reasonableresults. The essential features of the Engesser solutionmay, however, be retained by considering all restraintsto the chord from the web members to be distributedcontinuously as in a special sort of elastic medium.It is thus possible to make a sufficiently accurateallowance for all the restraints which have been con-sidered in more elaboratemanner by he manyauthors who havewrittennheubject.hisgeneralised solution to the buckling of a comprcssionchord in a truss is the subject of the present paper.

I t is a common feature of the treatments previouslygiven that they consider the feet of the web membersto be completely or partially restrained, without anyallowance for the behaviour of the tension chord.Because of the high rotational estraint at he feetof the web members in bridge trusses, he neglect ofthe tension chord is justified, but trusses forming

part of a building tructuremay be n a differentcategory.Subsidiary members such asshectingrails,connected to he tension chord, will usually sufficeto restrain it in position laterally, but may not havelarge enough flexural stiffness to offer significantrestraintagainst wisting. If tht: tension chord is oftubular section, i t will then ontribute ppreciablyto the stability of the truss . A solution which allowsfor the resistance to twisting of the tension chordmust necessarily treat the truss as a whole, and thisis the basis of what follows. The analysis is basedon the energymethod, allowance being made forthe resistance to bending and twisting of all the mem-bers, and also for partial wisting estraint applicdtohe tension chord.henalyticalesultsre

summarised in Table 2 on pages 150 and 151. Th esolutions arenterpreted specifically in relation tosix arrangements of vertical and diagonal members,butarenot restricted to these arrangements.



M a y , 1960 149

Notation

The notation is summarised in Figs. l (a ) and 2 andinTable 1. The flexural rigidities B2, Bs , B4 and

B5 are the values of E I for bending out of the planeof the russ ( E modulus of elasticity, I moment ofinertia).The torsional rigidities C1 to CS arehevalues of GJ (G elastic shear modulus and J the St.Venant torsion constant). The flexural stiffness ofthe members (not shown) which restrain he tension

chord CD against twisting is defined by the quanti tyA , which has the dimensions of the flexural andtorsional rigidities ((force) X (distance)2). If theexternal members attachedohe tension chordproduce a torque of T at eachpanelpoint for unit

angle of twist of the chord, then A =- where H is

the depthbetween centres of chords and S is the distancebetween panel ointsFig. (a)).The symbol 0

denotes the angle between the diagonal membersand he tension and compression chords. The engthof the truss, between points a t which the compressionchord is restrained gainstateraldisplacement, isdenoted by L, while l is the half-wavelength for the

compression chord n the buckled state.The hrustin the compression chord is denoted by P.

Various symbols, ( p , q, B , C, D, ) are used todenoteunctions of the abovequantities, nd redefined as occasion arises in Table 2. In the analysiswe take axis 02 along the centre of the tension chord(Fig. 2(a)) , OY in the plane of the truss perpendicular

to 02, and OX perpendicular to OY and OZ. Theangle of twist of the tension chord is denoted by 8,and of the compression chord by 02, while the de-flection of a pointon the compression chordout ofthe plane OYZ is denoted by U . Taking any particularweb member FG n he deformed state (Fig. 2(b )),the angles between the tangents at F and G and thestraight line FG are denoted by P1 and p2 respectively.

H2

S

General Analysis

It will be assumed that , in he buckled state , thelateral deflection of the compression chord (in directionXO) is given by

xzU =H 0 sin- . , . .

l? * (3)

where z is measured from one end. I t is assumedthathe tension chordemainstraight. Thesedeflected forms neglect the local distortions producedin the chords by the bending and twisting resistanceof the web members, andare justified provided theflexural rigidities of the chords are large compared with

the flexural and torsional rigidities of the web members.The angles of twist of the tension and cornpressionchords (clockwise about 02) arerepresented by

81 =alesin-

O2 =a&in -

. . . . . .l ’ (4)

I ’ (5 ). . . . . .

The external work, per length l of the truss, due tothe hrust P in the compression chord s Up where

1

0

Since the tension chord is assumed to remain straightno work is done by he force Q. Theexternal workUp has to be equated to the total stra in energy dueto buckling in the members of the truss.

The tension chord has strain energy due to twistingUcl where

l

The compression chord has strain energies uB2 due tobending and Uc2 due to twisting where

l

l

We consider now the wistingand flexure of any

diagonal member FG (Fig. 2). The line FlF2 is takenthrough F perpendicular to FG and in the plane OYZ.The line GlG2 passes through G and is parallel to F1F2.

Since there is continuity between the tension chord

and diagonal FG at F, the angent o FG rotatesduring buckling about FlF2 throughhe angle

(a10 sin 7. in o where z is the distance of F from

the origin. Similarly, the tangent to FG a t G rotatesabout GIG2 through the angle

)

( l )z ~s ine),i n@ + 0 cos x (z + . o s 0l!

Thechord FG rotates about FlF2 through he angle

Fig. 2(b) are thus

We now make the assumption that H and S are smallcompared with l , and ignore S in comparison with z.

Hence

p1=( 1 - l) 8 sin-.in 0 , . . . .z

l(10)

If we denote the bending moments at F and G inthe member F G by M1 and M2 respectively as shownin Fig. 2(b), hen he slope-deflection equations give

(12)



MW, 1960 151

t

I =’ I

+B = 6+Sln3 9 , B 3 = 0

C = C,SlN3 $, c 3= o

l - - - - - s 4--.t

= F L E X U R A L STIFFNESS OF T R A N S V E R S ET T A C H M E N T S A h2,-.AT EACH LOWER P A N E L POINT. 5

A S {c, } WITH C, R E P L A C E D BY Cl .-

IP - -- H2

r - l

SAFE RESULT, ALL LENGTHS,l

A = 0

c,= 0

A = Oc2- 0

c ,=0c2=0

t For- ead -x2q 2 7 iz



152 The Structural Engineer

If there isone diagonal number for each length S of thetruss (as n Fig. 2), the mean bending energy in the

diagonals per unit length of the truss is

Substituting for 8 1 and p 2 from equations ( 10 )and ( 1 1 )

in equation ( 1 4 ) ,and thence for A UB4in equation ( 15 ) ,the value of UB4 becomes, on ntegration,

{(I -a112 + ( I - a l ) ( l - 4 ) + . ( I -a2)2 3 tan 0

Theend F of the diagonalFG wists aboutFG

through the angle

twists through the angle

Again, neglecting S in comparison with z, the twisting

energy A U @ in the diagonal is

AUC4=m{4 (a1- 2) 8 sin-.XZ co s 0l

The total twisting nergy in the diagonals is Uc4 where

C

whence

The bending and twisting energies in he verticalmembers of the t russ m ay be derived from the valuesfor the diagonalmembers as follows. It is assumedthat th e vertical members are spaced at intervals of

S =H ot 0 > as in Fig. 2(a). The angles p 1 and p 2 for

any vertical F K areobtainedbyputting 0 =-2

in equations (10) an d ( 1 ) , and he bendingenergyA U,3 is then obtained from equation ( 1 4 ) by putting

0 = 'z and replacing B4 by B3. The otal bending

energy u B 3 in all the erticals is obtained from equation(15), but tan @ must here be retained since it corres-

ponds to-and defines the spacing of the members.

Hence

UB3= (( a1)2+ 1 ..--- a l ) ( l -- a d +(1- ~ ) ~ }

(20)

X

x

H

Similarly we obtain he twisting energy U@ in theverticals,

Finally we require the energy stored in thesub-sidiary members which restrainhe tension chordagainstwisting. heestrainingorquecrnit

T x2 T AS l s H2

length is- 1 8 sin f nd since -=- the total re-

straining nergy U A over the buckling length I is

l2

0

I

The en erw equation for the length I of the truss is

Upon substituting for the expressions for the separate

terms, we have

P H z = u 1 2 Cl+a22 C2+4B co t 0 +( C + C 3 ) tu n 0

+ +n2a12A +n2(a1 -. a2)2 C co t 02

+4n2[(1-- a1)2+(1- - -a l ) 1 - a z + ( l - a423 (B+ B3) tan 0

where . . . . D . . ( % )

I7tH '

n = - -

B =B4 sin3 0 ,

C =C4 sin3 0 .

L

The buckling lengthI

may assume any value -mwhere m is a digit.For each I , the corresponding

buckling load P is obtained by putting - = 0 and

=0 , so that P is a minimum with respect to the

ap

aa 1

aa2

arbitrarily chosen coefficients a1 and a2.

Hence

2(3- 2al-- Q) ( B +B3) fan 0 - - (a 1- 2) C co t 0

Equations (25) and (26) may be solved for a1 and a2

and he resulting values of a1 and a2 substituted inequation (24). If P = P1 corresponds to m = 1 ,

P =P2 to m =2 etc., t will usually be most con-venient first to calculate P1 and P2 ; if P1 <P2, thenP1 is the requiredsolution. If P1 > P2, then P3 iscslculatcd ; if then P2 <P S , P2 is the solution, andso on. Thc complete equations for this generalsolution are given at he beginning of Table 2 on

page 150. Thediagrams at he head of the ableshow how the results may be applied to trusses withsome common arrangements of web members by'suitably adjusting the ignificance of the termsB and C.



M a y , 1960 153

The above procedure is lengthy , and in some casesa more convenient solution, on the safe side, may beobtained if i t is assumed tha t I may possess any value

less thanL. We then have- 0 or, sincep

a j

Hence

a12 A +(a1- 2 co t 0

+4[(1- 1- (1 -Q) (1- 2)+ (1- 2121 ( B+B3) tan 0

- B2 . . - (27)-n4

The valueof P now appears as theolution of equations

(24)to (27),provided I <L or n <2 . The general

solution for P is not explicit, bu t an expression for Pmay be obtained for the following special cases.

( l ) A =0 , c1=c2,(2) A =0, c1=0 ,

(3) A =0 , c2 =0 ,(4) c1=0, c2 =0.

Resultsfor hesefoursets of conditions aresum-marised in Table 2. Their derivation will be illustratedby considering the second case, A =0, C1 =0.

TheCase A =0 C1 =0

If we put y =(7)an20,

equation (25) becomes

1 -2 y(1 --l) =

1 + 4 y(1- 2 ) (28)

and if ~2 =12 tan@- ' $- equations (26)

and (27) become respectively1 +4y'

B2n432 ( 1- 2 =-+B3

. . .(30)

Hence

By substituting for a l , a2 and n in equation (24), weobtain

Equation (33) applies provided the length I =n x Hobtained romequation (32) is less than he lengthof the truss L, that is provided

l L32 1 I, . 1,

When L is smaller than he limit given by he in-equality (34), the buckling length I must be takenequal to L, and the value of P is obtained by sub-

stituting n

Hence it is

L

7EH

found

=-

1+ F

in quations (24), (25)

that

and (26).

1+4 B c o t 0 +(C+C3) t a n 0 l

A safe solution for P will result if the member isassumed to be infinitely long, whatever itsactuallengthmay be. For sucha afe olution,equation(33) is appropriate provided the value of n given byequation (32) is real, that is, provided

2/w.t3+B3) <1.'1

c2

When thiscondition is not satisfied, it denotes that the bucklinglength is infinitely long, and a safevalue or P isobtained from equation (35) by putting L =m, that is

1

The interesting feature of equations (33) and (36) isthat they provide a safe estimate of the buckling loadof the truss without reference to the length betweenlaterallyestrainedoints. These safe estimates

are given for all the special cases in Table 2.NumericalExample

TheWarren truss n Fig. 4(a), 60 feet long and5 feetdeep between chordcentres, s composed ofcircular teelubes of the sections indicated.Themodulus of elasticity, E , is 13,000 tons/in.2 and heelastichear modulus, G, is 5,000 tons/in.2 Eachpanel point on the lower chord is partially restrainedagainst wistingabout the longitudinalaxis to giveA =150 x l o 3 tons inz., corresponding to a stiffnessat each panel point of 3,500 tons ins.

Thevalue of P which would cause instability,allmembers being assumed to remain elastic, may beestimatedn arious ways, andhe solutions re

summarisednTable 3. The first resultquotedsbased on the general solution in Table 2, it being foundthat three half-waves gives the minimum value of P

(313 tons). With wo half-waves, P =364 tons, andwithfour half-waves, P =347 tons. A cross-sectionof the truss n the deflected state when three half-wavesform is shown in Fig. 4(b).

Theanalytical solutions given inTable 2 for thethree cases ( A =0, C1 =Cz), ( A =0, C1 =0) an d(Cl =0, C2 =0) have been used in lines 2 to 4 ofTable 3 to provide safe estimates of P. The valuesof the various flexural and torsional stiffnesses assumedin ach case arequoted, as also are he bucklinglengths 1. It is interesting to note t hat heactualbuckling length (20 feet, line 1 of Table 3) is smallerthanan y of thoseobtained n hesubsequent safeestimates of P, eing slightly less than he shortest(26 feet, line 4) . The shortest buckling length obtainedfrom the safe estimates may, in fact, always be used



154

P

-7-5'

i

l CCOMPRESSION CHORD

W E B MEMBER S .

4%' '. x 0.176"

1 = 5 . 6 0 IN^

I

ILTENSION CHORD 6%" D. x 0 ~ 2 1 2 " . I = 22.0 IN^

( 4

WARREN T R U S S

( b ) DEFORMATION AT

(G) DIMENSIONS.

BUCKLING L O A D .

F I G . 4.

P-

l TABLE 3 . l

4 l7 66634550, = O , C 2 = 0

5 I33634550NGESSER

toobtainan approximation o heactual bucklinglength.Finally, if the Engesser formula is applied

(line 5) , allowance being made forhe incompletetorsional restraint on the lower chord, hen P =133tons. The Engesserormula thusnderestimatcsthe critical load in this case by some 57 per cent.

Application to TrusseswithNon-UniformThrust inCompressionChord

The analysis ontainedn this rticle as beenderived by reference to the case of uniform thrus t inthe compression chord.The "safe " results con-tained nTable 2 may, however, be used toobtainan approximatecriterion of safety or russes withnon-uniform thrusts,nd also non-uniform cross-

sections, since these "safe " results do not involvethe ength between points of support. A truss willbe stable provided the thrust P does not exceed thevalue given by the appropriate "safe " equations in

Table 2 at any section along the russ. The memberproperties assumed in any such calculation are those

corresponding to heparticular section of the russconsidered, certainroperties being reduced unt ilone of the cases contained in Table 2 becomes relevant.

An exception to the validity of the above criterionmay occur when the axial thrusts in the web membersbecome appreciablen relation toheir individualbuckling loads as pin-ended members. Hrennikoff7in his treatment considered the effect of such thrusts,but his is asubject equiring more detailed study.It may be noted t hat in many russes, the effect ofaxial hrusts n some of the web members will belargely compensated by equal tensions n djacentmembcrs. Moreover, the most critical section of thetruss for buckling of the compression chord will

usually correspond to ha t section where theaxialloads in the web members are small. I t may thereforebe concluded tha t the effect of axial hrusts n webmembers may usually be neglected.



M ay, 1960 155

1.

2.

3.

4.

5.

6.

References 7.Bleich, F., “Buckling Strength of Metal Structures.” Chap.VIII. McGraw-Hill,952. 8.

Hrennikoff, A . , “Elastic Stability of a Pony Truss,” Pub .I n t . A s s . Bridge and Structural Eng., Vol. 111, 1935, p. 192.Holt, E. C., ‘ I The Stability of Bridge Chords Without LateralBracing,” Reports 1 to 4, Colum n Research Council. U.S.A.Handa, V. K., ‘‘Pony Truss Instability,” M.Sc. Thes is ,Queen’s University, Kingston, Ontario, 1959.

Engesser, F. , “Die SicherungoffenerBriicken gegen Aus-knicken,” Zentralblatt der Bauverwaltung, 1884. p. 415: 1885, 9.

Zimmermann, H., “Die Knickfestigkeit eines Stabes mit

Elastischer Querstiitzung,” W. Ernst & Sohn, Berlin, 1906.Chwalla, E., “Die Seitensteifigkeit offenerParrallel-und-Trapeztragerbriicken,” DerBauingenieur, Vol. 10, p. 443,1929. The Counciloulde glad to consider the publication

p. 93.

Discussion

Timoshenko, S., “Theory of Elastic Stability,” Chap. 11.

McGraw Hill, 1936.Budiansky, B., Seide, P. and Weinberger,R. A., “The Buck-ling of a Columnon Equally Spaced Deflectional and Rota-tional Springs,” N A C A Tech. Note 1519,1948.

of correspondence in connection with the a6ove paper.Communications on this subject intendedor publicationshould be forwarded to reach the Institution by he

30th July, 1960.

Book ReviewsAdvancedStructuralDesign, by Cyril S. Benson.

(London : Batsford, 1959). gin. X 6in., 329 plusxiii pp. 50s.

This interesting nd practical book consists essentiallyof the material one would compile in designing seven-

teen structures. Of these ten deal with structural steelprojects, five with reinforced concrete structures andwowith brickwork construction.The problems analysedare diverse : included are he complete designs for agrain silo, highway bridge, tank tructure, 120 feetspan shed, two-bay portal plant house, theatre balcony,multi-storey office building,bunkers,gantries and achimney.

The work s clearly presented in a digestible form,andhe calculations re skilfully augmentedwithexplanation where necessary. I t should help to broadenth estructu ral horizons of the many designers who,unfortunately, are only trained in dealing adequatelywith one structural mater ial and rarely get t he chanceto see what the other fellow does in practice.

Unfortunately, he book must lose a great deal ofimpact in that BS.449 : 1948 is used for the steelworkdesigns. It may also be said tha t the designs tend tobe dated and manywill regret tha t there is no referenceto the Plastic Theory in steelwork, or to prestressedconcrete and tha t ittle guidance is given to the studentdesigning welded structures.

Despite these shortcomings, the book may prove auseful stimulant for students planning t o take heAssociate membership examination. R.H.

CivilEngineeringContractsOrganization, by JohnC. Maxwell-Cook. (London : Cleaver-Hume Press,1959). S+ in. x 54 in., 220 plus viii pp. 22s. 6d.

The Author sets out comprehensively the procedurefrom the inceptionof a scheme by he employer toits development in the ontract stage, giving therelationships between the variousparties concerned,advisory and executive, for the successful completionof a civil engineering project. A survey is given of thecontract documents in general use with definitions ofcontract ermsand useful comments on the clauses,and some piquantobservations on financial arrange-ments.

The personnel required for the execution of the com-plete scheme is described from top level to the abourerand detailed as regards their duties and relationships.The mportance to he client andcontractor of apreliminary site survey and repor t by the consulting

engineer on ground conditions has not, perhaps, beengiven sufficient place.

Thehapters n specifications areerhapsotsufficiently up to da te particularly regarding concrete

which is mostly specified by quality controland strengthnowadays, nor is there anymention of materials testing.Most large contracts are usually now equipped with asite laboratory for this purpose.

The site organization section contains many useful

suggestions and aids t o economy and smooth runningof work on a construction site, and stresses the desirefor design to be allied to easy and rapid execution a tsite. The Glossary forms a welcome appendix.

The book will be interesting and informative t o allconcerned in he development and execution of civilandtructural engineering projects. F.T .B.

LinearStructural Analysis, by P. B. Morice,D.Sc.’,Ph.D., A.M.T.C.E., A.M.1.Struct.E. (London :Thames& Hudson, 1959). 9& n. X 6g in, 170 plus xii pp.,35s.

Since the 193O’s,methods such as strain-energy andleast work have steadily been giving way to methods

of successive approximation for the analysis of manyforms of elasticstructures. Quite recently, however,the introduction of matrix algebra to st ruc tural nalysishasgreat ly increased the usefulness of the classicalapproach. Moreover, electronic computers are availableto perform the tedious numerical work involved inprocesses such asmatrix inversion, thusmakingpossible the solution of problems having a high degreeof indeterminacy.

This book forms an excellent introduction to hesubject. The first two chapters are largely devoted tothe basic concepts of strain energy, Castigliano’stheorems, and influence oefficients, and include avarie ty of examples. There follows a trea tment of the

question of the degree of indeterminacy o f structuresby a method which is sta ted tobe without exception inits application to skeletal tructures nd, s uch,represents a notable advance on previous methods.

Matrix algebra is then introduced, particular at ten -tion being paid to computational procedures, thedescription of each type of matrix operation beingaccompanied by a simple worked example. The follow-ing chapters deal with scale factors, transformation ofco-ordinate systems and a number of points concerningrelease systems (of which the uitable choice maysimplify the analysis).The final chapter deals verybriefly with the programming of an electronic digitalcomputer for structural analysis, and a n appendixgives four worked examples which nicely illustrate the

methods developed in the main text . There arenumerous references which will help the student who

wishes to pursue fur ther any spect of the subject.

E. M.

the elastic lateral stability of truss_1960

Documents