Analysis of Deflection Behavior in Tapered Cantilever Composite Girders with Corrugated Steel Webs

A widely adopted assumption in both design codes and conventional analytical models is that the shear force resisted by the CSWs is equivalent to the total shear force acting on the entire cross-section under external loads, which directly determines the corresponding shear deflection. Furthermore, the shear stress distribution across the CSWs is conventionally presumed to be uniform over their height, with the corresponding shear stress calculation expressed by the following formula: (1) τ=QAw \tau = {Q \over {{A_w}}} Where:

• τ - the shear stress acting on the CSWs,

• Q - the shear force on the entire cross-section,

• A_w - the effective shear area of the CSWs.

However, recent studies have demonstrated that the shear stress formula mentioned above is applicable only to prismatic BGCSWs. The shear stress distribution in tapered BGCSWs differs significantly. Specifically, the applied bending moment in a tapered girder redistributes the shear forces within the cross-section (Zhou et.al, 2016). Notably, this redistribution not only changes the shear force carried by the CSWs but also results in the development of shear deflection in the inclined bottom flange and the CSWs. Accordingly, the expression for shear stress in tapered BGCSWs, derived from previous studies, is expressed as: (2) τ=1bdDdx=1bdMSaIdx=1bIdMSaI−MSadIdxI2=1b1IMdSadx−MSaI2dIdx=QSaIb+MIbdSadx−SaIdIdx=τQ+τM \matrix{ {\tau = {1 \over b}{{dD} \over {dx}} = {1 \over b}\left[ {{{d\left( {{{M{S_a}} \over I}} \right)} \over {dx}}} \right] = {1 \over b}\left[ {{{Id\left( {{{M{S_a}} \over I}} \right) - M{S_a}{{dI} \over {dx}}} \over {{I^2}}}} \right]} \hfill \cr { = {1 \over b}\left[ {{1 \over I}M{{d{S_a}} \over {dx}} - {{M{S_a}} \over {{I^2}}}{{dI} \over {dx}}} \right] = {{Q{S_a}} \over {Ib}} + {M \over {Ib}}\left( {{{d{S_a}} \over {dx}} - {{{S_a}} \over I}{{dI} \over {dx}}} \right) = {\tau _Q} + {\tau _M}} \hfill \cr }

In Equation (2), M represents the bending moment at the calculated cross-section, Q denotes the vertical shear force over the entire section, and b is the width of the section at the point where shear stress is calculated. I indicate the moment of inertia of the cross-sectional area about the centroidal axis, and S_a refers to the static moment of the cross-sectional area about the centroidal axis of the full section. The term τ_Q represents the shear stress induced by the sectional shear force, which is the conventional formula adopted in existing design codes and most theoretical models. In contrast, the term corresponds to the shear stress components generated by the bending moment – a distinctive mechanical feature unique to tapered BGCSWs. Consequently, the deflection behaviour of tapered BGCSWs differs fundamentally from that of prismatic girders.

Figure 1:

Force and deflection analysis models for a cantilever tapered BGCSWs: (a) Force analysis model; (b) Deflection analysis model

To elucidate the mechanical phenomenon described by Equation (2), consider the force analysis of a tapered BGCSWs subjected to a concentrated load P, as illustrated in Figure 1(a). Under this load, the cross-section of the girder resists the combined action of bending moments and shear forces. Owing to the accordion effect, the CSWs provide negligible resistance to longitudinal bending moments. The applied shear force Q induces the corresponding shear stress within the CSWs. Based on Equation (2), it is assumed that the bending moment M induces an additional shear stress in the CSWs. Integrating over the web area yields an additional shear force Q_M resulting from the bending moment M in the CSWs. (3) QM=∫AτMdA=∫AMIbdSadx−SaIdIdxdA {Q_M} = \int_A {{\tau _M}dA} = \int_A {{M \over {Ib}}\left( {{{d{S_a}} \over {dx}} - {{{S_a}} \over I}{{dI} \over {dx}}} \right)dA}

Studies have demonstrated the self-equilibrating nature of the additional shear force induced by bending moments (Su et.al 2024; Su and Zhou, 2023) This means that the internal force redistributes shear across the cross-section without altering the magnitude of the total resultant shear. Consequently, a reactive shear force Q_M of equal magnitude and opposite direction is correspondingly imposed on the bottom flange (Figure 1(a)). It follows that the effective shear force governing the CSWs is governed by Equation (3). (4) QE=Q−QM {Q_E} = Q - {Q_M}

The preceding analysis indicates that in tapered BGCSWs, the shear force in the CSWs can no longer be calculated based on conventional code assumptions. Rather, the applied bending moment redistributes the internal shear forces, inducing an additional shear force within the cross-section. As a result, the bottom flange also resists a significant portion of the total shear. Furthermore, Equation (3) reveals that both the magnitude and direction of this additional shear force depend on those of the bending moment M at the section. As illustrated in Figure 1(b), these additional shear forces, in turn, generate corresponding shear deflections in their respective components. Consequently, a comprehensive calculation of the total deflection must incorporate the contributions from these additional shear deflections. The total deflection can thus be expressed by the following equation: (5) y=yb+ys y = {y_b} + {y_s} (6) ys=yws+ybs {y_s} = {y_{ws}} + {y_{bs}}

In the above equations, y_b represents the bending deflection generated by bending moment, y_s represents the total shear defections, y_ws represents the total shear deflection in the CSWs, which is the sum of the deflections induced by both the bending moment and shear force, y_bs signifies the additional shear deflection in the inclined bottom flange caused by the bending moment.

As illustrated in Figure 1(a), the load-transfer mechanism in tapered BGCSWs diverges from that in prismatic sections. Specifically, the bending moment M triggers a redistribution of shear forces, giving rise to an additional shear component Q_M within the cross-section. Consequently, the total shear deflection comprises two parts: the deflection of the CSWs under the direct shear force Q, and the additional deflection induced by Q_M in both the CSWs and the bottom flange. This combined action demonstrates that the method for calculating shear deflection in tapered BGCSWs differs substantially from existing code provisions, necessitating further refinement. Fundamentally, the shear deflection in tapered BGCSWs is the cumulative result of the interactive effects of both bending moment and shear force. Specifically, the shear deflection mechanism, detailed in Figure 1(b), can be decomposed into two distinct parts. Firstly, the shear force Q from the applied load P is carried solely by the CSWs due to their folding effect, resulting in shear deflection γswQ \gamma _{sw}^Q confined to the CSWs. Secondly, the bending moment induces a shear force Q_M in the bottom flange and the CSWs, which in turn causes separate shear deflections γswM \gamma _{sw}^M and γcbM \gamma _{cb}^M in each of these elements. Therefore, a complete expression for the shear deflection in a tapered BGCSWs must account for contributions from both shear forces and the bending moments, as follows: (7) dysdx=γswQx+γcbMx−γswMx=αswQxGseqAswx+αcbQMxGcAcbx−αswQMxGseqAswx {{d{y_s}} \over {dx}} = \gamma _{sw}^Q\left( x \right) + \gamma _{cb}^M\left( x \right) - \gamma _{sw}^M\left( x \right) = {{{\alpha _{sw}}Q\left( x \right)} \over {{G_{seq}}{A_{sw}}\left( x \right)}} + {{{\alpha _{cb}}{Q_M}\left( x \right)} \over {{G_c}{A_{cb}}\left( x \right)}} - {{{\alpha _{sw}}{Q_M}\left( x \right)} \over {{G_{seq}}{A_{sw}}\left( x \right)}}

In Equation (7), γswQ \gamma _{sw}^Q corresponds to the shear strain of the CSWs induced by the shear force. Terms γcbM \gamma _{cb}^M and γswM \gamma _{sw}^M , respectively, denote the shear strains of the bottom flange and the CSWs induced by the bending moment. The parameters α_sw and α_cb quantify the non-uniformity of shear stress distributions across the CSWs and concrete bottom flange, respectively. Numerically, these parameters represent the ratio of the shear stress at the neutral axis to the average shear stress on their respective cross-sections. In BGCSWs, shear stress is assumed to be uniformly distributed across the webs, whereas the bending-induced additional shear stress in the bottom flange can be approximated as a linear distribution along its height. This leads to the assignment of a value of 1.0 for both α_sw and α_cb. A_sw and A_cb are the effective cross-sectional areas of the CSWs and the concrete bottom flange, respectively. G_c denotes the shear modulus of concrete, and G_seq represents the equivalent shear modulus of the CSWs. The equivalent shear modulus G_seq is determined using the following expression (Johnson and Cafolla 1997): (8) Gseq=a1+a2a1+a3G0 {G_{seq}} = {{{a_1} + {a_2}} \over {{a_1} + {a_3}}}{G_0}

In Equation (8), a₁ refers to the length of the flat sub-panel of the CSWs, while a₂ and a₃ represent the projection length and inclined length of the inclined sub-panel of CSWs, respectively. G₀ is the shear modulus of the steel material. Integrating Equation (5) yields: (9) ys=∫QxGseqAswxdx+∫QMxGcAcbx−QMxGseqAswxdx+C1 {y_s} = \int {{{Q\left( x \right)} \over {{G_{seq}}{A_{sw}}\left( x \right)}}} dx + \int {\left[ {{{{Q_M}\left( x \right)} \over {{G_c}{A_{cb}}\left( x \right)}} - {{{Q_M}\left( x \right)} \over {{G_{seq}}{A_{sw}}\left( x \right)}}} \right]dx + {C_1}}

Equation (9) provides the analytical solution for the shear deflection of tapered BGCSWs. For cantilever girders, with the boundary condition y_s |_x=0 = 0, the constant term C₁ in Equation (9) can be determined by applying this condition.

Given that the flexural rigidity of a tapered BGCSWs is not constant along its span, it is therefore impractical to determine its bending deflection directly using standard formulas derived from material mechanics. To address this limitation and adapt classical computational methods for tapered BGCSWs, this study introduces the concept of an equivalent moment of inertia based on Simpson’s rule. As shown in Figure 2, a cantilever tapered BGCSWs of length l is longitudinally divided into n equal segments (with n being an even number). The moment of inertia at each segment division is denoted as I_j (j=0,1, 2, …, n, I₀=I_min, I_n=I_max).

Figure 2:

Segment division of the tapered cantilever BGCSWs

According to Simpson’s integration formula: (10) ∫abfξdx=b−a3ny0+4×y1+y3+⋯+yn−1+2×y2+y4+⋯+yn−2+yn \int_a^b {f\left( \xi \right)dx} = {{b - a} \over {3n}}\left[ {{y_0} + 4 \times \left( {{y_1} + {y_3} + \cdots + {y_{n - 1}}} \right) + 2 \times \left( {{y_2} + {y_4} + \cdots + {y_{n - 2}}} \right) + {y_n}} \right]

Within the numerical integration framework, the term y_i = f(ξ_i) (where i=0,1,2..., n) denotes the function value of the integrand f(ξ_i) evaluated at the discrete sampling point ξ = ξ_i. The parameter n specifies the total number of subintervals resulting from the discretization of the integration domain [a, b]. A key requirement for the applied method is that n must be an even integer, a condition typically imposed by composite Newton-Cotes formulas (e. g., Simpson’s rule) to ensure proper polynomial fitting over successive pairs of subintervals.

In accordance with Castigliano’s second theorem, the bending deflection for a tapered BGCSWs subjected to a concentrated force is formulated as follows: (11) δ1=∫olPξ2EIξdξ {\delta _1} = \int_o^l {{{P{\xi ^2}} \over {EI\left( \xi \right)}}} d\xi

In Equation (11), set n = 4, a = 0, b = l, fξ=Pξ2EIξ f\left( \xi \right) = {{P{\xi ^2}} \over {EI\left( \xi \right)}} . Then Equation (8) can be written as: (12) δ1=∫0lPξ2EIξdξ=Pl12E0+4×l42I1+3l42I3+2×l22I2+l2I4 =Pl312E14I1+12I2+94I3+1I4 \matrix{ {{\delta _1} = \int_0^l {{{P{\xi ^2}} \over {EI\left( \xi \right)}}d\xi } = {{Pl} \over {12E}}\left\{ {0 + 4 \times \left[ {{{{{\left( {{l \over 4}} \right)}^2}} \over {{I_1}}} + {{{{\left( {{{3l} \over 4}} \right)}^2}} \over {{I_3}}}} \right] + 2 \times {{{{\left( {{l \over 2}} \right)}^2}} \over {{I_2}}} + {{{l^2}} \over {{I_4}}}} \right\}} \hfill \cr {\;\;\;\; = {{P{l^3}} \over {12E}}\left( {{1 \over {4{I_1}}} + {1 \over {2{I_2}}} + {9 \over {4{I_3}}} + {1 \over {{I_4}}}} \right)} \hfill \cr }

A common analytical simplification involves substituting the tapered BGCSWs with an equivalent prismatic member characterized by a constant equivalent moment of inertia, I_eq. The deflection of this equivalent girder under a concentrated force is then described by: (13) δeq=∫olPξ2EIeqdξ=Pl33EIeq {\delta _{eq}} = \int_o^l {{{P{\xi ^2}} \over {E{I_{eq}}}}d\xi } = {{P{l^3}} \over {3E{I_{eq}}}}

Based on the principle of equivalent action effect, equating the displacements from Equation (12) and (13) yields: (14) Pl312E14I1+12I2+94I3+1I4=Pl33EIeq {{P{l^3}} \over {12E}}\left( {{1 \over {4{I_1}}} + {1 \over {2{I_2}}} + {9 \over {4{I_3}}} + {1 \over {{I_4}}}} \right) = {{P{l^3}} \over {3E{I_{eq}}}}

Simplifying Equation (14) yields: (15) Ieq=161I1+2I2+9I3+4I4 {I_{eq}} = {{16} \over {{1 \over {{I_1}}} + {2 \over {{I_2}}} + {9 \over {{I_3}}} + {4 \over {{I_4}}}}}

The introduction of an equivalent moment of inertia enables the application of established classical mechanics of materials methods to analyse the bending deflection in tapered BGCSWs. This approach effectively simplifies the analytical procedure by transforming the variable-section problem into one governed by equivalent prismatic beam theory. (16) d2wdx2=−MEI {{{d^2}w} \over {d{x^2}}} = - {M \over {EI}}

Based on the formulations given in Equation (15) and (16), the approximate differential equation for the deflection curve of a tapered BGCSW subjected to a concentrated load can be further derived and expressed in the following form: (17) EcIeqyb″=−M=Pl−x {E_c}{I_{eq}}y_b^{''} = - M = P\left( {l - x} \right)

Given the corrugated configuration, the flexural contribution of the CSWs can be omitted. The elastic modulus and equivalent moment of inertia are thereby determined using the concrete section alone.

Integrating Equation (17) yields: (18) EcIeqyb′=−P2(l−x)2+C1 {E_c}{I_{eq}}y_b^\prime = - {P \over 2}{(l - x)^2} + {C_1} Where:

• C₁ - the constant of integration, which is determined as C1=Pl22 {C_1} = {{P{l^2}} \over 2} from the boundary condition γb′0=0 \gamma _b^\prime\left( 0 \right) = 0 . Consequently, Equation (18) can be written as: (19) yb′=PEcIeqlx−x22 y_b^\prime = {P \over {{E_c}{I_{eq}}}}\left( {lx - {{{x^2}} \over 2}} \right)

Integrating Equation (19) again yields: (20) yb=PEcIeql2x2−x36+C2 {y_b} = {P \over {{E_c}{I_{eq}}}}\left( {{l \over 2}{x^2} - {{{x^3}} \over 6}} \right) + {C_2}

From the boundary condition at the fixed end of the cantilever girder, y_b(0) = 0, it obtains C₂ = 0. Thus, the formula for calculating the bending deflection of a tapered BGCSWs under a concentrated load is: (21) yb=PEcIeql2x2−x36 {y_b} = {P \over {{E_c}{I_{eq}}}}\left( {{l \over 2}{x^2} - {{{x^3}} \over 6}} \right)

By substituting Equation (9) and (21) into Equation (2), the total deflection of the cantilever girder can be calculated.

Aimed at investigating the deformation behaviour of tapered BGCSWs and verifying the accuracy of the theoretical approach, a full-scale experimental girder in a double-cantilever configuration was utilized, with concentrated loads acting at its free ends (Chen 2023). As depicted in Figure 3, the test specimen had a total length of 4700 mm, consisting of two cantilever girders, each with a span of 2100 mm, connected by a central crossbeam measuring 500 mm in length. The depth of the girder tapered linearly from 1030 mm at the fixed support to 740 mm at the free end. The top flange width was 1200 mm, and the bottom slab width was 650 mm. The crossbeam had a height of 1030 mm. Both the central crossbeam and the concrete flanges were reinforced with steel bars, and no prestressing was applied throughout the specimen.

Figure 3:

Geometric dimensions of the tapered BGCSWs (units: mm)

A distinct design difference was implemented between the two sides: the CSWs on the right side were reinforced with horizontal diaphragms and partially filled with concrete, whereas those on the left side were directly embedded into the flanges without any additional encasement. The trapezoidal CSWs were fabricated from 3-mm-thick steel plates with a trapezoidal profile. Full composite action between the CSWs and the concrete top/bottom flanges was ensured using embedment connectors, which guaranteed deflection compatibility and structural integrity. All concrete components were cast using C50 high-performance concrete. The CSWs were made of Q345D-grade steel, with material properties determined from four sets of coupon tests. The average measured yield strength, ultimate tensile strength, and elastic modulus were 365 MPa, 479 MPa, and 206 GPa, respectively.

This experimental investigation employed a static loading scheme to evaluate the structural response. As illustrated in Figure 4, the girder was cast monolithically with the pier cap and a lower crossbeam, which was anchored to the strong floor via ground anchors. This configuration simulated the fixed-end restraint condition representative of a cantilever construction stage. A symmetric loading protocol was implemented to ensure uniform transverse stress distribution. Two synchronized hydraulic jacks, each reacting against a stiff reaction frame, applied concentrated loads at points located 30 cm from the free end of each cantilever girder. Load was transferred from each jack to a distribution beam and then to the test specimen through steel bearing blocks.

Figure 4:

Experimental loading

The entire loading process was controlled by pressure sensors, with strict synchronization maintained between the two jacks to control the loading rate at both ends. The loading regime consisted of two phases: elastic loading followed by destructive loading. The loading was applied monotonically in discrete steps of 30 kN. A preload of 20 kN was first applied to ensure proper contact between the loading apparatus and the specimen, eliminating any gaps and reducing system-compliance effects. To investigate the stress distribution on the CSWs, strain gauges were installed at eight representative sections (A to H). In addition, a displacement transducer was placed at the bottom flange of Section A (Figure 4) to monitor the deflection of the test specimen during loading. After each load increment, a stabilization period of 5 minutes was allowed before recording the measurements. Data from the transducer was then collected twice at 1-minute intervals after readings stabilized. The average of these two readings was recorded as the final deflection value for that load level to ensure data accuracy and reliability.

The validation of the analytical and numerical models is demonstrated in Figure 6 through a direct comparison with experimental deflection data at two discrete control points under stepwise loading from 30 to 120 kN. The near-perfect overlap of the three datasets (experimental, theoretical and FE) across the entire load history confirms the models’ fidelity. Notably, the theoretical model maintains a high level of accuracy, with a maximum discrepancy of 8.2% compared to experiments, while its difference from the FE simulation is only 4.5%. This minor variance between the two computational approaches primarily stems from the idealized boundary conditions and perfect composite action assumed in the analytical model, whereas the FE model incorporates discrete constraints. The strong convergence of all three independent results not only verifies the core hypothesis – that the moment-induced shear contribution must be included – but also establishes the proposed method as a dependable tool for the deflection analysis and design of tapered BGCSWs.

Figure 6:

Deflection results derived from three independent approaches: FE simulation, theoretical computation, and experimental validation

The close agreement among the FE predictions and experimental measurements validates the proposed theoretical formulas, confirming their accuracy in predicting the deflection of tapered BGCSWs. This finding underscores the necessity of accounting for the contribution of bending moments to shear deflection in the deflection analysis of such girders. To further dissect the total deflection, the individual components were quantified using the derived analytical expressions, namely Eq. (9) and Eq. (21). The resulting breakdown is comprehensively summarized in Figure 7.

Figure 7:

Shear and bending deflections of the tapered BGCSWs

The results reveal that shear deflection accounts for a significant portion of the total deflection in BGCSWs. Under concentrated loading, the inclusion of shear deflection leads to a measurable increase in total deflection, with its contribution becoming particularly significant and non-negligible at the free end of the cantilever BGCSWs. Crucially, in tapered BGCSWs, shear deflection originates from both the direct shear force and the additional shear forces induced by bending moments in the bottom flange and CSWs.

Figure 8:

Proportional contributions of deflections from bending moment and shear force

To elucidate the interplay of bending and shear on deflection in tapered BGCSWs, the total strain was decomposed into components from direct shear and bending-induced mechanisms and quantified across multiple cross-sections of the experimental girder under a 300 kN load. Figure 8 illustrates the proportional distribution of each strain component relative to the total strain. A key finding is the development of considerable shear strain within the CSWs due to bending moments, whereas the corresponding strain in the bottom flange remains minimal. Furthermore, the proportion of this bending moment-induced shear strain exhibits a distinct spatial trend, increasing monotonically from the free end towards the fixed support - closely following the gradient of the bending moment diagram. Specifically, its contribution is nearly negligible near the free end but rises to a significant 29.36% in sections adjacent to the fixed support. These results irrefutably confirm that the contribution of bending moment-induced shear strain to the total deflection in tapered BGCSWs is substantial and non-negligible. This has considerable practical implications, indicating that prevailing design approaches, which overlook this coupling effect, require revision to explicitly incorporate it, thereby enhancing predictive accuracy and ensuring structural safety.