1.3.1
GENERAL — Formulas provided in the following sections are listed for reference purposes. Sign conventions generally accepted in their use are that quantities associated with tension action (loads, stresses, strains, etc.), are usually considered as positive and quantities associated with compressive action are considered as negative. When compressive action is of primary interest, it is sometimes convenient to identify associated properties with a positive sign. Formulas for all statistical computations relating to allowables development are presented in Chapter 9.
1.3.2 SIMPLE UNIT STRESSES —
| ft = | P / A (tension) | [1.3.2(a)] |
| fc = | P / A (compression) | [1.3.2(b)] |
| fb = | My / I = M / Z (bending) | [1.3.2(c)] |
| fs = | S / A (average direct shear stress) | [1.3.2(d)] |
| fx = | SQ / Ib (longitudinal or transverse shear stress) | [1.3.2(e)] |
| fx = | Ty / Ip (shear stress in round tubes due to torsion) | [1.3.2(f)] |
| fs = | (T/2At) (shear stress due to torsion in thin-walled structures of closed section. Note that A is the area enclosed by the median line of the section.) | [1.3.2(g)] |
| fA = | BfH ; fT = BfL (axial and tangential stresses, where B = biaxial ratio) | [1.3.2(h)] |
1.3.3 COMBINED STRESSES (SEE SECTION 1.5.3.4) —
| fA = fc + fb (compression and bending) | [1.3.3(a)] |
| fs max = [fs2 + (fn/2)2]1/2 (compression, bending, and torsion) | [1.3.3(b)] |
| fn max = fn/2 + fs max | [1.3.3(c)] |
1.3.4 DEFLECTIONS (AXIAL) —
| e = δ / L (unit deformation or strain) | [1.3.4(a)] |
| E = f/e (This equation applied when E is obtained from the same tests in which f and e are measured.) | [1.3.4(b)] |
| δ = eL = (f / E)L | [1.3.4(c)] |
| = PL / (AE) (This equation applies when the deflection is to be calculated using a known value of E.) | [1.3.4(d)] |
1.3.5 DEFLECTIONS (BENDING) —
(Change of slope per unit length of a beam; radians per unit length) [1.3.5(a)]
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1.3.6 DEFLECTIONS (TORSION) —
(Change of angular deflection or twist per unit length of a member, [1.3.6(a)] radians per unit length.)
— Total twist over a length from to . (This integral denotes the area under the curve of plotted against , between the limits of and .) [1.3.6(b)]
(Used when torque T/GJ is constant over length L.) [1.3.6(c)]
1.3.7 Biaxial Elastic Deformation
(Unit lateral deformation/unit axial deformation.) This identifies Poisson's ratio [1.3.7(a)] in uniaxial loading.
[1.3.7(b)]
[1.3.7(c)]
elastic modulus. [1.3.7(d)]
1.3.8 BASIC COLUMN FORMULAS —
| Fc = π² Et (L' / ρ)² where L' = L/√c — conservative using tangent modulus | [1.3.8(a)] |
| Fc = π² E (L' / ρ)² — standard Euler formula | [1.3.8(b)] |
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1.3.9 INELASTIC STRESS-STRAIN RESPONSE —
| etotal = f / E + ep (elastic strain response plus inelastic or plastic strain response) | [1.3.9(a)] |
| where | |
| ep = 0.002 * (f/f0.2ys)n, | [1.3.9(b)] |
| f0.2ys = the 0.2 percent yield stress and |
n = Ramberg-Osgood parameter
Equation [1.3.9(b)] implies a log-linear relationship between inelastic strain and stress, which is observed with many metallic materials, at least for inelastic strains ranging from the material’s proportional limit to its yield stress.