Predicting the Influence of Shear on the Seismic Response of Bridge Columns

Abstract

In the seismic design and evaluation of bridges, a method is required for determining the shear strength of reinforced concrete columns to avoid brittle shear failures. In addition, detailed design and evaluation often require predictions of the complete hysteretic response of bridge columns to accurately model the nonlinear dynamic response of the bridge. Predictions of the shear strength of columns using the provisions of the AASHTO Specifications are compared with the reversed-cyclic loading test results of shear-critical columns. It is found that the Simplified Procedure results in very conservative predictions of the seismic shear strength. The General Procedure provides conservative and more accurate predictions of the seismic shear strength. It is suggested that the AASHTO reduction factor on the concrete contribution resisting shear for low compressive axial load levels be removed. Nonlinear finite element analysis predictions are made for a selection of rectangular and circular columns tested in reversed-cyclic loading and are compared with the experimental results. The ability of nonlinear finite element analysis to predict the reversed-cyclic loading responses of columns with a wide range of variables and having different failure modes is demonstrated.

1. Introduction

In the seismic design and evaluation of reinforced concrete bridge columns, it is vital to have a method for determining the shear strength to avoid shear failures. Such a method should be capable of accounting for the key parameters affecting the shear strength.

The 2020 AASHTO LRFD Specifications [1] for designing bridge columns for shear strength under seismic actions are investigated using available test results of columns that failed in shear. The columns investigated were tested under reversed-cyclic loading and had a variety of axial load levels, amounts of shear and longitudinal reinforcement, cross-sectional shapes, aspect ratios, and material properties. Suggestions are made to improve the seismic strength predictions using the AASHTO Simplified and General Procedures for shear design.

While the AASHTO design provisions for new bridges provide excellent seismic performance, with adequate shear reinforcement and confinement reinforcement, care needs to be taken in evaluating existing bridge columns in older structures. To evaluate the performance of an older bridge, it may be necessary to carry out nonlinear dynamic analyses. To carry out such analyses, the complete hysteretic responses of the columns need to be modelled. To investigate the full seismic performance, nonlinear finite element analysis predictions are compared with the results of columns tested under reversed-cyclic loading and having different failure modes and levels of ductility. These analyses demonstrate the ability to predict the response of columns with ductile (flexural yielding), moderately ductile (flexure-shear), and brittle (shear) failure modes. This more detailed nonlinear finite element analysis provides the engineer with a means of assessing the expected seismic performance of existing bridge columns.

2. AASHTO Seismic Shear Design Provisions

The 2020 AASHTO LRFD Bridge Design Specifications [1] give procedures for the seismic design of reinforced concrete bridge columns. These procedures include a capacity design approach for determining the required shear strength, procedures for determining the amount of shear reinforcement, and the amount of confinement reinforcement in the plastic hinge region to ensure ductile response.

The design provisions for determining the shear capacity are discussed and the predictions using these methods are compared with experimental results on shear-critical columns subjected to reversed-cyclic loading. The experimental results include columns with various cross-sectional shapes (circular, octagonal, and rectangular), reinforcement details, material properties, and axial load levels.

In the 2020 AASHTO Specifications, the nominal shear resistance, $V_n$, has a concrete contribution, $V_c$, and a shear reinforcement contribution, $V_s$. The equations for determining the nominal shear resistance of reinforced concrete columns are:

$$V_n=V_c+V_s\le 0.25f_c^{\prime }{b}_v{d}_v$$

(1)

$$\mathrm{For} \frac{P}{A_g{f}_c^{\prime }}\ge 0.10: V_c=\beta \sqrt{f_c^{\prime } } b_v{d}_v$$

(2a)

$$\mathrm{For} \frac{P}{A_g{f}_c^{\prime }}<0.10: V_c=\beta \sqrt{f_c^{\prime } } b_v{d}_v\left(10\frac{P}{A_g{f}_c^{\prime }}\right)$$

(2b)

$$V_s=f_y{A}_v\frac{d_v\cot \mathsf{\theta }}{\mathrm{s}}$$

(3)

where the factor $\beta$ accounts for the shear resistance of the diagonally cracked concrete; $b_v$ is the effective web width; $d_v$ is the effective shear depth taken as the greater of $0.72h$ or $0.9d$, where $d$ is taken as the distance from the extreme compression fiber to the centroid of the longitudinal tension reinforcement in the tension half of the member; $s$ is the spacing of transverse shear reinforcement; $A_v$ is the area of transverse reinforcement within distance $s$; and $\theta$ is the angle of principal compressive stress with respect to the longitudinal axis of the member. It is noted that in the end regions of columns, Equation (2a) applies when the compressive axial load is greater than or equal to $0.10f_c^{\prime }{A}_g$. However, for compressive axial loads less than $0.10f_c^{\prime }{A}_g$, $V_c$ decreases linearly from the value given in Equation (2a) to zero when the compressive axial load is zero (see Equation (2b)). This reduction factor accounts for the influence of reversed-cyclic loading in reducing the tensile stresses in the concrete without the beneficial effects of significant compressive axial load.

The Specifications provide two different procedures for shear design. The Simplified Procedure assumes a value for the factor, $\beta$, of 0.167 MPa. It also assumes an angle of principal compression, $\theta$, of 45° in accordance with the traditional 45° truss model.

The General Procedure is based on the Modified Compression Field Theory [2,3,4] and is adapted from the General Method of the Canadian Highway Bridge Design Code (CHBDC) [5]. To determine $\beta$ and $\theta$, the longitudinal strain, ${\epsilon }_s$, at the centroid of the tension reinforcement must first be determined. The equations for determining $\beta$ and $\theta$ for a reinforced concrete column, containing at least minimum amounts of transverse reinforcement, may be written as the following in SI units:

$${\epsilon }_s=\frac{\frac{M_u}{d_v}+V_u+0.5N_u}{E_s{A}_s}$$

(4)

$$\beta =\frac{0.4}{1+750{\epsilon }_s }$$

(5)

$$\theta =29+3500{\epsilon }_s$$

(6)

where the moment, $M_u$ is taken as a positive value; $N_u$ is the axial load (positive for tension and negative for compression); $E_s$ is the elastic modulus of the longitudinal reinforcing steel; and $A_s$ is the area of the longitudinal reinforcing bars on the tension half of the member. The critical section for shear is taken at a distance $d_v$ from the face of the support and hence the moment, $M_u$, must be calculated at this section, but shall not be taken less than $V_u{d}_v$. The General Procedure takes account of the level of axial load and moment on the column, and also accounts for the amount of longitudinal reinforcement in the column. Although the equations above are expressed in terms of the strain, ${\epsilon }_s$, the General Method of the CHBDC uses the longitudinal strain at mid-depth of the section, ${\epsilon }_x$, as the indicator of the strain conditions in the web resisting shear. Hence, the equations of the CHBDC can be obtained by replacing ${\epsilon }_s$ in the equations above with 2${\epsilon }_x$.

3. Predicting Capacities of Shear-Critical Columns Using the AASHTO Procedures

To determine the accuracy of the Simplified and General Procedures for seismic shear design, the predicted shear strengths are compared with the shear strengths of reinforced concrete columns that failed in shear and were subjected to reversed-cyclic loading. Experimental results for columns failing in shear [6,7,8,9,10,11,12,13,14,15,16,17] were obtained from the PEER Structural Performance Database [18]. These column tests were supplemented with additional results for columns failing in shear from the literature [19,20,21,22,23,24,25]. The experimental results had a wide range of variables including: circular, octagonal, and rectangular columns; cross-sectional dimensions from 152 to 914 mm, axial compressive load level, $P/A_g{f}_c^{\prime }$ from zero to 0.506; concrete strengths from 16.0 to 56.2 MPa; yield stress of the longitudinal ($f_{yl}$) and transverse reinforcement ($f_{yv}$) from 249 to 496 MPa; transverse reinforcement ratio, ${\rho }_v=A_v/\left(b_{v}s\right)$, from zero to 0.761%; longitudinal reinforcement ratio, ${\rho }_l=A_{st}/A_g$ from 0.992 to 7.15%, and shear span-to-depth ratios, $a/D$ or $a/h$, from 1.09 to 3.22. The effective web width, $b_v$, for circular columns was taken as the diameter and the effective web width for the octagonal columns was taken as the flat-to-flat dimension of the column. The transverse reinforcement typically consists of hoops and seismic crossties in rectangular columns and spirals or circular hoops in circular or octagonal columns. In the application of the Simplified and General Procedures for circular and octagonal columns, the area of shear reinforcement, $A_v$, within distance $s$ was taken as twice the cross-sectional area of the spiral or circular hoop reinforcement. The cross-sectional dimensions and reinforcement details are given in Figure 1 and

Table 1. Details of circular and octagonal columns and shear strength predictions using the AASHTO Simplified and General Procedures.

Test	${\mathit{f}}_{\mathit{c}}^{\prime }$	$\frac{\mathit{P}}{{\mathit{A}}_{\mathit{g}}{\mathit{f}}_{\mathit{c}}^{\prime }}$	$\frac{\mathit{a}}{\mathit{D}}$	$\mathit{D}$	${\mathit{d}}_{\mathit{v}}$	${\mathit{\rho }}_{\mathit{v}}$	${\mathit{f}}_{\mathit{y}\mathit{v}}$	${\mathit{\rho }}_{\mathit{l}}$	${\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}$	${\mathit{V}}_{\mathit{s}\mathit{i}\mathit{m}}$	$\frac{{\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{s}\mathit{i}\mathit{m}}}$	${\mathit{V}}_{\mathit{g}\mathit{e}\mathit{n}}$	$\frac{{\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{g}\mathit{e}\mathit{n}}}$
	MPa			mm	mm	%	MPa	%	kN	kN		kN
Ang et al. 1985 & 1989 [6,7]
16	33.4	0.100	2.00	400	288	0.236	326	3.20	379	199	1.90	267	1.42
18	35.0	0.100	1.50	400	288	0.236	326	3.20	507	202	2.51	296	1.71
19	34.4	0.100	1.50	400	288	0.177	326	3.20	436	178	2.44	270	1.61
20	36.7	0.175	1.75	400	288	0.177	326	3.20	487	182	2.67	288	1.69
4	30.6	0.000	2.00	400	288	0.238	316	3.20	289	87	3.34	137	2.11
6	30.1	0.000	1.50	400	288	0.236	328	3.20	392	89	4.40	145	2.71
7	29.5	0.000	2.00	400	288	0.177	372	3.20	281	76	3.71	121	2.31
21	33.2	0.000	2.00	400	288	0.177	326	3.20	270	66	4.08	108	2.51
22	30.9	0.000	2.00	400	288	0.178	310	3.20	285	64	4.47	104	2.74
25	32.8	0.000	1.50	400	290	0.000	0	3.20	239	0	**	0	**
Arakawa et al. 1987 & 1988 [8,9]
3 *	28.6	0.120	1.09	275	198	0.000	0	3.85	158	48	3.27	93	1.69
4 *	29.8	0.115	1.09	275	198	0.206	368	3.85	196	91	2.16	143	1.37
6 *	28.6	0.120	1.09	275	198	0.411	368	3.85	225	131	1.72	190	1.18
8 *	31.4	0.109	1.09	275	198	0.588	368	3.85	216	168	1.28	231	0.93
9 *	30.5	0.113	1.09	275	198	0.411	368	5.13	228	132	1.72	202	1.13
11 *	28.7	0.239	1.09	275	198	0.000	0	3.85	188	48	3.89	113	1.67
12 *	27.8	0.247	1.09	275	198	0.206	368	3.85	192	89	2.16	158	1.22
13 *	30.5	0.225	1.09	275	198	0.411	368	3.85	238	132	1.80	207	1.15
14 *	31.3	0.219	1.09	275	198	0.588	368	3.85	279	168	1.66	246	1.13
1 *	28.8	0.000	1.09	275	198	0.206	368	3.85	176	41	4.28	69	2.57
2 *	29.3	0.000	1.09	275	198	0.411	368	3.85	204	82	2.47	128	1.59
17 *	31.3	0.110	1.09	275	198	0.274	381	3.85	247	107	2.29	163	1.51
18 *	31.1	0.110	1.64	275	198	0.000	0	3.85	132	50	2.62	92	1.43
19 *	31.2	0.110	1.64	275	198	0.274	381	3.85	186	107	1.74	157	1.19
22 *	20.5	0.167	1.64	275	198	0.274	381	3.85	171	98	1.75	148	1.16
24 *	31.1	0.221	1.09	275	198	0.274	381	3.85	234	107	2.18	179	1.31
25 *	29.7	0.231	1.64	275	198	0.274	381	3.85	201	106	1.90	170	1.18
27 *	18.9	0.363	1.64	275	198	0.274	381	3.85	176	96	1.83	159	1.11
28 *	41.3	0.166	1.64	275	198	0.274	381	3.85	231	115	2.01	179	1.29
Benzoni et al. 1996 [19]
1	29.3	0.347	1.98	460	331	0.146	369	2.41	524	226	2.32	379	1.38
Jaradat 1996 [20]
S1	29.0	0.058	2.00	254	183	0.089	210	2.04	79	36	2.23	48	1.64
McDaniel 1997 [10]
S1	29.8	0.002	2.00	610	439	0.061	200	1.36	405	38	10.71	65	6.24
S1-2	26.8	0.002	2.00	610	439	0.061	200	1.36	332	38	8.71	65	5.08
S2	31.2	0.002	2.00	610	439	0.061	200	1.36	332	38	8.80	65	5.13
Ranf et al. 2006 [21]
S3	56.2	0.100	3.00	508	366	0.062	414	0.99	266	287	0.93	257	1.04
C3R	52.7	0.100	3.00	508	366	0.062	414	0.99	267	281	0.95	249	1.07
Verma et al. 1993 [22]
5	35.9	0.057	2.00	610	439	0.083	324	2.53	614	224	2.74	335	1.83
Yalcin 1997 [23]
BR-C1	45.0	0.137	2.43	610	439	0.109	425	2.05	563	428	1.32	526	1.07
Yarandi 2007 [24]
CR-C	35.0	0.140	2.50	600	432	0.036	491	1.27	424	306	1.39	355	1.19
									Average	-	2.96	-	1.85
									Std. Dev.	-	2.12	-	1.18

* Indicates octagonal column. ** Column has no axial load and no transverse reinforcement and hence AASHTO predicts zero shear strength.

for the circular and octagonal columns, and Figure 2 and

Table 2. Details of rectangular columns and shear strength predictions using the AASHTO Simplified and General Procedures.

Test	${\mathit{f}}_{\mathit{c}}^{\mathit{\prime }}$	$\frac{\mathit{P}}{{\mathit{A}}_{\mathit{g}}{\mathit{f}}_{\mathit{c}}^{\mathit{\prime }}}$	$\frac{\mathit{a}}{\mathit{h}}$	$\mathit{b}$	$\mathit{h}$	${\mathit{d}}_{\mathit{v}}$	${\mathit{\rho }}_{\mathit{v}}$	${\mathit{f}}_{\mathit{y}\mathit{v}}$	${\mathit{\rho }}_{\mathit{l}}$	${\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}$	${\mathit{V}}_{\mathit{s}\mathit{i}\mathit{m}}$	$\frac{{\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{s}\mathit{i}\mathit{m}}}$	${\mathit{V}}_{\mathit{g}\mathit{e}\mathit{n}}$	$\frac{{\mathit{V}}_{\mathit{e}\mathit{x}\mathit{p}}}{{\mathit{V}}_{\mathit{g}\mathit{e}\mathit{n}}}$
	MPa			mm	mm	mm	%	MPa	%	kN	kN		kN
Aboutaha et al. 1999 [11]
SC3	21.9	0.000	2.67	914	457	397	0.095	400	1.88	450	125	3.61	199	2.26
SC9	16.0	0.000	1.33	457	914	684	0.076	400	1.88	643	92	7.00	157	4.10
Arakawa et al. 1989 [12]
OA2	31.8	0.184	1.25	180	180	142	0.217	249	3.13	131	34	3.79	64	2.04
OA5	33.0	0.445	1.25	180	180	142	0.217	249	3.13	134	35	3.84	76	1.75
Bett et al. 1985 [13]
No. 1-1	29.9	0.104	1.50	305	305	237	0.151	414	2.44	214	103	2.08	158	1.35
Imai and Yamamoto 1986 [14]
No. 1	27.1	0.072	1.65	400	500	406	0.318	336	2.66	471	248	1.90	359	1.31
Lynn et al. 1998 [15]
3CLH18	26.9	0.089	3.22	457	457	352	0.068	400	3.04	266	157	1.70	215	1.24
3CMH18	27.6	0.262	3.22	457	457	352	0.068	400	3.04	328	172	1.91	272	1.21
3CMD12	27.6	0.262	3.22	457	457	352	0.173	400	3.04	356	236	1.51	331	1.08
Massa et al. 2022 [25]
R1	38.6	0.130	1.50	250	400	315	0.285	332	3.93	323	142	2.27	235	1.37
R2	38.6	0.130	1.50	250	400	315	0.380	332	3.93	328	165	1.99	262	1.25
R3	35.9	0.139	1.50	250	400	315	0.571	332	3.93	334	208	1.61	312	1.07
R4	38.4	0.130	1.50	250	400	315	0.761	332	3.93	382	256	1.49	365	1.05
Umehara and Jirsa 1982 [16]
CUS	34.9	0.162	1.11	230	410	304	0.276	414	3.01	323	144	2.24	232	1.39
CUW	34.9	0.162	1.98	410	230	190	0.310	414	3.01	263	158	1.66	225	1.17
2CUS	42.0	0.270	1.11	230	410	304	0.276	414	3.01	405	151	2.69	292	1.39
Wight and Sozen 1973 [17]
No. 25.033 (East)	33.6	0.071	2.87	152	305	254	0.323	345	2.45	88	63	1.40	76	1.16
									Average		-	2.51	-	1.54
									Std. Dev.		-	1.36	-	0.72

for the rectangular columns.

Table 1 and Table 2 as well as Figure 3 and Figure 4 compare the shear strength predictions using the AASHTO Simplified ($V_{sim}$) and General Procedures ($V_{gen}$) with the experimentally obtained shear capacities ($V_{exp}$) for shear-critical rectangular and circular/octagonal columns. Table 1 and Table 2 as well as Figure 3a and Figure 4a include the reduction in $V_c$ for low axial load levels required in the AASHTO Specifications while Figure 3b and Figure 4b do not include this reduction for low axial load levels. The shear strength predictions using the AASHTO Simplified Procedure resulted in average $V_{exp}/V_{sim}$ of 2.96 (SD = 2.12) and 2.51 (SD = 1.36) for the circular/octagonal and rectangular columns, respectively. The AASHTO General Procedure is more accurate and resulted in an average $V_{exp}/V_{gen}$ of 1.85 (SD = 1.18) for the circular and octagonal columns, and 1.54 (SD = 0.72) for the rectangular columns.

If the reduction in $V_c$ is neglected, the Simplified Procedure results in average $V_{exp}/V_{sim}$ ratios of 1.89 (SD = 0.58) and 2.05 (SD = 0.75) for the circular/octagonal and rectangular columns, respectively. If the reduction in $V_c$ is neglected, the General Procedure results in average $V_{exp}/V_{gen}$ ratios of 1.30 (SD = 0.20), and 1.30 (SD = 0.26) for the circular/octagonal and rectangular columns, respectively. It is clear from Figure 3 and Figure 4 that excluding the reduction factor on $V_c$ results in generally conservative and more accurate predictions, with the General Procedure giving the best predictions. By accounting for the combined effects of moment, shear, and axial load, and the amount of longitudinal reinforcement, the General Procedure provides better strength predictions over a wider range of variables. As expected, increasing the axial compression or the amount of longitudinal reinforcement increases the predicted shear capacity.

It is noted that many of the columns tested have small shear span-to-depth ratios, $a/D$ or $a/h$ (see Figure 1 and Figure 2). These members that are subjected to an axial compressive load can develop an inclined compressive strut which helps to resist the shear along with the sectional shear resistance [26,27]. The sectional model predictions shown in Figure 3 do not include the beneficial influence of a compressive strut for columns with low $a/D$ or $a/h$ ratios and hence in some cases these predictions can be overly conservative. The predictions that are most conservative shown in Figure 3 are columns with very low shear span-to-depth ratios. These columns acted as wall-type piers and displayed strut action as indicated by the cracking pattern observed in the tests. For columns having $a/D$ or $a/h$ of 2.5 or greater, the sectional design approach using the General Procedure gives accurate predictions with an average $V_{exp}/V_{gen}$ of 1.11 and a standard deviation of 0.07.

The 2020 AASHTO Specifications have different requirements for columns and wall-type piers. If the ratio of the shear span to the dimension of the column in the direction of shear ($a/D$ or $a/h$) is equal to or greater than 2.5, the member is considered to be a column and the sectional design procedure is appropriate. For shear span-to-depth ratios less than 2.5, a rectangular member is considered to be a wall-type pier. The AASHTO Specifications give empirical equations for determining the seismic shear resistance of wall-type piers. For rectangular columns with $a/h$ less than 2.5, the use of these equations gives conservative predictions that are very similar to the predictions using the Simplified Procedure, while the General Procedure gives better predictions. More detailed predictions using nonlinear finite element analysis are given in Section 4.

4. Predicting Response of Reinforced Concrete Columns

To predict the nonlinear dynamic analysis of an existing bridge, the designer needs to develop appropriate hysteretic responses for the columns. To achieve this, 2D nonlinear finite element analysis provides a useful method for determining the complete hysteretic response and accounts for the different types of failure modes. This approach can be used for a wide range of variables and does not rely on empirically based equations that attempt to fit the experimental data. To demonstrate the capabilities of nonlinear finite element analysis in predicting the complex reversed-cyclic loading response, it is applied to selected experiments having a wide range of parameters including cross-section shape, axial load level, and amount and grade of longitudinal and transverse reinforcement.

4.1. Finite Element Modeling

The 2D nonlinear finite element analysis program, VecTor2 [28] was used to predict the response of columns experiencing a range of failure modes and levels of ductility. The program employs a smeared, rotating crack model which is based on the Modified Compression Field Theory and the Disturbed Stress Field Model [29] and hence is capable of predicting shear failures and shear distress.

The cracked concrete is treated as an orthotropic material with cracks distributed throughout the elements. The cracks can freely rotate to remain parallel with the principal compressive stresses in the material. Shear slip along the crack is accounted for, allowing the principal stresses to lag in their rotation relative to the principal strains.

The default material models were used in all the finite element predictions, except for tension stiffening, which was accounted for using the Vecchio 1982 model [2]. The effects of concrete confinement are accounted for by increasing the concrete strength and associated strains according to Kupfer [30] and Richart et al. [31]. The hysteretic response of the concrete is modelled with plastic strain offsets and nonlinear unloading [32]. The reinforcing steel model includes the effect of yielding, strain hardening, and bar rupture. The hysteretic response of the reinforcement includes the Bauschinger Effect [33]. In the finite element analysis, bar buckling was modeled using the Akkaya [34] model that was based on the model developed by Dhakal and Maekawa [35]. This bar buckling model accounts for the spacing of the transverse reinforcement, the diameter of the longitudinal reinforcing bars, and the stiffness of the transverse reinforcement that is controlling the dilation of the column in the plastic hinge region. The analysis considered spalling of the concrete cover by eliminating boundary elements based on their principal compressive strains, crack widths, and crack inclinations. Discrete truss elements were used to model the longitudinal reinforcement while smeared reinforcement ratios were used to model the in-plane and out-of-plane transverse reinforcement.

To model circular columns in VecTor2, the cross-section must be discretized into layers and the smeared reinforcement ratios determined as shown in Figure 5. The width and thickness of these elements vary according to their location in the cross-section. The layers were selected such that their boundaries occurred at the levels of longitudinal reinforcement. For each of these layers, the in-plane (${\rho }_x$) and out-of-plane (${\rho }_z$) smeared reinforcement ratios must be determined. The in-plane component resists the shear, while the out-of-plane component is used to determine the confinement effects on the concrete. These reinforcement ratios are dependent on the cross-sectional area of the hoop/spiral, $A_b$, and the angle, $\alpha$, and are given by Equations (7) and (8). For spirally reinforced columns, the pitch is typically small, such that the inclination of the spiral can be ignored [6].

$${\rho }_x=\frac{2A_b\cos \alpha }{ws}$$

(7)

$${\rho }_z=\frac{2A_b\sin \alpha }{ts}$$

(8)

The following sections provide details and response predictions of columns that displayed different failure modes. These predictions using nonlinear finite element analysis are compared with the experimental responses. In making the response predictions using nonlinear finite element analysis, the target displacements in the analysis were chosen to match, as closely as possible, the positive and negative peak displacements achieved in the reversed-cyclic loading tests.

4.2. Circular Columns Tested by Ang et al. 1989

Ang et al. [6,7] tested a large number of circular columns under reversed-cyclic loading. Figure 6 shows a typical cross-section and the finite element model for one of these columns. The various colors in the mesh indicate varying concrete thicknesses and amounts of smeared reinforcement (${\rho }_x$ and ${\rho }_z$). The constant axial load was applied first and then imposed displacements were applied in the direction of shear. The reversed-cyclic loading consisted of 5 cycles at each displacement level. The columns were loaded in single curvature with lateral displacements applied at the top of the columns and were supported by a heavily reinforced concrete beam that spanned between supports at its ends (see Figure 6). It was important to include the deformations of this beam in the model to realistically predict the displacements at the top of the column.

The predicted responses are compared with the experimental results in Figure 7. In addition, the shear span-to-depth ratio, $a/D$, the axial load ratio, $P/f_c^{\prime }{A}_g$, and the amount of transverse reinforcement are shown in the figure.

Column 9 had an axial load level of 0.2, a shear span-to-depth ratio of 2.5, and No. 2 spirals at a pitch of 30 mm. It displayed a very ductile response achieving a displacement ductility ratio of about 8 with little shear distress. The finite element analysis also predicted a ductile response with significant yielding of the flexural reinforcement (see Figure 7a). The precited response agrees well with the experimental response for this ductile column. The longitudinal reinforcement was predicted to yield at a displacement of 7.7 mm. For this ductile column, yielding of the transverse reinforcement occurred at an applied shear of about 350 kN during testing compared with a predicted shear of 314 kN. Much of the damage to the column was concentrated in the hinge at the column base, as predicted by the finite element analysis. The predicted strength was 382 kN compared with the test result of 393 kN.

Column 13 had a reduced axial load level of 0.1, $a/D$ of 2.0, and had the same transverse reinforcement as Column 9. Both the experimental response and the predicted response indicated flexural yielding followed by shear distress and then shear failure (see Figure 7b). The column was only able to reach a ductility ratio of about 4 (moderate ductility) before failing in shear. Yielding of the longitudinal reinforcement was predicted by the finite element analysis to occur at a displacement of 5.8 mm. The predicted strength and ductility at maximum load compare well with the experimental results. The predicted strength was 418 kN compared with the test result of 440 kN. The nonlinear finite element analysis was not able to converge after the brittle shear failure had occurred and hence result in conservative predictions after failure occurred.

Column 17 had an axial load level of 0.1 and the same shear span and half the amount of transverse reinforcement as Column 9. This column behaved in a similar manner to Column 13 but had a reduced displacement ductility ratio of 3 (limited ductility) that was limited by shear failure (see Figure 7c). Yielding of the longitudinal reinforcement was predicted to occur at a displacement of 7.9 mm. It is noted that the transverse reinforcement experienced high strains during testing that were also predicted by the finite element analysis. The predicted strength was 305 kN compared with the experimentally determined strength of 325 kN.

Column 16 was identical to Column 17 except that it was shorter, with a shear span-to-depth ratio of 2.0. It is also identical to Column 13 except that it has half the amount of transverse reinforcement. This column experienced brittle shear failure without general yielding of the longitudinal reinforcement. The maximum shear was predicted well by the finite element analysis with failure predicted in the negative cycle and the response displaying pinching of the hysteretic curves as can be seen in Figure 7d. The predicted strength was 363 kN compared with the experimentally determined strength of 372 kN.

4.3. Circular Columns Tested by Verma et al. 1993

Verma et al. [22] tested a series of circular columns under reversed-cyclic loading. The two columns for which predictions were made vary in their axial load level and grade of longitudinal reinforcement. Figure 8 shows a typical cross-section and the finite element model for one of these columns. The constant axial load was applied first. The reversed-cyclic loading consisted of 5 cycles at a load level corresponding to half the flexural strength followed by 3 cycles at each target displacement level. The columns were loaded in double curvature with lateral displacements applied on the top heavily reinforced end region, and the columns were supported by a heavily reinforced concrete end region anchored to the laboratory floor (see Figure 8). The hoops were anchored with 305 mm lap splices and hence to simulate this effect, strain hardening of the transverse reinforcement was neglected because of the uncertainty of developing stresses higher than the yield stress.

The finite element response predictions for the Verma et al. [22] columns are compared with the experimental results in Figure 9. The columns had a shear span-to-depth ratio of 2.0 and shear reinforcement consisting of No. 2 hoops at a spacing of 127 mm. The constant axial load ratios were 0.18, and 0.057 for Columns 3, and 5, respectively. The columns had different grades of longitudinal steel with Column 3 having Grade 40 (275 MPa) and Column 5 having Grade 60 (400 MPa).

Column 3 experienced a limited displacement ductility of 2 with yielding of the longitudinal reinforcement followed by a shear failure (see Figure 9a). First yielding of the longitudinal reinforcement was predicted by the finite element model at a displacement of 10.0 mm and occurred during testing at a displacement of 9.4 mm. Yield strains in the transverse reinforcement were also reasonably well predicted and occurred at a displacement of 12.0 mm compared with a displacement between 11.9 mm (ductility 1.0) and 17.8 mm (ductility 1.5) during testing. At failure, the predicted crack pattern shows the same steeply inclined flexure-shear cracks and splitting cracks in the central core as observed in the test. The predicted strength was 720 kN compared with the experimental strength of 746 kN.

Figure 9b shows the finite element predictions and experimental response of Column 5 that experienced a brittle shear failure with large shear cracks and fracture of hoops. This column had a higher moment capacity due to the higher yield strength of the longitudinal reinforcement and was predicted to have a shear failure. The sudden drop in strength and stiffness which occurred at a displacement of 18.5 mm was initiated by the formation of a wide diagonal shear crack from the top of the column. Yielding of the longitudinal reinforcement was predicted at a displacement of 15.0 mm which closely matched the 13.0 mm displacement at which yielding was observed during testing. Due to the very brittle shear failure, the finite element analysis failed to converge beyond the peak load. The predicted strength was 625 kN compared with the experimental strength of 599 kN.

4.4. Square Columns Tested by Lynn et al. 1996

Lynn et al. [15] tested a series of square columns under reversed-cyclic loading to investigate the behavior of columns with different levels of axial load, varying amounts of longitudinal reinforcement, different tie details, and large tie spacings typical of pre-1970’s construction. Figure 10 shows typical cross-sections and the finite element model for three columns with different failure modes that were selected for analysis. Each column was subjected to a constant axial compression with the reversed-cyclic loading consisting of 3 cycles at displacements corresponding to increments of the calculated yield displacement. The columns were loaded in double curvature and had heavily reinforced end regions (see Figure 10). To simulate the effect of 90° end anchorages on the ties, the stress in the ties was limited to the yield stress without strain hardening in the finite element analysis. The finite element analysis predicts that buckling of the longitudinal reinforcement occurred as was reported by the researchers.

The finite element response predictions for the Lynn et al. [15] columns analyzed are compared with the experimental results in Figure 11. The three columns had a shear span-to-depth ratio of 3.22 and shear reinforcement consisting of No. 3 square and diamond-shaped ties at a spacing of 457 mm or 305 mm (see Figure 10). The axial load ratios ($P/f_c^{\prime }{A}_g$) were 0.073 for column 2CLH18, and 0.262, for columns 3CMD12 and 2CMH18.

Column 2CLH18 achieved a ductility level of 4.2 and experienced a predominately flexural failure with yielding and buckling of the longitudinal reinforcement occurring at displacements of 15 mm and 46 mm, respectively. The finite element model gives reasonable predictions of the response of the longitudinal reinforcement with yielding and buckling occurring at displacements of 17 mm and 36 mm, respectively. The concentrated inelastic rotations reported within the bottom 102 mm of the column are also present in the finite element model (see Figure 11a). The influence of buckling can be seen in Figure 11. The predicted strength was 257 kN compared with the experimental strength of 241 kN.

Column 3CMD12 exhibited a flexure-shear failure with yielding of the longitudinal reinforcement occurring first, at a displacement of 30 mm followed by transverse reinforcement yielding at 46 mm. All ties except one indicated yielding prior to failure. The response is predicted well by the finite element model which indicates a drop in lateral load at a displacement target of 32 mm followed by yielding of the longitudinal reinforcement in the following cycle (see Figure 11b). The predicted strength was 367 kN compared with the experimental strength of 356 kN.

Column 2CMH18 experienced a brittle shear failure during the last cycle at a displacement of 30 mm. The failure occurred due to the formation of an inclined crack and yielding of the tie crossing the crack. Longitudinal reinforcement at the column base had yielded just prior to failure but flexural hinging had not occurred. The finite element analysis gives reasonable predictions of the column shear-displacement response, and correctly predicted that a brittle shear failure would occur (see Figure 11c). The analysis also predicts that the longitudinal reinforcement just reaches yield at a displacement of 30 mm. The predicted strength was 301 kN compared with the experimental strength of 306 kN.

5. Conclusions

The conclusions from this study are as follows:

• The use of the 2020 AASHTO Simplified Procedure for seismic shear design results in very conservative predictions of the shear strength of shear-critical columns tested under reversed-cyclic loading;
• The use of the 2020 AASHTO General Procedure for seismic shear design results in more accurate predictions of the shear strength of shear-critical columns tested under reversed-cyclic loading;
• Predictions using the 2020 AASHTO Simplified and General Procedures for seismic shear design are more accurate if the reduction factor on the concrete contribution to shear strength ($V_c$) for low compressive axial load levels is ignored. It is suggested that this reduction factor be removed from the AASHTO provisions.
• Nonlinear finite element analysis relies on behavioral models rather than empirical fitting equations to predict the reversed-cyclic loading response and hence is well suited to predicting the effects of opening and closing of cracks, strain softening, crack slip, cover spalling, confinement, bar buckling, yield penetration, shear-moment-axial load interaction, shear span-to-depth ratio, and level of axial load. The use of nonlinear finite element analyses enables the prediction of the reversed-cyclic loading responses of circular and rectangular columns with a wide range of variables as well as modes of failure including brittle shear failures, moderately ductile failures, and ductile flexural responses.