Selective Paste Intrusion – Shadowing Effects from Rebar Protrusion in the Particle Bed and Their Impact on Bond Strength

1. Introduction

Selective Paste Intrusion (SPI) is an additive manufacturing method for producing concrete elements with complex geometries. In the combined process investigated here, reinforcement bars are printed segment-wise by Wire Arc Additive Manufacturing (WAAM) directly in the growing particle bed of aggregates (Figure 1a). The process then involves sequentially spreading a layer of aggregates to form a particle bed (Figure 1b), followed by the selective intrusion of cement paste into the voids between the particles (fig:processc). The unbound surrounding aggregates act as a temporary support structure and enable the fabrication of freeform components without conventional formwork (Figure 1d, e) [1,2].

Figure 1. Combined SPI and WAAM process, modified after [3]. (a) Printing of a reinforcement bar, (b) spreading of an aggregate layer, (c) application of cement paste, (d) curing of the finished reinforced concrete element, (e) excavated component.

Figure 1. Combined SPI and WAAM process, modified after [3]. (a) Printing of a reinforcement bar, (b) spreading of an aggregate layer, (c) application of cement paste, (d) curing of the finished reinforced concrete element, (e) excavated component.

A key research focus in Selective Paste Intrusion is the integration of reinforcement, since this remains a central obstacle for structural applications of concrete additive manufacturing in general, and its interaction with process and form is increasingly treated as a governing design principle [4,5]. Combining Selective Paste Intrusion with WAAM has been identified as a strategy to produce custom-shaped reinforcement structures within the particle bed, and its feasibility has been demonstrated for printed specimens with integrated WAAM bars [6]. Initial investigations have shown that WAAM reinforcements, when tested in conventionally cast concrete, reach bond strengths comparable to reinforcing steel of grade B500B [7]. B500B denotes a characteristic yield strength $f_{yk}$ of 500 $\mathrm{M}$$\mathrm{Pa}$ and ductility class B per EN 1992-1-1 [8], the latter defined by a strain-hardening ratio $k={(f_t/f_y)}_k\ge 1.08$ at a uniform elongation of at least $5.0$ . By characteristic yield strength, B500B lies between ASTM A615 Grades 60 and 75. The tensile strength of the WAAM bars (454$\mathrm{M}$$\mathrm{Pa}$–481$\mathrm{M}$$\mathrm{Pa}$ against 647$\mathrm{M}$$\mathrm{Pa}$–658$\mathrm{M}$$\mathrm{Pa}$ for the tested B500B bars) remained lower at a higher uniform elongation [7].

The high temperatures generated during the WAAM process can alter the rheological properties of the cement paste. Yield stress and plastic viscosity of cement paste change with temperature, with the largest changes reported below 20 °C and nearly constant values between 30 °C and 40 °C [9]. For cement-based suspensions, temperature alters the apparent viscosity and the power-law flow parameters over the range 20 °C–80 °C [10], and the coupled effects of time and temperature in superplasticised pastes have been quantified for representative formulations [11].

For the SPI system used here, the relevant thermal envelope was established in a sequence of preceding studies. The fresh-state limit was identified by measuring how rising paste temperatures affect yield stress and viscosity: paste penetration into the particle bed remains sufficient up to 60 °C and deteriorates within a transition range of 60°C–70°C [12]. The hardened-state limit was confirmed by examining the effect of fresh-state thermal exposure on the compressive and flexural strength of the resulting concrete, which showed no reduction (rather a slight increase) up to 70 °C, with degradation setting in between 70 °C and 80 °C [13]. The effect of elevated temperatures on bond behaviour was investigated separately by pull-out tests in which WAAM reinforcement bars embedded in fresh concrete were heated to 60 °C, 80 °C, and 200 °C immediately after casting, with unheated specimens at 20 °C serving as reference: moderate exposure (60 °C–80 °C) led to a slight reduction in maximum bond strength, while substantial degradation occurred at 200 °C [14]. The actual temperatures occurring during WAAM at varying distances from the welding point were then quantified, and corresponding cooling strategies were evaluated [15]. One such strategy is increasing the vertical distance between the welding point and the particle bed, which reduces the thermal load on the bed. A subsequent study recommended a maximum nozzle-to-bed distance of 50 $\mathrm{m}$$\mathrm{m}$ as a balance between mechanical properties and print quality: shape accuracy remained consistent up to 40 $\mathrm{m}$$\mathrm{m}$, whereas at 50 $\mathrm{m}$$\mathrm{m}$ the compressive strength declined by about 16 [3]. A vertical protrusion of 40 $\mathrm{m}$$\mathrm{m}$, more conservative than the recommended 50 $\mathrm{m}$$\mathrm{m}$ maximum, was therefore adopted as the largest distance at which neither shape accuracy nor compressive strength was impaired. The shadowing effects investigated in the present study arise as a direct consequence of this 40 $\mathrm{m}$$\mathrm{m}$ process limit. To isolate the geometric component, thermal effects are not considered in the present experimental program.

A protruding reinforcement bar may influence the local material deposition during layer application and overprinting. An elevated bar partially obstructs the transport of aggregates and cement paste into the region beneath it, leading to so-called shadowing effects, see fig:concept. The result is a local inhomogeneity in the concrete matrix and an impaired interface quality between reinforcement and surrounding material, both of which can affect bond behaviour adversely.

The term “shadowing” has previously been used for related deposition-shielding phenomena around reinforcement in concrete additive manufacturing [16,17]. In this study, “shadowing” is defined as a process-induced disturbance of material deposition in the vicinity of a protruding reinforcement bar. Two related manifestations are considered, which can act simultaneously as degradation mechanisms.

The first is a geometrically projected shadowed region in the particle bed. As the bar protrudes above the bed, aggregates and paste applied from above cannot reach the volume that lies in the geometric projection beneath the bar. This region depends on bar diameter and inclination and is most pronounced for larger diameters and shallow inclinations.

The second is a layer-wise modification of the contact zone along the bar surface in the overprinted region. As the structure is built up layer by layer from above, the protruding part of the bar lies above the most recently deposited material at every step of the printing sequence. This generates a zone along the lower half of the bar surface in which material deposition is repeatedly disturbed. Over the printing sequence, this disturbance extends along the full bond length and is largely independent of inclination (Section 2.4).

Figure 2. Schematic of the two shadowing manifestations (cross-section, not to scale). (a) Projected shadow: while the bar protrudes by up to h above the current bed surface, material applied from above cannot reach the strip of the bed surface with horizontal reach $r\left(\theta \right)$ beneath the bar (red, momentary shadowed area $a_{\mathrm{shadow}}\left(\theta \right)=d·r\left(\theta \right)$). The union of these momentary shadows over the printing sequence forms the projected shadowed area $A_{\mathrm{shadow}}$ (sec:descriptors). The projected shadow grows with bar diameter d and with decreasing inclination angle $\theta$ and vanishes at $\theta =90$. (b) Contact zone: the lower half of the bar surface within the bond length $l_b$ (orange, $A_{\mathrm{contact}}$) experiences repeatedly disturbed material deposition during overprinting, independent of $\theta$ for $0\le \theta <90$.

Figure 2. Schematic of the two shadowing manifestations (cross-section, not to scale). (a) Projected shadow: while the bar protrudes by up to h above the current bed surface, material applied from above cannot reach the strip of the bed surface with horizontal reach $r\left(\theta \right)$ beneath the bar (red, momentary shadowed area $a_{\mathrm{shadow}}\left(\theta \right)=d·r\left(\theta \right)$). The union of these momentary shadows over the printing sequence forms the projected shadowed area $A_{\mathrm{shadow}}$ (sec:descriptors). The projected shadow grows with bar diameter d and with decreasing inclination angle $\theta$ and vanishes at $\theta =90$. (b) Contact zone: the lower half of the bar surface within the bond length $l_b$ (orange, $A_{\mathrm{contact}}$) experiences repeatedly disturbed material deposition during overprinting, independent of $\theta$ for $0\le \theta <90$.

The mechanistic plausibility of both manifestations follows from established phenomena. Penetration models for cement pastes into sand packings show that the intrusion depth depends on the pore structure of the packing, in particular its porosity and grain size [1,18,19], so that a region in the particle bed shielded from aggregate and paste deposition will exhibit a structurally distinct matrix. For the contact zone along the bar surface, the local packing in granular beds at a cylindrical contact differs from the bulk [20,21]: the porosity increases towards the contact [22,23], so that the lower side of the bar is, on purely geometric grounds and prior to any deposition-process consideration, a zone of altered packing. Whether the resulting void volume is later compensated by paste intrusion or remains partially unfilled depends on the local intrusion conditions [18,19]. Either case alters the steel-to-concrete contact compared to a conventionally cast geometry.

Bond stresses are transferred along the interface between reinforcement and surrounding concrete and are commonly described in terms of the bar surface area within the bond length. The bond behaviour is governed by adhesion, mechanical interlock at the ribs, and friction due to relative displacement [24,25,26]. At the onset of loading, adhesion and local contact mechanisms dominate, but they are lost at very small relative displacements. The bond is then primarily governed by mechanical interlock at the ribs, the dominant load-transfer mechanism over a wide displacement range. With increasing load, damage accumulates and the concrete between the ribs is progressively sheared off, leading to the maximum bond stress. Once this resistance is exhausted, friction along the sheared concrete surface remains as the only relevant load-transfer mechanism. Disturbances in material distribution near the reinforcement may therefore affect both early bond mobilisation and the maximum achievable bond capacity. In addition, sufficient concrete cover or radial confinement is required to ensure that the measured response is governed by pull-out rather than splitting failure [27].

The primary influencing parameters are the bar diameter d and the bar orientation relative to the particle bed surface, described by the inclination angle $\theta$ between the bar axis and the bed surface ($\theta =0$: bar parallel to the bed surface, $\theta =90$: bar orthogonal to it, see fig:concept). As the diameter increases, both the projected shadowed region and the layer-wise affected contact zone along the bar surface enlarge. As the inclination angle decreases from 90 toward 0, the projected shadow area increases, while the layer-wise contact-zone effect is expected to remain largely unchanged. Based on this, the working hypothesis is that larger diameters and smaller inclination angles lead to less favourable material deposition beneath the bar and consequently to reduced bond properties.

2. Materials and Methods

2.1. Objective and Experimental Design

The investigation is based on a systematic test series with varying bar diameters and inclination angles. Bond behaviour was tested in a push-through configuration that adopts the bond length, the loading rate, and the bond stress evaluation from RILEM RC6 [28]. In contrast to the pull-out setup defined there, the bar was loaded in compression (sec:setup). RILEM RC6 formulates its procedure for bar diameters of at least 10 $\mathrm{m}$$\mathrm{m}$ and for comparisons between bars of approximately equal diameter. The present series extends the configuration to 8 $\mathrm{m}$$\mathrm{m}$ and to comparisons across diameters, so the RC6-derived values serve as internally consistent comparative measures rather than normative bond strengths.

2.2. Materials

Specimens were produced using the Selective Paste Intrusion process. Aggregate layers with a thickness of 3 $\mathrm{m}$$\mathrm{m}$ were deposited successively and selectively infiltrated with cement paste until the target height was reached.

Quartz sand with a grain size of 1.0$\mathrm{m}$$\mathrm{m}$–2.2$\mathrm{m}$$\mathrm{m}$ was used as aggregate. The cement paste consisted of an ordinary Portland cement with strength class 42.5 (CEM I 42.5 N per EN 197-1 [29]) with a water-to-cement ratio of 0.40 and a polycarboxylate-based superplasticiser. The flowability of the cement paste was characterised by its spread-flow, measured with a Haegermann cone (cone geometry as specified in EN 1015-3 [30]) placed on a flat glass plate and lifted vertically. In contrast to the jolting procedure of the standard, the paste was allowed to spread under gravity without any jolts, so that the spread is governed by the paste yield stress alone. The spread was adjusted to 400$\mathrm{m}$$\mathrm{m}$–410$\mathrm{m}$$\mathrm{m}$ by varying the superplasticiser dosage. At this flow level, the SPI concrete can be considered isotropic in compressive strength with respect to the layer orientation, as demonstrated for a paste with the same aggregate and the same spread-flow level ( 400 $\mathrm{m}$$\mathrm{m}$), albeit with a water-to-cement ratio of 0.30 [2].

Ribbed reinforcing steel bars of grade B500B [31] (characteristic yield strength $f_{yk}=500\mathrm{M}\mathrm{Pa}$, ductility class B) served as reinforcement. Their diameters, geometry, and configuration are described in sec:setup.

After 24 $\mathrm{h}$, the specimens were excavated from the particle bed and subsequently stored for at least 28 days under constant ambient conditions ( 20 °C, 65 relative humidity) prior to testing.

2.3. Specimen Geometry and Reinforcement Configuration

Ribbed reinforcing bars of grade B500B with diameters of 8 $\mathrm{m}$$\mathrm{m}$, 16 $\mathrm{m}$$\mathrm{m}$, and 25 $\mathrm{m}$$\mathrm{m}$ were used. The bars were embedded at inclination angles $\theta$ between 0 (parallel to the particle bed surface) and 90 (orthogonal to the particle bed surface) in 15 increments. Five specimens were produced for each combination of diameter and angle.

The maximum protrusion of the reinforcement was limited to 40 $\mathrm{m}$$\mathrm{m}$ above the particle bed surface, measured vertically, corresponding to the largest nozzle-to-bed distance at which shape accuracy remained consistent in [3].

According to RILEM RC6 [28], a bond length of $5d$ is defined, corresponding to 40 $\mathrm{m}$$\mathrm{m}$, 80 $\mathrm{m}$$\mathrm{m}$, and 125 $\mathrm{m}$$\mathrm{m}$ for bar diameters of 8 $\mathrm{m}$$\mathrm{m}$, 16 $\mathrm{m}$$\mathrm{m}$, and 25 $\mathrm{m}$$\mathrm{m}$, respectively. With a constant vertical protrusion of $h=40\mathrm{m}\mathrm{m}$, measured to the upper tip of the bar, the required bond length exceeds the available bar length above the particle bed for most diameter–angle combinations. The bars must therefore be segmented and integrated stepwise (sec:fabrication). The maximum bar length per segment follows from the condition that the upper tip of a protruding segment must not exceed the protrusion limit h. The vertical extent of a protruding segment consists of the axial rise $L_{\mathrm{eff}}sin\theta$ and the vertical width $dcos\theta$ of the inclined bar cross-section, measured between the end of the lower bar surface and the upper tip. Setting $L_{\mathrm{eff}}sin\theta +dcos\theta =h$ and solving for the segment length gives $L_{\mathrm{eff}}=(h-dcos\theta )/sin\theta$ for $\theta >0$ (all lengths in $\mathrm{m}\mathrm{m}$). In descriptive terms, the correction $dcos\theta$ reflects that the bar is not an infinitely thin axis: the inclined cross-section itself occupies part of the admissible protrusion height (see the dimension chain in fig:shadowingb). As a consequence, $L_{\mathrm{eff}}$ does not vary monotonically with the inclination angle. Starting from 40 $\mathrm{m}$$\mathrm{m}$ at $\theta =90$, it first decreases with decreasing angle, reaches a minimum between 45 and 75 depending on the diameter, and only then increases again, because a flatter bar needs more axial length for the same vertical rise (tab:segmentation). This non-monotonic behaviour carries over to the segmentation, since no segment may exceed $L_{\mathrm{eff}}$. The required bar length above the bed is the bond length plus 10 $\mathrm{m}$$\mathrm{m}$ ($\pm 2\mathrm{m}\mathrm{m}$) of free bar end for the displacement sensor. For all configurations with more than one segment, the as-built segment lengths were chosen as practical lengths that do not exceed $L_{\mathrm{eff}}$ beyond the cutting tolerance of $\pm 2\mathrm{m}\mathrm{m}$, and the last segment of each bar was shortened to the remaining length required to complete the bond length plus the free bar end. If $L_{\mathrm{eff}}$ falls below a round length, the segments follow $L_{\mathrm{eff}}$ directly: at 60 and 75 for 16 $\mathrm{m}$$\mathrm{m}$, for example, $L_{\mathrm{eff}}=37.0\mathrm{m}\mathrm{m}$ and $37.1$ $\mathrm{m}$$\mathrm{m}$ limit the segments to 37 $\mathrm{m}$$\mathrm{m}$, so the as-built division is $37+37+16$. The same rule produces the divisions of the 25 $\mathrm{m}$$\mathrm{m}$ series, for example $30+30+30+30+15$ at 45, where $L_{\mathrm{eff}}=31.6\mathrm{m}\mathrm{m}$. tab:segmentation compares the available and required lengths, gives the resulting number of segments, and lists the as-built segment lengths.

Table 1. Maximum effective bar length $L_{\mathrm{eff}}=(h-dcos\theta )/sin\left(\theta \right)$ above the particle bed at $h=40\mathrm{m}\mathrm{m}$ vertical protrusion, number of segments n, and as-built segment lengths for all diameter–angle combinations. The required total length is the bond length $5d$ plus 10$\mathrm{m}$$\mathrm{m}$ of free bar end for the displacement sensor ($\pm 2\mathrm{m}\mathrm{m}$). Segments were cut to millimetre precision within a tolerance of $\pm 2\mathrm{m}\mathrm{m}$. Rounding to full millimetres and the continuous 45 configuration of the 8$\mathrm{m}$$\mathrm{m}$ series can marginally exceed the nominal $L_{\mathrm{eff}}$ (by up to $1.4$$\mathrm{m}$$\mathrm{m}$) and hence the momentary protrusion, within this tolerance. The last segment includes the free bar end.

d (mm)	req. (mm)	$\mathit{\theta }$	${\mathit{L}}_{\mathbf{eff}}$ (mm)	n	Segment lengths (mm)
8	50	0	—a	1	continuous
		15	124.7	1	continuous
		30	66.1	1	continuous
		45	48.6	1	continuous
		60	41.6	2	$40+10$
		75	39.3	2	$40+10$
		90	40.0	2	$40+10$
16	90	0	—a	1	continuous
		15	94.8	1	continuous
		30	52.3	2	$50+40$
		45	40.6	3	$40+40+10$
		60	37.0	3	$37+37+16$
		75	37.1	3	$37+37+16$
		90	40.0	3	$40+40+10$
25	135	0	—a	1	continuous
		15	61.2	3	$60+60+15$
		30	36.7	4	$35+35+35+30$
		45	31.6	5	$30+30+30+30+15$
		60	31.8	5	$30+30+30+30+15$
		75	34.7	4	$35+35+35+30$
		90	40.0	4	$40+40+40+15$
a At 0 the bar lies parallel to the particle bed at constant offset of 40 $\mathrm{m}$$\mathrm{m}$. Length is then governed by the bond length plus the bar ends for load application and displacement measurement.

Table 1. Maximum effective bar length $L_{\mathrm{eff}}=(h-dcos\theta )/sin\left(\theta \right)$ above the particle bed at $h=40\mathrm{m}\mathrm{m}$ vertical protrusion, number of segments n, and as-built segment lengths for all diameter–angle combinations. The required total length is the bond length $5d$ plus 10$\mathrm{m}$$\mathrm{m}$ of free bar end for the displacement sensor ($\pm 2\mathrm{m}\mathrm{m}$). Segments were cut to millimetre precision within a tolerance of $\pm 2\mathrm{m}\mathrm{m}$. Rounding to full millimetres and the continuous 45 configuration of the 8$\mathrm{m}$$\mathrm{m}$ series can marginally exceed the nominal $L_{\mathrm{eff}}$ (by up to $1.4$$\mathrm{m}$$\mathrm{m}$) and hence the momentary protrusion, within this tolerance. The last segment includes the free bar end.

d (mm)	req. (mm)	$\mathit{\theta }$	${\mathit{L}}_{\mathbf{eff}}$ (mm)	n	Segment lengths (mm)
8	50	0	—a	1	continuous
		15	124.7	1	continuous
		30	66.1	1	continuous
		45	48.6	1	continuous
		60	41.6	2	$40+10$
		75	39.3	2	$40+10$
		90	40.0	2	$40+10$
16	90	0	—a	1	continuous
		15	94.8	1	continuous
		30	52.3	2	$50+40$
		45	40.6	3	$40+40+10$
		60	37.0	3	$37+37+16$
		75	37.1	3	$37+37+16$
		90	40.0	3	$40+40+10$
25	135	0	—a	1	continuous
		15	61.2	3	$60+60+15$
		30	36.7	4	$35+35+35+30$
		45	31.6	5	$30+30+30+30+15$
		60	31.8	5	$30+30+30+30+15$
		75	34.7	4	$35+35+35+30$
		90	40.0	4	$40+40+40+15$
a At 0 the bar lies parallel to the particle bed at constant offset of 40 $\mathrm{m}$$\mathrm{m}$. Length is then governed by the bond length plus the bar ends for load application and displacement measurement.

For the special case of 0, the bar was positioned entirely above the particle bed at a constant vertical offset of 40 $\mathrm{m}$$\mathrm{m}$, measured to the upper bar edge. In this configuration, the bar was longer than the bond length and extended beyond the printed specimen on both sides. Only the bond length was overprinted, while the protruding bar ends on both sides served for load application and displacement measurement.

The need for segmentation has implications for the choice of test method. The individual bar segments were joined with a thin layer of cyanoacrylate adhesive during specimen fabrication. The adhesive served only as temporary fixation and did not contribute to load transfer during testing. In a conventional pull-out test, tensile forces would act on the adhesive joints at the segment interfaces and could lead to premature failure at these locations rather than mobilising the bond between reinforcement and concrete. To avoid this, a modified push-through test was performed, in which the bar is pushed through the specimen rather than pulled. This results in purely compressive loading of the reassembled segmented bar, so that the adhesive joints do not lead to premature failure. The measured force–displacement behaviour therefore reflects the steel-to-concrete bond rather than the strength of the segment joints. Local irregularities of the bond in the immediate vicinity of the joints may nonetheless remain and constitute a process-related source of scatter.

2.4. Bond Stress and Geometric Shadowing Descriptors

The bond stress $\tau$ was calculated assuming a uniform bond stress over the bond length, following the evaluation formula of RILEM RC6 [28] without its normalisation to a reference concrete strength, as

$$\tau ={\displaystyle \frac{F}{\pi ·d·l_b}}$$

(1)

with F the recorded force in $\mathrm{N}$, d the bar diameter in $\mathrm{m}\mathrm{m}$, and $l_b=5d$ the bond length in $\mathrm{m}\mathrm{m}$, so that $\tau$ follows in $\mathrm{M}\mathrm{Pa}$.

To quantify the geometric component of shadowing, the momentary shadow cast by the protruding bar is considered first. At any stage of the printing sequence, a bar segment protrudes above the current bed surface by up to $h=40\mathrm{m}\mathrm{m}$, measured vertically to the upper tip of the bar (fig:shadowingb). Material applied vertically from above is largely prevented from reaching the strip of the bed surface that lies in the horizontal projection of the protruding part. For a bar of diameter d at inclination angle $\theta$, this momentary shadow extends from the point at which the lower bar surface exits the bed. Its horizontal extent, the reach $r\left(\theta \right)$ in $\mathrm{m}\mathrm{m}$, is

$$r\left(\theta \right)={\displaystyle \frac{h-dcos\theta }{tan\theta }}$$

(2)

valid for $0<\theta <90$, where the term $dcos\theta$ accounts for the vertical offset between the upper bar tip, at which the protrusion is measured, and the end of the lower bar surface, from which the shadow extends (h and d in $\mathrm{m}\mathrm{m}$). The two limiting angles are treated below. Both contributions are indicated by the dimension chain in fig:shadowingb. The momentary shadowed area on the bed surface is $a_{\mathrm{shadow}}\left(\theta \right)=d·r\left(\theta \right)$, in $\square \mathrm{m}\mathrm{m}$.

The momentary shadow is not stationary. As printing proceeds, the bed surface rises layer by layer, the protrusion of the current segment decreases, and the shadow shortens while its starting point travels up the bar. Once a segment is fully overprinted, the next segment restores the protrusion and the same process repeats. Each fully overprinted segment of length ${\ell }_i$ along the bar axis, with the running index $i=1,\dots ,n$ numbering the segments of tab:segmentation, advances the starting point of the shadow by its horizontal projection ${\ell }_{i}cos\theta$. The union of all momentary shadows over the printing sequence therefore forms a continuous strip on the bed surface beneath the bar within the concrete specimen. The accumulated shadowed area follows in three steps: the strip contributions of the n segments are summed, the segment lengths within the bond length add up to the bond length (${\sum }_{i=1}^n{\ell }_i=l_b$, with $n=1$ for the continuous configurations of tab:segmentation), and the RILEM bond length $l_b=5d$ is inserted:

$$A_{\mathrm{shadow}}=d·{\displaystyle \sum _{i=1}^n}{\ell }_{i}cos\theta =d·l_b·cos\theta =5d^{2}cos\theta$$

(3)

with d, ${\ell }_i$, and $l_b$ in $\mathrm{m}\mathrm{m}$ and $A_{\mathrm{shadow}}$ in $\square \mathrm{m}\mathrm{m}$. The result is independent of the number and lengths of the individual segments and of the protrusion height h, which enter only the momentary shadow. The finite-diameter correction $dcos\theta$ in eq:reach shifts only the front of the momentary shadow within the strip and does not alter the total swept area. eq:Ashadow thus depends only on d and $\theta$, not on the fabrication parameters h or the segmentation. This distinguishes $A_{\mathrm{shadow}}$ from $A_{\mathrm{contact}}$ (introduced below), which is independent of $\theta$. For $\theta =0$ (bar parallel to the particle bed) no segmentation occurs and the shadow is no longer described by eq:reach: the entire lower side of the bar shades the bed simultaneously, bounded by the overprinted bond length. In this limiting case the momentary shadow spans the entire strip at once, $a_{\mathrm{shadow}}$ and $A_{\mathrm{shadow}}$ coincide, and eq:Ashadow reduces to $d·l_b$. For all inclined configurations the momentary shadow is a shorter strip that travels along the bar during overprinting. For $\theta =90$ (bar orthogonal to the particle bed) the projected shadow vanishes.

To describe the layer-wise process-induced manifestation of shadowing, the affected region is taken to be the lower half of the bar surface within the bond length:

$$A_{\mathrm{contact}}={\displaystyle \frac{\pi ·d·l_b}{2}}={\displaystyle \frac{5}{2}}\pi d^2$$

(4)

where the second expression follows by inserting the RILEM bond length $l_b=5d$ (d and $l_b$ in $\mathrm{m}\mathrm{m}$, $A_{\mathrm{contact}}$ in $\square \mathrm{m}\mathrm{m}$).

The geometric reasoning rests on the sequential overprinting of the protruding bar segments. At every printing step a portion of the bar protrudes above the particle bed by up to 40 $\mathrm{m}$$\mathrm{m}$, and material applied from above, aggregate during spreading as well as cement paste during intrusion, is blocked from the lower side of this protruding portion. The degree to which trickling aggregate and locally flowing cement paste compensate this blockage is addressed in sec:results. Once a segment is fully overprinted, a new segment is added that again protrudes by 40 $\mathrm{m}$$\mathrm{m}$, and the same shielding mechanism repeats. For the special case $\theta =0$, where the bar lies parallel to the bed and is not segmented, the lower side is shielded continuously rather than segment-wise. The affected region, the lower half of the bar surface, is the same. Over the sequence of segments, the lower half of the bar surface remains continuously shielded and the resulting disturbance extends along the full bond length. Owing to the cylindrical symmetry of the bar, the shielded half always covers half of the lateral surface area, regardless of inclination, as long as a defined upper and lower side of the bar exists. The descriptor is therefore taken as constant in $\theta$ for $0\le \theta <90$. At $\theta =90$ the bar axis becomes parallel to the deposition direction, no upper or lower side can be defined, and only a small front-facing cross-section of area $\pi d^2/4$ (plus the local rib protrusions) obstructs the material flow. This end face is not part of the bond-transferring bar surface. The mechanism described by $A_{\mathrm{contact}}$ therefore ceases to apply at this angle, and the configuration is treated as a special case in the discussion. The $d^2$ form of eq:Ashadow,eq:Acontact is a direct consequence of the RILEM bond length convention $l_b=5d$ [28] rather than a mechanistic prediction in itself. Both descriptors are illustrated in fig:shadowing: the momentary shadow, whose union over the printing sequence forms $A_{\mathrm{shadow}}$, is shown in red on the particle bed surface beneath the protruding bar, and $A_{\mathrm{contact}}$ is shown in orange along the lower half of the bar surface within the bond length.

2.5. Specimen Fabrication and Setup

The specimens were inclined cylinders with an as-printed diameter of 100 $\mathrm{m}$$\mathrm{m}$, their longitudinal axis aligned with the prescribed bar inclination angle. To realise this geometry within the SPI process, a cubic holder with an edge length of 100 $\mathrm{m}$$\mathrm{m}$ was placed on the base plate of the SPI printer. The holder contained a through-hole whose axis followed the prescribed inclination angle and whose diameter matched the bar diameter. Figure 3 shows a cross-sectional schematic of the specimen geometry and the fabrication process.

Figure 3. Cross-sectional schematic of the specimen geometry and the two shadowing manifestations. (a) Printing phase I: the lower specimen portion (yellow) is produced without the bar, and the particle bed is filled up to the holder top. (b) Printing phase II, the overprinting with the bar in place (blue area): the bar protrudes by up to $h=40\mathrm{m}\mathrm{m}$ above the current bed surface, and the momentary shadow (red outline, $a_{\mathrm{shadow}}\left(\theta \right)$) marks the geometrically shielded region. The shadowed area itself lies on the bed surface. The red patch marks its trace in the cross-sectional plane. The union of these momentary shadows over the printing sequence forms $A_{\mathrm{shadow}}$. (c) Completed specimen: the holder (slate-grey) supports the bar, and $A_{\mathrm{contact}}$ (orange) marks the affected contact zone along the lower half of the bar surface within the bond length $5d$.

Figure 3. Cross-sectional schematic of the specimen geometry and the two shadowing manifestations. (a) Printing phase I: the lower specimen portion (yellow) is produced without the bar, and the particle bed is filled up to the holder top. (b) Printing phase II, the overprinting with the bar in place (blue area): the bar protrudes by up to $h=40\mathrm{m}\mathrm{m}$ above the current bed surface, and the momentary shadow (red outline, $a_{\mathrm{shadow}}\left(\theta \right)$) marks the geometrically shielded region. The shadowed area itself lies on the bed surface. The red patch marks its trace in the cross-sectional plane. The union of these momentary shadows over the printing sequence forms $A_{\mathrm{shadow}}$. (c) Completed specimen: the holder (slate-grey) supports the bar, and $A_{\mathrm{contact}}$ (orange) marks the affected contact zone along the lower half of the bar surface within the bond length $5d$.

Printing proceeded in two phases. In phase I, the lower part of the specimen (yellow area) was printed without the bar. Because of the inclination, this part lay below the holder top and therefore had to be produced before the bar was inserted. Before printing, the upper opening of the hole was sealed with adhesive tape to prevent aggregate from entering during the first printing phase.

After phase I, the adhesive tape was removed and the reinforcement bar was inserted through the hole in the holder. In phase II, printing continued with the bar in place (blue area). The print head was kept at a constant offset of 40 $\mathrm{m}$$\mathrm{m}$ above the current bed surface [3], so that the bar initially protruded by this amount above the bed surface. With each successive layer, the vertical protrusion decreased by the layer thickness until the lower edge of the bar at its tip reached the level of the bed surface. At this stage, the full bar cross-section was exposed and the next bar segment was butt-joined with cyanoacrylate adhesive (Loxeal 32 Ethyl, Cesano Maderno, Italy), restoring the original protrusion of 40 $\mathrm{m}$$\mathrm{m}$ to the upper tip, and printing continued. This procedure was repeated until the upper part of the specimen reached the height corresponding to the bond length above the holder top. The length of the final segment was chosen such that it did not exceed 40 $\mathrm{m}$$\mathrm{m}$ and that 10 $\mathrm{m}$$\mathrm{m}$ ($\pm 2\mathrm{m}\mathrm{m}$) of free bar remained above the printed specimen for mounting the displacement sensor. For configurations at 15 to 90, the portion of the bar inside the holder constituted the unbonded length on the loading side during the subsequent push-through test.

To minimise geometric irregularities at the segment joints, the segments were prepared in advance by cutting a continuous bar with thin cutting blades and were then reassembled in their original order and orientation.

Figure 4 shows the configuration during printing.

2.6. Post-Processing and Specimen Preparation

To prevent splitting failure across the diameter range, the specimens were inserted into pre-cut steel confinement rings. For this purpose, all specimens were first reduced from their initial diameter of 100 $\mathrm{m}$$\mathrm{m}$ to a uniform outer diameter of 80 $\mathrm{m}$$\mathrm{m}$ by core drilling, which ensured a defined specimen geometry and a uniform lateral surface for bonding to the ring. The rings were dimensioned with a height equal to the bond length $5d$, so that the height of the embedded section after trimming corresponds exactly to the bond length. The steel rings were sandblasted prior to bonding to improve adhesion and bonded using an epoxy adhesive (Hilti HIT-RE 500, Schaan, Liechtenstein). The combined effect of the residual concrete cover and the steel ring was dimensioned to provide radial confinement at all bar diameters, so that the measured bond response is not dominated by splitting.

After adhesive curing for 48 $\mathrm{h}$ at 20 °C and 65 relative humidity, excess concrete above and below the rings was removed using a fine tile saw, so that the resulting specimens have a height equal to the defined bond length.

2.7. Mechanical Testing

Bond behaviour was determined using a modified push-through test on a universal testing machine (Zwick Z600, Ulm, Germany), see Figure 5a. The specimen was placed on a spherical seat to compensate for minor angular deviations introduced during core drilling and to ensure axial load application along the bar axis.

The bar protruded beyond the bonded section on the loading side. This protruding length was deliberately not trimmed after specimen preparation so as not to disturb the bond zone at the specimen face.

The test was run in force-controlled mode. The loading rate was adopted from RILEM RC6 [28] and computed as $\dot{F}=0.5·d^2$, with the bar diameter d inserted numerically in mm and the resulting loading rate in $\mathrm{N}/\mathrm{s}$. This gives nominal loading rates of 32 $\mathrm{N}$/$\mathrm{s}$, 128 $\mathrm{N}$/$\mathrm{s}$, and $312.5$ $\mathrm{N}$/$\mathrm{s}$ for the 8 $\mathrm{m}$$\mathrm{m}$, 16 $\mathrm{m}$$\mathrm{m}$, and 25 $\mathrm{m}$$\mathrm{m}$ bars, respectively.

Slip was measured at the unloaded bar end (the free end opposite to the load application point) using a linear displacement transducer (Ahlborn FWA025TR, Holzkirchen, Germany) mounted in a polymer holder clamped to the steel confinement ring, see Figure 5b. Force was recorded using the internal load cell of the testing machine.

2.8. Data Evaluation

For each diameter–angle combination, between three and five replicate specimens were tested. The nominal count was five, with deviations resulting from individual specimen exclusions due to fabrication defects. Bond stresses were extracted at four characteristic slip levels, ${\tau }_{0.001}$, ${\tau }_{0.01}$, ${\tau }_{0.1}$, and ${\tau }_{\max }$. The numeric subscript denotes the slip in mm at which the bond stress is evaluated: ${\tau }_{0.001}$ is read at a slip of $0.001$ $\mathrm{m}$$\mathrm{m}$ ( 1 $\mu$$\mathrm{m}$), ${\tau }_{0.01}$ at $0.01$ $\mathrm{m}$$\mathrm{m}$, and ${\tau }_{0.1}$ at $0.1$ $\mathrm{m}$$\mathrm{m}$, while ${\tau }_{\max }$ is the maximum bond stress, taken at the slip $s\left({\tau }_{\max }\right)$ at which it occurs. These four levels capture the response at very small slip, the early interlock stage, the developed interlock stage, and the bond capacity, respectively. At a slip of 1 $\mu$$\mathrm{m}$, ${\tau }_{0.001}$ lies at the resolution limit of the test setup (bedding-in of the spherical seat, compliance of the load path and of the segment joints) and is therefore best read as the system response at very small slip rather than as a pure chemical-adhesion measure.

3. Results and Discussion

The bond stress–slip curves for the three diameters are shown in Figure 6, Figure 7 and Figure 8. Each figure presents the individual curves for all replicates per inclination angle, the mean bond stresses at the four characteristic slip levels, and the angle dependence in the bottom-right panel.

Table 2 summarises the mean bond stresses and standard deviations at the four characteristic slip levels for all diameter–angle combinations.

Across all investigated parameters, the bond stress–slip curves exhibit the same characteristic shape. A shallow initial increase at very small displacements is followed by a steeper rise as mechanical interlock develops, before ${\tau }_{\max }$ is reached. A subsequent decrease occurs due to progressive damage within the bond zone. This sequence is in line with the classical adhesion–interlock–friction phasing reported for ribbed bars in conventional concrete [24,25].

The mean bond parameters ${\tau }_{0.001}$, ${\tau }_{0.01}$, ${\tau }_{0.1}$, and ${\tau }_{\max }$ as a function of inclination angle for the three diameters are shown in Figure 9.

The bar diameter dominates the bond response. The 8 $\mathrm{m}$$\mathrm{m}$ bars reach the highest bond stresses at every angle and every slip level. At the developed-interlock and capacity levels (${\tau }_{0.1}$ and ${\tau }_{\max }$) the full ordering $\tau \left(8\mathrm{m}\mathrm{m}\right)>\tau \left(16\mathrm{m}\mathrm{m}\right)>\tau \left(25\mathrm{m}\mathrm{m}\right)$ holds in the angle means, with a single interchange of the two larger diameters at 60 for ${\tau }_{\max }$. For ${\tau }_{\max }$, mean values for the 8 $\mathrm{m}$$\mathrm{m}$ bar range from approximately 33 $\mathrm{M}$$\mathrm{Pa}$ to 45 $\mathrm{M}$$\mathrm{Pa}$ depending on the angle, while values between approximately 12 $\mathrm{M}$$\mathrm{Pa}$ and 18 $\mathrm{M}$$\mathrm{Pa}$ are measured for 16 $\mathrm{m}$$\mathrm{m}$ and between approximately 6 $\mathrm{M}$$\mathrm{Pa}$ and 15 $\mathrm{M}$$\mathrm{Pa}$ for 25 $\mathrm{m}$$\mathrm{m}$. At the two early-slip levels the separation between the two larger diameters closes and partly inverts. For ${\tau }_{0.01}$, mean values range from approximately 7 $\mathrm{M}$$\mathrm{Pa}$ to 14 $\mathrm{M}$$\mathrm{Pa}$ for 8 $\mathrm{m}$$\mathrm{m}$, from approximately 3 $\mathrm{M}$$\mathrm{Pa}$ to 7 $\mathrm{M}$$\mathrm{Pa}$ for 16 $\mathrm{m}$$\mathrm{m}$, and from approximately 3 $\mathrm{M}$$\mathrm{Pa}$ to 9 $\mathrm{M}$$\mathrm{Pa}$ for 25 $\mathrm{m}$$\mathrm{m}$. The diameter effect is therefore most pronounced near the bond capacity.

The magnitude of this effect exceeds the conventional bar-diameter dependence. Bond tests on conventional concrete indicate a reduction of roughly 25–40 in bond strength as the bar diameter increases from 10 $\mathrm{m}$$\mathrm{m}$ to 50 $\mathrm{m}$$\mathrm{m}$[26], and for 10$\mathrm{m}$$\mathrm{m}$–16$\mathrm{m}$$\mathrm{m}$ bars at constant failure mode the diameter effect was even found to be insignificant [32]. The classical size effect is primarily driven by brittle splitting and diminishes with increasing confinement [33], although a size-dependent bond response has also been reported under confined conditions [34]. Since splitting is suppressed here by the steel confinement rings, only a minor contribution of this mechanism can be expected. The mean ${\tau }_{\max }$ differs by a factor of about 3.9 between the 8 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ bars and thus clearly exceeds the magnitude of conventional size effects. An additional process-related contribution acting more strongly on larger bars must therefore be present. This contribution is identified with the layer-wise contact-zone manifestation of shadowing below. A comparison with extrusion-based 3D concrete printing underlines that the direction of the diameter effect is process-specific: there, the bond strength of bars placed between layers was higher for 10 $\mathrm{m}$$\mathrm{m}$ than for 6 $\mathrm{m}$$\mathrm{m}$ bars, attributed to the larger absolute rib height of the tested bars, while the porosity at the steel–concrete interface correlated linearly and negatively with the achievable bond [35]. The direction of the diameter effect thus reflects the respective interface formation and rib geometry rather than a universal trend.

By contrast, the inclination angle does not show a consistent monotonic trend over the range from 0 to 90. Local extrema occur for individual angles, but no systematic ordering can be identified by visual inspection.

The projected-shadow component of the working hypothesis formulated in the introduction predicts a decrease in bond stress with increasing $A_{\mathrm{shadow}}$. Figure 10 shows ${\tau }_{0.001}$, ${\tau }_{0.01}$, ${\tau }_{0.1}$, and ${\tau }_{\max }$ as a function of $A_{\mathrm{shadow}}$.

The data points are scattered over the entire range of the shadowed area $A_{\mathrm{shadow}}$. Rather than a uniform $\tau$–$A_{\mathrm{shadow}}$ relationship, distinct clusters emerge according to bar diameter. Within each cluster, no clear monotonic structure with $A_{\mathrm{shadow}}$ can be identified. In regions of overlapping shadowed areas, where combinations of different diameters and angles lead to similar values of the shadowed area $A_{\mathrm{shadow}}$, the corresponding bond stresses remain separated by diameter, at ${\tau }_{\max }$ by up to a factor of about three, and no convergence of bond parameters occurs. A small secondary contribution of $A_{\mathrm{shadow}}$ cannot be ruled out by the present data: a region in the particle bed shielded against deposition can be expected to produce a locally weakened concrete matrix [18,19]. The projected shadowed area, although demonstrably present, therefore acts at most as a secondary factor, and the dominant mechanism must lie elsewhere.

The phenomenon is directly visible in the printed specimens. Figure 11 shows a specimen with a clearly identifiable aggregate-depleted zone in the particle bed beneath the protruding bar, exactly in the geometric position predicted by eq:reach. During each spreading step, aggregate trickles from above and partially re-fills the region beneath the bar, but the deposited sand layer remains locally thinner there. The dark patch in Figure 11 is the cement paste of the underlying, already intruded layer, which shines through this locally thinner sand cover. Cement paste, unlike the aggregate, is not restricted to deposition from above: it wets the bar, flows around its circumference, and can drip from the underside, as the paste accumulations visible on the bar underside in Figure 11 confirm. This asymmetry between blocked granular transport and liquid paste transport renders the zone beneath the bar paste-rich but aggregate-poor. The limited weight of $A_{\mathrm{shadow}}$ in the bond data is therefore not a question of whether the phenomenon exists, but of how much it contributes relative to other mechanisms. Inserting eq:Ashadow and the bar surface area within the bond length, $\pi d{l}_b$, the shadowed fraction of the bond surface becomes $A_{\mathrm{shadow}}/\left(\pi d{l}_b\right)=(d{l}_{b}cos\theta )/\left(\pi d{l}_b\right)=cos\theta /\pi$. This ratio is independent of bar diameter and bond length and reaches its maximum of $1/\pi \approx 32\%$ at $\theta =0$. The shadowed volume affects only the matrix in a thin lateral strip beneath the bar, while the load-bearing concrete around the rest of the bar circumference remains unaffected. The concrete compression cones that form at the ribs during mechanical interlock bear partly on this weakened strip, but the affected fraction of each cone is small relative to the full bearing area around the bar circumference. The independence of $A_{\mathrm{shadow}}/\left(\pi d{l}_b\right)$ from diameter is consistent with the observation that $A_{\mathrm{shadow}}$ does not account for the pronounced diameter effect in the data.

The complementary descriptor $A_{\mathrm{contact}}$ predicts a disturbed contact zone along the lower half of the bar surface within the bond length, formed by the sequential overprinting of protruding bar segments. Several process-related effects act simultaneously within this zone. Aggregate particles can trickle and rearrange beneath the bar during layer deposition, and cement paste can flow into the locally less accessible regions. Additional rearrangement processes within the particle bed modify the local packing state. These mechanisms partially counteract a purely geometric shadowing. The affected region should therefore not be interpreted as a material-free void, but as a structurally modified contact zone. A comparable partial filling of the disturbed zone has been reported for paste-coated bar penetrations in 3D concrete printing, where the deliberately applied paste filled the induced voids over the upper 47–56 of the penetration depth and increased the flexural capacity of the reinforced sections by about 50 [36]. At the lower part of such penetrations, the surrounding printed material was observed to compact against the bar even without coating, producing high bond [36,37]. How much paste actually flows into the region beneath the bar cannot be quantified from the present data. A geometric bound nonetheless exists. The redistribution length required to reach the region beneath a bar is of the order of the bar radius: about 4 $\mathrm{m}$$\mathrm{m}$, or two to four grain diameters, for the 8 $\mathrm{m}$$\mathrm{m}$ bars, but more than 12 $\mathrm{m}$$\mathrm{m}$ for the 25 $\mathrm{m}$$\mathrm{m}$ bars. Aggregate trickling and paste flow over a few grain diameters can therefore compensate a substantial part of the disturbance beneath the small bars, while the same short-range redistribution leaves most of the zone beneath the large bars unfilled. Because the required transport lengths remain in the range of a few millimetres, this compensation is also consistent with the shape fidelity of the printed specimens, for which paste spreading beyond the intrusion front is limited to a comparable scale [18,19]. Its direct measurement by sectioning and computed tomography is addressed in sec:conclusion. The observation that the diameter effect dominates the response across all slip levels is consistent with a disturbance whose severity increases with d, and the absence of a monotonic angle trend matches the constancy of $A_{\mathrm{contact}}$ in $\theta$.

The push-through evaluation normalises the recorded force over the full surface area of the embedded bar via eq:tau. The geometric size of the bar is therefore explicitly accounted for in the evaluation, and a straightforward expectation would be that differences in diameter normalise out and yield comparable bond stresses. The data show a pronounced diameter effect nonetheless. This indicates that the geometric bar surface does not contribute uniformly to load transfer: a fraction of the contact area is structurally impaired by the layer-wise manufacturing process, and the impaired fraction, or the disturbance intensity within it, increases with bar diameter. The layer-wise contact-zone disturbance, the second manifestation of shadowing, is therefore identified as the primary driver of the observed ordering $\tau \left(8\mathrm{m}\mathrm{m}\right)>\tau \left(16\mathrm{m}\mathrm{m}\right)>\tau \left(25\mathrm{m}\mathrm{m}\right)$ at the developed-interlock and capacity levels (${\tau }_{0.1}$ and ${\tau }_{\max }$). Such an obstacle-size scaling is supported by the physics of granular deposition around a cylindrical obstacle: in controlled experiments on dense granular flow past an immersed cylinder, the drag force on the obstacle grows nearly linearly with the cylinder diameter, decreases with grain size, and is independent of the flow rate [38]. The closest additive-manufacturing analogue points in the same direction: the air-void field forming beneath and around an integrated reinforcement bar in 3D concrete printing enlarges with increasing bar diameter at fixed paste, shown numerically for bar diameters of 6$\mathrm{m}$$\mathrm{m}$–12$\mathrm{m}$$\mathrm{m}$ [39], and the interfacial transition zone around larger aggregate inclusions is more porous at fixed water-to-cement ratio and identical cement [40], a size scaling that can be expected to transfer to cylindrical bars. Normalisation over the full bar surface removes the geometric scaling but not this process-induced inhomogeneity of the contact zone. The observed diameter effect is thus a structural, not a purely geometric, effect of bond formation. These analogues support the direction of the diameter dependence. Its magnitude here, the factor of about 3.9 between the 8 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ bars (tab:resultssummary), exceeds what any of them quantifies.

In addition, smaller bars can be more effectively surrounded by aggregate and paste during the deposition process, which facilitates material flow into the contact zone and can partially compensate for the disturbance. This grain-scale compensation acts on a different length scale from the obstacle-scale disturbance discussed above and does not contradict it: the near-contact packing perturbation scales with the grain size and is therefore relatively larger, in proportion to the bar circumference, for the smaller bars, whereas the obstacle-induced disturbance scales with the bar diameter. The two effects act in opposite senses. The grain-size scaling of the near-contact perturbation is consistent with wall-effect packing studies at cylindrical boundaries, which report a disturbed layer extending about two particle diameters from the wall [20], with the wall effect that reinforcement exerts on the packing density of concrete, accounted for explicitly in mix-design models [41], with sphere-packing simulations [22] and X-ray CT measurements [23] showing elevated near-wall porosity, and with paste-penetration models for sand packings, which predict easier intrusion with increasing packing porosity and grain size [18,19].

The special case of 90 acts as a qualitative consistency check for the contact-zone interpretation. As established in Section 2.4, the mechanism captured by $A_{\mathrm{contact}}$ does not apply at this angle, material accumulates rotationally symmetrically around the vertical bar, and a higher bond would be expected. The data show this trend. The highest or among the highest mean values of ${\tau }_{\max }$ within each diameter occur at 90: $45.1$ $\mathrm{M}$$\mathrm{Pa}$ for 8 $\mathrm{m}$$\mathrm{m}$, $17.7$ $\mathrm{M}$$\mathrm{Pa}$ for 16 $\mathrm{m}$$\mathrm{m}$, and $14.8$ $\mathrm{M}$$\mathrm{Pa}$ for 25 $\mathrm{m}$$\mathrm{m}$. The trend is clearest for the 16 $\mathrm{m}$$\mathrm{m}$ bar, where 90 is distinctly the maximum. For the 8 $\mathrm{m}$$\mathrm{m}$ bar the 90 value is essentially indistinguishable from the 45 value ( $44.9$ $\mathrm{M}$$\mathrm{Pa}$), and for the 25 $\mathrm{m}$$\mathrm{m}$ bar the 60 value ( $15.1$ $\mathrm{M}$$\mathrm{Pa}$) marginally exceeds the 90 value, both differences lying within the scatter of the respective configurations.

Three influences of the test configuration require examination as potential confounders of this interpretation: the Poisson effect of the push-through loading, the orientation of the layer planes relative to the loading direction, and the radial confinement configuration.

The push-through configuration subjects the bar to axial compression, which causes a lateral expansion of the bar cross-section due to the Poisson effect and thereby increases the radial contact pressure between bar and concrete. In a pull-out test, the bar contracts and the radial pressure decreases. Push-through tests therefore tend to produce higher absolute bond stresses than pull-out tests on otherwise identical specimens: for bars under externally applied lateral pressure, an increased radial pressure at the interface has been shown to raise the bond resistance, in those tests by up to 200 [42].

The size of this contribution can be estimated from the elastic constants of the bar. The axial stress in the bar follows from the recorded force as $\sigma =F/(\pi d^2/4)$. Expressing F through eq:tau as $F=\tau ·\pi ·d·l_b$ and inserting $l_b=5d$ gives

$$\sigma ={\displaystyle \frac{\tau ·\pi ·d·5d}{\pi d^2/4}}=20\tau .$$

(5)

The unconstrained lateral expansion of the bar diameter under this stress is $\Delta d=\nu \sigma d/E_s$, with the Poisson ratio $\nu =0.3$ and the elastic modulus $E_s=200\mathrm{G}\mathrm{Pa}$ of reinforcing steel. Within the elastic range, this expansion amounts to approximately 5$\mu$$\mathrm{m}$–11$\mu$$\mathrm{m}$ across the three diameters, below $1.5$ of the respective mean rib heights (Table 3, Table 4 and Table 5). The radial pressure that this expansion generates against the surrounding concrete scales linearly with $\sigma$ and hence with $\tau$ itself. Modelling the specimen as a thick-walled cylinder (inner radius $d/2$, outer radius 40 $\mathrm{m}$$\mathrm{m}$) with an assumed matrix modulus of 20$\mathrm{G}$$\mathrm{Pa}$–30$\mathrm{G}$$\mathrm{Pa}$ yields a radial pressure of approximately 2–3.5 of $\sigma$, with a geometry factor that varies by less than 20 between the 8 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ configurations (the outer steel ring stiffens this response similarly for all configurations). For the 16 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ series, whose axial stresses remain elastic throughout (at most 354 $\mathrm{M}$$\mathrm{Pa}$ and 302 $\mathrm{M}$$\mathrm{Pa}$, respectively), the Poisson-induced self-confinement amplifies the response by a comparable relative amount and cannot generate their separation. For the 8 $\mathrm{m}$$\mathrm{m}$ series the elastic estimate is a lower bound, since the implied axial stresses exceed the elastic range at ${\tau }_{\max }$ (see below). It raises the absolute level of the measured bond stresses, and it acts most strongly on the small bars, which reach the highest bond stresses and hence the highest axial stresses.

One restriction on the absolute values follows from eq:axialstress. With $\sigma =20{\tau }_{\max }$, the 8 $\mathrm{m}$$\mathrm{m}$ bars imply axial compressive stresses of 670$\mathrm{M}$$\mathrm{Pa}$–900$\mathrm{M}$$\mathrm{Pa}$ at ${\tau }_{\max }$, at or above the nominal yield strength of B500B ( 500 $\mathrm{M}$$\mathrm{Pa}$). Local plastic compression of the bar, together with the associated increased lateral expansion, may thus contribute to the high ${\tau }_{\max }$ recorded for this diameter and to a part of the factor of about 3.9 between the 8 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ series. The ordering of the three diameters is unaffected, since it is already present at ${\tau }_{0.1}$, where all series remain elastic, but the absolute ${\tau }_{\max }$ values of the 8 $\mathrm{m}$$\mathrm{m}$ series should be read with this restriction in mind.

The layer-wise fabrication introduces a systematic variation of the angle between the loading direction and the layer planes across the test series. At $\theta =0$ the push-through force acts parallel to the layers, at $\theta =90$ perpendicular to them. If the SPI matrix exhibited pronounced mechanical anisotropy, this variation would constitute a confounder correlated with the inclination angle. For an SPI concrete with the same aggregate and spread-flow level ( 400 $\mathrm{m}$$\mathrm{m}$, cf. sec:materials), albeit produced with a paste of water-to-cement ratio 0.30, no direction dependence of the compressive strength relative to the layer orientation was found [2]. Since the load is transferred primarily through the directly contacting grain skeleton [2], this finding should carry over to the mixture used here, so that the layer orientation is not expected to enter as a confounder of the observed angle dependence.

The third influence is the radial confinement configuration. The concrete cover between the bar surface and the outer specimen surface follows directly from the specimen geometry as $(80-d)/2$ in mm, owing to the constant outer specimen diameter of 80 $\mathrm{m}$$\mathrm{m}$. The cover and its ratio to the bar diameter therefore vary across the test series, from 36 $\mathrm{m}$$\mathrm{m}$ (ratio $\approx 4.5$) for 8 $\mathrm{m}$$\mathrm{m}$ to $27.5$ $\mathrm{m}$$\mathrm{m}$ (ratio $\approx 1.1$) for 25 $\mathrm{m}$$\mathrm{m}$. The steel confinement rings suppress splitting at all diameters, so that the measured response is governed by bond. If the shift of radial stiffness towards the steel ring at the larger diameters governed the diameter effect, higher bond stresses would be expected at the larger diameters, where the stiff ring lies closest to the bar. The data show the opposite trend. The confinement configuration therefore cannot account for the observed diameter dependence.

The phenomenology described above shares a parallel with the top-bar (or top-cast) effect in conventionally cast reinforced concrete. In gravity-cast members, horizontally placed bars near the top of the cast section experience reduced bond owing to the settlement of the fresh concrete and the accumulation of bleed water beneath the bar [43,44,45]. The reduction increases with casting depth and with the water content of the mixture [44], and it has also been reported for self-compacting concretes (SCC), in which the local bond strength of top-cast bars was about 20 lower than in comparable normal concrete [46]. For SCC, the static stability of the mixture has been identified as the governing parameter, with a locally weaker interfacial zone of reduced modulus and micro-strength reported beneath the bar, although this reduction is less pronounced than for vibrated concrete [47]. In the present data, the bar diameter takes over the role that the casting position plays in these studies, since the severity of the underside disturbance grows with the obstacle size. Reported bond reductions range from approximately zero to more than 50 depending on specimen geometry, bond length, and the slip range considered, magnitudes similar to the relative differences observed here between the 8 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$ bars [48].

The parallel is phenomenological only: in gravity-cast concrete, the underside disturbance is driven by settlement and bleed water in a fluid concrete, promoted in particular by vibration during compaction. In the SPI process considered here, no fluid concrete bath exists. The matrix is built up layer-wise from a dry particle bed selectively infiltrated by paste, and the bar is encased by repeated deposition events rather than floated through a sedimenting medium. The disturbance described by $A_{\mathrm{contact}}$ arises from this layer-wise build-up combined with the wall-effect-related packing perturbation at the cylindrical inclusion.

Besides the test configuration, the bars themselves differ in rib geometry, which is a further candidate explanation for the diameter ordering. The relative rib area $f_R$ was determined for all three diameters from the rib geometry measured and evaluated according to DIN EN ISO 15630-1 [49], and amounts to $f_R=0.079$, $0.094$, and $0.072$ for the 8 $\mathrm{m}$$\mathrm{m}$, 16 $\mathrm{m}$$\mathrm{m}$, and 25 $\mathrm{m}$$\mathrm{m}$ bars, respectively. All values exceed the characteristic values required by DIN 488-2 [50] (0.045 for 8 $\mathrm{m}$$\mathrm{m}$, 0.056 for 16 $\mathrm{m}$$\mathrm{m}$ and 25 $\mathrm{m}$$\mathrm{m}$). The measurements are summarised in tab:rib8mm,tab:rib16mm,tab:rib25mm. In these tables, $h_m$ is the transverse-rib height at mid-length and $h_{1/4}$, $h_{3/4}$ the rib heights at the quarter points, c the rib spacing, $\alpha$ and $\beta$ the rib flank and inclination angles, e the rib-row spacing, b the rib head width, and l the rib length. In the mean row, e is reported as the sum over the two rib rows, as this sum enters the $f_R$ evaluation, whereas the other mean-row entries are arithmetic means.

The rib geometry was assessed on two ribs per diameter, so these values are indicative rather than statistically robust.

A consistent monotonic relationship between $f_R$ and the bond parameters is not evident. If the rib geometry governed the response, the 16 $\mathrm{m}$$\mathrm{m}$ bars with the highest $f_R$ (0.094) would be expected to reach the highest bond stresses. Instead, the 8 $\mathrm{m}$$\mathrm{m}$ bars reach the highest bond stresses with an intermediate $f_R$ (0.079), the 16 $\mathrm{m}$$\mathrm{m}$ bars lie in between, and the 25 $\mathrm{m}$$\mathrm{m}$ bars combine the smallest $f_R$ (0.072) with the lowest bond stresses. The effect of $f_R$ on bond is reported to depend on the confinement and on the considered slip range: without confinement, bond strength is largely independent of the deformation pattern, whereas confined bars show bond strengths increasing with $f_R$ [51]. In a parametric study, bond stress showed small sensitivity to $f_R$ between 0.10 and 0.15 and a strong increase above 0.16, a range above the values measured here [52], and an interplay of rib width and clear rib spacing has been reported [53]. The lack of a clear $f_R$ correlation despite the confined configuration indicates that local rib geometry is secondary to the process-induced effects along the contact zone in the SPI configuration considered here.

4. Conclusions and Outlook

This study investigates bond behaviour at protruding reinforcement bars in the Selective Paste Intrusion (SPI) process for varying bar diameters (8, 16 and 25$\mathrm{m}$$\mathrm{m}$) and inclination angles (0 to 90 in 15 steps). The bar diameter dominates the bond response, most pronounced at the developed-interlock and capacity levels, and the magnitude of the diameter effect exceeds what a conventional pull-out comparison would predict. The inclination angle produces no monotonic trend, with individual angle contrasts remaining largely within the experimental scatter, and plays a secondary role compared to the diameter.

The projected shadowed area $A_{\mathrm{shadow}}$ acts at most as a secondary factor. Combinations of diameter and angle that lead to similar shadowed areas produce clearly separated bond parameters, and a consistent $\tau$–$A_{\mathrm{shadow}}$ structure is not observed. The layer-wise contact-zone descriptor $A_{\mathrm{contact}}$ provides a coherent interpretation of the data. It localises the affected region on the lower half of the bar surface within the bond length, where the disturbance is built up by the sequential overprinting of protruding bar segments. The descriptor is independent of inclination angle for $0\le \theta <90$ and ceases to apply at $\theta =90$. The observed elevation of bond capacity at 90 is qualitatively consistent with this loss of the contact-zone mechanism. The normalisation of bond stress over the full bar surface removes the purely geometric scaling but does not capture the process-induced inhomogeneity of the contact zone, so that the observed diameter effect is best read as a structural rather than as a purely geometric effect of bond formation. The required increase of the contact-zone disturbance with bar diameter is consistent with the diameter scaling of granular-obstacle disturbances [38], of air-void formation beneath and around integrated bars [39], and of interfacial porosity around embedded inclusions [40], while its pronounced magnitude in the present data remains to be verified directly. The phenomenology shares a parallel with the conventional top-bar effect, but the causal driver is different. The disturbance in the SPI case arises from layer-wise deposition combined with obstacle-scale and packing-related disturbances, not from sedimentation in a fluid concrete bath.

Future work should focus on a more direct characterisation of the contact zone. Sectioning of specimens along the bar axis and X-ray computed tomography can visualise the underside contact zone directly and quantify whether the process-induced void volume remains partially unfilled or is compensated by paste intrusion. Process parameters that govern deposition geometry and intrusion behaviour, such as layer thickness and the aggregate-to-bar diameter ratio, are candidates for systematic variation, since the contact-zone interpretation suggests that they directly modulate the affected region. Coupled experimental and numerical modelling of paste intrusion in disturbed packings beneath cylindrical inclusions would close the remaining gap between the layer-wise contact-zone interpretation and a fully mechanistically validated bond model for reinforced SPI components.