Finite Element Analysis and Optimization of the Rotational Stiffness of Semi-Rigid Base Connection under Simultaneous Moment and Tension

Abstract

The base connection is flexible, not fully pinned/fixed, implying a nonlinear moment–rotation relationship. This deviates from a linear response, where rotation is not directly proportional to the applied moment. Numerical investigations using the commercial software ABAQUS were conducted to analyze the steel base plate connections. The finite element (FE) models were verified against previous experimental results. Moreover, numerical findings of a comprehensive parametric investigation were conducted. The studied connections were examined with different configurations, including variations in the diameter, spacing, and number of the anchor bolts; the thickness of the base plate; and the applied axial force. The current study aims to use numerical results combined with the whale optimization algorithm (WOA) and classical genetic algorithm (GA) to derive a formulation for the moment–rotation (M-θr) relationship. The distinctive aspect of this formulation is that it aims to simulate the nonlinear rotational behavior exhibited by flexible base connections under combined moment and tension loads, while also considering various parameters such as bolt number/diameter and plate thickness. The findings indicate that the WOA is capable of obtaining an optimal equation for accurately simulating the M-Ɵr relationship. This underscores the ability of the WOA to effectively address the complexity of the problem and provide a reliable equation for predicting the rotational behavior of such connections. Consequently, the WOA method can be utilized to calculate the rotational stiffness at H/150, offering valuable support for engineering design processes.

1. Introduction

The majority of design methodologies simplify steel base connections by assuming either a completely hinged or entirely rigid connection. However, it should be noted that steel-hinged base plate connections exhibit a semi-rigid or flexible behavior in practice, as they possess rotational stiffness. Consequently, the actual behavior of these connections is intricate and nonlinear, influenced by various factors. In reality, their response lies somewhere between the two extreme assumptions. Therefore, relying on these oversimplified assumptions for base connections fails to provide an accurate depiction or a realistic understanding of the behavior of steel structures. In contrast to a fully rigid connection, a flexible base connection introduces a decrease in the overall stiffness of the structure. This decrease in stiffness results in an increase in drift and displacement within the structure under the same applied load. The amplified frame drift exacerbates the P-∆ effect, necessitating the consideration of this effect, as well as geometric nonlinearity (i.e., changes in joint coordination), through a stepped nonlinear analysis approach. Numerous research studies focus on establishing an equation for the rotational stiffness in this kind of connection, taking into account various types of parameters that have significant impacts [1,2,3,4,5,6,7,8,9]. Each study employed a specific formula to simulate the moment and rotation (M-θr) relationship and then the rotational stiffness was explored. However, these formulas typically focused solely on the moment as an externally applied strain, neglecting the effects of axial tensile stresses. Therefore, the objective of the present study is to derive a rotational stiffness equation for this kind of connection that encompasses both moment and tension loads. This will enable the establishment of a comprehensive relationship between the relative rotation of the connection and the applied moment and tension loads. The present study builds upon the findings of a parametric investigation conducted by Nawar et al. [10] that utilized ABAQUS to simulate the M-θr curve for semi-rigid connections subjected to simultaneous moment and tension loads.

An optimization algorithm technique is employed to deduce the optimal M-θr relationship for these semi-rigid connections under the combined influence of moment and tension loads. Two advanced algorithms are employed, with the first one being the whale optimization algorithm (WOA). The WOA has gained significant attention in recent years and has been successfully applied to various optimization problems across different engineering disciplines [11]. Inspired by the bubble-net scavenging behavior of humpback whales in oceans, this algorithm demonstrates promising capabilities in optimizing diverse engineering topics [12]. The second algorithm utilized in this study is the genetic algorithm (GA) [13], which is based on the principles of Darwinian theory [14]. The GA is widely regarded as a foundational optimization algorithm and has demonstrated significant efficiency in solving optimization problems across various fields. Its effectiveness, particularly in the field of structural engineering, has been well-established. The GA’s ability to explore a wide solution space, handle nonlinearity, and efficiently converge toward optimal solutions has made it a popular choice for addressing complex optimization challenges in structural engineering and related disciplines. Its versatility and demonstrated success have solidified its status as a fundamental and influential algorithm in the realm of optimization [15,16].

Although both algorithms possess their strengths and weaknesses, the WOA provides several distinct advantages compared to the GA. One of the primary advantages of the WOA over the GA is its requirement for fewer user-defined parameters. While the GA necessitates the specification of various parameters such as population size, mutation rate, and crossover probability, among others, the WOA simplifies this by only requiring the initial population and the maximum number of iterations. This reduced parameter dependency makes the WOA more user-friendly and accessible, particularly for beginners [12]. Another notable advantage of the WOA is its faster convergence rate. The WOA exhibits a more rapid convergence, requiring fewer iterations to reach the optimal solution compared to the GA. This expedited convergence is attributed to the WOA’s efficient and effective search strategy, which enables it to explore the search space more proficiently. As a result, the WOA offers a time-saving advantage by finding the optimal solution in a shorter timeframe. Moreover, the WOA has a reduced susceptibility to becoming trapped in local optima. Local optima represent points in the search space where the algorithm becomes stuck and cannot find superior solutions. The WOA mitigates this issue by incorporating a mechanism that promotes the exploration of the search space, making it less prone to being trapped in local optima. By encouraging a broader search, WOA enhances its ability to discover global optima, resulting in more robust and optimal solutions [12].

Previous research and experiments have focused on studying the behavior and rotational stiffness of fixed bases. However, the hinged base, particularly under axial tension loads, has received comparatively less attention. A gap in the literature regarding parametric studies that consider varying anchor bolt diameters, their configurations, and base plate thicknesses still exists. Despite the significance of previous studies in understanding this type of base connection, further exploration is necessary. Moreover, simulating the moment–rotation (M-θr) relationship of flexible steel baseplate connections under simultaneous tension and moment is a fascinating area of research. Therefore, numerical investigations using the commercial software ABAQUS were conducted to analyze the steel base plate connections. The finite element (FE) models were verified against previous experimental results. Moreover, numerical findings of a comprehensive parametric investigation were conducted. The studied connections were examined with different configurations, including variations in the number of anchor bolts, diameters of anchor bolts, thickness of the base plate, spacing between the anchor bolts, and applied tension force values. In addition, the results were utilized to employ the whale optimization algorithm (WOA) for deriving a formulation of the moment–rotation (M-Ɵr) relationship. This formulation aimed to simulate the nonlinear rotational behavior exhibited by flexible base connections under combined moment and tension loads. The formulation took into account all the variables considered in the parametric study, enabling a comprehensive representation of the connection’s rotational behavior. The WOA, being a relatively new and promising algorithm, has been applied to various optimization problems. To further validate its effectiveness, the classical genetic algorithm (GA) was utilized for comparison with the WOA results.

2. Finite Element Modeling

The FE models that were developed by Nawar et al. [17], using the commercial software ABAQUS [18], were used in these investigations.

2.1. Selection of Elements and Boundary Conditions

The base plate as well as the anchor bolts were modeled using the C3D8 element [16], while the S4R element was employed for modeling the flanges and web of the steel column [17]. Contact surfaces between the FE parts were utilized, all characterized by “Hard” normal behavior allowing separation and “penalty friction” for tangential behavior [16]. A coefficient of friction of 0.45 was assigned to the base plates [19], while 0.8 was used for the washers and base plates [20]. The steel–steel interaction was modeled using surface-to-surface discretization, which employed a finite sliding formulation and considered a friction coefficient of 0.4.

2.2. Modeling of Material

A bi-linear stress–strain relationship was adopted to represent the stress–strain curve for all structural steel materials [17]. Each steel component was characterized by its unique stress–strain curve based on these mechanical properties. The other steel components were modeled as elastic and rigid materials [17]. Rebar reinforcement was represented with elastic perfectly plastic behavior, with a defined yield stress [17]. In modeling the concrete pedestal, the concrete damage plasticity (CDP) model as outlined in [21] was employed. The failure stresses were defined according to [21] in terms of the compressive and tensile strengths. The stress–strain curves of concrete under compression and tension were defined according to [19].

2.3. FE Verifications

The applied moment (M) was calculated as a function of the applied lateral force multiplied by the column clear height and the corresponding rotation (θ) was calculated by dividing the lateral displacement of the column-free end by the column clear height. The experimental work conducted by Guisse et al. [22] was used to validate the developed FE model. The experimental work was carried out on 12 hinged base plate connections. Six specimens contain four anchor bolts outside column flanges as shown in Figure 1a, and the other six specimens contain two anchor bolts inside flanges as shown in Figure 1b.

Table 1. Specimens’ parameters [22].

Specimen Name	No of Anchor Bolts	Base Plate Thickness (mm)	Axial Comp. Force (kN)
PC2.15.100	2	15	100
PC2.15.600	2	15	600
PC2.15.1000	2	15	1000
PC2.30.100	2	30	100
PC2.30.600	2	30	600
PC2.30.1000	2	30	1000

lists the specimens’ names and variables for each specimen. Each specimen consisted of a steel column with a section HEB 160 and a length of 1050 mm from the lower level of the base plate to the center line of the lateral loading. The steel anchor bolts were 20 mm in diameter and threaded with a 70 mm length and were made from high-strength steel grade 10.9. The mechanical properties of the steel columns, base plates, and anchor bolts are listed in

Table 2. Material properties of the steel components [22].

	Steel Grade	Yield Stress (MPa)	Ultimate Stress (MPa)	Elasticity E (MPa)	Poisson Ratio
Steel Column	S355	464	580	200,000	0.3
Base Plate	S235	280	412	200,000	0.3
Anchor Bolt	10.9	830	1000	200,000	0.3

. The compressive strength of the concrete pedestal was 100 MPa.

Figure 2 shows the deformed shape of the analyzed connections, which contain only two anchor bolts inside the flanges. The failure mode was stress failure on the anchor bolt, which matched with the failure modes of the experimental specimens. Figure 3 provides validations of the FE results in terms of the moment–rotation curves of the column base. The shown results reveal an excellent agreement between the FE and experimental results. The initial stiffness of the moment–rotation curve as well as the progressive yielding of the column base and the ultimate load were accurately predicted by the model.

Nawar et al. conducted experiments on full-scale steel columns featuring hinged base plate connections subjected to axial tension force and bending moments [17]. The details of the tested specimens are depicted in Figure 4. Various configurations of each specimen are detailed in

Table 3. Parameters of each connection [17].

Specimen	Column Profile	N	D (mm)	T (mm)
Connection 1	IPE400	6	20	20
Connection 2		6	20	15
Connection 3		6	25	20
Connection 4		4	25	20
Connection 5		4	20	20
Connection 6		4	20	15

N: Number of the anchor bolt; D: Diameter of the anchor bolt; T: Thickness of the base plate.

.

Table 4. Mechanical characteristics of the steel parts [17].

		E (GPa)	Fu (MPa)	Fy (MPa)	Elongation (%)
Base plate	T = 15 mm	201	443	280	35
	T = 20 mm	198	403	226	37
Anchor bolt	D = 20 mm	208	633	361	28
	D = 25 mm	21	599	346	29

presents the average mechanical properties derived from the tensile tests conducted on the steel components.

In Figure 5, the FE M-Ɵr curves are juxtaposed with those obtained from experimental data. Across the elastic range and leading up to failure, the profiles of the numerical and experimental curves closely resemble each other. Remarkably, there is substantial alignment between the moment–rotation characteristics and the deformed shapes observed in both experimental data and finite element analyses. Therefore, these verified FE models were used to conduct further numerical analysis.

3. Discussions of the FE Results

This section delves into a comprehensive discussion of the FE results. Figure 6 and

Table 5. Parameters of the analyzed connections.

Parameter	Value Range
Thickness of the base plates (T) [mm]	12–30
Diameter of the anchor bolts (D) [mm]	12–30
Number of the anchor bolts (N) [-]	2–6
Spacing (S) [mm]	0–150
Axial tension force (F) [kN]	13, 26, and 38

provide detailed listings of all parameters along with their respective ranges. To encompass the breadth of parameters under scrutiny, each specimen was assigned a distinctive label. For instance, the axial tension ranged from 13 kN to 38 kN, with labels such as Ten1 representing 13 kN, Ten2 representing 26 kN, and Ten3 representing 38 kN. Conversely, the label Comp meant 60 kN axial compression force.

Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 show the FE results in terms of the M-Ɵr curves as well as the deformations and von Misses stresses. The results of the moment–rotation curves show that the beginning of the curve was almost a horizontal segment, which means that there is no rotational resistance at the start of loading. This phenomenon happened because of the gap, which was formed between the steel base plate and the RC pedestal because of the initial tension force as shown in Figure 8. The gap between the base plate and the RC pedestal was formed because of two reasons. The first reason was because of the elongation in the anchor rods. The second one was because of the deformation of the base plate where the force in the column section represented a concentrated load on the base plate thickness.

Figure 9 shows that the gap between the steel and RC pedestal decreased because of the increase in the anchor bolt diameter. The increase in bolt diameter, bolt number, and base plate thickness led to a reduction in the initial gap between the steel base plate and the RC pedestal. Figure 10 shows that the beginning of the curve has more resistance than the one in Figure 8 because of the increase in the anchor diameter, which reduced the gap before rotation.

Not only base plate dimension and anchor bolt diameter control the value of the gap between the steel base and the RC pedestal but also the number of anchor bolts and the distance between them. Figure 11 shows that the increase in the number of bolts with the same diameter and the increase in the space between them led to a reduction in the gap and subsequently increased the initial stiffness. Therefore, the relationship was initiated with a horizontal line, but with some resistance. Figure 12 also confirms that the increase in the bolt number and spacing makes the connection exhibit more resistance which affects the rotational stiffness and moment capacity.

Table 6. FE results for the analyzed specimens under axial tension = 3.8 ton.

Connection	Rotational Stiffness (kN.m/rad)				* M (KN.m)	* E ×10−3 (kN.m.rad)	Tension Force in the Anchor Bolt (kN)
	K0	K1	K2	* K			Comp. Side	Middle	Tension Side
B2-d12-S000-t12	73	138	163	64	0.43	1	-	40.9	-
B4-d12-S050-t12	482	4178	1553	1761	11.74	32	32.7	-	36.2
B4-d12-S100-t12	1532	3298	881	1524	10.16	31	30.5	-	37.1
B4-d12-S150-t12	3472	3484	476	1677	11.18	54	17.5	-	36.7
B2-d18-S000-t12	119	4040	2822	2118	14.12	29	-	84.9	-
B4-d18-S050-t12	2021	6802	3602	4990	33.27	98	72.3	-	94.9
B4-d18-S100-t12	2681	9000	2485	5946	39.64	143	45.2	-	85.4
B4-d18-S150-t12	6533	13,344	7041	6386	42.58	191	24.0	-	84.4
B2-d24-S000-t12	186	4621	3005	3137	20.92	54	-	152.3	-
B4-d24-S050-t12	1234	8823	4581	6653	44.35	142	96.4	-	158.8
B4-d24-S100-t12	3740	11,758	7173	9108	60.72	206	47.6	-	164.2
B4-d24-S150-t12	10,414	17,149	7329	12,577	83.85	328	24.1	-	160.7
B2-d30-S000-t12	319	6587	3454	4166	27.77	79	-	223.2	-
B4-d30-S050-t12	1664	11,304	5404	8359	55.73	185	91.2	-	237.6
B4-d30-S100-t12	6224	16,257	9722	11,134	74.23	261	53.2	-	255.9
B4-d30-S150-t12	12,583	23,925	16,457	16,183	107.89	422	24.6	-	263.2
B2-d12-S000-t18	75	166	163	75	0.50	2	-	41.3	-
B4-d12-S050-t18	566	4280	1985	1988	13.25	45	35.6	-	35.7
B4-d12-S100-t18	1951	6273	1146	1822	12.15	47	32.3	-	35.5
B4-d12-S150-t18	10	4034	1635	1647	10.98	45	16.8	-	36.2
B2-d18-S000-t18	120	4684	1263	2406	16.04	48	-	81.7	-
B4-d18-S050-t18	1056	9921	3153	6391	42.61	142	74.7	-	82.6
B4-d18-S100-t18	4222	12,736	1872	6289	41.93	171	53.4	-	82.3
B4-d18-S150-t18	7802	16,216	6738	6437	42.91	204	29.6	-	82.5
B2-d24-S000-t18	978	7009	829	5486	36.58	101	-	149.9	-
B4-d24-S050-t18	2501	12,995	9163	10,211	68.08	217	85.0	-	152.2
B4-d24-S100-t18	7404	16,646	5156	11,633	77.55	290	55.9	-	151.7
B4-d24-S150-t18	13,894	24,320	15,758	12,769	85.13	381	30.1	-	149.2
B2-d30-S000-t18	297	9597	5898	7163	47.76	140	-	227.4	-
B4-d30-S050-t18	2262	15,512	8675	12,537	83.58	275	89.7	-	239.2
B4-d30-S100-t18	6676	19,768	11,599	16,494	109.96	383	51.4	-	246.0
B4-d30-S150-t18	15,664	27,923	6443	20,224	134.83	545	18.5	-	237.1
B2-d12-S000-t24	75	146	161	75	0.50	2	-	38.3	-
B4-d12-S050-t24	374	5068	2088	1960	13.07	35	34.9	-	35.0
B4-d12-S100-t24	1162	4431	1174	1723	11.49	38	29.0	-	35.1
B4-d12-S150-t24	10	4104	1292	1660	11.06	45	17.0	-	35.1
B2-d18-S000-t24	115	5967	1918	2483	16.55	55	-	79.8	-
B4-d18-S050-t24	1151	11,040	4351	6734	44.89	159	67.6	-	79.6
B4-d18-S100-t24	3819	13,435	3432	6461	43.07	181	56.3	-	79.1
B4-d18-S150-t24	8855	18,751	5118	6474	43.16	208	30.8	-	80.8
B2-d24-S000-t24	179	7843	540	6121	40.81	127	-	144.9	-
B4-d24-S050-t24	4879	15,013	8476	11,430	76.20	257	89.2	-	140.2
B4-d24-S100-t24	5843	19,327	6091	12,050	80.33	322	64.3	-	146.5
B4-d24-S150-t24	12,968	24,584	4419	12,821	85.47	397	31.8	-	143.0
B2-d30-S000-t24	292	10,489	2618	9042	60.28	180	-	206.1	-
B4-d30-S050-t24	2621	18,152	13,632	15,303	102.02	335	80.1	-	213.7
B4-d30-S100-t24	7940	23,439	14,072	18,488	123.25	454	54.8	-	234.3
B4-d30-S150-t24	18,210	33,780	24,963	20,511	136.74	587	18.3	-	231.3
B2-d12-S000-t30	75	224	163	64	0.42	1	-	39.0	-
B4-d12-S050-t30	594	6872	1985	1982	13.22	49	35.1	-	35.3
B4-d12-S100-t30	1186	3273	1086	1728	11.52	38	28.7	-	35.0
B4-d12-S150-t30	0	4062	556	1664	11.10	45	18.3	-	35.0
B2-d18-S000-t30	111	6564	1265	2484	16.56	58	-	78.3	-
B4-d18-S050-t30	1179	11,390	3744	6883	45.89	168	75.6	-	78.7
B4-d18-S100-t30	4008	14,383	6637	6519	43.46	183	56.5	-	78.4
B4-d18-S150-t30	8724	18,944	6614	6534	43.56	213	31.7	-	79.0
B2-d24-S000-t30	170	9183	3305	5976	39.84	133	-	140.5	-
B4-d24-S050-t30	1935	16,106	5590	11,932	79.55	278	96.1	-	140.2
B4-d24-S100-t30	6309	19,730	4005	12,200	81.34	338	67.0	-	141.0
B4-d24-S150-t30	13,675	26,838	7193	12,842	85.61	399	32.6	-	142.2
B2-d30-S000-t30	1539	12,460	9433	9866	65.78	203	-	210.2	-
B4-d30-S050-t30	2835	19,536	14,265	16,620	110.80	369	86.8	-	212.6
B4-d30-S100-t30	9434	25,440	6117	18,664	124.43	483	59.2	-	223.4
B4-d30-S150-t30	20,392	35,450	23,153	20,573	137.15	604	17.8	-	223.9
B6-d12-S100-t12	1027	7093	3095	3230	21.53	56	17.0	40.4	38.4
B6-d18-S100-t12	2700	10,900	6327	8085	53.90	184	30.7	74.3	88.2
B6-d24-S100-t12	5322	15,201	7782	10,710	71.40	250	36.5	115.9	163.8
B6-d30-S100-t12	9114	18,001	9113	13,042	86.94	313	39.1	127.4	251.4
B6-d12-S100-t18	1223	8936	3842	3594	23.96	72	21.9	35.3	35.6
B6-d18-S100-t18	2428	14,051	5812	9122	60.82	177	26.9	65.4	83.6
B6-d24-S100-t18	10,106	17,561	5213	14,910	99.40	352	31.9	100.2	158.6
B6-d30-S100-t18	14,482	20,691	10,953	18,421	122.81	433	32.5	128.5	245.4
B6-d12-S100-t24	1105	7806	2990	3690	24.60	77	25.0	53.9	61.2
B6-d18-S100-t24	2612	16,966	6955	9591	63.94	195	29.5	68.8	79.6
B6-d24-S100-t24	13,179	21,523	8037	16,724	111.49	404	31.1	123.8	141.7
B6-d30-S100-t24	18,184	24,185	11,609	21,556	143.70	509	24.1	118.2	233.6
B6-d12-S100-t30	1129	10,017	3993	3721	24.81	79	41.3	50.6	50.9
B6-d18-S100-t30	6819	17,692	8018	10,553	70.35	277	39.8	77.5	78.9
B6-d24-S100-t30	14,969	23,751	15,093	17,224	114.83	427	33.0	109.6	140.5
B6-d30-S100-t30	20,552	26,036	8840	23,137	154.25	553	21.1	124.0	217.5
B6-d12-S150-t12	3818	10,166	3074	3655	24.36	100	13.3	79.5	85.3
B6-d18-S150-t12	6919	16,626	5905	9759	65.06	256	14.6	91.5	95.1
B6-d24-S150-t12	13,393	21,640	13,099	15,151	101.00	385	14.4	120.1	165.8
B6-d30-S150-t12	19,621	26,488	12,075	19,297	128.65	490	10.2	136.7	265.7
B6-d12-S150-t18	2565	11,873	3726	3694	24.63	87	20.7	55.4	55.6
B6-d18-S150-t18	10,559	19,933	6623	10,672	71.15	297	16.1	82.3	82.5
B6-d24-S150-t18	18,537	26,489	12,442	17,787	118.58	468	8.5	116.9	150.1
B6-d30-S150-t18	25,410	27,597	5928	24,506	163.38	621	1.9	120.0	242.5
B6-d12-S150-t24	2412	12,637	4741	3715	24.77	89	15.2	63.5	63.6
B6-d18-S150-t24	8778	21,818	12,410	10,732	71.55	308	16.7	76.1	80.7
B6-d24-S150-t24	13,598	28,692	7757	18,729	124.86	508	5.0	125.5	146.9
B6-d30-S150-t24	28,957	34,418	10,600	26,251	175.01	684	0.4	124.4	234.2
B6-d12-S150-t30	2677	12,449	3888	3721	24.80	91	14.3	58.1	58.3
B6-d18-S150-t30	8776	22,284	7052	10,899	72.66	317	17.4	78.1	78.6
B6-d24-S150-t30	13,834	28,879	8096	19,124	127.49	526	3.9	123.0	141.9
B6-d30-S150-t30	31,040	36,153	21,764	27,039	180.26	715	0.3	128.8	224.6

* K: Rotational stiffness at H/150 of drift; * M: Moment at H/150 of drift; * E: Energy at H/150 of drift.

presents the calculated elastic rotational stiffness, moment resistance, and energy absorption for each analyzed connection [23]. The elastic performances of the analyzed connections were examined by fitting two piecewise linear functions to the M-Ɵr curves [24], as illustrated in Figure 13. In the initial branch, the stiffness K0 represents the rotational stiffness of the initial curve, occurring just before the steel and RC pedestal fully re-contact during lateral drift. This phenomenon happened because the axial strain in the anchor bolt and the bending deformation of the base plate due to the initial axial force before the bending moment make a separation between the base plate and the RC pedestal. Therefore, the re-contact between them needs a lateral deformation with small resistance. This resistance depends on the thickness of the base plate, anchor number, and anchor bolt diameter. The stiffness K1 is determined by the elastic rotational stiffness after the contact. The stiffness K0 is always less than the value of K1 and in some samples may be just equal. Nonetheless, the optimal fit, which minimizes the discrepancy between the idealized curve and the FE one, is employed to ascertain the stiffness of K2.

At the tension side of the moment, the anchor bolts exhibited dissimilarities in the tensile stresses compared to the specimens under axial compression, despite sharing identical arrangements of bolts and base plate thicknesses but having varying anchor bolt diameters, as illustrated in Figure 14a. The initial tension force made different gap distances between the steel base and the RC pedestal due to the diameter and the number of anchor bolts. Conversely, at the compression side, the anchor bolts exhibited reduced tensile stresses with increasing diameters, as depicted in Figure 14b. The tensile behavior of the anchor bolts for both four- and six-bolt arrangements remained consistent (see Figure 15). However, there was a notable difference: the middle anchor in the connection with a six-bolt arrangement demonstrated higher stiffness, resulting in a 66% increase in elastic moment resistance.

The rotational stiffness experienced its highest surge, reaching 137%, with an anchor bolt diameter of 30 mm when the base plate thickness expanded from 12 mm to 30 mm, as demonstrated in Figure 16a. However, the rate of increase diminishes as the anchor bolt diameter decreases, a trend observed across various anchor diameters and base plate thicknesses. Figure 16b–d highlight that widening the spacing between anchors diminishes the impact of base plate thickness (in the case of four-bolt arrangements), while Figure 16e,f depict the same trend for six-anchor bolt configurations. This decline occurs because the column flange provides support to the base plate against bending as the anchors draw closer to the column flanges.

Figure 17 illustrates how the rotational stiffness is influenced by both the arrangement and diameter of anchor bolts at a drift level of H/150, considering various base plate thicknesses. These findings underscore the substantial impact of anchor bolt diameter and arrangement on rotational stiffness values. Enlarging the number of bolts and adjusting their spacing resulted in notable improvements in rotational stiffness.

4. Optimization Algorithms

The whale optimization algorithm (WOA) employed in this formulation is inspired by the unique bubble-net foraging behavior observed in humpback whales [12]. Humpback whales are known to employ a distinctive feeding technique where they encircle and hunt fish swarms in the ocean. This specialized technique involves the whales forming a spiral-shaped path around the target fish school, while simultaneously creating a net of bubbles to corral and capture the fish, as illustrated in Figure 18.

Notably, during this hunting process, the humpback whales exhibit an intriguing behavior where they lead the fish school in a spiraling trajectory towards the water surface. Throughout this spiral path, the whales gradually reduce the radius of the spiral, slowly enclosing the fish school. When the fish school reaches proximity to the water surface, the humpback whales then initiate their final attack and capture the prey.

The WOA algorithm employed in this formulation directly emulates this unique hunting mechanism of humpback whales through three primary stages or operators: encircling the prey, bubble-net attacking, and searching for the prey. These stages closely mimic the observed behavior of humpback whales during their bubble-net feeding process. By replicating this spiral optimization approach, the WOA is able to effectively capture the complex nonlinear rotational response of the flexible base connections being modeled.

In the WOA, whales navigate through an n-dimensional search space, where the value of n corresponds to the number of variables being considered. In essence, each whale represents a potential solution comprising n variables. The objective is to obtain the global solution within the search space, which involves updating the position of each solution using the following equations:

$$\overrightarrow{X}(t+1)=\overrightarrow{X*}(t)-\overrightarrow{A}·\left|\overrightarrow{C}·\overrightarrow{X}*(t)-\overrightarrow{X}(t)\right|$$

(1)

where t indicates the current iteration, $\overrightarrow{X}*$ represents the prey’s best solution position vector obtained so far, $\overrightarrow{X}$ is the prey’s probable position vector, · signifies an element-by-element multiplication, and ∣ ∣ represents the absolute value. Moreover, $\overrightarrow{A}$ and $\overrightarrow{C}$ are the coefficient vectors that are calculated and modified for each dimension and at every iteration, as demonstrated by the subsequent equations:

$$\overrightarrow{A}=2\overrightarrow{a}·\overrightarrow{r}-\overrightarrow{a}$$

(2)

$$\overrightarrow{C}=2·\overrightarrow{r}$$

(3)

where $\overrightarrow{a}$ undergoes a linear decrease from a value of two to zero throughout the optimization process (yielding values for each dimension) and $\overrightarrow{r}$ represents a random vector, with values ranging from zero to one (generated independently for each dimension).

Equation (1) facilitates the movement of whales or solutions to nearby areas in the proximity of prey. The components of the equation possess random values, which result in a movement pattern resembling a hyper-rectangle path around a fish school, as illustrated in Figure 19.

Additionally, the spiral movement of the whales in the WOA can be mathematically represented by the following equation:

$$\overrightarrow{X}(t+1)=\overrightarrow{D^{\prime }·}{e}^{bt}\cos (2a\pi l)+\overrightarrow{X*}(t)$$

(4)

The variable $\overrightarrow{D^{\prime }}=\left|\overrightarrow{X}*(t)-\overrightarrow{X}(t)\right|$ represents the distance between the t-th whale and the prey, which can be interpreted as the best solution obtained thus far. The constant “b” is used to define the shape of the logarithmic spiral. The operator “.” denotes element-by-element multiplication, and “l” represents a random number ranging between −1 and 1. Figure 20 depicts the impact of the spiral motion on the movement of whales. The spiral pattern influences the trajectory and direction of whale movement within the optimization process.

The process of simulating the spiral movement and encircling of the prey in the WOA algorithm is achieved through the utilization of the following equation:

$$\mathrm{X}\left(\mathrm{t}+1\right)=\left\{\begin{array}{cc}\mathrm{X}*\left(\mathrm{t}+1\right)-\mathrm{A}·\mathrm{D}& P<0.5\\ \overline{\mathrm{D}}·{\mathrm{e}}^{\mathrm{b}\mathrm{t}}·\cos \left(2\mathrm{a}\mathsf{\pi }\mathrm{l}\right)+\overline{\mathrm{x}\mathrm{*}}\left(\mathrm{t}\right)& \mathrm{p}\ge 0.5\end{array}\right.$$

(5)

The variable “p” represents a randomly generated number within the range of zero to one. By incorporating this equation, there is a 50% probability assigned to the position updating for each value in every dimension and iteration.

Figure 21 illustrates a summary of the classical GA. It provides an overview of the key components and steps involved in the GA methodology, showcasing its fundamental structure and operation. To enhance verification and assess the capabilities of the WOA and GA, a comparative analysis is conducted, examining the results obtained from both algorithms. This comparison aims to provide further validation and insights into the performance and effectiveness of the WOA and GA in solving the given problem or optimization task.

5. Deducting the Rotational Stiffness Equation

The current research endeavors to develop a comprehensive equation that describes the relationship between the moment M, relative rotation Ɵr, and other relevant parameters. The objective is to establish a unified equation that can effectively predict and analyze the M-Ɵr behavior in base connections.

In the present study, the optimal formula format is derived from the research conducted by Nawar et al. [16]. This optimal formula format was utilized and updated to obtain the M-Ɵr formulas for flexible base connections subjected to tension and moment loads as shown in Equation (6). The investigation involved the application of both the GA and WOA to derive these formulas. The objective was to explore the capabilities and performance of both algorithms in obtaining accurate M-Ɵr formulas for flexible base connections under the specified loading conditions.

$${\mathsf{\theta }}_{{\mathrm{r}}^{″}}={\mathsf{\theta }}_{\mathrm{r}}(1+\frac{{\mathrm{C}}_{12}}{{\mathrm{C}}_{13}}\ast \mathrm{K}\ast \mathrm{F})$$

(6)

$${\mathsf{\theta }}_{\mathrm{r}}=100\ast [{\mathrm{C}}_1{\ast {10}^{{-\mathrm{C}}_4{10}^{-3}}\left(\mathrm{K}\mathrm{M}\right)}^1+{\mathrm{C}}_2\ast {10}^{-{\mathrm{C}}_5{10}^{-3}}{\left(\mathrm{K}\mathrm{M}\right)}^3+{\mathrm{C}}_3{\ast {10}^{{-\mathrm{C}}_6{10}^{-3}}\left(\mathrm{K}\mathrm{M}\right)}^5$$

(7)

$$\mathrm{K}={\mathrm{C}}_{11}{10}^{-3}\ast {\mathrm{N}}^{{-\mathrm{C}}_7{10}^{-3}}\ast {\mathrm{D}}^{{-\mathrm{C}}_8{10}^{-3}}\ast {\mathrm{S}}^{{-\mathrm{C}}_9{10}^{-3}}\ast {\mathrm{T}}^{{-\mathrm{C}}_{10}{10}^{-3}}$$

(8)

where ${\theta }_{r^{″}}$ is the relative rotation considering moment and tension loads simultaneously; ${\theta }_r$ is the relative rotation considering moment only; F is the tension force value in kN; M is the applied moment value in kN.m; C1 to C13 are constants; K is a standardization constant depending on the considered variables, i.e., bolt’s diameter and number of bolts, spacing between bolts, and the base plate thickness as shown in the following equation; N is the number of bolts; D is the bolt diameter in mm; S is the spaces between bolt rows in mm, where S = 1 for two bolts only in the “one-row” case; and T is the thickness of the base plate in mm. The algorithms utilize variables C₁ to C₁₃ to obtain optimal values for these constants and form the final optimal equation that governs the M-Ɵr relationship. This relationship represents the nonlinear rotational stiffness behavior of the flexible base connection exposed to simultaneous tension and moment loads.

Each of the optimization algorithms, the WOA and GA, work by iteratively suggesting different values for the variables within a predetermined range. These variable values are then substituted into the previously mentioned moment–rotation (M-θr) equation to generate a corresponding moment–rotation curve. The algorithms compare this generated M-θr curve against the M-θr curve obtained from the previous numerical study [16]. This comparison allows for the algorithms to evaluate the fitness or accuracy of the suggested variable values in terms of how well they are able to replicate the reference M-θr relationship. The algorithms then proceed through their respective iterative processes, using techniques such as updating position vectors (WOA) or performing crossover and mutation operations (GA), to generate new candidate variable values. This cycle of substituting values, generating M-θr curves, evaluating fitness, and updating variables continues until the algorithms converge on the optimal variable values that best fit the reference moment–rotation behavior.

The variables in the optimization algorithms are constrained within the range of 0 to 10,000. These algorithms aim to identify the optimal values for these variables within this specified limit. By searching and evaluating different combinations of variable values within this range, the algorithms strive to determine the most favorable configuration that maximizes the desired objective or minimizes the defined cost function. The adopted fitness function that requires minimization by the optimization algorithms is represented by the following equation:

$$\mathrm{F}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}=\sum _{\mathrm{n}=1}^{\mathrm{n}=282}\sum _{\mathrm{i}=1}^{\mathrm{i}=\mathrm{M}}\left|{\mathsf{\theta }}_{\mathrm{r}\ast }-{\mathsf{\theta }}_{{\mathrm{r}}^{″}}\right|$$

(9)

where n represents the connection number, i is the data number of the moment–rotation relation for each connection in the parametric study, and M signifies the total amount of data in each connection. Additionally, the relative rotations observed in the parametric study are denoted as ${\mathsf{\theta }}_{\mathrm{r}\ast }$, while the proposed equation captures these rotations as ${\mathsf{\theta }}_{{\mathrm{r}}^{″}}$. The studied relationship between M and Ɵr is confined to the interval of 0 to 15 radians for Ɵr. This range is deemed suitable to encompass the relative rotation experienced during the loading procedure. In the case of the algorithms, the population size or the initial number of search agents is set to 5000, while the maximum number of iterations or generations is limited to 100. For the genetic algorithm (GA) specifically, the crossover rate is defined as 0.9, and the mutation rate is set at 0.05. As evident, the minimization of the fitness function plays a crucial role in identifying the optimal variables that result in the smallest disparity between the proposed formula and the parametric study.

Table 7. The optimum variable values that resulted from algorithms.

Constant	GA	WOA
C1	15,481	706
C2	864	1052
C3	15,157	1046
C4	8293	5316
C5	3967	3855
C6	6173	14,837
C7	767	862
C8	1831	1886
C9	36	32
C10	63	20
C11	8665	8696
C12	8182	12,453
C13	6401	8976

displays the values of the optimal variables obtained from both the GA and the WOA when considering various formulas. On the other hand,

Table 8. The fitness value of each algorithm.

Algorithm	Fitness Value
WOA	29,189
GA	30,403

presents the corresponding fitness values for each case. Based on the optimum variables specified in Table 7, the polynomial formula obtained through the WOA can be represented in the following equations:

$${\mathsf{\theta }}_{{\mathrm{r}}^{″}}={\mathsf{\theta }}_{\mathrm{r}}(1+1.387\ast \mathrm{K}\ast \mathrm{F})$$

(10)

$${\mathsf{\theta }}_{\mathrm{r}}=0.341\ast {\left(\mathrm{K}\mathrm{M}\right)}^1+14.69\ast {\left(\mathrm{K}\mathrm{M}\right)}^3+1.52\mathrm{E}-10\ast {\left(\mathrm{K}\mathrm{M}\right)}^5$$

(11)

$$\mathrm{K}=8.696\ast {\mathrm{N}}^{-0.862}\ast {\mathrm{D}}^{-1.886}\ast {\mathrm{S}}^{-3.20\mathrm{E}-02}\ast {\mathrm{T}}^{-2.0\mathrm{E}-02}$$

(12)

Figure 22 illustrates the correlation between M-Ɵr for a specific group of base connections, which were selected from a larger pool of 282 connections examined in the study. The figures incorporate connection labels that are assigned to each base connection represented. These labels fulfill the purpose of identifying and associating each base connection with its corresponding data, enabling a clear reference for analysis and comparison within the figures. By including these labels, the figures provide a convenient way to differentiate and track individual base connections, facilitating a comprehensive understanding of the rotational stiffness behavior across different connections. The beam label format is BA-DB-SC-TD-FE, where B indicates “base connection”, A represents the number of bolts, D refers to “diameter”, B represents the diameter value in mm, S signifies “space”, C is the space value in mm, T represents “thickness of base plate”, D is the base plate thickness in mm, F refers the tension force, and E is the tension value in kN.

These results display the rotational stiffness curves of flexible base connections under combined moment and tension loads. Each figure includes the M-Ɵr curve obtained from the parametric study (FE analysis), as well as the M-Ɵr curve derived from the WOA and GA. Furthermore, to assess the validity of the modified equation (Equation (10)) that incorporates the tension force in the M-Ɵr relationship, the M-Ɵr curve for the WOA with a tension force of F = 0 is also provided. This additional curve evaluates the impact of tension on the rotational stiffness of the connections. Moreover, these figures aim to visually compare the performance of the WOA and GA in representing the rotational stiffness behavior of flexible base connections under combined moment and tension forces. These comparisons provide insights into which algorithm better represents the rotational stiffness behavior and can guide the selection of the most suitable algorithm for accurately predicting the rotational stiffness in practical scenarios involving combined moment and tension loading conditions.

Figure 23 presents the fitness values for each of the 282 connections analyzed using both the GA and WOA. By examining these values, the performance and effectiveness of both the GA and WOA in optimizing the connections and determining the suitability of each connection in relation to the objectives set by the algorithms can be evaluated. Based on these findings, the formulas derived from the WOA and GA algorithms are effective in simulating the M-Ɵr relationship, as compared to the curves obtained from the parametric study (FE analysis). These results validate the accuracy and reliability of the suggested formula in capturing the rotational stiffness behavior of flexible base connections under combined moment and tension loads. The developed results visually demonstrate the effectiveness of the proposed formula, particularly when considering the value of the tension force. Furthermore, the WOA algorithm demonstrates superior performance when compared to the GA algorithm.

Upon comparing the WOA and WOA (F = 0) curves, it becomes evident that the presence of tension force leads to a reduction in the rotational stiffness of the base connection, rendering it more flexible. The inclusion of tension force in the analysis influences the behavior of the connection, resulting in a reduction in its resistance to rotation. This observation highlights the significant impact of tension force on the overall flexibility and rotational behavior of base connections.

The findings suggest that the optimal variables listed in Table 7 are not consistent across algorithms, as demonstrated by the varying slopes in Figure 22. This can be attributed to the extensive search space and the multitude of potential combinations of variable values needed to achieve a satisfactory solution. If anything, this highlights the complexity involved in developing a formula that accurately simulates connection stiffness. The results emphasize the challenges associated with capturing the intricate behavior of connection stiffness and the need for comprehensive analysis and consideration of multiple factors.

6. Conclusions

The present research focuses on analyzing the rotational stiffness characteristics of a flexible base connection subjected to simultaneous moment and tension forces. Building upon the findings of an extensive parametric study, the rotational stiffness behavior of the flexible base connection is deduced through the M-Ɵr relation. The study explores the application of the WOA and GA to obtain the M-Ɵr relation, taking into account various variables examined in the previous parametric study, including the number of bolts, bolt diameter, spacing between bolts, base plate thickness, and tension force magnitude.

• The WOA and GA have demonstrated their effectiveness as optimization algorithms for deriving an acceptable M-Ɵr equation for flexible base connections, taking into account multiple parameters, including the applied tension force.
• The WOA algorithm exhibits superior performance when compared to the results obtained from the GA.
• The actual flexible behavior of the base plate connections is a complex and intriguing area of research in structural engineering, as it is complicated and affected by many different factors.
• The presence of tension force plays a crucial role in the M-Ɵr equation of base connections. It leads to a reduction in the rotational stiffness of the connection, making it more flexible.
• The rotational stiffness, as depicted in the M-Ɵr relation, exhibits high sensitivity to various parameters, including the number of bolts, bolt diameter, and other relevant factors.
• The formula derived through the application of the WOA can be utilized as a design aid in engineering to calculate the rotational stiffness for a base connection subjected to a combined moment and tension force to facilitate the design process.