Yield limit interaction relations for biaxially loaded non-sway steel beam-columns with applied torsion

Journal of Constructional Steel Research 156 (2019) 182–191

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Journal of Constructional Steel Research

Yield limit interaction relations for biaxially loaded non-sway steel beam-columns with applied torsion Mamadou Konate a,⁎, Zia Razzaq b a b

School of Engineering Science and Technology, Norfolk State Univ., Norfolk, Virginia 23504, United States Dept. of Civil and Environmental Engineering, Old Dominion Univ., Norfolk, Virginia 23529, United States

a r t i c l e

i n f o

Article history: Received 26 November 2018 Received in revised form 1 February 2019 Accepted 1 February 2019 Available online 16 February 2019 Keywords: Yield limit load-moment-torsion interaction relations Combined axial load, biaxial bending and torsion New interaction expressions

a b s t r a c t This paper presents yield limit interaction relations for biaxially loaded non-sway steel beam-columns with a concentrated torsional moment applied at the bottom end. The interaction relations are based on an experimentally verified general analysis for hollow square section and then utilized for predicting the behavior of both rectangular and I-shaped section members. The analysis includes p-delta effects and initial residual stresses. A set of new yield limit interaction expressions are finally developed, including the influence of applied torsion. The use of interaction relations is demonstrated by means of analysis and design examples. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Hollow rectangular and I-shaped section members used in steel structures can be subjected simultaneously to an axial compression load, biaxial bending, and torsion. Razzaq and Galambos [1] presented a theoretical and experimental study of biaxial bending of beam with torsion. The semi-analytic method was found to be the most efficient one among the six techniques considered for solving the three simultaneous governing differential equations for bending and torsion. Trahair and Bild [2] conducted a theoretical investigation of elastic biaxial bending and torsion of thin-walled open section members. It was concluded that the classical equations developed by Timoshenko, Vlasov and others were too complex for hand solution; therefore, numerical methods such as finite integral technique were used. Trahair and Pi [3] studied the behavior, analysis and design of compressed members under the combined actions of torsion and bending, but their research was also limited to the elastic range. Interaction equations were formulated and could be used for a design purpose with this type of loading; however, the assumption was that the member will remain elastic. Nethercot et al. [4] studied the design of members under combined bending and torsion. El-Khenfas and Nethercot [5] studied the behavior of open section steel members under combined bending and torsion, however, their research was based only on numerical analysis. Trahair [6,7] studied the buckling and torsion of steel angle section beams, but this research did not include biaxial bending. In another study, Trahair ⁎ Corresponding author. E-mail addresses: [email protected] (M. Konate), [email protected] (Z. Razzaq).

https://doi.org/10.1016/j.jcsr.2019.02.001 0143-974X/© 2019 Elsevier Ltd. All rights reserved.

[8] conducted experiments with biaxial bending and torsion applied to steel angle section beams. Furthermore, Trahair [9] investigated the non-linear biaxial bending response of steel Z-beams including induced torsional effects. These studies and other previously published works do not account for the combined action of biaxial bending, axial load, and applied torsion including the second-order p-delta effects and residual stresses due to manufacturing. The current paper presents a set of interaction expressions based on an investigation of the yield limit response of biaxially-loaded steel beam-columns including applied torsion while accounting for p-delta effects as well as residual stresses. 2. Problem and analysis Fig. 1 shows an imperfect steel member BT with z as its longitudinal axis and subjected to a concentrated torsional moment Mz applied at the bottom end, an axial compressive load P, and biaxial bending moments MBx, MBy, MTx, and MTy. Here, the subscripts B and T refer to the member bottom and top ends. The boundaries of the member have partial rotational restraints about the x and y axes. The terms kBx, kBy, kTx and kTy represent partial rotational stiffness values at B and T about the x and y axes. The initial geometric imperfection is taken in the form of half-sine wave functions ui and vi in the xz and yz planes as shown in Fig. 1. The total displacement U is the sum of u and ui, where u is the displacement due to the applied loads. Similarly, V is the sum of v and vi, where v is the displacement caused by the applied loads. The crosssectional centroidal axes, x and y, and the longitudinal member axis z are also indicated in this figure. The induced reactions Rx and Ry shown in the figure are developed due to unequal applied end-bending

M. Konate, Z. Razzaq / Journal of Constructional Steel Research 156 (2019) 182–191

moments. Fig. 2a and b show the residual stress distributions for Ishaped and hollow rectangular cross sections, respectively. The maximum compressive and tensile residual stresses, σrc and σrt, respectively, are also shown in these figures. Depending on the loading condition, yielding may occur at the tips of flanges or at the middle of the web for an I-shaped cross section. For a hollow rectangular cross-section, yielding may occur either at the middle of a typical side of the cross section or at a typical corner. Also, for the type of loading considered in this paper, yielding is initiated either at the member's midspan or at the end where the external uniaxial or biaxial bending moment is applied. Equilibrium between internal and external loads results in the following differential equations governing the member behavior [10]: Bxx v00 þ Bxy u00 þ Bxω ψ00 −Ae M Rz u0 þ Ae Pv    z
þ Ae ψ mBy −MBy þ M By þ MTy þ mTy −mBy −Pxo −Ae mBx L z þ Ae ð−mTx þ mBx Þ L z ¼ −Ae Pvi þ Ae MBx −Ae ðMBx þ M Tx Þ−Sxe P−Sxe P re L þ Ae M xre −Sxe P p þ Ae M xp

Fig. 1. Torsionally loaded imperfect beam-column.

183

Byx v00 þ Byy u00 þ Byω ψ00 þ Ae MRz v0 þ Ae Pu   z þ Ae ψ −mBx −M Bx þ ðMBx þ M Tx −mTx þ mBx Þ þ Pxo L   z
z
−Ae mBy −Ae mTy −mBy ¼ −Ae P ui −Ae MBy þ Ae M By þ M Ty L L −Sye P−Sye P re þ Ae M yre −Sye P p þ Ae M yp ð2Þ

ð1Þ

Fig. 2. Residual stress patterns for hot rolled sections.

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 C ωe ψ000 − C te þ K ψ0 h i  z
þ v0 mBy −M By þ M By þ MTy þ mTy −mBy −Px0 L h i z þ u0 −mBx −M Bx þ ðMBx þ MTx −mTx þ mBx Þ þ Py0 L  u v
− M By þ M Ty þ mTy −mBy − ðMBx þ MTx −mTx þ mBx Þ L L ¼ −M Rz þ M zp

ð3Þ

in which: Bxx

Bxy

  a11 a23 −a21 a13 ¼ a11

Bxω

  a11 a24 −a21 a14 ¼ a11

ð6Þ

  a11 a32 −a12 a31 a11

ð7Þ

  a11 a33 −a13 a31 a11

ð8Þ

  a11 a34 −a14 a31 a11

ð9Þ

Byy ¼

Byω ¼

ð4Þ

ð5Þ

in which the primes indicate differentiation with respect to z; Pre is the residual axial force; Mxre and Myre are the residual bending moments about the x and y axes; Pp is the internal plastic axial force; Mxp and Myp are the internal plastic bending moments about the x and y axes; MRz is the reacting torsional moment; Mzp is the internal torsional moment for the plastic portion of the cross section; and the internal forces, moments, and the aij terms are defined as follows: Z P re ¼

σ r dA

ð10Þ

σ Y dA

ð11Þ

Ae

Z Pp ¼

Ap

Z Mxre ¼

σ r y dA

ð12Þ

σ r x dA

ð13Þ

σ Y y dA

ð14Þ

Ae

Z Myre ¼

Ae

Z Mxp ¼

Ap

Ap

a31 ¼ E Sye

ð24Þ

a32 ¼ E I xye

ð25Þ

a33 ¼ E I ye

ð26Þ

a34 ¼ E I ωxe

ð27Þ

Z Ae ¼

dA

ð28Þ

y dA

ð29Þ

x dA

ð30Þ

Ae

Z Sxe ¼

Ae

Z Sye ¼

Ae

Z Sωne ¼

Ae

ωn dA

ð31Þ

Z Ixe ¼

y2 dA

ð32Þ

x2 dA

ð33Þ

Ae

Z Iye ¼

Ae

Z Iωxe ¼

ωx dA

ð34Þ

ωy dA

ð35Þ

xy dA

ð36Þ

Ae

Z Iωye ¼

Ae

Z Ixye ¼

Ae

In these equations, dA is an elemental area of the cross section; σr is the residual stress; σY is the normal yield stress; ∫Ae and ∫Ap represent integrals over elastic and plastic regions; and the other terms in the equations above are defined in the Appendix. In the member study presented in this paper, the displacements and the angle of torsional rotation at the boundaries are all equal to zero. An iterative finite integral procedure is developed to solve Eqs. 1-3 for combined loads and is given in Reference 10. The analysis utilizes the following well-known von Mises yield criterion [11]: σ 2 þ 3τ 2 ≤ σ Y 2

ð37Þ

In this expression, σ is the normal stress; τ is the shear stress; and σY is the normal yield stress. Expression 37 can also be written as follows:

Z Myp ¼

ð23Þ

in which:

  a11 a22 −a12 a21 ¼ a11

Byx ¼

a24 ¼ E I ωye

σ Y x dA

ð15Þ



σ σY

2

 þ

τ τY

2 ≤1

ð38Þ

a11 ¼ E Ae

ð16Þ

a12 ¼ E Sxe

ð17Þ

a13 ¼ E Sye

ð18Þ

in which, τY is the shear yield stress. Also, the warping terms given in Eqs. 1-3 are only considered for I-shaped sections and are practically negligible for hollow rectangular sections.

a14 ¼ E Sωne

ð19Þ

3. Experimental investigation and analysis

a21 ¼ ESxe

ð20Þ

a22 ¼ E Ixe

ð21Þ

a23 ¼ E Ixye

ð22Þ

A 1.5 × 1.5 x 0.125 in. (38.1 × 38.1 × 3.175 mm) hollow square steel section is used for the experimental study. As schematically shown in Fig. 3, each test member has a clear length of 34 in. (863.6 mm). However, the distance between the centerlines of the end gimbals, that is, including both the actual member length and the solid portions of the end

M. Konate, Z. Razzaq / Journal of Constructional Steel Research 156 (2019) 182–191

fixtures, is 37.75 in. (958.85 mm), which is the length adopted in the analysis. Fig. 4 shows the experimental setup. The torsional moment is applied at the bottom end by means of an eccentric force F applied with Hydraulic Jack C, however, transmitted to the member by means of a chain as shown in Fig. 4. shows lower and upper-end gimbals at the ends of the test specimen as well as a steel casing with Hydraulic Jack A and Load Cell A. As shown in Fig. 5, the lower-end gimbal is attached to Steel Plate 1, which has a gliding steel chamber below it. The gliding steel chamber rides on the outer surface of a steel casing which houses a 50-kip capacity compression Load Cell A mounted on Hydraulic Jack A. The steel casing is welded to a floor steel plate which in turn is anchored to the laboratory test bed. The upper-end gimbal is identical to the lower-end gimbal; however, it is mounted in an upside down position. The upper-end gimbal is attached to a steel cross-beam which is bolted at its ends to steel columns. The end columns are anchored to the laboratory test bed, thus forming a large reaction frame while the cross-beam supports the upper-end gimbal. The axial load is applied using Hydraulic Jack A and measured with Load Cell A. The Load Cell A pushes Steel Plate 1, which in turn transmits the axial load to the lower gimbal outer box through a pair of outer bearings and

Fig. 4. Schematic of a portion of apparatus with torque device.

Fig. 3. Schematic of specimen with inner part of the gimbals.

Fig. 5. Schematic of lower portion of setup for axial load and bending tests.

185

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shafts. The load is finally transferred to the test specimen through the gimbal inner box. The bending moment is applied at the top end of the member by means of a moment arm bolted to the upper gimbal inner box as shown in Fig. 6. The moment arm is 1.0 × 2.0 × 24.0 in. solid rectangle steel section. The load W is applied through two 0.75in. diameter tie rods. These rods are 75 in. long each and separated by 12-in. long 0.5-in thick steel plates forming a closed ring at the top and bottom. By means of ball and socket arrangement, the Top Plate B sits on the machined arm. Using similar arrangement, the Bottom Plate B is attached to a 5-kip capacity compression Load Cell B. The Load Cell B is mounted on Hydraulic Jack B, which is bolted to a small reaction frame in an upside-down position as illustrated in Fig. 7. The small steel reaction frame is mounted to the laboratory test bed. The load W can finally be produced by manually controlling the Hydraulic Jack B. Based on ASTM tension tests, the following values of Young's modulus, E, and the normal yield stress, σY, for the steel specimens were E = 29, 599 ksi (204,077,921 kPa) and σY = 58.999 ksi (406,783,785 kPa). The shear modulus, G, and the shear yield stress, τY, for the steel, were found using a torsion test conducted on a segment of a 1.5 × 1.5 × 0.125 in. (38.1 × 38.1 × 3.175 mm) hollow square section. The values were found to be G = 11,200 ksi (77,221,282 kPa) and τY = 32.301 ksi (222,707,555 kPa). The member initial uniaxial and biaxial crookedness functions are taken as L/1000. In this paper, the above material properties mentioned are used in the analysis. The yield limit loading is defined as that set of external loads which cause initiation of yielding at any point in a member. The yield load values used are dimensionless and are defined as follows: p¼

P Pc

ð46Þ

σ c ¼ ðσ Y −σ r Þ

ð47Þ

τc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v ffi u u σ 2 −σ 2 t y r

ð48Þ

3

In these expressions, Pc, Mcx, Mcy and Tc are the yield limit axial load, bending moment about x and y axes, and torsional moment, respectively; Sx and Sy are section of modulie about x and y axis, respectively; σr is the residual stress, however, taken as σrt or σrc with reference to Fig. 2; σc is the yield limit normal stress; and τc is the yield limit shear stress. Nine combined axial load, biaxial bending and torsion tests are performed on member with hollow square sections with pinned boundary conditions by first applying the axial load and the biaxial bending moments and keeping them constant, followed by a gradually increasing torsional moment up to the yield limit. Fig. 8 shows dimensionless applied torsion, t, versus angle of twist, ψ, relations with constant axial loads and biaxial bending moments for Tests 1 through 9. As the applied axial load and biaxial bending moments become larger, the torsional moment capacity has a corresponding reduction. The constant loads M and P, magnitude of p-delta effects Pu and Pv, dimensionless predicted

ð39Þ

mx ¼

Mx Mcx

ð40Þ

my ¼

My Mcy

ð41Þ

t¼

T c ¼ 2AE t ðτc Þ

T Tc

ð42Þ

where: P c ¼ Aðσ Y −σ r Þ

ð43Þ

Mcx ¼ Sx ðσ Y −σ r Þ

ð44Þ

Mcy ¼ Sy ðσ Y −σ r Þ

ð45Þ

Fig. 6. Schematic of upper end gimbal with moment arm.

Fig. 7. Schematic of bending device.

M. Konate, Z. Razzaq / Journal of Constructional Steel Research 156 (2019) 182–191

Fig. 9. Dimensionless stress interaction curves for combined loads.

Fig. 8. Dimensionless applied torsion moment versus angle of twist curves with constant Mx, My and P.

torsional moment ttheo, dimensionless experimental yield torsional moment texp, and angle of twist ψexp, respectively, are presented in Table 1. In this table, Mx, My, Pc, Mcx, Mcy represent the bending moments about x and y axes, axial yield limit load; and yield bending moments about x and y axes, respectively. A computer program based on a finite integral solution to the system of governing differential Eqs. 1-3 for the beam-column subjected to a torsional moment at one end is developed [10]. The cross-sectional normal stresses are computed iteratively using the governing differential equations. The yield limit is determined by invoking the von Mises criterion given in Eq. 37. In the analysis, the p-delta effects, residual stresses shown in Fig. 2, and initial crookedness in the form of a halfsine wave with a midspan amplitude of L/1000 only in the xz plane are accounted for. 4. Yield load-moment-torsion interaction formulation In order to develop yield load-moment-torsion interaction relationships, the normal stress σ and the shear stress τ at any point on a given cross section can be expressed in the following dimensionless forms: My σ P Mx ¼ þ þ σ c P c Mcx M cy

ð49Þ

axis represents the dimensionless normal stress and the vertical axis represents the dimensionless shear stress. Bresler [13], and Parme et al. [14] used a two-segment linear and load contour approximation approaches to curve fit. In this paper, bilinear and trilinear approximaton approaches are used to curve fit as well as an approximation with an exponent α. In addition, a combination of the two approaches is presented. 4.1. Piecewise Linear Approximations The following two and three segment linear approximation approaches are used to curve fit the experimental dimensionless interaction shown in Fig. 9 and presented as bilinear and trilinear interaction, respectively. 4.1.1. Bilinear Yield Limit Interaction Expressions for Biaxial Loading and Torsion Fig. 10 shows the experimental dimensionless interaction curve and a bilinear approximation using lines AB and BC. The linear expressions for the two segments are obtained as:   6 P Mx My T þ þ þ ≤1 9 P c M cx Mcy Tc 

τ T ¼ τc T c

ð50Þ

In these expressions, P, Mx, My, and T are the applied axial load, bending moment about x and y axes, and torsional moment, respectively; The terms Pc, Mc, Mc, Tc, σc, and τc are previously defined in Eqs. 43-48. Fig. 9 shows the theoretically predicted and experimental dimensionless interaction curves for the nine combined load tests. The horizontal

187

My P Mx þ þ P c Mcx Mcy

 þ

1T ≤1 4 Tc

if

if

T N 0:4 Tc T ≤0:4 Tc

Mx/Mcx

My/Mcy

Test 1 2 3 4 5 6 7 8 9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Pu

Pv

k-in (x 113 kN-mm)

k-in (x 113 kN-mm)

0.019 0.076 0.172 0.308 0.485 0.705 0.969 1.280 1.640

0.003 0.012 0.029 0.054 0.089 0.135 0.193 0.267 0.358

ttheo

texp

0.738 0.727 0.707 0.680 0.642 0.593 0.530 0.445 0.323

0.707 0.696 0.681 0.650 0.614 0.584 0.522 0.426 0.296

4.1.2. Trilinear yield limit interaction expressions for biaxial loading and torsion Fig. 11 shows the experimental dimensionless interaction curve and trilinear approximation using three lines DE, EF and FG. The linear

ψexp

radians 0.095 0.089 0.083 0.083 0.082 0.082 0.071 0.057 0.037

ð52Þ

In these expressions, P, Mx, and My are the applied axial load, and biaxial bending moments about the x and y axis, respectively.

Table 1 Torsional Moments and Angle of twist with Constant M and P and p-delta effects [L = 37.75 in., 1.5 × 1.5 × 0.125 in section]. P/Pc

ð51Þ

Fig. 10. Bilinear approximation approach.

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approximated as follows: 8 s
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > σ 2Y −σ 2rc > > > > > τ2c > > sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >  <
2 σ Y −σ 2rmax α ¼ larger of > τ2c > > s
ffiffiffiffiffiffiffiffiffiffi > ffi > > > σ 2Y > > > : τ2

ð57Þ

c

Fig. 11. Trilinear approximation approach.

expressions for the three segments are obtained as:   My T 1 P Mx þ þ þ ≤1 T c 4 P c Mcx M cy 

if

T ≥0:9 Tc

maining portion.



My 5T 5 P Mx ≤1 þ þ þ 6T c 8 P c M cx M cy 

My P Mx þ þ P c Mcx Mcy

 þ

1T ≤1 3 Tc

ð53Þ

T b 0:9 Tc

ð54Þ

T ≤ 0:6 ð55Þ Tc

ð55Þ

if 0:6 b

if

Fig. 12 shows the experimental dimensionless interaction relationships between dimensionless normal and shear stresses corresponding to the initiation of yielding for the nine combined load tests and those based on both bilinear and trilinear approximations. As seen in this figure, the trilinear approach more closely approximates the experimental curve. 4.2. Exponent α Approximation Keeping the experimental dimensionless normal stress the same as in Eq. 49 and applying an exponent α to the dimensionless shear stress given in Eq. 50, a new dimensionless torsion term is introduced here in beam-column interaction expression corresponding to yield limit as follows: 

My P Mx þ þ P c Mcx Mcy

 þ

in which σrmax is the larger of the absolute values of σrc and σrt. This definition of α is also used throughout this paper. Fig. 13 shows the experimental dimensionless interaction relationships between dimensionless normal and shear stresses corresponding to the initiation of yielding for the nine combined load tests and those based on Expression 56 with various α values named as Biaxial Loading Interaction Expressions with Torsion at Yield Limit (BLIET). As seen in this figure, the interaction curve for α = 2.5 is close to the experimental curve for TTc b 0.5 and on the convervative side for its re-

 α T ≤1 Tc

ð56Þ

To account for the presence of residual stresses, the relationship

4.3. Combined piecewise linear and exponent α approximations Combining the bilinear expressions with an exponent α approximation approach, the Yield Limit Interaction Expressions with Exponent α named as IEE are as follows:    α My 6 P Mx T þ þ ≤1 þ 9 P c M cx M cy Tc 

My P Mx þ þ P c Mcx Mcy

 þ

  1 T α ≤1 4 Tc

if

T N0:4 Tc

ð58Þ

if

T ≤ 0:4 Tc

ð59Þ

In Expressions 58 and 59, the α value is calculated based on Eq. 57. Fig. 14 shows the experimental dimensionless interaction relationships between dimensionless normal and shear stresses corresponding to the initiation of yielding for the nine combined load tests and those based on Expressions 58–59 with various α values. For 0.82 ≤ TTc ≤ 0.4, the experimental curve is conservative of all curves and is close to that for TTc ≤ 0.4. Thus, it is concluded that α should be taken as a value between 1.2 and 2.0 to be on the conservative side everywhere. Fig. 15 compares the various dimensionless interaction relationships between normal and shear stresses corresponding to the initiation of yielding. The BLIET curve based on Expression 56 and the IEE curve based on Expressions 58–59 are each developed using an α value of 2.0. Except for the curve based on Expressions 58–59 (IEE), all curves are on the conservative side as compared with the experimental curve.

between σ and τ is expressed as σ 2 þ α2 τ2 ¼ σ Y 2 , where α is

Fig. 12. Comparison of straight line interaction approximation.

Fig. 13. Dimensionless shear and normal stress interaction curves for various α values using Expression 56.

M. Konate, Z. Razzaq / Journal of Constructional Steel Research 156 (2019) 182–191

189

For lateral-torsional buckling: P mLT MLT my My þ þ ≤1 P cy Mb Sy σ Y

Fig. 14. Dimensionless shear and normal stress interaction curves for various α values using Expressions 58–59.

in which Pc is the smaller of Pcx and Pcy; Pcx and Pcy are the compressive resistance about x and y axes, respectively; mx and my are the equivalent uniform moment factors for x and y axes, respectively, obtained from Table 26 of British Standard Institution (BS 5950); Sx and Sy are the elastic moduli about x and y axes, respectively; mLT is the equivalent uniform moment factor for y axis flexural buckling obtained from Table 18 of BS 5950; MLT is the maximum major axis moment in the segment length Lx governing Pcx; and Mb is the buckling resistance moment. The AISC-LRFD beam-column interaction expressions [16] used in USA are as follows: Pu þ ϕc P n

4.4. Discussion

ð62Þ

  M uy 8 Mux þ ≤1 9 ϕb M nx ϕb Mny

  Pu Mux Muy þ þ ≤1 2ϕc P n ϕb Mnx ϕb Mny

if

Pu ≥ 0:2 ϕc P n

ð63Þ

if

Pu b 0:2 ϕc P n

ð64Þ

As seen from Fig. 15, the trilinear interaction expressions represent a better approximation of the experimental results. One can also use Expressions 53–55 if a piecewise-linear approximation is desired. Expression 56 is more conservative than Expressions 58–59 and results in a better curve fit as shown in Fig. 12. Since the governing differential equations are developed for an arbitrary cross section, it can therefore be concluded that the general load-moment-torsion interaction expressions developed in this investigation can be applied to hollow rectangular and I-shaped sections.

in which Pu is the applied compressive axial load; Pn is the factored axial compressive capacity; Mux and Muy are the applied bending moments about the strong and weak axis; Mnx and Mny are the factored moment capacity with respect to the x and y axis, respectively; ϕc is the resistance factor for compression; and ϕb is the resistance factor for bending. A modulus of elasticity of 29,000 ksi (199,947,962 kPa) is also used.

5. Analysis examples

5.1. Analysis Example 1

The interaction relationships presented in this paper are developed based on the condition that no member instability occurs for the given loads. Therefore, the beam-column stability check must be performed utilizing international steel beam-column expressions. The British beam-column interaction expressions [15] are as follows: For cross-section capacity check:

A simply-supported steel member of length 12 ft. (3.658 m) has a “W21x48” I-shaped section having a nominal depth of 21 in. (533.4 mm) and a self-weight of 48 lb./ft. (700.507 N/m). It is subjected to a factored axial live load of 150 kips (667 kN), and factored live bending moments of magnitudes 45 k-in. (5085 kN-mm.), and 10 k-in. (1130 kN-mm.), respectively, about the x and y axes. Determine the value of the maximum torsional moment that the member can be subjected to at its bottom end corresponding to the yield limit. The steel normal yield stress value is 50 ksi (344,738 kPa).

P Mx My þ þ ≤1 P Y Mpx M py

ð60Þ

5.1.1. Solution 1) Stability check (BS 5950) using Eqs. 60–62 and taking mx = m1 = 1:

For in-plane ‘buckling’: P mx M x my My þ þ ≤1 Pc Sx σ Y Sy σ Y

ð61Þ

M

P PY

þ MMpxx þ Mpyy ≤1 gives 0.475

P Pc

þ mSxxσMYx þ

P P cy

þ

mLT MLT Mb

my My Sy σ Y ≤ m M þ Syyσ Yy

1 gives 0.806 ≤ 1 gives 0.855

Thus, the member is stable as a beam-column under the given loads. 2) Using Eq. 57, α is found to be 1.816. Finally, Expression 51 gives T = 16.086 k-in. (1817.718 kN-mm.) since TTc N 0.4; Expression 55 gives T = 18.367 k-in. (2075.471 kN.mm.) since

T Tc

b 0.6; Expression 56

gives T = 13.571 k-in. (1533.523 kN.mm.); and Expression 58 gives T = 23.104 k-in. (2610.752 kN.mm.) since TTc N 0.4. One should use the smallest T value since it leads to the smallest interaction expression value. Thus, the maximum torsional moment is T = 13.571 k-in (1533.523 kN.mm).

5.2. Analysis example 2

Fig. 15. Comparison of various interaction curves.

A simply-supported steel member of length 14 ft. (4.267 m) has a 7x5x0.375 in. (178 × 127 × 9.525 mm.) hollow rectangular section. It is subjected to a factored axial live load of 62.211 kips (277 kN), and

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factored live bending moments of magnitudes 88.263 k-in. (9972 kNmm.), and 63.263 k-in. (7148 kN-mm.), respectively, about the x and y axes. Determine the value of the maximum torsional moment that the member can be subjected to at its bottom end corresponding to the yield limit. The steel normal yield stress value is 46 ksi (317,159 kPa). 5.2.1. SOLUTION 1) Stability check (AISC-LRFD) using Eqs. 63–64: Since ϕ PPn = 0.264, use Expression 63: c

Pu 8 Mux ϕc P n 9 ðϕb Mnx

Muy Þ≤ b M ny

þϕ

1 gives 0.722 b 1. Thus, the member is stable as

a beam-column under the given loads. 2) Using Eq. 57, α is found to be 2.000. Finally, Expression 51 gives T = 254.101 k-in. (28,713.413 kN-mm.) since TTc N 0.4; Expression 54 gives T = 325.153 k-in. (36,742.289 kN-mm.) since 0.6b

T Tc

b0.9;

Expression 56 gives T = 249.912 k-in. (28,240.056 kN-mm.); and Expression 58 gives T = 364.846 k-in. (41,227.598 kN-mm.) since T T c N 0.4. One should use the smallest T value since it leads to the smallest interaction expression value. Thus, the maximum torsional moment is T = 249.912 k-in. (28,240.056 kN-mm).

5.3. Design Example A steel member of length 14 ft. (4.267 m) is expected to carry a factored axial live load of 124.604 kips (554.266 kN), and factored live bending moments of magnitude 154.257 k-in. (17,429 kN-mm.) about each of the major x and y axes. The member must also support a torsional moment of 349.381 k-in (39,475 kN-mm). The steel normal yield stress value is 46 ksi (317,159 kPa). 5.3.1. SOLUTION Try a 7x7x0.375 in. (179x179x9.525 mm) hollow square section. Since ϕ PPn = 0.389, use Expression 63: c

Pu ϕc P n

Muy Þ≤ b M ny

þ 89 ðϕ MMuxnx þ ϕ b

1 gives 1.033 b 1. Thus, the section is unsat-

isfactory since the member is unstable even in the absence of an applied torsional moment. Try an 8x8x0.375 in. (203x203x9.525 mm) hollow square section. This section gives: P ϕ P n = 0.327, c

Thus Expression 63 must be used which provides: Pu ϕc P n

þ

Mux 8 9 ðϕb Mnx

Muy Þ≤ b M ny

þϕ

1 resulting in 0.756 b 1. Thus, the section is

satisfactory when subjected to an axial load and biaxial bending. However, its adequacy also needs to be evaluated when a torsional moment is also applied. The self-weight of the 8x8x0.375 in. (203x203x9.525 mm.) hollow square section is 37.69 lb./ft. (550 N/m). Using Eqs. 46 and 57, Tc and α are found to be 998 k-in (112,785 kNmm) and 2.000, respectively. Thus, the ratioTTc is found to be 0.35. Finally, Expression 52 gives 0.656; Expression 55 gives 0.927; Expression 56 gives 0.932; and Expression 59 gives 0.841. Therefore, the member is satisfactory even in the presence of applied torsion. 6. Conclusions The interaction relationships presented in this paper are in good agreement with those based on the experimental results. Various types of yield limit interaction expressions for a steel beam-column with an applied torsional moment are presented. The interaction expressions are applicable to steel members with both I-shaped and hollow rectangular sections.

Appendix: Nomenclature

dA Elemental area. E Modulus of elasticity. G Shear modulus. Ix, Iy Moment of inertia about x-axis and y-axis. Ixy Product of inertia relative to x-axis and y-axis. Iωx, Iωx Warping product of inertia about x-axis and y-axis. K Wagner term. kBx, kTx End rotational stiffness about x axis. kBy, kTy End rotational stiffness about y axis. L Member length. mBx, mTx Restraint moments at bottom and top end about x axis. mBy, mTy Restraint moments at bottom and top end about y axis. MBx, MTx Applied moments at bottom and top end about x axis. MBy, MTy Applied moments at bottom and top end about y axis. Mx, My, Applied bending moments about the x and y axes respectively. Mxp, Myp, Bending moments due to plastification. Mxre, Myre Bending moments due to residual stress. Mze Torsional moment due to residual stress. Mzp Torsional moment due to plastification. P Applied axial load. Pp Axial load due to plastified elements in the member. Pr Axial load due to residual stress. Rx, Ry Reaction at bottom of the beam-column. Sx, Sy Elastic section of modulus about x-axis and y-axis. Sωx, Sωy Warping section of modulus about x-axis and y-axis. T Applied torque. U Total deflection in x-direction. V Total deflection in y-direction. u Deflection due to load in x-direction. v Deflection due to load in y-direction. uo Midspan initial member crookedness in x-direction. vo Midspan initial member crookedness in y-direction. uoi Initial member crookedness in x-direction. voi Initial member crookedness in y-direction. u′′ The second order derivative of u. v′′ The second order derivative of v Zx, Zy Plastic section of modulus about x-axis and y-axis β Angle between the column and the base plate ε Normal strain εr Residual strain ε0 Average axial strain εω Normal strain due to warping Φx,Φy Bending curvatures σY, Yield normal stress τY, Yield shear stress σrt,σrc Compressive and tensile residual stress ѱ Angle of twist

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