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(BQ) Part 2 book Mechanics of materials has contents: Stresses in beams (advanced topics), applications of plane stress (pressure vessels, beams, and combined loadings), analysis of stress and strain, statically indeterminate beams, deflections of beams, columns, review of centroids and moments of inertia.
A more advanced theory is required for analysis and design of composite beams and beams with unsymmetric cross sections Stresses in Beams (Advanced Topics) CHAPTER OVERVIEW In Chapter 6, we will consider a number of advanced topics related to shear and bending of beams of arbitrary cross section First, stresses and strains in composite beams, that is beams fabricated of more than one material, is discussed in Section 6.2 First, we locate the neutral axis then find the flexure formula for a composite beam made up of two different materials We then study the transformed-section method as an alternative procedure for analyzing the bending stresses in a composite beam in Section 6.3 Next, we study bending of doubly symmetric beams acted on by inclined loads having a line of action through the centroid of the cross section (Section 6.4) In this case, there are bending moments (My, Mz) about each of the principal axes of the cross section, and the neutral axis is no longer perpendicular to the longitudinal plane containing the applied loads The final normal stresses are obtained by superposing the stresses obtained from the flexure formulas for each of the separate axes of the cross section Next, we investigate the general case of unsymmetric beams in pure bending, removing the restriction of at least one axis of symmetry in the cross section (Section 6.5) We develop a general procedure for analyzing an unsymmetric beam subjected to any bending moment M resolved into components along the principal centroidal axes of the cross section Of course, symmetric beams are special cases of unsymmetric beams, and therefore, the discussions also apply to symmetric beams If the restriction of pure bending is removed and transverse loads are allowed, we note that these loads must act through the shear center of the cross section so that twisting of the beam about a longitudinal axis can be avoided (Sections 6.6 and 6.9) The distributions of shear stresses in the elements of the cross sections of a number of beams of thin-walled open section (such as channels, angles, and Z shapes) are calculated and then used to locate the shear center for each particular cross-sectional shape (Sections 6.7, 6.8 and 6.9) As the final topic in the chapter, the bending of elastoplastic beams is described in which the normal stresses go beyond the linear elastic range of behavior (Section 6.10) 455 456 CHAPTER Stresses in Beams (Advanced Topics) Chapter is organized as follows: 6.1 Introduction 457 6.2 Composite Beams 6.3 6.4 6.5 6.6 6.7 6.8 6.9 *6.10 457 Transformed-Section Method 466 Doubly Symmetric Beams with Inclined Loads 472 Bending of Unsymmetric Beams 479 The Shear-Center Concept 487 Shear Stresses in Beams of Thin-Walled Open Cross Sections 489 Shear Stresses in Wide-Flange Beams 492 Shear Centers of Thin-Walled Open Sections 496 Elastoplastic Bending 504 Chapter Summary & Review 514 Problems 516 * Advanced Topics SECTION 6.2 Composite Beams 457 6.1 INTRODUCTION In this chapter, we continue our study of the bending of beams by examining several specialized topics These subjects are based upon the fundamental topics discussed previously in Chapter 5—topics such as curvature, normal stresses in beams (including the flexure formula), and shear stresses in beams However, we will no longer require that beams be composed of one material only; and we will also remove the restriction that the beams have a plane of symmetry in which transverse loads must be applied Finally, we will extend the performance into the inelastic range of behavior for beams made of elastoplastic materials Later, in Chapters and 10, we will discuss two additional subjects of fundamental importance in beam design—deflections of beams and statically indeterminate beams 6.2 COMPOSITE BEAMS (a) (b) Beams that are fabricated of more than one material are called composite beams Examples are bimetallic beams (such as those used in thermostats), plastic coated pipes, and wood beams with steel reinforcing plates (see Fig 6-1) Many other types of composite beams have been developed in recent years, primarily to save material and reduce weight For instance, sandwich beams are widely used in the aviation and aerospace industries, where light weight plus high strength and rigidity are required Such familiar objects as skis, doors, wall panels, book shelves, and cardboard boxes are also manufactured in sandwich style A typical sandwich beam (Fig 6-2) consists of two thin faces of relatively high-strength material (such as aluminum) separated by a thick core of lightweight, low-strength material Since the faces are at the greatest distance from the neutral axis (where the bending stresses are highest), they function somewhat like the flanges of an I-beam The core serves as a filler and provides support for the faces, stabilizing them against wrinkling or buckling Lightweight plastics and foams, as well as honeycombs and corrugations, are often used for cores Strains and Stresses (c) (c) FIG 6-1 Examples of composite beams: (a) bimetallic beam, (b) plastic-coated steel pipe, and (c) wood beam reinforced with a steel plate The strains in composite beams are determined from the same basic axiom that we used for finding the strains in beams of one material, namely, cross sections remain plane during bending This axiom is valid for pure bending regardless of the nature of the material (see Section 5.4) Therefore, the longitudinal strains ex in a composite beam vary linearly from top to bottom of the beam, as expressed by Eq (5-4), which is repeated here: y ex ϭ Ϫ ᎏᎏ ϭ Ϫky r (6-1) 458 CHAPTER Stresses in Beams (Advanced Topics) (a) (b) (c) In this equation, y is the distance from the neutral axis, r is the radius of curvature, and k is the curvature Beginning with the linear strain distribution represented by Eq (6-1), we can determine the strains and stresses in any composite beam To show how this is accomplished, consider the composite beam shown in Fig 6-3 This beam consists of two materials, labeled and in the figure, which are securely bonded so that they act as a single solid beam As in our previous discussions of beams (Chapter 5), we assume that the xy plane is a plane of symmetry and that the xz plane is the neutral plane of the beam However, the neutral axis (the z axis in Fig 6-3b) does not pass through the centroid of the cross-sectional area when the beam is made of two different materials If the beam is bent with positive curvature, the strains ex will vary as shown in Fig 6-3c, where eA is the compressive strain at the top of the beam, eB is the tensile strain at the bottom, and eC is the strain at the contact surface of the two materials Of course, the strain is zero at the neutral axis (the z axis) The normal stresses acting on the cross section can be obtained from the strains by using the stress-strain relationships for the two materials Let us assume that both materials behave in a linearly elastic manner so that Hooke’s law for uniaxial stress is valid Then the stresses in the materials are obtained by multiplying the strains by the appropriate modulus of elasticity Denoting the moduli of elasticity for materials and as E1 and E2, respectively, and also assuming that E2 Ͼ E1, we obtain the stress diagram shown in Fig 6-3d The compressive stress at the top of the beam is sA ϭ E1eA and the tensile stress at the bottom is sB ϭ E2eB FIG 6-2 Sandwich beams with: y (a) plastic core, (b) honeycomb core, and (c) corrugated core z (a) x y eA A eC s1C FIG 6-3 (a) Composite beam of two materials, (b) cross section of beam, (c) distribution of strains ex throughout the height of the beam, and (d) distribution of stresses sx in the beam for the case where E2 Ͼ E1 sA = E1eA C z O (b) B eB (c) s2C sB = E2eB (d) SECTION 6.2 Composite Beams 459 At the contact surface (C ) the stresses in the two materials are different because their moduli are different In material the stress is s1C ϭ E1eC and in material it is s2C ϭ E2eC Using Hooke’s law and Eq (6-1), we can express the normal stresses at distance y from the neutral axis in terms of the curvature: sx1 ϭ ϪE1ky sx2 ϭ ϪE2ky (6-2a,b) in which sx1 is the stress in material and sx2 is the stress in material With the aid of these equations, we can locate the neutral axis and obtain the moment-curvature relationship Neutral Axis The position of the neutral axis (the z axis) is found from the condition that the resultant axial force acting on the cross section is zero (see Section 5.5); therefore, ͵ sx1 d A ϩ ͵ sx2 d A ϭ (a) where it is understood that the first integral is evaluated over the crosssectional area of material and the second integral is evaluated over the cross-sectional area of material Replacing sx1 and sx2 in the preceding equation by their expressions from Eqs (6-2a) and (6-2b), we get ͵ Ϫ E1kydA Ϫ ͵ E2kydA ϭ Since the curvature is a constant at any given cross section, it is not involved in the integrations and can be cancelled from the equation; thus, the equation for locating the neutral axis becomes ͵ ͵ E1 ydA ϩ E2 ydA ϭ y t h — z h O h — t FIG 6-4 Doubly symmetric cross section (6-3) The integrals in this equation represent the first moments of the two parts of the cross-sectional area with respect to the neutral axis (If there are more than two materials—a rare condition—additional terms are required in the equation.) Equation (6-3) is a generalized form of the analogous equation for a beam of one material (Eq 5-8) The details of the procedure for locating the neutral axis with the aid of Eq (6-3) are illustrated later in Example 6-1 If the cross section of a beam is doubly symmetric, as in the case of a wood beam with steel cover plates on the top and bottom (Fig 6-4), the neutral axis is located at the midheight of the cross section and Eq (6-3) is not needed 460 CHAPTER Stresses in Beams (Advanced Topics) Moment-Curvature Relationship The moment-curvature relationship for a composite beam of two materials (Fig 6-3) may be determined from the condition that the moment resultant of the bending stresses is equal to the bending moment M acting at the cross section Following the same steps as for a beam of one material (see Eqs 5-9 through 5-12), and also using Eqs (6-2a) and (6-2b), we obtain ͵ ͵ MϭϪ ͵ ͵ sx ydA ϭ Ϫ sx1 ydA Ϫ A ͵ sx2 ydA ϭ kE1 y 2dA ϩ kE2 y 2dA (b) This equation can be written in the simpler form M ϭ k(E1I1 ϩ E2I2) (6-4) in which I1 and I2 are the moments of inertia about the neutral axis (the z axis) of the cross-sectional areas of materials and 2, respectively Note that I ϭ I1 ϩ I2, where I is the moment of inertia of the entire cross-sectional area about the neutral axis Equation (6-4) can now be solved for the curvature in terms of the bending moment: M k ϭ ᎏᎏ ϭ ᎏᎏ r E1I1 ϩ E2I2 (6-5) This equation is the moment-curvature relationship for a beam of two materials (compare with Eq 5-12 for a beam of one material) The denominator on the right-hand side is the flexural rigidity of the composite beam Normal Stresses (Flexure Formulas) The normal stresses (or bending stresses) in the beam are obtained by substituting the expression for curvature (Eq 6-5) into the expressions for sx1 and sx2 (Eqs 6-2a and 6-2b); thus, MyE1 sx1 ϭ Ϫ ᎏᎏ E1I1 ϩ E2I2 MyE2 sx2 ϭ Ϫ ᎏᎏ E1I1 ϩ E 2I2 (6-6a,b) These expressions, known as the flexure formulas for a composite beam, give the normal stresses in materials and 2, respectively If the two materials have the same modulus of elasticity (E1 ϭ E2 ϭ E), then both equations reduce to the flexure formula for a beam of one material (Eq 5-13) The analysis of composite beams, using Eqs (6-3) through (6-6), is illustrated in Examples 6-1 and 6-2 at the end of this section SECTION 6.2 Composite Beams 461 y t z hc O h FIG 6-5 Cross section of a sandwich beam having two axes of symmetry (doubly symmetric cross section) t b Approximate Theory for Bending of Sandwich Beams Sandwich beams having doubly symmetric cross sections and composed of two linearly elastic materials (Fig 6-5) can be analyzed for bending using Eqs (6-5) and (6-6), as described previously However, we can also develop an approximate theory for bending of sandwich beams by introducing some simplifying assumptions If the material of the faces (material 1) has a much larger modulus of elasticity than does the material of the core (material 2), it is reasonable to disregard the normal stresses in the core and assume that the faces resist all of the longitudinal bending stresses This assumption is equivalent to saying that the modulus of elasticity E2 of the core is zero Under these conditions the flexure formula for material (Eq 6-6b) gives sx2 ϭ (as expected), and the flexure formula for material (Eq 6-6a) gives My sx1 ϭ Ϫᎏᎏ I1 (6-7) which is similar to the ordinary flexure formula (Eq 5-13) The quantity I1 is the moment of inertia of the two faces evaluated with respect to the neutral axis; thus, b I1 ϭ ᎏᎏ h3 Ϫ h3c 12 (6-8) in which b is the width of the beam, h is the overall height of the beam, and hc is the height of the core Note that hc ϭ h Ϫ 2t where t is the thickness of the faces The maximum normal stresses in the sandwich beam occur at the top and bottom of the cross section where y ϭ h/2 and Ϫh/2, respectively Thus, from Eq (6-7), we obtain Mh stop ϭ Ϫ ᎏᎏ 2I1 Mh s bottom ϭ ᎏᎏ 2I1 (6-9a,b) 462 CHAPTER Stresses in Beams (Advanced Topics) If the bending moment M is positive, the upper face is in compression and the lower face is in tension (These equations are conservative because they give stresses in the faces that are higher than those obtained from Eqs 6-6a and 6-6b.) If the faces are thin compared to the thickness of the core (that is, if t is small compared to hc), we can disregard the shear stresses in the faces and assume that the core carries all of the shear stresses Under these conditions the average shear stress and average shear strain in the core are, respectively, V taver ϭ ᎏᎏ bhc V gaver ϭ ᎏᎏ bhcGc (6-10a,b) in which V is the shear force acting on the cross section and Gc is the shear modulus of elasticity for the core material (Although the maximum shear stress and maximum shear strain are larger than the average values, the average values are often used for design purposes.) Limitations FIG 6-6 Reinforced concrete beam with longitudinal reinforcing bars and vertical stirrups Throughout the preceding discussion of composite beams, we assumed that both materials followed Hooke’s law and that the two parts of the beam were adequately bonded so that they acted as a single unit Thus, our analysis is highly idealized and represents only a first step in understanding the behavior of composite beams and composite materials Methods for dealing with nonhomogeneous and nonlinear materials, bond stresses between the parts, shear stresses on the cross sections, buckling of the faces, and other such matters are treated in reference books dealing specifically with composite construction Reinforced concrete beams are one of the most complex types of composite construction (Fig 6-6), and their behavior differs significantly from that of the composite beams discussed in this section Concrete is strong in compression but extremely weak in tension Consequently, its tensile strength is usually disregarded entirely Under those conditions, the formulas given in this section not apply Furthermore, most reinforced concrete beams are not designed on the basis of linearly elastic behavior—instead, more realistic design methods (based upon load-carrying capacity instead of allowable stresses) are used The design of reinforced concrete members is a highly specialized subject that is presented in courses and textbooks devoted solely to that subject SECTION 6.2 Composite Beams 463 Example 6-1 A composite beam (Fig 6-7) is constructed from a wood beam (4.0 in ϫ 6.0 in actual dimensions) and a steel reinforcing plate (4.0 in wide and 0.5 in thick) The wood and steel are securely fastened to act as a single beam The beam is subjected to a positive bending moment M ϭ 60 k-in Calculate the largest tensile and compressive stresses in the wood (material 1) and the maximum and minimum tensile stresses in the steel (material 2) if E1 ϭ 1500 ksi and E2 ϭ 30,000 ksi y A h1 in Solution z h2 O C 0.5 in B in Neutral axis The first step in the analysis is to locate the neutral axis of the cross section For that purpose, let us denote the distances from the neutral axis to the top and bottom of the beam as h1 and h2, respectively To obtain these distances, we use Eq (6-3) The integrals in that equation are evaluated by taking the first moments of areas and about the z axis, as follows: ͵ FIG 6-7 Example 6-1 Cross section of a composite beam of wood and steel ͵ ydA ϭ ෆy1A1 ϭ (h1 Ϫ in.)(4 in ϫ in.) ϭ (h1 Ϫ in.)(24 in.2) ydA ϭ ෆy2 A2 ϭ Ϫ(6.25 in Ϫ h1)(4 in ϫ 0.5 in.) ϭ (h1 Ϫ 6.25 in.)(2 in.2) in which A1 and A2 are the areas of parts and of the cross section, ෆy1 and ෆy2 are the y coordinates of the centroids of the respective areas, and h1 has units of inches Substituting the preceding expressions into Eq (6-3) gives the equation for locating the neutral axis, as follows: E1 ͵ ydA ϩ E2 ͵ ydA ϭ or (1500 ksi)(h1 Ϫ in.)(24 in.2) ϩ (30,000 ksi)(h1 Ϫ 6.25 in.)(2 in.2) ϭ Solving this equation, we obtain the distance h1 from the neutral axis to the top of the beam: hl ϭ 5.031 in Also, the distance h2 from the neutral axis to the bottom of the beam is h2 ϭ 6.5 in Ϫ hl ϭ 1.469 in Thus, the position of the neutral axis is established Moments of inertia The moments of inertia I1 and I2 of areas A1 and A2 with respect to the neutral axis can be found by using the parallel-axis theorem (see Section 12.5 of Chapter 12) Beginning with area (Fig 6-7), we get Il ϭ ᎏᎏ(4 in.)(6 in.) ϩ (4 in.)(6 in.)(h1 Ϫ in.) ϭ 171.0 in.4 12 1010 9.5-19 9.5-20 9.5-21 9.5-22 9.7-23 9.5-24 9.6-4 9.6-5 9.6-6 9.6-8 9.6-9 9.6-10 9.6-11 9.7-1 9.7-2 9.7-3 9.7-4 9.7-5 9.7-6 Answers to Problems dC ϭ 0.120 in q ϭ 16cEI/7L4 dh ϭ Pcb2/2EI, dv ϭ Pc2(c ϩ 3b)/3EI d ϭ PL2(2L ϩ 3a)/3EI (a) b/L ϭ 0.4030; (b) dC ϭ 0.002870qL4/EI a ϭ 22.5°, 112.5°, Ϫ67.5°, or Ϫ157.5° uB ϭ 7qL3/162EI, dB ϭ 23qL4/648EI dB ϭ 0.443 in., dC ϭ 0.137 in dB ϭ 11.8 mm, dC ϭ 4.10 mm P ϭ 64 kN uA ϭ M0L/6EI, uB ϭ M0L/3EI, d ϭ M0L2/16EI uA ϭ Pa(L Ϫ a)(L Ϫ 2a)/6LEI, d1 ϭ Pa2(L Ϫ 2a)2/6LEI, d2 ϭ uA ϭ M0L/6EI, uB ϭ 0, d ϭ M0L2/27EI (downward) (a) dB ϭ PL3(1 ϩ 7I1/I2)/24EI1; (b) r ϭ (1 ϩ 7I1/I2)/8 (a) dB ϭ qL4(1 ϩ 15I1/I2)/128EI1; (b) r ϭ (1 ϩ 15I1/I2)/16 (a) dc =0.31 in (upward) (b) dc =0.75 in (downward) v ϭ Ϫqx(21L3 Ϫ 64Lx2 ϩ 32x3)/768EI for Յ x Յ L/4; v ϭ Ϫq(13L4 ϩ 256L3x Ϫ 512L x3 ϩ 256x4)/12,288EI for L/4 Յ x Յ L/2; uA ϭ 7qL3/256EI; dmax ϭ 31qL4/4096EI uA ϭ 8PL2/243EI, dB ϭ 8PL3/729EI, dmax ϭ 0.01363PL3/EI v ϭ Ϫ2Px(19L2 Ϫ 27x2)/729EI for Յ x Յ L/3; v ϭ P(13L Ϫ 175L x ϩ 243Lx Ϫ 81x )/1458EI for L/3 Յ x Յ L; uA ϭ 38PL2/729EI, uC ϭ 34PL2/729EI, dB ϭ 32PL3/2187EI PL3 Lϩx L 3x v ϭ ᎏᎏ ᎏᎏ Ϫ ᎏᎏ ϩ ᎏᎏ ϩ ln ᎏᎏ ; EIA 2(L ϩ x) 8L 2L PL dA ϭ ᎏᎏ(8 ln Ϫ 5) 8EIA 4L(2L ϩ 3x) 2x PL3 v ϭ ᎏᎏ Ϫ ᎏᎏ Ϫ ᎏᎏ ; (L ϩ x)2 L 24EIA PL dA ϭ ᎏᎏ 24EIA 8PL3 L 2x 2L ϩ x v ϭ ᎏᎏ ᎏᎏ Ϫ ᎏᎏ Ϫ ᎏᎏ ϩ ln ᎏᎏ ; EIA 2L ϩ x 9L 3L 8PL dA ϭ ᎏᎏ ln ᎏᎏ Ϫ ᎏᎏ EIA 18 9.7-7 9.7-8 9.7-9 2 ΄ ΅ ΅ ΅ v (x) ϭ 19683PL3 ⎛ 81L ϩ ln 2000 EI A ⎜⎝ 81L ϩ 40 x 40 x ⎞ 6440xx 3361 ⎞ ⎛ 81 ⎜⎝ 121 ϩ 121L ⎟⎠ Ϫ 14641L Ϫ 14641⎟⎠ dA ϭ 9.7-11 19683PL3 ⎛ ⎛ 11⎞ ⎞ Ϫ2820 ϩ 14641 ln ⎜ ⎟ ⎟ ⎝ ⎠⎠ 7320500 EI A ⎜⎝ v (x) ϭ Ϫ 19683PL3 ⎛ 81L 40 x ⎞ ⎛ ϩ ln ⎜ ϩ Ϫ ⎜ ⎝ 2000 EI A ⎝ 81L ϩ 40 x 81L ⎟⎠ Ϫ dB ϭ 9.7-12 ⎞ 6440 x Ϫ 1⎟ ⎠ 146441L 19683PL3 ⎛ ⎛ 11⎞ ⎞ Ϫ2820 ϩ 14641 ln ⎜ ⎟ ⎟ ⎜ ⎝ ⎠⎠ 7320500 EI A ⎝ ΄ ΅ qL3 8Lx2 (a) vЈ ϭ Ϫ ᎏᎏ Ϫ ᎏᎏ for Յ x Յ L, 16EIA (L ϩ x)3 qL4 (9L2 ϩ 14Lx ϩ x2)x x v ϭ Ϫ ᎏᎏ ᎏᎏᎏ Ϫ ln ϩ ᎏᎏ 8L(L ϩ x)2 2EIA L ΄ ΅ for Յ x Յ L; 9.8-1 9.8-2 9.8-3 9.8-4 9.8-5 ΄ ΄ 9.7-10 9.8-6 9.8-7 9.9-2 9.9-3 9.9-6 9.9-7 9.9-8 9.9-9 9.9-10 9.9-11 9.9-12 9.10-1 9.10-2 qL3 qL4(3 Ϫ ln2) (b) uA ϭ ᎏᎏ, dC ϭ ᎏᎏ 16EIA 8EIA U ϭ 4bhL s 2max /45E (a) and (b) U ϭ P 2L3/96EI; (c) d ϭ PL3/48EI q L5 (a) and (b) U ϭ 15EI (a) U ϭ 32EId 2/L3; (b) U ϭ p4EId 2/4L3 (a) U ϭ P 2a2(L ϩ a)/6EI; (b) dC ϭ Pa2(L ϩ a)/3EI; (c) U ϭ 241 in.-lb, dC ϭ 0.133 in L Uϭ 17L4q ϩ 280qL2 M ϩ 2560 M 15360 EI dB ϭ 2PL3/3EI ϩ 8͙2ෆPL/EA dD ϭ Pa2b2/3LEI dC ϭ Pa2(L ϩ a)/3EI dC ϭ L3(2P1 ϩ 5P2)/48EI, dB ϭ L3(5P1 ϩ 16P2)/48EI uA ϭ 7qL3/48EI dC ϭ Pb2(b ϩ 3h)/3EI, uC ϭ Pb(b ϩ 2h)/2EI dC ϭ 31qL4/4096EI uA ϭ MA(L ϩ 3a)/3EI, dA ϭ MAa(2L ϩ 3a)/6EI dC ϭ Pa2(L ϩ a)/3EI ϩ P(L ϩ a)2/kL2 dD ϭ 37qL4/6144EI (upward) smax ϭ sst1[ ϩ (1 ϩ 2h/dst)1/2] smax ϭ ͙18WEh ෆ/AL ෆ ( ) Answers to Problems dmax ϭ 0.302 in., smax ϭ 21,700 psi d ϭ 281 mm W 14 ϫ 53 h ϭ 360 mm R ϭ ͙ෆ 3EIImv2ෆ /L3 v ϭ Ϫa(T2 Ϫ T1)(x)(L Ϫ x)/2h (pos upward); uA ϭ aL(T2 Ϫ T1)/2h (clockwise); dmax ϭ aL2(T2 Ϫ T1)/8h (downward) 9.11-2 v ϭ a(T2 Ϫ T1)(x )/2h (upward); uB ϭ aL(T2 Ϫ T1)/h (counterclockwise); dB ϭ aL2(T2 Ϫ T1)/2h (upward) 9.10-3 9.10-4 9.10-5 9.10-6 9.10-7 9.11-1 9.11-3 v (x) ϭ uC ϭ ( ) a (T2 Ϫ T1 ) x Ϫ L2 2h a (T2 Ϫ T1 ) ( L ϩ a ) (counterclockwise) h a (T2 Ϫ T1 ) La ϩ a (upward) dC ϭ 2h aT0 L3 9.11-4 (a) dmax ϭ (downward) 3h ( (b) dmax ϭ ( ) ) aT0 L4 2 Ϫ 48h (downward) MA ϭ aT0 L (downward) 6h aT L3 (b) dmax ϭ (downward) 12h (c) dmax ϭ aT0 L (downward) 6h aT0 L3 dmax ϭ (downward) 12h (a) dmax ϭ (a) RA ϭ V (0) ϭ ( 10.3-7 CHAPTER 10 10.3-1 10.3-5 q0 L 40 11 RB ϭ ϪV ( L ) ϭ q0 L 40 R A ϭ V (0 ) ϭ 10.3-8 ) 24 q0 L p4 24 RB ϭ ϪV ( L ) ϭ Ϫ q0 L p 1⎞ ⎛ 12 M A ϭ ⎜ Ϫ ⎟ q0 L2 (counterclockwise) ⎝p p ⎠ 1⎞ ⎛ 12 M B ϭ ⎜ Ϫ ⎟ q0 L2 (counterclockwise) ⎝p p ⎠ (a) RA ϭ V (0) ϭ vϭ RA ϭ ϪRB ϭ 3M0 /2L, MA ϭ M0 /2; v ϭ ϪM0 x2(L Ϫ x)/4LEI 10.3-2 RA ϭ RB ϭ qL/2, MA ϭ MB ϭ qL /12; 2 v ϭ Ϫqx (L Ϫ x) /24EI 10.3-3 RA ϭ RB ϭ 3EIdB /L , MA ϭ 3EIdB /L ; v ϭ ϪdB x (3L Ϫ x)/2L k R qL5 qL3 10.3-4 u B ϭ dB = − qL4 + 12(k R L − EI ) 6(k R L Ϫ EI ) q0 L2 120 q0 L 60 13 RB = −V ( L ) = q0 L 60 MA ϭ q0 L2 30 q0 vϭ Ϫx ϩ 7L3 x Ϫ 6q0 L4 x 360 L2 EI (b) ⎛2 p Ϫ 4p ϩ 8⎞ RA ϭ V (0) ϭ 0.31 ⋅ q0 L ϭ ⎜ Ϫ ⎟⎠ ⋅ q0 L p4 ⎝p ⎛ p Ϫ 4p ϩ 8⎞ RB ϭ ϪV ( L ) ϭ 0.327q0 L ϭ ⎜ ⎟⎠ ⋅ q0 L p4 ⎝ p Ϫ 12 p ϩ 24 M A ϭ Ϫ2q0 L2 p4 ⎡ p2 Ϫ 4p ϩ x ⎤ ⎛ 2L ⎞ ⎛ px⎞ ⋅ ⎥ ⎢Ϫq0 ⎜ ⎟ sin ⎜ ⎟ Ϫ 6q0 L ⎝p⎠ ⎝ 2L ⎠ ⎢ p4 6⎥ vϭ 2 ⎥ EI ⎢ p Ϫ 12p ϩ 24 x ⎛ 2L ⎞ ⎢ ϩ 2q0 L2 ⋅ ⋅ ϩ q0 ⎜ ⎟ x ⎥ ⎝ p⎠ p ⎢⎣ ⎥⎦ 10.3-6 9.11-5 1011 ⎡ ⎛ px⎞ Ϫq0 L4 cos ⎜ ⎟ ⎝ L ⎠ p EI ⎢⎣ ⎤ ϩ 4q0 Lx Ϫ 6q0 L2 x ϩ q0 L4 ⎥ ⎦ (b) RA ϭ RB ϭ q0L/p, MA ϭ MB ϭ 2q0L2/p 3; v ϭ Ϫq0L2(L2 sin px/L ϩ p x Ϫ pLx)/p4EI 48(4 Ϫ p ) ⋅ q0 L (a) RA ϭ V (0) ϭ p4 48(4 Ϫ p ) ⎞ ⎛2 RB ϭ ϪV ( L ) ϭ ⎜ Ϫ ⎟⎠ ⋅ q0 L ⎝p p4 16 (6 Ϫ p) ⎛ 2L ⎞ M A ϭ Ϫq0 ⎜ ⎟ ϩ q0 L ⎝p⎠ p4 32(p Ϫ 3) MB ϭ Ϫ q0 L2 p4 1012 Answers to Problems vϭ ⎡ ⎛ px⎞ Ϫ16q0 L4 cos ⎜ ⎟ ϩ 8(4 Ϫ p )q0 Lx ⎝ 2L ⎠ p EI ⎢⎣ 10.4-5 ⎤ Ϫ 8(6 Ϫ p)q0 L2 x ϩ 16q0 L4 ⎥ ⎦ 13 q0 L (b) RA ϭ V (0) ϭ 30 RB ϭ ϪV ( L ) ϭ q0 L 30 M A ϭ q0 L2 (counterclockwise) 15 M B ϭ Ϫ q0 L2 (counterclockwise) 20 q0 ⎡ x Ϫ 15L2 x ϩ 26 L3 x vϭ 360 L2 EI ⎣ Ϫ 12 L4 x ⎤⎦ 10.3-9 R A ϭ V (0 ) ϭ q0 L 20 RB ϭϪV ( L ) ϭ q0 L 20 MA ϭ q0 L 30 vϭ Ϫq0 x ϩ 3q0 Lx Ϫ 2q0 L2 x 120 LEI 10.3-10 RA ϭ ϪRB ϭ 3M0 /2L, MA ϭ ϪMB ϭ M0/4; v ϭ ϪM0x2(L Ϫ 2x)/8LEI for Յ x Յ L/2 M0 10.3-11 R ϭ Ϫ B L M0 RA ϭ L M0 MA ϭ L L⎞ ⎛ 9M0 M0 ⎞ ⎛ vϭ x Ϫ x ⎟ ⎜⎝ Յ x Յ ⎟⎠ ⎜ ⎝ ⎠ EI 48L 16 M L2 ⎞ ⎛ 9M0 9M0 M0 L v= x Ϫ x ϩ x Ϫ EI ⎜⎝ 48L 16 ⎟⎠ ( ) 10.4-6 10.4-7 10.4-8 10.4-9 10.4-10 10.4-11 10.4-12 10.4-13 10.4-14 10.4-15 10.4-16 10.4-17 10.4-18 10.4-19 10.4-20 10.4-21 10.4-22 10.4-23 10.4-24 10.4-25 ⎛L ⎞ ⎜⎝ Յ x Յ L ⎟⎠ 10.4-1 10.4-2 10.4-3 10.4-4 RA ϭ Pb(3L2 Ϫ b2)/2L3, RB ϭ Pa 2(3L Ϫ a)/2L3, MA ϭ Pab(L ϩ b)/2L2 qL2 qL2 M ϭ M ϭ B RA ϭ qL , A , 17 1 RA ϭ Ϫ qL , RB ϭ qL , M A ϭ Ϫ qL2 8 tAB/tCD ϭ LAB /LCD 10.4-26 10.5-1 10.5-2 7 17 qL , RB ϭ qL , M A ϭ qL2 12 12 12 M B ϭ qL2 RA ϭ 2qL , 12 RA ϭ RB ϭ q0L/4, MA ϭ MB ϭ 5q0L2/96 RA ϭ qL/8, RB ϭ 33qL/16, RC ϭ 13qL/16 RA ϭ 1100 lb (downward), RB ϭ 2800 lb (upward), MA ϭ 30,000 lb-in (clockwise) RB ϭ 6.436 kN (a) The tension force in the tie rod ϭ RD ϭ 604.3 lb (b) RA ϭ 795.7 lb MA ϭ 1307.5 lb-ft ϭ 1.567 ϫ 104 lb-in RA ϭ 31qL/48, RB ϭ 17qL/48, MA ϭ 7qL2/48 (a) RA ϭ Ϫ23P/17, RD ϭ RE ϭ 20P/17, MA ϭ 3PL/17; (b) Mmax ϭ PL/2 RA ϭ RD ϭ 2qL/5, RB ϭ RC ϭ 11qL/10 M B (q ) ϭ (Ϫ800 ⋅ q ) lb-in for q Ͻ 250 lb րin M B (q ) ϭ (Ϫ200 ⋅ q Ϫ 150000 ) lb-in RA ϭ for q Ն 250 lb րin RA ϭ ϪRB ϭ 6M0ab/L3; MA ϭ M0b(3a Ϫ L)/L2, MB ϭ ϪM0a(3b Ϫ L)/L2 s ϭ 509 psi (MAB)max ϭ 121qL2/2048 ϭ 6.05 kNиm; (MCD)max ϭ 5qL2/64 ϭ 8.0 kNиm F ϭ 3,160 lb, MAB ϭ 18,960 lb-ft, MDE ϭ 7,320 lb-ft k ϭ 48EI(6 ϩ ͙2ෆ )/7L3 ϭ 89.63EI/L3 (a) VA ϭ VC ϭ 3P/32, HA ϭ P, MA ϭ 13PL/32; (b) Mmax ϭ 13PL/32 35 29 35 H A ϭ Ϫ P, HC ϭ Ϫ P, M max ϭ PL 64 64 128 RA ϭ RB ϭ 3000 lb, RC ϭ (a) MA ϭ MB ϭ qb(3L2 Ϫ b2)/24L; (b) b/L ϭ 1.0, MA ϭ qL2/12; (c) For a ϭ b ϭ L/3, (Mmax)pos ϭ 19qL2/648 (a) d2/d1 ϭ ͙8ෆ ϭ 1.682; (b) Mmax ϭ qL2(3 Ϫ ͙2ෆ)/2 ϭ 0.08579qL2; (c) Point C is below points A and B by the amount 0.01307qL4/EI Mmax ϭ 19q0L 2/256, smax ϭ 13.4 MPa, dmax ϭ 19q0L4/7680EI ϭ 0.00891 mm 243ES EW IAHa(⌬T ) Sϭ AL3 ES ϩ 243IHEW (a) R ϭ Ϫ a(T2 Ϫ T1 )L ⋅ ⎛ 3EI ⋅ k ⎞ A ⎜⎝ 3EI ϩ L3 ⋅ k ⎟⎠ 2h Answers to Problems a(T2 Ϫ T1 )L2 ⎛ 3EI ⋅ k ⎞ ⋅⎜ ⎝ 3EI ϩ L3 ⋅ k ⎟⎠ 2h a(T2 − T1 )L3 ⎛ 3EI ⋅ k ⎞ M A = RB L = ⋅⎜ ⎝ 3EI + L3 ⋅ k ⎟⎠ 2h (b) R ϭ ϪR ϭ Ϫ 3EI a(T2 Ϫ T1 ) (upward) A B 2hL 3EI a(T2 Ϫ T1 ) (downward) RB ϭ 2hL 3EI a(T2 − T1 ) M A = RB L = (counterclockwise) 2h RB ϭ 10.5-3 a(T2 Ϫ T1 )L2 2h ⎛ 3EI ⋅ k ⎞ (upward) ⋅⎜ ⎝ 3EI ϩ L3 ⋅ k ⎟⎠ R A ϭ ϪR B ϭ Ϫ a(T2 Ϫ T1 )L ⎛ 3EI ⋅ k ⎞ (downward) ⋅⎜ ⎝ 3EI ϩ L3 ⋅ k ⎟⎠ 2h a(T2 Ϫ T1 )L3 M A ϭ RB L ϭ 2h ⎛ 3EI ⋅ k ⎞ (counterclockwise) ⋅⎜ ⎝ 3EI ϩ L3 ⋅ k ⎟⎠ 10.5-4 (a) R ϭ Ϫ a(T1 Ϫ T2 ) L B h EI ⋅ k ⎞ (downward) ⎛ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ RB ϭ a(T1 Ϫ T2 )L R A ϭ Ϫ RB ϭ 2h 3EI ⋅ k ⎞ (upward) ⎛ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ a(T1 Ϫ T2 )L2 R C ϭ Ϫ RB ϭ 2h 9EI ⋅ k ⎞ (upward) ⎛ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ (b) R B ϭ Ϫ EI a(T1 Ϫ T2 ) (downward) Lh 3EI a(T1 Ϫ T2 ) (upward) RA ϭ Lh 9EI a(T1 Ϫ T2 ) (upward) RC ϭ Lh a(T Ϫ T2 )L2 ⎛ EI ⋅ k ⎞ (downward) RB ϭϪ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ h a(T1 Ϫ T2 )L2 R Aϭ Ϫ R B ϭ 2h 3EI ⋅ k ⎞ (upward) ⎛ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ a(T1 Ϫ T2 )L2 R Cϭ Ϫ R B ϭ 2h 9EI ⋅ k ⎞ (upward) ⎛ ⋅⎜ ⎝ 36 EI ϩ L3 ⋅ k ⎟⎠ 2 2 2 10.6-1 (a) H ϭ p EAd /4L , st ϭ p Ed /4L ; (b) st ϭ 617, 154, and 69 psi 2 10.6-2 (a) l ϭ 17q L /40,320E I ; sb ϭ qhL /16I; (b) st ϭ 17q L /40,320EI ; (c) l ϭ 0.01112 mm, sb ϭ 117.2 MPa, st ϭ 0.741 MPa CHAPTER 11 11.2-1 11.2-2 11.2-3 11.2-4 11.2-5 11.2-6 11.2-7 11.3-1 10.5-5 1013 11.3-2 11.3-3 11.3-4 11.3-5 11.3-6 11.3-7 11.3-8 11.3-9 11.3-10 11.3-11 11.3-12 11.3-13 11.3-14 11.3-15 11.3-16 Pcr ϭ bR /L ba ϩ bR ba ϩ bR (a) Pcr ϭ (b) Pcr ϭ L L Pcr ϭ 6bR /L ( L Ϫ a ) ba ϩ bR bL2 ϩ 20 bR (a) Pcr ϭ (b) Pcr ϭ aL 4L 3bR Pcr ϭ L Pcr ϭ bL Pcr ϭ bL (a) Pcr ϭ 453 k; (b) Pcr ϭ 152 k (a) Pcr ϭ 2803 kN; (b) Pcr ϭ 953 kN (a) Pcr ϭ 650 k; (b) Pcr ϭ 140 k Mallow ϭ 1143 kNиm Qallow ϭ 23.8 k p EI (a) Qcr ϭ L2 2p EI (b) Qcr ϭ 9L2 2p EI (a) Qcr ϭ L2 3dp EI (b) M cr ϭ L2 ⌬T ϭ p 2I/aAL2 h/b ϭ (a) Pcr ϭ 3p 3Er 4/4L2; (b) Pcr ϭ 11p 3Er4/4L2 P1 : P2 : P3 ϭ 1.000 :1.047 :1.209 Pallow ϭ 604 kN Fallow ϭ 54.40 k Wmax ϭ 124 kN tmin ϭ 0.165 in Pcr ϭ 497 kN ( ) 1014 11.3-17 11.3-18 11.3-19 11.4-1 11.4-2 11.4-3 11.4-4 11.4-5 11.4-6 11.4-7 11.4-8 11.4-9 11.4-10 11.4-11 11.5-1 11.5-2 11.5-3 11.5-4 11.5-5 11.5-6 11.5-7 11.5-8 11.5-9 11.5-10 11.5-11 11.5-12 11.5-13 11.6-1 11.6-2 11.6-3 11.6-4 11.6-5 11.6-6 11.6-7 11.6-8 11.6-9 11.6-10 11.6-11 11.6-12 11.6-13 Answers to Problems Wcr ϭ 51.90 k u ϭ arctan 0.5 ϭ 26.57° (a) qmax ϭ 142.4 lb/ft (b) Ib,min ϭ 38.52 in4 (c) s ϭ 0.264 ft, 2.424 ft Pcr ϭ 235 k, 58.7 k, 480 k, 939 k Pcr ϭ 62.2 kN, 15.6 kN, 127 kN, 249 kN Pallow ϭ 253 k, 63.2 k, 517 k, 1011 k Pallow ϭ 678.0 kN, 169.5 kN, 1387 kN, 2712 kN Pcr ϭ 229 k Tallow ϭ 18.1 kN (a) Qcr ϭ 4575 lb; (b) Qcr ϭ 10065 lb, a ϭ in Pcr ϭ 447 kN, 875 kN, 54.7 kN, 219 kN Pcr ϭ 4p 2EI/L2, v ϭ d(1 Ϫ cos 2px/L)/2 tmin ϭ 10.0 mm (b) Pcr ϭ 13.89EI/L2 d ϭ 0.112 in., Mmax ϭ 1710 lb-in d ϭ 8.87 mm, Mmax ϭ 2.03 kNиm For P ϭ 0.3Pcr: M/Pe ϭ 1.162(sin 1.721x/L) ϩ cos 1.721x/L P ϭ 583.33 {arccos [5/(5 ϩ d)]} 2, in which P ϭ kN and d ϭ mm; P ϭ 884 kN when d ϭ 10 mm P ϭ 125.58 a{rccos 0[ 2/(0.2 ϩ d)]} 2, in which P ϭ kips and d ϭ in.; P ϭ 190 k when d ϭ 0.4 in Pallow ϭ 49.91 kN Lmax ϭ 150.5 in ϭ 12.5 ft Lmax ϭ 3.14 m d ϭ e(sec kL Ϫ 1), Mmax ϭ Pe sec kL Lmax ϭ 2.21 m Lmax ϭ 130.3 in ϭ 10.9 ft Tmax ϭ 8.29 kN (a) q0 ϭ 2230 lb/ft ϭ 186 lb/in.; (b) Mmax ϭ 37.7 k-in., ratio ϭ 0.47 (a) smax ϭ 17.3 ksi; (b) Lmax ϭ 46.2 in Pallow ϭ 37.2 kN bmin ϭ 4.10 in (a) smax ϭ 38.8 MPa; (b) Lmax ϭ 5.03 m (a) smax ϭ 9.65 ksi; (b) Pallow ϭ 3.59 k d2 ϭ 131 mm (a) smax ϭ 10.9 ksi; (b) Pallow ϭ 160 k (a) smax ϭ 104.5 MPa; (b) Lmax ϭ 3.66 m (a) smax ϭ 9.60 ksi; (b) Pallow ϭ 53.6 k (a) smax ϭ 47.6 MPa; (b) n ϭ 5.49 (a) smax ϭ 13.4 ksi; (b) n ϭ 2.61 (a) smax ϭ 120.4 MPa; (b) P2 ϭ 387 kN (a) smax ϭ 17.6 ksi; (b) n ϭ 1.89 11.6-14 11.9-1 11.9-2 11.9-3 11.9-4 11.9-5 11.9-6 11.9-7 11.9-8 11.9-9 11.9-10 11.9-11 11.9-12 11.9-13 11.9-14 11.9-15 11.9-16 11.9-17 11.9-18 11.9-19 11.9-20 11.9-21 11.9-22 11.9-23 11.9-24 11.9-25 11.9-26 11.9-27 11.9-28 11.9-29 11.9-30 11.9-31 11.9-32 11.9-33 11.9-34 11.9-35 11.9-36 (a) smax ϭ 106.7 MPa; (b) P2 ϭ 314 kN Pallow ϭ 247 k, 180 k, 96.7 k, 54.4 k Pallow ϭ 2927 kN, 2213 kN, 1276 kN, 718 kN Pallow ϭ 328 k, 243 k, 134 k, 75.3 k W 250 ϫ 67 W 12 ϫ 87 W 360 ϫ 122 Pallow ϭ 58.9 k, 43.0 k, 23.1 k, 13.0 k Pallow ϭ 1070 kN, 906 kN, 692 kN, 438 kN Pallow ϭ 95 k, 75 k, 51 k, 32 k Pallow ϭ 235 kN, 211 kN, 186 kN, 163 kN Lmax ϭ 5.23 ft Lmax ϭ 3.59 m Lmax ϭ 166.3 in ϭ 13.9 ft Pallow ϭ 5634 kN (a) Lmax ϭ 254.6 in ϭ 21.2 ft; (b) Lmax ϭ 173.0 in ϭ 14.4 ft (a) Lmax ϭ 6.41 m, (b) Lmax ϭ 4.76 m d ϭ 4.89 in d ϭ 99 mm d ϭ 5.23 in d ϭ 194 mm Pallow ϭ 142 k, 122 k, 83 k, 58 k Pallow ϭ 312 kN, 242 kN, 124 kN, 70 kN Pallow ϭ 18.1 k, 14.7 k, 8.3 k, 5.3 k Pallow ϭ 96 kN, 84 kN, 61 kN, 42 kN (a) Lmax ϭ 25.2 in.; (b) dmin ϭ 2.12 in (a) Lmax ϭ 457 mm; (b) dmin ϭ 43.1 mm (a) Lmax ϭ 14.8 in.; (b) dmin ϭ 1.12 in (a) Lmax ϭ 469 mm; (b) dmin ϭ 33.4 mm Pallow ϭ 25.4 k, 14.1 k, 8.4 k Pallow ϭ 154 kN, 110 kN, 77 kN Pallow ϭ 16.8 k, 11.3 k, 7.7 k Pallow ϭ 212 kN, 127 kN, 81 kN (a) Lmax ϭ 75.2 in ϭ 6.27 ft; (b) bmin ϭ 6.71 in (a) Lmax ϭ 2.08 m; (b) bmin ϭ 184 mm (a) Lmax ϭ 74.3 in ϭ 6.19 ft; (b) bmin ϭ 4.20 in (a) Lmax ϭ 1.51 m; (b) bmin ϭ 165 mm CHAPTER 12 12.3-2 12.3-3 12.3-4 12.3-5 12.3-6 12.3-7 12.3-8 xෆ ϭ ෆy ϭ 5a/12 ෆy ϭ 1.10 in 2c2 ϭ ab ෆy ϭ 13.94 in ෆy ϭ 52.5 mm xෆ ϭ 0.99 in., ෆy ϭ 1.99 in xෆ ϭ 137 mm, ෆy ϭ 132 mm Answers to Problems 12.4-6 12.4-7 12.4-8 12.4-9 12.5-1 12.5-2 12.5-3 12.5-4 12.5-5 12.5-6 12.5-7 12.5-8 12.6-1 12.6-2 12.6-3 12.6-4 12.6-5 12.7-2 12.7-3 12.7-4 12.7-5 12.7-6 12.7-7 12.8-1 12.8-2 Ix ϭ 518 ϫ 103 mm4 Ix ϭ 36.1 in.4, Iy ϭ 10.9 in.4 Ix ϭ Iy ϭ 194.6 ϫ 106 mm4, rx ϭ ry ϭ 80.1 mm I1 ϭ 1480 in.4, I2 ϭ 186 in.4, r1 ϭ 7.10 in., r2 ϭ 2.52 in Ib ϭ 940 in.4 Ic ϭ 11a4/192 Ixc ϭ 7.23 in.4 I2 ϭ 405 ϫ 103 mm4 Ixc ϭ 6050 in.4, Iyc ϭ 475 in.4 Ixc ϭ 106 ϫ 106 mm4 Ixc ϭ 17.40 in.4, Iyc ϭ 6.27 in.4 b ϭ 250 mm IP ϭ bh(b2 ϩ 12h2)/48 (IP)C ϭ r 4(9a Ϫ sin2 a)/18a IP ϭ 233 in.4 IP ϭ bh(b2 ϩ h2)/24 (IP)C ϭ r 4(176 Ϫ 84p ϩ 9p 2)/[72(4 Ϫ p)] Ixy ϭ r 4/24 b ϭ 2r Ixy ϭ t 2(2b2 Ϫ t 2)/4 I12 ϭ Ϫ20.5 in.4 Ixy ϭ 24.3 ϫ 106 mm4 Ixcyc ϭ Ϫ6.079 in.4 Ix1 ϭ Iy1 ϭ b4/12, Ix1y1 ϭ bh(b4 ϩ h4) b3h3 Ix1 ϭ ᎏ Iy1 ϭ ᎏ ᎏ, ᎏ, 12(b2 ϩ h2) 6(b ϩ h ) b2h2(h2 Ϫ b2) ᎏ Ix1y1 ϭ ᎏ 12(b2 ϩ h2) 12.8-3 12.8-4 12.8-5 12.8-6 12.9-1 12.9-2 12.9-3 12.9-4 12.9-5 12.9-6 12.9-7 12.9-8 12.9-9 1015 Id ϭ 159 in.4 Ix1 ϭ 12.44 ϫ 106 mm4, Iy1 ϭ 9.68 ϫ 106 mm4, Ix1y1 ϭ6.03 ϫ 106 mm4 Ix1 ϭ 13.50 in.4, Iy1 ϭ 3.84 in.4, Ix1y1 ϭ 4.76 in.4 Ix1 ϭ 8.75 ϫ 106 mm4, Iy1 ϭ 1.02 ϫ 106 mm4, Ix1y1 ϭ Ϫ0.356 ϫ 106 mm4 (a) c ϭ ͙ෆ a2 Ϫ b2ෆ/2; (b) a/b ϭ ͙5 ෆ; (c) Յ a/b Ͻ ͙5 ෆ Show that two different sets of principal axes exist at each point up1 ϭ Ϫ29.87°, up2 ϭ 60.13°, I1 ϭ 311.1 in.4, I2 ϭ 88.9 in.4 up1 ϭ Ϫ8.54°, up2 ϭ 81.46°, I1 ϭ 17.24 ϫ 106 mm4, I2 ϭ 4.88 ϫ 106 mm4 up1 ϭ 37.73°, up2 ϭ 127.73°, I1 ϭ 15.45 in.4, I2 ϭ 1.89 in.4 up1 ϭ 32.63°, up2 ϭ 122.63°, I1 ϭ 8.76 ϫ 106 mm4, I2 ϭ 1.00 ϫ 106 mm4 up1 ϭ 16.85°, up2 ϭ 106.85°, I1 ϭ 0.2390b4, I2 ϭ 0.0387b4 up1 ϭ 74.08°, up2 ϭ Ϫ15.92°, I1 ϭ 8.29 ϫ 106 mm4, I2 ϭ 1.00 ϫ 106 mm4 up1 ϭ 75.73°, up2 ϭ Ϫ14.27°, I1 ϭ 20.07 in.4, I2 ϭ 2.12 in.4 Name Index A G P Andrews, E S., 940 (9-4) Galilei, Galileo, 5, 5n, 366n, 937–938 (5-2*, 5-3) Goodier, James Norman, 936 (2-1*) Parent, Antoine, 366n, 938 (5-3) Pascal, Blaise, 948n Pearson, Karl, 935 (1-2*) Pilkey, W D., 940 (9-10) Piobert, Guillaume, 135, 936 (2-5*) Poisson, Siméon Denis, 29, 936 (1-8*) Poncelet, Jean Victor, 18n, 157n, 935 (1-4*), 937 (2-8) B Bernoulli, Daniel, 939–940 (9-1*) Bernoulli, Jacob, 18n, 366n, 684n, 935 (1-4*), 937 (5-1), 938 (5-3), 939 (9-1) Bernoulli, Jacob (James), 939–940 (9-1*) Bernoulli, John, 939–940 (9-1*) Bredt, Rudolph, 276n, 937 (3-2*) Budynas, R., 937 (2-9) H Hauff, E., 940 (9-3) Hertz, Heinrich Rudolf, 948n Hoff, N J., 863, 942 (11-17, 11-18) Hooke, Robert, 27, 935 (1-6*) R J C Castigliano, Carlos Albert Pio, 733n, 940 (9-2*, 9-3, 9-4, 9-5, 9-6) Cauchy, Augustin Louis, 551n, 939 (7-1*) Celsius, Anders, 952n Clapeyron, Benoit Paul Emile, 142n, 936–937 (2-7*) Clebsch, Rudolph Frederich Alfred, 940 (9-8*) Considère, Armand Gabriel, 863, 942 (11-7*) Coulomb, Charles Augustin, de, 231n, 937 (3-1*), 938 (5-3) Crew, Henry, 937 (5-2) Culmann, Karl, 637n, 939 (8-1*) Jasinsky, Félix S., 863, 942 (11-10*) Johnston, B G., 863, 942 (11-19) Joule, James Prescott, 945n Jourawski, D J., 392n, 938 (5-7*, 5-8) S K Keller, J B., 941 (11-4) Kelvin, William Thomas, Lord, 952n Kuenzi, E W., 938 (5-10) E Elsevir, Louis, 937 (5-2) Engesser, Friedrich, 863, 942 (11-8*, 11-9, 11-11) Euler, Leonhard, 5, 109n, 366n, 684n, 828, 835n, 862, 936 (2-2*), 938 (5-3), 940–941 (11-1*, 11-2) Saint–Venant, Barré de, 166n, 231n, 366n, 551n, 937 (2-10*), 938 (5-3), 939 (7-2), 940 (9-8) Shanley, Francis Reynolds, 861, 862–863, 942 (11-15*, 11-16) L Lamarle, Anatole Henri Ernest, 862–863, 942 (11-6*) L’Hôpital, G F A de, 939 (9-1) Love, Augustus Edward Hough, 935 (1-3*) Lüders, W., 135, 936 (2-6*) D da Vinci, Leonardo, 5, 5n De Salvio, Alfonso, 937 (5-2) Didion, I., 936 (2-5*) Duleau, Alphonse, 231n, 937 (3-1) Ramberg, W A., 937 (2-12) Rankine, William John Macquorn, 551n, 939 (7-3*), 953n Roark, R J., 937 (2-9) T Timoshenko, Stephen P., 502n, 935 (1-1*), 936 (1-9), 938–939 (6-1), 941 (11-5) Todhunter, Isaac, 935 (1-2*) V M Macaulay, William Henry, 940 (9-7*) Maki, A C., 938 (5-10) Mariotte, Edme, 366n, 938 (5-3) Maxwell, James Clerk, 939 (7-4) McLean, L., 936 (2-3) Mohr, Otto Christian, 558n, 939 (7-4*, 7-5) Morin, A.-J., 936 (2-5*) Van den Broek, J A., 941 (11-3) von Kármán, Theodore, 863, 942 (11-12*, 11-13, 11-14) W Watt, James, 948n Williot, Joseph Victor, 939 (7-4) Y N Navier, Louise Marie Henri, 109n, 936 (2-4*), 937 (5-1), 938 (5-3) Newton, Isaac, Sir, 954n Young, Donovan Harold, 941 (11-5*) Young, Thomas, 28, 231n, 936 (1-7*), 937 (3-1) Young, W C., 937 (2-9) O Z Oravas, Gunhard A., 936 (2-3), 940 (9-2, 9-6) Osgood, W R., 937 (2-12) Zazlavsky, A., 937 (2-11), 938 (5-9), 940 (10-1) F Fahrenheit, Gabriel Daniel, 953n Fazekas, G A., 937 (5-1) Föppl, August, 940 (9-9*) Notes: Numbers in parenthesis are reference numbers An asterisk indicates a reference containing bibliographical information The letter n indicates material in a footnote 1016 Index Acceleration of gravity (g), 153, 947–948 Alternating (reverse) loads, 162–164 Aluminum, 21–22, 375, 868–869 beam design, 375 column design, 868–869 material properties of, 21–22 Aluminum Association, 375, 868 American Forest and Paper Association, 375 American Institute of Steel Construction (AISC), 375 Analysis of stress and strain, see Plane stress Angle of rotation ( or ), 254, 681–683, 711–712 Angle of twist (), 223–226, 229, 239, 275–276 bars, 223–226, 229, 239 per unit length (rate of) (), 224–225 thin-walled tubes, 275–276 Angle sections of beams, 500–501 Angular speed (), 254–255 Anisotropic materials, 29 Area (A), 8–9, 33, 93–94, 902–911, 966–971 bearing (Ab), 33 centroids of, 902–908 composite, 905–908 cross-sectional, 8–9 effective (metallic), 93–94 moment of inertia of, 909–911 plane, 902–904, 909–911, 966–971 symmetry of, 903 Axial force (N), 309, 412–413 Axial loads (P), 11–14, 49–54, 88–219, 412–417, 845–849 bars, 92–93, 100–115, 142–143 beams, 412–417 cables, 93–94 columns, 845–849 combined stresses from, 412 direct shear and, 49–54 dynamic loads (P) and, 153–164 eccentric (Pe), 413–414, 845–849 elastoplastic materials, 170–171, 175–180 elongation (␦) from, 91–106, 154–155, 171–172, 175–180 fatigue and, 162–164 impact loads (P) and, 153–161 inclined sections (), 128–139 length changes of, 91–106, 171–172 line of action for, 11–14 linearly elastic materials, 91–106, 142 misfits and, 124–127 neutral axis for, 414 nonlinear behavior, 170–175 nonuniform conditions and, 100–106 prestrains and, 124–127 prismatic bars, 11–14, 92–93 repeated loads (P) and, 162–164 springs, 91–92, 142 static loads (P) and, 140–152 statically determinate structures, 91–107 statically indeterminate structures, 107–115 strain energy (U) from, 140–152 stress concentrations, 164–169 structural design and, 49–54, 169 structural members, 88–219 thermal effects and, 116–123 Axial rigidity (EA), 92 Bars, 92–93, 100–115, 142–143, 171–172, 223–252 See also Prismatic bars angle of twist (), 223–226, 229, 239 axially loaded, 92–93, 100–115, 142–143, 171–172 circular, 223–252 elongation (␦) of, 92–93, 100–106, 171–172 linearly elastic, 142, 226–237 nonuniform, 100–106, 143–144, 238–244 pure shear and, 223–225, 245–252 segmented, 101, 238–244 statically determinate, 92–93, 100–107 statically indeterminate, 107–115 strain energy (U), 142–143 tapered, 101–102, 171–172 torsional deformations of, 223–252 Beam-columns, 848 Beams, 304–349, 350–453, 454–535, 635–644, 676–769, 770–815, 984–989 axial loads and, 412–417 bending, 351–354, 374–382, 418–420, 466–487, 504–513 built-up, 352, 408–411 cantilever, 306–307, 310–312, 773–775, 976–978 circular, 366, 376–377, 397–399 composite, 455, 457–466 cross sections of, 356–360, 361–373, 376–377, 387–399 curvature () of, 351, 353–356, 362–363, 679–684 deflection (v) of, 353–356, 676–769, 984–989 design of, 351, 374–382 drawing symbol conventions for, 306–307 designations of, 375–376 doubly symmetric, 455, 459, 461–462, 472–478 free-body diagrams (FBD), 309 fully stressed, 383, 386 idealized model of, 308 inclined loads and, 455, 472–478 linearly elastic, 361–373 loads (P) on, 308, 320–325 longitudinal displacements at ends of, 801–804 longitudinal strains (x) in, 356–360, 457–458 neutral axis, 357, 361–362, 414, 418–419, 459, 467–468, 479–481 nonprismatic, 383–388, 683, 720–742 plane stresses in, 635–644 principal stresses in, 637–640 prismatic, 683–684 reactions of, 308–313, 803 rectangular, 365, 376, 387–396, 635–639 shear center concept for, 455, 487–489 shear flow ( f ) and, 408–411 shear stresses () in, 387–407, 489–496, 637 shear-force and bending-moment diagrams for, 325–336 slopes of, 984–989 statically indeterminate, 770–815 stress concentrations in, 352, 418–420 stress resultants for, 305, 313–320, 361, 412–413 stresses () in, 350–453, 454–535 thin-walled open cross section, 489–492, 496–504 transformed-section method of analysis, 455, 466–471 unsymmetric, 455, 479–487, 488–489 webs of, 400–407, 494–495 wide-flange, 376–377, 400–407, 492–496, 639–640 Bearing stress (b), 32–34 Bending, 306, 308, 351–354, 374–382, 418–420, 454–535, 725–730 approximate theory of, 461–462 beams, 351–354, 374–382, 418–420, 454–535, 504–513 composite beams, 455, 457–466 deflections (v) by, 725–730 doubly symmetric beams, 455, 459, 461–462, 472–478, 488 elastoplastic, 504–513 flexure formula, 351–352, 364, 460, 468–469 inclined loads and, 455, 472–478 nonuniform, 353–354 plane of, 306, 308, 352–353 pure, 353–354 shear center (S), 455, 487–489, 496–504 strain energy (U ) of, 725–730 stress concentrations in, 418–420 stresses (normal), 364, 374–382, 454–535 thin-walled open cross-section beams, 489–492, 496–504 transformed-section method of analysis, 455, 466–471 unsymmetric beams, 455, 479–489 wide-flange beams, 376–377, 492–496 Bending moments (M ), 304–349, 351, 362–364, 374, 412–413, 472–473, 510, 685–695 axial force (N ) and, 309, 412–413 beams, 304–349, 351, 362–364, 412–413, 685–695 deflections (v) by integration of, 685–695 diagrams, 325–336 doubly symmetric beams, 472–473 elastoplastic materials, 510 loads and, 320–325 maximum (Mmax), 328, 330, 351, 374 moment-curvature relationship, 362–364 shear forces (V ) and, 304–349 sign conventions for, 314–315, 472 Biaxial stress, 545, 550, 566, 576, 578, 579, 582 Hooke’s law for, 576, 578, 582 plane stress and, 545, 550, 566, 576, 578, 579, 582 strain-energy density (u) for, 579 Bifurcation point, 822 Bolted connections, 31–35 Bolts, misfits and prestrains of, 125 Brittle materials, 22–23 Buckling, 891–823, 856–863 columns, 891–823, 856–863 1017 1018 INDEX Buckling (Continued ) critical loads for, 820–821, 828–829 differential equations for, 824–827, 835–837, 840, 845 elastic behavior, 856–858 equilibrium and, 822–824 Euler, 823–834 idealized structures, 819–823 inelastic, 856–863 reduced-modulus theory for, 860–861 Shanely theory for, 861–863 stress–strain diagrams for, 858–860 tangent-modulus theory, 858–860 Built-up beams, 352, 408–411 first (integral) moment (Q), 410 glulam (glued laminated), 408 plate girder, 408, 410 shear flow ( f ) in, 408–411 wide-flange, 410 wood box, 408, 410–411 Camber, 26 Cantilever beams, 306–307, 310–312, 773–775, 976–978 deflections and slopes of, 976–978 fixed support for, 307–308 propped, 773–775 reactions, 310–312 Castigliano’s theorem, 731–743 applications of, 733–736 deflections (v) by, 731–743 derivation of, 731–733 integral signs for, 736–738 Celsius (°C), unit of degrees, 116, 952–953 Center of curvature, 355 Centroid (C ), 362, 487–489, 900–933 composite areas, 905–908 moments of inertia and, 900–933 neutral axis and, 362 plane areas, 902–904 shear center (S ) and, 487–489 Channel sections of beams, 497–500 Circular beams, 366, 376–377, 397–399 design of, 376–377 hollow cross sections of, 398 linearly elastic, 366 shear stresses () in, 397–399 Circular members, 223–237, 254–258, 270–279 bars, 223–225, 226–229, 232–237 shafts, 254–258 tubes, 225–226, 230–231, 270–279 Circumferential (hoop) stress, 627–628 Columns, 816–899 allowable loads, 864–865 allowable stress, 43–48, 864–871 buckling, 819–844, 856–863 critical loads for, 820–821, 828–829 critical stress of, 830 design formulas for, 863–881 eccentric axial loads on, 845–849 effective length (L) of, 837–838 elastic behavior, 856–858 Euler buckling, 823–834, 870–871 inelastic behavior of, 831–832, 856–858 inelastic buckling, 856, 858–863 optimum shapes of, 832 pinned ends, 823–834 secant formula for, 850–855 slenderness ratio, 830, 851, 856–858, 870 stability of, 891–823 various support conditions for, 834–844 Combined loads, 645–660 critical points of, 647–648 plane stress and, 645–660 Compatibility, equations of, 107–115, 176, 259–260, 784 Compliance, see Flexibility ( f ) Composite areas, 905–908 Composite beams, 455, 457–466 bending, approximate theory of, 461–462 doubly symmetric cross sections of, 459, 461–462 strains () and stresses () in, 457–459 Compression, 17–18, 23–24, 91, 134–135, 144 axially loaded members, 91, 144 maximum stresses in, 134–134 strain energy (U) and, 144 stress–strain diagrams, 23–24 tests, 17–18 Concentrated loads, 308, 324, 325–327, 328–330, 727 deflection (v) and, 727 diagrams for, 325–327, 328–330 several acting on beams, 328–330 shear-force and bending-moments of, 324 Conservation of energy, principle of, 154 Continuous beams, 776 Coordinate axes, 353 Couples, 36, 222, 308, 325 bending moments, 325 loading moments, 308, 325 shear forces (magnitude) of, 36, 325 torsional moment of, 222 Creep, 26 Critical loads, 820–821, 828–829, 842 Critical points, 647–648 Cross sections, 7–8, 92, 270–273, 356–360, 361–373, 376–377, 387–399, 459, 461–462, 489–492 See also Neutral axis; Shear center beams, 356–373, 376–377, 387–399, 459, 461–462, 489–492 centroid, 362 circular beams, 366, 376–377, 397–399 composite beams, 459, 461–462 doubly symmetric, 365–366, 459, 461–462 element, 388–389 hollow circular, 398 ideal shapes, 377 linearly elastic materials, 361–373 median line, 272–273 neutral axis, 357, 361–362, 459 neutral surface, 357–358 normal stresses () and, 361–373 rectangular beams, 365, 376, 388–391, 387–396 section moduli, 365 structural members, 92 subelement, 389–391 thin-walled beams, 489–492 thin-walled tubes, 270–273 Curvature () of, 351, 353–356, 358, 362–363, 679–684, 801–803 See also Deflection beams, 351, 353–356, 358, 362–363, 679–684, 801–803 center of, 680 deflection curve for, 353–354, 679–684 equations for, 355, 684 moment-curvature relation, 351, 362–363 moment of inertia and, 363 radius () of, 355, 680 shortening, 801–803 sign convention for, 356, 680–681 small deflections, 355–356 strain-curvature relation, 358 Cylindrical pressure vessels, 627–635 Deflection (v), 353–356, 676–769, 770–815, 816–899, 984–989 See also Buckling; Curvature angle of rotation () and, 681–683, 711–713 beams and, 353–356, 676–769, 770–815, 984–989 bending-moment equation, integration of, 685–695 boundary conditions, 685 Castigliano’s theorem, 731–743 columns and, 816–899 concentrated loads and, 727 continuity conditions, 686 curve, 351, 353–354, 679–684, 713–716, 777–783 differential equations for, 679–684, 748, 777–783, 824–827, 835–837, 840, 845 impact loads and, 744–746 maximum, 846–848 moment-area method, 711–719 nonprismatic beams, 683, 720–742 prismatic beams, 683–684 shear-force (V) and load (q) equations, integration of, 696–701 sign conventions for, 682–683 slope of curve, 681, 984–989 statically indeterminate beams, 770–815 strain energy (U) of bending, 725–730 successive integrations, method of, 686 superposition, method of, 702–710, 784–796, 798–799 symmetry conditions, 686 temperature effects on, 746–748, 799–700 Deformation sign conventions, 314–315 Differential equations of deflection (v), 679–684, 748, 777–783, 799–800, 824–827, 835–837, 840, 845 column buckling, 824–827, 835–837, 840, 845 constants of integration, 826–827 statically determinate beams, 679–684, 748 statically indeterminate beams, 777–783, 799–800 temperature effects and, 748, 799–800 Dimensionless quantities, 10–11 Direct shear, 35, 49–54 Displacements, 91–106, 117–123, 140–152, 175–180, 221, 259–262, 784–796 See also Bending; Deflection; Elongation (␦) diagrams, 97–99 elastoplastic analysis and, 175–180 force (P) relations, 107–115, 176–178, 784–796 length changes in axially loaded members, 91–106 load-displacement diagrams, 140–142 plastic (␦P), 177–180 single-load, 144–145 strain energy (U) and, 140–152 superposition, method of for, 784–796 temperature (T ) relations, 117–123 torque (T ) relations, 221, 259–262 yield (␦Y), 175–180 Distributed loads, 11–12, 308, 321–324, 327–328 beams and, 308, 321–324 bending moments of, 323–324 line of action for, 11–12 shear force of, 321–323 uniform, 11–12, 308, 327–328 Doubly symmetric beams, 455, 459, 461–462, 472–478, 488 bending moments (M), 472–473 bending of, 461–462 bending stresses in, 472–473 cross sections of, 459, 461–462 inclined loads and, 472–478 neutral axis of, 459, 473–474 shear center of, 488 Ductile materials, 21 Dynamic loads, 153–164 Dynamic test, 17 Eccentric axial loads (Pe), 413–414, 845–849 Eccentricity ratio, 851 Effective length (L), 837–838, 842 Effective modulus, 94 Elastic core of beams, 505, 509–510 Elastic limit, 25 Elasticity (E), 19, 24–25, 38, 27–28, 94, 252–253, 584, 992 See also Yielding cables and, 94 material properties of, 24–25 modulus of (E), 19, 27–28, 38, 94, 992 rigidity (G), relationship to, 252–253 shear and, 38 volume (bulk) modulus of (K), 584 Elastoplastic materials, 170–171, 175–180, 504–513 analysis for, 171, 175–180 axially loaded members of, 170–171, 175–180 bending of beams, 504–513 force-displacement relations, 176–178 INDEX load-displacement diagrams, 175–178 neutral axis of, 505–507 plastic displacement (␦P), 177–180 plastic modulus (Z), 508–510 plastic moment (MP), 505–508 shape factor ( f ), 508–509 stress–strain diagrams for, 170–171 yield displacement (␦Y), 175–180 yield moment (MY), 505, 508–509 Elongation (␦), 91–106, 117–123, 154–155, 171–172, 175–180 See also Displacement axially loaded members, 91–106, 154–155, 171–172, 175–180 bars, 92–93, 100–106, 171–172 cables, 93–94 conservation of energy, principle of, 154 elastoplastic analysis, 175–180 force-displacement relations and, impact loading and, 154–155 linearly elastic materials, 91–99 maximum (␦max), 154–155 nonlinear behavior and, 171–172 nonuniform conditions and, 100–106 plastic displacement (␦P), 177–180 prismatic bars, 92–93 sign conventions for, 92–93 springs, 91–92 temperature-displacement relations and, 117–123 yield displacement (␦Y), 175–180 Endurance limit, 163–164 Energy, see Strain Energy (U) Equilibrium, 35–36, 107–115, 176, 259, 784, 822–824 columns, 822–824 equations of, 107–115, 176, 259, 784 neutral, 822, 824 shear stresses () on perpendicular planes, 35–36 stable, 822–824 statically indeterminate structures and, 107–115, 176, 259, 784 unstable, 822, 824 Euler buckling, 823–834, 870–871 Euler load, 828, 865 Euler’s curve, 830, 852 Extensometer, 16 Factors of safety (n), 43–44, 858 Fahrenheit (°F), unit of degrees, 116, 953 Fatigue, 162–164 Filament-reinforced materials, 23 Fillets, stress-concentration factors for, 168, 279–281 Fixed-end beams, 307–308, 775 Flexibility ( f ), 91–93, 229 prismatic bars, 92–93 springs, 91–92 torsional ( fT), 229 Flexural rigidity (EI), 351, 363, 460 Flexure formula, 351–352, 364, 460, 468–469, 635 bending stresses and, 351–352 composite beams, 460 linearly elastic beams, 364 stress analysis using, 635 transformed-section method of analysis, 468–469 Fluctuating loads, 153, 162 Force-displacement relations, 107–115, 176–178, 784–796 Force reaction (RB) redundants, 784–786 Free-body diagrams (FBD), 7–8, 32–35, 50–54, 309 Frequency of rotation ( f ), 255, 948 Gage length, 16–17 Gage pressure, 621 Glass, material properties of, 23 Glass fibers, material properties of, 23 Glulam (glued laminated) beam, 408 Hertz (Hz), unit of, 255, 948 Holes at neutral axis of beams, 418–419 Hollow circular cross sections, 398 Homogeneous material, 11, 29 Hooke’s law, 27–28, 38, 221, 226–227, 537, 575–580, 581–582 biaxial stress, 576, 578, 582 linear elasticity and, 27–28 modulus of elasticity (E), 27–28, 38 modulus of rigidity (G ), 38 plane stress and, 537, 575–580, 581–582 shear, in, 38, 226–227, 575–576 strain-energy density (u) and, 578–580 torsional deformation and, 221, 226–227 triaxial stress and, 581–582 uniaxial stress and, 576, 578 volume change and, 577–578 Horsepower (hp), unit of, 255 Hydrostatic stress, 584 I-beams, 375, 487 Impact factor, 156–157 Impact loads, 153–162, 744–746 deflections (v) by, 744–746 maximum elongation (␦max), 154–155 maximum shear (max), 153 maximum stress (max), 155–156 suddenly applied, 157 Inclined loads (P), doubly symmetric beams with, 472–478 Inclined sections (), 128–139, 246–250, 539, 541–547 maximum shear (max), 133–135 maximum stress (max), 133–135 orientation of, 130–131 plane stress and (), 539, 541–547 planes, 246–250 shear () on, 132–135, 246–250 sign conventions for, 132, 246 strains () on, 249–250 stresses () on, 128–139, 246–249 torsional deformation and, 246–250 uniaxial stress on, 135 Inertia (I ), 228, 230, 363, 900–933 moment-curvature relationship, 363 moments of, 363, 900–933 parallel-axis theorem for, 912–915, 918–920 plane areas, 909–911 polar (Ip) moments of, 228, 230, 916–917 principal moments of, 923–926 products of, 918–920 radius of gyration (r) and, 910 rotation of axes and, 921–922 Inner-surface stresses, 624, 629–630 Integral (first) moment (Q), 391, 410 Integral signs, differentiation using, 736–738 International System (SI) of units, 375, 944–951 Isotropic materials, 29 Joule (J), unit of, 141, 145, 263 Kelvin (K), unit of, 116, 952 Kinetic energy, 153–154 Lateral contraction, 20, 28 Length (L), 91–106, 171–172, 837–838 See also Elongation (␦) axially loaded members, changes of in, 91–106, 171–172 compression and, 91 effective, 837–838 linearly elastic materials, 91–106 natural, 91 nonlinear behavior and, 171–172 nonuniform conditions and, 100–106 tension and, 91 Line of action, 11–12 Linearly elastic materials, 27–31, 91–106, 142–146, 226–237, 361–373 angle of twist (), 229 axially loaded members, 91–106, 142–146 cross-sectional beam stresses, 361–373 doubly symmetric cross sections, 365–366 flexibility (f ), 91–93, 229 1019 Hooke’s law, 27–28 226–227 moment-curvature relationship, 362–363 normal stresses () and, 361–373 Poisson’s ratio, 28–29 properties of, 27–31 stiffness (k), 91–93, 229 strain energy (U), 142–146 torsional deformation of, 226–237 Load-deflection diagrams, 847–848 Load-displacement diagrams, 140–142, 175–178 Load tests, 858 Loads (P), 43–48, 49–54, 88–219, 308, 320–336, 412–417, 472–478, 487–489, 645–660, 727, 744–746, 820–821, 828–829, 845–849, 864–865, 948–949 allowable, 45–48, 864–865 alternating, 162 axial, 49–54, 88–219, 412–417 beams and, 308, 320–325, 412–417, 472–478, 487–489 columns, 820–821, 828–829, 845–849, 864–865 combined, 645–660 concentrated, 308, 324, 325–327, 328–330, 727 couple (moment), 308, 325 critical, 820–821, 828–829 deflection (v) and, 696–701, 727, 744–746 distributed, 308, 321–324 dynamic, 153–164 eccentric axial (Pe), 413–414, 845–849 equations of shear-force and intensity, 645–660 factors of safety, 43–44 fluctuating, 153, 162 impact, 153–162, 744–746 inclined, 472–478 intensity (q), 308, 645–660 linearly varying, 308 one-directional, 162 plane stress and, 645–660 repeated, 162–164 shear center concept using, 487–489 shear-force and bending-moment diagrams for, 325–336 sign conventions for, 320–321 single displacements, 144–145 static, 140–153 strain energy (U ) and, 144–145 structural design and, 50 uniform, 308, 327–328 units of, 948–949 yield (PY), 175–180 Longitudinal (axial) stress, 628–629 Longitudinal strain (x), 356–360, 457–458 Lüders’ bands, 135 Magnitude, 36, 153 Margin of safety, 44 Mechanics of materials, 2–87, 956–944 deflections, 984–989 dimensional homogeneity, 958–959 linearly elastic materials, 27–31 mathematical formulas for, 962–965 numerical problems in, 6, 956–961 properties of materials, 15–31, 972–983, 990–994 rounding numbers for, 961 shear, 32–42, 49–54 significant digits, 959–961 strain (), 7, 10–11 stress (), 7–10, 11–14, 43–48 stress–strain diagrams, 17–24 structural design, 49–54 symbolic problems in, 6, 956–959 tests for, 15–18 Median line, 272–273 Membrane stresses, 622 Misfits, 124–127 Modulus of elasticity (E), 19, 27–28, 38, 94, 252–253, 992 Modulus of resilience (ur), 145–146 Modulus of rigidity (G), 38, 252–253, 992 1020 INDEX Modulus of toughness (ut), 146 Mohr’s circle, 558–574, 581, 592–593 construction of, 560–562 equations of, 558–559 inclined stress elements and, 562–564 maximum shear stresses (max) and, 565 plane strain and, 592–593 plane stress and, 588–574 principal stresses and, 564–565 triaxial stress and, 581 Moment-area method of deflection, 711–719 angle of rotation (), 711–713 tangential deviation, 713–715 Moment-curvature relationship, 351, 362–363, 460, 468 bending moments, 362–363 composite beams, 460 flexural rigidity (EI ), 351, 363, 460 transformed-section method of analysis, 468 Moment reaction (MA) redundants, 786–787 Moments, 222, 304–349, 391, 410, 505–510, 943 See also Bending moments (M); Inertia (I) couple, of a, 222 elastoplastic bending and, 505–510 integral (first) (Q), 391, 410 plastic (MP), 505–508 twisting, 222 yield (MY), 505, 508–509 Necking, 20 Neutral axis, 357, 361–362, 414, 418–419, 459, 467–468, 473–474, 479–481, 505–507 composite beams, 459 doubly symmetric beams, 459, 473–474 eccentric axial loads (Pe) and, 414 elastoplastic bending and, 505–507 holes at, 418–419 inclined loads, relationship to, 473–474 linearly elastic beams, 361–362 stress concentrations in bending at, 418–419 transformed-section method of analysis, 467–468 unsymmetric beams, 479–481 Neutral surface, 357–358 Nonlinear behavior of axially loaded members, 170–174 elastoplastic materials, 170–171 elongation (␦) and, 171–172 Ramberg-Osgood equation for, 172–174 statically indeterminate structures and, 173 stress–strain diagrams for, 170–173 Notches, rectangular beams with, 419–420 Offset method, 21–22 Orientation of inclined sections, 130–131 Overhangs, beams with, 306–307, 312–313 Parallel-axis theorem, 912–915, 918–920 Partially elastic state, 25 Pascals (Pa), unit of, 145, 228, 266, 948 Percent elongation, 22 Percent reduction in area, 22 Permanent set, 25 Pin support, simply supported beams, 306–307 Pinned-end columns, 823–834 Piobert’s bands, 135 Pitch of threads, 95, 125 Plane areas, 902–904, 909–911, 966–971 centroid (C ) of, 902–904 moment of inertia of, 909–911 properties of, 966–971 Plane of bending, 306, 308, 352–353 Plane strain (), 584–599 calculation of stresses from, 594 maximum shear strains (␥max), 592 measurements of, 593–594 Mohr’s circle for, 592–593 plane stress versus, 585–586 principal strains, 591–592 transformation equations for, 587–591 Plane stress (), 536–617, 618–675 analysis of stress and strain as, 536–617 applications of, 618–675 beams and, 635–644 biaxial stress and, 545, 550, 566, 576–579, 582 combined loads and, 645–660 Hooke’s law for, 537, 575–582 inclined sections (), 539, 541–547 maximum shear stresses (max), 552–554, 565, 580–581 Mohr’s circle for, 558–574 plane strain and, 584–599 pressure vessels and, 621–635 principal stresses, 548–557, 564–565 pure shear and, 545, 551, 566, 576, 580 spherical stress, 583–584 strain-energy density (u), 578–580, 582–583 transformation equations for, 539, 543–544 triaxial stress, 537–538, 580–584 uniaxial stress and, 544–545, 550, 566, 576–579 unit volume change and, 577–578, 582 Planes, 35–36, 246–250 inclined (), 246–250 perpendicular, 35–36 shear strains (␥) on inclined, 249–250 shear stresses (), equality of on, 35–36 stresses (), on inclined, 246–249 Plastic, 20, 23, 25–26, 170, 177–180, 505–510 See also Elastoplastic materials beams, elastoplastic bending of, 505–510 displacement (␦P), 177–180 flow, 25 loads (PP), 177–180 material properties of, 22, 25–26 modulus (Z), 508–510 moment (MP), 505–508 perfectly, state of, 20, 170 Plate girder, 408, 410 Poisson’s ratio, 28–29 Polar moment of inertia (IP), 228, 230, 916–917 Potential energy, 144, 153 Power transmitted by shafts, 222, 254–258 Pressure vessels, 621–635 circumferential (hoop) stress, 627–628 cylindrical, 627–635 gage pressure, 621 inner-surface stresses, 624, 629–630 longitudinal (axial) stress, 628–629 outer-surface stresses, 623–624, 629 plane stress and, 621–635 spherical, 621–627 tensile stresses, 622 Prestrains, 124 Prestresses, 124 Principal angles, 548, 550 Principal axes, 923 Principal moments of inertia, 923–926 Principal point, 924 Principal strains, 591–592 Principal stresses, 548–557, 564–565, 637–640 beams, 637–640 eigenvalue analysis for, 551 in-plane, 551–554 maximum shear (max), 552–554 Mohr’s circle for, 564–565 out-of-plane, 554 plane stress (), 548–557, 564–565, 637–640 stress contours for, 638 stress trajectories for, 637–638 Prismatic bars, 7–14, 92–93, 100–101, 223–226, 238–239 angle of twist (), 223–226, 239 axial loads and, 11–12, 92–93, 100–101 cross sections of, 7–8 elongation (␦), 92–93, 100–101 nonuniform axial loads, 100–101 normal stress and strain in, 7–14 segmented, 101, 239 torsional deformation of, 223–226, 238–239 uniform stress distribution of, 7–14 Progressive fracture, 162–164 Properties of materials, 15–31, 972–983, 990–994 aluminum alloys, 21–22 creep, 26 elasticity, 24–26 Hooke’s law for, 27–28 linear elasticity, 27–31 lumber (structural), 983 mechanical, 15–24, 993–994 plasticity, 25–26 Poisson’s ratio for, 28–29, 992 stress–strain diagrams for, 17–24 structural-steel, 19–21, 972–982 thermal expansion (␣), 994 weight (␥) and mass densities (), 991 Proportional limit, 19–20 Pure shear (␥), 34, 223–225, 245–252, 263–270, 545, 551, 566, 576, 580 bars and, 223–225, 245–252 plane stress and, 545, 551, 566, 576, 580 strain () in, 249–250 strain energy (U) and, 263–270, 580 stresses () in, 245–249 torsion and, 223–225, 245–252, 263–270 Radius of curvature (), 355, 680 Radius of gyration (r), 830, 851, 910 Ramberg-Osgood equation, 172–174 Reactions in beams, 50, 308–313, 803 horizontal, 803 releases and, 309–313 structural design and, 50 Rectangular beams, 365, 376, 387–396, 419–420, 508–510, 635–639 cross sections of, 365, 388–391 design of, 376 elastoplastic bending of, 508–510 integral (first) moment (Q), 391 notches in, 419–420 plane stresses in, 635–639 shear strains (␥), effects of on, 393 shear stresses () in, 387–396 stress concentrations in bending of, 418–419 Rectangular tubes, 270–279 Reduced-modulus theory for column buckling, 860–861 Redundants, 774, 784–787 force reaction (RB), 784–786 moment reaction (MA), 786–787 static, 774 superposition, method of using, 784–787 Reinforced concrete beams, 462 Relaxation of materials, 26 Release (primary) structures, 774 Reloaded materials, 25–26 Repeated loads, fatigue and, 162–164 Residual strain, 24–25 Restoring moment, 820 Revolutions per minute (rpm), unit of, 255, 948 Right-hand rule, 11, 222 Rigidity, 38, 229, 252–253, 276, 351, 363, 460, 992 axial (EA), 92 elasticity (E), relationship to, 252–253 flexural (EI), 351, 363, 460 modulus of (G), 38, 252–253, 992 torsional (GIP and GJ), 229, 276 Roller support, simply supported beams, 307 Rotation, 254–255, 681–683, 711–712, 838–839, 921–922 angles of ( or ), 254, 681–683, 711–712 axes, 921–922 columns fixed against, 838–839 frequency of ( f ), 255 moments of inertia and, 921–922 Rubber, material properties of, 22 Sandwich beams, 457–458, 461–462 Secant formula, 850–855 INDEX Section moduli (S), 365, 374, 376–382 Shafts, power transmitted by, 222, 254–258 Shanely theory, 861–863 Shape factor ( f ), 508–509 Shear, 32–42, 49–54, 132–135, 223–227, 245–252, 263–273, 387–407, 489–496, 540–541, 545, 550, 552–554, 565–566, 580–581 See also Pure shear beams and, 387–407, 489–496 bearing stress (b), 32–34 bolted connections, 31–35 direct, 35, 49–54 distortion, 249 double, 33 equilibrium of, 35–34 formula, 388–391 hollow circular cross-sections, 398 Hooke’s law in, 38, 226–227, 575–576 inclined sections (), on, 132–135, 246–250 internal strain (␥), 225 maximum strains (␥max), 592 maximum stress (max), 133–135, 402–403, 552–554, 565, 580–581 minimum stress (min), 402–403 Mohr’s circle for, 565–566 outer-surface strain (␥), 223–225 plane stress and, 540–541, 545, 550, 552–554, 565–566, 580–581, 592 sign conventions for, 37–38, 245, 541, 565 single, 34 strain-energy density (u) in, 265–266 strains (␥), 37, 223–226, 249–250, 393 stresses (), 32–42, 245–249, 270–272, 387–407, 489–496, 450–541, 550 stress–strain diagrams, 38 thin-walled tubes and, 270–272 torsional deformation and, 223–227, 245–252, 263–273 triaxial stress and, 580–581 units of, 32, 38 webs of wide-flange beams, 400–407 Shear center (S), 455, 487–489, 496–504 angle sections, 500–501 beam cross sections and, 455, 487–489 centroid (C) and, 487–489 channel sections, 497–500 intersecting narrow rectangles, 501–502 symmetric cross sections, 488 thin-walled open cross sections, 489, 496–504 unsymmetrical cross sections, 488–489 Z-sections, 502 Shear flow ( f ), 270–273, 408–411, 492 built-up beams and, 408–411 thin-walled open cross-section beams and, 492 thin-walled tubes and, 270–273 Shear forces (V ), 33–35, 304–349, 403–404, 696–701 beams, 304–349, 403–404 bending moments (M) and, 304–349 deflections by integration of, 696–701 diagrams, 325–336 free-body diagrams, 33–35 loads and, 320–325, 696–701 sign conventions for, 314–315 webs (Vweb) in wide-flange beams, 403–404 Shell structures, see Pressure vessels Shoulders, stress-concentration factors for, 168 Simply supported (simple) beams, 306–307, 309–310, 979–981 deflections and slopes of, 979–981 reactions, 309–310 supports for, 306–307 Skew directions, 581 Slender beams, 415 Slenderness ratio, 830, 851, 856–858, 870 Slip bands, 135 Slopes and deflections of beams, 681, 984–989 S-N (endurance curve) diagrams, 163–164 Spherical pressure vessels, 621–627 Spherical stress, 583–584 Spring constant, 91–92 Springs, 91–92, 142 Static loads (P), 140–141, 153 Static sign conventions, 315 Static test, 17 Statically determinate structures, 91–107, 118, 124, 175 axial loads (P) and, 91–107 elastoplastic analysis, 175 misfits and, 124 thermal effects, 118 Statically indeterminate structures, 107–115, 118–123, 124–125, 173, 175–180, 259–262, 770–815 axial loads (P) and, 107–115 bars, 107–115 beams, 770–815 curvature shortening, 771, 801–803 deflections of, 770–815 degree of determinacy, 771, 774 differential equations of deflection curve for, 679–684, 748, 777–783, 799–800 elastoplastic analysis of, 175–180 equations of compatibility, 107–115, 176, 259–260, 771 equations of equilibrium, 107–115, 176, 259, 771 force-displacement relations of, 107–115, 176–178, 771, 784–796 longitudinal displacements of, 801–804 misfits and 124–125 nonlinear behavior and, 173 superposition, method of for, 784–796, 798–799 temperature effects on, 771, 797–800 thermal effects on, 118–123 torque-displacement relations of, 259–262 Stiffness (k), 49–50, 91–93, 229 linearly elastic materials, 91–93, 229 prismatic bars, 93 spring constant, 91–92 structural design and, 49–50 torsional (kT), 229 Stocky beams, 415 Strain (), 7, 10–11, 17–18, 28, 116–117, 175, 223–225, 249–250, 356–360, 457–459, 536–617 See also Plane strain; Shear analysis of, 536–617 axially loaded members and, 116–117, 175 beams and, 356–360, 457–459 inclined planes and, 249–250 lateral (Ј), 28 longitudinal (x), 356–360, 457–458 nominal, 17–18 normal (), 7, 10–11, 357–358 pure shear, in, 249–250 sign conventions for, 116 thermal (T), 116–117 torsional deformation and, 223–225, 249–250 true, 17 uniaxial, 11 yield (Y), 175 Strain-curvature relationship, 351 Strain energy (U), 140–152, 263–270, 274–275, 578–580, 582–583, 725–730 axially loaded members, 140–152 biaxial stress and, 579 bending, by, 725–730 density (u), 145–146, 265–266, 578–580, 582–583 deflections (v) and, 725–730 elastic, 141 inelastic, 141 linearly elastic behavior, 142–146 load-displacement diagrams, 140–142 load displacements and, 144–145 modulus of resilience (ur), 145–146 modulus of toughness (ut), 146 nonuniform bars and, 143–144, 263–264 nonuniform torsion and, 263–264 plane stress and, 578–580, 582–583 pure shear (␥) and, 265–270, 580 1021 static loads (P) and, 140–152 thin-walled tubes (U), 274–275 torsion and, 263–270, 274–275 triaxial stress and, 582–583 uniaxial stress and, 145–146, 579 units of, 141, 145, 266 work (W) and, 140–141, 263 Strain gage, 593–594 Strain hardening, 20 Strain rosette, 594 Strength, 20, 43–48, 49 allowable loads, 45–46 allowable stresses, 44–45 factors of safety, 43–44 structural design and, 49 Stress (), 7–14, 17–18, 43–48, 116–123, 128–139, 175, 245–249, 350–453, 454–535, 536–617, 618–675, 864–871 See also Flexural formulas; Plane Stress; Principal stresses; Shear allowable, 43–48, 864–871 analysis of, 536–617 axially loaded members and, 116–123, 128–139, 175 beams and, 350–453, 454–535, 635–644 bending (normal), 364, 374–382, 454–535 biaxial, 545, 550 columns and, 830, 864–871 compressive, contours, 638 critical, 830 element, 129, 539 factors of safety (n), 43–44 impact loading and, 155–156 inclined sections (), on, 128–139, 246–249 linearly elastic beams, 361–373 maximum (max), 133–135, 155–156, 364–365, 635–644 nominal, 17–18 normal, 7–14, 540 pressure vessels, 621–635 pure shear, in, 245–246, 551 sign conventions for, 8, 132, 246, 358, 540 tensile, 8, 622 thermal, 116–118 torsional deformation and, 245–249 trajectories, 637–638 true, 17 ultimate, 20 uniaxial (plane), 11, 135, 145–146, 545, 550 uniform distribution, 11–12 units of, 8–9 yield (Y), 20–22, 175 Stress concentrations, 9–10, 162, 164–169, 279–281, 418–420 axially loaded members, 162, 164–175 beams, 418–420 bending and, 418–420 factors (K), 166–168, 279–281, 420 fatigue and, 162 fillets, 162, 279–281 Saint-Venant’s principle, 164–166 stress raisers, 164 structural design for, 169 torsion, in, 279–281 Stress resultants, 305, 313–320, 361, 412–413 axial loads and, 412–413 beams, 313–320, 361, 412–413 deformation sign conventions for, 314–315 internal, 305, 313–320 static sign conventions for, 315 Stress–strain diagrams, 17–26, 35, 38, 170–173, 858–860 bilinear, 171 compression and, 23–24 curves, 17–24 elastoplastic materials and, 170–171 idealized curves, 170 inelastic buckling of columns, 858–860 1022 INDEX Stress–strain diagrams (Continued ) loaded materials, 17–24 modulus of elasticity (E), 19 nonlinear behavior, 170–173 properties of materials and, 17–26 proportional limit, 19–20 Ramberg-Osgood equation, 172–173 shear, 35 tangent-modulus theory, 858–860 tension and, 17–23 unloaded materials, 24–26 Structural analysis, 49 Structural design, 49–54 Structural steel, 19–21, 375, 865–867, 972–982 beam design, 375 column design, 865–867 material properties of, 19–21 properties of shapes of, 972–982 Superposition, method of, 702–710, 784–796, 798–799 deflection by, 702–710, 784–796 force reaction (RB) redundants for, 784–786 moment reaction (MA) redundants for, 786–787 statically determinate beams, 702–710 statically indeterminate beams, 784–796, 798–799 temperature differential and, 798–799 Tangential deviation, 713–715 Tangent-modulus theory for column buckling, 858–860 Temperature-displacement relations, 117–123 Temperature effects on deflection (v), 746–748, 797–800 See also Thermal effects Temperature scales (units) of, 952–953 Tensile test, 15–17 Tension, 91, 133–134 Tensors, 539 Thermal effects, 116–123 coefficient of thermal expansion (␣), 109 sign convention, 116–117 stress () and strain (T) and, 116–123 temperature-displacement relations, 117–123 Thermal expansion, coefficient of (␣), 116 Thin-walled open cross-section beams, 489–492, 496–504 angle sections, 500–501 channel sections, 497–500 intersecting narrow rectangles, 501–502 shear center of, 489, 496–504 shear flow ( f ) of, 492 shear stresses () in, 489–492 Z-sections, 502 Thin-walled tubes, torsion and, 270–279 See also Tubes Torque (T ), 222, 227–230, 238–245, 948 distributed, 240 internal (Ti), 238–239, 264 nonuniform torsion and, 238–245 torsional formula and, 227–230 Torque-displacement relations, 221, 259–262 Torsion, 220–303 angle of twist (), 223–226, 229, 239, 275–276 circular members, 223–237, 254–258, 270–279 constant (J), 274–275 deformations, 223–237 elasticity, moduli (E and G) of, 252–253 formula, 221, 227–229, 272–273 Hooke’s law, 221, 226–227 nonuniform, 221, 238–244 power transmitted by, 222, 254–258 pure shear and, 223–224, 245–252, 263–270 statically indeterminate members, 221, 259–262 strain energy (U), 263–270, 274–275 stress concentrations in, 279–281 thin-walled tubes and, 270–279 tubes, 225–226, 230–231, 270–279 uniform, 221 units of, 222, 228 Torsional flexibility ( fT), 229 Torsional rigidity (GIp), 229, 276 Torsional stiffness (kT), 229 Trail-an-error procedure for columns, 864–865 Transformation equations, 539, 543–544, 587–591 application of, 587 plane strain (), 587–591 plane stress (), 539, 543–544, 587 Transformed-section method of analysis, 455, 466–471 bending stresses, 468–471 moment-curvature relationship for, 468 neutral axis and, 467–468 Triaxial stress, 537–538, 580–584 Hooke’s law for, 581–582 maximum shear stresses (max), 580–581 plane stress and, 537–538, 580–584 spherical stress, 583–584 strain-energy density (u), 582–583 unit volume change and, 582 Tubes, 225–226, 230–231, 270–279 angle of twist () for, 225, 275–276 linearly elastic, 230–231 shear flow ( f ) in, 270–273 shear strains (␥) in, 225–226 shear stresses () in, 270–272 strain energy (U), 274–275 thin-walled, 270–279 torsion constant (J), 274–275 torsional deformation of, 225–226, 230–231, 270–279 Turnbuckles, misfits and prestrains of, 125 Twisting moments, see Torque U.S Customary System (USCS), 375–376, 951–952 Uniaxial stress, 11, 135, 145–146, 544–545, 550, 566, 576, 578, 579 Hooke’s law for, 576, 578 inclined sections and, 135, 544–545 plane stress and, 544–545, 550, 566, 576, 578, 579 strain-energy density (u) and, 145–146, 579 Uniform (distributed) loads, 11–12, 308, 327–328 beams and, 308 line of action for, 11–12 shear-force and bending-moment diagrams, 328–330 Units, 375–376, 943–955 beam section designations, 375–376 conversions between, 953–955 International System (SI), 375, 944–951 systems of, 943–944 temperature scales, 952–953 U.S Customary System (USCS), 375–376, 951–952 Unsymmetric beams, 455, 479–487, 488–489 analysis of, 481–487 neutral axis of, 479–481 shear center of, 488–489 Vectors as representation of moments, 222 Velocity (v), 153 Volume, 577–578, 582, 584 bulk modulus of elasticity (K), 584 change (dilatation), 577–578, 582 plane stress and, 577–578 triaxial stress and, 582, 584 Watts (W), unit of, 255, 943 Web shear force (Vweb), 403–404 Webs in wide-flange beams, 377, 400–407, 494–495 Wedge-shaped stress elements, 542–543 Wide-flange beams, 376–377, 400–407, 410, 492–496, 510, 639–640 built-up, 410 design of, 376–377 lower flanges of, 495 plane stresses in, 639–640 plastic modulus (Z), 510 shear stresses () in, 400–407, 492–496 upper flanges of, 493–494 webs of, 377, 400–407, 494–495 Wood, 375–376, 408, 410–411, 869–872, 983 beam design, 375–376 column design, 869–872 box beam, 408, 410–411 lumber (structural), properties of, 983 Work (W ), 140–141, 263, 948, 952 Yield displacement (␦Y), 175–180 Yield load (PY), 175–180 Yield moment (MY), 505, 508–509 Yield point, 20 Yield strain (Y), 175 Yield stress (Y), 20–22, 175 Yielding, 19–21, 175–180 strength, 20 stress–strain diagrams for, 19–21 elastoplastic analysis and, 175–180 Young’s modulus, 28 Z-sections of beams, 502 PRINCIPAL UNITS USED IN MECHANICS International System (SI) U.S Customary System (USCS) Quantity Unit Symbol Formula Unit Symbol Formula Acceleration (angular) radian per second squared rad/s2 radian per second squared rad/s2 Acceleration (linear) meter per second squared m/s2 foot per second squared ft/s2 Area square meter m2 square foot ft2 Density (mass) (Specific mass) kilogram per cubic meter kg/m3 slug per cubic foot slug/ft3 Density (weight) (Specific weight) newton per cubic meter N/m3 pound per cubic foot Energy; work joule Force newton Force per unit length (Intensity of force) newton per meter Frequency hertz Length Mass J N Nиm pcf foot-pound lb/ft3 ft-lb kgиm/s pound N/m pound per foot Hz sϪ1 hertz meter m (base unit) foot kilogram kg (base unit) slug lb-s2/ft Nиm pound-foot lb-ft Moment of a force; torque newton meter (base unit) lb/ft Hz sϪ1 ft (base unit) Moment of inertia (area) meter to fourth power m inch to fourth power in.4 Moment of inertia (mass) kilogram meter squared kgиm2 slug foot squared slug-ft2 Power watt W J/s (Nиm/s) foot-pound per second ft-lb/s Pressure pascal Pa N/m2 pound per square foot Section modulus meter to third power m3 inch to third power Stress pascal Pa N/m2 pound per square inch Time second s (base unit) second Velocity (angular) radian per second rad/s Velocity (linear) meter per second m/s Volume (liquids) liter Volume (solids) cubic meter lb L 10 Ϫ3 m3 psf in.3 psi lb/in.2 s (base unit) radian per second m lb/ft2 rad/s foot per second fps ft/s gallon gal 231 in.3 cubic foot cf ft3 SELECTED PHYSICAL PROPERTIES Property SI USCS Water (fresh) weight density mass density 9.81 kN/m3 1000 kg/m3 62.4 lb/ft3 1.94 slugs/ft3 Sea water weight density mass density 10.0 kN/m3 1020 kg/m3 63.8 lb/ft3 1.98 slugs/ft3 Aluminum (structural alloys) weight density mass density 28 kN/m3 2800 kg/m3 175 lb/ft3 5.4 slugs/ft3 weight density mass density 77.0 kN/m3 7850 kg/m3 490 lb/ft3 15.2 slugs/ft3 Reinforced concrete weight density mass density 24 kN/m3 2400 kg/m3 150 lb/ft3 4.7 slugs/ft3 Atmospheric pressure (sea level) Recommended value Standard international value 101 kPa 101.325 kPa 14.7 psi 14.6959 psi Acceleration of gravity (sea level, approx 45° latitude) Recommended value Standard international value 9.81 m/s2 9.80665 m/s2 32.2 ft/s2 32.1740 ft/s2 Steel SI PREFIXES Prefix tera giga mega kilo hecto deka deci centi milli micro nano pico Symbol T G M k h da d c m n p Multiplication factor 1012 109 106 103 102 101 10Ϫ1 10Ϫ2 10Ϫ3 10Ϫ6 10Ϫ9 10Ϫ12 ϭ 000 000 000 000 ϭ 000 000 000 ϭ 000 000 ϭ 000 ϭ 100 ϭ 10 ϭ 0.1 ϭ 0.01 ϭ 0.001 ϭ 0.000 001 ϭ 0.000 000 001 ϭ 0.000 000 000 001 Note: The use of the prefixes hecto, deka, deci, and centi is not recommended in SI ... section of a composite beam of wood and steel ͵ ydA ϭ ෆy1A1 ϭ (h1 Ϫ in.)(4 in ϫ in.) ϭ (h1 Ϫ in.) (24 in .2) ydA ϭ ෆy2 A2 ϭ Ϫ(6 .25 in Ϫ h1)(4 in ϫ 0.5 in.) ϭ (h1 Ϫ 6 .25 in.) (2 in .2) in which A1 and A2... and the moment of inertia I2 of the plastic core is b 20 0 mm I2 ϭ ᎏᎏ(h3c) ϭ ᎏᎏ (150 mm)3 ϭ 56 .25 0 ϫ 106 mm4 12 12 As a check on these results, note that the moment of inertia of the entire crosssectional... curvature (Eq 6-5) into the expressions for sx1 and sx2 (Eqs 6-2a and 6-2b); thus, MyE1 sx1 ϭ Ϫ ᎏᎏ E1I1 ϩ E2I2 MyE2 sx2 ϭ Ϫ ᎏᎏ E1I1 ϩ E 2I2 (6-6a,b) These expressions, known as the flexure formulas