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bending
bending (adj.)
1.not remaining rigid or straight"tried to support his weight on a bending cane"
bending (n.)
1.the act of bending something
2.the property of being bent or deflected
3.movement that causes the formation of a curve
bend (v. trans.)
1.bend or cause to bend"He crooked his index finger" "the road curved sharply"
bend (n.)
1.curved segment (of a road or river or railroad track etc.)
2.diagonal line traversing a shield from the upper right corner to the lower left
3.movement that causes the formation of a curve
4.a circular segment of a curve"a bend in the road" "a crook in the path"
5.an angular or rounded shape made by folding"a fold in the napkin" "a crease in his trousers" "a plication on her blouse" "a flexure of the colon" "a bend of his elbow"
Bend (n.)
1.a town in central Oregon at the eastern foot of the Cascade Range
bend (v.)
1.cause (a plastic object) to assume a crooked or angular form"bend the rod" "twist the dough into a braid" "the strong man could turn an iron bar"
2.bend a joint"flex your wrists" "bend your knees"
3.change direction"The road bends"
4.turn from a straight course, fixed direction, or line of interest
5.form a curve"The stick does not bend"
6.bend one's back forward from the waist on down"he crouched down" "She bowed before the Queen" "The young man stooped to pick up the girl's purse"
7.win a victory over"You must overcome all difficulties" "defeat your enemies" "He overcame his shyness" "He overcame his infirmity" "Her anger got the better of her and she blew up"
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Merriam Webster
BendingBend"ing, n. The marking of the clothes with stripes or horizontal bands. [Obs.] Chaucer.
BendBend (�), v. t. [imp. & p. p. Bended or Bent (�); p. pr. & vb. n. Bending.] [AS. bendan to bend, fr. bend a band, bond, fr. bindan to bind. See Bind, v. t., and cf. 3d & 4th Bend.]
1. To strain or move out of a straight line; to crook by straining; to make crooked; to curve; to make ready for use by drawing into a curve; as, to bend a bow; to bend the knee.
2. To turn toward some certain point; to direct; to incline. “Bend thine ear to supplication.” Milton.
Towards Coventry bend we our course. Shak.
Bending her eyes . . . upon her parent. Sir W. Scott.
3. To apply closely or with interest; to direct.
To bend his mind to any public business. Temple.
But when to mischief mortals bend their will. Pope.
4. To cause to yield; to render submissive; to subdue. “Except she bend her humor.” Shak.
5. (Naut.) To fasten, as one rope to another, or as a sail to its yard or stay; or as a cable to the ring of an anchor. Totten.
To bend the brow, to knit the brow, as in deep thought or in anger; to scowl; to frown. Camden.
Syn. -- To lean; stoop; deflect; bow; yield.
BendBend, v. i.
1. To be moved or strained out of a straight line; to crook or be curving; to bow.
The green earth's end
Where the bowed welkin slow doth bend. Milton.
2. To jut over; to overhang.
There is a cliff, whose high and bending head
Looks fearfully in the confined deep. Shak.
3. To be inclined; to be directed.
To whom our vows and wished bend. Milton.
4. To bow in prayer, or in token of submission.
While each to his great Father bends. Coleridge.
BendBend, n. [See Bend, v. t., and cf. Bent, n.]
1. A turn or deflection from a straight line or from the proper direction or normal position; a curve; a crook; as, a slight bend of the body; a bend in a road.
2. Turn; purpose; inclination; ends. [Obs.]
Farewell, poor swain; thou art not for my bend. Fletcher.
3. (Naut.) A knot by which one rope is fastened to another or to an anchor, spar, or post. Totten.
4. (Leather Trade) The best quality of sole leather; a butt. See Butt.
5. (Mining) Hard, indurated clay; bind.
6. pl. (Med.) same as caisson disease. Usually referred to as the bends.
Bends of a ship, the thickest and strongest planks in her sides, more generally called wales. They have the beams, knees, and foothooks bolted to them. Also, the frames or ribs that form the ship's body from the keel to the top of the sides; as, the midship bend.
BendBend, n. [AS. bend. See Band, and cf. the preceding noun.]
1. A band. [Obs.] Spenser.
2. [OF. bende, bande, F. bande. See Band.] (Her.) One of the honorable ordinaries, containing a third or a fifth part of the field. It crosses the field diagonally from the dexter chief to the sinister base.
Bend sinister (Her.), an honorable ordinary drawn from the sinister chief to the dexter base.
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⇨ definición de bending (Wikipedia)
bending (adj.)
bending (n.)
bend (n.)
bend dexter, bending, coil, crease, crimp, crook, curve, flexure, fold, loop, meander, plication, turn, twist
bend (v.)
arch, be first past the post, bend away, bend back, bend down, bend over, be superior, be victorious, bow, branch off, compel, crack, crouch, curve, defeat, deflect, deform, dispose, diverge, flex, get the best of, get the better of, have/get/gain the upper hand, incline, influence, kink, make a bow, overcome, persuade, snap, stoop, subdue, submit, sway, swerve, triumph, turn, turn away, twist, veer, warp, win
bend (v. intr.)
Ver también
bend (v. trans.)
↘ bending ↗ crooked, hunched, hunched up, round-backed, round-shouldered, stooped, stooping
bend (v.)
↘ flexion, flexure ≠ straighten, unbend
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⇨ Backward bending supply curve of labour • Band bending • Bender Bending Rodríguez • Bending (disambiguation) • Bending (metalworking) • Bending Science • Bending moment • Bending of starlight • Bending of the knee • Bending over • Bending radii • Bending spoons • Bending stiffness • Bending the Landscape • Brake (sheet metal bending) • Circuit bending • Gender-bending • Heat bending of wood • Joggle bending • Light bending • Patent Bending • Plastic bending • Pole bending • Pure bending • Race Bending • Spoon bending • Steam bending • Tube bending
⇨ Adjustable bend • Alpine butterfly bend • Anchor bend • Around the Bend • Ashley bend • Ashley's bend • Austin High School (Fort Bend County, Texas) • Babylon Bend Bridge • Battle of Horseshoe Bend (1814) • Battle of Horseshoe Bend (1832) • Battle of Irish Bend • Battle of Plum Point Bend • Battle of Samara Bend • Battle of Tebbs Bend • Beavers Bend Resort Park • Beech Bend Park • Bend (knot) • Bend Bandits • Bend Her • Bend It Like Beckham • Bend Or • Bend Ova • Bend Senior High School • Bend Sinister (album) • Bend and Break • Bend knot • Bend of Islands, Victoria • Bend of the River • Bend, Oregon • Bend, Texas • Bend-Or spots • Bessemer Bend, Wyoming • Beyond the Bend • Big Bend • Big Bend (Florida) • Big Bend (Texas) • Big Bend Canyon Lizard • Big Bend Community College • Big Bend Cowboy Hall of Fame • Big Bend Gold Rush • Big Bend National Park • Big Bend Ranch State Park • Big Bend Region • Big Bend Slider • Big Bend State Recreation Area • Big Bend Township, Chippewa County, Minnesota • Big Bend region • Big Bend, California • Big Bend, Rusk County, Wisconsin • Big Bend, South Australia • Big Bend, Waukesha County, Wisconsin • Big Bend, Wisconsin • Bourbaki dangerous bend symbol • Bradys Bend Township, Armstrong County, Pennsylvania • Brazos Bend State Park • Brazos Bend, Texas • Carrick bend • Cedar Creek Golf Course at Beavers Bend • Century Center (South Bend) • Chicago South Shore and South Bend Railroad • Chicago, Lake Shore and South Bend Railway • Coastal Bend Aviators • Coastal Bend College • Coastal Bend Council of Governments • Davis Bend, Mississippi • De Cordova Bend Dam • Do Not Bend • Eagle Bend • Eagle Bend, Minnesota • East Bend Township, Champaign County, Illinois • East Bend, Kentucky • East Bend, North Carolina • Education in South Bend, Indiana • Farewell Bend State Recreation Area • Flemish bend • Fort Bend • Fort Bend County Libraries • Fort Bend County Toll Road Authority • Fort Bend County, Texas • Fort Bend Independent School District • Fort Bend Parkway Toll Road • Fort Bend Sun • Four Corners, Fort Bend County, Texas • Gee's Bend • Gee's Bend, AL • Gee's Bend, Alabama • Gila Bend, Arizona • Grand Bend Airport • Grand Bend Motorplex • Great Bend Municipal Airport • Great Bend Township, Cottonwood County, Minnesota • Great Bend Township, Susquehanna County, Pennsylvania • Great Bend, Kansas • Great Bend, New York • Great Bend, North Dakota • Great Bend, Pennsylvania • Grecian bend • Heaving line bend • Holla Bend National Wildlife Refuge • Horseshoe Bend • Horseshoe Bend (Arizona) • Horseshoe Bend High School • Horseshoe Bend, Arkansas • Horseshoe Bend, Idaho • Horseshoe Bend, Shirehampton • Hudson Bend, Texas • Hunter's bend • Illinois Bend, Texas • Irishtown Bend Archeological District • Liberty Bend Bridge • List of bend knots • Lumber Exchange Building (South Bend, Washington) • Malta Bend, Missouri • McCord Bend, Missouri • McGee Bend Reservoir • Memorial Bend, Houston • Mission Bend • Mission Bend (Houston) • Mission Bend, Texas • Moccasin Bend • Mountain View High School (Bend, Oregon) • Neely's Bend • Never Bend • Nintendo North Bend • North Bend • North Bend Plantation • North Bend Rail Trail • North Bend State Park • North Bend, Nebraska • North Bend, Ohio • North Bend, Oregon • North Bend, Washington • North Bend, Wisconsin • Orange Bend, Florida • Peewee Bend • Peewee Bend, Queensland • Pine Bend Refinery • Post Oak Bend City, Texas • RCAF Detachment Grand Bend • Racking bend • River Bend (Illinois) • River Bend Lodge, California • River Bend Nature Center • River Bend, Gauteng • River Bend, Missouri • River Bend, North Carolina • River Bend, South Africa • Round the Bend • Round the Bend (1951 novel) • Samara Bend • Shoot-Out at Medicine Bend • Single Carrick bend • Sivells Bend Independent School District • Sivells Bend, Texas • Six Bend Trap • Sony Bend • South Bend (Amtrak station) • South Bend Blue Sox • South Bend Community School Corporation • South Bend Conservatory • South Bend Lathe Works • South Bend Museum of Art • South Bend Regional Airport • South Bend Silver Hawks • South Bend Township, Armstrong County, Pennsylvania • South Bend Township, Blue Earth County, Minnesota • South Bend Tribune • South Bend, Indiana • South Bend, Nebraska • South Bend, Washington • South Toledo Bend, Texas • St. Joseph's High School (South Bend, Indiana) • Staple Bend Tunnel • Strong Enough to Bend • Summit High School (Bend, Oregon) • Taylor High School (North Bend, Ohio) • Texas Coastal Bend • The Shops at Willow Bend • Toledo Bend Reservoir • Travis High School (Fort Bend County, Texas) • Triple Bend Invitational Handicap • Twenty Mile Bend, Florida • U-bend • UL Bend National Wildlife Refuge • UL Bend Wilderness • Union Station (South Bend, Indiana) • University City-Big Bend (St. Louis MetroLink) • Walnut Bend • Walnut Bend Independent School District • Walnut Bend, Houston • West Bend • West Bend (town), Wisconsin • West Bend Housewares • West Bend Municipal Airport • West Bend high schools • West Bend, Iowa • West Bend, Wisconsin • Yarra Bend Park, Melbourne • Zeppelin bend
bending (n.)
bend, bend back, bend double, bend down, bow, crook, curve[Nominalisation]
bending (n.)
action de plier, mouvement de flexion (fr)[Classe]
ployer (fr)[Nominalisation]
bending (n.)
action de plier, mouvement de flexion (fr)[Classe]
recourber (fr)[Nominalisation]
bend, bend back, bend double, bend down, bow, crook, curve[Nominalisation]
bending (n.)
change of shape[Hyper.]
bending (n.)
refraction[Classe]
changement de direction dans l'espace (fr)[Classe]
action, fait de suivre une direction (fr)[Classe]
écart concret ou abstrait de direction (fr)[Classe]
aérodynamique (fr)[termes liés]
défléchir (fr)[Nominalisation]
physical property[Hyper.]
bending (n.)
motion, movement[Hyper.]
Bend (n.)
town[Hyper.]
Beaver State, OR, Oregon[Desc]
bend (n.)
tracé d'un chemin indirect (fr)[Classe]
bend (n.)
endroit où une voie tourne (fr)[Classe]
zigzag (fr)[Classe]
ligne courbe (fr)[Classe]
ornement architectural en bande (fr)[Classe]
forêt (fr)[DomainDescrip.]
bend (n.)
endroit où une voie tourne (fr)[ClasseHyper.]
autre élément d'une route (fr)[DomainDescrip.]
bend (n.)
ordinary[Hyper.]
bend (n.)
motion, movement[Hyper.]
bend (n.)
endroit où une voie tourne (fr)[Classe]
bend (n.)
bend (v.)
bend (v.)
bend (v.)
rendre courbe (fr)[Classe]
move, throw[Hyper.]
flexion, flexure - flection, flexion, flexure[Dérivé]
bend (v.)
bend (v.)
bend (v.)
bend (v.)
adopter une attitude corporelle (fr)[Classe...]
devenir moins haut (fr)[Classe]
incliner son corps avec respect (fr)[Classe]
saluer (fr)[Classe]
courber le dos (fr)[Classe]
bend, flex[Hyper.]
stoop - crouch - bow, bowing, curtsey, obeisance - stooper[Dérivé]
change posture[Domaine]
bend (v.)
dominer (être meilleur pour une personne) (fr)[Classe]
être meilleur dans une confrontation (fr)[Classe]
soumettre (fr)[Classe]
overcome; best; outdo; beat; beat out; crush; shell; trounce; vanquish[Classe]
jeu d'échecs (fr)[DomaineCollocation]
overcomer, subduer, surmounter[Dérivé]
bend (v. intr.)
bend (v. intr.)
bend (v. tr.)
rendre courbe (fr)[Classe]
faire plier (fr)[Classe]
bend, flex[Hyper.]
crooked, hunched, hunched up, round-backed, round-shouldered, stooped, stooping[Rendre+Attrib.]
bend, crook, turn, twist - curvature[Dérivé]
Wikipedia
This article needs additional citations for verification. (April 2009) |
In engineering mechanics, bending (also known as flexure) characterizes the behavior of a slender structural element subjected to an external load applied perpendicularly to a longitudinal axis of the element. The structural element is assumed to be such that at least one of its dimensions is a small fraction, typically 1/10 or less, of the other two.[1] When the length is considerably longer than the width and the thickness, the element is called a beam. A closet rod sagging under the weight of clothes on clothes hangers is an example of a beam experiencing bending. On the other hand, a shell is a structure of any geometric form where the length and the width are of the same order of magnitude but the thickness of the structure (known as the 'wall') is considerably smaller. A large diameter, but thin-walled, short tube supported at its ends and loaded laterally is an example of a shell experiencing bending.
In the absence of a qualifier, the term bending is ambiguous because bending can occur locally in all objects. To make the usage of the term more precise, engineers refer to the bending of rods,[2] the bending of beams,[1] the bending of plates,[3] the bending of shells[2] and so on.
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A beam deforms and stresses develop inside it when a transverse load is applied on it. In the quasistatic case, the amount of bending deflection and the stresses that develop are assumed not to change over time. In a horizontal beam supported at the ends and loaded downwards in the middle, the material at the over-side of the beam is compressed while the material at the underside is stretched. There are two forms of internal stresses caused by lateral loads:
These last two forces form a couple or moment as they are equal in magnitude and opposite in direction. This bending moment resists the sagging deformation characteristic of a beam experiencing bending. The stress distribution in a beam can be predicted quite accurately even when some simplifying assumptions are used.[1]
In the Euler-Bernoulli theory of slender beams, a major assumption is that 'plane sections remain plane'. In other words, any deformation due to shear across the section is not accounted for (no shear deformation). Also, this linear distribution is only applicable if the maximum stress is less than the yield stress of the material. For stresses that exceed yield, refer to article plastic bending. At yield, the maximum stress experienced in the section (at the furthest points from the neutral axis of the beam) is defined as the flexural strength.
The Euler-Bernoulli equation for the quasistatic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load is[1]
where is the Young's modulus, is the area moment of inertia of the cross-section, and is the deflection of the neutral axis of the beam.
After a solution for the displacement of the beam has been obtained, the bending moment () and shear force () in the beam can be calculated using the relations
Simple beam bending is often analyzed with the Euler-Bernoulli beam equation. The conditions for using simple bending theory are [4]:
Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam. Since the stresses between these two opposing maxima vary linearly, there therefore exists a point on the linear path between them where there is no bending stress. The locus of these points is the neutral axis. Because of this area with no stress and the adjacent areas with low stress, using uniform cross section beams in bending is not a particularly efficient means of supporting a load as it does not use the full capacity of the beam until it is on the brink of collapse. Wide-flange beams (I-beams) and truss girders effectively address this inefficiency as they minimize the amount of material in this under-stressed region.
The classic formula for determining the bending stress in a beam under simple bending is[5]:
where
The equation is valid only when the stress at the extreme fiber (i.e. the portion of the beam farthest from the neutral axis) is below the yield stress of the material from which it is constructed. At higher loadings the stress distribution becomes non-linear, and ductile materials will eventually enter a plastic hinge state where the magnitude of the stress is equal to the yield stress everywhere in the beam, with a discontinuity at the neutral axis where the stress changes from tensile to compressive. This plastic hinge state is typically used as a limit state in the design of steel structures.
The equation above is only valid if the cross-section is symmetrical. For homogeneous beams with asymmetrical sections, the axial stress in the beam is given by
where are the coordinates of a point on the cross section at which the stress is to be determined as shown to the right, and are the bending moments about the y and z centroid axes, and are the second moments of area (distinct from moments of inertia) about the y and z axes, and is the product of moments of area. Using this equation it is possible to calculate the bending stress at any point on the beam cross section regardless of moment orientation or cross-sectional shape. Note that do not change from one point to another on the cross section.
For large deformations of the body, the stress in the cross-section is calculated using an extended version of this formula. First the following assumptions must be made:
Large bending considerations should be implemented when the bending radius is smaller than ten section heights h:
With those assumptions the stress in large bending is calculated as:
where
When bending radius approaches infinity and , the original formula is back:
In 1921, Timoshenko improved upon the Euler-Bernoulli theory of beams by adding the effect of shear into the beam equation. The kinematic assumptions of the Timoshenko theory are
However, normals to the axis are not required to remain perpendicular to the axis after deformation.
The equation for the quasistatic bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is[7]
where is the area moment of inertia of the cross-section, is the cross-sectional area, is the shear modulus, and is a shear correction factor. For materials with Poisson's ratios () close to 0.3, the shear correction factor for a rectangular cross-section is approximately
The rotation () of the normal is described by the equation
The bending moment () and the shear force () are given by
The dynamic bending of beams,[8] also known as flexural vibrations of beams, was first investigated by Daniel Bernoulli in the late 18th century. Bernoulli's equation of motion of a vibrating beam tended to overestimate the natural frequencies of beams and was improved marginally by Rayleigh in 1877 by the addition of a mid-plane rotation. In 1921 Stephen Timoshenko improved the theory further by incorporating the effect of shear on the dynamic response of bending beams. This allowed the theory to be used for problems involving high frequencies of vibration where the dynamic Euler-Bernoulli theory is inadequate. The Euler-Bernoulli and Timoshenko theories for the dynamic bending of beams continue to be used widely by engineers.
The Euler-Bernoulli equation for the dynamic bending of slender, isotropic, homogeneous beams of constant cross-section under an applied transverse load is[7]
where is the Young's modulus, is the area moment of inertia of the cross-section, is the deflection of the neutral axis of the beam, and is mass per unit length of the beam.
For the situation where there is no transverse load on the beam, the bending equation takes the form
Free, harmonic vibrations of the beam can then be expressed as
and the bending equation can be written as
The general solution of the above equation is
where are constants and
The mode shapes of a cantilevered I-beam | ||
---|---|---|
In 1877, Rayleigh proposed an improvement to the dynamic Euler-Bernoulli beam theory by including the effect of rotational inertia of the cross-section of the beam. Timoshenko improved upon that theory in 1922 by adding the effect of shear into the beam equation. Shear deformations of the normal to the mid-surface of the beam are allowed in the Timoshenko-Rayleigh theory.
The equation for the bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam under these assumptions is [7][9]
where is the polar moment of inertia of the cross-section, is the mass per unit length of the beam, is the density of the beam, is the cross-sectional area, is the shear modulus, and is a shear correction factor. For materials with Poisson's ratios () close to 0.3, the shear correction factor are approximately
For free, harmonic vibrations the Timoshenko-Rayleigh equations take the form
This equation can be solved by noting that all the derivatives of must have the same form to cancel out and hence as solution of the form may be expected. This observation leads to the characteristic equation
The solutions of this quartic equation are
where
The general solution of the Timoshenko-Rayleigh beam equation for free vibrations can then be written as
The defining feature of beams is that one of the dimensions is much larger than the other two. A structure is called a plate when it is flat and one of its dimensions is much smaller than the other two. There are several theories that attempt to describe the deformation and stress in a plate under applied loads two of which have been used widely. These are
The assumptions of Kirchhoff-Love theory are
These assumptions imply that
where is the displacement of a point in the plate and is the displacement of the mid-surface.
The strain-displacement relations are
The equilibrium equations are
where is an applied load normal to the surface of the plate.
In terms of displacements, the equilibrium equations for an isotropic, linear elastic plate in the absence of external load can be written as
In direct tensor notation,
The special assumption of this theory is that normals to the mid-surface remain straight and inextensible but not necessarily normal to the mid-surface after deformation. The displacements of the plate are given by
where are the rotations of the normal.
The strain-displacement relations that result from these assumptions are
where is a shear correction factor.
The equilibrium equations are
where
The dynamic theory of plates determines the propagation of waves in the plates, and the study of standing waves and vibration modes. The equations that govern the dynamic bending of Kirchhoff plates are
where, for a plate with density ,
and
The figures below show some vibrational modes of a circular plate.
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