Rigid Frame Research Articles

Experimental study on the axial tensile properties of polypropylene fiber reinforced concrete

In order to study the axial tensile properties of polypropylene fiber reinforced concrete, an axial tensile test device for concrete is developed in this paper. The device is composed of three parts: rigid frame, spherical hinge and puller, and specimen fabrication part. The test device can accurately measure the tensile strength and peak tensile strain of concrete, and perfectly solves the eccentricity problem of concrete specimens under tension. It can measure the post peak segment tensile strain, such that the whole process tensile stress–strain curve can be obtained. The axial tensile test of polypropylene fiber concrete was carried out using the above test device, and the results show that the tensile strength of concrete can be clearly improved by adding polypropylene fiber, which makes the tensile failure of concrete show certain plastic characteristics. The test results show that with the increase in fiber content, the tensile strength of concrete increases first and then decreases. The effects of polypropylene fiber content and curing age on the tensile properties of concrete were studied and the optimum polypropylene fiber content was determined. When the fiber content is 0.9 kg/m3, the tensile strength of concrete reaches the maximum value. The splitting tensile test of concrete under the same condition was carried out simultaneously. The damage phenomenon and test results of the axial tensile test and splitting tensile test of concrete were compared and analyzed, and the applicability of the new developed device in the concrete axial tensile test was verified.

Static Load Test Detection of Continuous Rigid Frame Railway Bridge

The quality of the continuous rigid frame railway bridge is related to the safety of train operation, so it is necessary to test its stiffness, strength and other indicators. Static load test is a common technique for bridge inspection. This article summarizes the purpose of static load test of continuous rigid frame railway bridge, required equipment, operation methods, etc., and lists examples to analyze the operation process and precautions of static load test, hoping to provide reference information for relevant personnel.

Blast Absorption Capacity of Brittle Mineral Foam: an Experimental Evaluation

Cellular materials, such as aluminum foam, have proven to be effective energy absorbents. They can be used as the crushable core for blast mitigation sacrificial cladding. In this paper insights into the blast response of a brittle mineral foam-based sacrificial cladding are presented. The experimental set-up used consists of a rigid steel frame (1000 mm × 1000 mm × 15 mm) with a square cavity of 300 mm x 300 mm in the center. An aluminum plate, representing the structure, with a total surface of 400 mm × 400 mm and a thickness of 2 mm is clamped into the steel frame. Two synchronized high-speed cameras in a stereoscopic configuration are used to capture the dynamic response of the aluminum plate at a frame rate of 20000 frames per second. The transient deformation fields are computed using a three-dimensional digital image correlation technique. The blast load is obtained by detonating 20 g of C4 placed at a distance of 250 mm from the center of the tested aluminum plate. The absorption capacity of the brittle mineral foam is assessed by comparing the out-of-plane displacement, the velocity, the acceleration and the major principal strain of the thin aluminum plate with and without the protective mineral foam. Two foam configurations with different thicknesses are considered: 60 mm and 120 mm. It is shown that adding the brittle mineral foam reduces the out-of-plane displacement together with the displacement velocity and acceleration of the aluminum plate at least by a factor of two. Post-mortem analysis of the foams shows that mitigation of the blast load in the present set-up and under the considered loading is partially obtained by crushing of the cell walls but mostly though the growth of cracks in the specimen.

Stability and Sommerfeld effect in a multi-resonant types vibrating system with isolated rigid frame driven by four exciters

The previous studies on the vibrating system with multiple exciters less considered the Sommerfeld effect, which were mainly focused on synchronization and stability problems of the system in the whole resonant region. A dynamical model is proposed in this paper to study the synchronization, stability and the Sommerfeld effect of four exciters in the multi-resonant types vibrating system (MRTVS). In order to improve the power of the system, the vibrating system with two rigid frames is driven by four especially distributed exciters. The motion differential equations of the system are given by Lagrange’s equations, and the dimensionless coupling equations and synchronization criterion are constructed as well by the average method. The theory condition for stability of four exciters in synchronous states is derived from the Hamilton principle. The coupling dynamic characteristics, stability and Sommerfeld effect of the system are discussed numerically. The whole resonant region of the system is divided into three segments by natural frequencies, and the synchronous and stable states in the different resonant types are analyzed respectively. The diversity phenomenon of the nonlinear system is observed, in other words, the system exists multiple stable equilibrium solutions at a certain particular resonant region. Then the Sommerfeld effect around the natural frequencies (NFs) is further revealed. The correctness of theoretical analyses is examined by simulation and the experiment. It’s shown that the reasonable working point in engineering should be selected in the sub-resonant region with respect to the natural frequency of the main vibrating system. The present work can provide theoretical guidance for designing some new types of vibrating machines with high driving power.

Study on the member fracture and collapse performance of single-layer grid structures using discrete solid element method

In this study, the discrete solid element method (DSEM) incorporating a softening model is proposed, to accurately simulate the dynamic crack propagation and the member fracture of the single-layer grid structure. Firstly, based on the fracture criteria of strain energy release rate, the bilinear softening model and the trilinear softening model are proposed to simulate the fracture in both elastic and elastic-plastic materials. Secondly, the validity of the proposed approach is assessed through a comparison with empirical results. Furthermore, the fracture of the rectangular beam, dynamic crack propagation of the circular tube and member fracture of the hexagonal spatial rigid frame are simulated and compared with other numerical methods. Finally, the DSEM with the softening model is employed to investigate the collapse performance of single-layer grid structures, and the whole process of member fracture is captured by the DSEM. A thorough analysis is conducted to evaluate the influence of material parameters, loading methods, and rise-span ratios on the bearing capacity and fracture occurrence of single-layer grid structures. The results show that the material property of steel is superior to aluminum for single-layer grid structures. Moreover, the single-layer grid structures adopting full node loading or high rise-span ratios are shown to exhibit premature member fracture and rapid structural collapse. The obtained results exhibit remarkable agreement with the experimental results and are superior to the finite particle method (FPM) and the extended finite element method (XFEM). Therefore, DSEM with the softening model possesses the potential to evolve into an effective numerical tool capable of solving complex problems such as dynamic crack propagation and structural collapse.

Sensitivity Analysis of Structural Parameters of Unequal-Span Continuous Rigid Frame Bridge with Corrugated Steel Webs

To investigate the effect of structural parameters of bridges with unequal spans on the bridge alignment, the finite element model simulating the full-scale bridge was developed, considering the construction process. For ease of finite element modeling and investigation, the section of composite beam with corrugated steel web was first converted into the section composed of the same material. For this purpose, an equivalent method of replacing corrugated steel webs with concrete webs was proposed based on theoretical derivation. After equivalent replacement, the influences of material bulk density, internal prestress, pipe friction coefficient, and pipe deviation coefficient on the main beam at the maximum cantilever stage were analyzed, and the influences of external prestress on the main beam after bridge construction were analyzed. The results show that the most sensitive parameter to structural response is bulk density, subsequently the external prestress, internal prestress, pipe friction coefficient, and pipe deviation coefficient. Among them, the bulk density, internal prestress, and external prestress are all sensitive parameters, while pipe friction coefficient and pipe deviation coefficient are non-sensitive parameters.

(Invited) Investigation on Dynamics of Expansion Forces and Dimensions Changes of the Pilot Scale Electrodialysis Stack

Electrodialysis (ED) represents a matured and well-established desalination and electromembrane separation technology with typical plate-and-frame configuration. In order to satisfy application demands, there is a trend for an increasing dimension and operating temperature of the industrial ED capable of high performance and efficiency in view of process time and system volume. These activities are motivated mainly by process cost reduction and exploiting new application fields benefiting from higher operating temperature and higher performance-to-volume ratio. This is also the case of reverse electrodialysis, currently meeting a growing popularity as a promising green energy source. However, experiences in a development of larger scale electromembrane devices show that scale-up from laboratory or pilot system to industrial ones is not straightforward. The scale-up simply based on increasing membrane active area or number of membranes, without appropriate modifications of entire unit design, often lead to accentuation of negative effects that are normally negligible on a small scale.One of the critical issues in larger scale represents dimensional changes of the ED stack, mainly due to membranes swelling and changes of mechanical properties of the stack materials as a respond to variation of operating conditions (temperature, electric current, salt concentration). When a robust rigid frame is used for stack assembly, such stack expansion may lead to an increase of the internal expansion forces causing irreversible mechanical damage of the materials used (membranes, spacers). On the contrary, the stack shrinking (e.g. due to membrane drying and deswelling) negatively affects the tightness. Consequently, the performance, reliability and lifetime of the apparatuses can be significantly exacerbated. Therefore, an autonomous (passive or active) system of compression force control has to be employed, which at each moment spontaneously adapts to stack dimensions keeping compression/expansion force at an optimal value. However, prior to a development of such system a deeper fundamental understanding of the stack behaviour is required. We are mainly interested at a) a qualitative and quantitative respond of the stack internal forces upon variation of different operating parameters, b) uniformity of distribution of internal expansion forces, c) dynamics (characteristic times) and reversibility of the stack dimensional changes, d) effect of different compression forces and stack compression-expansion cycles to longer-term operation of the stack and also e) stack tightness.In the first part of presented contribution the basic principle of electrodialysis process, system design and critical aspects of the ED process operation in industrial practice will be discussed. Next, the results of the research aimed at investigation of the dimensional and expansion forces changes in the pilot electrodialysis stack will be presented. The pilot-scale ED unit consists of 100 membrane pairs equipped by heterogeneous anion- and cation-selective membranes. Working area is of 0.4 × 0.1 m2. The stack is orientated horizontally assembled into the robust frame made of aluminum Item® profiles. A rigid endplate is attached to one side of the stack, whereas the opposite end-plate is segmented and equipped by 6 tensiometers measuring local stack expansion force. This compression force is applied to the stack by six pneumatic cylinders with equal pressure pressurized by a one standard compressor. The aspects of the stack behavior mentioned above are investigated as a dynamic respond of the stack to sudden changes in operating temperature (20 – 65 °C), electric current, salt concentration (0 – 160 g/dm3) and compression force (up to 45 kN). Different regimes are investigated: a) a regime with constant compression force in the pneumatic cylinders, or b) constant stack dimension (rigid compression frame) during the entire experiment. It was found that the effect of temperature was significantly more pronounced than the effect of salt concentration and electrical current. Moreover, up to 3 different characteristic times in the stack dynamic respond to temperature change were found.The financial support from grant agency of Ministry of Trade and Industry of the Czech Republic within program TRIO (project TOM2022 no. FV40053) is gratefully acknowledged.

Frequency Selectivity via Inner Boundary Conditions for A Self‐Powered Multiresonant Acoustic Sensing Array with Broad Bandwidth

AbstractThis study investigates and proposes innovative approaches to achieve frequency selectivity within a limited space. Traditional multiresonant acoustic devices use individual sensing elements of varying sizes to achieve resonance frequency (fr), leading to an inability to sense focused acoustic waves, unlike the human ear. A miniaturized, self‐powered artificial basilar membrane that incorporates multiresonant features is introduced. Multiple fr of the diaphragms are developed using inner boundary conditions (iBCs) defined by an adjustable micropatterned elastomeric support (µ‐support) and a porous nanofiber (NF) mat. This new approach offers the advantage of all‐in‐one fabrication, eliminating the need for device area variation or an additional rigid frame typically required in conventional multiresonant acoustic devices. The efficacy of the iBCs in shifting fr within the vocal frequency ranges is verified via a laser Doppler vibrometer, simulation, and triboelectric output. With its self‐powering capabilities based on triboelectric principles, this artificial basilar membrane holds promise for accurately recognizing musical and vocal signals with specific frequency characteristics. With four different iBCs in a total device area of 23 × 23 mm2, a tunable four‐channel system with fr ranging from 400 to 3000 Hz is achieved. This advancement enables the sensing of focused acoustic waves, simulating the functionality of an artificial human ear model.

Exploration and Innovative Research on Ideological and Political Education in Architectural Mechanics Course

The emphasis and implementation of ideological and political education in courses lie within the courses themselves, ensuring effective education through the process of course implementation and playing a role in nurturing students. This paper explores the elements of ideological and political education in the architectural mechanics course. It provides a detailed design of ideological and political education in accordance with the course content and characteristics. Additionally, innovative designs incorporating ideological elements are introduced for parts of the course. Finally, an application exploration is conducted within the sections of the course, specifically focusing on the subtopic of “rigid frames”, with the hope of providing reference value to related courses.

Study on the pier bottom self-centering seismic isolation structure of the high-pier continuous rigid frame bridges

Four seismic isolation specimens were designed, fabricated, and subjected to physical loading tests to enhance the seismic performance of a high-pier continuous rigid-frame bridge. A finite element model of the seismic isolation structure of the high-pier was established using numerical simulation software. The entire bridge model was established based on the multi-scale modeling method to investigate the seismic response of the high-pier continuous rigid frame bridge under different seismic waves. The results indicate that the seismic isolation pier has self-centering and energy dissipation capabilities. The seismic isolation pier has satisfactory energy dissipation capacity and ductility when the ratio of the long-axis and short-axis radii is 1.5. The results of the finite-element simulation corresponded well with the experimental results. Significant damage occurred to the concrete at the bottom and corner of the pier, and the reinforcement did not reach its yield strength throughout the loading process. Therefore, measures must be implemented to ensure the seismic performance of the structure. The pier bottom self-centering seismic isolation structure effectively absorbs shocks, as evidenced by reduced absolute acceleration and relative displacement at the pier top, decreased shear force and bending moment at the pier bottom, and minimized girder displacement. It was recommended that the ratio of the long-axis and short-axis radii be adopted as 1.5–2.0 and that the pier height be selected as 60 m–100 m. These results provide a reference for the research and applications of this type of structure.

Optimum Semirigid Connections in Hybrid Frames for Effective Seismic Performance

The beneficial effect of using hybrid frames is realized in terms of the reduction of some desired engineering demand parameters (EDPs). However, this is achieved at the cost of increasing the values of some other EDPs as compared to the rigid frame. A good balance between the two can be achieved by selecting the optimum number of semirigid (SR) connections. In the present study, two schemes of the selection of the optimum number of SR connections are presented to obtain the desired benefit of the hybrid frame. One scheme provides the optimum number of SR connections which provides the desired reductions in the base shear (BS) and total number of plastic hinges formed (THmax) with the marginal increase in the maximum inter-story drift ratio (IDRmax). The other scheme provides the number of SR connections to be used for achieving the limiting drift allowed in the hybrid frame and accordingly, obtains the percentage reductions of the BS and THmax. Both schemes are developed using a graphical technique. Three illustrative examples are considered to demonstrate the efficacy of the two schemes using the nonlinear time history analysis under three types of earthquakes (i.e., near and far-field) at three peak ground accelaration (PGA) levels. The results of the study show that in both schemes of providing the optimum number of SR connections, significant reductions in the BS and THmax can be achieved as compared to the fully rigid frame. The corresponding IDRmax in the hybrid frames remains less than equal to the limiting value of the IDRmax prescribed in the literature.

On Woven Fabric Sound Absorption Prediction

For building applications, woven fabrics have been widely used as finishing elements of room interior but not in particular aimed for sound absorbers. Considering the micro perforation of the woven fabrics, they should have potential to be used as micro-perforated panel (MPP) absorbers; some measurement results indicated such absorption ability. Hence, it is of importance to have a sound absorption model of the woven fabrics to enable us predicting their sound absorption characteristic that is beneficial in engineering design phase. Treating the woven fabric as a rigid frame, a fluid equivalent model is employed based on the formulation of Johnson-Champoux-Allard (JCA). The model obtained is then validated by measurement results where three kinds of commercially available woven fabrics are evaluated by considering their perforation properties. It is found that the model can reasonably predict their sound absorption coefficients. However, the presence of perturbations in pores give rise to inaccuracy of resistive component of the predicted surface impedance. The use of measured static flow resistive and corrected viscous length in the calculations are useful to cope with such a situation. Otherwise, the use of an optimized simple model as a function of flow resistivity is also applicable for this case.

EFFCT OF HOOPS AND COLUMN AXIAL LOAD ON SHEAR STRENGTH OF HIGH-STRENGTH FIBER REINFORCED BEAM-COLUMN JOINTS

A reinforced concrete frame is referred as "RIGID FRAMES". However, researches indicate that the Beam-Column joint (BCJ) is definitely not rigid. In addition, extensive research shows that failure may occur at the joint instead of in the beam or the column. Joint failure is known to be a catastrophic type which is difficult to repair. This study was carried out to investigate the effect of hoops and column axial load on the shear strength of high-strength fiber reinforced Beam-Column Joints by using a numerical model based on finite element method using computer program ANSYS (Version 11.0). The variables are: diameter of hoops and magnitude of column axial load. The theoretical results obtained from ANSYS program are in a good agreement with previous experimental results.

Judicial indifference in criminal sentencing: Explaining inequality of the Thai Fines

Abstract Courts in many jurisdictions remain indifferent to criticisms for their overly harsh or unequal treatments. There has been a debate whether this is attributed to judges’ individual dispositions or rather their environments. This article contributes to this debate by offering evidence from Thai courts about their indifference to inequality generated by the wealth-insensitive fine and fine-default custody. It argues that judges are situationally driven to adopt rigid framing about justice when performing duties, as a result of which judges develop indifference to the ‘side-effects’ of their frame-influenced decisions. The findings imply the possibility that the same mechanisms may exist in other jurisdictions and underline the need to address indifference to prevent failure in reforming for a more egalitarian system.

Seismic fragility of non-ductile RC frames for pounding risk assessment

Building structures subjected to seismic excitation tend to vibrate with a unique set of dynamic characteristics, such as natural frequencies, fundamental mode shapes etc. In series-configured structural systems, a difference in such characteristics induces a phase difference between the structures’ oscillations, leading to structural collisions or pounding. Pounding often becomes a cause of catastrophic failures, hence an accurate estimation of structural integrity with respect to pounding is of paramount importance. This study aims to examine different fragility curve development methodologies available in the literature to assess the seismic performance of adjacent Reinforced Concrete (RC) structures subjected to structural pounding. The study primarily focuses on displacement-based fragility curves for the highly critical floor-to-column interactions of an eight-storey non-ductile RC frame, non-eccentrically adjacent to a three-storey rigid RC frame having different storey heights. Nonlinear time–history analyses have been performed to simulate pounding, and fragility curves have been generated using five methodologies. The direct Monte-Carlo Simulation technique has been used to benchmark against the obtained results to identify the suitability of each fragility assessment technique. Results indicate that even though there is no effect of the choice of assessment methodology on the estimated risk at low performance levels and intensity measures, the disparity becomes clearly visible at higher levels. The study presents alternate ways of developing accurate fragility curves by keeping in mind the need for computational efficiency without compromising the accuracy of results. The High Dimensional Model Representation meta-model is found to efficiently represent target pounding fragilities while incurring minimal computational costs.

Construction-Monitoring Analysis of a Symmetrical Rigid Frame Tied Steel Box Arch Bridge in Southwest China Based on Segmental Assembly Technique

Tied steel box arch bridges are increasingly being used due to their attractive appearance, high load-bearing capacity, and good stress performance. Their construction involves multiple processes and factors. Construction monitoring can ensure that such a bridge remains in its intended stress and linear states during and after construction. This helps to minimize deviations from the design state at every stage of construction. Using the segmental assembly construction technique, this study conducted construction monitoring of the alignment and force at each stage of the reconstruction of bridges using MIDAS Civil software. The construction monitoring analysis indicated that the arch rib and lattice beam were correctly placed, thereby meeting the specified requirements for arch rib closure. Displacement errors between the measured and theoretical values at each stage of construction fell within an allowable range, resulting in overall smooth bridge alignment. The measured stress in the main arch and the lattice beam generally corresponded to the theoretical stress derived from the control section stress of the entire bridge. The deviation between the cable force of the suspender and the tie rod and theoretical value fell within 10%, indicating good stress reserve. The symmetrical monitoring points in the analyzed rigid-frame tied steel box arch bridges exhibited symmetrical displacement, stress, and cable force results under various working conditions. This observation further confirms the effectiveness of construction monitoring using the segmental assembly technique.

Light ray fluctuation and lattice refinement of simplicial quantum gravity

In several approaches of non-perturbative quantum gravity, a major outstanding problem is to obtain results valid at the infinite lattice refinement limit. Working with Lorentzian simplicial quantum gravity, we compute light ray fluctuation probabilities in 3D and 4D across different lattices. In a simplified refined box model with the Einstein–Hilbert action, numerical results show that lattice refinement does not simply suppress or simply enhance light ray fluctuations, but actually drives very wide and very narrow light probability distributions towards intermediate ones. A comparison across lattices and across couplings reveals numerical hints at a lattice refinement fixed point associated with a universality class of couplings. The results fit the intuition that quantum spacetime fluctuations reflected by light ray fluctuations start wild microscopically and become mild macroscopically. The refined box model is limited by the assumption of a rigid frame at all scales. The present results suggest further studies around the zero-coupling limit to relax the simplifying assumptions of the model.

Controlling wheel loader motion along a desired trajectory

Problem. Rigid-frame skid steer front loaders are popular material-handling machines widely used in various industries. However, such loaders experience frequent breakdowns due to high dynamic loads; they are often subjected to intensive tyre wear and excessive fuel consumption. Frequently, these issues arise from the improper actions of an operator. Given the labour intensity of the workflow, an operator, regardless of qualification, cannot ensure optimal loader control at an acceptable speed for a long time. In addition, an operator usually works in harsh environments with a high risk of injury, e.g. due to loss of stability. Manual control of loaders also limits the use of these machines in places that are dangerous for humans, for example, during the response to disasters. The solution to these issues is to automate the loader and exclude the operator from the machine work. One of the tasks of loaders automation is to ensure their movement along a given trajectory from the face to the place of unloading the material and back in the opposite direction. Goal. This work aims to increase the productivity and energy efficiency of a rigid-frame skid steer loader by ensuring its following along a given trajectory. Methodology. Based on the kinematics model of the loader, a two-loop control system for motion along a desired trajectory is designed. The control system combines feedforward and feedback loops. The feed-forward control is based on the desired speeds and accelerations. The closed loop is a state controller that uses the loader distance from the desired trajectory at a given time and the course angle as the state variables. Results. According to the simulation results, the proposed two-loop control system ensures the movement of the loader along a desired trajectory with an accuracy of up to 0.3 m. Originality. Further development of the up-to-date control theory was achieved by applying it to a new object, namely a skid steer front loader with a rigid frame. Practical value. The implementation of the proposed control system will increase the productivity of the loader, as well as reduce the number of accidents at construction sites.

Structural system yielding minimum differences between ordinary and staged analyses

Structural engineers should appropriately design concrete structures to resist lateral loads. Determining the adequate system for resisting the expected lateral loads is important to control the building drift. Choosing the appropriate system is usually conducted assuming the predicted forces are applied to completed concrete buildings at one step which is commonly known as ordinary analysis (OA). Nevertheless, these structures are constructed sequentially which requires using staged analysis (SA) instead of OA. In this paper, a comprehensive numerical model for SA of concrete buildings, which accounts for time dependent effects, is utilized using a well-validated commercial software. Six reinforced concrete buildings with 10 and 20 storeys are analyzed using the developed model. Three various structural systems are considered (Rigid Frame (RF), Shear Wall (SW), and Wall Frame (WF). A comparison is conducted between the displacements and internal forces in beams and slabs obtained from the SA and OA. For a 10-storeys RF building, maximum bending moment from SA is 29.9% higher than that from OA. The same conclusion was observed for the maximum shearing force with a percentage of 19.6%. Moreover, maximum bending moments and shearing forces from SA for the 20-storeys RF building are, respectively, 35.0% and 23.5% larger than those from OA. The RF and WF systems provided the minimum difference in differential displacement between the OA and SA analyses. The RF system produced the least differences in internal forces from OA and SA for all studied buildings.

Research on Loading Scheme for Large-Scale Model Tests of Super-Long-Span Arch Bridge

A reasonable and efficient loading scheme is needed to guarantee the success of large-scale bridge tests. In this study, an array-type, self-balancing pulley-group loading system was designed based on the world’s largest spanning arch bridge using a 1:10 scale model test. Automatic statistics of the required load at each loading point were realized using ANSYS, and a load optimization algorithm for loading points at different construction stages was proposed. Tests were carried out separately for the loading system using a single set of pulley groups and an array-type pulley group. Finite element models of the model bridge and the original bridge were established separately using ANSYS, and the stress results of different components during different construction stages of the main arch ring were compared. The research results show the following: (1) The load magnification factor of the single-pulley-group loading device is approximately 6.6 times, with a mechanical efficiency of 94.26%. (2) In the array-type loading system, the actual load at each loading point can reach the design value. The self-balancing characteristic of this system can eliminate the impact of vertical deformation of the structure on loading accuracy, verifying the reliability of the system. (3) The simulation results of the original bridge and the model bridge coincide well, and the stress of each component during the construction process has the same trend. At key construction stages, the maximum relative errors of the stress results of the rigid steel frame and the concrete inside the pipe of the two bridges are 8.33% and 9.34%, respectively, and the maximum absolute error of the bottom plate concrete is 0.66 MPa, verifying the correctness of the counterweight-optimization method. The loading scheme proposed in this paper can provide a reference for the design of loading systems with the same type of scale model test.