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Inelastic analysis of steel frames for multistory buildings
0045-7949185 $3.00 + .M) Pwgamon Press Ltd.
Computers & Structures Vol. 20, No. 4, pp. 707-713, 1985 Printed in Great Britain
INELASTIC
ANALYSIS OF STEEL FRAMES MULTISTORY BUILDINGS
FOR
E. Bozzo and L. GAMBAROTTA Istituto di Scienza defle Costmzio~, U~versi~ di Genova, 16145 Italy (Received 4 August 1983; received for publication 5 December
1983)
Abstract-A method has been developed directed towards the analysis of multistory buildings beyond the elastic limit and also considering geometrical non-linearities. The plastic zones are concentrated at the ends of the members and a single stiffness matrix has been developed which makes it possible to describe the behaviour of the member in the presence of plastic hinges at one or the other or at both ends. The solutions are obtained using an incremental-iterative method and a special algorithm, which takes into account the iteraction between the member end forces during the elasto-plastic phase, guarantees the equilibrium of each element and at the same time compels the stress point to respect the yield condition. The method has been codified in a computer program even for the study of very large structures. Some examples have been given to demonstrate the validity of the method and the reliability of the code. 1. ~RODU~ION
beyond the elastic limit of framed structures for tall buildings performed with general purpose finite element codes is still as of today cumbersome, even when considering the development of new computers, because of the large number of degrees of freedom and of the simultaneous intluence. of geometric and material non-linearities. One may obtain substantial simplifications with an “oriented” analysis, that is considering the peculiarities of such structures. The assumption of rigid floors on their own plane brings about a marked reduction in the degrees of freedom, the geometric non-linearity may be properly appro~ma~d by the P - d effects, and the particular internal forces distribution in columns allows further simplifications. In fact for the determination of the elasto-plastic stiffness, the characteristic linear distribution of bending moments makes it possible to obtain the shape function by means of a constitutive law of the Ramberg~good typefl-3] or justifies the hypothesis that concentrates the plastic zones in the member ends[49]. This last hypothesis is the one used in this work, and with it we arrive at the definition of only one elastoplastic stiffness matrix to describe the behaviour of the member when either one or both of the ends are in a plastic state. This simplification, disregarding the transverse shear influence on the yield condition, permits the description of the member behaviour considering material and geometrical nonlinearities separately. The solution to the nonlinear problem is obtained by employing either incremental[6] or mixed increments-iterative methods[lO]. In both cases one proceeds through finite increments, obtaining stress state that does not comply with the plastic condition, because of the convexity of the yield surface. The best known methods that respect the yield condition use either iterative procedures, heavy on the computational side, or one step approximate force corrections, which introduce errors, sometimes important, in the member forces[7]. The method proposed in this work follows instead from Ref. [I 1] and considers that plastic flows occur when the force point is contained between the yield The
analysis
CAS 20:4-o
707
surface and a second one, omothetic, internal and close to the first. In this case it is possible to account for the interaction due to the making of plastic hinges at the member end to ensure the element equilibrium, but with a limited increase in calculation load. A computer program for the analysis of multistory buildings1 12, 131was implements to allow for material non linearities in the hypothesis of elasto-ideally plastic behaviour. The solutions is obtained with an incremental-iterative method. Some examples to show the capabilities and the reliability of the code are then discussed.
2. TANGENT
STIFFNESS FOR
ELASTO-PLASTIC
BEAM
The elasto-plastic behaviour of a twelve degrees of freedom prismatic member is defined supposing that plastic deformations may occur only in the end sections of the same. In this case the member and displacement rate (elastic displacement) {z&ie> differs from the joint displacement rate {ti} because of the plastic deformation rate {tip} (ai) = {tiJ + (ic,).
(2.1)
The element is conceived as an elastic part as long as the element itself, and for which the elastic constitutive law is considered @ = r&rlcli,]
(2.2)
and by two zones of small relative length, at the ends, in which the possible plastic flows will appear. These are present in the ith end of the beam when the member end force {f} satisfies the yield condition expressed by MM, Ix]) = 0 i = f,2. (2.3) M_% {x]) = 0 Plastic flows are in the direction normal to the yield surface rr$ at the force point {J}. conforms to the
E. Bozzo and L.
708
associated flow rule, and are equal to
GAMBAROTTA
to calculate the matrix I (2.4)
with 1, being a positive scalar proportionality factor, as yet undetermined, related to the degree of freedom released by the generalized plastic hinge. The determination of the unknown vector {A), necessary for the evaluation of the elasto-plastic stiffness matrix, is made using the consistency relation given by the second of the formulas (2.3) which, expressed in matrix form, reads
I$, = [Gl’{f} + [rl?W = (01,
(2.5)
where [G] is the gradient matrix of the yield surface C$at the force points v}, and [r] is the matrix of the derivatives of the yield function with respect to hardening parameters {X}. The determination of {i} proceeds taking in consideration the ends which become plastic. In this way one obtains in general three elasto-plastic stiffness matrices[4,7-91. To obtain one single expression for these matrices we must consider the hardening plastic material. It will be possible at the end to remove this last assumption, as will be shown later. We define the hardening matrix rH_l with its ith term[lO]
(‘U+‘H1)m’=G;-(l r**(l -PJ i -
-p,)(l
- $;
~I,(1 - PdU
‘{i).
(2.6)
1
(2.14)
In this way the stiffness matrix may be calculated considering both ends in the plastic state, and afterwards by varying the value of the H;s or p,‘s, define which ones of these are really plastic. In fact the value of p, ranges from 0, to which corresponds the behaviour of the ideally plastic material (Hi = 0) and 1 to which corresponds an infinite hardening parameter that is in the case absence of plastic phenomena. The elasto ideally plastic behaviour hypothesis demands that pi’s take only the value 0 or 1, while for an hardening behaviour the pis will take positive values between 0 and 1. To summarize one calculates the values of the p,, assigning the value of 1 if the 4, or i, are negative, the value of 0 for an ideally plastic material, or an intermediate value given by (2.6) and (2.10) for hardening materials. If at least one of the two pis is different by 1, then is necessary to calculate the elasto-plastic correction matrix [S,]. ANALYSIS
OF
FRAME STRUCTURES
(2.8)
(2.9)
{Au’] = lSr,l- ‘{rU’]
(2.7)
Rewriting (2.4), by the definition of [G], {li,} = [G](i)
-P2)
The elasto-plastic analysis is developed using a mixed incremental-iterative method. This technique is justified by the path dependent behaviour of the structure described as successive increments. We obtain the solution for each increment using the iterative Newton-Raphson modified method, that demands the assembly process of the stiffness matrix only once at the beginning of the increment. The iterative sequence of this algorithm is as follows
Then, given (2.5), the following relation holds:
[GIT@ = [HJ{i).
:;:I”
- P2)
3. ELASTO-PLASTIC
Hi =
-p*)?J,2
through (2.8), (2.1), (2.2) and (2.5) one obtains {i} = ([GITIS,,l[Gl + [HJI-1[G17LS,,lb&
where, as defined before, {A} must be positive (a negative value for {i) indicates a condition of elastic unloading). Finally, by means of (2.8), (2.9), (2.2), (2.1), one obtains the force-displacement relation in the elastoplastic state If] = ([&,I -+ [S,l)M = [S,lb4, the elasto-plastic
(2.10)
correction matrix [S,] is
IS,1= - E.,l[Gl([Gl%.,l[Gl + WI)-‘tGlT[U.
(2.11)
It is worth while at this point to define t VI= ~GlTM[Gl Hi =
Vii
*
I
i=l,2
(2.12) (2.13)
(3.1)
{u”‘} = {uJ} + (AU’} wherej is the number of the iteration, {df’} the vector of increments of internal resisting loads, function of the (u’} nodal displacements, {df,} the vector of the increments of the external loads and {ri} the vector of the unbalanced loads. Figure 1 shows how in this procedure, the iterations are repeated until the increment of the nodal displacements becomes negligible. For each iteration one must evaluate {df’} as a function of the nodal forces at the beginning of the increment and of the ones corresponding to the displacement {~j}. The contribution integral
of each element is given by the
{Af} = 1;;’
lSrt{f])l{du],
(3.2)
Inelastic analysis of steel frames for multistory buildings
709
Af,
Fig. 1. Modified Newton-Raphson method. that may be written as
{V} =
s’
[&@-}l
0
drl *{Au’)
(3.3)
in the hypothesis of linearly growing displacements. The integral (3.3) may be calculated by subdividing the integration interval in n sub-intervals of width vi in which one may assume that the stiffness matrix remains constant. This matrix is calculated from (2.11) in the assumption of elasto-ideally plastic material and of a yield condition which can be expressed in an analytic form, function of the axial force and of the bending moments. Such a hypothesis, together with that of considering the geometric non-linearity as a P -A effect, is certainly suitable for the three dimensional structure of multistory buildings and makes it possible, as it is an important simplification, to evaluate 6rst the contribution to {Afj} from the elasto-plastic behaviour and then, as a function of the former, the contribution of the variation of P -A effects. In this last case, by substituting the matrix [S,,] = [S,] + [S,] in the (2.1 l), one verifies that the geometric stiffness matrix [S,] doesn’t have any effect on the matrix [S,]. Since the yield surface 1s convex the force-point moves, because of (2.10), along the tangent to the surface violating the yield condition. This force point drift from the yield surfaces becomes larger as the interval vi grows. It then becomes necessary to introduce an algorithm to maintain the force-point on the yield surface. This algorithm must be able to complete the increment of displacement {Ad] in few steps, and above all must at the same time fulfill the plasticity condition and guarantee the equilibrium of the member forces, i.e. account for the interaction between the variations of the conditions of elastic and/or plastic state. Among the various methods known[5,7] we have chosen the one proposed by Hodge[l I], that has the characteristics mentioned above, to calculate the vi intervals and the member end forces and then make possible the integration of eqn (3.3). The method starts from the consideration that the yield surface cannot in general be defined in a deterministic way and so is reasonable to think of a zone in which all the plastic flows occur. The zone has well defined dimensions and is between two omothetic surfaces with the shape of the yield surface. This treatment of the problem allows the deter-
Fig. 2. Motion of the force point in the yield zone. mination, with only one algorithm, of the amplitude of the subintervals vi corresponding to the member being in its elastic phase in one or both its ends. The activation of a plastic hinge at the end Qstarts as soon as the force point crosses the internal surface and then is forced to the interior of the plastic zone (unless it is a case of elastic unloading). Figure 2 shows how the point is forced to bounce between the two surfaces moving on a line normal to the surface if the impact point is on the external one, or on a tangent to the surface if the impact point is on the inner one. The intersection of the line AB with the above mentioned tangent determines the value of the subinterval qz. Figure 2(a) shows more in detail the passage from an elastic to a plastic state and Figs. 2(b, c) show the evolution of the process when one remains in a plastic state, Fig. 2(c) showing an example of tl. = 0. In the latter case the process is repeated using the elasto-plastic matrix calculated with the yield function gradients at point C. The correlation of state variations at the element ends is accounted for by the following procedure: after evaluating tl. at both ends one considers the stress increments as obtained with the smaller of the two intervals. This process is then repeated until one has completed the entire displacement {Auj}. Then, after the determination of the increments for the stresses due to the elasto-plastic behaviour, one calculates the ones corresponding to P - A effects. 4. COMPUTER PROGRAhI AND EXAMPLES
The above outlined method has been coded in the program CSSEM for multistory building analysis[ 12, 131. The program allows the analysis of 3-D frame structures with beams and columns without any limitations on the geometry and with the assumption of floors rigid in their own plane. Loads are applied at joints and distributed load conditions
710
E. Bozzo and L.
GAMBAROTTA
I -load
50
factor
10.0
15.0
20 0
Top story lateral
25 0
30 II
deflection,
35 0
40 0
(inches)
Fig. 4. Load vs deflection diagram of Example 1.
_
12’
-
24’
Fig. 3. Example l-twelve
_
P = 1 Klps w= 4 Kips q i 36 KSI E = 29000 KSI
story frame.
on the beams are simulated by defining intermediate joints to which equivalent forces are applied so as not to violate the hypothesis of a linear distribution of bending moments. A special algorithm has developed to avoid the appearance of two adjacent plastic hinges on the same beam, thus eliminating a possible unreal situations of lability. The incremental-iterative procedure is started by assigning an initial load increment value which will be automatically revaluated in the course of the analysis. This to follow more closely the real load vs displacement curve and to reduce the number of iterations needed. Actually when the maximum allowable number of iterations is reached the analysis continues with halved load increments, and so on until the load increment value becomes smaller than a predelined fraction of the applied load. This value is then considered as that at “incipient collapse” one. The program is coded using FORTRAN IV and the cases discussed here were computed on an IBM 370/168 computer. In order to verify the validity of the program we first study a two-dimensional steel frame, 12-stories high, already studied in [7] and of which the characteristics are those depicted in Fig. 3. The analysis was performed considering also geometrical non-linearities. In this case however this has little influence on the collapse load since the critical elastic load is much larger than the rigid-plastic one[l4, 151. Figure 4 shows the load-displacement curve obtained with an initial loading step value equal to l/5 of the applied load and assuming as
minimum value l/l00 of the load obtained. Plastic hinges position, numbered with the number of the load step in which they appeared, are shown in Fig. 3. The results obtained with a fairly large initial loading step (as inferred from the large number of subdivision needed) are in good agreement with [7], also because the plastic hinges appear mainly on the beams. We have also performed analyses with a smaller initial loading step, which required less computer time, given a smaller number of iterations. It is worth noting that the best value for the loading step is not known a priori and that the availability of an automatic algorithm to evaluate it while moving on the load-displacement curve is very useful. The structure chosen for the 3-D analysis example has been analyzed already by various authors[7,16] and is shown in Fig. 5. The loads considered here are uniform vertical forces applied at joints and equivalent to uniform load of 100 psf and horizontal forces equivalent to uniform pressure of 20 psf on the larger surface. To evaluate the stiffness of the frame each member was considered as coated with 1 in. thick concrete layer, while for the determination of the plastic characteristics only the steel section was taken into account. Moreover we considered the yield surface obtained in Ref. [7] and the yield stress 6 = 50 ksi. The analysis has been performed both in small and large displacements. The corresponding load displacements curves are shown in Fig. 6 and contain for each step the number of plastic hinges. The position of these for the incipient collapse condition in the case of small displacements analysis is shown in Fig. 7. Curves of Fig. 6 may not be compared with the ones in [7] beyond the elastic limit since the plastic characteristics of structural elements are not given. As one may see in Fig. 6 the effect of geometric non-linearities is important and causes a reduction of more than 20% in the collapse load. The collapse condition is reached through the elastic instability of the deteriorated structure and then the number of plastic hinges is smaller one. Since in the previous examples the loads were assumed monotonically
Inelastic analysis of steel frames for multistory buildings
711
-20 -19
T-
-19 -17 -16 -15 -14 -13 -12 - 11 -10
-9 -6 -7
I
: 3
21 WF55
Cc:
16WF36
-6
16WF36
1 9
7
-5 -4
i
-3 -2
-L
-1
10WF35
1 10
8 24’
24’ I-
Fig. 5. Example 2-twenty
c
I-load
story frame.
factor
--la6
61
l
plastic
hinges
--On6
Il.
-
large
displacamant
analysis
-----
small
displacement
analysis
20.
10.
V -
diaplacamsnt
30.
- node
1
top
40.
story
Fig. 6. Load vs deflection diagram of example 2.
60.
b
inches
E. hozzo and L.
GAMBAROTTA
growing, no marked movements of the stress point on the yield surface were recorded. Therefore in order to show up better the characteristics of the advancing algorithm proposed, the analysis was carried out on a 3-D one story frame, subjected to a prefixed load path as shown in Fig. 8. The results of the analysis are synthetized in Figs. 9 and 10 having made reference to a spherical yield surface what, after the first step, because of the load path adopted, did not record marked variations of the axial force.
Fig. 7. Plastic hinges position in the twenty stories frame.
5. CONCLUSIONS We have considered the post-elastic analysis of framed structures also taking into account geometrical non linearities, and we have proposed a method to evaluate only one elasto-plastic stiffness matrix for two-node prismatic members in the hypothesis that plastic zones may appear at either or both ends of the element. Solutions are obtained by means of an incremental-iterative method. An algorithm to determine automatically the step amplitude was developed, both to follow better the load-displacement curve with a limited number of increments and to limit the number of iterations. A method which is able to take into account the interaction between the member end forces during the elasto-plastic phases has been proposed to guarantee the equilibrium in each member and also to speed up the convergence of the iterative process. The code already developed for elastic analysis[l2, 131 was modified for the study of large structures beyond the elastic limit. The reliability of the code is demonstrated by the results of the first example, a twelve stories plane frame, while its capability to analyze large structures with a limited computational effort is shown by the second case, a three-dimensional twenty stories high structure. This example proves the importance of performing the analysis in successive loading steps, given the S t
e D
r--
3 4
mh
I=500cm
h=250
cm
P=130KN
H=43.3KN
beams
IPE 330
columns
HE 180A
Fig, 8. Example 3-one story frame.
5 11
0.65
1.1
0.675
8
1.1
07
9
11
0.725
Inelastic analysis of steel frames for multistory buildings
0.7
4
*UY
.-ux
(i’,---/
-:
--5
//’
-*
.i
91 0.4
V
-4
-
/x.
a.2
/
-
.
I!: --8. 2
0.
/
1.
:
:
:
:
2.
3.
4.
5.
10s4 rad t
cm
Fig. 9. Diagram of loads vs displacements of point A.
f
influence of large displacements on the the collapse load. Finally, the last example shows as the code makes it possible to obtain accurate solutions with large load increments even in the presence of marked variations of the internal forces during the plastic phase. REFERENCES
0.6
0.0
713
m2
Fig. 10. Motion of the internal forces in section A.
1. J. L. Bockolt and W. Weaver, Jr., Inelastic dynamic analysis of tier buildings. Comput. Structures 4,627+X (1974). 2. B. L. Gunnin, F. N. Rad and R. W. Furlong, A general non linear analysis of concrete structures and comparison with frame tests. Comput. Structures 7, 257-265 (1977). 3. F. Y. Cheng, Inelastic analysis of 3-D mixed steel and reinforced concrete seismic building systems. Comput. Structures 13, 189-196 (1981). 4. N. C. Nigam, Yielding in framed structures under dynamic loads, J. Engng Mech. Div. ASCE 96, (EM5), 687-709 (1970). 5. R. K. Wen and F. Farhomand, Dynamic analysis of inelastic space frames. J. Engng Mech. Div. AXE 96 (EMS), 667-686 (1970). 6. F. L. Porter and G. H. Powell, Static and dynamic analysis of inelastic frame structures. Rep. EERC 71-3, Earthquake Engng Res. Center, University of California, Berkeley, CA (1971). 7. J. G. Orbison, W. McGuire and J. F. Abel, Yield surface applications in nonlinear steel frame analysis. Comput. iUeth. Appl. Mech. Engng 33, 557-573, (1982). 8. Y. Ueda and T. Yao, The plastic node method: a new method of plastic analysis. Comput. Meth. Appl. Mech. Engng 34, 1089-I 104 (1982). 9. J. H. Argyris, B. Boni, U. Hindenlang and M. Kleiber, Finite element analysis of two and three-dimensional elasto-plastic framesthe natural approach. Comput. Mefh. Appf. Mech. Engng 35, 221-248, (1982). 10. 0. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London (1971). 11. P. G. Hodge, Jr., Automatic piecewise linearization in ideal plasticity. Comput. Meth. Appl. Mech. Engng 10, 249-272 (1977). 12. E. Bozxo and L. Gambarotta, Linear analysis of tridimensional structures for multistory buildings. Costruzioni Met&&e, 2 (198 1), 13. E. Bozzo and L. Gambarotta, Large displacement analysis of tridimensional structures for multistory buildings. Costruzioni Metalliche, 4 (198 1). 14. C. E. Massonet and M. A. Save, Plastic analysis and design. Beams and Frames, Vol. 1. Blaisdell, New York (1965). 15. E. Bozzo and L. Gambarotta, On the overall stability of steel frames. Giomate Italiane delle Costruzione in Acciaio, Perugia (1983). 16. W. Weaver and M. F. Nelson, Three-dimensional analysis of tier buildings, J. Struct. Div., ASCE 92 (ST6), 385-404. (1966).
Computers & Structures Vol. 20, No. 4, pp. 707-713, 1985 Printed in Great Britain
INELASTIC
ANALYSIS OF STEEL FRAMES MULTISTORY BUILDINGS
FOR
E. Bozzo and L. GAMBAROTTA Istituto di Scienza defle Costmzio~, U~versi~ di Genova, 16145 Italy (Received 4 August 1983; received for publication 5 December
1983)
Abstract-A method has been developed directed towards the analysis of multistory buildings beyond the elastic limit and also considering geometrical non-linearities. The plastic zones are concentrated at the ends of the members and a single stiffness matrix has been developed which makes it possible to describe the behaviour of the member in the presence of plastic hinges at one or the other or at both ends. The solutions are obtained using an incremental-iterative method and a special algorithm, which takes into account the iteraction between the member end forces during the elasto-plastic phase, guarantees the equilibrium of each element and at the same time compels the stress point to respect the yield condition. The method has been codified in a computer program even for the study of very large structures. Some examples have been given to demonstrate the validity of the method and the reliability of the code. 1. ~RODU~ION
beyond the elastic limit of framed structures for tall buildings performed with general purpose finite element codes is still as of today cumbersome, even when considering the development of new computers, because of the large number of degrees of freedom and of the simultaneous intluence. of geometric and material non-linearities. One may obtain substantial simplifications with an “oriented” analysis, that is considering the peculiarities of such structures. The assumption of rigid floors on their own plane brings about a marked reduction in the degrees of freedom, the geometric non-linearity may be properly appro~ma~d by the P - d effects, and the particular internal forces distribution in columns allows further simplifications. In fact for the determination of the elasto-plastic stiffness, the characteristic linear distribution of bending moments makes it possible to obtain the shape function by means of a constitutive law of the Ramberg~good typefl-3] or justifies the hypothesis that concentrates the plastic zones in the member ends[49]. This last hypothesis is the one used in this work, and with it we arrive at the definition of only one elastoplastic stiffness matrix to describe the behaviour of the member when either one or both of the ends are in a plastic state. This simplification, disregarding the transverse shear influence on the yield condition, permits the description of the member behaviour considering material and geometrical nonlinearities separately. The solution to the nonlinear problem is obtained by employing either incremental[6] or mixed increments-iterative methods[lO]. In both cases one proceeds through finite increments, obtaining stress state that does not comply with the plastic condition, because of the convexity of the yield surface. The best known methods that respect the yield condition use either iterative procedures, heavy on the computational side, or one step approximate force corrections, which introduce errors, sometimes important, in the member forces[7]. The method proposed in this work follows instead from Ref. [I 1] and considers that plastic flows occur when the force point is contained between the yield The
analysis
CAS 20:4-o
707
surface and a second one, omothetic, internal and close to the first. In this case it is possible to account for the interaction due to the making of plastic hinges at the member end to ensure the element equilibrium, but with a limited increase in calculation load. A computer program for the analysis of multistory buildings1 12, 131was implements to allow for material non linearities in the hypothesis of elasto-ideally plastic behaviour. The solutions is obtained with an incremental-iterative method. Some examples to show the capabilities and the reliability of the code are then discussed.
2. TANGENT
STIFFNESS FOR
ELASTO-PLASTIC
BEAM
The elasto-plastic behaviour of a twelve degrees of freedom prismatic member is defined supposing that plastic deformations may occur only in the end sections of the same. In this case the member and displacement rate (elastic displacement) {z&ie> differs from the joint displacement rate {ti} because of the plastic deformation rate {tip} (ai) = {tiJ + (ic,).
(2.1)
The element is conceived as an elastic part as long as the element itself, and for which the elastic constitutive law is considered @ = r&rlcli,]
(2.2)
and by two zones of small relative length, at the ends, in which the possible plastic flows will appear. These are present in the ith end of the beam when the member end force {f} satisfies the yield condition expressed by MM, Ix]) = 0 i = f,2. (2.3) M_% {x]) = 0 Plastic flows are in the direction normal to the yield surface rr$ at the force point {J}. conforms to the
E. Bozzo and L.
708
associated flow rule, and are equal to
GAMBAROTTA
to calculate the matrix I (2.4)
with 1, being a positive scalar proportionality factor, as yet undetermined, related to the degree of freedom released by the generalized plastic hinge. The determination of the unknown vector {A), necessary for the evaluation of the elasto-plastic stiffness matrix, is made using the consistency relation given by the second of the formulas (2.3) which, expressed in matrix form, reads
I$, = [Gl’{f} + [rl?W = (01,
(2.5)
where [G] is the gradient matrix of the yield surface C$at the force points v}, and [r] is the matrix of the derivatives of the yield function with respect to hardening parameters {X}. The determination of {i} proceeds taking in consideration the ends which become plastic. In this way one obtains in general three elasto-plastic stiffness matrices[4,7-91. To obtain one single expression for these matrices we must consider the hardening plastic material. It will be possible at the end to remove this last assumption, as will be shown later. We define the hardening matrix rH_l with its ith term[lO]
(‘U+‘H1)m’=G;-(l r**(l -PJ i -
-p,)(l
- $;
~I,(1 - PdU
‘{i).
(2.6)
1
(2.14)
In this way the stiffness matrix may be calculated considering both ends in the plastic state, and afterwards by varying the value of the H;s or p,‘s, define which ones of these are really plastic. In fact the value of p, ranges from 0, to which corresponds the behaviour of the ideally plastic material (Hi = 0) and 1 to which corresponds an infinite hardening parameter that is in the case absence of plastic phenomena. The elasto ideally plastic behaviour hypothesis demands that pi’s take only the value 0 or 1, while for an hardening behaviour the pis will take positive values between 0 and 1. To summarize one calculates the values of the p,, assigning the value of 1 if the 4, or i, are negative, the value of 0 for an ideally plastic material, or an intermediate value given by (2.6) and (2.10) for hardening materials. If at least one of the two pis is different by 1, then is necessary to calculate the elasto-plastic correction matrix [S,]. ANALYSIS
OF
FRAME STRUCTURES
(2.8)
(2.9)
{Au’] = lSr,l- ‘{rU’]
(2.7)
Rewriting (2.4), by the definition of [G], {li,} = [G](i)
-P2)
The elasto-plastic analysis is developed using a mixed incremental-iterative method. This technique is justified by the path dependent behaviour of the structure described as successive increments. We obtain the solution for each increment using the iterative Newton-Raphson modified method, that demands the assembly process of the stiffness matrix only once at the beginning of the increment. The iterative sequence of this algorithm is as follows
Then, given (2.5), the following relation holds:
[GIT@ = [HJ{i).
:;:I”
- P2)
3. ELASTO-PLASTIC
Hi =
-p*)?J,2
through (2.8), (2.1), (2.2) and (2.5) one obtains {i} = ([GITIS,,l[Gl + [HJI-1[G17LS,,lb&
where, as defined before, {A} must be positive (a negative value for {i) indicates a condition of elastic unloading). Finally, by means of (2.8), (2.9), (2.2), (2.1), one obtains the force-displacement relation in the elastoplastic state If] = ([&,I -+ [S,l)M = [S,lb4, the elasto-plastic
(2.10)
correction matrix [S,] is
IS,1= - E.,l[Gl([Gl%.,l[Gl + WI)-‘tGlT[U.
(2.11)
It is worth while at this point to define t VI= ~GlTM[Gl Hi =
Vii
*
I
i=l,2
(2.12) (2.13)
(3.1)
{u”‘} = {uJ} + (AU’} wherej is the number of the iteration, {df’} the vector of increments of internal resisting loads, function of the (u’} nodal displacements, {df,} the vector of the increments of the external loads and {ri} the vector of the unbalanced loads. Figure 1 shows how in this procedure, the iterations are repeated until the increment of the nodal displacements becomes negligible. For each iteration one must evaluate {df’} as a function of the nodal forces at the beginning of the increment and of the ones corresponding to the displacement {~j}. The contribution integral
of each element is given by the
{Af} = 1;;’
lSrt{f])l{du],
(3.2)
Inelastic analysis of steel frames for multistory buildings
709
Af,
Fig. 1. Modified Newton-Raphson method. that may be written as
{V} =
s’
[&@-}l
0
drl *{Au’)
(3.3)
in the hypothesis of linearly growing displacements. The integral (3.3) may be calculated by subdividing the integration interval in n sub-intervals of width vi in which one may assume that the stiffness matrix remains constant. This matrix is calculated from (2.11) in the assumption of elasto-ideally plastic material and of a yield condition which can be expressed in an analytic form, function of the axial force and of the bending moments. Such a hypothesis, together with that of considering the geometric non-linearity as a P -A effect, is certainly suitable for the three dimensional structure of multistory buildings and makes it possible, as it is an important simplification, to evaluate 6rst the contribution to {Afj} from the elasto-plastic behaviour and then, as a function of the former, the contribution of the variation of P -A effects. In this last case, by substituting the matrix [S,,] = [S,] + [S,] in the (2.1 l), one verifies that the geometric stiffness matrix [S,] doesn’t have any effect on the matrix [S,]. Since the yield surface 1s convex the force-point moves, because of (2.10), along the tangent to the surface violating the yield condition. This force point drift from the yield surfaces becomes larger as the interval vi grows. It then becomes necessary to introduce an algorithm to maintain the force-point on the yield surface. This algorithm must be able to complete the increment of displacement {Ad] in few steps, and above all must at the same time fulfill the plasticity condition and guarantee the equilibrium of the member forces, i.e. account for the interaction between the variations of the conditions of elastic and/or plastic state. Among the various methods known[5,7] we have chosen the one proposed by Hodge[l I], that has the characteristics mentioned above, to calculate the vi intervals and the member end forces and then make possible the integration of eqn (3.3). The method starts from the consideration that the yield surface cannot in general be defined in a deterministic way and so is reasonable to think of a zone in which all the plastic flows occur. The zone has well defined dimensions and is between two omothetic surfaces with the shape of the yield surface. This treatment of the problem allows the deter-
Fig. 2. Motion of the force point in the yield zone. mination, with only one algorithm, of the amplitude of the subintervals vi corresponding to the member being in its elastic phase in one or both its ends. The activation of a plastic hinge at the end Qstarts as soon as the force point crosses the internal surface and then is forced to the interior of the plastic zone (unless it is a case of elastic unloading). Figure 2 shows how the point is forced to bounce between the two surfaces moving on a line normal to the surface if the impact point is on the external one, or on a tangent to the surface if the impact point is on the inner one. The intersection of the line AB with the above mentioned tangent determines the value of the subinterval qz. Figure 2(a) shows more in detail the passage from an elastic to a plastic state and Figs. 2(b, c) show the evolution of the process when one remains in a plastic state, Fig. 2(c) showing an example of tl. = 0. In the latter case the process is repeated using the elasto-plastic matrix calculated with the yield function gradients at point C. The correlation of state variations at the element ends is accounted for by the following procedure: after evaluating tl. at both ends one considers the stress increments as obtained with the smaller of the two intervals. This process is then repeated until one has completed the entire displacement {Auj}. Then, after the determination of the increments for the stresses due to the elasto-plastic behaviour, one calculates the ones corresponding to P - A effects. 4. COMPUTER PROGRAhI AND EXAMPLES
The above outlined method has been coded in the program CSSEM for multistory building analysis[ 12, 131. The program allows the analysis of 3-D frame structures with beams and columns without any limitations on the geometry and with the assumption of floors rigid in their own plane. Loads are applied at joints and distributed load conditions
710
E. Bozzo and L.
GAMBAROTTA
I -load
50
factor
10.0
15.0
20 0
Top story lateral
25 0
30 II
deflection,
35 0
40 0
(inches)
Fig. 4. Load vs deflection diagram of Example 1.
_
12’
-
24’
Fig. 3. Example l-twelve
_
P = 1 Klps w= 4 Kips q i 36 KSI E = 29000 KSI
story frame.
on the beams are simulated by defining intermediate joints to which equivalent forces are applied so as not to violate the hypothesis of a linear distribution of bending moments. A special algorithm has developed to avoid the appearance of two adjacent plastic hinges on the same beam, thus eliminating a possible unreal situations of lability. The incremental-iterative procedure is started by assigning an initial load increment value which will be automatically revaluated in the course of the analysis. This to follow more closely the real load vs displacement curve and to reduce the number of iterations needed. Actually when the maximum allowable number of iterations is reached the analysis continues with halved load increments, and so on until the load increment value becomes smaller than a predelined fraction of the applied load. This value is then considered as that at “incipient collapse” one. The program is coded using FORTRAN IV and the cases discussed here were computed on an IBM 370/168 computer. In order to verify the validity of the program we first study a two-dimensional steel frame, 12-stories high, already studied in [7] and of which the characteristics are those depicted in Fig. 3. The analysis was performed considering also geometrical non-linearities. In this case however this has little influence on the collapse load since the critical elastic load is much larger than the rigid-plastic one[l4, 151. Figure 4 shows the load-displacement curve obtained with an initial loading step value equal to l/5 of the applied load and assuming as
minimum value l/l00 of the load obtained. Plastic hinges position, numbered with the number of the load step in which they appeared, are shown in Fig. 3. The results obtained with a fairly large initial loading step (as inferred from the large number of subdivision needed) are in good agreement with [7], also because the plastic hinges appear mainly on the beams. We have also performed analyses with a smaller initial loading step, which required less computer time, given a smaller number of iterations. It is worth noting that the best value for the loading step is not known a priori and that the availability of an automatic algorithm to evaluate it while moving on the load-displacement curve is very useful. The structure chosen for the 3-D analysis example has been analyzed already by various authors[7,16] and is shown in Fig. 5. The loads considered here are uniform vertical forces applied at joints and equivalent to uniform load of 100 psf and horizontal forces equivalent to uniform pressure of 20 psf on the larger surface. To evaluate the stiffness of the frame each member was considered as coated with 1 in. thick concrete layer, while for the determination of the plastic characteristics only the steel section was taken into account. Moreover we considered the yield surface obtained in Ref. [7] and the yield stress 6 = 50 ksi. The analysis has been performed both in small and large displacements. The corresponding load displacements curves are shown in Fig. 6 and contain for each step the number of plastic hinges. The position of these for the incipient collapse condition in the case of small displacements analysis is shown in Fig. 7. Curves of Fig. 6 may not be compared with the ones in [7] beyond the elastic limit since the plastic characteristics of structural elements are not given. As one may see in Fig. 6 the effect of geometric non-linearities is important and causes a reduction of more than 20% in the collapse load. The collapse condition is reached through the elastic instability of the deteriorated structure and then the number of plastic hinges is smaller one. Since in the previous examples the loads were assumed monotonically
Inelastic analysis of steel frames for multistory buildings
711
-20 -19
T-
-19 -17 -16 -15 -14 -13 -12 - 11 -10
-9 -6 -7
I
: 3
21 WF55
Cc:
16WF36
-6
16WF36
1 9
7
-5 -4
i
-3 -2
-L
-1
10WF35
1 10
8 24’
24’ I-
Fig. 5. Example 2-twenty
c
I-load
story frame.
factor
--la6
61
l
plastic
hinges
--On6
Il.
-
large
displacamant
analysis
-----
small
displacement
analysis
20.
10.
V -
diaplacamsnt
30.
- node
1
top
40.
story
Fig. 6. Load vs deflection diagram of example 2.
60.
b
inches
E. hozzo and L.
GAMBAROTTA
growing, no marked movements of the stress point on the yield surface were recorded. Therefore in order to show up better the characteristics of the advancing algorithm proposed, the analysis was carried out on a 3-D one story frame, subjected to a prefixed load path as shown in Fig. 8. The results of the analysis are synthetized in Figs. 9 and 10 having made reference to a spherical yield surface what, after the first step, because of the load path adopted, did not record marked variations of the axial force.
Fig. 7. Plastic hinges position in the twenty stories frame.
5. CONCLUSIONS We have considered the post-elastic analysis of framed structures also taking into account geometrical non linearities, and we have proposed a method to evaluate only one elasto-plastic stiffness matrix for two-node prismatic members in the hypothesis that plastic zones may appear at either or both ends of the element. Solutions are obtained by means of an incremental-iterative method. An algorithm to determine automatically the step amplitude was developed, both to follow better the load-displacement curve with a limited number of increments and to limit the number of iterations. A method which is able to take into account the interaction between the member end forces during the elasto-plastic phases has been proposed to guarantee the equilibrium in each member and also to speed up the convergence of the iterative process. The code already developed for elastic analysis[l2, 131 was modified for the study of large structures beyond the elastic limit. The reliability of the code is demonstrated by the results of the first example, a twelve stories plane frame, while its capability to analyze large structures with a limited computational effort is shown by the second case, a three-dimensional twenty stories high structure. This example proves the importance of performing the analysis in successive loading steps, given the S t
e D
r--
3 4
mh
I=500cm
h=250
cm
P=130KN
H=43.3KN
beams
IPE 330
columns
HE 180A
Fig, 8. Example 3-one story frame.
5 11
0.65
1.1
0.675
8
1.1
07
9
11
0.725
Inelastic analysis of steel frames for multistory buildings
0.7
4
*UY
.-ux
(i’,---/
-:
--5
//’
-*
.i
91 0.4
V
-4
-
/x.
a.2
/
-
.
I!: --8. 2
0.
/
1.
:
:
:
:
2.
3.
4.
5.
10s4 rad t
cm
Fig. 9. Diagram of loads vs displacements of point A.
f
influence of large displacements on the the collapse load. Finally, the last example shows as the code makes it possible to obtain accurate solutions with large load increments even in the presence of marked variations of the internal forces during the plastic phase. REFERENCES
0.6
0.0
713
m2
Fig. 10. Motion of the internal forces in section A.
1. J. L. Bockolt and W. Weaver, Jr., Inelastic dynamic analysis of tier buildings. Comput. Structures 4,627+X (1974). 2. B. L. Gunnin, F. N. Rad and R. W. Furlong, A general non linear analysis of concrete structures and comparison with frame tests. Comput. Structures 7, 257-265 (1977). 3. F. Y. Cheng, Inelastic analysis of 3-D mixed steel and reinforced concrete seismic building systems. Comput. Structures 13, 189-196 (1981). 4. N. C. Nigam, Yielding in framed structures under dynamic loads, J. Engng Mech. Div. ASCE 96, (EM5), 687-709 (1970). 5. R. K. Wen and F. Farhomand, Dynamic analysis of inelastic space frames. J. Engng Mech. Div. AXE 96 (EMS), 667-686 (1970). 6. F. L. Porter and G. H. Powell, Static and dynamic analysis of inelastic frame structures. Rep. EERC 71-3, Earthquake Engng Res. Center, University of California, Berkeley, CA (1971). 7. J. G. Orbison, W. McGuire and J. F. Abel, Yield surface applications in nonlinear steel frame analysis. Comput. iUeth. Appl. Mech. Engng 33, 557-573, (1982). 8. Y. Ueda and T. Yao, The plastic node method: a new method of plastic analysis. Comput. Meth. Appl. Mech. Engng 34, 1089-I 104 (1982). 9. J. H. Argyris, B. Boni, U. Hindenlang and M. Kleiber, Finite element analysis of two and three-dimensional elasto-plastic framesthe natural approach. Comput. Mefh. Appf. Mech. Engng 35, 221-248, (1982). 10. 0. C. Zienkiewicz, The Finite Element Method, McGraw-Hill, London (1971). 11. P. G. Hodge, Jr., Automatic piecewise linearization in ideal plasticity. Comput. Meth. Appl. Mech. Engng 10, 249-272 (1977). 12. E. Bozxo and L. Gambarotta, Linear analysis of tridimensional structures for multistory buildings. Costruzioni Met&&e, 2 (198 1), 13. E. Bozzo and L. Gambarotta, Large displacement analysis of tridimensional structures for multistory buildings. Costruzioni Metalliche, 4 (198 1). 14. C. E. Massonet and M. A. Save, Plastic analysis and design. Beams and Frames, Vol. 1. Blaisdell, New York (1965). 15. E. Bozzo and L. Gambarotta, On the overall stability of steel frames. Giomate Italiane delle Costruzione in Acciaio, Perugia (1983). 16. W. Weaver and M. F. Nelson, Three-dimensional analysis of tier buildings, J. Struct. Div., ASCE 92 (ST6), 385-404. (1966).