In the previous Sec. 4.2.4, one of four column specimens, SA35, showed a vertical splitting crack at the second cycle of deformation angle amplitude of 1.5 percent. It was a specimen subjected to antisymmetric bending. The vertical crack appeared along the plane of central axial reinforcement placed at a perpendicular location to the loading direction, and caused drastic loss of load carrying capacity of the column. After the conclusion of the loading test the specimen was cut along the loading direction, and internal crack distribution was examined, to find that the vertical splitting crack extended to the central portion of the section, and that it virtually divided the specimen into two pieces vertically.
This kind of vertical splitting crack had been observed in past experiments, but the phenomenon had not been completely explained. A study was con- ducted therefore to give some more lights to the mechanism of formation of this kind of vertical splitting crack, and to the expected strength of columns against this cracking, utilizing available test results as well.
Figure 4.22(a) shows idealized deformation of a column in which yield hinges have formed at both ends and splitting crack has appeared along the
Fig. 4.22. Idealized deformation and assumed forces.
center line. The column is subjected to axial load N and tensile yield force in the section is T, hence the compression resultant is N + T acting at the compression side of the yield hinge. Q is the column shear force. The forces acting in the tensile hinge zone may be expressed as in Fig. 4.22(b), where A T denotes bond forces along the tensile reinforcement, Tw is the resultant of forces in the lateral reinforcement, and Cp is the resultant of compression forces in the concrete struts, all in the tensile hinge zone. If we assume AT is zero considering that concrete cover in the hinge zone has spalled off all around the section due to cyclic reversal of loading, the resultant of Tw and Cp
must be zero from the equilibrium of the tensile hinge zone. Then we obtain simplified assumption of forces shown in Fig. 4.22(c), before formation of the splitting crack.
Thus we define forces acting along the potential splitting crack plane of a column in antisymmetric bending to be, as shown in Fig. 4.23(a), shear force N + 2T and normal force Q. Splitting crack plane has the area of column height minus depth D times the width of core concrete as shown in Fig. 4.23(b).
Then the average shear stress acting on the potential crack plane can be expressed by the following equation
TS = (N + 2T)/AC (4.4)
where
TS : the average shear stress N + 2T : shear force acting on the plane
Q N+2T'
T
^
V ,N+2T
Q N+2T - \ J "ằpotei
D " t \ : cracl
5 T—
|N+T
• potential si cracking plane
\7////7/W////\
\W///W///A
W/////////////A
(a) Assumed model with forces (b) Assumed section of splitting crack Fig. 4.23. Assumed model and crack section.
Ac : area of crack plane as explained above.
In case of cantilever type bending of previous subsection, this equation has to be modified appropriately.
The splitting crack strength of the vertical plane may be expressed by the combination of concrete term and normal force term. They are assumed as follows
where
OB
Q Ac
P
Tsu = ay/cTB~ + P(Q/AC + p)
splitting crack strength concrete compressive strength column shear force
area of crack plane
confining stress of concrete from lateral reinforcement numerical coefficents.
(4.5)
a, 13
The first term on the right hand side corresponds to shear cracking strength of plane concrete expressed as to be proportional to the square root of com- pressive strength, and the coefficient a may be assumed to be about 1.8 which gives shear strength of 30 to 40 percent of compressive strength to the ordi- nary strength concrete. The second term has a form of normal stress acting on the plane multiplied by a friction coefficient /?. Friction coefficient of a con- crete crack may involve aggregate interlock, and (5 may be assumed as high as 1.0. As to the confining stress p from lateral reinforcement, it is a well known
2.0 -
1.0
O ' No sub-hoops, uncracked
• I No sub-hoops, cracked /^v * Sub-hoops, uncracked J)L * Sub-hoops, cracked
— SA35
O O
CB60
C A 3 5 ^ C B 3 5
5r- X X
^
_L 0 "03 0.4 0.5 0.6
axial load ratio
Fig. 4.24. Splitting crack stress vs. axial load ratio.
observed fact that lateral confining stress is high in the hinge zones but it is low outside the hinge zones including point of contraflexure. Hence we may assume p = 0.
Figure 4.24 shows relationship between splitting crack stress vs. axial load ratio for specimens in the previous subsection as well as other existing data.
Splitting crack stress is the average shear stress of Eq. (4.4), and it is nor- malized by the splitting crack strength of Eq. (4.5), where a = 1.8, /3 = 1.0 and p = 0 were assumed. Axial load ratio is same as Eq. (4.3). Plotted in Fig. 4.24 are specimens without or with subhoops, which correspond to circle or triangle marks in the figure. Specimens that formed vertical splitting crack are marked black, while those that did not form crack are marked white. It is clear that columns tend to form vertical splitting cracks when rs approaches or exceeds rs u, and whether the column is provided with subhoops or not does not make much difference. Also the axial load ratio is irrelevant as long as it is incorporated in the form of Eq. (4.4). Needless to say that the area Ac in Eq. (4.4) depends on the column height, and TS becomes larger for shorter columns. On the other hand, rs is small for cantilever type columns in the preceding subsections.
It may be concluded that the mechanism of the formation of vertical split- ting crack may be explained by the proposed model of Fig. 4.23 and Eqs. (4.4) and (4.5), however crude it is.
4.2.6. Shear Strength of Columns
Shear strength of beams and columns of a moment resisting frame plays dou- ble roles, one in the pre-yield (elastic) range and another in the post-yield (inelastic) range. For members in which yield hinges are not expected to occur, premature shear failure must be prevented. For this purpose it only suffices to equate the shear force associated with the formation of yield mechanism to the shear strength of the member in the elastic range, i.e. shear strength at the pre-yield shear failure, which may be referred to as "elastic" shear strength.
On the other hand, for members in which yield hinges are expected to occur, hinge rotation corresponding to the maximum anticipated deformation must be ensured. According to the recent knowledge of shear strength in the inelas- tic range as explained in Sec. 4.2.3, shear strength of a member is not a unique constant but is a decreasing function of the inelastic deformation of yield hinge.
It is necessary to find out shear strength corresponding to the required inelastic
Fig. 4.25. Shape and size of column specimen.
deformation, which may be termed as "inelastic" shear strength. By equating the shear force associated with the formation of yield mechanism to this inelas- tic shear strength, inelastic deformation corresponding to the inelastic shear strength is ensured to occur to the member.
The study in this subsection is related to the elastic shear strength of columns of high strength RC, while the one in the next subsection is related to the inelastic shear strength of beams of high strength RC. Experimental program of columns involve eight column specimens made of 60 MPa concrete, as shown in Fig. 4.25. Column section is 300 mm square and clear height is 900 mm. Figure 4.26 shows two sections of column, reinforced laterally with D6 or D10 bars, both fabricated into closed form by flush-butt welding.
Axial re-bars are USD685 12-D19 bars with actual yield strength of 757 MPa, while lateral re-bars of two different grades, SD345 and SBPR 785/930 are used. Table 4.3 lists parameters for eight column specimens, and actual yield strength of lateral reinforcement are shown. As seen in Table 4.3, major testing parameters are axial load ratio, with the definition of Eq. (4.3), of 1/6 and 1/3,
.. W_Jm_L_<m._mv
-i—i—^-
J I i
•to-i—ff
- ^ - < N
=q
dDh=et
40 I 70 J40 ! 40! 70 140
(a) Using D10 lateral bars 3S_
4 1—I—I ^
(b) Using D6 lateral bars
Fig. 4.26. Section of column specimen.
Table 4.3. Column specimens for shear strength test.
Specimen 6-1 6-2 6-3 6-4 3-1 3-2 3-3 3-4
Axial Load Ratio
1/6
1/3
Lateral Bar D6 D10 D6 D10 D6 D10 D6 D10
Pv, (%) 0.53 1.19 0.53 1.19 0.53 1.19 0.53 1.19
(MPa) 402 409 931 1091
402 409 931 1091
(MPa) 2.13 4.87 4.93 12.98
2.13 4.87 4.93 12.98 Pw: lateral reinforcement ratio (%)
awy: yield strength of lateral reinforcement (MPa)
and amount and yield strength of lateral reinforcement. Compressive strength of concrete at the testing age was 73.5 MPa, and tensile strength was 4.9 MPa.
All columns were tested under constant axial load and incremental reversal of lateral load while keeping the top and bottom stubs of column in the parallel position. Actual axial load was 1100 kN and 1950 kN for two levels of axial load ratio.
Figure 4.27 indicates lateral load (column shear) vs. deformation angle relationship of four columns tested under axial load ratio of 1/3 (more exactly it was 0.30), together with points of flexural and diagonal cracking and maxi- mum load. Lines of P-delta effect and computed strengths, as explained later, are also shown. General appearance of load-deformation curves of other four specimens under axial load ratio of 1/6 was quite similar to these four columns.
All specimens first showed flexural cracks at the critical sections, followed by diagonal cracks in the central portion. Load at cracking was affected by the axial load ratio. Flexural cracks appeared for axial load ratio of 1/6 at 225-275 kN, and for 1/3 at 350-425 kN. Diagonal cracks appeared for axial load ratio of 1/6 at 350-390 kN, and for 1/3 at 475-500 kN. The increase of cracking loads due to axial load was in good accordance with the analysis based on fundamental theory of strength of materials.
All specimens failed in shear before formation of flexural yield hinges.
However the failure mode was significantly affected by the lateral reinforce- ment strength denoted by pw • awy as listed in Table 4.3. When the lateral reinforcement strength is low, columns failed in a typical shear failure, but those with high lateral reinforcement strength failed in bond-splitting failure
in the flexural compression zones. Deformations at the maximum load and at the failure were slightly larger for higher lateral reinforcement strength. Axial load ratio did not affect the mode of failure.
Table 4.4 lists measured and calculated maximum loads. Looking at measured loads, it is clear that axial load ratio has relatively small influence;
the maximum load is more significantly influenced by the amount of lateral
1000 800 600 400 2
& 200 a>
M o
u jS -200
tn
-400 -600 -800
-2 1 0 1 2 3 4 5 6
deformation angle (%) (a) Specimen 3-1
c5 1^
<8
she
1000 800 600 400 200 0 -200 -400 -600 -800
- 2 - 1 0 1 2 3 4 5 6
deformation angle (%) (b) Specimen 3-2
Note: V : flexural crack, T : diagonal crack, M: max. load
F i g . 4 . 2 7 . L o a d - d e f o r m a t i o n c u r v e s for four c o l u m n s p e c i m e n s failing i n s h e a r .
deformation angle (%) (c) Specimen 3 3 10001 ; 1 i ;
deformation angle (%) (d) Specimen 3-4
Note: V: flexural crack, T: diagonal crack, M: max. load Pig. 4.27. (Continued)
reinforcement and its yield strength. In terms of lateral reinforcement strength, Pw • &Wy, it is interesting to note that the second and the third specimen in each axial load group, provided with approximately same amount of pw • awy, showed different maximum load. The second specimen with large amount of weak re-bars is always stronger than the third specimen with small amount of strong steel. This fact will be investigated in the more general study of Sec. 4.5.
Table 4.4. Comparison of measured vs. calculated max. load.
Specimen 6-1 6-2 6-3 6-4 3-1 3-2 3-3 3-4
Measured Max. Load
Q ( k N ) 465.0 665.5 570.0 704.5 532.0 706.5 585.0 744.0
Calculated Flex. Strength
QF (kN) 787.8
898.3
Shear Strength Qs (kN)
321.6 507.3 510.0 649.0 321.6 507.3 510.0 649.0
Bond Strength
QB (kN) 851.9 919.1 816.8 926.3 851.9 919.1 816.8 926.3
Table 4.4 also lists various calculated values of the column strength.
Flexural strength was obtained by ordinary ultimate strength theory, and it was much greater than measured strength for all specimens. Shear strength calculated by the ultimate strength guidelines (Ref. 4.3) was found to be on the safe side for all specimens, particularly so for those with low yield strength lateral re-bars. Bond strength in Table 4.4 was calculated by Fujii-Morita equa- tion which is the original form of bond strength equation in Sec. 4.5. Calculated bond strength was much higher than observed maximum load, indicating that the observed bond-splitting failure in the flexural compression zone was dif- ferent from the bond-splitting failure along the axial bars in entire column length.
Figure 4.28 shows relationship between maximum load and lateral reinforce- ment strength or axial load. The ordinate shows maximum load Q, and positive abscissa shows lateral reinforcement strength pw • awy and negative abscissa indicates axial load N. Open circles and squares denote observed values, while flexural, shear and bond strengths as listed in Table 4.4 are shown by variety of lines. Flexural strength QF is not a function of pw -awy, and shear strength Qs
is not a function of N. Bond strength QB is shown only on the right hand side.
Also entered is another shear strength prediction Q's, calculated by a theory by Wakabayashi and Minami (Ref. 4.4), which is more complicated than the ultimate strength guidelines (Ref. 4.3) as it considers effect of axial load also.
When the observed values are compared with calculated lines, it can be con- cluded that Qs estimates the shear strength generally on the safe side without reflecting the effect of axial load, and that Q's estimates the shear strength on the unsafe side, though reflecting duly the effect of axial load.
I l l I I l I 3000 2000 1000 0 5.0 10.0 15.0
axial load N (kN) pw • or wy(MPa)
Fig. 4.28. Relationship between max. load Q and lateral reinforcement strength pw • o-wy
or axial load N.
Summarizing the findings from this experiment, one might note as follows.
(1) Shear strength of columns with 60 MPa concrete increases with pw • awy but not in proportion to it.
(2) For the same amount of pw • awy, it appears that greater pw is more advantageous than higher awy.
(3) Prediction by ultimate strength guidelines is on the safe side, and axial load effect, though small, is not reflected. Wakabayashi-Minami theory accounts for the axial load correctly, but overestimates the test results in general.
4.2.7. Shear Strength of Beams
In a preliminary investigation of the New RC project it became clear that the shear strength of a beam made of high strength concrete up to 120 MPa can be estimated fairly accurately by the equation in AIJ guidelines (Ref. 4.3), if the effective compressive strength of concrete is appropriately evaluated. Effective compressive strength for pre-yield shear strength, or "elastic" shear strength, of high strength concrete may be evaluated by an equation proposed by CEB, as explained later. On the other hand, beams in a building are mostly designed as
yielding members, and appropriate ductility is ensured by considering "inelas- tic" shear strength which is a decreasing function with respect to the inelastic deformation. The equation in AIJ guidelines is provided with two decreasing elements for this purpose; one is the inclination angle of concrete struts in the truss mechanism, and another is the effective compressive strength of con- crete. The study introduced in this subsection was conducted with the aim of establishing a unified expression of effective compressive strength of concrete for elastic as well as inelastic shear strength of beams made of high strength concrete.
In order to evaluate the effective compressive strength continuously from elastic to inelastic range, four beam specimens were tested under flexural shear, with the same sectional dimensions, same shear reinforcement, same concrete, and with the only difference in yield strength and amount of axial reinforce- ment. As shown in Fig. 4.29, specimens have section of 150 mm by 300 mm, and the first specimen BE-1 has top and bottom axial re-bars of 5-D16 SD980 steel (actual yield point was 970 MPa). Other three specimens have top and bottom axial re-bars of 4-D16, with different yield strength; BE-2 is provided with SD980 steel same as above, BE-3 with SD685 steel (actual yield point was 654 MPa), BE-4 with SD 390 steel (actual yield point was 424 MPa). Shear reinforcement of four specimens are identical, consisting of four legs of closed
2400
Reflection measuring point
specimen BE-1
^ - 5 - D l 6
Fig. 4.29. Dimension and reinforcement of beam specimens.
stirrups of D6 SD295 bars (actual yield point was 337 MPa) spaced at 70 mm, with shear reinforcement ratio of 1.22 percent. Concrete with nominal strength of 60 MPa was used for all specimens, whose actual strength was 69.3 MPa.
Beams were loaded in a test rig in such a way that the point of contraflexure came to a point shown in Fig. 4.29, i.e. 600 mm from the critical section at the left end of 900 mm clear span. Under this loading the shear span ratio was 2.0, and a yield hinge would form at the left end only. Deflection of the point of contraflexure was measured relative to the fixed left stub of the specimen.
Figure 4.30 shows load-deflection curves for four specimens. Specimen BE-1 with the largest flexural strength, shown in Fig. 4.30(a), had shear cracks at
rotation angle 0.5% 1% 1.5% 2%
15 -10 -5 0 5 deflection (mm) (a) Specimen BE-1
rotation angle 0.5% 1% 1.5% 2%
Xl' Kll^'l'
1
-5 0 5 deflection (mm) (b) Specimen BE-2
Fig. 4.30. Load-deflection curves for beam specimens.
3 0 0 i i i i i ) i • i i | i i i i
rotation angle 0,5% .1% L 5 % 2 % 4%
-30 -20 10 0 10 deflection (mm) (c) Specimen BE-3
20 30
rotation angle 0.5% 1% 1.5% 2% 4%
- J. j . i : :
u i j . 1 1 i i i i i i i i i
\lT£j^Cliit' i wFriJr-4
mffltf^^uL^i f&jhr—_^/
"8E-4 T !
200 100
z
M
ô
jo 100 -200
-30 -20 10 0 10 20 30 deflection (mm)
(d) Specimen BE-4 Fig. 4.30. {Continued)
the load of about 70 kN, then reached the maximum load at the deflection an- gle of 1 percent which was much lower than the calculated flexural capacity of 387 kN. After the deflection angle of 1.5 percent was exceeded, the load dropped quite rapidly without forming a yield hinge. Specimen BE-2 in Fig. 4.30(b) also had shear cracks at about the same time as BE-1, and the load at 0.5 percent deflection angle, axial re-bars being in the elastic range, was much smaller than BE-1, owing to smaller stiffness resulting from smaller amount of rein- forcement. The load was kept increasing up to 1.5 percent deflection, barely lower than the calculated flexural capacity of 314 kN. After the first cycle of 2 percent deflection, the load dropped quite abruptly without forming a yield hinge.
Contrary to above, specimens BE-3 and BE-4 in Figs. 4.30(c) and (d), respectively, reached the flexural capacity although they also had shear cracks at about the same time as above specimens. BE-3 in Fig. 4.30(c) experienced the axial re-bar yielding at 1 percent deflection under the load almost equal to calculated flexural capacity of 212 kN, and the load was kept going up until 2 percent deflection angle, and after that load was decreased gradually. Note that the scale on both axes of Figs. 4.30(c) and (d) is different from those of Figs. 4.30(a) and (b). Rapid load drop after 4 percent deflection occurred due to bond failure of top axial reinforcement. BE-4 with the weakest steel showed yielding at 0.5 percent deflection and reached the maximum load that exceeded the calculated flexural capacity of 137 kN. The load did not drop up to the deflection angle of 4 percent as shown in Fig. 4.30(d).
The shear strength equation of AIJ Guidelines (Ref. 4.3) is shown below.
Here the shear strength is expressed by the sum of forces carried by a truss mechanism and an arch (or a strut) mechanism
Vu = bjtPwPwy cot <j> + tan 0(1 - (3)bDvaB/2 (4.6)
tan 9 = y/(L/D)2 + 1 -L/D (4.7)
f3 = (1 + cot2 $)pw<rwyl{vaB) (4.8)
where
<JB • compressive strength of concrete
awy : yield strength of web reinforcement (to be taken 25<TB if awy exceeds 25a B)
b : width of the member
jt : distance between top and bottom axial re-bars D : total depth of the member
L : clear span of the member
pw : web reinforcement ratio (pwawy should be taken to be equal to VCB/2
if it exceeds vas/2)
8 : angle of concrete strut in the arch (strut) mechanism
(3 : the ratio of compressive stress in the concrete strut of truss mecha- nism to the effective concrete strength.
The coefficient v is the coefficient for the effective compressive strength of concrete and is expressed as follows. Before yielding, it is a constant equal to