EVALUATION OF SHEAR AND FLEXURAL DEFORMATIONS OF FLEXURAL TYPE SHEAR

In order to predict the history of inelastic response of reinforced concrete structures under dynamic earthquake loading, hysteretic behaviour must be known for each seismic member, that is, beams, columns and shear walls. Hysteretic behaviour of beams and columns has been made clear through many experimental and analytical studies, and some hysteretic load versus deformation models, such as the Takeda model (which shows good agreement with experimental results of flexural members), have already been proposed. However, a proper practical hysteretic load versus deformation model for shear walls has not yet been proposed.


INTRODUCTION
In order to predict the history of inelastic response of reinforced concrete structures under dynamic earthquake loading, hysteretic behaviour must be known for each seismic member, that is, beams, columns and shear walls.Hysteretic behaviour of beams and columns has been made clear through many experimental and analytical studies, and some hysteretic load versus deformation models, such as the Takeda model (which shows good agreement with experimental results of flexural members), have already been proposed.
However, a proper practical hysteretic load versus deformation model for shear walls has not yet been proposed.
Many experimental studies on shear walls have been carried out, but most of their load-deformation data are presented in terms of the load versus total deformation.
Very few data refer to the shear and flexural deformations primarily because of the difficulty in separating the total deformation into these two deformations.This paper reports on an evaluation method of distributing the total deformation of a shear wall into the flexural and shear deformations.
For this evaluation, rotation at the storey mid-height of a shear wall is used in addition to the horizontal and vertical displacements at four corners of a shear wall.Furthermore, this paper describes analytical method of evaluating flexural and shear deformations of shear walls whose flexural reinforcing bars of boundary column under tension yield.
In the analysis, such a shear wall is represented as a truss system having a non-prismatic truss member based on test results of shear walls conducted by US-Japan Cooperative Seismic Research Program.
This paper also proposes an hysteretic model of each component of deformation.

FLEXURAL DEFORMATION AND SHEAR DEFORMATION
In .shearwall tests, shear deformation is sometimes conventionally estimates from changes in the length of the two diagonals.However, the shear deformation given by this method contains flexural deformation because of the existence of a moment gradient along the height of shear walls .This chapter describes the relationship between horizontal and vertical displacements at the four corners of a shear wall, and the relationships between flexural deformation, shear deformation, and expansion.
It also proposes a simple method to evaluate each deformation.

Components of Deformation
Displacements of a shear wall subjected to a lateral load are illustrated in Figure 1.With an aim towards simplification of the development of equations, horizontal and vertical displacements at the base are modified to be zero.
It is also assumed that these displacements can be represented by three components, that is, shear deformation (which includes slip), flexural deformation, and expansion.
The sign for horizontal and vertical displacements is positive in the upper and right hand side directions and elongation is positive for diagonal displacements.
The following relations are assumed: There are relationships between these displacements and those shown in Figure 1 as follows: (5) (6) EARTHQUAKE ENGINEERING, VOL.17, NO. 2, JUNE 1984 From equations ( 5) and (6), horizontal displacement due to expansion and the total of flexural and shear deformations can be given by equations ( 10) and (11), respectively, in terms of nodal horizontal displacements .
Equation ( 11) indicates that the deformation of the hysteretic load versus deformation relationship which is used in dynamic response analysis must be expressed not by each nodal horizontal displacement but by their average value.
Thus, the shear deformation can be given by equation (12) from equation (9) where, I'he shear deformation is also given by equation (14) from equation (11) in a different expression: Expression of Flexural Deformation and Shear Deformation in Terms of Rotation Equations ( 12) and (14) imply that in order for shear deformation to be estimated, flexural deformation must inevitably be estimated with sufficient accuracy.Flexural deformation can be assumed to be given by equation (15): In equation (15), 9 , v Ly and v R y represent the rotation of a shear wall ana the vertical displacements of right-hand side and left-hand side boundary columns at y in the y direction, respectively.An example of the distribution of 6 along the height of a cantilever shear wall is illustrated in Figure 3.
Curvature is dramatically increased around the base where the occurrence of cracks and yielding of steel is usually observed.Equation ( 15) can be rewritten by equation (17) using a new factor a-defined by equation ( 16): ; h e dy o y y eh (16) a is the ratio surrounded by Therefore, it of this ratio tion.
With re a relationship when the point above the subj of the shaded area to that solid lines ABCD in Figure 3. is reduced to the prediction to evaluate flexural deformagard to a, there is generally as shown in equation ( 18), of contraflexure is located ect storey of a shear wall.
By substituting equation ( 17) into equation (12) or equation ( 14), shear deformation is given as follows: Equation ( 19) proves that shear deformation given approximately by only changes in the length of diagonals , corresponds to shear deformation given by 1/2 of a.However, if 1/2 of a is assumed, the shaded area in Figure 3 would be approximately represented by the area of the triangle ABC, which occurs only in the case of pure bending, and this therefore results in an overestimation of shear deformation.
Figure 3 is nothing but a conceptual illustration, but it does not seem to be difficult to evaluate a with certain accuracy because of the general behaviour of rotation.
For example, if either rotation 6 M , or right-hand side and left-hand side vertical displacements at mid-height of the subject storey of a shear wall were measured, considerable improvement of accuracy in evaluation of a could be expected as follows: in Figure 4, case 1 shows the case where rotation distribution is represented by the lines connecting the measured point M and points A and C, whereas case 2 shows the case where that is represented by the line connecting 6^/2 and (0 + ©M)/2 whose tangent is equal to that of the line connecting points A and C and by the lines connecting both ends and points A and C. The ratios 04 and a 2 of the area surrounded by these approximate lines to the area ABCD are given by equations ( 21) and ( 22), respectively.
There is generally relationships given by equations ( 23) and (24) between approximate values ot]_ and a 2 and the exact value a.
The maximum possible error amounts to 25 percent for 04 and 16.7 percent for a?.However, a 2 seems to estimate a with good accuracy as is expected from equation (24) and Figure 4, and the excellent accuracy of was proved through nonlinear analytical case studies on single and seven storey shear walls.As a result, both the shear and flexural deformations increase significantly (1).After these cracks take place, the hypothesis of a plane section remaining plane is no longer suitable.H Bachmann already reported experimental results that hinging region of beams increased according to development of shear cracks, and also presented its analytical proof based on the stress of flexural reinforcing bars under tension given by equilibrium of forces at an inclined cracked surface (2) .
In this chapter, the stress of flexural reinforcing bars of boundary column under tension is expressed by H Bachmann's theory.The shear wall is represented as a truss system having a non-prismatic truss member whose cross-sectional area is determined by the stress along the height of boundary column.
Then flexural and shear deformations are evaluated.Figure 7 shows forces and moment at an inclined cracked surface which is modified based on Figures 5 and 6, and those at the base.

Stress of Reinforcing Bars of Boundary Column Under Tension
In these figures, the following new notations are used: The analysis is not applied to the region where stress given by equation ( 32) is much less than that given by assuming a plane section remaining plane.

Truss Model of Shear Walls
Shear walls after yielding of flexural reinforcing bars of the boundary column under tension is expressed as a truss model shown in Figure 8.
In this paper, the main object of the analysis is placed to deformations at the first storey, so truss members above the first storey are assumed to be rigid.
It is also assumed that shortening of the boundary column under compression is negligible, and tensile chord member (boundary column under tension) is expressed as a non-prismatic member whose crosssectional area is given by equation ( 34) so as for existing stress to satisfy Specimen FW Specimen Wl The deformation of this truss system consists of deformation due to stretching of the tension side column, that due to shortening of the diagonal compression member, and that due to stretching of the beam (horizontal tensile member).Figures 9 and 10 show their components.There are following relationships between these components.
U s + U B Figure 11 shows the envelope curves of load versus deformation relationships, during positive loading, of the first storey of the shear wall tested in PGA in the US-Japan Cooperative Research Program (5).
The vertical displacement of the column under compression is little amount as it is assumed in the analysis, and the sum of the displacements u s2 and u g3 is small enough.
In the case of such small displacements of u g2 and u s O as this, there must be the relationship of u/h=6 according to the truss model shown in Figures 8-10.
This relationship is obviously observed in the test result shown in Figure 12.The similar results for this relationship also seem to be found in the test results of the nine flexural type shear walls reported by Mr R G Oesterle et al in 1976 (1).
Consequently, in order to evaluate deformations of flexural type shear walls, the truss model proposed in this paper seems to be a suitable representation.Furthermore, as the deformations of u g2 and u s 3 are little amount as shown in Figure 11, the dramatical increase in shear deformation after yielding is explained not by deterioration of shear rigidity due to flexural cracks, which have been pointed out by V V Bertero and R G Oesterle, but by a rotation mechanism having rotation origin at the base of the column under compression.

Vertical
where strain of tensile chord member at y height The stress of tensile chord member at y height is determined based on the existing stress which is given by equation (32).The stress versus strain relationship for reinforcing bars of the tension side column is represented as shown in Figure 13.The aspect ratio of the first storey height to width between centres of boundary columns, and material constants for steel are assumed as follows: ug and u i increase according to tip drift angle.
In the large deformation region, u sl is nearly equal to u B when a w /a^ is small, and u s i is a fairly large amount even when a w /at is large.
It is found, from Figure 15, that a has the maximum value when m/h-1/400, then it gradually decreases, and it converges into some constant value after yielding of flexural reinforcing bars of the boundary column under tension at top of the first storey.The symbol o in Figures 14 and 15  In this model, the relationships derived in preceding sections are taken into consideration.
A sufficient test data has not been presented even for the load versus flexural deformation relationship.
The proposed hysteretic model is, therefore, nothing but an idea.Further studies are highly recommended.

Primary Curve of the Load Versus Deformation Relationship
The primary curve for the load versus flexural deformation relationship has three breaking points as shown in Figure 16.These breaking points are flexural cracking, yielding and maximum strengths.
The primary curve for the load versus shear deformation relationship before yielding is represented by bilinear, and that after yielding in flexure is done by a curved line which is given by equation (47) by considering the u versus u« relationship shown in equation (41), and The Takeda model (6) seems to be suitable for the hysteretic load versus flexural deformation relationship.One of possible representations for the hysteretic load versus shear deformation relationship may be the Shina model (7) which was proposed by M A Sozen et al for the members including pinching phenomena.Figures 18 and 19 show these hysteretic relationships.

CONCLUSIONS
The following conclusions have been reached: 1.
Shear deformation is overestimated, and consequently flexural deformation is underestimated if the shear deformation is determined simply as a difference in length of two diagonals.

2.
Flexural and shear deformations are estimated with excellent accuracy by using the rotation at the storey midheight of a shear wall.

3.
Shear deformation increases by the rotational mechanism having a rotation centre at the base of the column under compression.

4.
Shear deformation significantly increases after reinforcing bars of the boundary column under tension yield.
The ratio of the shear deformation to the flexural deformation is analytically determined by a truss model which has a non-prismatic member.
^Senior Research Engineer of Structure Division, Building Research Institute, Ministry of Construction, Japanese Government, BULLETIN OF THE NEW ZEALAND NATIONAL SOCIETY FOR

Fig. 1 Fig. 3
Fig. 1 Deformation of Shear Wall Subjected to Lateral Load

2 Fig a 4
Fig a 4 Methods of Estimation for a

Figures 5
Figures 5 and 6 show crack patterns of three shear walls tested in US-Japan Cooperative Research Program.Figure7shows forces and moment at an inclined cracked surface which is modified based on Figures5 and 6, and those at the base.In these figures, the following new notations are used:

T = resultant tensile
force of reinforcing n bars of boundary column under tension at y height.T Q -resultant tensile force of reinforcing bars of boundary column under tension at the base T^ = resultant tensile force of vertical reinforcing bars, across an inclined Cracked surface, of the in-filled panel wall T^0 = resultant tensile force of vertical reinforcing bars of the in-filled panel wall at the base Q w = lateral force acting on the shear wall Q = shear force of boundary column under compression at an inclined cracked surface C Q = axial force of boundary column under compression at an inclined cracked surface M Q = moment at the base N = axial force of the shear wall n = y/h under tension is given by equation (26), Assuming that T , 0 acting on an lined surface are w concentrated at 1/2 and nh/2, respectively, the resultant tensile force T of the boundary column mc (26) Assuming that T _ at the base is concentrated at 1/2, equation (27) is obtained: (27) Substituting equation (27) into equation (26), and assuming equations (28) and (29), equation (31) is obtained: of vertical reinforcement area to the gross concrete area of a horizontal section of the in-filled panel wall t sum of the cross-sectional area of reinforcing bars of the boundary column under tension a = stress of reinforcing bars of the boundary column under tension at the base a = stress of reinforcing bars of the

Fig. 5 Fig. 7
Fig. 5 Crack Pattern of Lower Three Story Shear Wall of a Full-Scale Seven Story Structure Test 3) at Tip Drift Angle of 1/64 of the right-hand side and left-hand side horizontal displacements of the first storey (sum of flexural and shear deformations) tip horizontal displacement due to stretching of the tension side column tip horizontal displacement due to shortening of the diagonal compression member right-hand side tip horizontal displacement due to stretching of the beam u ?, u ^ = shear deformation due to sfiretcMng of the tension side column, that due to shortening of the diagonal compression member, and that due to stretching of the beam, respectively rotation at y height v = vertical tip displacement of the tension side column sl ; vertical displacement of the tension side column at y height sl It is clear from equation (41) and Figure 10 that the shear deformation u increases according to increase in tip rotation (flexural deformation), because of the general condition: 1/2 < a < 1.
Displacement of the Boundary Column Under Tension and Flexural DeformationBVertical displacement v of the boundary column under tension is given by equation (44), and flexural deformation u 1 and a are given by equation (45) and (46)

Fig. 9
Fig. 9 Deformations due to Stretching and Shortening of Each Truss Member means this condition.HYSTERETIC MODEL OF THE LOAD VERSUS DEFORMATION RELATIONSHIPThis chapter proposes a hysteretic model of the load versus deformation relationships of the first storey for flexural type shear walls.