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Inelastic analysis of reinforced concrete space frames influenced by axial, torsional and bending interaction
Computers & Structures Vol. 46, No. 1, pp. 83-97. 1993 Printed in Great Britain.
0
CVM-7949193 s6.00 + 0.00 1992 Pergamon Press Ltd
INELASTIC ANALYSIS OF REINFORCED CONCRETE SPACE FRAMES INFLUENCED BY AXIAL, TORSIONAL AND BENDING INTERACTION G. M. CQWHI and F. CAPPELLO Universitri di Ancona, Facolta di Ingegneria, Istituto di Scienza e Tecnica delle Costruxioni, Ancona 60131, Italy (Received 7 October 1991) Abstract-The inelastic analysis of spatial reinforced concrete frames under static loads is performed according to a peculiar discrete truss model of the beam element behaviour having any mechanic, geometric and reinforcing characteristics, based on the diagonal compression field theory, in the presence of axial, bending and torsional interaction. The authors propose a theoretical iterative method of calculating the nodal element actions at various inelastic nodal displacements and supply a suitable computer procedure, based on the residual redistribution concept, in order to analyse the deformational and stress state of any spatial frame at various load levels. Numerical analyses by a suitable computer program show that the theoretical results are good in comparison with the experimental results and the influence of the interaction effects on the cracking and ultimate behaviour of the structure is quite significant in many cases.
1. INTRODUCTION
moment is resisted by the circulatory shear flow in the wall of the tube. Each element of wall works like a plane truss with concrete compression struts inclined at an angle a with the longitudinal axis of the tube.. Longitudinal steel reinforcement and closed stirrups are the tension struts (Fig. 1). By means of the theory of diagonal compression fields [3], this model is extended to combined axial, biaxial bending and torsional components [4-61. The effect of the bending component about the y and z axes and of the axial component action along x, is that the values of a and t, become functions of the coordinates x, y and z (Fig. 2b). The problem can be faced, as indicated in [4], discretizing the hollow cross-section in a sufficient number of rectangular elements, having height bi. The elements must be fairly small to ensure that ai and ti are constant in each element. Figures 2(a and b) show the difference between the pure torsion model (constant thickness t and angle a) and the combined axial, bending and torsional components model (variable thickness ti and angle ai). The proposed model, as an improvement of that in [4], is also based on the following assumptions: (a) no tension in concrete after cracking; (b) cross-sections remaining plane (the longitudinal strains vary linearly over the cross-section and the shear strain is neglected); (c) absence of warping restraints (the torsional moment is resisted by the circulatory shear flow); (d) small deformations. Given the stress-strain relationship of concrete and steel, the method consists of determining the axial, torsional and bending components N(x), M,(x), M,(x) and M,(x) as functions of the axial defor-
investigations in the field of inelastic analysis of reinforced concrete frames, subject to static actions involve significant problems and a considerable computational effort, since the stiffness of the structure depends on the stress and strain state. In a previous study [l] the authors faced the problem related to calculating the stiffness, above all in space frames, using a suitable iterative procedure. The nonlinear force displacement relations were built up by a polynomial interpolation algorithm. This paper, as an improvement and extension of the previous study, presents a suitable method for analysis of reinforced concrete frames, which takes into account the nonlinear effects of axial, torsional and bending interactions, according to the geometric characteristics of the cross-sections and the mechanical characteristics of the materials. Only the viscous effects are not taken into account. Useful comparisons with the results of experimental tests are also provided. As indicated in [l], a general approach to the problem has been adopted by identifying three behaviour models of the cross-section, element (beam) and structure. Theoretical
2. ANALYSIS OF THE CROSS-SECTION BEHAVIOUR MODEL The behaviour of a reinforced concrete cross-section subject to torsion is provided, before cracking, by the linear elastic Saint-Venant theory with reasonable accuracy. After cracking, literature and codes provide models that assume the cross-section acts like a tube of constant thickness t [2]. The torsional
a3
84
G. M. Coccm and F. CAPPELU)
Fig. 1. Space truss in pure torsion. mation and curvatures about the x, y and z axes p(x). L(X), a(x) and X,(X). For each element as many equations of equilibrium and compatibility as the number of unknowns relating to the same must be written. The following equation of compatibility can be obtained for each element i of the generic crosssection of abscissa x, according to assumption (b): Gilx)
=
VCx)
-
L(xl.Yt(x)
-
Xy(xk(x)s
(1)
relating the strain state to the cu~atures and axial deformation. Q(X) is the longitudinal strain in the element i, the same being defined as positive for tension, q(x), X,(X) and x,(x) being positive according to Fig. 2(b). J+(X) and .ai(x) are the coordinates of the average point of the external edge of the element. The compressed diagonal struts angle ai is assumed to he the same as the local principal strain direction. It may be easily obtained [2] from the stirrup strain EJx), the same being positive for tension, from the longitudinal strain c,,(x) and the compressed concrete strut strain eci(x) the same being positive for compression
where &(x) is a suitable coefficient which accounts for the application point of the compression stress resultant on element i. The same expression may he obtained [2] calculating the angular distortions due to ~,~(x), cSi(x) and ~Jx) in a rectangular wall element having unit height and cotan a width (Figs 3a, band c) at the application point of the compression resultant on element i. The longitudinal component c,,(x) cotan a&), due to E,~(x),and 6,(X)(1 -/Ii(x)) cotan a,(x), due to cd(x), must be equated to the transverse component Ebb tan CQ(X), due to LJx). and c&x)(1 -&(x)) tan ai( due to Ebb. This corresponds to a minimum of the total angular distortion. The following equation of compatibility relates the torsional curvature L(X) to strains Q(X), gSi(x) and eci(x). It may be obtained by applying the principle of virtual work to the equilibrated set of M,(x) component and the I”i(x) circulatory shear flow to the congruent set of x,(x) curvature and r,(x) total angular distortion, relative to the above strains in every element i
Mx(xkr(x) = i
Ti(x)ri(x),
i=l
tanz
a.(x)
i
=
Mx) + ~AX)(l - B,(x))1 I%tx> +
%itx)tl
-
Bdx))l
’
(2)
where n is the total number of elements.
(3)
85
Inelastic analysis of reinforced concrete space frames (4 Y f
Fig. 2. (a) Pure torsion model (constant thickness t and angle a). (b) Bending-torsion model (variable thickness ti and angle a,). Summing the above contributions due to Q(X), E&) and Eli (Fig. 3a-c) the r,(x) total angular distortion is T,(x) = c,,(x)cotan ai + [l -
flj(X)]Cci(X)[COtan
where the coefficient yi(x) accounts for the effect of
stress variation in the thickness. The following results from rotational equilibrium
the
cross-section
+ &(X)tan ai(x) a,(x) + tan ai(X
(4)
The circulatory shear flow Tt(x) is obtained from equilibrium along y or z axis of the compression stress resultants on element i (Fig. 4) T,(X) = 2~,(x)y,(X)t,(X)U,,(X)b,
COS
a,(x)sin cLi(X), (5)
Mx(x)= i: Ti(x)[4 - Bi(x)ti(xll* i-l
(6)
where d, is the distance of the element edge from origin 0. Substituting the values of r!(x) from eqn (4) Tj(x) from eqn (5) and M,(x) from eqn (6) in eqn (3) and by equating the left-hand terms to the right-hand
G. M. Coccyx and F. CHPELLO
86 (a)
@I
‘------wtg r ----_---_-----------
a(
.#4
Fig. 3. (a) Longitudinal angular distortion due to Q. @) Transverse angular distortion due to 4. (c) Longitudinal and transverse angular distortions due to cc,.
terms of the sum, after some algebra, the following results are obtained &)cotan
a&) + &x)tan
a,(x) x [cos a&)sin a,(x)/cotan a,(x)],
+ r1 - Mx)l&) sin a,(x)cos ai =[ddi-
Pi(~x)ti(x)lXx(xx)*
from the equilibrium along y or z axes of the stress resultants in the truss (Fig. 4)
(7)
The equation relating the Q&S) stirrup stress to the compressed concrete strut stress may be obtained
(8)
where A, is the area of the stirrup. As shown in 133, the r,(x) thickness must be calculated by mounting for the beading of compression struts caused by xX(x), X,(X) and x,(x) curvatures. Figures 5 and 6 show this effect on
87
Inelastic analysis of reinforced concrete space frames
Fig. 4. Stress resultant equilibrium at the comer element.
element i due to curvature in direction a,(x) and stress and strain fields along the t,(x) thickness. The pi(x) and y,(x) coefficients, previously indicated, may be calculated, according to the strain and stress compression diagrams along the t,(x) thickness, by the well-known compatibility, rotational and translational equilibrium relations [7), which account
for the inelastic behaviour of diagonal compression concrete struts. For small values of actions, the compression diagram is triangular and the coefficient values are pi(x) = l/3 and n(x) = 3/4 (elastic behaviour). The curvature about the y, axis, related to the above effect, may be obtained by the first and the second partial derivatives, with respect to the variable
261(x)71 (x)t, b.)%I(~)c~~~I(x)
I
Fig. 5. Stress and strain fields due to curvature in direction ai(
88
G.
M. Coccm and F.
CAPPELLO
Fig. 6. Effect of flexural and torsional curvatures in direction ai(
xi, of the displacement element xY,(x) = xY(x)cos2 ai
function
along z of the
+ 2XX(x)sin a,(x)cos ai(
axes and to the signs of Fig. 2(b), the equilibrium yields N(x) = 2 Ajao - i j=1
(9) According to the strain diagram of Fig. 5 and from eqn (9) it follows thatt
cci(x)= ~dx)[&(x)co~* a,(x) + x,(x)sin 2%(x)1. (10) For assigned values of ?I, x,(x), x,(x) and X,(X), eqns (l), (2), (7), (8) and (10) together represents a nonlinear system in the five unknowns ai( r,(x), +(x), Q(X) and eci(x) for every element i, because tensions a,,(x) and uSi may be obtained from the uniaxial stress-strain curves for concrete and steel. The internal actions in the generic cross-section of abscissa x may be obtained, from the solution of the above system, by means of eqn (6) and the following flexural translational and rotational equilibrium equations. According to the xyz Cartesian
7 For a nonrectangular cross-section, as shown in Fig. 7(a), the following value of the curvature about the parallel axis to the element edge: x+,(x) = ~,(x)cos 4, + X,(x)sin Qi, must substitute the x,(x) value in eqn (9).
M,(x)
T,(x)cotan ai(
i=l
= - 2 Ajarizj+ i T,(x)cotan ai j= 1 i= I x Izi - Bi(X)fi(X)CoS
M,(x) = 2 j-I
4il9
Ajouyj - i T,(x)cotan a,(x) i=l
X [Yi - MxMx)sin
4il9
(11)
where Aj is the area, uri is the stress and zj, yj are the coordinates of the n, longitudinal bars in the crosssection $i is the angle between the outward normal to the element edge and the positive direction of the z axis. The stress state corresponding to assigned values of longitudinal deformation and curvatures may be cross-section, calculated in a nonrectangular described by any irregular set of vertices (Fig. 7a), by means of the same equations. In fact, eqns (l), (2), (7), (8) and (10) are related only to the ith element, while eqns (11) are related to the element subdivision number of the cross-section perimeter. In particular, it is possible to apply this method to hollow cross-sections provided that the ti(x) thickness is checked for values greater than the real values.
Inelastic analysis of reinforced concrete space frames
89
6)
Fig. 7. (a) Discretization of nonrectangular cross-section. (b) Discretization of composite cross-section. More complex T or double T shaped cross-sections can be accurately analysed, according to the hypotheses of undeformed transversal cross-section profile and absence of warping restraints, subdividing these into many rectangular elements as shown in Fig. 7(b). Each may be analysed according to the rectangular cross-section of Fig. 2(b) and its contribution to internal actions may be calculated by means of eqns (11) and summed to the contribution of the other elements. However a unique cross-section having many vertices may be considered because, as it is well known, the effect of shear flow is nearly negligible along the common edges of various rectangular elements. The reader should note that the nonlinear equation system in the five unknowns a,(x), ti(x), Q(X), Gus and Gus, for each element i, cannot have a solution for any q, xX(x), x,(x) and x,(x) values, as in the presence of bending or biaxial bending with small or null torsion, or in absence of cracking. If a solution does not exist, the well-known expressions,
accounting for the tension stiffening or cracking of concrete [7], may be utilized for calculating the internal actions. 3. SOLUTION
ALGORITHM OF THE NONLINEAR SYSTEM
As is well known, the solution of a nonlinear algebraic system of n equations in n unknowns may be obtained by means of Newton iterative method, or other similar current methods, according to the degree of nonlinearity of the system. The authors have noticed that calculating the solution in this way requires a considerable computational effort, which is highly influenced by the accuracy of the initial trial solution, related to the pure torsion hypothesis, as shown in [4]. Therefore an additional computational effort for calculating the initial solution is required. At the same time prevailing bending or biaxial bending effects cannot in any way be drawn from this initial trial solution.
G. M. Coocm and F. CAPPELLO
90
In order to avoid the above shortcomings the authors have developed an iterative solution algorithm, eliminating a priori the unknown values, which were inconsistent with the real physical problem. Initially t&) is derived from eqn (7)
~ ,txj = [4- Bi(x)ti(x)lXx(x) SI
6lii(x)
x’, is considered. The beam is subjected to the set of end displacements and forces shown and is loaded by a uniform set of pX,, p,,, and pzPcomponents. According to the previous hypotheses (Sec. 2) and of small displacements, the strain state of the generic cross-section may be obtained from the nodal displacement components [1]:
-Gz$g
tan ai
- 11- Bi(Xki(X)~ sin2 ai Deriving ti(x) from eqn (10) and substituting above expression in eqn (2), Q(X) is obtained:
(12)
the
x = _ c&d - 41x,) I L ' ~,,(x')=a(x')~,,,+6(x')~~~,+c(x')(W2-
&i(X)
d,XJx)tan ai
-
-26,(x)
ai
Bi(xMxNan
~,(x)cos2 a,(x) + x,(x)sin 2a,(x)
+
x,,(x')=a(x')~,,,+b(x')~,,-
211-ml
ai
c(x')(u2-4),
a(x’)=
-g
b(x’)=
-1 +f$,
results from eqn (8)
= arcsin
J
usi(X)As
2/?i(x)yi(x)ti(x)uCi(x)~x .
+s,
(14)
(16)
The method consists of assigning small discrete increments to the unknown ai in the O-(x/2) range (for instance l/1000 of the interval) according to the desired accuracy. For each trial value of ai( Q(X) is calculated from eqn (1) according to q, xX(x), x,(x) and x,(x). Q(X) is calculated from eqn (13), &(x) from eqn (10) and +(x) from eqn (12), assuming the pi(x) value found in the previous trial and checking the physical conditions Q(X) >, 0, t,(x) >, 0 and Q(X) 2 0. If at least one of these conditions is not verified, the current a,(x) is discarded. In the first trial (at(x) = 0), the elastic value of the coefficients pi(x) and y,(x) is assumed. The tension aCi(x) and the coefficients /Ii(x) and yi(x) are calculated according to the uniaxial stress-strain relationship of concrete. Then ai is drawn again from eqn (14), having previously defined the tension a,,(x) according to the uniaxial stress-strain relationship of steel. If the value found differs, by a negligible amount, from the trial value, this is assumed as the ith element solution and the contributions to the internal actions are calculated by means of eqn (11). If the above condition is not verified in any trial, according to the proposed behaviour model, the solution does not exist and the internal actions are calculated as previously indicated. The iterative algorithm is applied, in the same way, to all elements of every cross-section. 4. DISPLACEMENT BEHAVIOUR BEAM ELEMENT
(15)
where (13)
The value ai
41,
ANALYSIS
OF
The beam element of Fig. 8, having length L, mechanical and geometric properties constant along
As shown in [1], the smaller the length L of the beam element, the more eqn (15) can be utilized successfully for a nonlinear analysis, because they represent linear compatibility relationships. The internal actions can therefore be drawn applying the proposed method to any set of nodal displacement components. The nodal forces can be drawn from the following according to the principle of virtual work 0’ = 1,2):
qx.=qj
N(x’) dx’ + fp,. L,
0
Map.=
L
I
-(-lpi
M,.(x’) dx’,
0
s L
qy,=
-(-
1)’
M,.(x’)c(x’)
dx’+$p,.L
0
L
Mi,,<=
M,,,(x’)a(x’)
dx’ - ( - l)j$pz,L2,
s 0 L
M,.(x’)c(x’)
Q=(-ly’
dx’ + $p?L
s0
Mjzv=
L M,.(x’)a(x’) s0
dx’ + ( - l)j$p,,,L’.
(17)
For every set of inelastic nodal displacements, the nodal forces can be calculated by applying a suitable numerical integration method to the first term of the right-hand side of eqn (17). In this way the overall
Inelastic analysis of reinforced concrete space frames
91
Fig. 8. Loads, displacements and forces in the beam element.
equilibrium of the beam element is violated owing to the numerical approximations of the integration method. In order to avoid this shortcoming, a heavy iterative procedure may be used to correct the initial displacement values in order to satisfy the equilibrium and compatibility relations at the same time. Alternatively, as shown in [l], a method of linearly interpolating the internal actions of the two end cross-sections, may be utilized. The advantages of this method consist in applying the procedure shown in Sec. 2 only to the two end cross-sections and in globally accounting for linear noncracked behaviour of intermediate cross-sections. 5. DIBPLACEMENTBEHAVIOUR OF
THE WHOLE
STRUCTURE The displacement inelastic analysis of space frames can be performed discretizing the whole structure in a suitable number of beam elements and solving the matrix algebraic linear equation system [l] by
means of the usual iterative Newton-Raphson nique
tech-
K,s(”+ ‘) = P + P*(“), n = 0, . . . , N,
W-0
where s(”+ ‘) is the nodal displacement vector related to n + 1 iteration, K, is the elastic stiffness matrix of the structure and P is the nodal load vector. P*(“) is the fictitious nodal load vector, relevant to the previous iteration n, expressed as p*Q) = K, i
i-0
s;],
(19)
where $1 is the nonlinear nodal displacement vector relevant to the ith iteration. As is well known, the latter can be determined in every iteration by applying an unbalanced nodal load vector to the structure. The components of this vector are calculated according to the difference between the elastic nodal forces
92
G.
M. Coccm and F.
and the inelastic nodal forces, relevant to the ~(“1 component vector of the previous iteration. The procedure is applied through N iterative cycles up to convergence. In this respect the authors have noticed that the application of the above procedure to the present analysis may involve a considerable number of iterative cycles owing to the great influence of axial internal action on the flexural and torsional stress state. This influence is emphasized in structures having a low degree of redundancy, owing to the low capacity of stress redistribution. To obviate this shortcoming it is possible to apply a convergence acceleration algorithm, based on the Aitken secant method [8], which utilizes suitable amplification coefficients of the s$ components. According to the nonlinear uniaxial relationship between the load component P and displacement component s, shown in Fig. 9, as is well known, the following amplification coefficient is obtained: 6(”
=
P Pci-l) ~(0 _ p(i- U ’
CAPPELLO
A suitable choice of the amplification matrix is the product 6’01, where gcn
=
coefficient
IIF-F”- I)11 IIF” _ Ii+
UII ’
(21)
I is the identity matrix, F(‘) and F(‘- ‘) are, respectively, the vectors of nodal forces relevant to the ith and (i - 1)th iterations. The above expression is a very good approximation and extension of eqn (20) to the multidimensional case, because the coefficient is proportional to the ratio between the unbalanced force vector modulus, at the (i - 1)th iteration, and that relevant to the variation of forces at the ith and (i - 1)th iterations. The above procedure may be applied to the analysis of geometric nonlinear effects, by utilizing suitable corrective coefficients in the elastic stiffness matrix of the beam element, as functions of the action values, as shown in [9] and in [l].
-
which agrees with the ratio of the diagram secants at the i - 1th and ith iterations respectively. In the present case of multidimensional nonlinear relationship between P and s, it is necessary to determine a matrix of coefficients [8], calculated according to the mutual differences among elastic and inelastic nodal forces at the (i - 1) and (i) iterations.
6. COMPUTER PROGRAMMING PROCEDURE
The proposed method can be easily applied to structural analysis by a suitable computer program according to the following steps 1. the structure is discretized by beam elements having length L, which permits assuming the crosssection, the longitudinal and transverse reinforce-
P
P P’ ptt-1
Fig. 9. Iterative technique using amplification coefficient.
Inelasticanalysisof reinforcedconcrete space frames
ment, the mechanical properties of concrete and steel constant in each of these; 2. for each heam element, or groups, the shape of the section, by means of the vertex coordinates, the dimensions and the placement of lon~tudin~ bars, the area and the spacing of stirrups, the elastic moduli, the ultimate strength of materials and the number of elements, which discretize the section, are given; 3. the elastic nodal displacement vector at 0 initial iteration {s(O) = s, = K;‘p) is calculated after having assembled the stiffness matrix of the structure, according to the geometric elastic characteristic of beam elements. If the analysis of geometric nonlinear effects is required, the calculation of vector s(O)is performed as shown in [9]; 4. according to the proposed method the nodal forces are calculated in the ith iterative cycle, utilizing the uniaxial stress-strain curve for concrete, which accounts for softening and tension stiffening effects [2], and the perfectly plastic uniaxial elastic
93
stress-strain curve for steel. In this step, the existence of the system solution is verified. If the solution does not exist the internal actions are obtained by discretizing the whole cross-section by means of a large number of elements in which the stresses and their resultant are calculated according to eqn (1) and to the stress-strain curves; 5. the unbalanced nodal loads and the s$ vector, relevant to these, are calculated. In order to limit the computational effort the 6” amplitication coefficient matrix is calculated according to eqn (21); 6. at the end of ith iteration, the global nodal displacement vector sci) is increased by the S%$ contribution; 7. steps 4,5 and 6 are repeated up to convergence, which is assumed as reached, according to the following inequality:
Y
St
-
As*
0.31
cm21lS
5.06+5.06
cm
II
II 43&n
cm2 Aa =
Eb
=250,000 kg cm2 -246 kg/cm l Rbk Rbk lono =4.400 kg/cm2 Rbc -=3.000 kg/cm* Fig. 10. Analysed structural frame.
1.98+1.98
cm
G. M. CoccHl and F. CAPPELU)
94
Porg) 30000
Q +
Tests (71 Theory
- VS (cm)
0
0
2
4
6
8
10
12
Fig. 11. Load deflection for specimen Sl. with 6 = 0.0001. The above inequality agrees with the minimization of the ratio between the norm of nonlinear iterative displacements and the norm of initial elastic displacements; 8. cci(x), cli(x) and csi(x) strains are compared with ultimate conventional strain of concrete and steel; 9. the end nodal forces of the beam elements are obtained according to the calculated global displacement vector s; 10. the above steps are repeated for every load condition. The authors have noticed that the reliability of the above procedure is perfectly guaranteed as regards the solution of nonlinear equation systems for every cross-section, owing to the restricted existence inter-
val of the unknowns ui which can be obtained quite accurately by means of a reduced number of trials. The reliability is also guaranteed as regards the solution of equation system (18), relevant to the whole structure, because the above procedure consists in subsequent elastic analyses at fictitious load conditions, caused by inelastic behaviour of beam elements.
7. NUMERICAL RESULTS AND COMPARISON THEORY WITH TESTS
The structure in Fig. 10 ([lo] specimen S l), consisting of a floor beam of length L, and a spandrel beam of length L,, is analysed. Both beams have rectangu-
&kg) 30000 1
Q
OF
Test (71
*Theory + Gross + Cracked
I---
I
1
2
Fig. 12. Load twist for specimen Sl.
Inelastic analysis of reinforced concrete space frames
95
pw 30000 -
20000 -
0
Q
Test 171
* 4
m-Y
-9
Cracked
Gross
2000
1000 Fig. 13. Load torque for specimen Sl.
lar cross-sections. The elastic, geometric and reinforcement properties are shown in Fig. 10. The structure is discretized by means of 17 nodes and 8 beam elements, of constant length, for the floor beam, and 8 beam elements for the spandrel. The end restraints are described as those carried out in [lo]: the allowable (1) and nonallowable (0) nodal displacement components, for every restraint, are given in Fig. 10. The diagrams in Fig. 11 show the greatest vertical displacement (- u of node 5) at various p load values, given by theoretical analysis and tests. There is good agreement between theoretical and experimental results. The typical slope variation at the cracking load P = 4500 kg, up to the ultimate load
P = 18,000 kg, utilized in [lo] to design the cross-sections, is noticed. Beyond this value both diagrams show identical behaviour of large displacements at small increments of load, with a low percentage of shiftings. It should be noticed that it is necessary to remove the controls of the ultimate conventional strains of concrete and steel, shown at step 8 in Sec. 6, in order to describe the structural behaviour at higher loads than the above, because the tests are performed up to the condition of concrete crushing. The diagrams in Fig. 12 show the greatest torsional rotation (4, of node 9) at various P load values. There is good agreement between theoretical and experimental results. A slope variation according to the
&kg) 30000
1
Q +
Test [7]
Theory
-VS(em) 1
2
3
Fig. 14. Load deflection for specimen S6. CAS w-3
4
5
G. M. Coccm and F.
96
CMPELL.O
ww 30000 1
20000 -
10000 -)
+
0
Gross Cracked
100 200 300 Fig. 15. Load torque for specimen S6.
previous behaviour is noticed. In the same figure, the straight lines relevant to the elastic linear behaviour of gross and cracked cross-sections, described in [lo], are shown. Inspection of the theoretical diagram slopes shows that the torsional stiffness of the spandrel initially agrees with the elastic gross stiffness up to P = 4500 kg. It then approaches the cracked stiffness and finally is almost null for values of loads higher than the ultimate. The comparison between the theoretical and experimental diagram shows a lower stiffness of the truss model with respect to the real model with a small percentage of shiftings. Figure 13 shows the diagram of the ratio between the torsional continuity moment X at node 9 and the length L, of the floor beam, as a function of the P value. This diagram is indicative of the flexural state interaction with the torsional state of the spandrel beam. It is shown that for a P value lower than the ultimate, the theoretical X values are included among those relevant to the cracked model and those relevant to the elastic model. Beyond this limit the diagram approaches the last model as confirmed by experimental results, owing to the variation of the ratio between the flexural stiffness of the floor beam and the torsional stiffness of the spandrel. As pointed out previously, the shiftings are caused by lower torsional stiffness of the truss model. The proposed method is applied to the analysis of the structure shown in Fig. 10, reversing the dimensional ratio between the lengths (L,= 290 cm, L, = 457 cm) under the same geometric crosssections and material strength characteristics. For the floor beam, the longitudinal steel areas are As = 0.71 + 0.71 cm*, As = 5.06 + 5.06 cm*, stirrups 0.31/19cm2/cm, for the spandrel As = 0.71 + 0.71 cm*, As = 4 x 1.98 cm*, stirrups 0.31/11 cm*/cm (specimen S6). The diagrams in Fig. 14 show the theoretical and experimental values of the vertical displacement (-v of node 5) at various
400
P load values. Unlike the results of specimen S 1, the theoretical and experimental diagrams agree well at every load value, owing to the lower influence of the torsional behaviour of the spandrel as a consequence of the different length ratio. As regards the variation of the diagrams themselves, the observations made regarding Fig. 11, apply. The effect of variation of the ratios between the flexural and torsional stiffness of the beams is shown in the diagrams of Fig. 15, similar to those of Fig. 13. Contrary to the results obtained for Sl, the theoretical behaviour model is closer to the linear elastic behaviour than the experimental test.
8. CONCLUSIONS
In this paper the authors studied the problem of the theoretical analysis of reinforced concrete space frames accounting for the interaction effects among the stress components, by means of a suitable extension of the diagonal compression field theory, for pure torsion, applied to the general case of coupled axial, torsional and bending actions. Major attention was devoted to numerical problems in calculating the stresses and the strains in reinforced concrete crosssections, subject to coupled actions, having any geometric and reinforcing characteristics, and to the inelastic analysis of the whole structure, by means of suitable iterative algorithms. The numerical results, obtained by means of a computer program, have proved the reliability and the versatility of the proposed procedure and the good agreement between theoretical and experimental analysis of typical structures. The comparison of the results has likewise proved the weight of a correct prevision of the above interaction effects, not only at the cracking state but also at the ultimate state.
Inelastic analysis of reinforced concrete space frames REFERJSNCES
1. G. M. Cocchi and F. Cappello, Sul calcolo delle ossature spaziali in regime non lineare. Giomale de1 Genie Civile 7, 8, 9 (1987). 2. T. T. C. Hsu and Y. L. MO, Softening of concrete in torsional members-theory and tests. ACZ JI, Paper No. 82-25, May/June (1985). 3. D. Mitchell and M. P. Collins, Diagonal compression field theory-a rational model for structural concrete in pure torsion. ACZ Jl, 71, 396-408 (1974). 4. M. B. Leung and W. C. Schnobrich, Reinforced concrete beams subiected to bending and torsion. J Srruct. Eegsg,ASCE i13,No. 2 (1987). 5.P. Lampert and B. ~mlim~n, Ultimate strength and design of reinforced concrete beams in torsion and bending. International Association for Bridge and Structural En8ineering, 31-1, Zurich (1971).
97
6. A. E. McMullen and J. Wanvaruk, Concrete beams in bending, torsion and shear. J. Sttuct. Engng, ASCE May (1970). 7. R. Park and T. Paulay, Reinforced Concrete Structures. John Wiley (1979). 8. K. J. Bathe and A. P. Cimento, Some practical procedures for the solution of non-linear finite element equations. Comput. Meth. Appf. Meek Engng 22,59-85 (1980). 9. G. M. Cocchi and F. Cappello, Instabilita’ laterale di strutture reticolari prefabbricate in regime transitorio e di esercizio. Atti de1 Congresso CTE sull’Industriahzzazione Edilizia, Firenze, 9-l 1, November (1984). 10. M. P. Collins and P. Lampert, R~st~bution of moments at cracking-the key to simpler torsion design. AC1 Publication SP35, March (1971).
0
CVM-7949193 s6.00 + 0.00 1992 Pergamon Press Ltd
INELASTIC ANALYSIS OF REINFORCED CONCRETE SPACE FRAMES INFLUENCED BY AXIAL, TORSIONAL AND BENDING INTERACTION G. M. CQWHI and F. CAPPELLO Universitri di Ancona, Facolta di Ingegneria, Istituto di Scienza e Tecnica delle Costruxioni, Ancona 60131, Italy (Received 7 October 1991) Abstract-The inelastic analysis of spatial reinforced concrete frames under static loads is performed according to a peculiar discrete truss model of the beam element behaviour having any mechanic, geometric and reinforcing characteristics, based on the diagonal compression field theory, in the presence of axial, bending and torsional interaction. The authors propose a theoretical iterative method of calculating the nodal element actions at various inelastic nodal displacements and supply a suitable computer procedure, based on the residual redistribution concept, in order to analyse the deformational and stress state of any spatial frame at various load levels. Numerical analyses by a suitable computer program show that the theoretical results are good in comparison with the experimental results and the influence of the interaction effects on the cracking and ultimate behaviour of the structure is quite significant in many cases.
1. INTRODUCTION
moment is resisted by the circulatory shear flow in the wall of the tube. Each element of wall works like a plane truss with concrete compression struts inclined at an angle a with the longitudinal axis of the tube.. Longitudinal steel reinforcement and closed stirrups are the tension struts (Fig. 1). By means of the theory of diagonal compression fields [3], this model is extended to combined axial, biaxial bending and torsional components [4-61. The effect of the bending component about the y and z axes and of the axial component action along x, is that the values of a and t, become functions of the coordinates x, y and z (Fig. 2b). The problem can be faced, as indicated in [4], discretizing the hollow cross-section in a sufficient number of rectangular elements, having height bi. The elements must be fairly small to ensure that ai and ti are constant in each element. Figures 2(a and b) show the difference between the pure torsion model (constant thickness t and angle a) and the combined axial, bending and torsional components model (variable thickness ti and angle ai). The proposed model, as an improvement of that in [4], is also based on the following assumptions: (a) no tension in concrete after cracking; (b) cross-sections remaining plane (the longitudinal strains vary linearly over the cross-section and the shear strain is neglected); (c) absence of warping restraints (the torsional moment is resisted by the circulatory shear flow); (d) small deformations. Given the stress-strain relationship of concrete and steel, the method consists of determining the axial, torsional and bending components N(x), M,(x), M,(x) and M,(x) as functions of the axial defor-
investigations in the field of inelastic analysis of reinforced concrete frames, subject to static actions involve significant problems and a considerable computational effort, since the stiffness of the structure depends on the stress and strain state. In a previous study [l] the authors faced the problem related to calculating the stiffness, above all in space frames, using a suitable iterative procedure. The nonlinear force displacement relations were built up by a polynomial interpolation algorithm. This paper, as an improvement and extension of the previous study, presents a suitable method for analysis of reinforced concrete frames, which takes into account the nonlinear effects of axial, torsional and bending interactions, according to the geometric characteristics of the cross-sections and the mechanical characteristics of the materials. Only the viscous effects are not taken into account. Useful comparisons with the results of experimental tests are also provided. As indicated in [l], a general approach to the problem has been adopted by identifying three behaviour models of the cross-section, element (beam) and structure. Theoretical
2. ANALYSIS OF THE CROSS-SECTION BEHAVIOUR MODEL The behaviour of a reinforced concrete cross-section subject to torsion is provided, before cracking, by the linear elastic Saint-Venant theory with reasonable accuracy. After cracking, literature and codes provide models that assume the cross-section acts like a tube of constant thickness t [2]. The torsional
a3
84
G. M. Coccm and F. CAPPELU)
Fig. 1. Space truss in pure torsion. mation and curvatures about the x, y and z axes p(x). L(X), a(x) and X,(X). For each element as many equations of equilibrium and compatibility as the number of unknowns relating to the same must be written. The following equation of compatibility can be obtained for each element i of the generic crosssection of abscissa x, according to assumption (b): Gilx)
=
VCx)
-
L(xl.Yt(x)
-
Xy(xk(x)s
(1)
relating the strain state to the cu~atures and axial deformation. Q(X) is the longitudinal strain in the element i, the same being defined as positive for tension, q(x), X,(X) and x,(x) being positive according to Fig. 2(b). J+(X) and .ai(x) are the coordinates of the average point of the external edge of the element. The compressed diagonal struts angle ai is assumed to he the same as the local principal strain direction. It may be easily obtained [2] from the stirrup strain EJx), the same being positive for tension, from the longitudinal strain c,,(x) and the compressed concrete strut strain eci(x) the same being positive for compression
where &(x) is a suitable coefficient which accounts for the application point of the compression stress resultant on element i. The same expression may he obtained [2] calculating the angular distortions due to ~,~(x), cSi(x) and ~Jx) in a rectangular wall element having unit height and cotan a width (Figs 3a, band c) at the application point of the compression resultant on element i. The longitudinal component c,,(x) cotan a&), due to E,~(x),and 6,(X)(1 -/Ii(x)) cotan a,(x), due to cd(x), must be equated to the transverse component Ebb tan CQ(X), due to LJx). and c&x)(1 -&(x)) tan ai( due to Ebb. This corresponds to a minimum of the total angular distortion. The following equation of compatibility relates the torsional curvature L(X) to strains Q(X), gSi(x) and eci(x). It may be obtained by applying the principle of virtual work to the equilibrated set of M,(x) component and the I”i(x) circulatory shear flow to the congruent set of x,(x) curvature and r,(x) total angular distortion, relative to the above strains in every element i
Mx(xkr(x) = i
Ti(x)ri(x),
i=l
tanz
a.(x)
i
=
Mx) + ~AX)(l - B,(x))1 I%tx> +
%itx)tl
-
Bdx))l
’
(2)
where n is the total number of elements.
(3)
85
Inelastic analysis of reinforced concrete space frames (4 Y f
Fig. 2. (a) Pure torsion model (constant thickness t and angle a). (b) Bending-torsion model (variable thickness ti and angle a,). Summing the above contributions due to Q(X), E&) and Eli (Fig. 3a-c) the r,(x) total angular distortion is T,(x) = c,,(x)cotan ai + [l -
flj(X)]Cci(X)[COtan
where the coefficient yi(x) accounts for the effect of
stress variation in the thickness. The following results from rotational equilibrium
the
cross-section
+ &(X)tan ai(x) a,(x) + tan ai(X
(4)
The circulatory shear flow Tt(x) is obtained from equilibrium along y or z axis of the compression stress resultants on element i (Fig. 4) T,(X) = 2~,(x)y,(X)t,(X)U,,(X)b,
COS
a,(x)sin cLi(X), (5)
Mx(x)= i: Ti(x)[4 - Bi(x)ti(xll* i-l
(6)
where d, is the distance of the element edge from origin 0. Substituting the values of r!(x) from eqn (4) Tj(x) from eqn (5) and M,(x) from eqn (6) in eqn (3) and by equating the left-hand terms to the right-hand
G. M. Coccyx and F. CHPELLO
86 (a)
@I
‘------wtg r ----_---_-----------
a(
.#4
Fig. 3. (a) Longitudinal angular distortion due to Q. @) Transverse angular distortion due to 4. (c) Longitudinal and transverse angular distortions due to cc,.
terms of the sum, after some algebra, the following results are obtained &)cotan
a&) + &x)tan
a,(x) x [cos a&)sin a,(x)/cotan a,(x)],
+ r1 - Mx)l&) sin a,(x)cos ai =[ddi-
Pi(~x)ti(x)lXx(xx)*
from the equilibrium along y or z axes of the stress resultants in the truss (Fig. 4)
(7)
The equation relating the Q&S) stirrup stress to the compressed concrete strut stress may be obtained
(8)
where A, is the area of the stirrup. As shown in 133, the r,(x) thickness must be calculated by mounting for the beading of compression struts caused by xX(x), X,(X) and x,(x) curvatures. Figures 5 and 6 show this effect on
87
Inelastic analysis of reinforced concrete space frames
Fig. 4. Stress resultant equilibrium at the comer element.
element i due to curvature in direction a,(x) and stress and strain fields along the t,(x) thickness. The pi(x) and y,(x) coefficients, previously indicated, may be calculated, according to the strain and stress compression diagrams along the t,(x) thickness, by the well-known compatibility, rotational and translational equilibrium relations [7), which account
for the inelastic behaviour of diagonal compression concrete struts. For small values of actions, the compression diagram is triangular and the coefficient values are pi(x) = l/3 and n(x) = 3/4 (elastic behaviour). The curvature about the y, axis, related to the above effect, may be obtained by the first and the second partial derivatives, with respect to the variable
261(x)71 (x)t, b.)%I(~)c~~~I(x)
I
Fig. 5. Stress and strain fields due to curvature in direction ai(
88
G.
M. Coccm and F.
CAPPELLO
Fig. 6. Effect of flexural and torsional curvatures in direction ai(
xi, of the displacement element xY,(x) = xY(x)cos2 ai
function
along z of the
+ 2XX(x)sin a,(x)cos ai(
axes and to the signs of Fig. 2(b), the equilibrium yields N(x) = 2 Ajao - i j=1
(9) According to the strain diagram of Fig. 5 and from eqn (9) it follows thatt
cci(x)= ~dx)[&(x)co~* a,(x) + x,(x)sin 2%(x)1. (10) For assigned values of ?I, x,(x), x,(x) and X,(X), eqns (l), (2), (7), (8) and (10) together represents a nonlinear system in the five unknowns ai( r,(x), +(x), Q(X) and eci(x) for every element i, because tensions a,,(x) and uSi may be obtained from the uniaxial stress-strain curves for concrete and steel. The internal actions in the generic cross-section of abscissa x may be obtained, from the solution of the above system, by means of eqn (6) and the following flexural translational and rotational equilibrium equations. According to the xyz Cartesian
7 For a nonrectangular cross-section, as shown in Fig. 7(a), the following value of the curvature about the parallel axis to the element edge: x+,(x) = ~,(x)cos 4, + X,(x)sin Qi, must substitute the x,(x) value in eqn (9).
M,(x)
T,(x)cotan ai(
i=l
= - 2 Ajarizj+ i T,(x)cotan ai j= 1 i= I x Izi - Bi(X)fi(X)CoS
M,(x) = 2 j-I
4il9
Ajouyj - i T,(x)cotan a,(x) i=l
X [Yi - MxMx)sin
4il9
(11)
where Aj is the area, uri is the stress and zj, yj are the coordinates of the n, longitudinal bars in the crosssection $i is the angle between the outward normal to the element edge and the positive direction of the z axis. The stress state corresponding to assigned values of longitudinal deformation and curvatures may be cross-section, calculated in a nonrectangular described by any irregular set of vertices (Fig. 7a), by means of the same equations. In fact, eqns (l), (2), (7), (8) and (10) are related only to the ith element, while eqns (11) are related to the element subdivision number of the cross-section perimeter. In particular, it is possible to apply this method to hollow cross-sections provided that the ti(x) thickness is checked for values greater than the real values.
Inelastic analysis of reinforced concrete space frames
89
6)
Fig. 7. (a) Discretization of nonrectangular cross-section. (b) Discretization of composite cross-section. More complex T or double T shaped cross-sections can be accurately analysed, according to the hypotheses of undeformed transversal cross-section profile and absence of warping restraints, subdividing these into many rectangular elements as shown in Fig. 7(b). Each may be analysed according to the rectangular cross-section of Fig. 2(b) and its contribution to internal actions may be calculated by means of eqns (11) and summed to the contribution of the other elements. However a unique cross-section having many vertices may be considered because, as it is well known, the effect of shear flow is nearly negligible along the common edges of various rectangular elements. The reader should note that the nonlinear equation system in the five unknowns a,(x), ti(x), Q(X), Gus and Gus, for each element i, cannot have a solution for any q, xX(x), x,(x) and x,(x) values, as in the presence of bending or biaxial bending with small or null torsion, or in absence of cracking. If a solution does not exist, the well-known expressions,
accounting for the tension stiffening or cracking of concrete [7], may be utilized for calculating the internal actions. 3. SOLUTION
ALGORITHM OF THE NONLINEAR SYSTEM
As is well known, the solution of a nonlinear algebraic system of n equations in n unknowns may be obtained by means of Newton iterative method, or other similar current methods, according to the degree of nonlinearity of the system. The authors have noticed that calculating the solution in this way requires a considerable computational effort, which is highly influenced by the accuracy of the initial trial solution, related to the pure torsion hypothesis, as shown in [4]. Therefore an additional computational effort for calculating the initial solution is required. At the same time prevailing bending or biaxial bending effects cannot in any way be drawn from this initial trial solution.
G. M. Coocm and F. CAPPELLO
90
In order to avoid the above shortcomings the authors have developed an iterative solution algorithm, eliminating a priori the unknown values, which were inconsistent with the real physical problem. Initially t&) is derived from eqn (7)
~ ,txj = [4- Bi(x)ti(x)lXx(x) SI
6lii(x)
x’, is considered. The beam is subjected to the set of end displacements and forces shown and is loaded by a uniform set of pX,, p,,, and pzPcomponents. According to the previous hypotheses (Sec. 2) and of small displacements, the strain state of the generic cross-section may be obtained from the nodal displacement components [1]:
-Gz$g
tan ai
- 11- Bi(Xki(X)~ sin2 ai Deriving ti(x) from eqn (10) and substituting above expression in eqn (2), Q(X) is obtained:
(12)
the
x = _ c&d - 41x,) I L ' ~,,(x')=a(x')~,,,+6(x')~~~,+c(x')(W2-
&i(X)
d,XJx)tan ai
-
-26,(x)
ai
Bi(xMxNan
~,(x)cos2 a,(x) + x,(x)sin 2a,(x)
+
x,,(x')=a(x')~,,,+b(x')~,,-
211-ml
ai
c(x')(u2-4),
a(x’)=
-g
b(x’)=
-1 +f$,
results from eqn (8)
= arcsin
J
usi(X)As
2/?i(x)yi(x)ti(x)uCi(x)~x .
+s,
(14)
(16)
The method consists of assigning small discrete increments to the unknown ai in the O-(x/2) range (for instance l/1000 of the interval) according to the desired accuracy. For each trial value of ai( Q(X) is calculated from eqn (1) according to q, xX(x), x,(x) and x,(x). Q(X) is calculated from eqn (13), &(x) from eqn (10) and +(x) from eqn (12), assuming the pi(x) value found in the previous trial and checking the physical conditions Q(X) >, 0, t,(x) >, 0 and Q(X) 2 0. If at least one of these conditions is not verified, the current a,(x) is discarded. In the first trial (at(x) = 0), the elastic value of the coefficients pi(x) and y,(x) is assumed. The tension aCi(x) and the coefficients /Ii(x) and yi(x) are calculated according to the uniaxial stress-strain relationship of concrete. Then ai is drawn again from eqn (14), having previously defined the tension a,,(x) according to the uniaxial stress-strain relationship of steel. If the value found differs, by a negligible amount, from the trial value, this is assumed as the ith element solution and the contributions to the internal actions are calculated by means of eqn (11). If the above condition is not verified in any trial, according to the proposed behaviour model, the solution does not exist and the internal actions are calculated as previously indicated. The iterative algorithm is applied, in the same way, to all elements of every cross-section. 4. DISPLACEMENT BEHAVIOUR BEAM ELEMENT
(15)
where (13)
The value ai
41,
ANALYSIS
OF
The beam element of Fig. 8, having length L, mechanical and geometric properties constant along
As shown in [1], the smaller the length L of the beam element, the more eqn (15) can be utilized successfully for a nonlinear analysis, because they represent linear compatibility relationships. The internal actions can therefore be drawn applying the proposed method to any set of nodal displacement components. The nodal forces can be drawn from the following according to the principle of virtual work 0’ = 1,2):
qx.=qj
N(x’) dx’ + fp,. L,
0
Map.=
L
I
-(-lpi
M,.(x’) dx’,
0
s L
qy,=
-(-
1)’
M,.(x’)c(x’)
dx’+$p,.L
0
L
Mi,,<=
M,,,(x’)a(x’)
dx’ - ( - l)j$pz,L2,
s 0 L
M,.(x’)c(x’)
Q=(-ly’
dx’ + $p?L
s0
Mjzv=
L M,.(x’)a(x’) s0
dx’ + ( - l)j$p,,,L’.
(17)
For every set of inelastic nodal displacements, the nodal forces can be calculated by applying a suitable numerical integration method to the first term of the right-hand side of eqn (17). In this way the overall
Inelastic analysis of reinforced concrete space frames
91
Fig. 8. Loads, displacements and forces in the beam element.
equilibrium of the beam element is violated owing to the numerical approximations of the integration method. In order to avoid this shortcoming, a heavy iterative procedure may be used to correct the initial displacement values in order to satisfy the equilibrium and compatibility relations at the same time. Alternatively, as shown in [l], a method of linearly interpolating the internal actions of the two end cross-sections, may be utilized. The advantages of this method consist in applying the procedure shown in Sec. 2 only to the two end cross-sections and in globally accounting for linear noncracked behaviour of intermediate cross-sections. 5. DIBPLACEMENTBEHAVIOUR OF
THE WHOLE
STRUCTURE The displacement inelastic analysis of space frames can be performed discretizing the whole structure in a suitable number of beam elements and solving the matrix algebraic linear equation system [l] by
means of the usual iterative Newton-Raphson nique
tech-
K,s(”+ ‘) = P + P*(“), n = 0, . . . , N,
W-0
where s(”+ ‘) is the nodal displacement vector related to n + 1 iteration, K, is the elastic stiffness matrix of the structure and P is the nodal load vector. P*(“) is the fictitious nodal load vector, relevant to the previous iteration n, expressed as p*Q) = K, i
i-0
s;],
(19)
where $1 is the nonlinear nodal displacement vector relevant to the ith iteration. As is well known, the latter can be determined in every iteration by applying an unbalanced nodal load vector to the structure. The components of this vector are calculated according to the difference between the elastic nodal forces
92
G.
M. Coccm and F.
and the inelastic nodal forces, relevant to the ~(“1 component vector of the previous iteration. The procedure is applied through N iterative cycles up to convergence. In this respect the authors have noticed that the application of the above procedure to the present analysis may involve a considerable number of iterative cycles owing to the great influence of axial internal action on the flexural and torsional stress state. This influence is emphasized in structures having a low degree of redundancy, owing to the low capacity of stress redistribution. To obviate this shortcoming it is possible to apply a convergence acceleration algorithm, based on the Aitken secant method [8], which utilizes suitable amplification coefficients of the s$ components. According to the nonlinear uniaxial relationship between the load component P and displacement component s, shown in Fig. 9, as is well known, the following amplification coefficient is obtained: 6(”
=
P Pci-l) ~(0 _ p(i- U ’
CAPPELLO
A suitable choice of the amplification matrix is the product 6’01, where gcn
=
coefficient
IIF-F”- I)11 IIF” _ Ii+
UII ’
(21)
I is the identity matrix, F(‘) and F(‘- ‘) are, respectively, the vectors of nodal forces relevant to the ith and (i - 1)th iterations. The above expression is a very good approximation and extension of eqn (20) to the multidimensional case, because the coefficient is proportional to the ratio between the unbalanced force vector modulus, at the (i - 1)th iteration, and that relevant to the variation of forces at the ith and (i - 1)th iterations. The above procedure may be applied to the analysis of geometric nonlinear effects, by utilizing suitable corrective coefficients in the elastic stiffness matrix of the beam element, as functions of the action values, as shown in [9] and in [l].
-
which agrees with the ratio of the diagram secants at the i - 1th and ith iterations respectively. In the present case of multidimensional nonlinear relationship between P and s, it is necessary to determine a matrix of coefficients [8], calculated according to the mutual differences among elastic and inelastic nodal forces at the (i - 1) and (i) iterations.
6. COMPUTER PROGRAMMING PROCEDURE
The proposed method can be easily applied to structural analysis by a suitable computer program according to the following steps 1. the structure is discretized by beam elements having length L, which permits assuming the crosssection, the longitudinal and transverse reinforce-
P
P P’ ptt-1
Fig. 9. Iterative technique using amplification coefficient.
Inelasticanalysisof reinforcedconcrete space frames
ment, the mechanical properties of concrete and steel constant in each of these; 2. for each heam element, or groups, the shape of the section, by means of the vertex coordinates, the dimensions and the placement of lon~tudin~ bars, the area and the spacing of stirrups, the elastic moduli, the ultimate strength of materials and the number of elements, which discretize the section, are given; 3. the elastic nodal displacement vector at 0 initial iteration {s(O) = s, = K;‘p) is calculated after having assembled the stiffness matrix of the structure, according to the geometric elastic characteristic of beam elements. If the analysis of geometric nonlinear effects is required, the calculation of vector s(O)is performed as shown in [9]; 4. according to the proposed method the nodal forces are calculated in the ith iterative cycle, utilizing the uniaxial stress-strain curve for concrete, which accounts for softening and tension stiffening effects [2], and the perfectly plastic uniaxial elastic
93
stress-strain curve for steel. In this step, the existence of the system solution is verified. If the solution does not exist the internal actions are obtained by discretizing the whole cross-section by means of a large number of elements in which the stresses and their resultant are calculated according to eqn (1) and to the stress-strain curves; 5. the unbalanced nodal loads and the s$ vector, relevant to these, are calculated. In order to limit the computational effort the 6” amplitication coefficient matrix is calculated according to eqn (21); 6. at the end of ith iteration, the global nodal displacement vector sci) is increased by the S%$ contribution; 7. steps 4,5 and 6 are repeated up to convergence, which is assumed as reached, according to the following inequality:
Y
St
-
As*
0.31
cm21lS
5.06+5.06
cm
II
II 43&n
cm2 Aa =
Eb
=250,000 kg cm2 -246 kg/cm l Rbk Rbk lono =4.400 kg/cm2 Rbc -=3.000 kg/cm* Fig. 10. Analysed structural frame.
1.98+1.98
cm
G. M. CoccHl and F. CAPPELU)
94
Porg) 30000
Q +
Tests (71 Theory
- VS (cm)
0
0
2
4
6
8
10
12
Fig. 11. Load deflection for specimen Sl. with 6 = 0.0001. The above inequality agrees with the minimization of the ratio between the norm of nonlinear iterative displacements and the norm of initial elastic displacements; 8. cci(x), cli(x) and csi(x) strains are compared with ultimate conventional strain of concrete and steel; 9. the end nodal forces of the beam elements are obtained according to the calculated global displacement vector s; 10. the above steps are repeated for every load condition. The authors have noticed that the reliability of the above procedure is perfectly guaranteed as regards the solution of nonlinear equation systems for every cross-section, owing to the restricted existence inter-
val of the unknowns ui which can be obtained quite accurately by means of a reduced number of trials. The reliability is also guaranteed as regards the solution of equation system (18), relevant to the whole structure, because the above procedure consists in subsequent elastic analyses at fictitious load conditions, caused by inelastic behaviour of beam elements.
7. NUMERICAL RESULTS AND COMPARISON THEORY WITH TESTS
The structure in Fig. 10 ([lo] specimen S l), consisting of a floor beam of length L, and a spandrel beam of length L,, is analysed. Both beams have rectangu-
&kg) 30000 1
Q
OF
Test (71
*Theory + Gross + Cracked
I---
I
1
2
Fig. 12. Load twist for specimen Sl.
Inelastic analysis of reinforced concrete space frames
95
pw 30000 -
20000 -
0
Q
Test 171
* 4
m-Y
-9
Cracked
Gross
2000
1000 Fig. 13. Load torque for specimen Sl.
lar cross-sections. The elastic, geometric and reinforcement properties are shown in Fig. 10. The structure is discretized by means of 17 nodes and 8 beam elements, of constant length, for the floor beam, and 8 beam elements for the spandrel. The end restraints are described as those carried out in [lo]: the allowable (1) and nonallowable (0) nodal displacement components, for every restraint, are given in Fig. 10. The diagrams in Fig. 11 show the greatest vertical displacement (- u of node 5) at various p load values, given by theoretical analysis and tests. There is good agreement between theoretical and experimental results. The typical slope variation at the cracking load P = 4500 kg, up to the ultimate load
P = 18,000 kg, utilized in [lo] to design the cross-sections, is noticed. Beyond this value both diagrams show identical behaviour of large displacements at small increments of load, with a low percentage of shiftings. It should be noticed that it is necessary to remove the controls of the ultimate conventional strains of concrete and steel, shown at step 8 in Sec. 6, in order to describe the structural behaviour at higher loads than the above, because the tests are performed up to the condition of concrete crushing. The diagrams in Fig. 12 show the greatest torsional rotation (4, of node 9) at various P load values. There is good agreement between theoretical and experimental results. A slope variation according to the
&kg) 30000
1
Q +
Test [7]
Theory
-VS(em) 1
2
3
Fig. 14. Load deflection for specimen S6. CAS w-3
4
5
G. M. Coccm and F.
96
CMPELL.O
ww 30000 1
20000 -
10000 -)
+
0
Gross Cracked
100 200 300 Fig. 15. Load torque for specimen S6.
previous behaviour is noticed. In the same figure, the straight lines relevant to the elastic linear behaviour of gross and cracked cross-sections, described in [lo], are shown. Inspection of the theoretical diagram slopes shows that the torsional stiffness of the spandrel initially agrees with the elastic gross stiffness up to P = 4500 kg. It then approaches the cracked stiffness and finally is almost null for values of loads higher than the ultimate. The comparison between the theoretical and experimental diagram shows a lower stiffness of the truss model with respect to the real model with a small percentage of shiftings. Figure 13 shows the diagram of the ratio between the torsional continuity moment X at node 9 and the length L, of the floor beam, as a function of the P value. This diagram is indicative of the flexural state interaction with the torsional state of the spandrel beam. It is shown that for a P value lower than the ultimate, the theoretical X values are included among those relevant to the cracked model and those relevant to the elastic model. Beyond this limit the diagram approaches the last model as confirmed by experimental results, owing to the variation of the ratio between the flexural stiffness of the floor beam and the torsional stiffness of the spandrel. As pointed out previously, the shiftings are caused by lower torsional stiffness of the truss model. The proposed method is applied to the analysis of the structure shown in Fig. 10, reversing the dimensional ratio between the lengths (L,= 290 cm, L, = 457 cm) under the same geometric crosssections and material strength characteristics. For the floor beam, the longitudinal steel areas are As = 0.71 + 0.71 cm*, As = 5.06 + 5.06 cm*, stirrups 0.31/19cm2/cm, for the spandrel As = 0.71 + 0.71 cm*, As = 4 x 1.98 cm*, stirrups 0.31/11 cm*/cm (specimen S6). The diagrams in Fig. 14 show the theoretical and experimental values of the vertical displacement (-v of node 5) at various
400
P load values. Unlike the results of specimen S 1, the theoretical and experimental diagrams agree well at every load value, owing to the lower influence of the torsional behaviour of the spandrel as a consequence of the different length ratio. As regards the variation of the diagrams themselves, the observations made regarding Fig. 11, apply. The effect of variation of the ratios between the flexural and torsional stiffness of the beams is shown in the diagrams of Fig. 15, similar to those of Fig. 13. Contrary to the results obtained for Sl, the theoretical behaviour model is closer to the linear elastic behaviour than the experimental test.
8. CONCLUSIONS
In this paper the authors studied the problem of the theoretical analysis of reinforced concrete space frames accounting for the interaction effects among the stress components, by means of a suitable extension of the diagonal compression field theory, for pure torsion, applied to the general case of coupled axial, torsional and bending actions. Major attention was devoted to numerical problems in calculating the stresses and the strains in reinforced concrete crosssections, subject to coupled actions, having any geometric and reinforcing characteristics, and to the inelastic analysis of the whole structure, by means of suitable iterative algorithms. The numerical results, obtained by means of a computer program, have proved the reliability and the versatility of the proposed procedure and the good agreement between theoretical and experimental analysis of typical structures. The comparison of the results has likewise proved the weight of a correct prevision of the above interaction effects, not only at the cracking state but also at the ultimate state.
Inelastic analysis of reinforced concrete space frames REFERJSNCES
1. G. M. Cocchi and F. Cappello, Sul calcolo delle ossature spaziali in regime non lineare. Giomale de1 Genie Civile 7, 8, 9 (1987). 2. T. T. C. Hsu and Y. L. MO, Softening of concrete in torsional members-theory and tests. ACZ JI, Paper No. 82-25, May/June (1985). 3. D. Mitchell and M. P. Collins, Diagonal compression field theory-a rational model for structural concrete in pure torsion. ACZ Jl, 71, 396-408 (1974). 4. M. B. Leung and W. C. Schnobrich, Reinforced concrete beams subiected to bending and torsion. J Srruct. Eegsg,ASCE i13,No. 2 (1987). 5.P. Lampert and B. ~mlim~n, Ultimate strength and design of reinforced concrete beams in torsion and bending. International Association for Bridge and Structural En8ineering, 31-1, Zurich (1971).
97
6. A. E. McMullen and J. Wanvaruk, Concrete beams in bending, torsion and shear. J. Sttuct. Engng, ASCE May (1970). 7. R. Park and T. Paulay, Reinforced Concrete Structures. John Wiley (1979). 8. K. J. Bathe and A. P. Cimento, Some practical procedures for the solution of non-linear finite element equations. Comput. Meth. Appf. Meek Engng 22,59-85 (1980). 9. G. M. Cocchi and F. Cappello, Instabilita’ laterale di strutture reticolari prefabbricate in regime transitorio e di esercizio. Atti de1 Congresso CTE sull’Industriahzzazione Edilizia, Firenze, 9-l 1, November (1984). 10. M. P. Collins and P. Lampert, R~st~bution of moments at cracking-the key to simpler torsion design. AC1 Publication SP35, March (1971).