ANALYSIS AND DESIGN OF MEMBERS FOR BENDING

Beams are structural members that support transverse loads and are therefore subjected primarily to flexure, or bending. If a substantial amount of axial load is also present, the member is referred to as a beam–column. Although some degree of axial load will be present in any structural member, in many practical situations this effect is negligible and the member can be treated as a beam. Beams are usually thought of as being oriented horizontally and subjected to vertical loads, but that is not necessarily the case. A structural member is considered to be a beam if it is loaded so as to cause bending.

flexural limit states

Increasing the loading on the beam will eventually cause failure of the beam by either

1.Yielding of the material and formation of a plastic hinge and a collapse mechanism

2. Lateral-torsional buckling of the beam.

3. Local buckling of the flange or web of a beam that is not compact

lateral bracing of beams

Based from the module given, a compact beam can collapse in one of three possible modes

The plastic mode occurs when the beam is adequately braced to prevent lateral torsional buckling.The limiting laterally unbraced length for the limit state of yielding is denoted by Lp. When the distance between braces does not exceed Lp (Lb<Lp), yielding of the steel occurs over the full depth of the beam at the point of maximum moment. The nominal flexural strength of the beam is given by Mn=Mp=FyZx
When the distance between braces exceeds Lp and is less than Lr, the limiting laterally unbraced length for the limit state of inelastic lateral-torsional buckling, collapse occurs prior to the development of the full plastic moment. When Lb = Lr the nominal flexural strength of the beam is given by the equation as shown below

3. When the distance between braces exceeds Lr collapse occurs by elastic lateral torsional buckling. The nominal flexural strength of the beam is given by Mn=Mcr=FcrSx, where Fcr is given by the equation below.

design and allowable flexural strength

For load and resistance factor design (LRFD), the equation are shown on the left where Mu is required moment strength, phi sub b is the resistance factor for bending = 0.90, and Mn is the nominal moment strength
On the other hand, allowable strength design is also provided by the equation as shown where Ma is the required moment strength, and omega sub b as the safety factor for bending = 1.67.

plastic moment of resistance (yielding)

The bending moment applied to the central portion of the beam shown in the right is given by the equation M = WL/3 = fbSx
Further increase in the load causes the plasticity to spread toward the center of the beam shown at (c). Eventually, as shown at (d), all fibers in the cross section have yielded, a plastic hinge has formed, and collapse occurs. The nominal flexural strength of the section is provided by the equation Mn = Mp = FyZx, where Zx is plastic section modulus and Mp is the plastic moment of resistance.

SHAPE FACTOR AND ASD

The shape factor is defined as Mp/My = Zx/Sx, which is approximately equal to 1.1 to 1.3 for a W-shaped beam]
Using a shape factor of 1.1, the nominal flexural strength is given by Mn = FyZx = 1.1FySx
The allowable flexural strength and flexural stress are provided using the equation as shown in the left.

lateral torsional buckling modification factor

The lateral-torsional buckling modification factor accounts for the effect that a variation in bending moment has on the lateral-torsional buckling of a beam. Beams in which the applied moments cause reversed curvature have a greater resistance to lateral-torsional buckling than beams subjected to a uniform bending moment.
The figure shown on the right illustrates derivation of terms used in determing Cb
The modification factor Cb is given by the equation as shown in the right

compact, non-compact and slender sections

The flexural capacity of an adequately braced beam depends on the slenderness ratio of the compression flange and the web. When the slenderness ratios are sufficiently small, the beam can attain its full plastic moment and the cross section is classified as compact. When the slenderness ratios are larger, the compression flange or the web may buckle locally before a full plastic moment is attained and the cross section is classified as noncompact. When the slenderness ratios are sufficiently large, local buckling will occur before the yield stress of the material is reached and the cross section is classified as slender.
Flexural response of these three classifications are shown in the figure on the left.

A compact section is one that can develop a plastic hinge prior to local buckling of the flange or web, provided that adequate lateral bracing is provided. The criteria for determining compactness of flange and web of rolled beams are shown on the right figure

A noncompact section is one that can, prior to local buckling of the flange or web, develop a nominal flexural strength Mn given by Mr ≤ Mn < Mp.
The criteria for determining non-compactness of flange and compactness of web is defined using the equations shown in the left.
The nominal flexural strength of a section with compact web and noncompact flange with adequate lateral bracing, is given by AISC 360 Eq. (F3-1) as shown.

A slender section is one that cannot develop the yield stress prior to web or flange local buckling. A section is classified as slender when the slenderness ratio of the flange or web exceeds the limiting slenderness parameters for a noncompact section.
A slender section for flange and web local buckling is defined as shown on the right. Also, the nominal flexural strength of section with slender flanges are also shown.

SUPPLEMENTARY MATERIALS

20221227 - CENG131 Module 5.pdf

20221230 - CENG131 Module 5B.pdf

final exam

CamScanner 01-05-2023 20.59.pdf

reflection

The last module that was uploaded recently tackled the analysis and design of steel members subjected to flexure. Although the formulas used for determining the flexural strength for non-slender and slender elements were too lengthy and complex enough, I was able to memorize them before the final examination. Also, I have grasped lots of concepts and was able to understand the general provisions for doubly symmetric I shaped members. The step-by-step procedure was not too cumbersome at all, since they were quite similar to the previous module that I read, the analysis and design for steel members subjected to compression. Using LRFD and ASD was too helpful for me so that I can use either of the two in analysis and design of structures. With this, I am very hopeful that the concepts that I have learned in this subject course will be very useful in my future profession as structural engineer.

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