Frame structures of buildings usually assume weak-beam and strong-column type collapse mechanism, where yield hinges would form at the first story column bases and beam ends of all upper floors. The restoring force charac- teristics of beams would dictate the overall behavior of frames under earth- quake excitation, and therefore it must be accurately evaluated in the design.
In the New RC project, experimental force displacement relationship was sup- posed to be idealized as shown in Fig. 4.58.
It may be clear from the figure how to construct trilinear idealization of an experimental curve. Firstly, a point corresponding to the first cracking is determined from the initial stiffness and observed crack load. Secondly, mea- sured yield load is confirmed at maximum strength or the load at 2 percent drift angle, and a load corresponding to the three quarters along the way from
eQ, yielding
01 o
cQo
c<yằ<
maximum load or load at 2% drift angle
S7
y~cQbc) +cQbc
^
—y cracking
V*^ \ idealized Q' 8 experimental Q- 6
deformation
Fig. 4.58. Idealized force-deformation relationship.
300.0
I
2 200.0 3
.§ 100.0 o.
0 rectangular D T-beams
• rectangular (New RC)
• T-beams (New RC)
100.0 200.0 calculated values (MN/m)
300.0
Fig. 4.59. Experimental vs. calculated initial stiffness.
cracking to yielding is determined. A point marked "A" is found for this load, and a straight line connecting the crack point and point "A" and its extension is drawn to find the deformation at yielding. Characteristic points of this method are that the yield load corresponds to ultimate flexural load carrying capacity, and that the stiffness after yield is zero. How to determine each parameter in Fig. 4.58 will follow.
4.5.1.1. Initial Stiffness
Initial uncracked stiffness is calculated considering flexural and shear deforma- tion of members. Figure 4.59 shows relationship of observed vs. calculated initial stiffness, considering rigid zones at the ends of each member as spec- ified in the Calculation Standards of Reinforced Concrete Structures of AIJ (Ref. 4.3).
4.5.1.2. Flexural Cracking
Cracking moment is calculated from the tensile strength of concrete and the equivalent section modulus including effect of reinforcement. Either splitting tensile strength or 0.56 times the square root of concrete strength in MPa may be used as tensile strength. Figure 4.60 shows the relationship of observed vs. calculated cracking load of beams where the tensile strength was evaluated by the latter method. Large scatter is conspicuous as it is inherent to pheno- mena like cracking, but it will be agreed that the calculated values generally
98
& 78
CO <u J3
1 59
73 c 2 39
0 * i 1 i i i
0 20 39 59 78 98 C a l c u l a t e d v a l u e s (kN)
Fig. 4.60. Experimental vs. calculated cracking load.
shoot the average of observed values, and no different trends are seen between previous tests and New RC tests.
4.5.1.3. Yield Deflection
Figure 4.61 shows experimental values of yield deflection of New RC beams as contrasted to those of conventional RC beams. New RC beams with high strength material clearly show greater yield deflection. This is due, first, to larger yield strain of high strength steel, and secondly, due to increased pull- out displacement of beam bars from columns or loading stubs, and increased yield hinge length, inherent to relatively poor bond behavior of high strength concrete.
In terms of yield stiffness reduction factor, however, the commonly used Sugano's equation may be applied to New RC members. Yield stiffness re- duction factor is defined as the ratio of secant modulus at yield point of RC members to the initial uncracked stiffness, and is expressed based on a statis- tical survey as follows
ay = (0.043 + 1.64npt + 0.043a/D + 0.33?7o)(d/D)2 (4.36) where
ay : yield stiffness reduction factor
n : Young's modulus ratio of steel and concrete
Pt : tensile reinforcement ratio bared on gross concrete section
0 rectangular Q T-beams
• rectangular (New RC)
• T-beams (New RC)
j
I
2.0
1.5 -
I t
1.0
0.5
0
°a
I:
•
O rectangular Q T-beams
• rectangular (New RC)
• T-beams (New RC) 30 _L _J_ _L
40 50 60 70 80 90 concrete s t r e n g t h (MPa) (a) Relation to concrete s t r e n g t h
2.0 r- •
1.5
1.0
0.5
0
O O O a
*
-L.
O rectangular
• T-beams
• rectangular (New RC)
• T-beams (New RC)
300 800
Fig. 4
400 500 600 700 steel yield point(MPa) (b) Relation to steel s t r e n g t h
61. Experimental yield deformation and material strength.
a : shear s p a n length d e t e r m i n e d from t h e r a t i o of m a x i m u m b e n d i n g m o m e n t t o t h e m a x i m u m s h e a r force
D : d e p t h of t h e m e m b e r
770 : axial stress r a t i o d e t e r m i n e d from t h e axial stress considering concrete area only divided by t h e concrete s t r e n g t h (rjo = 0 for b e a m s )
0.4 r
3.0r
2.0
1.0
O rectangular D T-beams
• rectangular (New RC)
• T-beams (New RC) 0.1 0.2 0.3
calculated values (a) Exp. vs. calc. values
-cS=
O rectangular D T-beams
• rectangular (New RC)
• T-beams (New RC)
Al.
x;
j _ _i_
30 40 50 60 70 80 concrete strength (MPa) (b) Accuracy vs. concrete strength
90
Fig. 4.62. Yield stiffness reduction factor.
d : effective depth of the member i.e. depth from the most compressive fiber to the centroid of tensile reinforcement.
Figure 4.62 is the comparison of experimental values of yield stiffness re- duction factor to the values from Eq. (4.36). Although large scatter is seen, it may be concluded that Sugano's equation is equally effective to the New RC beams as those of conventional material.
4.5.1.4. Flexural Strength
Flexural strength of a beam may be obtained by one of the following three methods. The first is to use approximate equation given by the Building Center of Japan (Ref. 4.7)
Mu = 0.9Zat<Tyd (4.37)
where
Mu : is flexural ultimate moment of a beam at : is tensile reinforcement area
ay : is yield strength of tensile reinforcement
d : is effective depth of the beam to the tensile reinforcement S : is for different tensile reinforcement groups.
The second method is to conduct theoretical analysis assuming rectangular stress block of American Concrete Institute, and ultimate compressive strain of 0.3 percent.
The third method is to use stress block proposed by the high strength steel committee of New RC project, and ultimate compressive strain of 0.3 percent, same as the ACI method.
It was shown that the difference between results of the second and third methods was small, and they predicted the observed flexural strength of both rectangular and tee beams reasonably well, provided for the latter that the entire slab width is taken effective. The approximate equation in the first method gave approximately 10 percent smaller values both for rectangular and tee beams, even for the latter if the entire slab width is taken.
4.5.1.5. Limiting Deflection
In order to avoid shear failure in the yield hinge zone, and to secure the plastic hinge rotation capacity, it was found necessary to follow the same procedure as will be described later for columns.
4.5.1.6. Equivalent Viscous Damping
Flexural deformation usually dominates the beam deformation. For such beams it was shown that the equivalent viscous damping at yielding is about 5 to 10 percent, and it increases as deformation gets larger. It was about 10 to 15 percent at the drift angle of 2 percent.
Thus it was concluded that the restoring force characteristics of New RC beams can be formulated generally by the same methods as the conventional beams. However in case where a more precise idealization is required by the analysis software such as separation of flexural and shear deformations, it will be necessary to refer to the original research data.