Showing papers on "Bending moment published in 1986"

Concrete structures: Stresses and deformations

Abstract: Preface to the third edition Acknowledgements Note The SI system of units and British equivalents Notation 1 Creep and shrinkage of concrete and relaxation of steel 11 Introduction 12 Creep of Concrete 13 Shrinkage of Concrete 14 Relaxation of prestressed steel 15 Reduced relaxation 16 Creep superposition 11 The aging coefficient ?: definition 18 Equation for the aging coefficient ? 19 Relaxation of concrete 110 Step-by Step calculation of the relaxation function for concrete 111 Age-adjusted elasticity modulus 112 General 2 Stress and strain of uncracked sections 21 Introduction 22 Sign convention 23 Strain, stress and curvature in composite and homogeneous cross-sections 24 Strain and stress due to non-linear temperature variation 25 Time-dependent stress and strain in a composite section 26 Summary of analysis of time-dependent strain and stress 27 Examples worked out in British units 28 General 3 Special cases of uncracked sections and calculation of displacements 31 Introduction 32 Prestress loss in a section with one layer of reinforcement 33 Effects of presence of non-prestressed steel 34 Reinforced concrete section without prestress: effects of creep and shrinkage 35 Approximate equations for axial strain and curvature due to creep 36 Graphs for rectangular sections 37 Multi-stage prestressing 38 Calculation of displacements 39 Example worked out in British units 310 General 4 Time-dependent internal forces in uncracked structures: analysis by the force method 41 Introduction 42 The force method 43 Analysis of time-dependent changes of internal forces by the force method 44 Movement of supports of continuous structures 45 Accounting for the reinforcement 46 Step-by-step analysis by the force method 47 Example worked out in British units 48 General 5 Time-dependent internal forces in uncracked structures: analysis by the displacement method 51 Introduction 52 The displacement method 53 Time-dependent changes in fixed-end forces in a homogeneous member 54 Analysis of time-dependent changes in internal forces in continuous structures 55 Continuous composite structures 56 Time-dependent changes in the fixed-end forces in a composite member 57 Artificial restraining forces 58 Step-by-step analysis by the displacement method 59 General 6 Analysis of time-dependent internal forces with conventional computer programs 61 Introduction 62 Assumptions and limitations 63 Problem statement 64 Computer programs 65 Two computer runs 66 Equivalent temperature parameters 67 Multi-stage loading 68 Examples 69 General 7 Stress and strain of cracked sections 71 Introduction 72 Basic assumptions 73 Sign convention 74 Instantaneous stress and strain 75 Effects of creep and shrinkage on a reinforced concrete section without prestress 76 Partial prestressed sections 77 Flow chart 78 Example worked out in British units 79 General8 Displacements of cracked members 81 Introduction 82 Basic assumptions 83 Strain due to axial tension 84 Curvature due to bending 85 Curvature due to a bending moment combined with an axial force 86 Summary and idealized model for calculation of deformations of cracked members subjected to N and/or M 87 Time-dependent deformations of cracked members 88 Shear deformations 89 Angle of twist due to torsion 810 Examples worked out in British units 811 General 9 Simplified prediction of deflections 91 Introduction 92 Curvature coefficients, k 93 Deflection prediction by interpolation between uncracked and cracked states 94 Interpolation procedure: the 'bilinear method' 95 Effective moment of inertia 96 Simplified procedure for calculation of curvature at a section subjected to M and N 97 Deflections by bilinear method: members subjected to M and N 98 Estimation of probable defection: method of 'global coefficients' 99 Deflection of two-way s

Ultimate axial load capacity of a delaminated beam-plate

Abstract: The paper presents elastic buckling and postbuckling analysis of an axially loaded beam-plate with an across-the-width delamination symmetrically located at an arbitrary depth. It is found that, for a relatively short and thick delamination, the buckling load of the delaminated plate is a close lower bound of the ultimate axial load capacity. In the case of a relatively slender delamination, the postbuckling axial load can be considerably greater than the buckling load, while the failure of the plate may or may not be governed by delamination growth. The energy-release rate associated with delamination growth is systematically computed for various combinations of the delamination length and axial load. The resulting curves determine the possibility and the stability characteristics of delamination growth and provide a basis for finding the ultimate axial load capacity. Nomenclature = half-length of delamination = thickness of delamination = half-length of the plate = thickness of the plate =a/t = a/h = axial forces = bending moments w = transverse deflections = bending stiffnesses Young's modulus in the axial direction amplitude of deflection a h / t a a Pi Mf u, v,

Joint Rotation and Geometric Nonlinear Analysis

Abstract: A geometric stiffness matrix for a beam element has been derived in a related paper. The derived matrix implies quasitangential behavior of bending moments and semitangential behavior of torques. Use of this element matrix in a three‐dimensional, large displacement analysis can result in an apparent lack of joint equilibrium. Alternative definitions for moments are studied for the purpose of resolving this problem. It is verified that equilibrium of a rotated angle joint can be achieved if each joint moment is defined as a semitangential moment.

Exact Bernoulli‐Euler static stiffness matrix for a range of tapered beam‐columns

Abstract: The static stiffnesses for torsional and flexural deformation of a tapered beam under axial loading are determined analytically, extending the Bernoulli-Euler/Bessel-function approach of Banerjee and Williams (1985). The derivation of the expressions is explained in detail, and numerical results for sample problems are presented in tables and shown to be in excellent agreement with those obtained using stepped beams of 400 or 500 elements.

Surface waves of large amplitude beneath an elastic sheet. Part 1. High-order series solution

Abstract: Two-dimensional periodic waves beneath an elastic sheet resting on the surface of an infinitely deep fluid are investigated using a high-order series-expansion technique. The solution is found to have certain features in common with capillary-gravity waves; specifically, there is a countably infinite set of values of the flexural rigidity of the sheet at which the series solution fails, and these values are conjectured to be bifurcation points of the solution. Limiting waves of maximum height are found at each value of the flexural rigidity investigated. These are characterized by a cusp singularity in the elastic bending moment at the wave crest, and infinite fluid pressure there. For at least one value of the flexural rigidity, the series solution shows that the wave of maximum height also travels with infinite speed.

Static Stability of Curved Thin‐Walled Beams

Abstract: Starting from a continuum mechanics basis, the principle of virtual displacements is used for deriving the differential equations of equilibrium for a horizontally curved I‐beam. In the present formulation, the effect of curvature is considered, while the condition of inextensibility, usually adopted in conventional analyses, is not required. The cross‐sectional displacements for the curved I‐beam are derived through employment of Vlasov's thin‐walled beam assumptions. Due to the inclusion of nonlinear strains in the virtual work equation, the instability effects caused by various loads, including the axial force, transverse shears, torque, bending moments, and bimoment are generally taken into account. The solutions are presented and compared with existing ones for two special cases.

A refined shear deformation theory for the analysis of laminated plates

Abstract: A refined, third-order plate theory that accounts for the transverse shear strains is presented, the Navier solutions are derived for certain simply supported cross-ply and antisymmetric angle-ply laminates, and finite-element models are developed for general laminates. The new theory does not require the shear correction factors of the first-order theory (i.e., the Reissner-Mindlin plate theory) because the transverse shear stresses are represented parabolically in the present theory. A mixed finite-element model that uses independent approximations of the generalized displacements and generalized moments, and a displacement model that uses only the generalized displacements as degrees of freedom are developed. The displacement model requires C sup 1-continuity of the transverse deflection across the inter-element boundaries, whereas the mixed model requires a C sup 0-element. Also, the mixed model does not require continuous approximations (between elements) of the bending moments. Numerical results are presented to show the accuracy of the present theory in predicting the transverse stresses. Numerical results are also presented for the nonlinear bending of plates, and the results compare well with the experimental results available in the literature.

Combined Membrane and Flexural Reinforcement in Plates and Shells

Abstract: A plate or shell element subjected to membrane forces Nx, Ny, Nxy and bending moments Mx, My, Mxy, is considered. Based on equilibrium considerations, equations for capacities of top and bottom reinforcements in two orthogonal directions have been derived. An iterative method is suggested for calculating the design capacities. The proposed equations are more general and rigorous than those derived for membrane reinforcement alone and those for flexure only. For the membrane alone case, the proposed equations degenerate into the previously derived equations. For the latter, it is shown that the present practice of designing flexural reinforcement may underestimate the required capacity.

Wind turbine gust load alleviation utilizing curved blades

Abstract: The aeroelastic response of curved (i.e., tip sweep in planform) wind turbine blades is analyzed. Numerical results for a 3-bladed, 20-ft radius wind turbine with fiberglass blades were obtained for various tip sweeps. Tip sweeps up to 5 ft appear to provide little tip deflection and bending moment reductions in response to gusts. However, blades with much lower torsional rigidities than the fiberglass blades were shown to provide up to 279b in tip deflections and bending moment relief during gusts.

Buckling of viscoelastic beam columns

Abstract: The paper presents a theoretical study of the creep buckling behaviour of viscoelastic beam-columns under general loading conditions. A detailed analysis is given for three particular cases, axially compressed columns with initial curvatures, laterally loaded beam-columns and beam-columns in bending. The creep buckling problem is formulated in terms of the constitutive equations of the linear hereditary viscoelasticity. It includes two types of viscoelastic materials, those with limited and unlimited creep. The general solution is derived by means of the quasi-elastic method and is examined in detail for two simple rheological material models. It is shown that the creep buckling behaviour of linearly viscoelastic beam-columns under various loading conditions is typically governed by the magnitude of the axial compressive force. For viscoelastic materials of the limited creep type there is a safe load limit below which the creep buckling characteristics of the structure are limited in time. The magnitude of the safe load limit as related to the Euler's elastic critical load solely depends upon the asymptotic value of the creep function of the material. Quasi-elastic approximations and the corresponding exact analytical solutions are compared in two simple problems. It is observed that the quasi-elastic technique is adequate for applications to the linear creep buckling analysis.

Shear and Bending Response of a Rigid-Plastic Beam to Partly Distributed Blast-Type Loading

Abstract: A study is undertaken on dynamic response of a simply supported rigid perfectly plastic beam that is subjected to partly distributed blast-type pressure loading. The beam material has finite shear strength and obeys a square yield criterion relating bending moment and transverse shear force. The transverse dynamic load is uniformly and symmetrically distributed over a middle portion of the span. Various patterns of deformation, which combine plastic bending and shear sliding, are obtained for a wide range of parameters, and the effects of transverse shear forces and time dependence of the dynamic pressure are examined.

Method for making a prestressed composite structure and structure made thereby

Abstract: A prestressed composite structure (48) and method for making same. The ferroconcrete prestressed structure (48) includes a tensile member (10) which includes a steel I-beam (12) which has on its upper flange (16) a plurality of shear connectors (14). The beam (12) is bent or bowed by pushing on the center region of the beam (12) with a screw jack (28) or the like while forces are applied to the first and second ends (21) and (23), respectively, of the beam (12). The end forces can simply be due to the weight of the beam (12) or can be supplemented, in one embodiment, through the use of threaded rods (32) and (38) which interconnect the beam (12) and a dummy beam (26). The bowed beam (12) has a convex surface (42) on which a compressive layer (46) is attached. Preferably, a concrete layer (46) is utilized with the concrete bonding to an upper flange (16) of the beam (12) and the concrete layer (46) encasing or enveloping the shear connectors (14) to make the composite unit (48) act as a single structural device. Once the concrete layer (46) has sufficiently cured, the bending moment created by the screw jack (28) and rods (32) and (38) is removed and the resulting composite structure (48) is prestressed and is therefore better able to withstand dead and live loading.

Bending machine for bending bars, channels, sections and the like

Abstract: A bending machine comprises a support to which first and second spaced cooperating bending devices are displaceably mounted. Each of the bending devices comprises first and second spaced cooperating bending components and the components of the first and second bending devices define a first axis extending between the bending devices and further define second and third axes extending generally transverse to the first axis and about which a length to be bent may pivot and each of the second and third axes are disposed between the components of one of the bending devices. A drive mechanism is operably associated with at least one of the components of each of the bending devices for causing displacement thereof. A guide system is operably associated with the support and with each of the bending devices so that displacement of the bending components by the drive system causes the bending components to move relative to the first axis and to engage the length to be bent extending between the devices. The guide system further permits the bending means to move along the guide system relative to each other so that the length is bent intermediate the bending devices.

Method of forming a composite structural member

Abstract: A composite structural member is formed by fixedly mounting a prestressed concrete slab having compressive stress acting along pc steel wires buried therein on a beam. The compressive stress is thereafter released from the prestressed concrete slab by loosening a turnbuckle or the like provided in the prestressed concrete slab, so as to produce in the beam a tensile force acting in the same direction as the direction of the compressive stress acting in the prestressed concrete slab and a bending moment. The slabs, placed in a side by side and end to end configuration, have undulated end surfaces spaced apart on a beam. The ends abut dowels fixed to the main beam. The space defined by the end surfaces with the dowels is filled with mortar to integrally fix the slab to the beam.

Finite element stress analysis of an equal diameter branch pipe intersection subjected to out-of-plane and twisting moments

Abstract: The authors have used the BERSAFE finite element computer program to model an equal diameter branch pipe intersection of mean diameter/thickness ratio 24.5. Previous results for internal pressure and the two in-plane bending moments are augmented by the present results for the two out-of-plane and the two twisting moment load categories. The predicted stresses are compared with results from tests on a 254 mm (10 inch) diameter welded branch junction, and also with the values from the current UK power piping code BS 806.

R/C Space Frames with Column Axial Force and Biaxial Bending Moment Interactions

Abstract: Hysteretic responses of single mass reinforced concrete columns subjected to seismic biaxial base motions with constant and varying axial load are studied using the triaxial spring model. Comparisons are made with the trilinear degrading and the biaxial bilinear models based on plasticity theory. Varying axial load on columns due to overturning effects of 10‐story space frames subjected to biaxial base motions are investigated. The results indicate the axial force and biaxial bending moment interaction on R/C column is important.

A Unified Approach for Serviceability Design of Prestressed and Non prestressed Reinforced Concrete Structure

Abstract: restressing is now widely used i reinforced concrete structures to improve their performance during service conditions without necessarily eliminating cracking. The total reinforcement, prestressed and nonprestressed, must be sufficient to satisfy requirements for ultimate strength. This aspect of the design is relatively simple and is not treated in this paper. The checks for serviceability are more difficult because they involve a determination of stress and strain distributions in a cross section at various loading stages. The analysis must account for the time-dependent effects of creep and shrinkage of concrete and relaxation of prestressed steel which include the effects of cracking when the tensile strength of concrete is exceeded. In this paper one set of equations is presented to determine the changes in stress and strain in concrete sections reinforced with prestressed and nonprestressed steels when subjected to normal forces and bending moments. The equations are applicable to statically determinate or indeterminate structures with or without cracking. A part of the tension applied at the jacking end of a post-tensioned tendon is lost by friction and anchor set. It is assumed that the initial tension excluding the loss due to these causes has been determined prior to starting the analysis presented herein. Other prestress losses

Prediction of blade stresses due to gust loading

Abstract: An analysis is developed for investigating the response of a rotor-fuselage system in a three-dimensional gust field wherein the gust velocity conponents can have arbitrary variation in space and time. Each rotor blade undergoes flap bending, lag bending and torsional deflections. The blades are divided into beam elements and each element consists of fifteen nodal degrees of freedom. Quasi-steady strip theory is used to obtain the aerodynamic loads. Unsteady aerodynamic effects are introduced through dynamic inflow modeling. Dynamic stall and reverse flow effects are also included. The fuselage is allowed five degrees of freedom: vertical, longitudinal, lateral, pitch and roll motions. The gust response equations are linearized about the vehicle trim state and the blade steady-state deflected position, and then solved by time integration. The blade bending moments, which determine blade stresses, are evaluated using the force summation technique. Systematic studies are made to identify the importance of several parameters including dynamic stall, forward speed, lag stiffness, gust profile, gust penetration rate and gust velocity direction.

Steel Stresses in Partially Prestressed Concrete Members

Abstract: P prestressed concrete members are usually analyzed for crack control by procedures similar to those for reinforced concrete. The criteria to insure that cracks in partially prestressed members do not exceed allowable limits should be based on a rational cracked section analysis. One way to control crack width is to Iimit the incremental stress in steel after decompression. This approach is reeomrnended by Bachmann,' Bruggeling;l Moustafa,3 Inomata' and others. Statistical analysis conducted on available test datas reinforces this opinion by suggesting a consistent but nonlinear relation between the stress in steel and the maximum crack width. Many other methods can be used to achieve crack control such as limiting the crack width predicted by an equation to a desired value. Naaman$ summarizes five such equations applicable to partially prestressed members in which all but one formula use the change of stress in the reinforcement as the main variable. The results of Naaman and Siriaksorn 7 and Tadros5 also confirm that the steel stresses are good indicators of crack width and that their determination is necessary for a satisfactory crack control design of partially prestressed members. The calculation of the exact value of the steel stress is complicated by the following factors; first, the cracked section is subjected to an axial force and a bending moment; second, the creep and the shrinkage of concrete cause redistribution of stresses by transferring compression from the concrete to the reinforcement, Tadrose proposed a rational method and furnished the necessary equations for calculating the cracked section properties and for determining the incre-

Fixed‐End Restraint Effect on Steel Arch Strength

Abstract: Fixed‐end restraint effect on the load carrying capacity of steel arches loaded into the inelastic and the finite deformation range is studied by an accurate nonlinear finite element approach. This approach takes into consideration the two important factors on material nonlinear phenomenon, namely, spread of yielding zones in the cross section and along the longitudinal axis of the arch and unloading caused by strain reversal. Special characteristics of the interaction between bending moment and axial thrust, the changes in deflectional mode, the progress of yielding and unloading zones with load level, and local failure at the springing are thoroughly examined. The load‐deflection behavior until the stability limit load is reached, has been analyzed by linear theory, the second order elastic theory, and an accurate nonlinear theory.