Steel Cross Sections. Structural Steel Design

Transcription

1 Steel Cross Sections Structural Steel Design

2 PROPERTIES OF SECTIONS Perhaps the most important properties of a beam are the depth and shape of its cross section. There are many to choose from, and there are standard shapes developed for particular materials. Each section has its own use, and each section behaves slightly differently under load.

3 PROPERTIES OF SECTIONS A problem arises when we must consider (1) the efficient use of material and (2) the comparison of complex shapes to other complex shapes in terms of strength and stiffness. Like stress and strain, engineers need to be able to reduce complexity into a single quantity, notably a single number, in order to make judgements about those complexities. ƒ( ) =? What is the function of a shape that leads to a quantity that can be used to compare shapes to each other?

4 MOMENT OF INERTIA Moment of inertia is a way of quantifying the relative strength of a beam s section. It calculates the effectiveness of a piece of the section to resist bending moment. # $!!! " "#! #! $ " "#!

5 CALCULATING MOMENT OF INERTIA x d b '( # $ %&%%%%%%%% ) *" Moment of inertia is always measured about an axis. For bending, this axis is always perpendicular to the line of force the section is intended to hold (for gravity loads, this is the x-axis, which is parallel to the ground). Some shapes have a clearly defined formula, like the rectangular section above left. Other shapes, such as the wide-flange shape to the right (W-shape), are complex enough that it is simply easier to look up the value in a table. The units are in 4. x

6 RECTANGULAR SECTIONS d d b '( #%&%%%%%%%% ) *" > b '( #%&%%%%%%%% ) *" The equation for moment of inertia for solid rectangular sections mathematically parallels the phenomenon that thin sections are stronger in one direction than the other. If a beam is placed vertically as on the left, the depth is cubed. So if the depth is doubled, the moment of inertia increases by a factor of 8. Since I is directly proportional to b, doubling the breadth only doubles I.

7 DISTRIBUTION OF MATERIAL Moment of inertia is really about the distribution of material in a section. Different sections of equal area may have vastly different moments of inertia simply because the shape, and thus the distribution, is different. ; 9+3:,255-+/%657+,42/58 Given our diagram of internal stresses inside a portion of a beam, we can see that the compressive and tensile stresses present are greatest towards the top and bottom of the beam. Because of the definition of stress, ƒ = P/A, in order to keep those stresses under the yield stress, it makes sense to increase the amount of material, and thus the cross sectional area, A, in the equation, so that the stress (ƒ) is minimized. " /2<4,01%0$-5! +,-.-/01%21232/4 42/5-+/%612/.472/58 "!

8 DISTRIBUTION OF MATERIAL Here are some sample values with the same area but different moments of inertia. =%&%">%-/ " %%%%%%% '( # $ %&%%%%%%%% )!"""""""" %%%%%%%% *" '( # $ %&%%%%%%%% )!"""""""" %%%%%%%% *" '( # $ %&%%%%%%%% )!"""""""" %%%%%%%% *" # $ %%%%%%% B*>$CD # $ %%%%%%% B*@$CD # $ %%%%%%% B"*$CD

9 WIDE FLANGE SECTION NOMENCLATURE flange W12x45 webbing centroid shape nominal depth in inches weight of 1 foot of the beam in pounds W = wide flange section Steel shapes are classified and cataloged by the AISC. Their are designations for the standard shapes that follow the pattern above. Wide-flange sections are the most common structural steel cross section. They are ideal for beams because their material is distributed as far away from the centroid of the section as possible.

10 ALLOWABLE STRESSES stress strain hardening ultimate stress 58 ksi failure pt elastic range plastic range yield stress 36 ksi Safety factor yield point A36 structural steel E = 29.6 x 10 6 psi, or 29,600 ksi Allowable stress ƒ allowable = ƒ yield / safety factor The allowable stress is a mechanism in some design methods to keep the stresses well below the yield stress. It attempts to use a safety factor for a particular type of stress (compresion, shear, etc.), to establish a maximum design value for stress. strain

11 MATERIAL PROPERTIES TABLES Tables such as these provide standardized data for finding the appropriate values for allowable stresses for particular design situations. In particular, look at the Allowable stress columns for steel in table A-15-1.

12 BEAM SELECTION Besides using the formulas for section modulus and moment of intertia to go from a loading condition (maximum moment) and allowable stress to a beam, we can use tables that have values for section modulus and moment of inertia of standard structural shapes. Using these tables, we can select a beam s cross section from standardized parts. These tables look similar to the one on the right.

13 STEEL WIDE-FLANGE BEAM COMPARISON If we were to use the Manual of Steel Construction, we would have hundreds of wide-flange options to choose from. This poses a problem, however, since even for a single value of S, we may find a dozen acceptable sections, all with a value of S just larger than the minimum acceptable value we calculated. What do we do? Let s first look at some sections that have a similar value for : W10x88 = 98.5 in 3 W12x72 = 97.4 in 3 W14x61 = 92.1 in 3 W16x57 = 92.2 in 3 W18x55 = 98.3 in 3 W21x48 = 93.0 in 3 There is a trade-off between the depth of the beam and its weight. The second number in each W??x?? designation is the section s weight in pounds per foot (lb/ft). So the last selection on the right, 30 ft long, would weigh 48 lb/ft 30ft = 1440 lbs, but the one on the right would weigh 88 lb/ft 30ft = 2640 lbs! Since we pay per pound for steel, the beams get cheaper as we move to the right, but also less desirable from an architectural standpoint.

14 STEEL WIDE-FLANGE BEAM COMPARISON W10x88 = 98.5 in 3 W12x72 = 97.4 in 3 W14x61 = 92.1 in 3 W16x57 = 92.2 in 3 W18x55 = 98.3 in 3 W21x48 = 93.0 in 3 Cheaper Deeper floors Lower ceilings and/or taller building More expensive Heavier structure underneath to support, can lead to even greater expense Higher ceilings and thinner floors possible Among other structural calculations, beam selection ultimately also has to balance these factors.

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