Parametric analysis and stability study of narrow steel box girders based on Midas FEA

2.1. Establishment of the Midas FEA model

Based on the “Zhuyong Expressway Reconstruction and Expansion Project”, a constant-section narrow steel box composite girder bridge (40+60+40 m) was modeled in Midas FEA. The single steel box girder has a height of 2200 mm, bottom plate width of 2000 mm, plate thicknesses of 16/14/20 mm (top/web/bottom), and longitudinal stiffeners of 12/14/18 mm (Fig. 1). Q345qD steel was modeled as linear elastic ($E=$ 206 GPa, $\rho =$ 7850 kg/m³, $\mu =$ 0.3) using shell elements (SHELL4). The structured mesh (aspect ratio ≤ 1:3, max size 200 mm) contains 366,170 elements and 409,630 nodes (Fig. 2).

Fig. 1Schematic diagram of the narrow steel box girder structure

Fig. 2Schematic diagram of the overall model of the narrow steel box girder

2.2. Boundary conditions and load case setup

At the main pier, all translations are constrained ($ux=uy=uz=$ 0) with rotations released; at the side piers, only vertical translation is constrained ($uz=$ 0) with rotations free. Loads include self-weight, secondary dead loads (bridge deck), Highway-Grade I vehicle load, and wind load.

Parameter variations are shown in Table 1. The basic model has girder height 2200 mm, bottom plate width 2000 mm, plate thicknesses of 16/14/20 mm (top/web/bottom), 6 U-ribs, and 7 diaphragms spaced at 20 m.

2.3. Model validation

To validate the shell-element model, the analytical solution for a three-span continuous beam was employed. With $A=$ 0.4425 m², $I=$ 0.3606 m⁴ and $q=$ 60.5 kN/m, the three-moment equation gives $M_2=M_3=$ 16288 kN·m, and the mid-span deflection is $\delta =5q{L}^4/384EI-M{L}^2/8EI$ = 38.8 mm. The FEA model predicted 38.2 mm (deviation 1.5 %), confirming the model's reliability in capturing global flexural behavior.

3.1. Influence of structural parameters on deformation and stress

3.1.1. Girder height and plate thickness

In the basic model, under the most unfavorable combination of dead load, vehicle load and wind load, the deformation and stress contours are shown in Fig. 3. The maximum horizontal and vertical displacements both occur at the mid-span top plate, while the maximum compressive stress occurs at the support bottom plate.

Fig. 3Deformation and stress distribution of the bridge

a) Vertical displacement under dead load + vehicle load combination in the typical case

b) Stress under dead load + vehicle load combination in the typical case

c) Horizontal displacement under wind load in the typical case

d) Stress distribution under wind load in the typical case

Table 2Influence of girder height and plate thickness on structural deformation and stress

Case group		Maximum vertical displacement /mm	Maximum horizontal displacement /mm	Maximum tensile stress/MPa	Maximum compressive stress/MPa
Girder height / mm	2600	–60.52	3.24	163.28	–228.06
	2200	–67.39	2.38	155.04	–221.77
	1800	–80.88	2.27	246.55	–264.54
Top plate thickness / mm	20	–65.68	2.35	150.08	214.86
	16	–67.39	2.38	155.04	221.77
	12	–67.67	2.39	160.05	228.70
Web plate thickness / mm	18	–67.23	1.87	150.08	214.86
	14	–67.39	2.38	155.04	221.77
	10	–67.62	2.57	160.05	228.70
Bottom plate thickness / mm	24	–66.73	2.38	150.08	214.86
	20	–67.39	2.38	155.04	221.77
	16	–72.28	2.38	160.05	228.70

Table 2 shows the effects of girder height and plate thickness. Increasing the girder height from 1800 to 2600 mm reduces mid-span vertical displacement by 25.2 % (80.88 to 60.52 mm), but raises horizontal displacement from 2.27 to 3.24 mm. Support compressive stress varies non-monotonically, peaking at 1800 mm (264.54 MPa), decreasing to 221.77 MPa at 2200 mm, then rising slightly to 228.06 MPa at 2600 mm.

Variations in top plate and web thickness within ±4 mm change vertical displacement by less than 3 %, while reducing the bottom plate thickness from 20 to 16 mm increases it by 7.3 %. Increasing the web thickness reduces horizontal displacement by 21.4 %, showing the web provides lateral restraint.

3.2. Structural buckling analysis

The critical load factors were obtained from linear eigenvalue buckling analysis (LBA) in Midas FEA, which solves $\left[K\right]\left\{\phi \right\}=\lambda \left[K_G\right]\left\{\phi \right\}$, where $\lambda$ is the critical load factor and [$K$], [$K_G$] are the elastic and geometric stiffness matrices. The first buckling mode – web panel buckling near the support – agrees with classical plate buckling theory. The basic model gave a critical load factor of 1.92. Among the single measures (Table 4), diaphragms are the most effective (1.92→6.43, +234.8 %), whereas increasing the top plate thickness slightly reduces the factor due to added self-weight.

Combined measures show varied effects: individual factors for diaphragms, web stiffeners, and bottom stiffeners are 6.43, 3.85, and 2.70. Combining diaphragms with bottom stiffeners raises the factor to 12.67, but with web stiffeners gives only 6.96 (+7.6 %); bottom + web stiffeners yield only 4.21. Thus, selecting the right combination is key.

Combining three measures (diaphragms, bottom plate stiffeners, and web stiffeners) raises the critical load factor from 12.67 to 30.20, an increase of 138.6 %. As a linear analysis, LBA ignores initial imperfections and material nonlinearity; the results should be treated as upper-bound estimates.

An efficiency index $\eta =\mathrm{\Delta }\lambda /\mathrm{\Delta }V$ (stability gain per added steel volume) shows the diaphragm-only scheme as the most efficient (+234.8 % in $\lambda$), while the three-factor combination ($\lambda =$ 30.20) achieves the highest stability at the greatest material cost.

3.3. Analysis of structural reinforcement sequence

Fig. 4 summarizes the buckling failure modes under the most effective single-, two-, and three-factor reinforcements. In the basic model, buckling occurs at the web near the support, so diaphragms or web stiffeners help. After adding diaphragms, the failure location shifts from the web to the bottom plate near the support, making further web strengthening less effective but bottom plate strengthening highly effective. After strengthening both, the failure location shifts to the diaphragm above the support. The recommended reinforcement sequence is therefore web → bottom plate → diaphragm. No buckling occurs at the top plate or U-ribs because the concrete deck suppresses top plate buckling.

Fig. 4First-order buckling failure under the influence of different reinforcement measures

a) Basic model

b) After adding diaphragms

c) After adding diaphragms + bottom plate stiffeners

d) After adding diaphragms + bottom plate stiffeners + web stiffeners