The elastic-plastic buckIing failure load of structures made of nonlinear elastic-plastic material

The approximate determination of the elastic-plastic buckling load parameter of structures was discussed. For the critical failure load parameter a lower and an upper bound has been established, with the aid of which the results of computer calculations can be checked.


Introduction
Nowadays we can perform the exact stability analysis of structures with the aid of computer techniques.The geometric, material and cross-sectional nonlinearities can be taken into account by the programmes (cross-sectional nonlinearity is caused for example by the cracking of the reinforced concrete).
Unfortunately, a mistake in the programme, an incorrect entry, or a change of sign may result in an entirely wrong result.
Hence the design engineer highly appreciates a method by means of which the upper and lower bounds of the criticalload Ackllowledgemellt parameter of the structure can be determined.

This paper was prodLtced by the Reinforced ConC/'ete Re-
In this way we can check the results of the computer calculasearch Group of Hungarian Academy of Sciences.
tion.Such a method will be presented in this paper.

The Upper and Lower Sound of the Elastic-Plastic SuckIing Load Parameter of Structures
On the basis of investigations on elastic-plastic structures (Bartha, 1972) [l], (Dulácska, 1972) [2], (Home, 1963) [6], (Home, 1965) [7] we can state some theorem: the elastic-plastic buckling failure load parameter ).F is smaller than the "elastic" criticalload parameter ).c; -).F is also smaller than the rigid-plastic failure load parameter ),p; -and }.F is greater than the Rankine-type straight line between points le and h, (see in Fig. 1).
Here }'E is the load parameter of the elastic limit state i.e. at which yielding starts in the structure.
Thus the elastic-plastic buckling failure load parameter },F of structure lies on the straight line running at 45° in the hatched domain of According to a more accurate analysis of structures with compact cross sections, ).Fcan be approximated by the formula: For laminated and reticulated structures, and for structures with I cross sections we can estimate as: l + (I.~/).p)]' We thus have to examine "only" how the values of the load parameters ).E, I.p, and ).C can be determinated.

The Load Parameter ).E of the Elastic Limit State of Structures
We determine the internal forces of the structure at the load parameter )'j and we calculate the stresses from the axial load and bending of the structure according to the elastic theory of the strength of materials.
After this we select the maximum stress ).j.max and we compute the load parameter ).E from Eq. (3),

Uj.max
Here Uy is the yield stress of the material.
4 Determination of the Rigid-Plastic Failure Load Parameter ).p of the Structure We may exactly compute the rigid-plastic failure load (collapse load) parameter ).p by the theory of plasticity (Kaliszky, 1984) [8].
We obtain the lower bound of p if we increase the load of the structure until the cross sectional internal force (moment) Y at any point of the structure reaches the value Y p that causes rigid-plastic failure.
Of course, the rigid-plastic failure load Y p may also mean coupled bending moment M and axial force N (that is the compression is usually eccentric).
The lower bound of the collapse load parameter may be computed from the equation The character ofthese curves can be seen in Fig. 2 for various materials, or may by computed with knowledge of the strength of the materials.

Estimation of the Critical Load Parameter i.c of the Structure
As the basis of the determination of the criticalload parameter ).c we use the "c1assical" criticalload parameter ).c.o which is computed by the sec ond-order theory, assuming small deformations (Croll-Walker, 1972) [2].This value may be determined from the worked-out cases of the stability theory (Petersen, 1982) [10].
If we do not know the c1assical critical parameter of the full load ).c. O of the structure, but we know the c1assical cri tic al load parameter ).~ of every component load separately, the critical load parameter of the complete load ).c,o can be computed by Dunkerley's approximate relationship In addition we have to analyse whether the post-buckling loadbearing capacity of our structure is decreasing, constant or increasing.Shells, shallow arches and some reticulated structures have, as a rule, decreasing post-buckling load-bearing capacity.
When examining such structures we must take the reduction of the critical load parameter, caused by imperfections and ec-

. Yp
).p :s ).j .K' centricities, into account.This can be done by the following ( 4 ) formula: The collapse load Y p of the cross section can be determined I.C = p .).c,o.(6) with the aid of the limit curves of the load-bearing capacity of For shells, the reduction factor , based on (Kollár-Dulácska, the cross sections.1984) [9] is shown in Fig. 3. Plates and frames have a .::onstant(or increasing) post-buckIing load bearing capacity, hence with these structures the imperfections do not inftuence the criticalload.
Reinforced concrete structures form, however an exception.With these, the cracks reduce the stifness, and, thus also the criticalload.
The critical load of structures made of reinforced concrete, timber and plastic is reduced by the creep as weIl.The inftuence of creep may be taken into account by reducing modul us of elasticity according to the formula: If the reinforced structure has a geometricaIly decreasing load -bearing character (e.g.shells), we must take also this effect, characterised by , into account.

Notation
The elastic-plastic buckling failure load parameter.
where Eo is the initial value of the modulus of elasticity and is the creep factor (Dulácska, 1981) [5].The effect ofthe variation of the modulus of elasticity with the stress in the case eo= O is taken into account by the Eg (1) (Dulácska, 1.972) [3].
We mayestimate the decrease of critical load parameter of a reinforced concrete structure with the aid of Fig. 4, based on (Dulácska, 1978) [4].Here eo is the eccentricity of the compressive force of the most onerous cross section of the structure, and }.~wer is the lower bound of the critical load parameter of the reinforced concrete structure, which is determined by the second order theory of elastic stability theory taking the effect of the decrease of stiffening caused by cracks and creeps.That is, we compute the value of the lower criticalload parameter with the bending stiffnesses of the cracked reinforced concrete cross section on the basis of stadium Il. (cracked elastic state), with the creep reduced modulus of elasticity.;.~.o 2 -eo,~m eo.~m . (8)  We may use the value J,~wer in the entire range of eo as a lower bound.

I.,
,i The rigid-plastic failure load parameter.The load parameter of the elastic-limit state.
The load parameter of the elastic state.
The load component parameter.The J.c value for reinforced concrete.The lower value of Xc.
The upper value of J.c.The ;.F for compact cross section.
The ).F for sandwich and I cross section.
The modulus of elasticity.The initial value of E. The bending moment.
The bending moment at the elastic limit state.The ben ding moment at the plastic limit state.The axial force.The creep factor.The eccentricity of the external force.The initial eccentricity of the external force.The height of the cross section.
Factor for shells.

Gy
The yield stress of the material.
G',max The cross sectional stress at the elastic state.y The cross sectional internal action effect.

Y,
The Y value at the elastic state.

Y p
The Y value at the rigid-plastic failure state.

Fig. 2 .
Fig. 2. Limit curves of cross sections made of various materials a: with tensile strength.b: without tensile strength

Fig. 4 .
Fig. 4. Decrease of the eritical load parameter i. c of reinforced concrete structures with increasing eccentricity eo The value }.~wer is valid in the range eo 2: eO,lim = ~ (1 -*lower J;c ) where t is the height of the cross section.In the range '•c.o ~ ~ O S eo S eO,lim the value Cmay be computed from the formula: } 10Wer) ( e) e ]