Negative Moment Research Articles

Optimization of Technical Calculation of Precast Wall Application for Kawo Secondary Canal Improvement in Sempor Irrigation Area, Kebumen, Central Java

This study aims to perform technical calculations using precast lining in the repair of secondary channels. The planning data used included a concrete grade of fc 18.68 MPa and a reinforcement grade of fy 500 MPa. The service condition analysis was carried out by taking into account the loading during the most critical condition, where the channel is empty and active soil pressure and external water pressure are the main loads. The fixed load calculation was the main focus, where the active earth pressure (PD) was calculated by considering the soil-specific gravity, soil shear angle, and soil coefficient. The calculation results show that the PD reaches 12.24 kN, which gives an idea of the load that the structure has to bear. Furthermore, the internal force analysis provides a deeper understanding of the moments occurring at specific points in the structure. For example, the maximum negative moment at point A was calculated by considering the moments due to fixed load and transient load, which reached 12.73 kNm. This research contributes to improving efficiency and safety in the application of precast lining for secondary channel improvement.

COMMERCE IN THE SHADOWS: EXPLORING DARK WEB BLACK MARKETS

In the twenty-first century, several negative moments have accompanied many positive technological progress events. With the development of the digital world, criminals with special knowledge – cybercriminals have become active and have developed many means in the depths of cyberspace, with the help of which they achieve their criminal and illegal goals. The presented paper aims to analyze in detail and in-depth the system and working principles of digital black markets created by cybercriminals on the dark side of the Internet. The paper also looks at what tools and products are available on the digital black market and why digital black markets are hazardous to operate successfully. The paper analyzes practical cases, legal studies, and other interdisciplinary studies to present the problems in the legal struggle against digital black markets as of the day it was written. The paper’s primary purpose is to show why the unimpeded functioning of digital black markets on the dark web is dangerous – both for the ordinary citizen and the state structures. In the final part of the article, based on the analysis of the processed literature and practical cases, recommendations will be presented, the use of which is of particular importance to make the legal fight against digital black markets successful and to prevent such a dangerous phenomenon as digital black markets from existing in any part of the digital world.

Experimental and numerical investigation on flexural behavior of steel-UHPC composite slabs with PBL shear connectors

A full-scale flexural test was conducted to investigate the flexural and cracking behavior of Steel-UHPC Composite Slabs (SUCS) with PBL (Perfobond rib) shear connectors, with a focus on analyzing the impact of the tensile reinforcement ratio on flexural performance. During the test, the flexural performance of SUCS was assessed through measurements of strain, mid-span deflection, relative slip, crack width, and crack morphology. A section analysis was performed to predict the flexural and cracking behaviors of SUCS, which was further verified by experimental results. The results show that the PBL shear key effectively ensures the overall force of steel plate and UHPC plate, resulting in good ductility of SUCS under negative bending moment. As tensile reinforcement ratio increased from 0.89 % to 1.69 %, the cracking load remained stable while the yield load and ultimate load demonstrated significant improvements of 54 % and 57 %, respectively. The cross-section analysis indicated that with the increment in tensile reinforcement ratio, the contribution of tensile reinforcement to flexural capacity increased, whereas the contribution of UHPC to tensile capacity notably decreased. The equivalent uniform stress reduction coefficient of the UHPC's tensile zone, determined through simplified negative flexural bearing capacity, was calculated to be 0.87. The failure mode of negative flexural resistance differs significantly from positive flexural resistance, as it involves the breaking of the tensile steel bar.

Международные тренды цифровизации валютно-финансовых отношений: от глобализации к децентрализации

The need for the use of new, digital forms of organization of monetary and financial settlements with foreign partners, as well as the formation of a national system of secure payments. The article presents the result of analysis of motivations of introduction of digital currencies of central banks (positive and negative moments of the considered process). The article gives an overview of existing international payment systems, including the Russian one, and identifies advantages and disadvantages. The main tendencies of the cryptocurrency market as a possible mechanism of formation of decentralized finance are considered.

Research Progress on Shear Characteristics and Rapid Post-Disaster Construction of Narrow-Width Steel Box–UHPC Composite Beams

The narrow-width steel box girder is an important type of steel–concrete composite bridge structure, which is usually composed of reinforced concrete wing plates, narrow steel boxes partially injected with concrete, and shear connectors that promote shear force transfer. The utilization of narrow-width steel box girders, augmented by partially filled concrete, embodies the synthesis of steel and concrete elements, fostering structural efficiency. Moreover, its attributes, including reduced structural weight, diminished vertical profile, enhanced load-bearing capacity, and augmented stiffness, have prompted its gradual integration into bridge engineering applications. In this study, the calculated values of shear strength under three current design codes were reviewed, and the shear failure phenomena and its determinants of narrow-width steel box–ultra-high-performance concrete (UHPC) composite beams under negative bending moment conditions were investigated, which were mainly determined by shear span ratio, concrete wing plate, UHPC steel fiber content, UHPC plate thickness, and transverse partition inside the box. Concurrently, this paper evaluates two innovative structural designs, including a double-narrow steel box girder and a three-narrow steel box girder. In addition, strategies to reduce crack formation under the negative bending moment of long-span continuous narrow and wide box girder abutments are discussed, and we show that this measure can effectively control the formation of cracks to support the negative bending moment zone. At the same time, the scope of the application of a narrow-width steel box girder composite bridge is reviewed, and the conclusion is that a narrow-width steel box girder is mainly used in small-radius flat-curved bridges or widened-ramp bridges with a span of 30 m or more in interworking areas and in the main line with a 60–100 m span in mountainous or urban areas. Finally, the research direction of the shear resistance of the UHPC–narrow steel box girder under negative bending moments is proposed.

Theoretical analysis on the behavior of SPM anchor piles embedded in sand under oblique pullout load with emphasis on the padeye depth

Single-point mooring (SPM) system has been used in offshore engineering for its superiority in overcoming severe sea conditions. However, researches focused on the interaction mechanism between the anchor pile and soil in SPM systems remains insufficient, especially regarding the calculation method for oblique pullout bearing capacity that considers padeye depth. Based on the p-y curve method and load transfer method, this paper presents a simple prediction method for the behavior of anchor piles embedded in sand under oblique pullout load. The theoretical predictions are validated through comparing with results from the three-dimensional finite element method and existing centrifuge model tests. The effect of the loading angle and the padeye depth are analyzed and discussed. A negative bending moment peak and axial force discontinuity phenomenon appear at the padeye depth. The optimal padeye depth for anchor piles in this study ranges from approximately 0.53 to 0.80 times the pile length. When the padeye depth is less than 0.33 times of the pile length, the ultimate bearing capacity of the anchor pile increases with increasing load angles. Conversely, when the padeye depth exceeds 0.33 times the pile length, the ultimate bearing capacity decreases with the increasing load angles.

Fatigue behavior and damage mechanism of steel plate-UHPC composite bridge deck subjected to negative bending moment

In order to investigate the fatigue behavior and failure mechanism of steel plate-UHPC composite bridge deck under hogging moment, a total of four full-scale specimens with two class of reinforcement ratio (Three specimens with reinforcement ratio of 1.58 %, 1.69 %, 1.86 %, respectively, which called longitudinal specimen. One specimen with reinforcement ratio of 0.89 %, which called transverse specimen.) were designed and tested under fatigue loading. Test results reveal that when the fatigue loading ratio was 0.25 with maximum load of 0.8 times of the static cracking load, taking the crack width of 0.10 mm as the fatigue failure criterion, the fatigue life of longitudinal specimen over 16 million cycles. However, the fatigue life of transverse specimen was only 9.08 million cycles. The longitudinal reinforcement restricts the propagation of fatigue cracks along the depth of the UHPC slab, and tensile fracture of the steel fibers occurred in the region between the top layer of longitudinal reinforcement and the UHPC surface. The fatigue damage mechanism resulting from fatigue degradation of the UHPC layer in the tension zone is identified. In addition, The fatigue design of the steel plate-UHPC composite bridge deck should be controlled by the durability rather than the strength. The established tensile S-N curve with 0.05 mm critical UHPC crack width could be used evaluate the fatigue cracking resistance of composite bridge deck.

Mechanical behavior improving study of concrete deck of main beam at pylon root of composite beam cable-stayed bridge

Steel-concrete composite beam has been increasingly applied to large span cable-stayed bridges. It takes full advantage of the material properties of steel and concrete. However, the concrete deck bears tension in the negative moment zone, such as zero block, which is disadvantageous to structures. Aiming at this problem, a finite element model of the zero block in the negative moment zone of a semi-floating cable-stayed bridge is built, and the local mechanical performance of the bridge deck under completed status is studied. Based on the analysis results, three improvement measures have been proposed. The improvement effect of each method and composed of three methods has been studied. The numerical results show that the whole zero block zone is in the compressed state under the combined action of the bending moment and axial force of the stay cable. However, the local negative moment effect in the zero block zone is very prominent under the support of the diaphragm plate. Removing parts of the diaphragm plate at the bearing position can significantly improve local mechanical behavior in the concrete deck, which transfers the local support to the adjacent two diaphragm plates. The composed improvement effect is prominent when the three measures are adopted simultaneously.

Local buckling of the bottom flange in steel-concrete composite beams with trapezoidal corrugated webs subjected to end restraint at high temperatures

Steel-concrete composite beams with trapezoidal corrugated webs (SCBCW) are prone to rapid failure at high temperatures due to local instability of the bottom flange. The failure mechanism of SCBCW, which is an integral part of the holistic structure and subject to axial and rotational restraints from adjacent members at elevated temperatures, differs from conventionally simply supported steel-concrete composite beams with straight webs. To investigate the evolution of local buckling in the compressed flange of SCBCW under high temperature conditions, a composite beam model with frame restraints was developed using ABAQUS finite element software. In this model, the composite beam is connected to restraint frame columns through high-strength bolts. The changes in axial stress, bending moment, and deflection of the compressed flange of SCBCW were analyzed as temperature increased. Results indicate that at elevated temperatures, additional bending moments occur in the flange plane due to axial restraints, section temperature gradients, section negative bending moments and accordion effect caused by corrugated web on compressed bottom flange of SCBCW. This leads to a gradual increase in axial compressive stress before local buckling occurs followed by a decrease due to internal force redistribution upon onset of buckling. Studies show that initial buckling of bottom flange occurs on one side of flat section of web where axial compressive stress is relatively high while restraint effect on web is low; Subsequent local buckling of the beam then appears on opposite side adjacent flat section of web after initial buckle occure on one side. After local buckling appears on both sides of web tension bands appear on both sides where bottom flange experiences localized bucking. The setting of longitudinal stiffeners (LS) can effectively improve the local stability of the bottom flange of the composite beam, while the setting of transverse stiffeners (TS) has no significant impact.

Negative discrete moments of the derivative of the Riemann zeta‐function

AbstractWe obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta‐function averaged over a subfamily of zeros of the zeta function that is expected to be arbitrarily close to full density inside the set of all zeros. For , our bounds for the ‐th moments are expected to be almost optimal. Assuming a conjecture about the maximum size of the argument of the zeta function on the critical line, we obtain upper bounds for these negative moments of the same strength while summing over a larger subfamily of zeta zeros.

Improving the local and distortional resistance of CFS trapezoidal self-supporting roof members

This paper reports the development, numerical implementation and application of a methodology to perform the multi-objective optimisation, with respect to local and distortional failures, of cold-formed steel (CFS) trapezoidal cross-section shapes used in members belonging to self-supporting roof systems and manufactured by bending the steel sheets with roll-forming machines. Initially, a brute-force approach is adopted to search for optimal unstiffened cross-section shapes of members subjected to positive (sagging) and negative (hogging) bending moments. Next, an optimisation procedure based on a Genetic Algorithm (GA) is employed to find stiffened cross-sections that (i) outperform the original (plain/unstiffened) ones, as far as the resistance against local and distortional failures is concerned, and (ii) can be fabricated with minimal additional cost, through the sole addition of small intermediate stiffeners, whose possible locations are pre-defined in accordance with a map of executable fold-line positions – the original cross-section typology, overall dimensions and wall angles are retained. The design approach adopted to estimate the member ultimate strengths is based on the Direct Strength Method (DSM), and Generalised Beam Theory (GBT) is used to calculate the elastic buckling stresses due to bending. The application and capabilities of the proposed methodology are illustrated, by means of numerical results presented in terms of cross-section families belonging to optimal Pareto Fronts – an excellent performance of the procedure was found in all cases.

Integrating Superconductivity in Cu Replace Lead Apatite by Nuclear Magnetic Moment Theory of RBL

Recently, scientists proclaimed superconductivity under ambient conditions of room temperature and 1 atmosphere pressure in Cu partial substituted lead apatite: Pb(10-x)Cux(PO4)6. This paper highlights the application of RBL’s stable isotope of positive and negative nuclear magnetic moments (NMMs) theory for explaining the heavy isotopic enrichment of this materials as {207Pb{10-x}63Cux(‘31PιO4)6}, where ι = 17 or 18 and the resulting superconductivity and novel room temperature atmospheric pressure superconductivity of this heavy isotopic enriched substance. On the basis of such analysis by RBL theory, the synthesis of high temperature, regular pressure superconductivity can be explained and this recent experimental observation of ambient pressure and temperature superconductivity in 207Pb{10-x}63Cux(‘31PιO4)6, where ι = 17 or 18, proves RBL’s NMMs theory of high temperature superconductivity. After 1 month on July 22, 2023 of the archiving of such data for superconductivity Pb(10-x)Cux(PO4)6 by Lee and coworkers, a huge experimental effort by hundreds of researchers around the world has attempted to replicate this room temperature, ambient pressure superconductivity in Pb(10-x)Cux(PO4)6 without success. The resulting inability to replicate the ambient superconductivity and note of some fractional component in the mixture causing the ambient superconductivity is explained by RBL’s theory as the fractional component may be clumped isotopes as in {207Pb{10-x}63Cux(‘31PιO4)6}, where ι = 17 or 18, and RBL here notes the vapor deposition process alleged by Lee and coworkers for producing the superconductivity was previously predicted by RBL to cause isotopic fractionation during solid to liquid and liquid to gas and gas to solid physical changes.

Flexural performance of full-scale two-span Nail-Laminated Timber Concrete composite slabs

This study examines the flexural performance of six 9-m full-scale two-span Nail-Laminated Timber Concrete (NLTC) composite slabs. The slabs were made with lumber beams edge-joined with double nailing, end-joined with butt joints, and the reinforced concrete topping connected with a set of notches, inclined screws, or a combination of both. The multi-span configuration of slabs reduces their deflections simply and effectively. Five-point monotonic bending tests were considered for all slabs. Before full-scale slabs, compressive and tensile pull-out tests of Timber-Concrete Composite (TCC) shear connections were performed, including notches and inclined screws. Tensile pull-out tests of shear connections were also included to emulate the negative bending moments that occur in the middle of the slabs. Failure modes, load–mid-span deflection relation, bending stiffness, and timber-concrete slip were evaluated for all slabs. A detailed 3D micro-Finite Element (FE) model of the shear connections was built in ANSYS software, whereas a macro-FE model of NLTC slabs was made in SAP2000, demonstrating a good fit for the timber-concrete interaction and the load-carrying capacity of the composite slab at the serviceability range. Moreover, an analytical elastic TCC beam with the Girhammar method was assessed and demonstrated as more precise than the traditional γ-method. Finally, an accurate prediction of the numerical and analytical (Girhammar) models for the bending stiffness at service loads up to 30% of capacity is observed, with errors in a range of 2–23% and 9–74%, respectively.

'Role Model Moments' and 'Troll Model Moments' in Surgical Residency: How Do They Influence Professional Identity Formation?

Role models are powerful contributors to residents' professional identity formation (PIF) by exhibiting the values and attributes of the community. While substantial knowledge on different attributes of role models exists, little is known about their influence on residents' PIF. The aim of this study was to explore surgical residents' experiences with role models and to understand how these contribute to residents' PIF. Adopting a social constructivist paradigm, the authors used a grounded theory approach to develop an explanatory model for residents' experiences with role models regarding PIF. Fourteen surgical residents participated in individual interviews. The authors iteratively performed data collection and analysis, and applied constant comparison to identify relevant themes. Role model behavior is highly situation dependent. Therefore, residents learn through specific 'role model moments'. These moments arise when residents (1) feel positive about a moment, e.g. "inspiration", (2) have a sense of involvement, and (3) identify with their role model. Negative role model moments ('troll model moments') are dominated by negative emotions and residents reject the modeled behavior. Residents learn through observation, reflection and adapting modeled behavior. As a result, residents negotiate their values, strengthen attributes, and learn to make choices on the individual path of becoming a surgeon. The authors suggest a nuance in the discussion on role modelling: from 'learning from role models' to 'learning from role model moments'. It is expected that residents' PIF will benefit from this approach since contextual factors and individual needs are emphasized. Residents need to develop antennae for both role model moments and troll model moments and acquire the skills to learn from them. Role model moments and troll model moments are strong catalysts of PIF as residents follow in the footsteps of their role models, yet learn to go their own way.

Investigation of Concrete Damage on a Prefabricated Steel Spring Floating Slab Track by Finite Element Modelling

AbstractTo investigate the concrete damage of prefabricated steel spring floating slab tracks (SSFST), a three-slab prefabricated SSFST system was established using the ABAQUS finite element software. Full trainload conditions and fatigue load conditions of a train passage were successively applied to the system. Plastic damage and fatigue damage of the floating slab were simulated based on concrete damage plasticity theory and model code, respectively. For comparison, a simulation of the fatigue experiment was conducted. Parametric analyses of the concrete strength and isolator stiffness were also performed. The results show that the maximum positive and negative bending moments of the floating slab throughout the loading stage are close in value. The positive bending moment causes stress concentration on the top slab surface which leads to plastic damage and low-cycle fatigue damage, while the negative bending moment causes middle-level elastic tensile stress on the bottom slab surface which leads to high-cycle fatigue damage. Under experimental conditions, damage on the bottom surface is much more severe, while the upper part is undamaged. Improving the concrete strength can reduce both kinds of damage, while increasing the isolator stiffness can only mitigate the high-cycle fatigue damage. Accordingly, recommendations are provided for improving fatigue experiments and structural design of prefabricated floating slabs.This study can inform the design and maintenance of the prefabricated SSFST system, ultimately enhancing their safety and longevity.

A first-principles study of bilayer 1T'-WTe2/CrI3: a candidate topological spin filter

The ability to manipulate electronic spin channels in 2D materials is crucial for realizing next-generation spintronics. Spin filters are spintronic components that polarize spins using external electromagnetic fields or intrinsic material properties like magnetism. Recently, topological protection from backscattering has emerged as an enticing feature that can be leveraged to enhance the robustness of 2D spin filters. In this work, we propose and then characterize one of the first 2D topological spin filters: bilayer CrI3/1T’-WTe2. To do so, we use a combination of density functional theory, maximally localized Wannier functions, and quantum transport calculations to demonstrate that a terraced bilayer satisfies the principal criteria for being a topological spin filter: namely, that it is gapless, exhibits spin-polarized charge transfer from WTe2 to CrI3 that renders the bilayer metallic, and has a topological boundary which retains the edge conductance of monolayer 1T’-WTe2. In particular, we observe that small negative ferromagnetic moments are induced on the W atoms in the bilayer, and the atomic magnetic moments on the Cr are approximately 3.2 μB/Cr compared to 2.9 μB/Cr in freestanding monolayer CrI3. Subtracting the charge and spin densities of the constituent monolayers from those of the bilayer further reveals spin-orbit coupling-enhanced spin-polarized charge transfer from WTe2 to CrI3. We demonstrate that the bilayer is topologically trivial by showing that its Chern number is zero. Lastly, we show that interfacial scattering at the boundary between the terraced materials does not remove WTe2’s edge conductance. Altogether, this evidence indicates that BL 1T’-WTe2/CrI3 is gapless, magnetic, and topologically trivial, meaning that a terraced WTe2/CrI3 bilayer heterostructure in which only a portion of a WTe2 monolayer is topped with CrI3 is a promising candidate for a 2D topological spin filter. Our results further suggest that 1D chiral edge states may be realized by stacking strongly ferromagnetic monolayers, like CrI3, atop 2D nonmagnetic Weyl semimetals like 1T’-WTe2.

Experimental Study on Flexural Resistance of UHPC Wet Joint Precast Reinforced Concrete Bridge Deck Slabs with Variable Cross-Section

With the convenient and fast requirements for construction in bridge engineering, prefabricated assembly technology is widely applied in engineering construction. Typically, prefabricated bridge decks are connected through cast-in-place wet joints. Wet joints, as the primary load-bearing parts of bridge decks, undergo complex stress and are prone to cracking and damage. Particularly in the negative bending moment region of bridges, the influence of tensile stress on wet joints is more severe, thus enhancing the mechanical performance and crack resistance of joints becomes crucial. This paper investigates the mechanical behavior of prefabricated reinforced concrete bridge deck panels with variable cross-sections under negative bending moments, focusing on the performance of Ultra High-Performance Concrete (UHPC) wet joints. Full-scale experimental tests were conducted on a 176 m steel truss composite continuous rigid bridge, employing C50 concrete panels with UHPC wet joints. Results show three distinct stages: elastic, crack initiation and propagation, and failure. The maximum failure load reached 822 kN, with a maximum displacement of 21.64 mm. Concrete strains indicate compressive stress near the wet joint and tensile stress near the loading positions. Cracks primarily develop at the wet joint interface and propagate under increasing load, ultimately leading to flexural–shear failure near the variable cross-section of the wet joint. Numerical simulations using ABAQUS/CAE (2020) corroborate experimental findings, closely matching load-displacement curves and identifying damage locations. The study demonstrates that UHPC wet joints significantly enhance crack resistance, meeting design requirements for improved mechanical performance in bridge structures subjected to negative bending moments.

Experimental and Theoretical Investigations of the Mechanical Behavior of Column-Free Quasi-Rectangular Segmental Tunnel Linings

To investigate the mechanical behavior and design methodology of column-free QRST (quasi-rectangular segmental tunnel) structures, a theoretical analysis based on prototype experiments and simulation models is conducted. Initially, a prototype experimental investigation is conducted to reveal the structural behavior at the service stage. Subsequently, the ESHR model (Equivalent Stiffness Homogeneous Ring), the BS model (Beam Spring), and the MBS model (Modified Beam Spring) are used to simulate structural behavior. For design purposes, the design methodology is explored based on the ESHR model, followed by a sensitivity analysis of several key load parameters. Based on the experimental results, weak parts of the column-free QRST structure are found to include several joints (Joint 1, Joint 5, Joint 3, and Joint 8), and corresponding optimization measures are proposed. By comparing the test results, the above-mentioned three models demonstrate their applicability in structural simulation, with the ESHR model having sufficient design accuracy. A model-based deformation mechanism analysis found that joints contribute approximately 2/3 of the structural deformation. For the structural design of the column-free QRST using the ESHR model, amplifying the calculated results of structures directly subjected to the service stage by 10% suffices to meet engineering requirements. Based on the test and study, special attention should be paid to the negative bending moment regions at the waists of the structure during both the design and service stages.

Negative moment of inertia and rotational instability of gluon plasma

Using first-principle numerical simulations of the lattice SU(3) gauge theory, we calculate the isothermal moment of inertia of the rigidly rotating gluon plasma. We find that the moment of inertia unexpectedly takes a negative value below the “supervortical temperature” Ts=1.50(10)Tc, vanishes at T=Ts, and becomes a positive quantity at higher temperatures. The negative moment of inertia indicates a thermodynamic instability of rigid rotation. We derive the condition of thermodynamic stability of the vortical plasma and show how it relates to the scale anomaly and the magnetic gluon condensate. The rotational instability of gluon plasma shares a striking similarity with the rotational instabilities of spinning Kerr and Myers-Perry black holes.

Flexural behavior of steel-UHPC composite beams with different connectors in negative moment

This paper investigates the flexural performance of steel-ultra-high performance concrete (UHPC) composite beams with different types of connectors (normal studs and rubber-sleeve studs) and concrete types (normal strength concrete, UHPC and slag UHPC) in the negative moment region. The feasibility of the application of environmentally friendly UHPC in composite beams is analyzed and a moment-curvature calculation model for composite beams is developed. The cracking moment calculation formula is proposed based on the plane section assumption, and the reliability of the method is verified. The results show that UHPC can significantly improve the flexural performance of the structure in the negative moment region, but the change in the type of UHPC has less effect on the flexural performance. Slag UHPC reduces price, reduces environmental impact and is viable for structural applications. The rubber connectors can change the strain distribution of composite beams in the early loading period and improve the cracking resistance of composite UHPC slabs. The moment-curvature model developed in this paper can effectively identify the change characteristics of the structure during the loading process, and then determine the characteristic moment of the structure. Comparison of the equations for cracking moments shows that the cracking loads on the structure are overestimated if the interface between the concrete slab and the steel beam is assumed to be free to move.