On the Application of the Simplified Crack Model in the Bending, Free Vibration and Buckling Analysis of Beams with Linear Variation of Widths

Adequate computational models are crucial for a reliable representation of the mechanical behaviour of structural elements and, therefore, numerous investigations are oriented towards the modelling of cracked structures. This paper studies the behaviour of transversely cracked beams of rectangular cross-sections with linearly-varying widths. The governing differential equations of bending are analytically solved for various beams with different boundary conditions. These simplified model’s solutions are further validated by the results from corresponding 3D finite models of the considered structures. Furthermore, strain and kinetic energy, as well as the work done by an external axial compressive force P are evaluated from the computed transverse displacement functions. These values allowed for estimations of the first eigenfrequency as well as the buckling load. These structural’s parameters were additionally evaluated by implementing dedicated polynomial functions for some cases considered.The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis. The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis

IZVOD MATRICE ELASTIČNE STABILNOSTI STUPA SA POPREČNOM PUKOTINOM NA OSNOVI TAYLOROVOG RAZVOJA

This paper progresses the implementation of a simplified computational model for predicting buckling load Pcr for slender beam ‒ type structures with transverse cracks. The work presents an upgrade of the finite element model that has already successfully proved itself as being capable of producing applicable results by exhibiting considerably good agreement with those values from more precise and complex computational models. The major handicap of the previously applied finite element was the absence of a clear analytical form as it was presented (due to its complex form) only by the main steps of the numerical procedure. For that reason, the main advance of the presented finite element lies in its clear analytical form. In order to achieve this improvement the location of the crack was limited to the mid ‒ span position. Although both (old and new) cracked beam ‒ column finite elements are derived at on the basis of a fairly accurate approximation of the governing differential equation’s solution (GDE), the novel finite element even produces slightly better results. It further allows for flexible utilization and also yields a small compact computational model, thus exhibiting itself as very suitable for inverse identification problems. Numerical examples covering several structures with different boundary conditions are briefly presented in order to confirm the competence of the newly derived finite element. The results obtained using the presented finite element are further compared with corresponding values from references thus clearly proving the quality of the newly derived finite element.Ovaj rad poboljšava uporabu pojednostavljenog komputacijskog modela u predviđanju sile izvijanja Pcr-za vitke konstrukcije grednog tipa s poprečnom pukotinom. Rad predstavlja nadogradnju modela konačnog elementa za koji je već uspješno dokazano da daje primjenjive rezultate pokazujući dobro slaganje s vrijednostima dobivenih preciznijim i složenijim komputacijskim modelima. Glavni nedostatak prethodno primijenjenog konačnog elementa je nepostojanje jasnog analitičkog oblika, jer je (zbog složenosti) predstavljen samo po glavnim koracima numeričkog postupka. Iz tog razloga, glavna prednost predstavljenog konačnog elementa leži u jasnom analitičkom obliku. Da bi se postiglo to poboljšanje mjesto pukotine je ograničeno na sredinu elementa. Iako su oba konačna elementa (stari i novi) za napuknuti stup izvedena na osnovi prilično točne aproksimacije rješenja diferencijalne jednadžbe za poprečne pomake, novi daje i nešto bolje rezultate. To dodatno omogućuje fleksibilnu uporabu i daje mali kompaktni komputacijski model, zbog čega je vrlo pogodan za probleme inverzne identifikacije. Numerički primjeri koji obuhvaćaju nekoliko konstrukcija s različitim rubnim uvjetima ukratko su prikazani kako bi potvrdili učinkovitost novo izvedenog konačnog elementa. Ovaj rad poboljšava uporabu pojednostavljenog komputacijskog modela u predviđanju sile izvijanja Pcr-za vitke konstrukcije grednog tipa s poprečnom pukotinom. Rad predstavlja nadogradnju modela konačnog elementa za koji je već uspješno dokazano da daje primjenjive rezultate pokazujući dobro slaganje s vrijednostima dobivenih preciznijim i složenijim komputacijskim modelima. Glavni nedostatak prethodno primijenjenog konačnog elementa je nepostojanje jasnog analitičkog oblika, jer je (zbog složenosti) predstavljen samo po glavnim koracima numeričkog postupka. Iz tog razloga, glavna prednost predstavljenog konačnog elementa leži u jasnom analitičkom obliku. Da bi se postiglo to poboljšanje mjesto pukotine je ograničeno na sredinu elementa. Iako su oba konačna elementa (stari i novi) za napuknuti stup izvedena na osnovi prilično točne aproksimacije rješenja diferencijalne jednadžbe za poprečne pomake, novi daje i nešto bolje rezultate. To dodatno omogućuje fleksibilnu uporabu i daje mali kompaktni komputacijski model, zbog čega je vrlo pogodan za probleme inverzne identifikacije. Numerički primjeri koji obuhvaćaju nekoliko konstrukcija s različitim rubnim uvjetima ukratko su prikazani kako bi potvrdili učinkovitost novo izvedenog konačnog elementa. Dobiveni rezultati dalje se uspoređuju s odgovarajućim vrijednostima iz literature čime se jasno dokazuje postignuta kvaliteta

On the Bending Analysis of Multi-Cracked Slender Beams with Continuous Height Variations

We studied the implementation of a multi-stepped multi-cracked beam finite element for analysing beams of various heights’ variations along the length. The genuine continuous variation of height was modelled by an adequate series of steps and for the cracks, the model of internal hinges endowed with rotational springs was applied. As the increased number of steps simultaneously increases the possibility that the crack location and step location may coincide, the continuous conditions at the steps were rewritten to a more general form in order to cover this combination. The newly presented form is also capable of taking a transverse concentrated force into account. Numerical evaluations were performed for several structures under various loads and the transverse displacements, nodal reactions, and inner forces were studied. The results obtained are positioned side by side with the results from more detailed 2D finite element meshes. The comparisons show excellent matching of the beam’s response providing the mesh of steps was dense enough. The examples demonstrate the potential of the present approach proving that the model is suitable for achieving computationally-efficient and truthful analyses

A simple FEM Beam Element with an Arbitrary Number of Cracks

To fulfil safety requirements the changes in the static and/or dynamic behaviour of the structure must be analysed with great care. These changes are often caused by local reduction of the stiffness of the structure caused by the irregularities in the structure, as for example cracks. In simple structures such analysis can be performed directly, by solving equations of motion, but for more complex structures a different approach, usually numerical, must be applied. The problem of crack implementation into the structure behaviour has been studied by many authors who have usually modelled the crack as a massless rotational spring of suitable stiffness placed at the beam at the location where the crack occurs. Recently, the numerical procedure for the computation of the stiffness matrix for a beam element with a single transverse crack has been replaced with the element stiffness matrix written in fully symbolic form. A detailed comparison of the results obtained by using 200 2D finite elements with those obtained with a single cracked beam element has confirmed the usefulness of such element

Computation of Stresses and Settlements under an arbitrary Point in Homogenous, elastic, isotropic Half-Space, under the Load described by a uniform Load over a general quadrilateral

The problem of the computation of stresses and settlements in the half-space under various types of loads is often presented in geotechnical engineering. In 1885 Boussinesq advanced theoretical expressions to determine stresses at a point within an ideal mass. His equation considers a point load on the surface of a semi-infinite, homogeneous, isotropic, weightless, elastic half-space. Newmark in 1942 performed the integration of Boussinesq's equations for the vertical stress under a corner of a rectangular area loaded with a uniform load. The problem of the determination of vertical stresses under a rectangular shaped footing has been satisfactorily solved with renewal integration of the Boussinesq's equation over the arbitrary rectangle on surface of the half-space, with a non-uniform load represented with piecewise linear interpolation functions. The problem of the determination of stresses in the case when the footing shape is an arbitrary quadrilateral however remains unsolved. The paper discusses an approach to the computation of vertical stresses and settlements in an arbitrary point of the half-space, loaded with a uniform load, which shape in the ground plan can be a general four noded form with straight edges. Since the form is transformed into a biunit square and all integrations are performed over this area, all solutions are valid also for an arbitrary triangle by the implementation of the degeneration rule

The Evaluation of the Dynamical Soil-Bridge Interaction

In the design of a structure, the implementation of reliable soil-foundation-structure interaction into the analysis process plays a very important role. The paper presents a determination of parameters of a suitably chosen soil-foundation model and their influence on the structure response. Since the mechanical data for the structure can be determined with satisfactory accuracy, the properties of the soil-foundation model were identified using measured dynamic response of the real structure. A simple model describing soil-foundation structure was incorporated into the classical 3-D finite element analysis of the structure with commercial software. Results obtained from the measured data on the pier were afterwards compared with those obtained with the finite model of the pier-foundation-soil structure. On the basis of this comparison the coefficients describing the properties in the soil-foundation model were adjusted until the calculated dynamic response coincided with the measured ones. In this way, the difference between both results was reduced to 1%. Full-scale tests measuring eigenmotion of the bridge were performed through all erection stages of the new bridge in Maribor. In this way an effective and experimentally verified 3-D model for a complex dynamic analysis of the bridge under the earthquake loading was obtained. The significant advantage of the obtained model is that it was updated on the basis of the dynamic measurements thus improving the model on the basis of in-situ geomechanical measurements. The model is very accurate in describing the upper structure and economical in describing the soil mass thus representing an optimal solution regarding computational efforts

Burosumab therapy in children with x-linked hypophosphatemia

BACKGROUND X-linked hypophosphatemia is characterized by increased secretion of fibroblast growth factor 23 (FGF-23), which leads to hypophosphatemia and consequently rickets, osteomalacia, and skeletal deformities. We investigated burosumab, a monoclonal antibody that targets FGF-23, in patients with X-linked hypophosphatemia. METHODS In an open-label, phase 2 trial, we randomly assigned 52 children with X-linked hypophosphatemia, in a 1:1 ratio, to receive subcutaneous burosumab either every 2 weeks or every 4 weeks; the dose was adjusted to achieve a serum phosphorus level at the low end of the normal range. The primary end point was the change from baseline to weeks 40 and 64 in the Thacher rickets severity total score (ranging from 0 to 10, with higher scores indicating greater disease severity). In addition, the Radiographic Global Impression of Change was used to evaluate rachitic changes from baseline to week 40 and to week 64. Additional end points were changes in pharmacodynamic markers, linear growth, physical ability, and patient-reported outcomes and the incidence of adverse events. RESULTS The mean Thacher rickets severity total score decreased from 1.9 at baseline to 0.8 at week 40 with every-2-week dosing and from 1.7 at baseline to 1.1 at week 40 with every-4-week dosing (P<0.001 for both comparisons); these improvements persisted at week 64. The mean serum phosphorus level increased after the first dose in both groups, and more than half the patients in both groups had levels within the normal range (3.2 to 6.1 mg per deciliter [1.0 to 2.0 mmol per liter]) by week 6. Stable serum phosphorus levels were maintained through week 64 with every-2-week dosing. Renal tubular phosphate reabsorption increased from baseline in both groups, with an overall mean increase of 0.98 mg per deciliter (0.32 mmol per liter). The mean dose of burosumab at week 40 was 0.98 mg per kilogram of body weight with every-2-week dosing and 1.50 mg per kilogram with every-4-week dosing. Across both groups, the mean serum alkaline phosphatase level decreased from 459 U per liter at baseline to 369 U per liter at week 64. The mean standing-height z score increased in both groups, with greater improvement seen at all time points with every-2-week dosing (an increase from baseline of 0.19 at week 64) than with every-4-week dosing (an increase from baseline of 0.12 at week 64). Physical ability improved and pain decreased. Nearly all the adverse events were mild or moderate in severity. CONCLUSIONS In children with X-linked hypophosphatemia, treatment with burosumab improved renal tubular phosphate reabsorption, serum phosphorus levels, linear growth, and physical function and reduced pain and the severity of rickets

Results from a 3-year Non-interventional, Observational Disease Monitoring Program in Adults with GNE Myopathy.

BACKGROUND: GNE myopathy is a rare, autosomal recessive, muscle disease caused by mutations in GNE and is characterized by rimmed vacuoles on muscle biopsy and progressive distal to proximal muscle weakness. OBJECTIVE: Investigate the clinical presentation and progression of GNE myopathy. METHODS: The GNE Myopathy Disease Monitoring Program was an international, prospective, observational study in subjects with GNE myopathy. Muscle strength was assessed with hand-held dynamometry (HHD), with upper extremity (UE) and lower extremity (LE) composite scores reflecting upper and lower extremity muscle groups, respectively. The GNE myopathy-Functional Activity Scale (GNEM-FAS) was used to further assess impairment in mobility, upper extremity function, and self-care. RESULTS: Eighty-seven of 101 enrolled subjects completed the trial until study closure by the sponsor; 60 completed 36 months. Mean (SD) HHD UE composite score decreased from 34.3 kg (32.0) at baseline to 29.4 kg (32.6) kg at month 36 (LS mean change [95%CI]: -3.8 kg [-5.9, -1.7]; P = 0.0005). Mean (SD) HHD LE composite score decreased from 32.0 kg (34.1) at baseline to 25.5 kg (31.2) at month 36 (LS mean change [95%CI]: -4.9 [-7.7, -2.2]; P = 0.0005). GNEM-FAS scores were more severe at baseline in subjects who walked <200 meters versus ≥200 meters in 6 minutes; in both groups, GNEM-FAS total, mobility, UE, and self-care scores decreased from baseline through month 36. CONCLUSIONS: These findings demonstrate progressive decline in muscle strength in GNE myopathy and provide insight into the appropriate tools to detect clinically meaningful changes in future GNE myopathy interventional trials

OR13-2 Burosumab Resulted in Greater Improvement in Rickets Than Conventional Therapy in Children with X-Linked Hypophosphatemia (XLH)

XLH is characterized by excess FGF23, hypophosphatemia, skeletal deformities, and growth impairment. For the last 40 years, XLH has been treated with multiple daily doses of oral phosphate and active vitamin D (Pi/D). Burosumab, a fully human monoclonal antibody to FGF23, has been approved by the FDA for the treatment of XLH in patients ≥1 year-old. In this Phase 3 trial (NCT02915705), 61 children with XLH (1-12 years old) were randomized (1:1) to receive subcutaneous burosumab starting at 0.8 mg/kg every 2 weeks or continue Pi/D titrated and individualized for each subject by investigators. Eligibility criteria included a Total Rickets Severity Score (RSS) ≥2.0 despite prior treatment with Pi/D (>7-day washout before baseline). The primary endpoint was healing of rickets at Week 40 assessed by radiologists blinded to treatment using the Radiographic Global Impression of Change (RGI-C). The mean ± SE daily oral phosphate dose from baseline to Week 40 was 37.8 ± 3.2 mg/kg, with >99% compliance reported based on days of dosing. Compared with Pi/D, 40 weeks of burosumab resulted in a greater LS mean ± SE increase in serum phosphorus (0.92 ± 0.08 vs 0.20 ± 0.06 mg/dL), TmP/GFR (1.19 ± 0.11 vs -0.16 ± 0.05 mg/dL), and 1,25(OH)2D (30 ± 4 vs 19 ± 4 pg/mL). At Week 40, rickets improved in both groups; RGI-C global score was significantly higher in burosumab subjects than in Pi/D subjects (LS mean ± SE: +1.9 ± 0.1 vs +0.8 ± 0.1; p<0.0001). More burosumab subjects had substantial healing (RGI-C ≥+2.0), compared with Pi/D subjects (21/29, 72% vs 2/32, 6%; odds ratio of 39.1, p<0.0001). Improvement in the RGI-C lower limb deformity score was greater with burosumab than with Pi/D (+0.62 ± 0.12 vs +0.21 ± 0.12; p=0.02). Alkaline phosphatase decreased more with burosumab compared with Pi/D (-131 ± 13 vs 35 ± 19; p<0.0001). Consistent with decreases in rickets severity, burosumab improved growth and mobility. Standing height Z-score increased by a LS mean change (95% CI) of +0.15 (0.05, 0.25) for burosumab and +0.08 (-0.02, 0.19) for Pi/D. The 6 Minute Walk Test percent predicted distance increased with burosumab (Baseline to Week 40: 62% to 72%) and was unchanged with Pi/D (76% to 75%). Nephrocalcinosis score (range 0-4) shifted 0 in 20 Pi/D and 24 burosumab subjects; +1 in 3 Pi/D and 0 burosumab subjects; and -1 in 3 Pi/D and 2 burosumab subjects. Pre-defined adverse events (AEs) of interest, including hypersensitivity and injection site reactions, were higher in the burosumab group and were mild to moderate in severity overall. There were 4 serious AEs (3 burosumab, 1 Pi/D); none were treatment-related and all resolved. No subject discontinued study drug in either group. Data after 64 weeks of treatment will be available at the time of presentation. In this randomized Phase 3 trial, burosumab resulted in increases in growth and mobility, and significantly greater improvements in rickets than Pi/D in 1-12 year-old children with XLH