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Publication - International Association for Bridge and Structural Engineering, 29-2,
pp. 19-27, 1970-07-01
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Safety factors for stress reversal = Facteurs de securité pour des
contraintes variables = Sicherheitsfaktoren fur spannungswechsel
Allen, D. E.
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Safety Factors for Stress Reversal
*)
Facteurs cle secu~ite' p o w des cont7.aintes variablesD. E. ALLEN
Building Structures Section, Division of Building Research, Nat,ional Research Council of Canada. Ottawa
Safety factors a t present used in the design of structures (usually embodied in the allowable stresses) have been empirically derived on the basis of trial and experience. When new forms of constructioll come into nse it callnot always be stated with certainty that existing rules will ensure adequate safety. I n other cases it may be that the old rules provide too much safety and there- fore are uneconomical. An important criterion in the design of structures, therefore, should be that of consistent safety, that is, the probability of failure for a given type of construction should not change when it is subject to different loads of the same type, such as dead, floor, and static wind loads, or these loads in combination.
This investigation shows that the desigu rules given in mally building codes are sometimes unsafe for structural members subject to stress reversal. Stress reversal occurs a t a critical section of a ineinber when loads from different sources counteract each other, such as in a truss nlenlber which is in tension under dead load but undergoes con~pression due to wind load or ilolluniformly distributed snow load. The explanation of the lack of safety against failure is simply that the design rule in which the usual safety factor is applied to the difference between two independent load effects of the same magnitude, is uilsatisfactory. Although this practice is avoided by some designers, it is over- looked by many building codes.
I n this discussion, code rules for stress reversal are investigated and com- pared on the basis of probability of failure. Calculations are limited to a critical section of a statically determinate structure subject to dead load and wind *) This paper was submitted as a contribution to the proparccl cliscussion on Thc~llc I a of the 8th Congress, Septombor 1968 in Ncxv York. By error, it was not published in the "Final Report".
effects only. Although this and other simplifyiiig assumptioils are made for the calculation of failure probabilities, it is considered t h a t t h e sigilificailce of the results remains valid.
Existing Design Rules
For a critical section of statically deterinillate structure, clesign rules call be expressed in the form:
R o > = F S o , ( l a ) where Ro is the design strength that is required,
So is the design load effect, and
P
is the factor of safety.Both Ro and So are expressed in terms of the primary force causing failure, e. g. tension or bending. When the critical section is subject only t o dead load and to wind effect opposite t o the dead load effect (stress reversal), the design negative strength
R;
is governed by:where Do is the design dead load effect,
k Do is the design wind effect,
and
P'
are load factors for wind and dead loacl respectively, and is a reductioil factor for material or structure.I n examples t h a t follow, a critical section is designed according to the rules given in the following building codes: National Building Code of Canada (NBC 1965) [l], AISC Specifications for Structural Steel in Buildings (AISC 1961) [2], ACI Building Code (ACI 1963) [3], lZecon~inendatioils for a n Inter- national Code of Practice for Reinforcecl Concrete (CEB 1964) [4]. Table I
lists the corresponding design rules in the form of E q . (1 b ) for the following cases.
Structural Steel. Design rules for the cases of sinlple bending, tension, and
compression of short members, are given in Table I according to NBC 1965 and AISC 1961. I t will be shown from the results t h a t a particularly important case for stress reversal in steel structures is a long member subject to buckling.
Reinforced Concrete. Critical cases for stress reversal in reinforced concrete
structures occur wllen the negative strength is governed inainly b y the rein- forcing steel alone. Examples are axial or membrane stress and bending of under-reinforced sections. Design rules for reinforciilg steel or for bending strength of reinforced concrete beanls are given in Table I in accordance with NBC 1965, ACI 1963, and CEB 1964.
SliFlCTY FACTORS F O R STRESS REVERSAL 2 1
Il'nble I . l l e s i q t ~ yules JOT sl~.ess ~ ~ e u e v s a l
\
Structurnl St,eclI
(Trnsion, brnding, C o ~ n l ) ~ ossion of Short ;\lcmbc~-3)Rcinfbrcctl Concrctc (Tension, Bentling)
Worliiirg strcs.; ant1 plast,ic tlc.iign
N H C 196.5 (ii;ecs%o~c 4.1)
If, 2 1.67 (I,- 1 ) U ,
Xli'C 1965 (Sec. 4.6); =IISC 1961
R, > 0.75 x 1.07 (I;-- 1 ) U,,
I ' r ~ o l ~ o ~ c d Design R~rlr
Ultimate strength design
AC'I 1963 0.9 R, 2 (1.1 1;-0.9) Do X B G 1965 o.!) R,; 2 (1.35 Ic-0.9) D, ('1$'4 I964 E ; / l . 1 5 2 (1.40 12-0.9) D,, Expected Loads
Deu'd Loud Effect
For this investigation t h e esl~ectccl dead load effect,
D ,
is assuined t o follow t h e norinal distribution with t h e follo\ving statistical parameters (see reference 1-41, 11. 6).DID,
15)
Steel coilstructioil 1.00 0.05 Reinforced Concrete 1.00 0.08 where D is the average of
D
andJ>,
its coefficient of variation.1;T'ilzcl Effect
Except for large or flexible structures, t h e xvind effect correspoi~ds approxi- mately t o t h e static load due to t h e maximum gust. 011t h e basis of gust mea- surements a t Ottawa Airport, Canada, t h e expected wind effect, JJT, is assumed t o follow t h e Extreme Value Distribution, Type 1, with t h e followiilg statistical parameters :
1 = 1.10 a i d
q,.
= 0.194.For t h e cleriva,tioi~ of these values i t is assunlecl t h a t t h e design wind effect, W,, corresponds t o t h e 30-year return period [ l ] a n d t h a t the design life of t h e structure is 30 years. Gust records a t other locations i n Canada give values of
W /
W, andVw
similar t o t h e above.2 2 1). E. ALLEN
Expected Strengths
The failure conditioil assumed for calculatioil corresponds to the yield point of steel; strain llardeiliilg and other effects such as corrosioil and fatigue are therefore neglected. This assumption correspoilds to the plastic momeilt in steel and is also used herein for reinforced coilcrete in teilsioil and bending (under-reinforced section). Expected strengths of a critical section also depend on the geometry of the cross-section; except for t l i n reinforced concrete members in bencling, however, geoizletric deviatioils are small conlpared to deviations in the yield point of steel.
On the basis of test information from Lehigh University [5] the expected yield point R, of structural aizcl reinforcing steel is assumed to follow the normal distributioil with t h e following statistical parameters:
\ R,/Ru 0 VR*
Strength under dead load 1.072 0.09SS
Negative streizgtlz during wiizcl effect (stress reversal) 1.1 S 0.0900 The derivation of tlzese values takes account of tlze effect of rate of straining oil the yield point of steel (see reference [6], Fig. 9).
Calculation of Probability of Failure
Given the probability distributioizs of tlze espected loads aizd strengths, and also the desigiz rules (including safety factors), the probability of failure call be calculated using probability calculus as developed for structural engi- neering by FREUDENTHAL [7] and others. The probability of failure is herein restricted to tlzqt for the critical section due to stress reversal only.
If the loads aizd strength are assumed to be statistically independent, the probability of failure is given b y an equation of the same form as Eq. (2.15) of reference [7]. This was integrated numerically with the aid of Simpson's Rule.
Cases Considered
I n practice i t often occurs t h a t althougll there is no ilecessity to design a critical section for stress reversal, there may be inherent izegative strength capacity. Cases representing different ratios of the iniilimum desigiz negative strength,
R,,,,,
to the design positive strength, R t , are considered as follows: 1. R,,,, = 0. Examples: reillforced concrete beams or unreinforced concreteS A F E T Y FACTORS FOR STRESS REVERSAL 2 3
2. RG,~,, = 0.03 R;I-. Example: reinforced concrete shell containiilg the mini- mum reillforcement required by the ACI Building Code (0.2 per cent for 40 k.s.i. steel).
3. RG,~,, = 0.2 R;I-. Example: long steel members relatively weak in com- pression.
4. RtInin = R$. Examples: horizoiltal steel I or W F beams; short steel members whose joints are as strong as the member themselves.
The results, shown in Fig. 1 for structural steel and in Fig. 2 for reinforcing steel, express the probability of failure of the critical section as a fuilctioil of Ic (design wind effect/design dead load effect).
- I - I I I I I I I I I I I 1 1 1 1 1 1 8 4 AlSC 1961 and 0 - NBC 1965 (sectlan 4.1) P I -
----___
- 6 a! 2 - Y neg. strength 3 - ~ 0 . 2 DOS strength neg slrenglh = 0 - 4 - - 0.003 5 - - 0 . 0 0 0 0 3 6 I I I I I I I I l l 0.1 0.2 0.4 0.6 0.8 1.0 2.0 4.0 6 0 8.0 10.0 des~gn wlnd effect k 'deslgn deod lood effect
Pig. 1. Failure probability for structural stccl: stress reversal (.ivind effect opposite to dead lond effect).
Pig. Failure
ferry bridge cool ...,
W5D;ACI 1963 ond NEC 1965 WSD: ACI 1963 and NBC 1965 "ll"l""hl0 c t . D c c \ y2
1
2.3---
-
-_______
g641ikSD:N8C I 9 6 5 0 - + 3 - '-- - 1 O.I4 2 0 p r o ~ o s e d ruleneg. strength = 0 . 0 3 Dos. strength neg. strength = 0
0.1 0 2 0.4 0.6 0.8 1.0 2.0 4 0 6.0 8.0 10.0 design wind effect
k =
design dead lood effect
probability for reinforcing steel: stress reversal (wind effect opposite . . load effect).
Discussion of Results
Figures 1 and 2 sllow t h a t lnaily existing desigii rules are unsafe whell the negative strength is sillall coillpared with t h e positive strength ancl when 1; is near t o one. Failure probabilities up t o 64 per cent occur when tlie negative strengtll is zero ailcl r';= I. This result is t o be expected since inost existing design rules are in error because they apply the usual safety factor t o tllc difference between two iildepeilclent load effects of t h e same magnitude.
Fig. 2 shows t h a t for stress reversal ultimate strength design (USD) loacl factors provide more consistent safety t h a n worliing stress design (WSD). It
appears, ho\vever, t h a t some revision of the loacl factors for stress reversal is badly needed.
Existing desigi~ rules for stress 1:eversal in structural steel (which are the same for both elastic and plastic clesign) are uiisafe when the negative strength is snlall compared with t h e positive strength (Fig. 1). This occurs in practice for long steel meinhers weali in compression, or joints weak in tension. Although the results given in Fig. 1 are based on statistical parailleters and safety factors applicable only t o yielcliilg of steel, they also apply approximately to buckling because the basic design rule for buclrling gives roughly t h e same safety as the one for yielding or plastic moment.
Recommenclations
One way t o avoid such inconsisteilt safety is for thc designer t o apply probability calculus clirectly t o check t h e probability of failure. I11most cases, however, there is neither timc nor sufficieilt illformation t o clo this. A rulc of thumb for stress reversal t h a t corresponds t o t h c one for overturning given in NBC 1965 (Article 4.1.4.3.) is as follows: apply t h e usual safety factor (1.5 t o 2.0) t o the design wincl effect illinus half the dead loacl effect (Table 1). Figures 1 and 2 sho\v t h a t for t h e coilditioiis assuilled for calculation, this rule gives consistellt safety for clifferent values of 1 ~ .
Another method of provicling more uniform safety for stress reversal is t o adopt a systenl of partial safety factors similar t o t h a t proposecl in CEB 1964. (This is partially done for ultimate strength clesign in ACT 1063, although erro~leously for stress reversal.) The aclvantage of t h e C E B system of safety factors is t h a t i t induces t h e clesigiler t o think more clearly because i t separates t h e safety factor into two co~nponents, one pertaining to t h e loacl effects (load factors) a n d the other pertaiiiiilg t o t h e inaterial or structure. 'iVhen cleacl load helps prevent failure (stress reversal, overturning, etc.), then a load factor less than 1.0 should be used.
S A F E T Y FACTORS F O R STRESS REVERSAL 25
Collapse of Ferrybridge Cooling Towers
If the situation is as serious as is indicated in Figures 1 and 2, i t is sur- prising t h a t it has not been corrected. There are, however, a number of reasons why there have not been more failures:
1. Some structures have as much negative strength (stress reversal) as positive strength. (See Fig. 1: negative strength = positive strength.)
2. Many structures are statically indeterminate and, because of stress redis- tribution, failure probabilities are not as severe a s indicated in Figs. 1 and 2.
3. Most structures built in the past have been heavy so t h a t k is not the critical range.
4. Designers have intuitively provided more strength for stress reversal than required by building codes.
I n an age of changing technology these reasons are no longer reliable; as reported by a Committee of Inquiry [9], the collapse of the cooling towers a t Ferrybridge, England was partly due t o incorrect design rules based on working stress hypothesis. On 1 November 1965 three cooling towers - reinforced
concrete shells
-
collapsed as a result of excess vertical tensile stress in the shell. The value ofk
for the design of these tower shells [9] a t the location of failure was about 1.18. Fig. 2 shows t h a t the probability of failure was 41 per cent accordiilg to working stress design in ACI 1963, and close to this according t o British design rules [S]. The high failure probability and a n error in the estimation of the wind effects appear to have been the main resaoils for the failure under wind speeds with a recurrence period of only two to five years [9]. The Committee of Inquiry recommended adoption of CEB 1966; this would result in a calculated failure probability for a Ferrybridge tower of 1.7 per cent (Fig. 2).Concluding Statement
The collapse of the Ferrybridge cooling towers and the results of probability calculations (Figs. 1 a n d 2) are evidence t h a t a revision of existing design rules for stress reversal is needed. Two approaches t o revision have been suggested in this discussion under the heading "RecommendatiolIs".
List of Symbols
dead load effect factor of safety
design wind effect
/
design dead load effect probability of failure of a critical section2 6 I).E. ALLEX
+
R, : So :Do, etc.
R,
S,
D, etc.vIt,
vy,
VD, etc.streligth
positive strength negative strength
strength as defined by tllc yielcl point load effect coefficient of variation wind effect CO N? 1 ' parameter defined by
P
- -- e 2 d z .-yJreduction factor for material or structure design values of
R:
S, D: etc.expected average values of R , S , D , etc.
expected coefficients of variation of
R,
S, D, etc.References
1. National Building Code of Canada 1965. Associate Committee on the National Building Code, National Rcsearch Council, Ottawa.
2. Specification for the Design, Fabrication and Ercction of Structural Steel for Buildings (adoptecl November 30, 1961). Amcrican Institi~te of Steel Construction, New Yorlr. 3. ACI Standard Building Code Recluireinents for Rcinforced Concrcte (ACI 318-63).
American Concret;e Institute, Detroit, June 1963.
4. Reconlmendations for an International Codc of Practicc for Reinforced Concreto. Coinitd EuropCen clu Bdton, Paris, 1964, published by the American Concrete Institute and the Cenlent and Concretc Association.
5. BEEDLE, L. S. ancl L. TALL: iiBasic Column Strength." Journal of thc Structural Division, Proceedings, American Socicty of Civil Engincers, Proc. Papcr 2555, Vol. 86, ST 7, July 1960.
6 . I ~ A o , N. R. N., L o ~ n s r a x ~ , N. ancl L. TALL: "EKccL of Strain Ratc on the Yielcl
Strcss of Strnctnral Steels." Journal of Materials, dlnerican Socicty for Tcsting and Materials, Vol. 1, No. 1, March 1966.
7. FREUDENTHAL, A. M . : "Critical Appraisal of Safety Criteria and Their Basic Con-
ccpts." International Association for Bridge and Structural Engineering, Eighth Congress, Septenlber 1968 (Preliminary Publication).
8. The Structural Use of Reinforced Concrcte in Buildings. British Standarc1 Code of Practice CP 114: 1967. The Council for Coclcs of Practicc, British Standards Institu- tion, Lonclon, 1965.
9. Rcport of the Comrnittcc of Incluiry into Collapsc of Cooling Towers a t F c ~ ~ y b r i d g e , Monday, 1 Nolrcn~bcr 1965. Central Electricity Gcncrating Board, London.
Summary
Probability calc~~lations for statically determinate structures subject only t o dead load and wind effects show t h a t existing design rules for stress reversal are soinetimes unsafe. A committee of inquiry collcluded t h a t the collapse of
SAFETY FACTORS FOR STRESS REVERSAL 27
the Ferrybridge cooliilg towers was to a considerable extent due to adoption of existing ~vorbing stress design rules for stress reversal. Chailges i11 the design rules are proposed which give more uilifornl safety.
Les calculs de probabilit6 dails le cas d'ossatures statiqueinent d6teriniilBes soumises uniquemeilt aux charges propres e t au vent nloiltreilt qne si les con- traiiltes changent de signe les r8gles usuelles de calculs n'assurent pas toujours une s6curit6 suffisante. Une conlit6 d7enqu_te a coilclu qne l'effondrement des tours cle refroidissement de Ferrybridge Btait dQ essentielleineilt & l'emploi des norines usuelles concerilailt le changement de signe cles contraintes. L'au- teur propose une nlodification de ces ilornles afin d'assurer une s6curit6 plus unifornle dens tous les cas.
Zusarnmenfassung
Wahrscheiillichkeitsrechilungen fur statisch bestimmte Tragwerke, welche nur durch Eigengewicht und Windkrafte belastet sind, deuten darauf hin, dalJ derzeit bestehende Berecl~nungsweisei fur die Spannungsuml~elu zu ganz uilgenugendeil Sicherheitsfaktoren fuluen konnen. Es ist bekannt, dalJ das Versagen der Hiihlturine Ferrybridge haupts%chlich auf Verwenduilg von bestehendeil Berechnungsweisei~ der Spannungsun~bel~r im Betriebsspannungs- bereich zuriicBzufiihren ist. E s werdeil ~ilderuilgeil cler Berechnungsweiseil vorgeschlagen, um eine einheitlichere Sicherheit zu erlangen.
