AT03 - Beam

CONTINUOUS BEAM

A beam extending over more than two supports in order to develop greater rigidity and smaller moments than a series of simple beams having similar spans and loading. Both fixed-end and continuous beams are indeterminate structures for which the values of all reactions, shears, and moments are dependent not only on span and loading but also on cross-sectional shape and material.

FIXED-END BEAM

A beam having both ends restrained against translation and rotation. The fixed ends transfer bending stresses, increase the rigidity of the beam, and reduce its maximum deflection.

CANTILEVER

A beam or other rigid structural member extending beyond a fulcrum and supported by a balancing member or a downward force behind the fulcrum.

SIMPLE BEAM

A beam resting on simple supports at both ends, which are free to rotate and have no moment resistance. As with any statically determinate structure, the values of all reactions, shears, and moments for a simple beam are independent of its cross-sectional shape and material.

POSITIVE MOMENT

A bending moment that produces a concave curvature at a section of a structure.

NEGATIVE MOMENT

A bending moment that produces a convex curvature at a section of a structure.

BENDING STRESS

A combination of compressive and tensile stresses developed at a cross section of a structural member to resist a transverse force, having a maximum value at the surface furthest from the neutral axis.

FLEXURE FORMULA

A formula defining the relationship between bending moment, bending stress, and the cross-sectional properties of a beam. Bending stress is directly proportional to bending moment and inversely proportional to the moment of inertia of a beam section.

SECTION MODULUS

A geometric property of a cross section defined as the moment of inertia of the section divided by the distance from the neutral axis to the most remote surface.

MOMENT DIAGRAM

A graphic representation of the variation in magnitude of the bending moments present in a structure for a given set of transverse loads and support conditions. The overall deflected shape of a structure subject to bending can often be inferred from the shape of its moment diagram.

SHEAR DIAGRAM

A graphic representation of the variation in magnitude of the external shears present in a structure for a given set of transverse loads and support conditions.

NEGATIVE SHEAR

A net resultant of shear forces that acts vertically downward on the left part of the structure being considered.

POSITIVE SHEAR

A net resultant of shear forces that acts vertically upward on the left part of the structure being considered.

INFLECTION POINT

A point at which a structure changes curvature from convex to concave or vice versa as it deflects under a transverse load; theoretically, an internal hinge and therefore a point of zero moment.

CANTILEVER BEAM

A projecting beam supported at only one fixed end.

BEAM

A rigid structural member designed to carry and transfer transverse loads across space to supporting elements.

TRANSVERSE SHEAR

A shear force at a cross section of a beam or other member subject to bending, equal to the algebraic sum of transverse forces on one side of the section.

DOUBLE OVERHANGING BEAM

A simple beam extending beyond both of its supports.

OVERHANGING BEAM

A simple beam extending beyond one of its supports. The overhang reduces the positive moment at midspan while developing a negative moment at the base of the cantilever over the support.

SUSPENDED-SPAN

A simple beam supported by the cantilevers of two adjoining spans with pinned construction joints at points of zero moment. Also called hung-span.

HUNG-SPAN

A simple beam supported by the cantilevers of two adjoining spans with pinned construction joints at points of zero moment. Also called suspended-span.

CAMBER

A slight convex curvature intentionally built into a beam, girder, or truss to compensate for an anticipated deflection.

BENDING MOMENT

An external moment tending to cause part of a structure to rotate or bend, equal to the algebraic sum of the moments about the neutral axis of the section under consideration.

NEUTRAL AXIS

An imaginary line passing through the centroid of the cross section of a beam or other member subject to bending, along which no bending stresses occur.

RESISTING MOMENT

An internal moment equal and opposite to a bending moment, generated by a force couple to maintain equilibrium of the section being considered.

fb = Mc/I where fb = extreme fiber stress in bending M = bending moment c = distance from neutral axis to the outermost surface in bending I = moment of inertia if I/c = S where S = section modulus then fb = M/S

Complete Flexure Formula

FACTOR OF 4

Doubling the beam depth reduces the bending stresses by what factor?

FACTOR OF 2

Doubling the beam width reduces the bending stresses by what factor?

fb = M/S

Flexure Formula in terms of fb, M, and S

fb = Mc/I

Flexure Formula in terms of fb, M, c, and I.

FACTOR OF 2

Halving the beam span reduces the bending stresses by what factor?

PROVIDE required I or S with the SMALLEST POSSIBLE AREA

How does the efficiency of a beam increased with regards to moment of inertia (I) and section modulus (S)?

MAKE SECTION DEEP with MOST of MATERIALS at EXTREMETIES where the maximum bending stresses occur.

How does the required moment of inertia (I) or section modulus (S) provided while at the same time minimizing the cross-sectional area of the beam.

STRESS TRAJECTORIES

Lines depicting the direction but not the magnitude of the principal stresses in a beam.

CONCENTRATED LOADS

Loads that produce bending moments that vary linearly between loads.

CONCENTRATED LOADS

Loads that produce external shears that are constant in magnitude between loads.

UNIFORMLY DISTRIBUTED LOADS

Loads that produce linearly varying shears.

UNIFORMLY DISTRIBUTED LOADS

Loads that produce parabolically varying moments.

S = I/c

Section Modulus Formula in term of S, I, and c.

COMPRESSION

Stress caused by bending surfaces inward.

TENSION

Stress caused by bending surfaces outward.

M

The bending moment on the Flexure Formula.

LATERAL BUCKLING

The buckling of a structural member induced by compressive stresses acting on a slender portion insufficiently rigid in the lateral direction.

EFFECTIVE SPAN

The center-to-center distance between the supports of a span.

EFFECTIVE LENGTH

The distance between inflection points in the span of a fixed-end or continuous beam, equivalent in nature to the actual length of a simply supported beam.

CLEAR SPAN

The distance between the inner faces of the supports of a span.

c

The distance from neutral axis to the outermost surface in bending on the Flexure Formula.

SPAN

The extent of space between two supports of a structure. Also the structure so supported.

fb

The extreme fiber stress in bending on the Flexure Formula.

I

The moment of inertia on the Flexure Formula.

HAUNCH

The part of a beam that is thickened or deepened to develop greater moment resistance. The efficiency of a beam can be increased by shaping its length in response to the moment and shear values, which typically vary along its longitudinal axis.

EXTREME SURFACES OF BEAM

The part of the beam in which only bending stresses exist and the principal stresses are equivalent to the tensile and compressive stresses resulting from bending.

NEUTRAL AXIS OF BEAM SECTION

The part of the beam in which only shear stresses exist and these can be resolved into tensile and compressive stresses acting at 45° angles to it.

INTERMEDIATE ELEMENTS OF BEAM

The part of the beam subject to both bending and shear stresses, the principal stresses have an inclination determined by the relative magnitudes of these stresses.

DEFLECTION

The perpendicular distance a spanning member deviates from a true course under transverse lading, increasing with load and span, and decreasing with an increase in the moment of inertia of the section or the modulus of elasticity of the material.

SHEAR CENTER

The point in the cross-sectional plane of a structural member through which a transverse load must pass in order to prevent torsion or twisting of the member about a longitudinal axis.

VERTICAL SHEARING STRESS

The shearing stress developed along a cross section of a beam to resist transverse shear, having a maximum value at the neutral axis and decreasing nonlinearly toward the outer faces.

LONGITUDINAL SHEARING STRESS

The shearing stress developed to prevent slippage along horizontal planes of a beam under transverse loading, equal at any point to the vertical shearing stress at that point. Also called horizontal shearing stress.

HORIZONTAL SHEARING STRESS

The shearing stress developed to prevent slippage along horizontal planes of a beam under transverse loading, equal at any point to the vertical shearing stress at that point. Also called longitudinal shearing stress.

MOMENT OF INERTIA

The sum of the products of each element of an area and the square of its distance from a coplanar axis of rotation. It is a geometric property that indicates how the cross-sectional area of a structural member is distributed and does not reflect the intrinsic physical properties of a material.

PRINCIPAL STRESSES

The tensile and compressive stresses resulting from the interaction of bending and shear stresses at a cross-section of a beam.