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Second-order axial force and midspan deflection in a simple supported beam axially restrained
Engineering Structures 30 (2008) 561–569 www.elsevier.com/locate/engstruct
Review article
Second-order axial force and midspan deflection in a simple supported beam axially restrained J. Paul Smith-Pardo a , J. Dario Aristizabal-Ochoa b,∗ a Berger/Abam Engrs. Inc., 33301 9th Ave. South, Suite 300, Federal Way, WA 98003, United States b National University, School of Mines, Medellin, Colombia
Received 28 November 2006; received in revised form 13 January 2007; accepted 14 March 2007 Available online 23 May 2007
Abstract Design aids for the calculation of the induced axial force and mid-span deflection in a simple supported beam axially restrained with an initial imperfection (camber) under transverse loads are presented. In addition, closed-form equations based on equilibrium developed by AristizabalOchoa [Aristizabal-Ochoa JD. Second-order axial deflections of imperfect 3-D beam–column. ASCE Journal of Engineering Mechanics 2000; 126(11): 1201–8] are also presented in order to verify the validity of the proposed method and design aids. Both, equations and design aids find great applicability in the analysis and design of slender flexible beams that are restrained axially. The method proposed and design aids are based on a general model presented by the authors in a previous paper [Smith-Pardo JP, Aristizabal-Ochoa JD. Buckling with reversals of axially restrained beam–column with initial imperfections. ASCE Journal of Engineering Mechanics 1999; 125(4): 401–9]. A comprehensive example is included to show the simplicity and validity of the proposed method. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Beams; Beam-columns; Buckling; Computer applications; Elastic behavior; Non-linear behavior; P–δ effects; Second-order analysis; Stability; Stiffness; Structural analysis
Contents 1. 2. 3.
4.
5. 6.
Introduction............................................................................................................................................................................ 561 Structural model ..................................................................................................................................................................... 562 Application–design aids ........................................................................................................................................................... 563 3.1. Beam–column under a uniformly distributed lateral load ................................................................................................... 563 3.2. Beam–column under concentrated load located at a distance c from the left support.............................................................. 563 Verification equations .............................................................................................................................................................. 564 4.1. Beam–column under a uniformly distributed load q .......................................................................................................... 564 4.2. Beam under concentrated load at abscissa c from the left support........................................................................................ 565 Comprehensive example .......................................................................................................................................................... 566 Summary and conclusions ........................................................................................................................................................ 568 References ............................................................................................................................................................................. 569
1. Introduction ∗ Corresponding address: National University of Colombia, Civil Engineering, Calle 9c # 15-165 Casa 2Urbanizacion Villaverde, Medellin, Antioquia, Colombia. Tel.: +57 42686218; fax: +57 4 4255152. E-mail addresses: [email protected], [email protected] (J.D. Aristizabal-Ochoa).
c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.03.011
Beams supporting lateral loads are usually analyzed as extensionless members under flexure and shear. However, if supports are restraining the relative elongation between the ends of a member, axial forces are induced. Such induced
562
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
Nomenclature A an bk ci dj E I i j k L mn m 01 n P q s S∆ uq
y α δt
Beam cross-sectional area; Coefficients of the Fourier’s series of the deflection of the beam; Distance from end A to point of application of concentrated moment Mk ; Distance from end A to point of application of concentrated load Q i ; Magnitude of the initial imperfection corresponding to mode j; Beam Young modulus; Beam moment of inertia; Subscript that numbers the concentrated loads (varying from 1 to m); Subscript that numbers the coefficients of the initial deflection (varying from 1 to s); Subscript that numbers the concentrated moments (varying from 1 to r ); Beam span; 4I /[A(ndn )2 ] = series of geometric parameters; m 1 [1 + (AE/L)/S∆ ]; Subscript to number the coefficients of a Fourier’s series (varying from 1 to ∞); Axial force (+compression); Distributed transverse load; Subscript to number the number of terms in Eq. (1); Axial stiffness of spring at B; 4q L 4 /(π 5 E I d1 ) for a uniformly distributed load or u Q = 2Q L 3 Sin(π c/L)/(π 4 E I d1 ) for a concentrated load at c from the left support; Beam total transverse deflection; P L 2 /(π 2 E I ); Beam total axial elongation.
forces affect the overall response of the beam. Well-known phenomena such as snap-through, snap-back and reversal of lateral deflections are caused in simple beams with initial camber that are axially restrained. The effects of transverse loads and deflections on the axial stiffness, pre- and postbuckling behavior, and stability including the phenomena of snap-through, snap-back and reversal of deflections of a beam–column have been studied extensively during the last two decades by several researchers. Aristizabal-Ochoa [1] developed a complete theory for the elastic axial stiffness of a doubly symmetric beam–column including the 3-D effects of lack of straightness, transverse forces and moments and sidesway. Complete details on these phenomena and the overall response of a simple beam with an arbitrary imperfection (camber) and under any transverse load have been given by Smith-Pardo and Aristizabal-Ochoa [2]. The main objective of this publication is to present a particular application of the formulation developed by the authors [2]. The application consist of a set of design aids for the calculation of the induced axial force and mid-span
deflection of an imperfect simply supported beam axially restrained. In addition, closed form equations based on an equilibrium method developed by Aristizabal-Ochoa [1] are also presented in order to verify the validity of the proposed design aids. A comprehensive example is included that shows the simplicity and validity of the proposed charts. 2. Structural model The structural model is identical to that previously presented by the authors [2] and will be described herein for easy reference. It consists of a prismatic beam–column with a single symmetrical cross section shown schematically in Fig. 1. The longitudinal x-axis is defined by the centroid of the crosssectional area at the extremes A and B. The beam AB is assumed to be: (1) Made of an isotropic homogeneous linear-elastic material with a modulus of elasticity, E, and axially restrained at the end B by a spring with stiffness S∆ . (2) Prismatic with a single-symmetrical cross section of area A, moment of inertia in the plane of bending I , and span L. (3) Slightly deformed in the plane of bending with an initial out-of-straightness modeled by a plane curve given by Eq. (1). yo =
s X j=1
d j sin
jπ x . L
(1)
Where: d1 , d2 , . . . ds = coefficients in a sine series describing the magnitudes of the initial imperfection. (4) Loaded and deformed in the plane of bending as indicated in Fig. 1. In addition to the axial load, P, three types of transverse loads were considered: (i) a uniformly distributed load q along its entire span; (ii) a series of concentrated loads Q i (with i varying from 1 to N Q) located at distances ci from the left support and; and (iii) a series of concentrated moments Mk (with k varying from 1 to NM) located at distances bk from the left support. Positive directions are indicated in Fig. 1. (5) The study beam–column undergoes small strains with plane sections remaining plane. Torsion and shear effects are neglected throughout. To determine the transverse deflections y, and the axial elongation between the ends of the member δt , Eqs. (2) and (3) were developed by Smith-Pardo and Aristizabal-Ochoa [2]. s ∞ X n 2 dn nπ x nπ x X − an sin sin 2 L L n −α n=1 n=1 ( " 2 # s X an n2 m1 δt 1 1− = − 2 P L/AE m 1 α n=1 m n dn n −α 2 ) ∞ X nan AE/L − − 1+ . d1 S∆ n=s+1
y=
(2)
(3)
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
563
Fig. 1. Structural model. P L2
Where: α = π 2 E I ; and m 1 , m 2 , . . . m n, . . . m s = nondimensional geometrical parameters given by: mn =
4I A(ndn )2
(4)
an = coefficients given by Eqs. (5) and (6). • For n = 1, 3, 5, 7, . . . (odd numbers) " NQ nπc 2L 2 L X i an = 3 Q i sin 2 L π E In (n − α) π n i=1 # NM X nπbk 2q L 2 − Mk cos + 2 2 . L π n k=1 • For n = 2, 4, 6, 8, . . . (even numbers) " NQ nπc 2L 2 L X i an = 3 Q i sin 2 L π E In (n − α) π n i=1 # NM X nπbk − Mk cos . L k=1
+
u q /25 25 − α
2
+ · · · = 1 − m 01 α
π x u /27 1 − uq y 3π x q = sin − sin d1 1−α L 9−α L u q /125 5π x − sin + ···. 25 − α L
(7)
(8)
4
Where: u q = π45 Eq ILd1 (non-dimensional parameter), m 01 = m 1 [1 + (AE/L)/S∆ ] and m 1 is as defined in Eq. (4).
(5)
(6)
Eqs. (2) and (3) were derived based on energy principles and are valid when the beam–column is either under axial tension or compression and for upward or downward initial imperfection. 3. Application–design aids In the two particular applications considered below it is assumed that the axial extension is zero (δt = 0) and that the initial beam imperfection is completely defined by the first term of the sine series in Eq. (1) only. 3.1. Beam–column under a uniformly distributed lateral load For a beam–column under a uniformly distributed transverse load q, Eqs. (2)–(6) are reduced to (7) and (8) from which the induced axial force P, and the corresponding transverse deflection y can be found, respectively. 1 − uq 2 u q /9 2 + 1−α 9−α
3.2. Beam–column under concentrated load located at a distance c from the left support For a beam–column under a single concentrated load Q located at a distance c from the left support, the transverse deflection y and the induced axial force P can be obtained from the simultaneous solution of Eqs. (9) and (10). 1 − u Q sin(π c/L) 2 u Q sin(2πc/L)/2 2 + 1−α 4−α 2 u Q sin(3πc/L)/3 + + · · · = 1 − m 01 α (9) 9−α πx π c u /9 y 1 − uQ 3π x Q = sin sin + sin d1 1−α L L 9−α L 3πc u Q /25 5π x 5πc × sin + sin sin + ··· L 25 − α L L (10) where: u Q =
2 Q L3 π 4 E I d1
(non-dimensional parameter).
Cases > 1 and d1 < 0 As shown by Timoshenko and Woinowsky-Krieger [3], Timoshenko and Gere [4] and also by Smith-Pardo and AristizabalOchoa [2], for m 01 > 1, the solutions of Eqs. (7)–(10) can be obtained with sufficient accuracy (<5% of the exact solution) by taking the first term of each series only. The authors also showed that this is also the case if the initial imperfection is downwards (d1 < 0) regardless of the value of m 01 . This results in simplified equations that have the same form for both loading cases. Thus, a single set of design aids can be formulated for m 01
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Fig. 2. Beam with initial downward imperfection under a uniformly distributed load q or a concentrated load Q at c from the left support: (a) induced axial force (tension); (b) transverse mid-span deflection.
both uniformly distributed and concentrated loads as shown in Figs. 2 and 3. Fig. 2(a) and (b) are provided to calculate the induced axial force (tension) P, and mid-span deflection ymidspan , in a beam with an initial downward imperfection under a uniformly distributed or a concentrated load. Fig. 3(a) and (b) are provided for the case when the initial imperfection is upwards and the induced axial force is compression. Case 1/15 < m 01 < 1 and d1 > 0 If the initial imperfection is upwards and 1/15 < m 01 < 1, Smith-Pardo and Aristizabal-Ochoa [2] showed that it is sufficiently accurate to take the first three non-zero terms in each series of Eq. (7) through (10) in order to obtain reasonable estimates of the induced axial force and mid-span deflection (<5% of the exact solution). Notice that for this range of values of m 01 , the resulting equations for uniformly distributed (Eqs. (7) and (8)) and concentrated loads (Eqs. (9) and (10)) do not have the same form, then separate design aids are necessary for each load case. Design aids, obtained from Eqs. (7)–(10) using three terms of each series, are provided by means of Figs. 4–7
for a beam with an initial upward imperfection under a uniformly distributed load and concentrated loads at L/2, L/3, and L/4 from the left support, respectively. For values of m 01 < 1/15, it becomes necessary to use more than three terms in the series of Eq. (7) through (10). However, for this particular range, design aids were not developed due to their limited use in practice (unrealistic initial imperfection of the beam–column). 4. Verification equations In order to verify the proposed design aids, closed-form equations obtained using equilibrium principles (AristizabalOchoa [1]) are presented below for the beam–column of Fig. 1. 4.1. Beam–column under a uniformly distributed load q In this case the deflection is given by Eqs. (11) and (12).
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
565
Fig. 3. Beam with initial upward imperfection under a uniformly distributed load q or concentrated load Q at c from the left support (m 01 > 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
• For compressive axial force √ q L 2 cos(π/2 αx/L) q x(L − x) y = 2 −1 − √ 2P π Pα cos(π/2 α) d1 πx + sin . 1−α L • For tensile axial force √ √ q L2 cosh(π/2 α − π αx/L) y = 2 1− √ π Pα cosh(π/2 α) −
d1 πx q x(L − x) + sin . 2P 1+α L
(11)
(12)
4.2. Beam under concentrated load at abscissa c from the left support In this case the deflection is given by Eqs. (13) and (14). • For compressive axial force For 0 ≤ x ≤ c d1 πx y = sin 1−α L
√ √ Q L sin[π α(1 − c/L)] sin(π αx/L) √ √ π P α sin(π α) Q c − 1− x. P L For c ≤ x ≤ L d1 πx y = sin 1−α L √ √ Q L sin(π αc/L) sin(π α(1 − x/L)) + √ √ π P α sin(π α) +
Q c (L − x) . P L • For tensile axial force For 0 ≤ x ≤ c d1 πx y = sin 1+α L √ √ Q L sinh[π α(1 − c/L)] sinh(π αx/L) − √ √ π P α sinh(π α) Q c + 1− x. P L −
(13a)
(13b)
(14a)
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J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
Fig. 4. Beam with initial upward imperfection under a uniformly distributed load q(1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
5. Comprehensive example
For c ≤ x ≤ L y =
πx d1 sin 1+α L √ √ Q L sinh(π αc/L) sinh[π α(1 − x/L)] − √ √ π P α sinh(π α) +
c Q (L − x) . P L
(14b)
In both loading cases the condition that δt = 0 given by Eq. (15) must satisfied. L
Z 0
"
dy dx
2
−
dyo dx
2 # dx −
PL P − = 0. AE S∆
(15)
After the integration of Eq. (15) is carried out, the solutions for P and y can obtained for loading cases (1) and (2) described above.
Consider bending of the weak axis of a 5000 mm simple supported standard steel beam W12 × 22 (I = 1.94 × 106 mm4 , A = 4.18×103 mm2 , E = 200 kN/mm2 ) that is fully restrained against axial displacement (S∆ → ∞). Assume that the initial imperfection can be described by the function yo = d1 sin(π x/L), with the coordinate system as depicted in Fig. 1. Assume further that the beam is laterally restrained so that any torsional instability is precluded. Calculate the induced axial force (P) and the corresponding transverse midspan deflection (ymid ), for the given initial camber (d1 ) and loading conditions indicated by the first 5 columns of Table 1. For comparison purposes, the selected concentrated load (Q) in the example was made equal to wL. It must be noticed that when S∆ → ∞, m 01 becomes equal to the geometrical parameter m 1 . Therefore, based on Eq. √ (4), the condition m 01 ≥ 1 is equivalent to d1 ≤ 2r , where r (= I /A) is the radius of gyration of the beam cross section. This implies that when the maximum initial upward imperfection is less than
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J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
Fig. 5. Beam with initial upward imperfection under a concentrated load Q at L/2 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
Table 1 Steel beam with initial camber under various transverse loadings d1 (mm) 1
m 01 2
Q (kN) 3
c/L 4
30 60 30 60 30 60 −30 −60
2.06 0.52 2.06 0.52 2.06 0.52 2.06 0.52
8 8 8 8
0.5 0.5 0.33 0.33
q (kN/mm) 5
uQ 6
uq 7
1.76 0.88 1.53 0.88 1.60E−03 1.60E−03 1.60E−03 1.60E−03
1.12 0.56 −1.12 −0.56
P(1) (kN) 8 18 129 36 107 70 75 −76 −64
ymid(1) (mm) 9 −27 44 −21 47 −6.7 52 −43 −66
P(2) (kN) 10 18 125 35 106 70 76 −75 −62
ymid(2) (mm) 11 −25 43 −21 46 −6.8 53 −44 −65
1. Subscript (1) indicates the results calculated using the equilibrium method developed by Aristizabal-Ochoa [1] and (2) using the proposed design aids. 2. Negative values correspond to tension forces and to downward displacements.
twice the radius of gyration of the section (2r = 43 mm), only Fig. 3 is needed to determine the induced axial load and midspan deflection of the beam. However, when the initial deflection exceeds the radius of gyration by a factor of more
than two (nonlinear effects are more significant), then Figs. 4–6, or 7 need to be used depending on whether the transverse load was uniform or concentrated at L/2, L/3, or L/4. For downward imperfections (d1 < 0 in Table 1), only Fig. 2 was
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J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
Fig. 6. Beam with initial upward imperfection under a concentrated load at L/3 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
needed regardless of the type of transverse. The range of initial imperfections considered in this numerical example included all these cases. For clarity, Table 1 also summarizes the calculated geometrical parameter m 1 (Eq. (4)) and the non-dimensional load parameters u Q and u q . These are the input values needed to use the proposed design aids (notice that for concentrated loads in Figs. 2 and 3, the user must enter the load parameter u Q multiplied by sin(π c/L). The results obtained using the design aids are listed in columns 8 and 9 of Table 1 and are compared with those obtained using the alternative theoretical formulation proposed by Aristizabal-Ochoa [1]. The results between the two methods are practically identical to each other. Some interesting observations can be made from this numerical example: (a) Increasing the initial imperfection by a factor of two (from 30 to 60 mm) can increase the induced axial compression by a factor as high as seven (for a concentrated load at midspan).
(b) The magnitude of the induced axial load can be as high as 16 times the applied transverse load. (c) The induced axial load does not increase as much with an increase in the initial imperfection for uniform loads as compared to equivalent concentrated loads. (d) Midspan deformations due to uniform loads are smaller than those with equivalent concentrated loads. 6. Summary and conclusions A straightforward approach that allows determining the induced axial force and the corresponding transverse deflection of a simple supported axially restrained-imperfect beam under a uniformly distributed load or a single transverse load is presented. Closed-form formulas based on classical beam theory are also included to allow comparison with the suggested design aids. The results obtained with the two methods are practically identical. The main advantage of the proposed formulation resides on its simplicity as compared to the
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
569
Fig. 7. Beam with initial upward imperfection under a concentrated load at L/4 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
equilibrium-based formulation, especially for practical cases (m 01 > 1/15); requiring only the use of a hand calculator. References [1] Aristizabal-Ochoa JD. Second-order axial deflections of imperfect 3-D beam–column. ASCE Journal of Engineering Mechanics 2000;126(11):
1201–8. [2] Smith-Pardo JP, Aristizabal-Ochoa JD. Buckling with reversals of axially restrained beam–column with initial imperfections. ASCE Journal of Engineering Mechanics 1999;125(4):401–9. [3] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. Engineering society monographs. New York: McGraw-Hill; 1959. p. 4–32. [4] Timoshenko SP, Gere JM. Theory of elastic stability. Engineering society monographs. NY: McGraw-Hill; 1961. p. 1–43, 279–317.
Review article
Second-order axial force and midspan deflection in a simple supported beam axially restrained J. Paul Smith-Pardo a , J. Dario Aristizabal-Ochoa b,∗ a Berger/Abam Engrs. Inc., 33301 9th Ave. South, Suite 300, Federal Way, WA 98003, United States b National University, School of Mines, Medellin, Colombia
Received 28 November 2006; received in revised form 13 January 2007; accepted 14 March 2007 Available online 23 May 2007
Abstract Design aids for the calculation of the induced axial force and mid-span deflection in a simple supported beam axially restrained with an initial imperfection (camber) under transverse loads are presented. In addition, closed-form equations based on equilibrium developed by AristizabalOchoa [Aristizabal-Ochoa JD. Second-order axial deflections of imperfect 3-D beam–column. ASCE Journal of Engineering Mechanics 2000; 126(11): 1201–8] are also presented in order to verify the validity of the proposed method and design aids. Both, equations and design aids find great applicability in the analysis and design of slender flexible beams that are restrained axially. The method proposed and design aids are based on a general model presented by the authors in a previous paper [Smith-Pardo JP, Aristizabal-Ochoa JD. Buckling with reversals of axially restrained beam–column with initial imperfections. ASCE Journal of Engineering Mechanics 1999; 125(4): 401–9]. A comprehensive example is included to show the simplicity and validity of the proposed method. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Beams; Beam-columns; Buckling; Computer applications; Elastic behavior; Non-linear behavior; P–δ effects; Second-order analysis; Stability; Stiffness; Structural analysis
Contents 1. 2. 3.
4.
5. 6.
Introduction............................................................................................................................................................................ 561 Structural model ..................................................................................................................................................................... 562 Application–design aids ........................................................................................................................................................... 563 3.1. Beam–column under a uniformly distributed lateral load ................................................................................................... 563 3.2. Beam–column under concentrated load located at a distance c from the left support.............................................................. 563 Verification equations .............................................................................................................................................................. 564 4.1. Beam–column under a uniformly distributed load q .......................................................................................................... 564 4.2. Beam under concentrated load at abscissa c from the left support........................................................................................ 565 Comprehensive example .......................................................................................................................................................... 566 Summary and conclusions ........................................................................................................................................................ 568 References ............................................................................................................................................................................. 569
1. Introduction ∗ Corresponding address: National University of Colombia, Civil Engineering, Calle 9c # 15-165 Casa 2Urbanizacion Villaverde, Medellin, Antioquia, Colombia. Tel.: +57 42686218; fax: +57 4 4255152. E-mail addresses: [email protected], [email protected] (J.D. Aristizabal-Ochoa).
c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.03.011
Beams supporting lateral loads are usually analyzed as extensionless members under flexure and shear. However, if supports are restraining the relative elongation between the ends of a member, axial forces are induced. Such induced
562
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
Nomenclature A an bk ci dj E I i j k L mn m 01 n P q s S∆ uq
y α δt
Beam cross-sectional area; Coefficients of the Fourier’s series of the deflection of the beam; Distance from end A to point of application of concentrated moment Mk ; Distance from end A to point of application of concentrated load Q i ; Magnitude of the initial imperfection corresponding to mode j; Beam Young modulus; Beam moment of inertia; Subscript that numbers the concentrated loads (varying from 1 to m); Subscript that numbers the coefficients of the initial deflection (varying from 1 to s); Subscript that numbers the concentrated moments (varying from 1 to r ); Beam span; 4I /[A(ndn )2 ] = series of geometric parameters; m 1 [1 + (AE/L)/S∆ ]; Subscript to number the coefficients of a Fourier’s series (varying from 1 to ∞); Axial force (+compression); Distributed transverse load; Subscript to number the number of terms in Eq. (1); Axial stiffness of spring at B; 4q L 4 /(π 5 E I d1 ) for a uniformly distributed load or u Q = 2Q L 3 Sin(π c/L)/(π 4 E I d1 ) for a concentrated load at c from the left support; Beam total transverse deflection; P L 2 /(π 2 E I ); Beam total axial elongation.
forces affect the overall response of the beam. Well-known phenomena such as snap-through, snap-back and reversal of lateral deflections are caused in simple beams with initial camber that are axially restrained. The effects of transverse loads and deflections on the axial stiffness, pre- and postbuckling behavior, and stability including the phenomena of snap-through, snap-back and reversal of deflections of a beam–column have been studied extensively during the last two decades by several researchers. Aristizabal-Ochoa [1] developed a complete theory for the elastic axial stiffness of a doubly symmetric beam–column including the 3-D effects of lack of straightness, transverse forces and moments and sidesway. Complete details on these phenomena and the overall response of a simple beam with an arbitrary imperfection (camber) and under any transverse load have been given by Smith-Pardo and Aristizabal-Ochoa [2]. The main objective of this publication is to present a particular application of the formulation developed by the authors [2]. The application consist of a set of design aids for the calculation of the induced axial force and mid-span
deflection of an imperfect simply supported beam axially restrained. In addition, closed form equations based on an equilibrium method developed by Aristizabal-Ochoa [1] are also presented in order to verify the validity of the proposed design aids. A comprehensive example is included that shows the simplicity and validity of the proposed charts. 2. Structural model The structural model is identical to that previously presented by the authors [2] and will be described herein for easy reference. It consists of a prismatic beam–column with a single symmetrical cross section shown schematically in Fig. 1. The longitudinal x-axis is defined by the centroid of the crosssectional area at the extremes A and B. The beam AB is assumed to be: (1) Made of an isotropic homogeneous linear-elastic material with a modulus of elasticity, E, and axially restrained at the end B by a spring with stiffness S∆ . (2) Prismatic with a single-symmetrical cross section of area A, moment of inertia in the plane of bending I , and span L. (3) Slightly deformed in the plane of bending with an initial out-of-straightness modeled by a plane curve given by Eq. (1). yo =
s X j=1
d j sin
jπ x . L
(1)
Where: d1 , d2 , . . . ds = coefficients in a sine series describing the magnitudes of the initial imperfection. (4) Loaded and deformed in the plane of bending as indicated in Fig. 1. In addition to the axial load, P, three types of transverse loads were considered: (i) a uniformly distributed load q along its entire span; (ii) a series of concentrated loads Q i (with i varying from 1 to N Q) located at distances ci from the left support and; and (iii) a series of concentrated moments Mk (with k varying from 1 to NM) located at distances bk from the left support. Positive directions are indicated in Fig. 1. (5) The study beam–column undergoes small strains with plane sections remaining plane. Torsion and shear effects are neglected throughout. To determine the transverse deflections y, and the axial elongation between the ends of the member δt , Eqs. (2) and (3) were developed by Smith-Pardo and Aristizabal-Ochoa [2]. s ∞ X n 2 dn nπ x nπ x X − an sin sin 2 L L n −α n=1 n=1 ( " 2 # s X an n2 m1 δt 1 1− = − 2 P L/AE m 1 α n=1 m n dn n −α 2 ) ∞ X nan AE/L − − 1+ . d1 S∆ n=s+1
y=
(2)
(3)
J.P. Smith-Pardo, J.D. Aristizabal-Ochoa / Engineering Structures 30 (2008) 561–569
563
Fig. 1. Structural model. P L2
Where: α = π 2 E I ; and m 1 , m 2 , . . . m n, . . . m s = nondimensional geometrical parameters given by: mn =
4I A(ndn )2
(4)
an = coefficients given by Eqs. (5) and (6). • For n = 1, 3, 5, 7, . . . (odd numbers) " NQ nπc 2L 2 L X i an = 3 Q i sin 2 L π E In (n − α) π n i=1 # NM X nπbk 2q L 2 − Mk cos + 2 2 . L π n k=1 • For n = 2, 4, 6, 8, . . . (even numbers) " NQ nπc 2L 2 L X i an = 3 Q i sin 2 L π E In (n − α) π n i=1 # NM X nπbk − Mk cos . L k=1
+
u q /25 25 − α
2
+ · · · = 1 − m 01 α
π x u /27 1 − uq y 3π x q = sin − sin d1 1−α L 9−α L u q /125 5π x − sin + ···. 25 − α L
(7)
(8)
4
Where: u q = π45 Eq ILd1 (non-dimensional parameter), m 01 = m 1 [1 + (AE/L)/S∆ ] and m 1 is as defined in Eq. (4).
(5)
(6)
Eqs. (2) and (3) were derived based on energy principles and are valid when the beam–column is either under axial tension or compression and for upward or downward initial imperfection. 3. Application–design aids In the two particular applications considered below it is assumed that the axial extension is zero (δt = 0) and that the initial beam imperfection is completely defined by the first term of the sine series in Eq. (1) only. 3.1. Beam–column under a uniformly distributed lateral load For a beam–column under a uniformly distributed transverse load q, Eqs. (2)–(6) are reduced to (7) and (8) from which the induced axial force P, and the corresponding transverse deflection y can be found, respectively. 1 − uq 2 u q /9 2 + 1−α 9−α
3.2. Beam–column under concentrated load located at a distance c from the left support For a beam–column under a single concentrated load Q located at a distance c from the left support, the transverse deflection y and the induced axial force P can be obtained from the simultaneous solution of Eqs. (9) and (10). 1 − u Q sin(π c/L) 2 u Q sin(2πc/L)/2 2 + 1−α 4−α 2 u Q sin(3πc/L)/3 + + · · · = 1 − m 01 α (9) 9−α πx π c u /9 y 1 − uQ 3π x Q = sin sin + sin d1 1−α L L 9−α L 3πc u Q /25 5π x 5πc × sin + sin sin + ··· L 25 − α L L (10) where: u Q =
2 Q L3 π 4 E I d1
(non-dimensional parameter).
Cases > 1 and d1 < 0 As shown by Timoshenko and Woinowsky-Krieger [3], Timoshenko and Gere [4] and also by Smith-Pardo and AristizabalOchoa [2], for m 01 > 1, the solutions of Eqs. (7)–(10) can be obtained with sufficient accuracy (<5% of the exact solution) by taking the first term of each series only. The authors also showed that this is also the case if the initial imperfection is downwards (d1 < 0) regardless of the value of m 01 . This results in simplified equations that have the same form for both loading cases. Thus, a single set of design aids can be formulated for m 01
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Fig. 2. Beam with initial downward imperfection under a uniformly distributed load q or a concentrated load Q at c from the left support: (a) induced axial force (tension); (b) transverse mid-span deflection.
both uniformly distributed and concentrated loads as shown in Figs. 2 and 3. Fig. 2(a) and (b) are provided to calculate the induced axial force (tension) P, and mid-span deflection ymidspan , in a beam with an initial downward imperfection under a uniformly distributed or a concentrated load. Fig. 3(a) and (b) are provided for the case when the initial imperfection is upwards and the induced axial force is compression. Case 1/15 < m 01 < 1 and d1 > 0 If the initial imperfection is upwards and 1/15 < m 01 < 1, Smith-Pardo and Aristizabal-Ochoa [2] showed that it is sufficiently accurate to take the first three non-zero terms in each series of Eq. (7) through (10) in order to obtain reasonable estimates of the induced axial force and mid-span deflection (<5% of the exact solution). Notice that for this range of values of m 01 , the resulting equations for uniformly distributed (Eqs. (7) and (8)) and concentrated loads (Eqs. (9) and (10)) do not have the same form, then separate design aids are necessary for each load case. Design aids, obtained from Eqs. (7)–(10) using three terms of each series, are provided by means of Figs. 4–7
for a beam with an initial upward imperfection under a uniformly distributed load and concentrated loads at L/2, L/3, and L/4 from the left support, respectively. For values of m 01 < 1/15, it becomes necessary to use more than three terms in the series of Eq. (7) through (10). However, for this particular range, design aids were not developed due to their limited use in practice (unrealistic initial imperfection of the beam–column). 4. Verification equations In order to verify the proposed design aids, closed-form equations obtained using equilibrium principles (AristizabalOchoa [1]) are presented below for the beam–column of Fig. 1. 4.1. Beam–column under a uniformly distributed load q In this case the deflection is given by Eqs. (11) and (12).
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Fig. 3. Beam with initial upward imperfection under a uniformly distributed load q or concentrated load Q at c from the left support (m 01 > 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
• For compressive axial force √ q L 2 cos(π/2 αx/L) q x(L − x) y = 2 −1 − √ 2P π Pα cos(π/2 α) d1 πx + sin . 1−α L • For tensile axial force √ √ q L2 cosh(π/2 α − π αx/L) y = 2 1− √ π Pα cosh(π/2 α) −
d1 πx q x(L − x) + sin . 2P 1+α L
(11)
(12)
4.2. Beam under concentrated load at abscissa c from the left support In this case the deflection is given by Eqs. (13) and (14). • For compressive axial force For 0 ≤ x ≤ c d1 πx y = sin 1−α L
√ √ Q L sin[π α(1 − c/L)] sin(π αx/L) √ √ π P α sin(π α) Q c − 1− x. P L For c ≤ x ≤ L d1 πx y = sin 1−α L √ √ Q L sin(π αc/L) sin(π α(1 − x/L)) + √ √ π P α sin(π α) +
Q c (L − x) . P L • For tensile axial force For 0 ≤ x ≤ c d1 πx y = sin 1+α L √ √ Q L sinh[π α(1 − c/L)] sinh(π αx/L) − √ √ π P α sinh(π α) Q c + 1− x. P L −
(13a)
(13b)
(14a)
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Fig. 4. Beam with initial upward imperfection under a uniformly distributed load q(1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
5. Comprehensive example
For c ≤ x ≤ L y =
πx d1 sin 1+α L √ √ Q L sinh(π αc/L) sinh[π α(1 − x/L)] − √ √ π P α sinh(π α) +
c Q (L − x) . P L
(14b)
In both loading cases the condition that δt = 0 given by Eq. (15) must satisfied. L
Z 0
"
dy dx
2
−
dyo dx
2 # dx −
PL P − = 0. AE S∆
(15)
After the integration of Eq. (15) is carried out, the solutions for P and y can obtained for loading cases (1) and (2) described above.
Consider bending of the weak axis of a 5000 mm simple supported standard steel beam W12 × 22 (I = 1.94 × 106 mm4 , A = 4.18×103 mm2 , E = 200 kN/mm2 ) that is fully restrained against axial displacement (S∆ → ∞). Assume that the initial imperfection can be described by the function yo = d1 sin(π x/L), with the coordinate system as depicted in Fig. 1. Assume further that the beam is laterally restrained so that any torsional instability is precluded. Calculate the induced axial force (P) and the corresponding transverse midspan deflection (ymid ), for the given initial camber (d1 ) and loading conditions indicated by the first 5 columns of Table 1. For comparison purposes, the selected concentrated load (Q) in the example was made equal to wL. It must be noticed that when S∆ → ∞, m 01 becomes equal to the geometrical parameter m 1 . Therefore, based on Eq. √ (4), the condition m 01 ≥ 1 is equivalent to d1 ≤ 2r , where r (= I /A) is the radius of gyration of the beam cross section. This implies that when the maximum initial upward imperfection is less than
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Fig. 5. Beam with initial upward imperfection under a concentrated load Q at L/2 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
Table 1 Steel beam with initial camber under various transverse loadings d1 (mm) 1
m 01 2
Q (kN) 3
c/L 4
30 60 30 60 30 60 −30 −60
2.06 0.52 2.06 0.52 2.06 0.52 2.06 0.52
8 8 8 8
0.5 0.5 0.33 0.33
q (kN/mm) 5
uQ 6
uq 7
1.76 0.88 1.53 0.88 1.60E−03 1.60E−03 1.60E−03 1.60E−03
1.12 0.56 −1.12 −0.56
P(1) (kN) 8 18 129 36 107 70 75 −76 −64
ymid(1) (mm) 9 −27 44 −21 47 −6.7 52 −43 −66
P(2) (kN) 10 18 125 35 106 70 76 −75 −62
ymid(2) (mm) 11 −25 43 −21 46 −6.8 53 −44 −65
1. Subscript (1) indicates the results calculated using the equilibrium method developed by Aristizabal-Ochoa [1] and (2) using the proposed design aids. 2. Negative values correspond to tension forces and to downward displacements.
twice the radius of gyration of the section (2r = 43 mm), only Fig. 3 is needed to determine the induced axial load and midspan deflection of the beam. However, when the initial deflection exceeds the radius of gyration by a factor of more
than two (nonlinear effects are more significant), then Figs. 4–6, or 7 need to be used depending on whether the transverse load was uniform or concentrated at L/2, L/3, or L/4. For downward imperfections (d1 < 0 in Table 1), only Fig. 2 was
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Fig. 6. Beam with initial upward imperfection under a concentrated load at L/3 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
needed regardless of the type of transverse. The range of initial imperfections considered in this numerical example included all these cases. For clarity, Table 1 also summarizes the calculated geometrical parameter m 1 (Eq. (4)) and the non-dimensional load parameters u Q and u q . These are the input values needed to use the proposed design aids (notice that for concentrated loads in Figs. 2 and 3, the user must enter the load parameter u Q multiplied by sin(π c/L). The results obtained using the design aids are listed in columns 8 and 9 of Table 1 and are compared with those obtained using the alternative theoretical formulation proposed by Aristizabal-Ochoa [1]. The results between the two methods are practically identical to each other. Some interesting observations can be made from this numerical example: (a) Increasing the initial imperfection by a factor of two (from 30 to 60 mm) can increase the induced axial compression by a factor as high as seven (for a concentrated load at midspan).
(b) The magnitude of the induced axial load can be as high as 16 times the applied transverse load. (c) The induced axial load does not increase as much with an increase in the initial imperfection for uniform loads as compared to equivalent concentrated loads. (d) Midspan deformations due to uniform loads are smaller than those with equivalent concentrated loads. 6. Summary and conclusions A straightforward approach that allows determining the induced axial force and the corresponding transverse deflection of a simple supported axially restrained-imperfect beam under a uniformly distributed load or a single transverse load is presented. Closed-form formulas based on classical beam theory are also included to allow comparison with the suggested design aids. The results obtained with the two methods are practically identical. The main advantage of the proposed formulation resides on its simplicity as compared to the
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Fig. 7. Beam with initial upward imperfection under a concentrated load at L/4 from the left support (1/15 < m 01 < 1): (a) induced axial force (compression); (b) transverse mid-span deflection.
equilibrium-based formulation, especially for practical cases (m 01 > 1/15); requiring only the use of a hand calculator. References [1] Aristizabal-Ochoa JD. Second-order axial deflections of imperfect 3-D beam–column. ASCE Journal of Engineering Mechanics 2000;126(11):
1201–8. [2] Smith-Pardo JP, Aristizabal-Ochoa JD. Buckling with reversals of axially restrained beam–column with initial imperfections. ASCE Journal of Engineering Mechanics 1999;125(4):401–9. [3] Timoshenko SP, Woinowsky-Krieger S. Theory of plates and shells. Engineering society monographs. New York: McGraw-Hill; 1959. p. 4–32. [4] Timoshenko SP, Gere JM. Theory of elastic stability. Engineering society monographs. NY: McGraw-Hill; 1961. p. 1–43, 279–317.