A Design Model for Urban Single-Lane Roundabouts with Offset Approaches

Abstract

The roundabout design procedures specified in current standards and guidelines presuppose that the centrelines of the approach legs intersect at right angles at the geometric centre of the roundabout. In urban areas, this requirement cannot always be met due to fixed structural constraints along the approaches. Here, a lateral or radial offset of the approach legs from the roundabout’s geometric centre is required. This offset plays an important role in roundabout design, as it affects the roundabout’s ability to control vehicle speed. This study investigates the effects of a lateral approach leg offset on the geometric design of urban single-lane roundabouts and the driving speeds through them. Accordingly, speed analyses were conducted for numerous theoretical roundabouts with outer radii between 15 and 25 m, designed based on swept path analysis results, were conducted. The research results showed that it is possible to offset approaches laterally on roundabouts with outer radii between 15 and 25 m, depending on the design vehicle, and that the allowable offset values increase proportionally with the roundabout outer radius. The analysis results were used to create a design model for urban single-lane roundabouts with lateral approach leg offsets enabling their adaptation to spatial constraints while maintaining safe operating speeds.

1. Introduction

The main advantages of single-lane roundabouts in road networks, compared to conventional intersections, are increased traffic safety, lower air pollution, and lower maintenance costs. Previous studies have reported that roundabouts, compared to signalised intersections, reduce the emissions of carbon monoxide by 21–42%, emissions of carbon dioxide by 16–59%, emissions of oxides of nitrogen by 20–48%, emissions of hydrocarbons by 18–65%, and the noise emission from 1 to 4 dB(A) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. Safety benefits of single-lane roundabouts stem from their reduced number of potential conflict points between vehicles compared to conventional intersections, and from their geometry, which results in deflected vehicle trajectories that force drivers to reduce their speed, thereby lowering both crash probability and severity [17,18]. A meta-analysis [19] of the impact of converting conventional unsignalised intersections to single-lane roundabouts on safety found that such conversions are associated with a reduction of about 65% in fatal accidents and about 40% in accidents with injuries. According to a Danish study [20], converting intersections to roundabouts led to reductions in of 27% crashes and 60% in injuries. The number of fatal accidents fell by 87%, and the number of crashes with property damage fell by 16%. The number of bicycle crashes and injured cyclists increased by 65% and 40%, respectively. According to studies conducted in the US, converting conventional intersections to single-lane roundabouts reduced the number of fatal and serious-injury accidents by over 80% [21]. It led to a 51% reduction in the total number of crashes [18]. The studies focused on the conversion of unsignalised intersections and have shown a significant reduction in the number and severity of accidents. The aim of the studies [20,22,23,24,25] was to close this gap by evaluating the safety effectiveness of converting signalised intersections into roundabouts. Despite the safety benefits that roundabouts offer, accidents still occur. A study [26] conducted in Italy at 15 urban roundabouts concluded that the factors contributing to accidents were primarily related to the roundabout’s geometric design, markings, and signage. This study emphasised the role of a moderate deflection radius and a large deviation angle in the design of new roundabouts, as improving these factors in existing roundabouts can be quite costly. Another study from Italy [27] was carried out to determine the geometric characteristics of roundabouts that influence the perception of safety during typical driving manoeuvres (entry, circulation, exit). The study concluded that users clearly prefer single-lane roundabouts. This can be described by the fact that users perceive the highest level of safety in configurations with one entry, one exit and one roundabout lane.

The roundabout design process is specified by national guidelines, standards, and recommendations [28,29,30,31,32,33,34,35,36,37,38,39]. As shown in Figure 1, the key elements of an urban single-lane roundabout are the splitter islands, pedestrian and cycle crossings, entry and exit radii, entry and exit widths, central island, apron, and circulatory roadway width.

The roundabout design process is highly iterative and consists of the initial design, performance checks (swept path analysis for the design vehicle movements, speed analysis, and visibility tests), and the final design of the elements. Swept path analysis is needed to ensure that design vehicles can safely navigate the roundabout. This analysis, usually conducted using specialised software, yields the required entry, exit, and circulatory lane widths based on the characteristics of the selected design vehicle.

Speed analysis is usually conducted by measuring the roundabout geometric characteristics and checking the obtained deflection [31,32,37,38], by measuring the geometric characteristics of the roundabout and calculating the path radii and vehicle speeds [30,39], or by drawing the fastest vehicle path, measuring the path radii and estimating the vehicle speed based on the radii [28,29,33,34,35,36]. If the initial roundabout geometric design does not meet the requirements of the selected design vehicle and required vehicle speed, the initial design is revised. Predicting speeds at roundabouts in urban areas, where severe spatial and environmental constraints dictate layouts that do not conform to design standards, is challenging and can lead to inconsistent speed analysis results. As a result, standardised speed prediction models have been tested in several studies. For instance, design and operating speeds were evaluated for several urban roundabouts in the Municipality of Cuneo, Italy [40]. The observed speeds were comparable to those derived from the US guidelines’ speed model [34], despite certain discrepancies that arose from local driver behaviour, vehicle performance, and design practices. Only one roundabout presented in this study strictly adhered to the Italian design standards [38]. Consequently, this was the only case in which the variance between observed speeds and those predicted by the US guidelines [34] remained below 10%. According to the findings of a study [41] conducted on urban roundabouts in Zagreb, Croatia, deviations between design speeds calculated using US guidelines [33] and measured speeds ranged from −45.69% to +13.07%. As in the Italian study, these deviations were attributed to various factors, including intersection geometry, vehicle characteristics, and driver behaviour. Another recent study [42] analysed connected vehicle (CV) trajectory speeds at 56 roundabouts in Carmel, Indiana. Comparing passenger vehicle speeds with theoretical models revealed that certain aspects of the US guideline speed model [36] require updating. Specifically, future models should account for lower entry speeds and higher speeds within the circulatory and departure sections for vehicles approaching slowly. Notably, however, deceleration and acceleration values remained consistent across both the theoretical approach and the sampled data, suggesting that current modelling of speed change rates remains accurate.

The initial geometric design of the roundabout elements begins with the horizontal alignment of the approaches (Figure 2, dash-dot lines) relative to the geometric centre of the intersection, i.e., the centre of the central island (Figure 2, point (C)). In real-life scenarios, this alignment can be radial or offset laterally [36,39]. With radial alignment, the centerline of each leg crosses the geometric centre of the roundabout, i.e., the centre of the central island, as shown in Figure 2a. Lateral alignment offset, shown in Figure 2b, can be applied to the left or right relative to the geometric centre. The approach centreline alignment influences vehicle paths through the intersection and is therefore an important input for swept path analyses. It also influences the deflection of the central island, providing the required vehicle speeds on entry, around the central island, and on the exit.

According to the analysed roundabout design guidelines, standards, and recommendations [28,29,30,31,32,33,34,35,36,37,38,39], optimal roundabout design requires that the centrelines of all approaches intersect at right angles at the roundabout’s geometric centre. In the design practice, this alignment cannot always be achieved due to spatial restrictions, especially in urban areas. In these cases, approach centrelines are offset; offset placement provides flexibility in roundabout design, allowing engineers to adapt the roundabout geometry to specific site conditions and traffic patterns. Approach offset should not result in a tangential entry and exit. A slight left offset is suitable, as it increases vehicle path deflection, thereby reducing vehicle speed. Still, a right offset could result in a tangential entry and a high entry speed. Tangential entry increases entry vehicle speed and crash severity between vehicles in the roundabout and those entering it. Generally, offsetting the approach centerline to the right should be avoided. Still, it can be applied if the correct entry speed can be achieved and other geometric elements of the roundabout can be formed [28,29,35,36,39]. The Croatian guidelines [39] recommend that the maximum offset be 15% of the outer roundabout radius and that the angle between two consecutive approach centrelines should allow the right carriageway edge to be designed with three distinct consecutive arcs, as shown in Figure 1.

Recently, multi-objective optimisation models for the design of single-lane roundabouts have been proposed in studies [43,44]. They yield optimal roundabout geometric dimensions, thereby reducing the number of design iterations. Nevertheless, the aforementioned performance checks should always be conducted, as these models have not yet been included in guidelines or standards, and their effectiveness still needs to be tested.

Design limits for intersection angles between approach legs are described in [45], while the influence of the radial offset on the geometric design of roundabouts is described in [46]. This investigation of radial offset was based on the analysis of the design vehicle path and the fastest vehicle path, and the results showed that it is possible to radially offset roundabout approaches to the right and left without compromising safety. The lateral offset of the approach was also analysed using predefined roundabout geometry in the study [47]. In this study, a model was developed for the geometric design of single-lane roundabouts with offset approaches. The presented optimisation methodology was intended to determine balanced minimum design values among the correlated parameters and rather than to identify an optimal roundabout design [47].

In this paper, the methodology and results of an investigation into the effects of the approach centreline offset on the geometric design and safety performance of urban single-lane roundabouts are presented. This research aimed to define the maximum lateral offset as a function of the design vehicle swept path and fastest path analyses, and to develop a model for the design of urban single-lane roundabouts with varying outer radii and approach centreline offsets to eliminate numerous iterations in the roundabout design process. As the influence of the approach offset on the geometric design of single-lane roundabouts is particularly problematic in urban and suburban locations due to spatial constraints, the analyses conducted in this research were focused on theoretical examples of four-legged single-lane roundabouts with geometric elements whose dimensions are in line with values given in roundabout design guidelines, standards, and recommendations [28,29,30,31,32,33,34,35,36,37,38,39] for urban and suburban roundabouts. This approach fills a gap in the fields of civil and transport engineering by eliminating the need for time-consuming, numerous iterations of swept path and fastest path analyses. These models enable the selection of outer roundabout radii or offsets beyond those specifically presented in this paper, with a high degree of certainty in the results. While most current guidelines merely suggest avoiding offset approaches without providing supporting numerical data, this study provides the specific parameters required for informed design to prevent inconsistent results, offers a clear design methodology, and shortens the time needed for quality roundabout design.

Following this introduction, Section 2 outlines the research methods and highlights considerations regarding approach alignment, the geometric design of roundabout elements based on swept path analyses, and the safety performance assessment. Section 3 presents the results of the analysis, i.e., maximum allowable lateral offsets of the approaches, a proposed model for the design of urban single-lane roundabouts with various lateral offsets of the approach alignments, and model validation. Section 4 discusses the results, highlights the key findings, and provides directions for future research and applications.

2. Materials and Methods

The roundabout design process is typically governed by national guidelines, standards, and recommendations, and it requires several iterations of geometric design based on the results of the operational analysis and safety assessment, such as the fastest paths and speeds, natural travel paths, sight distances, and visibility angles [35].

In this section, the approach alignment considerations, the geometric design of roundabout elements based on swept path analyses, and the fastest path analysis methods used in this research are described in detail. Key steps in the investigation process are shown in Figure 3.

2.1. The Approach Alignment

The approach centreline offset offers flexibility in roundabout design, enabling engineers to adapt the geometry to specific site conditions and traffic patterns. In this investigation, examples of single-lane, four-legged roundabouts with the angle between the centrelines of the approaches set at 90° and with various outer circle radii (Ro) of 15.0, 17.5, 20.0, 22.5 and 25.0 m (Figure 3) were constructed via AutoCAD 2025 software.

The first design step was to define the approach centreline offset. This was carried out in several stages. In the first stage (Figure 4a), roundabouts with an offset of 0, i.e., with approach centrelines intersecting at the roundabout centre, were investigated. In the second stage (Figure 4b), the approach (A) centreline was offset to the left (−) and right (+) in increments of 1 m. In the third stage (Figure 4c), the approach (A) and (C) centrelines were offset to the left (−) and right (+) in increments of 1 m. The above-mentioned increments were chosen to capture the dispersion of the results and to create a sample that is representative, manageable, and easy to present. From a geometric perspective, the maximum offset was linked to the distance between the entry (R_entry) and exit (R_exit) roadway edge curves of adjacent approaches, measured on the outer circle of the roundabout. This distance had to be at least 1 m (Figure 1). From a safety perspective, the maximum offset was determined by the fastest path analysis results, i.e., the optimal driving speed, which was set to 40 km/h.

2.2. Geometric Design of Roundabout Elements Based on Swept Path Analysis

The second step in the design process applied in this investigation involved the design of the following roundabout elements: outer radii (Ro), circulatory roadway width (u), entry and exit widths (e_entry, e_exit), entry and exit radii (R_entry, R_exit), apron width (a_w) and triangular splitter islands on each approach. The design was based on the results of the swept path analyses, while the initial dimensions of roundabout elements were in line with the values recommended by Croatian, Dutch, German, French and US guidelines, as well as Swiss standards [29,30,31,32,33,34,35,36,37,38,39].

Swept path analysis simulates the trajectories of a vehicle’s outermost points, including the wheels and body, as it turns or moves through a defined area. This process determines the space required for the vehicle to manoeuvre and identifies potential design issues. In this way, designers can optimise the geometry of a roundabout and ensure that design vehicles can move smoothly and efficiently. Modern swept path analysis uses specialised software, such as Autodesk Vehicle Tracking, although Autodesk disclaims responsibility for any potential errors or damage resulting from its use [48]. The reliability of Autodesk Vehicle Tracking 2019 software was verified in a study [49] through real drives of a two-axle truck with a three-axle semitrailer at the test site. The test vehicle performed a critical manoeuvre (a 270° left turn) 10 times, and its swept path widths were measured at 32 cross-sections using a precise GPS device (Two-frequency GNSS (GPS + GLONASS) device Trimble R8 with TSC3 controller, Trimble Inc., Westminster, CO, USA). The situation at the test site was then simulated using Autodesk Vehicle Tracking software. The accuracy of the simulation results was evaluated using a t-test at a 95% confidence interval to assess the significance of the differences between the means of two independent samples: real and simulated swept path widths. The t-test showed that the swept path widths determined by field measurements and those determined by vehicle simulation are, from a statistical point of view, the same.

Accordingly, all design vehicle trajectories in the roundabout were drawn using Autodesk Vehicle Tracking 2025 software [48] for all directions of movement: right turn, through movement, left turn, U-turn, and circular turn (Figure 5). The selected design vehicles were a 12.0 m long, 2.55 m wide two-axle bus and a 9.95 m long, 2.55 m wide three-axle refuse collection vehicle (RCV) with a self-steering last axle for improved manoeuvrability, as described in [50]. These vehicle types are common on the European road networks, particularly in urban areas. To ensure passenger comfort when navigating the roundabout, the bus was not permitted to use the traversable part of the central island (apron), whereas the RCV was. The RCV was selected to bridge the gap between various outer circle radii (Ro) of 15 and 20 m, where roundabouts designed solely for the bus would be too restrictive. The distance between the vehicles’ outermost points (trajectories) from the edges of the approaches and the circulatory roadway was in accordance with the applied lateral clearances.

In the investigation presented in this paper, the theoretical examples of urban four-legged single-lane roundabouts were designed following the previously mentioned guidelines, standards and recommendations [29,30,31,32,33,34,35,36,37,38,39], and using the following assumptions and input values (Figure 5):

• The selected outer radii (Ro) were between 15 m and 25 m, and the selected increment was 2.5 m.
• The selected triangular splitter islands on each approach were 29.5 m long, and the width of the cut-through islands, on which the pedestrian and cyclist crossings are located was 2.4–3.0 m. The pedestrian and cyclist crossings within the splitter islands were 5 m wide.
• The selected roadway lane widths at the beginning of the approaches were 3.5 m.
• The circulatory roadway width (u) was determined based on the swept path width (sp) by adding the selected lateral (inner and outer) clearances (zo was set to 1 m, and zi was set to 0.5 m) according to the procedure described in [39]. The width was rounded to 0.1 m.
• The entry and exit widths (e_entry, e_exit) were determined based on the swept path width of the selected design vehicle (sp) by adding a lateral clearance of 0.5 m on both sides of the path. The widths were rounded to 0.1 m.
• Based on the preliminary analysis, for roundabouts where the bus was the design vehicle, the entry radii (R_entry) and exit radii (R_exit) were set to 13 m and 15 m, respectively. In cases where the RCV was the design vehicle, the entry radii were 11 m, and the exit radii were 13 m.
• The outer edges of the entry and exit lanes were flared, i.e., they were not parallel to the edges of the splitter islands. The widened entry and exit began at the cross-section where the splitter island begins. In this way, the roadway edges follow the swept path of a design vehicle as closely as possible.
• The selected apron width (a_w) was 1 m.

As mentioned earlier, the crucial parameter governing the possibilities for lateral offsets of the approaches in the roundabout geometric design based on swept path analysis was the distance between the entry (R_entry) and exit (R_exit) roadway edge curves of adjacent approaches, measured along the outer circle of the roundabout, which had to be at least 1 m (Figure 5b). The resulting lengths of pedestrian and cyclist crossings over the entry and exit lanes were also checked, as excessive lengths can increase the time required for users to cross.

2.3. Fastest Path and Speed Analyses

The safety performance of a roundabout is often assessed using four distinct methods described in various design standards, guidelines [29,30,31,32,33,34,35,36,37,38,39], and practices [40,41,42,51,52]. The first method involves measuring the geometric characteristics of the roundabout and calculating the path radii and vehicle speeds, as outlined in the Dutch [30] and Croatian [39] guidelines and illustrated in Figure 6a. The radius of the vehicle’s fastest path is calculated using the following equation:

$$\mathrm{R} = {\displaystyle \frac{{(0.25·\mathrm{L})}^2 + (0.5{·(\mathrm{U} + 2\left)\right)}^2}{\mathrm{U} + 2}}$$

(1)

where R (m) is the radius of the vehicle’s fastest path, L (m) is the distance between the tangent of the entry radius and the tangent of the exit radius, and U (m) is the distance between the edge of the central island and the right side of the lane of the approach road, measured at the tangent point (see Figure 6a). The design of the roundabout is considered correct if the radius of the driving path is between 22 and 23 m. According to the Dutch guidelines [30], the acceptable vehicle speed through the roundabout should be between 30 and 35 km/h, while the Croatian guidelines [39] specify a maximum of 40 km/h. The speed is calculated using the following equation:

$$\mathrm{V} = 7.4·\sqrt{\mathrm{R}}$$

(2)

where V (km/h) is the vehicle speed and R (m) is the radius of the vehicle’s fastest path.

The second method includes measuring the geometric characteristics of the roundabout and checking the achieved deflection (as described in the German guidelines [31,32] and shown in Figure 6b). The deflection of the vehicle path while traversing around the central island should not be less than twice the lane width (Bz) measured on the roundabout approach.

The third method includes measuring the achieved deflection by the deviation angle (as described in the Swiss standards [36] and shown in Figure 6c), where deviation angle (β) must be at least 45°.

The fourth method includes drawing the fastest paths through the roundabout and measuring the path radii, as described in the French guidelines [29], and then calculating the vehicle speed, as described in the US guidelines [36] (Figure 7). According to [36], the fastest paths for all approaches and all movements (Figure 7a), including left turns, must be drawn (usually CAD-based), and the roundabout must be designed for vehicle speeds (e.g., 25 km/h to 40 km/h) throughout the roundabout. The acceptable entry vehicle speed is 40 km/h [35]. The construction of the fastest path should begin at least 50 m before the entry line, accounting for distances to geometric features [35]. It is assumed that a vehicle is 1.8 m wide [36]. The centreline of the vehicle path is drawn at the following distances from the geometric features (Figure 6a): a = 1.5 m, b = 1.5 m and c = 1.5 m (Figure 7b). According to [36], R₁ is the entry path radius, R₂ is the circulating path radius, R₃ is the exit path radius, R₄ is the left-turn path radius, and R₅ is the right-turn path radius. The R₁ radius should be measured as the smallest best-fit circular curve over a distance of at least 20 to 25 m near the entry line [35].

The relationship between travel speed and horizontal curvature is documented in the AASHTO “Green Book” [53]. Equation (3) provides a simplified relationship between speed and radius for any superelevation rate that incorporates the AASHTO [53] side friction factors:

$$\mathrm{V} = \sqrt{127·\mathrm{R}·(\mathrm{f} \pm \mathrm{e})}$$

(3)

where V (km/h) is the predicted speed, R (m) is radius of the vehicle’s fastest path, f (-) is the side friction factor, and e (m/m) is the superelevation rate. The US [36] and French [29] guidelines differ in the distances of the fastest path from the geometric features (Figure 5).

According to the French guidelines [29], the distances are a = 2.0 m, b = 1.5 m, and c = 2.0 m from a painted edge line (Figure 7b). Furthermore, according to the French guidelines [29], speed estimation is not conducted: appropriate vehicle speed is achieved by limiting the fastest path radii for vehicles travelling through the roundabout to 100 m.

Based on the above, the optimal driving speed through a roundabout is typically 25–40 km/h, with 40 km/h set as the limit in this study. Relative speeds between conflicting traffic streams and between consecutive geometric elements should be minimised so that the maximum speed differential between movements does not exceed approximately 15 to 25 km/h. This speed range is generally recommended to maintain safety and control when navigating the roundabout.

In this study, the safety performance assessment (deflection and speed analysis) for the offsets of approaches (A) and (C) was conducted in accordance with the Dutch [30], Croatian [39], German [31,32] and US [36] guidelines and Swiss [37] standards. The speed analysis for the US guidelines was carried out using superelevation values of +0.025 for entry and exit curves and −0.025 for curves around the central island, along with Formula (3). The French guidelines [29] were not used in the continuation of this research, as a 100 m radius results in a speed over 50 km/h. The most restrictive guidelines were selected for the final investigation of the maximum approach offset for both approaches (A) and (C).

3. Results

Following the iterative process of roundabout design, the results of the analysis, i.e., maximum allowable lateral offsets of the approaches, a model for the design of roundabouts with various lateral offsets of the approach alignments, and model evaluation results are given in this section.

3.1. Maximum Offset of the Approach (A)

The research results for the 12.0 m long bus, presented in

Table 1. Entry and exit widths for bus-based roundabouts with various approach centreline offsets, from −12 to 0 m, and for different outer radii Ro.

Offset (m)	−12	−11	−10	−9	−8	−7	−6	−5	−4	−3	−2	−1	0
Ro = 15.0 m
eentry (m)	-	-	-	-	-	-	-	-	-	-	-	-	4.5
eexit (m)	-	-	-	-	-	-	-	-	-	-	-	-	5.6
Ro = 17.5 m
eentry (m)	-	-	-	-	-	-	-	-	-	5.5	5.3	5.0	4.8
eexit (m)	-	-	-	-	-	-	-	-	-	5.4	5.4	5.5	5.7
Ro = 20 m
eentry (m)	-	-	-	-	-	-	6.5	6.3	6.0	5.8	5.5	5.3	5.1
eexit (m)	-	-	-	-	-	-	5.2	5.2	5.2	5.4	5.6	5.8	6.0
Ro = 22.5 m
eentry (m)	-	-	-	7.6	7.2	6.9	6.6	6.4	6.2	6.0	5.8	5.6	5.4
eexit (m)	-	-	-	5.1	5.1	5.1	5.1	5.2	5.4	5.6	5.8	6.0	6.2
Ro = 25.0 m
eentry (m)	8.9	8.5	8.1	7.7	7.3	7.0	6.8	6.6	6.4	6.2	6.0	5.8	5.6
eexit (m)	5.0	5.0	5.0	5.0	5.0	5.1	5.3	5.5	5.7	5.9	6.1	6.3	6.5

and

Table 2. Entry and exit widths for bus-based roundabouts with various approach centreline offsets, from 0 to 12 m, and for different outer radii Ro.

Offset (m)	0	1	2	3	4	5	6	7	8	9	10	11	12
Ro = 15 m
eentry (m)	4.5	-	-	-	-	-	-	-	-	-	-	-	-
eexit (m)	5.6	-	-	-	-	-	-	-	-	-	-	-	-
Ro = 17.5 m
eentry (m)	4.8	4.6	4.4	4.2	-	-	-	-	-	-	-	-	-
eexit (m)	5.7	5.9	6.1	6.4	-	-	-	-	-	-	-	-	-
Ro = 20 m
eentry (m)	5.1	4.9	4.7	4.5	4.3	4.1	4.1	-	-	-	-	-	-
eexit (m)	6.0	6.2	6.4	6.6	6.9	7.2	7.6	-	-	-	-	-	-
Ro = 22.5 m
eentry (m)	5.4	5.2	5.0	4.8	4.6	4.4	4.2	4.1	4.1	4.1	-	-	-
eexit (m)	6.2	6.4	6.6	6.8	7.1	7.4	7.7	8.0	8.4	8.7	-	-	-
Ro = 25 m
eentry (m)	5.6	5.4	5.2	5.0	4.8	4.6	4.4	4.3	4.2	4.2	4.2	4.2	4.2
eexit (m)	6.5	6.7	6.9	7.1	7.3	7.6	7.9	8.2	8.6	8.9	9.1	9.5	9.9

show that the offset of approach (A), from a geometric point of view, is proportional to the outer radius (Ro). The entry and exit widths (e_entry, e_exit) depend on the approach offset side and the value of the outer radii (Ro), as shown in Table 1 and Table 2.

These widths are proportional to the outer radius size (Ro). Larger entry widths are needed for offsets to the left (up to 8.9 m for the offset of 12 m to the left), while larger exit widths are needed for the offsets to the right (up to 9.9 m for the offset of 12 m to the right).

The results of the investigation, given in

Table 3. Circulatory roadway width for a 12.0 m long bus depending on the outer radii (Ro).

Ro (m)	15	17.5	20	22.5	25
sp (m)	5.6	5.0	4.7	4.5	4.3
u (m)	7.1	6.5	6.2	6.0	5.8

, show that the required circulatory roadway width (u), which depends on the outer radius (Ro) and the swept path width (sp) (Figure 5), is inversely proportional to the outer radius. The values for the circulatory roadway width (u) range from 5.8 to 7.1 m.

Roundabouts with an outer radius Ro = 15.0 m were excluded from further investigation (Table 2) when the bus was used as a design vehicle, as it was not possible to offset the approach due to the overlap of right carriageway arcs (R_entry and R_exit).

After the maximum offset of approach (A) has been determined (based on the 64 combinations), the speed and deflection analyses were conducted in accordance with Croatian [39], Dutch [30], German [31,32], and US [36] guidelines, and Swiss [37] standards. The results for the roundabouts designed for the bus, presented in Figure 8 and Appendix A.1 (

Table A1. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 17.5 m and different approach offsets (values marked in red indicate excessive deflection or vehicle speed).

Ro = 17.5 m
Offset (m)		−3	−2	−1	0	1	2	3
CH	βA-C (°)	59	57	54	52	49	47	44
	βC-A (°)	43	46	49	52	55	57	59
DE	Deflection A-C	3.4 Bz	2.9 Bz	2.7 Bz	2.5 Bz	2.1 Bz	1.9 Bz	1.6 Bz
	Deflection C-A	2.5 Bz	2.5 Bz	2.5 Bz	2.5 Bz	2.5 Bz	2.5 Bz	2.5 Bz
NL, HR	VA-C (km/h)	32	33	33	33	34	35	35
	VC-A (km/h)	36	35	34	33	33	33	32
US	V1A-C (km/h)	31	32	33	33	35	37	38
	V2A-C (km/h)	22	22	22	23	23	24	24
	V3A-C (km/h)	39	40	40	40	40	40	40
	V1C-A (km/h)	34	34	34	33	34	34	34
	V2C-A (km/h)	25	24	23	23	22	22	22
	V3C-A (km/h)	53	45	41	40	39	38	35

,

Table A2. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 20 m and different approach offsets (values marked in red indicate excessive deflection or vehicle speed).

Ro = 20 m
Offset (m)		−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6
CH	βA-C (°)	75	73	71	69	67	65	63	60	58	56	53	51	48
	βC-A (°)	47	50	53	56	58	60	63	65	67	69	71	73	75
DE	DA-C	4.7 Bz	4.4 Bz	4.2 Bz	3.7 Bz	3.7 Bz	3.3 Bz	3.1 Bz	2.8 Bz	2.5 Bz	2.3 Bz	2.1 Bz	1.8 Bz	1.5 Bz
	DC-A	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz
NL, HR	VA-C (km/h)	32	32	32	32	33	33	33	33	34	34	34	34	35
	VC-A (km/h)	36	35	34	33	33	33	33	33	33	32	32	32	32
US	V1A-C (km/h)	27	28	29	30	31	32	33	34	35	36	37	39	43
	V2A-C (km/h)	22	22	22	22	22	22	22	22	22	23	23	24	25
	V3A-C (km/h)	37	37	40	40	39	39	37	38	38	38	38	38	38
	V1C-A (km/h)	32	32	32	32	33	33	33	33	33	33	33	33	32
	V2C-A (km/h)	25	25	24	24	23	22	22	22	22	21	21	21	22
	V3C-A (km/h)	55	49	44	41	40	39	37	37	36	34	34	34	32

,

Table A3. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 22.5 m and different approach offsets between −9 and 0 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 22.5 m
Offset (m)		−9	−8	−7	−6	−5	−4	−3	−2	−1	0
CH	βA-C (°)	88	87	85	83	81	79	77	75	73	71
	βC-A (°)	50	53	56	59	61	63	65	67	69	71
DE	DA-C	6.2 Bz	6.1 Bz	5.8 Bz	5.6 Bz	5.2 Bz	4.8 Bz	4.4 Bz	4.2 Bz	3.8 Bz	3.6 Bz
	DC-A	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz
NL, HR	VA-C (km/h)	32	32	32	33	33	33	33	33	33	33
	VC-A (km/h)	37	36	35	35	34	34	34	34	33	33
US	V1A-C (km/h)	26	26	26	26	27	28	28	29	29	30
	V2A-C (km/h)	23	23	23	23	23	23	23	23	23	23
	V3A-C (km/h)	35	36	35	36	35	36	36	36	36	36
	V1C-A (km/h)	31	32	32	33	32	32	32	32	32	30
	V2C-A (km/h)	28	27	25	24	23	23	22	22	22	23
	V3C-A (km/h)	66	55	48	43	41	40	39	38	38	36

,

Table A4. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 22.5 m and different approach offsets between 0 and 9 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 22.5 m
Offset (m)		0	1	2	3	4	5	6	7	8	9
CH	βA-C (°)	71	69	67	65	63	61	59	56	54	51
	βC-A (°)	71	73	75	77	79	81	82	84	86	87
GER	DA-C	3.6 Bz	3.3 Bz	3.1 Bz	2.9 Bz	2.7 Bz	2.5 Bz	2.2 Bz	1.9 Bz	1.6 Bz	1.3 Bz
	DC-A	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6 Bz	3.6	3.6	3.6
NL, HR	VA-C (km/h)	33	33	34	34	34	34	35	35	36	37
	VC-A (km/h)	33	33	33	33	33	33	32	32	32	32
US	V1A-C (km/h)	30	33	34	35	35	36	38	39	41	45
	V2A-C (km/h)	23	22	22	22	23	23	23	24	24	25
	V3A-C (km/h)	36	37	37	37	37	37	37	37	37	37
	V1C-A (km/h)	30	30	30	30	30	30	30	30	30	30
	V2C-A (km/h)	23	23	23	23	23	24	23	24	24	24
	V3C-A (km/h)	36	35	34	33	33	30	30	29	29	28

,

Table A5. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 25 m and different approach offsets between −12 and 0 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 25 m
Offset (m)		−12	−11	−10	−9	−8	−7	−6	−5	−4	−3	−2	−1	0
CH	βA-C (°)	98	97	95	94	93	91	89	88	86	84	82	81	79
	βC-A (°)	53	55	58	61	64	66	68	70	71	73	75	77	79
DE	DA-C	6.8 Bz	6.7 Bz	6.6 Bz	6.6 Bz	6.7 Bz	6.4 Bz	6.0 Bz	5.6 Bz	5.2 Bz	5.0 Bz	4.6 Bz	4.4 Bz	4.1 Bz
	DC-A	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz
NL, HR	VA-C (km/h)	33	33	33	33	33	33	33	33	33	33	34	35	34
	VC-A (km/h)	38	37	36	36	35	35	35	34	34	34	34	35	34
US	V1A-C (km/h)	26	26	26	26	26	26	26	27	27	28	29	29	29
	V2A-C (km/h)	25	25	25	25	25	24	24	24	25	24	24	24	25
	V3A-C (km/h)	33	33	33	33	33	34	34	33	34	34	33	34	33
	V1C-A (km/h)	29	29	29	29	29	29	29	29	29	29	29	29	29
	V2C-A (km/h)	29	26	26	26	25	25	24	24	25	24	24	24	25
	V3C-A (km/h)	62	52	48	44	41	41	39	39	37	37	35	35	33

and

Table A6. Speed and deflection analysis for roundabouts designed for the 12.0 m long bus with radii Ro = 25 m and different approach offsets, between 0 and 12 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 25 m
Offset (m)		0	1	2	3	4	5	6	7	8	9	10	11	12
CH	βA-C (°)	79	77	75	73	72	70	68	66	63	61	58	55	52
	βC-A (°)	79	81	82	84	86	87	89	91	92	93	95	97	98
DE	DA-C	4.1 Bz	3.9 Bz	3.7 Bz	3.5 Bz	3.3 Bz	3.2 Bz	2.9 Bz	2.6 Bz	2.4 Bz	2.1 Bz	1.7 Bz	1.5 Bz	1.2 Bz
	DC-A	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz	4.1 Bz
NL, HR	VA-C (km/h)	34	34	34	34	34	34	35	35	35	36	36	37	38
	VC-A (km/h)	34	33	33	33	33	33	33	33	33	33	33	33	33
US	V1A-C (km/h)	29	30	31	31	32	34	35	36	38	40	42	46	49
	V2A-C (km/h)	25	24	24	24	24	24	24	25	25	26	26	27	27
	V3A-C (km/h)	33	33	33	33	33	33	33	33	33	33	33	33	33
	V1C-A (km/h)	29	29	29	29	29	29	29	29	29	29	29	29	29
	V2C-A (km/h)	25	25	24	24	25	25	25	25	25	25	25	25	25
	V3C-A (km/h)	33	32	32	31	30	30	29	29	29	29	28	28	27

), show that the limit values for the approach offset varied depending on the procedure applied. For all analysed roundabouts, the US guidelines [36] were the most restrictive, followed by the German guidelines [31,32]. The differences between the speeds calculated according to the Croatian [39] and Dutch [30] guidelines for different approach offsets in one direction of travel (A-C or C-A) are small; the largest difference was only 5 km/h, with the highest calculated speed being 38 km/h and the lowest 32 km/h. Values above 35 km/h would limit the offset in the Netherlands, but not in Croatia, where only values above 40 km/h would set the limit. German guidelines [31,32] indicate that, in some cases, a right offset is not applicable. The right-turn speed values determined in accordance with the US guidelines [36] are not relevant to determining the maximum approach offset, as the approach centrelines intersect at a 90° angle.

The research results for the 9.95 m long RCV, presented in

Table 4. Entry and exit widths for RCV-based roundabouts with various approach centreline offsets, from −8 to 8 m, and for different outer radii Ro.

Offset (m)	−8	−7	−6	−5	−4	−3	−2	−1	0	1	2	3	4	5	6	7	8
Ro = 15.0 m
eentry (m)	-	-	-	-	-	-	4.9	4.6	4.3	4.1	3.9	-	-	-	-	-	-
eexit (m)	-	-	-	-	-	-	4.4	4.6	4.8	5.0	5.3	-	-	-	-	-	-
Ro = 17.5 m
eentry (m)	-	-	-	-	5.5	5.2	5.0	4.8	4.6	4.4	4.2	4.1	4.1	-	-	-	-
eexit (m)	-	-	-	-	4.4	4.5	4.7	4.9	5.1	5.3	5.6	5.9	6.2	-	-	-	-
Ro = 20 m
eentry (m)	6.8	6.5	6.2	5.9	5.6	5.3	5.1	4.9	4.7	4.5	4.3	4.2	4.1	4.0	3.9	3.8	3.7
eexit (m)	4.2	4.2	4.3	4.4	4.5	4.7	4.9	5.1	5.3	5.5	5.7	6.0	6.2	6.5	6.8	7.1	7.5

, show that the offset of approach (A) from a geometric point of view, is proportional to the outer radius (Ro). The entry and exit widths (e_entry, e_exit) depend on the approach offset side and the value of the outer radii (Ro), as shown in Table 4. These widths are proportional to the outer radius size (Ro). Larger entry widths are needed for the offsets to the left (up to 6.8 m for the offset of 8 m to the left), while larger exit widths are needed for the offsets to the right (up to 7.5 m for the offset of 8 m to the right).

Circulatory roadway widths (u) for different outer radii (Ro) are derived from the swept path (sp) and lateral clearances (zo) and (zi), as shown in Figure 5c.

Table 5. Circulatory roadway width for a 9.95 m long RCV depending on the outer radii (Ro).

Ro (m)	15	17.5	20
sp (m)	4.1	3.5	3.3
u (m)	4.6	4.5	4.5

indicates that the (sp) for the 9.95 m long RCV is inversely related to the outer radius (Ro) and is notably narrower than the bus’s swept path (Table 3). Relying strictly on the (sp) values from Table 5 for radii of 17.5 m and 20.0 m would result in an insufficiently wide roadway. This is because the RCV is permitted to traverse the apron by 0.5 m, and the (zi) clearance is measured directly from the edge of the central island (Figure 5c). Consequently, a width of 4.5 m was adopted for these cases, as recommended by the Croatian guidelines [39].

After the maximum offset of the approach (A) was determined (based on 31 combinations), the speed and deflection analyses were conducted in accordance with Croatian [39], Dutch [30], German [31,32], and US [36] guidelines and Swiss [37] standards. The results for the RCV, presented in Figure 9 and Appendix A.1 (

Table A7. Speed and deflection analysis for roundabouts designed for 9.95 m long RCV with radii Ro = 15 m and different approach offsets (values marked in red indicate excessive deflection or vehicle speed).

Ro = 15 m
Offset (m)		−2	−1	0	1	2
CH	βA-C (°)	65	63	60	57	61
	βC-A (°)	54	57	60	63	73
DE	Deflection A-C	2.6 Bz	2.4 Bz	2.2 Bz	2.0 Bz	1.8 Bz
	Deflection C-A	2.2 Bz	2.2 Bz	2.2 Bz	2.2 Bz	2.2 Bz
NL, HR	VA-C (km/h)	29	30	30	31	30
	VC-A (km/h)	31	31	30	30	29
US	V1A-C (km/h)	31	32	33	34	34
	V2A-C (km/h)	20	20	20	21	20
	V3A-C (km/h)	35	34	34	34	33
	V1C-A (km/h)	32	32	33	32	32
	V2C-A (km/h)	21	21	20	20	20
	V3C-A (km/h)	38	37	35	33	32

,

Table A8. Speed and deflection analysis for roundabouts for the 9.95 m long RCV with radii Ro = 17.5 m and different approach offsets (values marked in red indicate excessive deflection or vehicle speed).

Ro = 17.5 m
Offset (m)		−4	−3	−2	−1	0	1	2	3	4
CH	βA-C (°)	87	85	83	80	77	75	72	69	66
	βC-A (°)	66	70	72	75	77	80	82	84	87
DE	DA-C	4.2 Bz	4.1 Bz	3.7 Bz	3.4 Bz	3.1 Bz	2.9 Bz	2.6 Bz	2.2 Bz	1.9 Bz
	DC-A	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz	3.1 Bz
NL, HR	VA-C (km/h)	29	29	29	29	29	29	30	30	30
	VC-A (km/h)	31	30	30	30	29	30	29	29	29
US	V1A-C (km/h)	25	26	27	30	31	31	32	33	35
	V2A-C (km/h)	20	20	20	20	20	20	20	20	20
	V3A-C (km/h)	32	32	32	32	32	32	32	32	33
	V1C-A (km/h)	31	31	31	31	31	31	30	30	30
	V2C-A (km/h)	20	20	20	20	20	19	20	20	20
	V3C-A (km/h)	36	35	34	33	32	31	29	28	28

,

Table A9. Speed and deflection analysis for the RCV-based roundabouts with radii Ro = 20 m and different approach offsets between −8 and 0 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 20 m
Offset (m)		−8	−7	−6	−5	−4	−3	−2	−1	0
CH	βA-C (°)	102	100	98	96	94	92	81	88	86
	βC-A (°)	64	68	71	74	76	79	90	83	86
DE	DA-C	5.8 Bz	5.4 Bz	5.2 Bz	5.0 Bz	4.8 Bz	4.6 Bz	4.3 Bz	4.1 Bz	3.8 Bz
	DC-A	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz
NL, HR	VA-C (km/h)	30	30	30	30	30	30	30	30	30
	VC-A (km/h)	32	32	33	31	31	31	31	31	30
US	V1A-C (km/h)	23	24	25	24	25	25	26	26	31
	V2A-C (km/h)	21	21	22	21	21	21	21	21	20
	V3A-C (km/h)	32	32	31	32	31	32	32	31	32
	V1C-A (km/h)	30	30	30	30	30	30	31	31	31
	V2C-A (km/h)	21	21	21	21	20	20	20	20	20
	V3C-A (km/h)	42	40	38	37	35	34	33	33	32

and

Table A10. Speed and deflection analysis for the RCV-based roundabouts with radii Ro = 20 m and different approach offsets between 0 and 8 m (values marked in red indicate excessive deflection or vehicle speed).

Ro = 20 m
Offset (m)		0	1	2	3	4	5	6	7	8
CH	βA-C (°)	86	83	81	78	76	73	70	67	64
	βC-A (°)	86	88	90	92	94	96	98	100	102
DE	DA-C	3.8 Bz	3.5 Bz	3.2 Bz	2.9 Bz	2.6 Bz	2.2 Bz	2.0 Bz	1.8 Bz	1.5 Bz
	DC-A	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz	3.8 Bz
NL, HR	VA-C (km/h)	30	31	30	31	31	31	32	32	33
	VC-A (km/h)	30	30	30	30	30	30	30	30	30
US	V1A-C (km/h)	31	31	31	32	33	35	36	37	40
	V2A-C (km/h)	20	20	20	20	20	21	21	21	21
	V3A-C (km/h)	32	32	33	32	32	32	32	32	32
	V1C-A (km/h)	31	30	30	30	30	30	30	30	30
	V2C-A (km/h)	20	20	20	20	21	21	21	21	21
	V3C-A (km/h)	32	29	28	27	26	26	26	25	25

), show that the limit values for the approach offset varied depending on the procedure applied.

For all analysed roundabouts, the German guidelines [31,32] were the most restrictive, followed by the US guidelines [36], while the Croatian [39], Dutch [30], and Swiss [37] standards were not restrictive at all. The German guidelines [31,32] were particularly restrictive regarding the right-side offset. The differences between the speeds calculated according to the Croatian [39] and Dutch [30] guidelines for various approach offsets in a single direction of travel (A–C or C–A) were minor; the largest difference was only 4 km/h, with the highest calculated speed being 33 km/h and the lowest 29 km/h. In this instance, the US guidelines [36] proved to be the most restrictive only for the roundabout with an outer radius (Ro) of 20 m and a left-side approach offset of 8 m.

The investigation results for both design vehicles show that the outer radius (Ro) and the approach offset influence the design of roundabout geometric elements and roundabout speeds. Increasing the approach offset to the left results in the following (Table 1, Table 2, Table 3, Table 4, Table 5, Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9 and Table A10):

• Entry: Required entry width (e_entry), deflection angle β_A-C, and the deflection increase, while path radii (R and R₁) and speeds (V_A-C and V1_A-C) decrease.
• Exit: Required exit width (e_exit) and deflection angle β_A-C decrease, deflection remains constant, while path radii (R and R₃) and speeds (V_C-A and V3_C-A) increase.
• Circulatory roadway: Variations in the outer radius (Ro) influence the passing-through speeds (V2_A-C and V2_C-A). Specifically, the bus-based roundabout speeds range between 20 and 25 km/h (V2_A-C) and 22–29 km/h (V2_C-A). Conversely, the RCV-based roundabout speeds remain nearly uniform, varying only between 20 and 21 km/h.

Increasing the approach offset to the right results in the following (Table 1, Table 2, Table 3, Table 4, Table 5, Table A1, Table A2, Table A3, Table A4, Table A5, Table A6, Table A7, Table A8, Table A9 and Table A10):

• Entry: Required entry width (e_entry), deflection angle (β_A-C) and deflection decrease, while path radii (R and R₁) and speeds (V_A-C and V1_A-C) increase.
• Exit: Required exit width (e_exit) and deflection increase, deflection angle (β_A-C) decrease, while path radii (R and R₃) and speeds (V_C-A and V3_C-A) decrease.
• Circulatory roadway: Outer radius (Ro) variations have a minimal impact on passing-through speeds (V2_A-C and V2_C-A). Both speeds range from 21 to 24 km/h for bus-based designs and from 19 to 21 km/h for RCV-based roundabouts.

3.2. Maximum Offset of the Approaches (A) and (C)

To determine the maximum offset for approaches (A) and (C) on the bus-based roundabouts, their centrelines were shifted to the right and left in various combinations. Design elements and the maximum initial offset values were taken from research results shown in Table 1, Table 2 and Table 3, and Appendix A.1 (Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6), following speed and deflection restrictions obtained from the US [36] and German [31,32] guidelines. As the deflection on the roundabout does not change regardless of the combination of approach offsets, no further investigation of deflection was necessary for those cases; consequently, roundabouts with maximum offsets were designed and tested solely for speed according to US guidelines [36].

The research results, given in Appendix A.2 (

Table A11. Speed analysis for bus-based roundabouts with radii Ro from 17.5 to 25 m and maximum approach offsets.

Ro (m)	17.5	20	22.5	25
Approach offset (m)	A = 0 C = 1	A = −2 C = 4	A = −4 C = 6	A = −6 C = 9
V1A-C (km/h)	31	31	28	28
V2A-C (km/h)	20	20	24	26
V3A-C (km/h)	32	34	30	29
V1C-A (km/h)	31	37	37	40
V2C-A (km/h)	19	24	26	25
V3C-A (km/h)	31	40	39	39

), show that the offset limits defined during the investigation of the maximum offset of approach (A) remain unchanged, even when combined with the maximum offsets of approaches (C). These roundabouts met all requirements for entry, exit, and circulatory roadway widths, as well as speeds and deflection according to the US [36] and German [31,32] guidelines. Larger offsets than those shown in Table A11 resulted in unacceptable speeds or deflections on the roundabout. The combinations with maximum offsets are as follows:

• For Ro = 17.5 m (Figure 10a), the centreline was offset 1 m to the right for only one approach (approach (A) or approach (C)).
• For Ro = 20 m (Figure 10b), the centreline was offset 2 m to the left for approach (A) and 4 m to the right for approach (C), and vice versa.
• For Ro = 22.5 m (Figure 10c), the centreline was offset 4 m to the left for approach (A) and 6 m to the right for approach (C), and vice versa.
• For Ro = 25 m (Figure 10d), the centreline was offset 6 m to the left for approach (A) and 9 m to the right for approach (C), and vice versa.

The maximum offsets for approaches (A) and (C) on the RCV-based roundabouts were determined in the same way as for the bus-based roundabouts, but using Table 4 and Table 5, and Appendix A.1 (Table A7, Table A8, Table A9 and Table A10). The research results, given in Appendix A.2 (

Table A12. Speed analysis for RCV-based roundabouts with radii Ro from 15 to 20 m and maximum approach offsets.

Ro (m)	15	17.5	20
Approach offset (m)	A = −2 C = 1	A = −4 C = 3	A = −7 C = 6
V1A-C (km/h)	30	25	24
V2A-C (km/h)	19	21	23
V3A-C (km/h)	31	31	26
V1C-A (km/h)	33	34	36
V2C-A (km/h)	22	23	25
V3C-A (km/h)	37	37	40

), show that the offset limits defined during the investigation of the maximum offset of approaches (A) remain unchanged, even when combining the maximum offsets on approaches (C). These roundabouts met all requirements for entry, exit, and circulatory roadway widths, as well as speeds and deflection according to the US [36] and German [31,32] guidelines. Larger offsets than those shown in Table A12 resulted in unacceptable speeds or deflections on the roundabout. The combinations with maximum offsets are as follows:

• For Ro = 15 m (Figure 11a), the centreline was offset 2 m to the left for approach (A) and 1 m to the right for approach (C), and vice versa.
• For Ro = 17.5 m (Figure 11b), the centreline was offset 4 m to the left for approach (A) and 3 m to the right for approach (C), and vice versa.
• For Ro = 20 m (Figure 11c), the centreline was offset 7 m to the left for approach (A) and 6 m to the right for approach (C), and vice versa.

The lengths of the pedestrian and cycle crossings were measured at the longest point and varied (Figure 10 and Figure 11). For the bus-based roundabouts the maximum value on the entry lane ranges between 3.8 m and 4.6 m, while the maximum value on the exit lane is between 4.7 m and 6.5 m, depending on the outer radii (Ro) and approach offset. For the RCV-based roundabouts the maximum value on the entry lane ranges between 3.8 m and 4.2 m, while the maximum value on the exit lane is between 4.1 m and 4.7 m, depending on the outer radii (Ro) and approach offset. The main reason for the different crossing widths is the use of widened (flared) entries and exits, i.e., the outer edges of the entry and exit lanes are not parallel to the edges of the splitter islands. It can also be seen (Figure 10 and Figure 11) that the maximum possible offset ensured that the distance (measured on the outer circle of the roundabout) between the entries and exits of the adjacent approaches is not less than 1.0 m.

3.3. Roundabout Design Model

Utilising the research findings and diagrams from Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16, the proposed design model is tailored for single-lane roundabouts featuring approach offsets (A) and/or (C). Developed around a two-axle bus and an RCV as the primary design vehicles, the model covers outer radii (Ro) from 15 to 25 m (15–20 m for RCVs; 17.5–25 m for buses) with perpendicular approach centrelines. Specific geometric elements comprise flared entries with radii of 11 m (RCVs) and 13 m (buses); flared exits with radii of 13 m (RCVs) and 15 m (buses); and the splitter island configurations detailed in Figure 4 and Figure 5b. It is critical to note that if a roundabout is designed specifically for RCVs, buses will experience significant traversing difficulties negotiating it; conversely, a bus-orientated design will seamlessly accommodate RCVs. Therefore, designers must comprehensively understand the local vehicle fleet structure. The proposed modelling steps are as follows (Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16):

• The centre of the selected outer radius (Ro) is placed depending on spatial constraints.
• The circulatory roadway width (u) is selected depending on the outer radius (Ro) and the design vehicle, according to the diagram shown in Figure 13 (rounded up to the nearest 0.1 m).
• The approach centrelines are placed according to the spatial constraints and the design vehicle. Offset values must be within the limits shown in Figure 14. If they are not, the procedure is repeated from step 1.
  • The splitter islands (shown in Figure 4 and Figure 5b) are placed along the centrelines.
  • The entry and exit widths (e_entry, e_exit) are selected according to the offsets from Figure 15 and Figure 16 and the design vehicle (rounded up to the nearest 0.1 m).
  • Flared entries and exits with an entry radius of 11 or 13 m and exit radius of 13 or 15 m depending on the design vehicle, are drawn.

3.4. Roundabout Design Model Validation

To validate the proposed roundabout design models, roundabouts with different outer radii (Ro) were randomly selected within the limits shown in Figure 13 and tested. These roundabouts were not included in the first part of the study. For the bus-based designs, the selected Ro values were 18.5, 21.5, and 23.5 m, while for the RCV-based designs, the selected values were 16 and 18.5 m. For these outer radii (Ro) values, the maximum offset values were interpolated from the diagrams in Figure 14. The design elements for these roundabouts were then selected from the diagrams in Figure 13, Figure 14, Figure 15 and Figure 16 and are listed in

Table 6. Design elements for bus-based roundabouts with various radii Ro and a maximum approach offset.

Ro (m)	18.5			21.5			23.5
u (m)	6.4			6.1			6
Approach offset (m)	A	B/D	C	A	B/D	C	A	B/D	C
	−0.9	0	2.2	−3.1	0	5.2	−4.8	0	7.2
eentry (m)	5.1	4.9	4.7	5.9	5.2	4.3	6.5	5.5	4.3
eexit (m)	5.6	5.8	6.3	5.5	6.1	7.4	5.4	6.4	8.2

and

Table 7. Design elements for RCV-based roundabouts with various radii Ro and a maximum approach offset.

Ro (m)	16			18.5
u (m)	4.6			4.5
Approach offset (m)	A	B/D	C	A	B/D	C
	−2.8	0	1.8	−5.2	0	4.2
eentry (m)	5.1	4.4	4.1	6.0	4.7	4.1
eexit (m)	4.5	4.9	5.4	4.4	5.2	6.3

. The final roundabout designs are shown in Figure 17 and Figure 18. These roundabouts were tested, and the evaluation included swept path, speed and deflection analyses.

The results of the swept path analysis showed that all approaches were correctly designed, and all lateral clearances were met. Namely, the roundabouts designed using this method satisfied all criteria for unobstructed vehicle movement, including the specified lateral clearances and the speed–deflection requirements according to the US [36] and German guidelines [31,32] (Appendix A.3,

Table A13. Speed and deflection analysis for bus-based roundabouts with different radii Ro = 18.5, 21.5 and 23.5 m and maximum approach offsets.

Ro (m)		18.5	21.5	23.5
Offset (m)		A = −0.9 C = 2.2	A = −3.1 C = 5.2	A = −4.8 C = 7.2
DE	DA-C	2.9 Bz	4.2 Bz	5.1 Bz
	DC-A	2.0 Bz	2.1 Bz	2.0 Bz
	DB-D,D-B	2.7 Bz	3.5 Bz	3.7 Bz
US	V1A-C (km/h)	32	28	27
	V2A-C (km/h)	22	23	25
	V3A-C (km/h)	35	31	30
	V1C-A (km/h)	38	37	39
	V2C-A (km/h)	25	25	25
	V3C-A (km/h)	40	40	40
	V1B-D,D-B (km/h)	33	32	33
	V2B-D,D-B (km/h)	23	22	22
	V3B-D,D-B (km/h)	38	37	37

and

Table A14. Speed and deflection analysis for RCV-based roundabouts with different radii Ro = 16 and 18.5 m and maximum approach offset.

Ro (m)		16	18.5
Offset (m)		A = −2.8 C = 1.8	A = −5.2 C = 4.2
DE	DA-C	3.4 Bz	4.5 Bz
	DC-A	2.1 Bz	2.0 Bz
	DB-D,D-B	2.6 Bz	3.2 Bz
US	V1A-C (km/h)	26	25
	V2A-C (km/h)	20	23
	V3A-C (km/h)	31	27
	V1C-A (km/h)	33	35
	V2C-A (km/h)	22	23
	V3C-A (km/h)	37	38
	V1B-D,D-B (km/h)	31	34
	V2B-D,D-B (km/h)	20	19
	V3B-D,D-B (km/h)	32	32

). The results of the speed and deflection analyses indicate that the roundabout design model is applicable within the limits shown in Figure 14. Furthermore, a separation of over 1.0 m between adjacent entry and exit roadway edges (R_entry) and (R_exit), was maintained, thereby ensuring that no overlap occurred (Figure 17 and Figure 18).

4. Discussion

This study establishes a unique design model for roundabouts with offset approaches and outer radii ranging from 15 to 25 m using two design vehicles; a bus and a refuse collection vehicle RCV. The model is based on an analysis of 95 single-approach offsets (64 for the bus and 31 for the RCV) and up to 377 dual-approach offset combinations (229 for the buses and 148 for the RCV), using 1.0 m increments. The results indicate that approach legs (A) and (C) can be offset further to the right and left for the RCV compared to the bus for identical outer radii. This is due to the fact that the RCV features a shorter front overhang and wheelbase, as well as a self-steering rear axle. Consequently, utilising the RCV as the design vehicle yielded larger approach offset limits for roundabouts with outer radii between 15 and 20 m, which are most prevalent in urban areas.

This study addresses a critical gap in civil and transport engineering by eliminating the requirement for time-consuming iterations of swept path and fastest path analyses. The proposed model allows for the selection of roundabout radii and offsets beyond the specific study combinations with high confidence in the outcomes. Unlike existing guidelines, which often discourage the use of offset approaches, this research provides the definitive numerical data necessary for informed design. By establishing specific parameters to prevent inconsistent results, this work represents a significant scientific advancement in the area of civil and transport engineering and offers a transparent, replicable methodology.

The approach offset values determined in this study are considerably higher for the outer radius (Ro) of 17.5–25 m than those recommended by the Croatian guidelines [39], which set the maximum offset at 0.15 Ro. These findings are based on speed and deflection analyses conducted in accordance with the US [36] and German guidelines [31,32], which are the most restrictive among the documents analysed in this research and provided the most comprehensive results by considering entry, through, and exit speeds and deflection. Additionally, the speed at the exit of the roundabout increases more rapidly when the approach is shifted to the left than the entry speed increases when the approach is shifted to the right, even though shifting the approach to the left reduces the required exit width. The design vehicle swept path naturally influences the determination of entry, exit, and circulatory roadway widths, and thus the driving speed. The aim was to design entry and exit widths to be as narrow as possible, with lateral clearances sufficient to ensure smooth movement of the chosen design vehicle.

Given that no single model perfectly predicts roundabout speeds, our study employed established European and US guidelines that have remained industry standards for over two decades. This study found the German [31,32] and US guidelines [36] to be robust indicators for bus and RCV related restrictions. Furthermore, this research provides the specific parameters required for the prevention of inconsistent design outcomes.

The results of the investigation presented in this paper could not be compared with the results of a similar study given in [47], because it used a predefined roundabout geometry and had the following limitations: the analysed roundabouts were designed without splitter islands, the design vehicle swept path was not considered, the safety performance was not tested, and the two adjacent approach edges at entry and exit could touch each other, i.e., there was no outer circular arc.

The design vehicle’s manoeuvres and swept paths at entry and exit are not identical, which directly affects the shape of the roadway edges. Because the rear wheels do not follow the front wheels, the vehicle requires a longer distance to realign upon exit; therefore, a larger roadway curvature radius is necessary. The applied entry radii of 11 or 13 m and exit radii of 13 or 15 m, depending on the design vehicle, combined with flared entries and exits, ideally follow the swept path with lateral clearances, aligning with results from previous studies [45,46].

In this study, roundabouts were designed so that the design vehicle, a two-axle, 12-metre-long bus, does not traverse the apron. This ensured passenger comfort, as recommended by the US guidelines [35]. For the analysed roundabouts with an outer radius of 15 m designed for this bus, swept path analyses showed that the adjacent approach roadway edges (entry and exit) were less than 1.0 m apart. When a shorter and more manoeuvrable refuse collection vehicle (RCV) was introduced, which could traverse the apron by 0.5 m, the roundabout entries and exits became narrower. Consequently, the potential for approach offsets in roundabouts with an outer radius (Ro) of 15 to 20 metres increased. Designers must choose between the two vehicles depending on the location and local vehicle fleet composition. If a roundabout is designed specifically for RCVs, buses will experience significant traversing difficulties negotiating it; conversely, a bus-oriented design will seamlessly accommodate RCVs.

The crossing lengths for pedestrians and cyclists across entry and exit lanes were evaluated (Figure 10 and Figure 11). At the entry lane, crossing lengths decrease when the approach offset is to the right and increase when it is to the left; for the exit lane, the opposite is true. The greatest variance was observed between roundabouts, designed for the bus, with no offset and those with a 9.0 m offset to the right at a 25 m outer radius. Specifically, with a 9.0 m offset, the entry crossing length decreases by 0.3 m, while the exit crossing length increases by 1.3 m. Similar trends were observed for other radii: for an outer radius of 17.5 m with a 3.0 m offset, crossings were 0.1 m shorter at the entry and 0.1 m longer at the exit. At a roundabout with an outer radius of 20 m with a 5.0 m offset, crossings were 0.2 m shorter at the entry and 0.4 m longer at the exit. In comparison, at a 22.5 m radius with a 7.0 m offset, the entry decreases by 0.3 m and the exit crossing increases by 0.9 m. Based on the previous research, a pedestrian crossing at a roundabout with offset approaches that is up to 2.0 m longer than at a standard roundabout layout would not pose a problem even for the slowest pedestrians, as it would require only approximately 2 extra s for them to cross. Namely, according to Giannoulaki and Christoforou [54], the slowest measured walking speeds range from 1.2 m/s to 1.4 m/s, while for seniors, the speed drops to 0.9 m/s. Furthermore, Gates et al. [55] analysed 1947 crossing events across eleven intersections in Wisconsin and recommended a walking speed of 0.88 m/s for areas where nearly all pedestrians are over age 65. They suggest that 1.2 m/s is only appropriate for locations with very few older or disabled pedestrians, such as college campuses, while 1.15 m/s is the general recommendation for timing clearance intervals. Consequently, an additional 1.5 to 2.0 m of crossing distance would only require approximately 2 extra seconds for even the slowest pedestrians. To reduce pedestrian and cyclist exposure, crossings can be positioned further back from, but not too far away from roundabout exits (splitter island should not be narrower than 2 m in that place) or supplemented with road markings such as ‘shark’s teeth’ (triangular yield markings). Widely used in the Netherlands, these markings legally prioritise vulnerable road users by requiring exiting motorists to slow down and yield to those crossing their path. For the roundabouts designed for the RCV, the maximum difference in crossing lengths on the entry lane is 0.4 m, while the maximum difference on the exit lane is 0.6 m, depending on the outer radius (Ro) and the approach offset. Consequently, there are no significant changes in crossing lengths between approaches with and without an offset.

The limitations of this study were that it was not conducted on real examples of urban single-lane roundabouts with offset approaches, that the approach axes on the analysed roundabouts were all positioned at right angles, and only the triangular splitter island was evaluated. To help identify potential shortcomings in the roundabout design model, future research will include analysis of roundabout schemes with approach angles ranging from 70° to 90° in 5° increments, as well as one-approach offsets, two design vehicles, and other splitter island types. Furthermore, the model will be applied to a real-world roundabout experiencing safety or operational issues due to an offset alignment, and the real-world parameters will be compared with those suggested by the model.

5. Conclusions

High-quality geometric design of intersections is essential for safe and uninterrupted traffic flow and for maintaining an adequate level of service. The study presented in this paper focuses on the geometric design of urban four-leg single-lane roundabouts, specifically the possibilities of offsetting the approaches and modifying the shapes of roundabout design elements (entry and exit widths, circulatory roadway width) to accommodate different offset values. The motivation for this study was the need to construct roundabouts with offset approaches, especially in urban and suburban road networks. This study aimed to determine the influence of approach offset on the geometric design of roundabouts. The research was conducted using theoretical roundabout examples with outer radii (Ro = 15, 17.5, 20, 22.5, and 25 m) designed to accommodate a 12 m long two-axle bus and 9.95 m long three-axle refuse collection vehicles (RCV). The results showed that offsetting an approach is possible for roundabouts with outer radii ranging from 15 to 25 m, depending on the design vehicle, and that this offset affects entry and exit widths. Offsetting the approach leg to the left increases the required entry width and decreases the required exit width. Offsetting the approach leg to the right decreases the required entry width and increases the required exit width. The offset value is proportional to the roundabout outer radius. The speed analysis results further restricted the approach leg offsets.

This study and the new model contribute to the consistent design of roundabouts, particularly in the redesign of standard four- and three-legged intersections, by providing a model that does not require swept path and speed analyses. The efficiency of roundabouts designed using the new model is demonstrated by the smooth and unobstructed movement of the design vehicle, ensuring sufficient lateral clearances.

Future research should focus on the effects of varying splitter island widths on the fastest path or deflection, as well as the combined effects of radial and lateral approach offsets in urban roundabout design. In addition, the results obtained in this study should be compared with those from real examples of roundabouts with approach offsets to reach conclusions. This study provides researchers, practitioners involved in roundabout design, and decision-makers with valuable insights into the impact of approach offset on roundabout design.