To reproduce the behavior of flexural and axial deformation of column element representing the interaction among bidirectional bending moments and axial load, multiaxial spring model, called MS model, has been developed and used
(Ref. 7.5).
The MS model has a line element and two multiaxial spring elements (MS element) at the column-ends, as shown in Fig. 7.12. The MS element consists of a number of uniaxial springs, at least four springs. The spring deformation conforms to the plane section assumption or linear strain distribution at a section. There are then two internal nodes between the line element and the MS elements. The line element is elastic in flexural behavior and axial deformation.
It may include inelastic shear deformation represented by shear spring or a shear element.
The number of springs in MS element depends on material properties, sec- tion shape and size, and reinforcing bar arrangement. For a reinforced concrete member, steel springs may be placed at the location of reinforcing steel bar center point, and concrete springs may be placed at the center of portions properly divided into for example, 2 x 2, 3 x 3, 4 x 4, or more. The number
l-spring
X, i
fa,d0 j '"J" -y IW do (b) MS element and the forces (a) Column with MS element and displacements (positive)
Fig. 7.12. Column member model by MS model.
of springs in MS element may affect the accuracy in simulating the column force-deformation relation. Calibration and reliability examination is given in Ref. 7.6. If the cover concrete is modeled separately, the different hysteresis models may be used for the cover concrete and the confined core concrete.
The MS element has elasto-plastic flexibility under moment and axial force but is rigid to shear force. The flexibility of a small portion, namely called as
"plastic zone" of the column is assigned to the spring as its initial flexibility, as shown in Fig. 7.13. In that case, the spring initial stiffness and strength- displacement are simply calculated as
K^ — • ^i (for ith spring) fc — acAi, dc — ec • r]Lo (for concrete) Fsy - asyAi, dsy - £sy • r/Lo (f°r steel)
where K\, is initial stiffness of ith spring, Ei is the material young's modulus, Ai is the spring governed area, and TJLQ is the length of assumed plastic zone.
<jc, £c are the concrete material compression strength and corresponding strain, and asy, esy are the steel material yielding stress and strain. Empirically, rjLo may be taken as D/2 or O.lLo, where Lo is the column clear length, and D is the depth of the column cross section. The plastic zone length riL0 can be selected by the user of analytical program. Different mathematical formula from above based on the curvature at the two ends and distribution along the member is also available.
Trilinear curves may be used to represent the force-deformation relation of steel and concrete springs shown in Fig. 7.14. The relations between tensile force and averaged strain of steel covered with concrete degrade before yielding due to cracking and bond deterioration. To allow for such stiffness degradation, the stiffness of the steel spring is reduced at a point lower than yielding. It is roughly determined by magnifying yielding displacement of the spring by the
, Assumed plastic zone, i]Lo
-XL
Column deformable part Lo
i
Fig. 7.13. Assumed plastic zone for determining spring initial stiffness.
City
(a) Steel spring
fc-OcAi dc=sc-rjLo
*-D
*-D (b) Concrete spring
Fig. 7.14. Skeleton curve of spring force-displacement relations.
factor of
1 . 0 + ^ ^ ° ko,D>1.0 h0/D
1.0 ho/D < 1.0
(7.3)
where ho is the shear span, and D is the depth of the cross section of the column.
To balance the initial stiffness of the line element and two MS elements, a flexibility reduction factor is considered for the line element to make total initial flexibility approximately equal to the original column member. The ini- tial rotational flexibility 5sr and axial flexibility <5s0 of the MS element (to its section centroid) can be calculated as
osr
$s0 =
T) • Lp ^ T] • Lp
ZEiAiY? ~ 0.9EI
T)- Lp _ Tj- Lp
ZEtAi ~~ EA
(7.4)
where EI is the initial flexural stiffness of original column section around concerned axis, and EA is the initial axial stiffness of the column.
Using flexibility reduction factors 71, 72, 70, the bending and axial flexi- bility of the line element can be expressed as
[*L] = 7 I A)
3EI L0
6EI
6EI I2L0
3EI J
It must be „ „ , , > u
ZEI 6EI , or 71 > 0.5
5o = ^ x (70>0)-
Including the flexibility of MS elements, the total bending flexibility of a column in unidirection is given by
U
3EI
U
6EI
6EI Lo 3EI
EA
l\Lp V- LO
3EI 0.9EI Lp
6EI
•yoLo T}- L0
LQ
6EI I2L0 n- Lo
3EI V LQ
0.9EI1
EA ' EA ' EA That gives the following flexibility reduction factors
71 > 0.5, 72 > 0.5 (for r) > 0.15)
7 l = 72 = 1.0 - ^ (for rj < 0.15)
(7.5)
(7.6)
2r,
(for T] > 0.5) (for rj < 0.5) That is, the initial stiffness may not be balanced if the plastic hinge zone parameter 77 is over certain value.
Stiffness matrix can be transformed into the relation between nodal dis- placements and nodal forces as in the same way as shown for the one compo- nent model.
7.3.3. Wall Model
In the architectural design point of view, a wall is an element used to partition the space, which may be either structural or nonstructural. A typical structural wall in Japan, which is called as "shear wall", is dumbbell shaped section with two boundary columns, as shown in Fig. 7.15(a). In this case, a wall can clearly
i r r i *
• 1 • • • • • • • • • • < • - 1 1
i 1 i t
i ằ ô ô t u j
(a) Wall with boundary columns
ằ W I M • ằ ô • • 1 > 1 — • • II I
(b) Wall with confined regions i p ; — p : — T B I 7m •
ằ •
(c) Wall-type column Fig. 7.15. Horizontal sections of wall.
be differentiated from "columns". However, a wall may be designed without boundary columns, as shown in Fig. 7.15(b). Instead, the boundary regions are designed with sufficient confinement to ensure flexural ductility. Further, the horizontal section of a vertical member could be with thick and short depth as shown in Fig. 7.15(c), which may be called as either of wall-type column or column-type wall. In this case, it is difficult to define the boundary between column and wall.
The only difference between the wall and the column is the shape of the horizontal section. Analytical models for the column may be used for a wall member, especially for slender wall. However, it may be preferable to use a special model for wall member, which is different from the member model for the column, because the characteristics of walls are generally different in the following points:
(1) The ratio of shear deformation to flexural deformation of the wall is relatively large. Not only flexural mode of failure but also shear failure need be considered in design and analysis.
(2) The wall member consists of different elements, i.e. wall panel, boun- dary columns and beams. The behavior is affected by their composite actions.
(3) Nonlinear axial elongation of the tensile boundary column is not negli- gibly small but is much larger than that of the compressive boundary column.
(4) Stiffness of wall is relatively large in a frame and greatly affects the results of overall structural analysis.
(5) Moment distribution is not antisymmetric in a story and depends on the structures.
Here, various macroscopic member models for walls are introduced. Finite element model is not included, although it will also be one of practical models in the future. Above characteristic behavior is simulated by models in which sev- eral nonlinear springs or line elements are used. For the multistory wall, linear strain distribution at horizontal section is usually assumed using rigid beams, though axial elongation of the boundary beam is important in some cases.
As shown in Fig. 7.16(a), one component model for column with flexural, shear and axial springs can be used as a simple model for wall members. The overall behavior of the wall can be simulated well if the nonlinear charac- teristics of the springs are determined appropriately. However, movement of centroid of strain in nonlinear range, i.e. larger axial elongation in tension side, is not considered in this model, because the rotation occurs around the center line. Therefore, for example, the analysis by this model does not express the difference of ductility of connecting beams on tension side and compression side.
The so-called fiber model, shown in Fig. 7.16(b), by which the flexural curvature distribution along the wall height is intended to be simulated as rigorously as possible, has been used not only for walls but also for beams and columns (Ref. 7.7). However, the model does not express nonlinear shear deformation or bond deterioration so that theoretical model does not simulate the experimental behavior, though the model requires relatively large number of degrees-of-freedom. Therefore, it is not so rational for frame analysis to use many fiber slices.
To consider the larger inelastic axial elongation on the tensile side, three vertical line element model (Ref. 7.8), as shown in Fig. 7.16(c) has often been used as a practical model. The boundary columns are idealized using line el- ement with nonlinear axial spring, which give flexural rigidity under symmet- ric bending moment. The panel element is idealized by one-component model with nonlinear flexural, shear and axial springs at the base. The central line element is intended to give shear and flexural rigidity under antisymmetric bending moment. Flexural spring must be evaluated so as to separate the ef- fects of boundary columns and panel. To separate the effects more clearly, flexural spring is removed and only shear and flexural springs are used in
(a) One-component model (b) Fiber model (c) TVELM model
(d) MVELM model (e) M-S model (f) Truss model
(g) Panel and boundary element model Fig. 7.16. Various wall models.
another model (Ref. 7.9), as shown in Fig. 7.16(d). This model, which may be called as multiple vertical line element model, is equivalent to the fiber model of one layer.
The MS model for column is also useful for walls as shown in Fig. 7.16(e).
This model is especially useful under biaxial bending. The method to release the unbalanced force is important in this model.
The truss model, shown in Fig. 7.16(f), which has been used for a model in elastic analysis, is also used for nonlinear analysis. The stiffness of each element is to be given so as to give equivalent stiffness of the whole section, which is easy in elastic range. The model is developed to meet with the resistance mechanism of truss and arch model for the evaluation of ultimate shear strength. However, it is a little difficult to determine the inelastic stiffness rationally, especially of
tensile truss element. More complicated model with additional truss element, shown with dotted lines in the figure, has also been proposed.
Based on the constitutive law for two-dimensional reinforced concrete el- ement for FEM analysis (Refs. 7.10 and 7.11), a simple model for the wall is also proposed, as shown in Fig. 7.16(g), which simulate shear and flexural behavior very well. Further study is needed to verify the stability of the model in numerical calculation to apply the method to practical design analysis.
The wall model need be sophisticated further for frame analysis on the following points:
(1) Axial deformation of beam with slab must be idealized.
(2) Column axial property must be idealized with the effect of confinement.
(3) Simple but rational 2-D constitutive model need be developed, espe- cially under cyclic loading.
(4) 3-D model need be developed, under skewed loading.
(5) Modelling of irregular walls should be developed.