TFIGURE C 2

(P /P )< = 0,452 c p

I I

FIGURE C3

i I

FIGURE i:4

From f i gu re C.4 there fore , the f a i lu r e load as a percentage o f the

p las t ic co l lapse load is 0,73, i * e . the expected f a i lu r e load

using Schol : ' in teract ion formula is 111 kg/bay.

Failure loads predicted by Scho l l ' method, fo r the frames tested,

were calculated in the same manner.

CALCULATION OF LOADING POINTS TO SIMULATF APPKNDIX D

A UNIFORMLY DISTRIBUTED LOAD

The loading pattern i l lu s t ra ted in f igu re D.l is used to simulate

a uniformly d is tr ibuted load applied to the r a f t e r s .

In the above diagram:

P » to ta l load on the span

* w x L

w = equivalent uniformly d is tr ibuted lead.

A comparison o f the f ree bending moment diagram, and f i .*oj-ended

moments wi l l show that the loading i l lu s t ra ted accurately

simulates a uniformly d is tr ibuted

Comparison of Free Bending Moment Diagrams

f i gu re D.2 g ives a comparison between bending moment under a

uniformly d is tr ibuted load and under the loading used in the

laboratory tes ts .

Comparison of Fixed-ended Moments

For a uniformly d is tr ibuted load, the f ixed-ended moment is

M = wL2 /l 2

= 0,0833wLl

Assuming that the point loads i l lu s t ra ted in f i gu re P .l also y i e ld

a fixed-ended moment o f 0.0*33*1- . rhe bending moment diagram

shown in f igure P .3 is obtained.

FIGL'RE DJ

For the f ixed-end momert to in fac t be 0 OP.HwL , area A must be

equal to area B, by the moment-area method.

Area A = 0,0079wL*

Area B * 0,007^wL

Hence the loading i l lu s t r a t ed in f igu re D.l g ives s im ilar f ree

bending moments, and the same f ixed-ended momenta as a uniformly

d is t r ibu ted load of magnitude w P/L. The loading mechanism was

there fore quite adequate to be used in the laboratory te s ts .

CALCULATION OF THE RAT/OS OF P J P AND APPENDIX Ec snap pP J P , USiNC THE APPROXIMATE ENERGY

c swajr pMETHOD AND HORNE'S ASSUMPTIONS

The equ. 'ised to ca lcu la te the ra t io o f P p is:’ c p

> 4 ( i ), « ? )

P Elastic buckling load o f the framec

P = P last ic co l lapse load P

R: Axial load in member at p las t ic col lapse (newtons)

Lj Length o f each membe * (m i l l im etres )

Q = Applied s'xternal forces which i n i t i a t e the required

mode of f a i lu r e i . e . snap-through or sway

A = Elast ic displacement as a result of Q, at the point

o f app l icat ion o f Q.

0, - Elastic ' t a t : n o f each member as a result o f Q.

The ax ia l forces at p la s t i c co l lapse are found by undertaking a

simple p last ic ana lyr is . Values of /3 and A are found by

undertaking an e la s t i c analys is o f the frame with Q as the applied

load. Values of R. l, Q, (3 and A are given in Table El. Also

illustrated is the calculation of P P for the case ail baysc sway p equally loaded, bases pinned. Other rat ios o f

calculated in the same way.

In ca lculat ing the ra t ios o f P P , the fo l low ing s im p l i f i ca t ion s

were made. These are the same s im p l i f ica t ions Horne used in his

research o f sway and snap-through s t a b i l i t y .

F irs t order s t i f fn e s se s , i . e . EI/'L were used. The

s t i f fn ess reducing e f f e c t of ax ia l forces was there­

fore ignored.

Axial r a f t e r forces were neglected e n t i r e l y in the

calcula t ion o f the P /P ra t ios .c sway p

Average axial rafter forces based on the undeformed

structure were used in ca lcu la t ing the P /Pc snap pvalues.

The e f f e c t o f r a f t e r shortening, as a result o f the

frames d e f l e c t io n under the applied load, was not.

considered.

Ax ial r a f t e r forces at the e la s t i c c r i t i c a l buculing

load were assumed to be proportional to those at

r i g id -p la s t i c co l lapse .

KXAMPLE OF THE INTKRACTION EQUATION IN

BS5950, TO PREVENT MEMBER INSTABILITY'

APPENDIX F

To prevent indiv idual member i n s t a b i l i t y , the fo l low ing in t e r ­

action equation must be applied to c r i t i c a l members within a

f rame.

Max May

m - uniform moment fac to r obtained from Tab3 .* 18 in BS5950

Mx = applied moment about the major a\is at the c r i t i c a l

section

My * applied moment about the minor axis at the c r i t i c a l

sect ion

Max = maximum al lowable buckling moment about the major axis in

the presence o f ax ia l load

May = maximum al lowable buckling moment about ihe minor axis in

the presence o f ax ia l load

Since a l l framei tested in the laboratory were bent about the

minor axis only, the in t * i 3 't.ion equation ’"educes to:

R * applied ax ia l loao in the c r i t i c a l section

Rcy = compressive res istance o f the sect ion about the minor axis

Mcy =■ p las t ic moment capacity o f the sect ion calculated from

m.Mx + m._Mv < 1( 1 )

m.Mv < 1 (2 )May

May is ca lculated usini; equation (3 )

May = Mcy( 1 - 3 7 W R c y » (3 )

clause 4.2.5 or 4-2.6 in BS5950.

Since the case a l l bays equally loaded a l l bases pinned, v,as

considered in the other appendices, the interaction equation w i l l

be applied to th is frame. The bending moment diagram under t.it

r i g id -p la s t i c co l lapse load is given in f i gu re F .1.

FIGL R E FI

The c r i t i c a l member in the frame is PF as i l lu s t r a t e d . The ax ia l

fo rce in this member is 739 N.

Calculation of the Numerator in Equation (2)

From Table 18 o f BS5950, m = 0,43

Since the greatest moment in the c r i t i c a l sect ion is Mp the

numerator of equation (2 ) is 0,43Mp

Calculation of the Denomenator in Equation (2)

Calculat ion o f Mcy:

The value o f Mcy depends on the shear capacity o f the sect ion.

Shear capacity 0.6*(7y*Av

Av s 0.9 An 0.0x20x5 90mnr

Hence shear capacity is:

0.6x3,34*90 1S036 N

I f the maximum shoar in the r a f t e r is < 0.6 times the shear

capacity o f the sect ion; Mcy = Mp.

The maximum shear force applied to member DF. from f igu re P . I , is

746 N which is smaller than 0.6 times the shear capacity o f the

ra f t e r .

Therefore Mcy = Mp, the p las t ic moment.

Calculation o f Rcy:

The compressive res is tance Rcy of the r a f t e r is obtained from:

Rcy = Ag.Ocy

<7cv = al lowable compressive stress o f the section about the miner

ax is .

Ocy depends on the y i e ld stress Oy and the slenderness o f the

member. Assuming an e f f e c t i v e length fa c to r o f 1, table 27(b) o f

BS505O g ives fo r a y i e ld stress o f 334 N/rrun* :

4 1 •5 ̂ mm

loe re fo re Rcy is:

30x41.? = 41 SO N

.Subst i tut inn R 739 N, Rcy = 41 SO N and Mcy ■■ Mp into equation

(3 ) g ives fo r May:

May * 0.75S Mp

Substituting May into equation (2 ) g ives:

0.43Mp « 0.S7 O.K.0.7S5Mp

From the in teract ion equation i t appears that member in s t a b i l i t y

is not a problem. This tvpe o f f a i lu r e should not occur in the

given frame.

REFERENCES

1.

2 .

3.

4- Scholz, H.

5. Scholz, H.

6. Merchant, W.

7 • Wood , fl. H.

8 . Scholl , H.

9. Kemp, A.R.

Structural lise of Steelwork in Bui ld ing Part1, BS 5O5O. London, Bri t ish .Standards Instl tu- t ion, 10^ ' .

Safeguards Against Frame In s ta b i l i t y in the Pl a s t i c Design o f Single Storey Pitched-rooT Frames, Conference or Slender Structures, City I’n i v e r s i t y , Sept 1977. unpublished paper.

SABS 0162-19*4, Code of Practice fo r__ t heStructural I'se of Sreel , f i r s t rev is ion , South A f r i c in Bureau o f Standards, Pretor ia , 1Q84*

A Re-apprais 1 of the Elastic Buckling Load of Pitched-root Frames, Part i o? Proceedings, Internat ional Conference on Steel Structures - Recent Research Advances, Budva, Sept 28-Oct1. 19*6, pp 35-42.

E last ic Snap-thrjugh Bu.'kling Load of Pirched- roof Steel Frames, I 'n ive ic i ty o7 the Wit- watersrand, Johannesburg, unpublished, 1986.

The Fai lure Load of R ig id- jo in ted Framework as influenced by S t a b i l i t y , The Structural Fngin- eer. Vol. ',2 1054 , pp""" 185-190.

E f f e c t i v e Lengths of Columns in Multi-storey Bui 1d i ngs. Proceed ings Inst i tu t ion oT C i v i l Engineers, 1074 , \^1. 52, p p 2 ^ , 29b. 341*

A New Mult i-curve Interaction Method f o r the P las t ic Ana lysis and Pesign o f ~ l ;nhraced and Part j 11ly-braced Frames, PhP Thesis, Ini ver­s i f y of1 the Witwatersrand. Johannesburg, Pec 19*1.

\ Consistent Mixed Approach to Computer Analysis of Frames, University of the Wit- watersrand. Johannesburg, unpublished, 19*6.

.

102

10. American Ins t i tu te of Stee l Construction,Spec i f i ca t ion fo r the Design, Fabricat ion and Erection of Structural Steel fo r Build ings , New Yorlc, August 197^*

11. Anderson, D. F i rs t -o rder P las t ic Design of Single StoreyPinned-base Frames, Document oT ECCS Working Group TW<5 5.2. May 19^6.

12. Commentary on SABS 0162-H84, Code of Pract icef o r the Structural Use of S t e e l , South African Ins t i tu te o f Steel Construct ion, Johannesburg, Republic o f South A f r ica , 1985.

Author Bryant John Spencer Name of thesis The Snap-through Stability Of Plastically Designed Steel Pitched-roof Portal Frames. 1987

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