formulas 1 - mathematical table and data

8/12/2019 Formulas 1 - Mathematical Table and Data

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PARTaneous Data and

Page NosMISCELLANEOUS DATA

ire and Sheet Metal Gagestandard Nomenclature for Flat Rolled Carbon Steelffect of Heat on Structural Steeloefficients of ExpansionWEIGHTS MEASURES AND CONVERSION FACTORS

eights and Specific Gravitieseights of uilding MaterialsWeights and Measuresngineering Conversion FactorsGEOMETRIC AND TRlGONOMETRlC DATA

racing Formulasroperties of the Parabola and Ellipseroperties of the Circleroperties of Geometric Sectionsrigonometric FormulasDECIMAL EQUIVALENTS

ecimals of an Inchecimals of a Foot


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mation only The product is commonly specified to

FLAT ROLLED CARBON STEEL

0 2299 o 0 20310 2030 o 0 18000 1799o 0 04490 0448 o 0 03440 0343o 0 0255 Hot rolled sheet and strip not generally0 0254 thinner produced in these widths and thicknesses


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EFFECT O HE T ON STRUCTUR L STEEL

Short-time elevated-temperature tensile tests on the constructional steels permittedby the AISC Specification indicate that the ratios of the elevated-temperature yieldand tensile strengths to their respective room-temperature strength values are-rea-sonably similar at any particular temperature for the various steels in the 300 to 700F. range, except for variations due to strain aging. (The tensile strength ratio may in-crease to a value greater than unity in the 300 to 700 F. range when strain aging oc-curs.) Above this range, the ratio of elevated-temperature to room-temperaturestrength decreases as the temperature increases.The composition of the steels is usually such that the carbon steels exhibit strainaging with attendant reduced notch toughness. The high-strength low-alloy and heat-treated constructional alloy steels exhibit less-pronounced or little strain aging.As examples of the decreased ratio levels obtained at elevated temperature, theyield strength ratios for carbon and high-strength low-alloy steels are approximately0.77 at 800 F., 0.63 at 1000 F. and 0.37 at 1200 F.FIRE RESISTANT CONSTRUCTIONASTM Specification E119, Standard Methods of Fire Tests of Building Constructionand Materials outlines the procedures of fire testing of structural elements locatedinside a building and exposed to fire within the compartment or room in which theyare located. The temperature criterion used requires that the average of the temper-ature readings not exceed 000 F. for columns and 1100 F. for beams. An individualtemperature reading may not exceed 1100 F. for columns and 1200 F. for beams.Steel buildings whose condition of exterior exposure and whose combustiblecontents under fire hazards will not produce a steel temperature greater than theforegoing criteria may therefore be considered fire-resistive without the provision ofinsulating protection for the steel.

A fire exposure of severity and duration sufficient to raise the temperature ofthe steel much above the fire test criteria temperature will seriously impair its tibilityto sustain loads at the unit stresses or plasticity load factors permitted by the AISCSpecification. In such cases, the members upon which the stability of the structuredepends should be insulated by fire-resistive materials or construction capable ofholding the average temperature of the steel to not more than that specified for thefire test standard.Under the El19 specification, each tested assembly is subjected to a standardfire of controlled extent and severity. The fire resistance rating is expressed as thetime, in hours, that the assembly is able to withstand the fire exposure before the firstcritical point in its behavior is reached. These tests indicate the minimum period oftime during which structural members, such as columns and beams, are capable ofmaintaining their strength and rigidity when subjected to the standard fire. They alsoestablish the minimum period of time during which floors, roofs, walls or partitionswill prevent fire spread by protecting against the passage of flame, hot gases and ex-cessive heat.Tables of fire resistance ratings for various insulating materials and construc-tions applied to structural elements are published in the AISI booklets Fire ResistantSteel Frame Constru ction Designing Fire Protection for Steel Columns and DesigningFire Protection for Steel Trusses. Ratings may also be found in publications of theUnderwriters' Laboratories, Hmc


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new rational fire-protection design procedure for exposed columns and beamsat building exteriors has been developed by the American Iron and Steel Institute,and is described in APSI publication no. FS3, Fire safe Structura l Steel Desig nGuide . The Design Guide provides a step by step procedure which enables buildingdesigners to estimate the maximum steel temperature that would occur during a fireat any location on a structural member located outside a building. The design proce-dure is accepted by some building codes and is under study for adoption by others.To judge the effect of a fire on structural steel, it is necessary to consider whathappens in such an exposure. Peculiarities of this exposure are: (1) temperature at-tained by the steel can only be estimated, (2) time of exposure at any given tempera-ture is unknown, 3) heating is uneven, (4) cooling rates vary and can only be esti-mated and (5) the steel is usually under load, and is sometimes restrained fromnormal expansion.Carbon and high-strength low-alloy steels that show no evidence of gross dam-age from exposure to high temperatures, or from sudden cooling from high tempera-tures, can usually be straightened as necessary and be reused without reduction ofworking stress. Quenched and tempered alloy steels should not be heated to temper-atures within 50 F of the tempering temperature used in heat treatment. Thus, forthe quenched and tempered constructional alloy steels approved by they AISC Spec-ification, i.e., ASTM A514, for which the tempering temperature is 1150 F., themaximum steel temperature should be 1100 F.Steel that has been exposed to very high temperatures can be identified by veryheavy scale, pitting, and surface erosion. Such temperatures may not only cause aloss of cross section, but may also result in metallurgical changes. Normally theseconditions will be accompanied by such severe deformation that the cost and diffi-culty of straightening such members, as compared to replacement, dictates that theybe discarded.Steel members that have suffered rapid cooling will usually be so severely dis-torted that straightening for reuse will seldom be considered practicable.In some cases, there may be some deformation in members whose normal ther-mal expansion is inhibited or prevented by the nature of the construction. Suchmembers may usually be straightened and reused.Connections require special attention to make sure that the stresses induced bya fire, and by subsequent cooling after the fire, have not sheared or loosened boltsor rivets, or cracked welds.COEFFICIENT OF EXP NSIONThe average coefficient of expansion for structural steel between 70 F. and 100 F.is 0.0000065 for each degree. For temperatures of 100 F. to 1200 F. the coefficientis given by the approximate formula:

(6.1 0.0019t)in which is the coefficient of expansion for each degree Fahrenheit and t is the tem-perature in degrees Fahrenheit.The modulus of elasticity of structural steel is approximately 29,000 ksi at 70 F.It decreases linearly to about 25,000 ksi at 900 F., and then begins to drop at an in-creasing rate at higher temperatures.EFFECT OF HE T UE TO WEL ING~ ~ ~ l i c a t i o nf heat by welding produces residual stresses, which are generally ac-companied by distortion of various amounts. Both the stresses and distortions are


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minimized by controlled welding procedures and fabrication methods. In normalstructural practice, it has not been found necessary or desirable to use heat treat-ment (stress-relieving) as a means of reducing residual stresses. Procedures normallyfollowed include: (1) proper positioning of the components of joints before welding,2) selection of welding sequences determined by experience, 3) deposition of aminimum volume of weld metal with a minimum number of passes for the designcondition and (4) preheating as determined by experience (usually above the speci-fied minimums).USE O HEAT TO STRAIGHTEN CAMBER OR CURVE MEMBERSWith modern fabrication techniques, a controlled application of heat can be effec-tively used to either straighten or to intentionally curve structural members. By thisprocess, the member is rapidly heated in selected areas; the heated areas tend to ex-pand, but are restrained by adjacent cooler areas. This action causes a permanentplastic deformation or upset of the heated areas and, thus, a change of shape is de-veloped in the cooled member.

Heat straightening is used in both normal shop fabrication operations and inthe field to remove relatively severe accidental bends in members. Conversely, heatcambering and heat curving of either rolled beams or welded girders are exam-ples of the use of heat to effect a desired curvature.

As with many other fabrication operations, the use of heat to straighten orcurve will cause residual stresses in the member as a result of plastic deformations.These stresses are similar to those that develop in rolled structural shapes as theycool from the rolling temperature; in this case, the stresses arise because all parts ofthe shape do not cool at the same rate. In like manner, welded members develop re-sidual stresses from the localized heat of welding.

In general, the residual stresses from heating operations do not affect the ulti-mate strength of structural members. Any reduction in column strength due to resid-ual stresses is incorporated in the present design provisions.

The mechanical properties of steels are largely unaffected by heating opera-tions, provided that the maximum temperature does not exceed 1100 F. forquenched and tempered alloy steels, and 1300 F. for other steels. The temperatureshould be carefully checked by temperature-indicating crayons or other suitablemeans during the heating process.

COEFFICIENTS OF EXP NSIONThe coefficient of linear expansion (E) s the change in length, per unit of length, fora change of one degree of temperature. The coefficient of surface expansion is ap-proximately two times the linear coefficient, and the coefficient of volume expan-sion, for solids, is approximately three times the linear coefficient.

A bar, free to move, will increase in length with an increase in temperature andwill decrease in length with a decrease in temperature. The change in length will be~ t l , here is the coefficient of linear expansion, t the change in temperature, andI the length. If the ends of a bar are fixed, a change in temperature t will cause achange in the unit stress of E E ~ ,nd in the total stress of AEet, where is the crosssectional area of the bar and the modulus of elasticity.


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The following table gives the coefficient of linear expansion for 100, or 100times the value indicated above.Example: piece of m edium steel is exactly 40 ft long a t 60' F. Find the lengtha t 90' F ., assuming the ends free to move.

.00065 x 30 x 40 .0078 f tChange of length ~ t 100

The length at 90' F. is 40.0078 ft.Exa mp le: piece of medium steel is exactly 40 ft long and th e ends are fixed.If the temperature increases 30 F. what is the resulting change in the unit stress?

29,000,000 x .00065 x 30Chang e in unit stress 100 5655 lbs. per sq. in.


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VARIOUS BUILDING

EARTH ETC. EXCAVATED


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Acoustical fiber tile

Wood Studs 2 xConcrete-Plain 1 in. 12-16 in. O C

Gypsum Board h in.

Ceramic or Quarry TileLinoleum in. Hollow Concrete BlockHardwood in.Softwood 3 4 in.

3-ply ready roofing3-ply felt and gravel5-ply felt and gravel

Glass Block in.


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Plane angle Radian radKilogram kg Solid angle Steradian srSecond s

DERIVED UNITS WITH SPECIAL NAMES)Symbol Formula

Pressure, stress

DERIVED UNITS WITHOUT SPECIAL NAMES)

Cubic metreMetre per secondMetre per second squaredKilogram per cubic metre

SI PREFIXESPrefix Symbol


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1.0 .08333 .02778 .0050505 .00012626 .0000157812.0 1.0 .33333 .0606061 .00151515 .0001893936.0 3.0 1.0 1818182 .00454545 .00056818198.0 16.57920.0 660.063360.0 =5280.0

SQUARE AND LAND MEASUREAcres Sq M

.O 006944 000772144.0 1.0

AVOIRDUPOIS WEIGHTS1.0 03657 002286 .000143 .0000000727.34375 .O .0625 .003906 .00000195437.5 16.0 1.0 .0625 .000031257000.0 256.014000000.0 =512000.0

DRY MEASUREPints Quarts Pecks Cubic Feet Bushels

1.0 .5 0625 .01945 .015632.0 1.0 .I25 .03891 0312516.0 8.064.0 =32.0

LIQUID MEASUREGills Pints Quarts U S Gallons Cubic Feet1.0 .25 125 03125 004184.0 =1.0 .5 .I25 .016718.0 2.0 1.0 250 0334232.0 8.0 4.0 .O

7.48052 .O


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Mile U S . Statute)

Square millimetre

Quart U S . liquid)Cubic millimetre

Quart U S . liquid)Ounce avoirdupois)Pound avoirdupois)

Pound avoirdupois) b av


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Newton-metre Pound-force-inchNewton-metre Pound-force-foot

lnch of Mercury (32 F

lnch of mercury (32 F)

Horsepower 550 ft. Ibfls)

British thermal unit


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a = T H + ( T + e + p )b = T h + ( T + e + p )c 4 1 / 2 T 1 2 e)' a2

log e k log Tlog f k log alog g log blog m log clog n k log dlog p log e

The above m ethod can be used for anynumber of panels.In the formulas for a and b the sum inparenthesis, which in the case shown inT + e + p), is always composed of allthe ho rizontal distances


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Circumference 6.28318 r = 3.1 4159 dDiameter 0.31831 circumference

Radius rhord c = 2d m 2 r sin

y = b - r + d F T ?= d r 2 - r + y - b ) 2

Diameter of circle of equal periphery as square 1.27324 side of squareSide of square of equal periphery as circle 0.78540 diameter of circle

r = radius of circle y angle ncp in degreesArea of Sector ncpo length of arc nop x r)

0.0087266 x r2 x y

r radius of circle chord b = riseArea of Segment nop = area of Sector ncpo area of triangle ncp

Length of arc nop x r) r b)Area of Segment nsp area of circle area of segment nop


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Axis of moments through center

Axis of mome nts on base d

Axis of moments on diagonal


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A bdthrough center of gravity b sin a d cosbd (b in2a d2 cos2a)bd (b2 sin2a d2 cos2a)

b sin a d cos a)

HOLLOW RECTANGLE A bd bldlAxis of moments through center


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EQUAL RECTANGLES = b d dl )Axis of m oments throughcenter of gravity

UNEQUAL RECTANGLES A = bt + b l t lAxis of m oments through l/ bt2 bltl d tl )center of gravity

center o gravity

TRIANGLEAxis of mom ents on base


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HALF IR LE 1.570796R575587

190687 3.264336R


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P R BOL

H LF P R BOL

COMPLEMENT OF H LFP R BOL

P R BOLIC FILLET INRIGHT NGLE

I = l


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HALF ELLIPSE

QUARTER ELLIPSE

ELLIPTIC COMPLEMENT

A = - nab2am = 37r

1 = a3b T -L8 9n1I2 = nab31l3 = - 7ra3b


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EGULAR POLYGON n = Number of sidesAxis of momentsthrough centera 22 R2 RI2

K = Product of Inertia about X-X

IZ = Ixsin20 ly cos20 K sin20

BEAMS AND CHANNELSTransverse force obliquethrough center of gravity

I Ixsin2 ly cos2I = Ixcos2 I sin2

where Mj is bending moment due to force F


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sln2 A + cos2 A sln A cosec Acos A sec A tan A cot A

cos A tan A V Bsln A 1Cos~ne tan A sec A sin A co t A Asln ATangent A s ~ n sec Acos A cot Acos ACotangent A cos A cosec Asln A tan A

Secant A

az cb2 CZ a2c2 a2 b2

a + b + c a2 b2 + c2 2 bc cos Ab2 a2 c2 2 ac cos B

sin B

formulas 1 - mathematical table and data

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