Computers and Structures 90–91 (2012) 28–41

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Computers and Structures

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Elastic sectional stress analysis of variable section piers subjectedto three-dimensional loads

Lucian Stefan, Pierre Léger ⇑Dept. of Civil, Geological, and Mining Engineering, École Polytechnique, P.O. Box 6079, Station CV, Montréal, Québec, Canada H3C 3A7

a r t i c l e i n f o

Article history:Received 12 April 2011Accepted 10 October 2011Available online 8 November 2011

Keywords:Elastic sectional analysisShear stressVariable sectionTapered sectionThree-dimensional loadsWarping function

0045-7949/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.compstruc.2011.10.012

⇑ Corresponding author. Tel.: +1 (514) 340 4711x37E-mail addresses: lucian.stefan@polymtl.ca (L. Ste

(P. Léger).

a b s t r a c t

The elastic stress analysis of beam-column structures of uniaxial symmetrical variable cross-sections isdeveloped using an extension of Euler–Bernoulli beam theory. The applied loads are general consideringaxial (P), flexure (Mx–My), shear (Vx–Vy), and torsion (T). Three-dimensional analytical solutions for nor-mal and shear stresses are derived using sectional analysis with warping functions for variable bound-aries in elevation. The differential equations of equilibrium and deformations are accounting for thevariations in the geometrical properties of the cross-section and related boundary conditions. The strongform solution is then written in a weak form that is implemented in a 2D sectional finite element (FE)code assuming linear normal stress distribution. Three application examples are presented to validatethe proposed sectional approach and illustrate its accuracy by comparing with results from full 3D FEanalyses: (a) a slender rectangular section pier with a sloped boundary, (b) a bulk rectangular section but-tress with unsymmetrical slopes, and (c) a pier (squat wall) for a hydraulic structure. When the assump-tion of linear distribution for normal flexural stress is satisfied, the proposed sectional approach producesresults within 1% of 3D FE with much reduced computational efforts. For bulk and squat walls the stressfield distribution are very similar to 3D FE while the stress intensity shows some variations. This is ofmajor practical significance because the proposed approach allows performing first a series of simplifiedyet acceptable sectional analyses in safety assessment of the three-dimensional type of structures consid-ered herein.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Several structures are made from the assembly of tapered ele-ments with sections possessing a uniaxial symmetrical axis witharbitrary boundaries (Fig. 1). These structures are subjected to 3Dload combinations. For spillway piers (Fig. 1(d)) those are arisingfrom operational conditions, earthquakes or lateral impact of icefloes. Thus, 3D actions imparted by P (axial), Mx, My (moments),Vx, Vy (shear forces) and T (torsion) loads must be considered instress analysis. Elastic normal and shear stress distributions arefirst computed in safety assessment. Solutions for elastic normalstresses arising from the PMM problem are well known for arbi-trary cross-sections [1–3]. Solutions for the shear stresses arisingfrom the VVT problem are more complex involving warping ofthe section [4–6]. For structures with variable sections along thevertical axis, such as typical hydraulic structures (Fig. 2), it is usefulto perform first series of sectional analyses along the height to as-sess the stress field for critical planes, such as concrete lift joints,before undergoing three dimensional finite element (FE) analyses,

ll rights reserved.

12; fax: +1 (514) 340 5881.fan), pierre.leger@polymtl.ca

if necessary. Mason and Herrmann [7] presented a general methodfor sectional analysis including warping of arbitrary constant sec-tion elements due to biaxial shear forces; for these elements, a FEformulation for Saint–Venant torsion was also proposed by Herr-mann [8]. Sapountzakis and Mokos [9] suggested an estimative for-mulation for sectional analysis of variable section elementssubjected to non-uniform torsion. However, there is no formulationfor sectional analysis of variable section columns subjected tothree-dimensional loads, P–Vx–Vy–Mx–My–T, having arbitrarydeformation patterns (coupled ‘‘wall’’/in-plane and ‘‘plate’’/out-of-plane behaviors).

1.1. Review of previous work

Shear stresses s are generally produced by the presence of ashear force or of a torsion moment which represents the resultantof these shear stresses. When stresses produce neither force normoment as resultant, the distribution is self-equilibrated. For exam-ple, this situation arises for the normal stresses r due to the non-uniform torsion when they have no resultant force or bendingmoment [9]. The shear distribution determination for variablesection elements can be seen as the search of a self-equilibratedstress field which is added to the shear distribution found for the

bs, hs sizes for rectangular analyzed sectionE, G, m material properties: Young modulus, shear modulus,

and Poisson ratiof1P, f2P, fIx, f1Iy, f2Iy, hv, hnx, hny right hand functions for strong form

of pure shear formulationJ, Cw torsion and warping constantsL, Ls column height and position of the reference sectionLn length of the in-plane outward normal of curve C;~nLns length of the outward 3D normal of surface X; ~ns

l, m, n x, y, and z components of the in-plane outward normalof curve C;~n

ls, ms, ns x, y, and z components of the outward 3D normal of sur-face X; ~ns

P, ex, ey, Mx, My, Vx, Vy, T direct applied loads: axial force with twoeccentricities, biaxial moments, biaxial shear forces, tor-sion moment

-RMx, RMy total resultant bending moments in the x and y direc-tion (sum of contribution of direct applied moments,eccentrically axial force and biaxial shear forces); notethat mathematically, RMx is the algebraic opposite ofthe bending moment in x direction

TB torsion due to the difference between the centroid andshear center of analyzed section accounting for 3D var-iation of element boundary

Ttot total torsion for analysis; sum of direct torsion, torsiondue to the difference between the centroid of top andanalyzed section, torsion due to the difference betweenthe centroid and shear center of analyzed section

u, v, w displacements in the x, y, and z direction

xDv, yDv coordinates of variable shear center (when shear forcesare not null)

xg, A, Ix, Iy, xgP geometrical properties of the analyzed section:centroid, area, x and y moments of inertia, and the cen-troid coordinate of the section where P is applied (topsection)

au,ad angle between the outward 3D normal of surface Xu, Xd

and the horizontalex, ey, ez, cxy, cxz, cyz normal strain in the x, y, and z direction and

respectively shearing strain in the xy, xz, and yz planesg 2D warping function for computation of normal stresses

in the x and y directionh twist anglerx, ry, rz, rW normal stresses in the x, y, and z direction; the

additional subscript W stands for ‘‘warping’’sxy, sxz, syz shear stresses in the xy, xz, and yz planes; the addi-

tional subscripts S, SV, W, T stand for ‘‘shear’’, ‘‘Saint–Ve-nant’’, ‘‘warping’’ and ‘‘total’’

w, wG 2D warping functions for pure shear formulation(wG = w � G)

C closed curve bounding the reference/analyzed section;the additional subscripts u or d stands for upstreamand downstream

X outward surface boundary of the pier having the equa-tion F1(x,y) + F2(z) with F1 and F2 being arbitrary func-tions; the additional subscripts u or d stands forupstream and downstream

W, WS 2D warping functions for pure torsion formulation: pri-mary and secondary warping function

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 29

constant section; thus, the self-equilibrated field can be inter-preted as a correction to the stress field for constant section [10].

The problem of finding the shear stress of constant section ele-ments is separated in two boundary value problems: one for theshear force without torsion, and the other for the pure torsion mo-ment [5,11,12]. The analytical solution for the wedge loaded at itsapex was employed to solve the shear problem for variable sectionelements [11,13,14]. When the hypotheses of plane stress are sat-isfied, based on equilibrium equations (neglecting Poisson ratio) ofthe Theory of Elasticity (TE) the analytical solution for shear stres-ses was determined for sections subjected to uniaxial shear force[15–17]. Analytical solutions using asymptotic techniques are alsoavailable for variable sections subjected to three-dimensionalloads as long as the beam length approaches infinity [18]. Severalauthors utilize the idea of the Jourawski formula for equilibrium

Fig. 1. Structural elements with variable sections made of (a) wood, (b) steel, and (c

conditions written for variable section elements to determine ana-lytically the shear stress [10,19–21]; this solution is well adaptedto design due to its accessible form and generalization for one-dimensional stress states. Another strategy to find shear stressesis to propose a shear distribution and to check it against equilib-rium conditions [10,22]. The above mentioned methods were val-idated experimentally or with finite element (FE) formulations[10,16,21]. For constant section elements, various FE solutions forfinding the shear stress distribution on arbitrary sections are avail-able either based on warping or on stress functions [4,5,7,12,23].An abundant literature is found for the matrix analysis of variablesection elements subjected to three-dimensional loads [10,24–28]. The matrix analysis is developed on energy principles whereonly the product between stress and strain is accounted for. Fromthis point of view, the actual stress distribution is not important

) reinforced concrete; (d1) unit slice of a concrete dam, and (d2) spillway pier.

Fig. 2. Geometry and notations: (a) 3D view; (b) notation for boundary conditions (linear slope variation): (b1) vertical cut, (b2) reference section, and (b3) relationshipbetween surface ~ns and curve ~n outside unit normal.

30 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

as long as the same energy is computed with two differentdistributions; the lack of identical satisfaction of boundary condi-tions at section level is also of no concern. If a particular distribu-tion is validated by using an energy indicator (e.g., displacement)then the distribution is considered acceptable [29–32]. This is thecase of the Timoshenko coefficient [6,33] or the transformationof continuously variable elements in stepped variable elementsas often done in practice. For sectional stress analysis this energeticequivalence is not directly useful because the accurate distributionis desired. However, none of the above methods deals with three-dimensional loading scenario P–Vx–Vy–Mx–My and arbitrarycolumn displacements.

The evaluation of shear stresses due to torsion T was addressedby many authors especially for thin-walled bars. The solution foruniform torsion (without normal stresses) was mathematicallysolved by Saint–Venant and later formulated in FE using warpingor stress functions [8,34–36]. The warping or non-uniform torsionproduces self-equilibrated normal stress along the bar as well asshear stresses due to the normal stress variation. The solution forwarping torsion was discussed for both constant and variable sec-tion elements [9,37–39]. However, the real 3D boundary unit nor-mal and the z variation of warping function were neglected.

This paper provides a uniform and general FE formulation forthe sectional analysis of variable sections structural elements sub-jected to three-dimensional loads. The element has arbitrary unidi-rectional longitudinal variability and arbitrary uniaxialsymmetrical cross section. It accounts for: (a) compatibility andequilibrium equations with respect to material law (initially lim-ited to isotropic linear elastic material); (b) boundary conditionsof the actual 3D element geometry; and (c) Poisson ratio. To vali-date the proposed sectional analysis method three applicationsare shown ranging from a typical tapered rectangular section pierto the more challenging analysis of a squat wall. The implementa-tion of this sectional approach for variable geometry in existentsectional analysis software of constant section elements can be di-rectly performed using the developments presented herein.

2. Problem formulation and solution strategy

The main idea of the proposed method is that the sectionalanalysis at elevation z of a column with variable section can be per-formed at the same elevation z of a constant section column byextending the differential formulation of Euler–Bernoulli beamtheory including warping deformation. There are three conditionsto satisfy: (a) the constant section must be identical with the sec-tion at elevation z; (b) the function in the right term of the differ-

ential equation must account for the variation of the geometricalproperties of the section, and (c) the boundary conditions must re-spect the spatial variable geometry. By doing this, existing soft-ware developed for sectional analysis of constant sectioncolumns can be modified to analyze variable section columns. Inthis paper, the meaning of ‘‘column’’ is ‘‘structural element’’ sub-jected to three-dimensional forces: P–Vx–Vy–Mx–My–T.

The following assumptions of the proposed method create theframework for the presentation. These assumptions are (Fig. 2):

(a) Plane sections before deformation for bending remain planeafter deformation (linear e,r).

(b) The section is symmetrical according to Ox; along the ele-ment height (Oz) the section size can vary in the Ox directionbut cannot change in the Oy.

(c) The centroidal Cartesian system Oxy is employed to computethe geometrical properties of the section as well as the pla-nar rz distribution (isotropic linear elastic material):

rzðx; y; zÞ ¼ �Eðr0zðzÞ þ r1zðzÞðx� xgðzÞÞ þ r2zðzÞyÞ ð1Þ

r0zðzÞ ¼ �1E� PAðzÞ ð2Þ

r1zðzÞ ¼ �1E�My þ Pex � VxðL� Ls � zÞ þ PðxgP � xgðzÞÞ

IyðzÞ

� �

¼ �1E� �RMyðzÞ

IyðzÞð3Þ

r2zðzÞ ¼ �1E

Mx þ Pey � VyðL� Ls � zÞIxðzÞ

� �¼ �1

E� �RMxðzÞ

IxðzÞ

ð4Þ

RMxðzÞ ¼ �Mx � Pey þ VyðL� Ls � zÞ ð5Þ

RMyðzÞ ¼ My � Pðex þ xgP � xgðzÞÞ þ VxðL� Ls � zÞ ð6Þ

where forces are applied on the centroid of the top section: P = axialforce; ex, ey = eccentricities of P with respect to the centroid of topsection; Mx, My = concentrated bending moments; Vx, Vy = shearforces; xg, A, Ix, Iy are the centroidal coordinates and geometricalproperties of the current section: the area, and the x and y momentsof inertia (they are z functions); xgP is the centroid coordinate of thetop section; L, Ls is the column height and respectively the positionof the reference section according to the column base; E is theYoung modulus; m is the Poisson ratio. Note that RMy representsthe resultant My moment on the section, while RMx is the algebraicopposite of the resultant Mx moment; they should be seen only asnotations.

Fig. 3. Algorithm to compute, for structural elements with variable section, theshear stresses due to three-dimensional loads.

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 31

(d) The contribution of normal stresses rx and ry to the normalstrain are neglected; the shear strain (stress) cxy(sxy) is zero.

These assumptions can be progressively eliminated by follow-ing the steps of the procedure described below (e.g., uniform dis-tributed loads, non-linear normal stress distribution, etc.). Adetailed flowchart of the proposed algorithm is given in Fig. 3.

2.1. Geometry and notations

A detailed view of the analysis model is shown in Fig. 2. Theoutside surface boundary of the column is denoted X; the columnis bounded to the left by a surface called Xu and to the right by Xd;the left side of the column is called ‘‘u’’ (upstream) and its rightside ‘‘d’’ (downstream), a terminology borrowed from hydraulicstructures. The plane containing the reference section is Oxy, at ele-vation Ls according to the column base; the closed curve whichbounds the reference section is C; the intersection between Oxyand Xu(Xd) is an open curve denoted Cu(Cd). The cross-sectionof the column is arbitrary with a single axis of symmetry Ox. Whenthe surface Xu(Xd) is planar, the angle between the plane and thevertical is called au(ad); see Fig. 2(b1) for the positive direction ofthis angle. These angles can be positive or negative.

The analyzed (current) section can be at any elevation z mea-sured from the reference section, but from a computational pointof view it is convenient to be at z = 0. To do this, as the referencesection can be anywhere on the column, this section is positionedat the elevation of the analyzed section. The displacements in x, y,and z direction are u, v and w; the normal and shearing strains aredenoted by the letters e and c (G is the shear modulus):

ex¼@u@x¼1

Eðrx�mðryþrzÞÞ; cxy¼

@u@yþ@v@x¼ sxy

G

ey¼@v@y¼1

Eðry�mðrxþrzÞÞ; cxz¼

@u@zþ@w@x¼ sxz

G; G¼ E

2ð1þmÞ

ez¼@w@z¼1

Eðrz�mðrxþryÞÞ; cyz¼

@v@zþ@w@y¼ syz

Gð7Þ

From the TE [11], the equilibrium equations and boundary condi-tions of a solid in a three-dimensional stress state are (the normaland shear stresses are denoted by r and s):

@rx@x þ

@sxy

@y þ@sxz@z ¼ 0

@sxy

@x þ@ry

@y þ@syz

@z ¼ 0@sxz@x þ

@syz

@y þ@rz@z ¼ 0

8>><>>: ð8Þ

rxls þ sxyms þ sxzns ¼ 0sxyls þ ryms þ syzns ¼ 0sxzls þ syzms þ rzns ¼ 0

8><>: ð9Þ

In (9) ls, ms, ns are the direction cosines of the external normal to thesurface of the body at the point under consideration (they are thecomponent of the unit normal vector ~ns ¼ hls;ms;nsiÞ; the unit out-ward normal vector of the boundary curve C is ~n ¼ hl;m;ni.

2.2. Strong form for shear without torsion

From (7), by imposing rx = ry = 0, the displacement distributionis determined by integration (the form of arbitrary functions /(y,z)and u(x,z) comes from the condition cxy = 0):

uðx;y;zÞ¼�mE

Zrzdxþ/ðy;zÞ; /ðy;zÞ¼�mr1z

y2

2�hðzÞyþbuðzÞ

ð10Þ

vðx;y;zÞ¼�mE

Zrzdyþuðx;zÞ; uðx;zÞ¼�mr2z

x2

2þhðzÞxþbvðzÞ

ð11Þ

wðx;y;zÞ¼�Zðr0zþr1zðx�xgÞþr2zyÞdzþwðx;yÞ ð12Þ

The function w(x,y) is the sought warping function. The shearstrains are computed with the above displacements substituted in(7):

cxz¼ m xdr0z

dzþ x2�y2

2�xgx

� �dr1z

dz�xr1z

dxg

dzþxy

dr2z

dz

� �

�ydhdzþdbu

dz�Z

r1zdzþ@w@x

ð13Þ

cyz¼ m ydr0z

dzþðx�xgÞy

dr1z

dz�yr1z

dxg

dzþy2�x2

2�dr2z

dz

� �

þxdhdzþdbv

dz�Z

r2zdzþ@w@y

ð14Þ

Until Eq. (12) there is no difference between the formulations forconstant and variable section columns. The difference appears onlyin (13) and (14) in the derivatives of riz and xg. The terms in h(z)pertain to torsion and are neglected in the determination of thewarping function for shear; they will be included in the hereinafterformulation for torsion.

32 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

The compatibility equations need not be considered as the dif-ferential formulation is based on an assumed displacement field,Eqs. (10)–(12) [11, p. 296]. When the normal stress rz (1) and shearstresses sxz, syz (7), (13), and (14) are used in the third equilibriumEq. (8) the following Poisson differential equation is derived(wG = w � G):

@sxz

@xþ @syz

@yþ @rz

@z

�� ����z¼0¼ 0) �r2wG ¼ f ðx; yÞjz¼0 ð15Þ

The analyzed section is located at the elevation z = 0. The functionf(x,y) is thus:

f ðx; yÞ ¼ �2Gdr0z

dzþ ðx� xgÞ

dr1z

dz� r1z

dxg

dzþ y

dr2z

dz

�� ����z¼0

ð16Þ

f ðx; yÞ ¼ 11þ m

x � Vx

Iyþ y � Vy

Ixþ x � f1IyP þ y � fIx þ f2IyP

�� ����z¼0

¼ 11þ m

� @rz

@zð17Þ

fIx ¼RMx

I2x

� dIx

dz; f 1Iy ¼

RMy

I2y

� dIy

dz; f 2Iy ¼

RMy

Iy� dxg

dzð18Þ

f1P ¼ �P

A2 �dAdz

; f 2P ¼ �PIy� dxg

dz; f 1IyP ¼ f1Iy þ f2P;

f 2IyP ¼ f2Iy þ f1P ð19Þ

The last three terms in (17), denoted fi, are detailed in (18) and (19).They are constant over the Oxy, as they are only z functions. For con-stant section columns, all fi functions are zero, and (17) becomes thefunction for shear analysis of prismatic bars. The functions fi can becomputed analytically or more general numerically as they arefunctions of only one variable, z. The solution of (15) accounts forthe displacements due to three-dimensional forces and momentsP–Vx–Vy–Mx–My: the ‘‘wall’’ behavior (u,w displacements and hy

rotation), the ‘‘plate’’ behavior (v,w,hx), as well as the coupledthree-dimensional behavior (u,v,w,hx,hy). The shear stresses dueto applied torsion and total shear stresses are detailed in followingsections as depicted in Fig. 3.

2.3. Limit boundary conditions

The boundary conditions (9) have to be included in the formu-lation of the differential problem (15). The domain for the bound-ary conditions and the differential equation is not the same: thedomain is 3D (Oxyz) for (9) and it is 2D (Oxy) for (15). Thus, a trans-formation of the boundary is needed. Let the outside surfaceboundary of the column be: X(x,y,z) = F1(x,y) + F2(z) = 0; most sur-faces representing real columns can be put in this form (plane,polynomial variation, etc.) The intersection between this surfaceand the plane Oxy is a cross-section boundary formed by the closedcurve C(x,y) = F1(x,y) + F2(Ls) = 0. The components of the outwardunit normal vector ~ns of the surface X and respectively ~n of thecurve C are, Fig. 2(b3), [40]:

~ns ¼F 01x

Lns|{z}nsx¼ls

;F 01y

Lns|{z}nsy¼ms

;F 02z

Lns|{z}nsz¼ns

* +; Lns ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF 021x þ F 021y þ F 022z

q;

~n ¼ F 01x

Ln|{z}nx¼l

;F 01y

Ln|{z}ny¼m

* +; Ln ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiF 021x þ F 021y

qð20Þ

The relationship between ~ns and~n is found by eliminating the deriv-atives F 01x and F 01y between both vectors in (20). The ratios Ln/Lns andF 02z=Lns represent the sine and cosine of the angle between the nor-mal and the horizontal, an (DNED, Fig. 2(b3)):

~ns ¼ lLn

Lns;m

Ln

Lns;F 02z

Lns

� �¼ l � cosðanÞ;m � cosðanÞ; sinðanÞh i ð21Þ

For the variable sides of the column, if the surface Xu(Xd) is planar,then the angle an is identical with au(ad), Fig. 2(b1). As the positivedirection for au and ad is not the same, in (9), the tangent of an

should be multiplied by l, the x component of the normal vectorof C. The third equation of the (9) as well as the boundary condi-tions for the differential equation (15) now become (the symbol‘‘�’’ means vector product when it multiplies two vectors):

ðsxzlþsyzm¼�rz � tanðanÞÞjz¼0)@wG

@~n¼ðhv ðx;yÞ� ~hnðx;yÞ �~nÞjz¼0 ð22Þ

hv ðx;yÞ ¼�rz � tanðanÞ; ~hnðx;yÞ¼ hhnx;hnyi ð23Þ

hnxðx;yÞ¼�m

2ð1þmÞx2�y2

2�Vx

Iyþxy

Vy

Ixþx � f2IyPþ

x2�y2

2� f1IyPþxy � fIx

� �ð24Þ

hnyðx;yÞ¼�m

2ð1þmÞ xyVx

Iy�x2�y2

2�Vy

Ixþy � f2IyPþxy � f1IyP�

x2�y2

2� fIx

� �ð25Þ

For constant sections, the functions fi are zero and (22) describes theboundary conditions for prismatic bars. Moreover, if the Poisson ra-tio m is assumed to be zero, then the vector function ~hn is zero andthe formulation is identical with the one when only equilibriumequations are integrated [15]. The terms containing the functionsh, bu, bd in (13) and (14) are neglected for the vector function~hnðx; yÞ because: (a) the terms in h relies on torsion formulation,which is discussed in a next paragraph; and (b) the derivatives ofbu and bd evaluated for z = 0 are some constants and they do notinfluence the value of shear stresses as they are known from equi-librium equations on the boundary (these functions play a role onthe boundary conditions at the column extremities).

To find the shear stresses distribution due to the application ofthree-dimensional forces P–Vx–Vy–Mx–My without torsion oneshould solve the strong form of the problem described by Eq.(15) with boundary conditions (22). Furthermore, with the helpof component hnx and hny the correspondent shear stresses are(the subscript S is used to indicate that shear stresses come froma differential formulation for shear without torsion):

sxzS ¼@wG

@xþ hnx; syzS ¼

@wG

@yþ hny ð26Þ

2.4. Torsion due to section variability

The shear distribution computed with Eq. (26) neglects thepresence of torsion. The torsion is considered in the next step.For variable section columns there are three sources for torsion:

(i) The externally applied torsion T.(ii) The internal torsion Vy � xgP due to the difference between

the centroid of the section where the shear force Vy isapplied and the reference section, Fig. 2(a).

(iii) The torsion TB due to the integral of shear stresses computedfor shear without torsion (previous paragraph). This torsioncan be interpreted conceptually as being produced by the dif-ference between the centroid and the shear center of the ref-erence section. The shear center Dv for a section being part of avariable section column is no longer a sectional property asclassically assumed by the Theory of Elasticity [9,11]. Thesame section part of different 3D configurations issues differ-ent ‘‘shear centers’’ as direct consequence of the boundaryconditions Eq. (22) that impose to the warping function adependency on the outside 3D normal of the section. Note thatTB can be non zero even if no shear force is applied. If the shearforces are zero the shear center is not geometrically well-

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 33

defined but TB can still exist. For example, if a variable sectionpier is subjected to pure axial force the normal stresses arecomputed with the formula r = P/A(z). Due to the z variabilityof the area (A(z)) the normal stresses present also a z variabil-ity (contrary to a constant section element where the applica-tion of pure axial force produces constant normal stresses);the z variation of the normal stresses generates shear stressesthat can furthermore generate a torque. Thus, even withoutapplying a shear force the torsion appears in the section (thisinternal torsion is self-equilibrated as no external torsion wasapplied). For this case TB is non zero but the shear center Dvclassically defined is not geometrically well-defined as itimplies a division by zero; e.g., xDv = TB/Vy and Vy = 0. Hence,for variable section elements the presence of internal torsionand associated shear stresses on a section can not be judgedbased only on the presence of shear forces [41].

The total applied torsion is the algebraic sum of the three pre-vious sources of torsion. As TB is produced by the integral of shearstresses s xzS and syzS, to reestablish the torsional equilibrium TB

must be applied in the opposite direction; thus, the sign of the ap-plied TB is negative. The component TB is computed numerically.

TB ¼Z

Aðz¼0Þð�sxzS � yþ syzS � xÞdA; xDv ¼

TB

Vy; yDv ¼ �

TB

Vx; ð27Þ

Ttot ¼ T þ Vy � xgP � TB ð28Þ

The total torsion Ttot is furthermore applied on the section to eval-uate the shear stresses due to pure torsion (sxzT and syzT).

2.5. Uniform and non-uniform torsion considerations

At sectional level, the total applied torsion Ttot is resisted by twomechanisms: (a) the Saint–Venant (uniform) torsion when the sec-tion is free to warp; and (b) the warping (non-uniform) torsionwhen the sectional warping is restrained. The main difference be-tween these mechanisms is that for the first one there are no nor-mal stresses on the section while for the second one the normalstresses are not null. In this paper, the evaluation of shear stressesdue to torsion for variable section follows the method described bySapountzakis and Mokos [9]. It consists in the separation of theuniform and non-uniform torsion in two separate problems relatedto two warping functions (the primary and the secondary one). Thestrong form of these two problems is (for more details see [9]):

�r2W ¼ 0@w@~n ¼ ðy � l� x �mÞjz¼0

(with

sxzSV ¼ Gdhdz

@W@x� y

� �;

syzSV ¼ Gdhdz

@W@yþ x

� �ð29Þ

�r2WS ¼ 2ð1þ mÞ � d3hdz3 �W

���z¼ 0

@wS@~n ¼ 0

8<: with

rzW ¼ Ed2h

dz2 � w; sxzW ¼ G@WS

@x;

syzW ¼ G@WS

@yð30Þ

The warping functions in Eqs. (29) and (30) are written with respectto longitudinally variable shear center [9]. The subscript ‘SV’ is em-ployed for Saint–Venant (uniform) torsion and ‘W’ for warping(non-uniform) torsion. The total shear stress due to torsion aggre-gates these two contributions:

sxzT ¼ sxzSV þ sxzW; syzT ¼ syzSV þ syzW ð31Þ

The torsional rotation h is determined as the solution of the follow-ing linear differential equation with appropriate boundary condi-tions at the bar extremities:

GJdhdz� E

ddz

Cwd2h

dz2

!¼ Ttot ð32Þ

In (32) J is the torsion constant JðzÞ ¼R

AðzÞ x2 þ y2 þ x @W@y � y @W

@x dA and

Cw is the warping constant CwðzÞ ¼R

AðzÞW2dA; W is the warping

function computed at elevation z. The contribution of the uniformand non-uniform torsion to total sectional resistance is [9]:

TSV ¼ GJdhdz

; TW ¼ �Eddz

Cwd2h

dz2

!; TSV þ TW ¼ Ttot ð33Þ

This method [9] does not consider that the warping functions have az variation and ignores the relationship between the normal of thesurface and the normal of the curve for the boundary conditions.However, even with these approximations, the error between theshear stress distribution computed with this method and a full 3DFE model is reasonable as shown latter in Section 3.

2.6. Computation of related quantities

The equilibrium equations at stress level are satisfied by theintrinsic formulation of the problems. The external equilibriumequations relating stresses and applied forces must also be satis-fied. Moreover, even if the contribution of rx and ry to normalstrain is neglected these normal stresses are not zero. The evalua-tion of these quantities is addressed in the following sections.

2.6.1. Stress resultantsFor the axial force, biaxial moments, and torsion the equilib-

rium equations are already satisfied because normal stress rz andshear stresses sxzT, syzT are obtained by using these equations;the remaining equations are those concerning sxzS,syzS and Vx, Vy.The total shear stresses include the contribution of pure shearstress and torsion:

sxz ¼ sxzS þ sxzT ; syz ¼ syzS þ syzT ð34Þ

To ensure equilibrium, the force resultants of sxzS, syzS must be Vx

and respectively Vy (for sxzT,syzT the force resultants are zero dueto the formulation). The main ideas of the proofs are given in thefollowing equations and they are detailed in Appendix A:

ZAðzÞ

sxzdAjz¼0 ¼Z

Asxz þ x

@sxz

@xþ @syz

@yþ @rz

@z

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼0ðequilibrium equationÞ

dA

¼Z

A

@ðxrzÞ@z|fflfflffl{zfflfflffl}H1

þ @ðxsxzÞ@x

þ @ðxsyzÞ@y|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

H2

dA ¼ Vx ð35Þ

ZAðzÞ

syzdAjz¼0 ¼Z

Asyz þ y

@sxz

@xþ @syz

@yþ @rz

@z

� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}¼0ðequilibrium equationÞ

dA

¼Z

A

@ðyrzÞ@z|fflfflffl{zfflfflffl}H3

þ @ðysxzÞ@x

þ @ðysyzÞ@y|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

H4

dA ¼ Vy ð36Þ

34 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

2.6.2. Determination of normal stresses rx and ry

The first two equilibrium Eq. (8) and boundary conditions (9)are coupled in a single differential equilibrium equation with onecoupled boundary condition:

@rx@x þ

@sxy

@y þ@sxz@z ¼ 0

@sxy

@x þ@ry

@y þ@syz

@z ¼ 0

8<: ������!

summing;sxy¼0

@rx

@xþ @ry

@y¼ � @ðsxz þ syzÞ

@zð37Þ

rxls þ sxyms þ sxzns ¼ 0

sxyls þ ryms þ syzns ¼ 0

(������!

summing;sxy¼0rxlþ rym

¼ �ðsxz þ syzÞ � tanðanÞ ð38Þ

This sum is weighted with an appropriate function into the FE for-mulation. Let us assume that the normal stresses rx and ry repre-sent the x and y derivatives of some function g(x,y). Byintroducing rx and ry as g derivatives in (37) and (38) the strongform of the problem is found:

�r2g ¼ @sz@z

��z¼0

@g@~n ¼ �ðsz tanðanÞÞjz¼0

(with rx ¼

@g@x

; ry ¼@g@y

; sz ¼ sxz þ syz

ð39Þ

To solve problem (39) the FE formulation presented in the next sec-tion can be employed. The derivation @sz/oz can be performednumerically (see also Appendix B).

2.7. Finite element formulation

The evaluation of the stress distribution was transformed in theprevious sections into a boundary value problem having the sym-bolical form:

�r2u ¼ f ðx; yÞ@u@~n ¼ hðx; yÞR

A udA ¼ 0

8>>><>>>: ð40Þ

If the last condition in (40) is not added, the unknown function ucan be determined only up to a constant because the boundary con-ditions are given only in terms of function derivatives. The constantdoes not influence the shear stresses because they use the deriva-tives of u. For a FE implementation the solution must be unique.To insure its uniqueness the additional condition

RA udA ¼ 0 was

added to the formulation (40). Because function u correspondsphysically to a warping function, the last condition in (40) can beinterpreted as a condition for the longitudinal displacement of thebar w to fulfill compatibility and equilibrium equations [5,6]. Threesteps are involved in the application of this additional condition inthe FE code: (i) assign a homogenous Dirichlet boundary conditionto only a single arbitrary point B on the FE mesh [u0(xB,yB) = 0]; (ii)solve the Eq. (40) and find u0; (iii) compute the final warping func-tion: u ¼ u0 � 1

A �R

A u0dA.The implementation in FE of the typical strong form (40) starts

with the determination of the weak form following the standardRitz–Galerkin procedure described in three steps [42]:

(i) Multiply differential Eq. (40) by a test function w(x,y) andintegrate over the domain:

Z

Aw � ð�r2uÞdA ¼

ZA

w � fdA ð41Þ

(ii) Integrate by parts the left term (use the divergence theoremand �r2u = �r � ru):

ZA

w � ð�r � ruÞdA ¼Z

A�wr �rudA

¼Z

Aru � rwdA�

ZC

w@u@~n

ds ð42Þ

(iii) Impose the boundary conditions from (40) in (42):

ZAru �rwdA�

ZC

w@u@~n

ds¼Z

Aru �rwdA�

ZC

h �wds ð43Þ

Thus, the weak form (or the variational form) is:ZAru � rwdA ¼

ZA

f �wdAþZ

Ch �wds ð44Þ

Suppose that the unknown field u(x, y) is approximated over a finiteelement Ae with n nodes by: uðx; yÞ ¼ uhðx; yÞ ¼

Pnj¼1ujwjðx; yÞ, and

write the weak form (44) over this element:

Xn

j¼1

uj

ZAe

rwj � rwdA ¼Z

Ae

f �wdAþZ

Ch �wds ð45Þ

The system of linear equations at element level is obtained by suc-cessively replacing the test function w(x,y) with the interpolationfunctions wi(x,y)i = 1,n:

Ke � ue ¼ fe þ he with Ki;j ¼Z

Ae

rwj � rwidA;

fi ¼Z

Ae

f � widA; hi ¼Z

Ch � wids

and ue is the vector of nodal values uj of the unknown functionu(x,y).

The solution is found by (a) assembling the elementary matrices(46) into the global linear system K � u = f + h, (b) imposing theDirichlet condition, and (c) solving the global system. The completedetails of the implementation of a strong form into FE are given inReddy [42].

3. The case of rectangular sections

For stress analysis of the variable section elements the rectan-gular section is probably the most studied section. The elementis subjected to P–Vx–My forces which produce the displacementspecific to the ‘‘wall’’ behavior (u,w displacements and hy rotation);the Poisson ratio is usually neglected. In the case of hydraulicstructures, the columns with variable sections are subjected to fullthree-dimensional load combinations, P–Vx–Vy–Mx–My–T (self-weight, hydrostatic pressure, ice impact, earthquake, uplift pres-sures, etc.). This paragraph covers the determination of the shearstresses due to the three-dimensional forces P–Vx–Vy–Mx–My pro-ducing an arbitrary displacement patterns. The torsion formulationfor variable section columns can be found in [9].

3.1. Section subjected to P–Vx–My

Suppose that a variable section column having rectangularcross-section is bounded to the u and d sides by two planes,Fig. 2(b1); the analyzed section bs � hs is located at elevation Ls.If only a shear force is applied in x direction (Vx – 0) it is logicalto expect that all stresses are zero with two exceptions: the normalstress rz and the shear stress sxz, Fig. 2. Because rz has a linear var-iation in x (planar distribution) from the third equation of (8) itcomes out, by integration, that sxz is a second degree polynomial(parabolic distribution, null Poisson ratio) [10,15]:

sxzðxÞ ¼ a � x2 þ b � xþ c ð47Þ

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 35

This idea is generalized for the case when the column is subjected toP–ex–Vx–My. The three constants in (47) need three conditions to beuniquely identified (all variables are evaluated at elevation Ls): (i)sxz(�hs/2) = rz(�hs/2) � tan(au), (ii) sxz(hs/2) = �rz(hs/2) � tan (ad),and (iii)

RA sxzdA ¼ Vx. These conditions represent the boundary

conditions (22) and the external equilibrium equation (the forceresultant of shear stresses is Vx). In the case when the Xu and Xd

are not planes, the boundary conditions should be changed accord-ingly. Replacing the above mentioned conditions in (47) andsolving:

a ¼ �6 � Vx

bsh3s

� 3

h2s

ðrMadþu þ rPad�uÞ ð48Þ

b ¼ � 1hsðrMad�u þ rPadþuÞ ð49Þ

c ¼ 32� Vx

bshsþ 1

4ðrMadþu þ rPad�uÞ ð50Þ

where

rP ¼P

bshs; rM ¼ �

6 � RMy

bsh2s

;

ad�u ¼ tanðadÞ � tanðauÞ; adþu ¼ tanðadÞ þ tanðauÞ ð51Þ

The first terms in Eqs. (48) and (50) represent the coefficients forshear distribution due to Vx, for constant section elements (Jouraw-ski formula); for this case b = 0. The advantage of this approach isthat the finding of the shear stress distribution involves no differen-tial equation. Eq. (47) was validated using the examples presentedin the next section as well as with similar models found in technicalliterature [15].

3.2. Consideration of three-dimensional loads

The general method (Section 2) is employed to compute theshear stress distribution of a rectangular variable section columnsubjected to P–Vx–Vy–Mx–My. The z variation of the geometricalproperties for the variable section column are, Fig. 2(b1):

hsðzÞ¼ hs�zðtanðauÞþ tanðadÞÞ; huðzÞ¼huþz � tanðauÞ;hdðzÞ¼hd� z � tanðadÞ ð52Þ

xgðzÞ¼zðtanðauÞ� tanðadÞÞ

2; xgP ¼

ðL�LsÞ � ðtanðauÞ� tanðadÞÞ2

ð53Þ

AðzÞ¼ bshsðzÞ; IxðzÞ¼b3

s hsðzÞ12

; IyðzÞ¼bshsðzÞ3

12ð54Þ

When the Eqs. (52)–(54) are inserted in (15)–(19) and (22)–(25) thestrong form of the problem is reached. The z derivatives of xg, A, Ix,and Iy can be computed analytically but also numerically. Analyticalrelationships for the fi functions, Eqs. (18) and (19), for rectangularsections have been developed by the authors for validationpurposes.

4. Applications

This section has two main goals: (a) validation of the proposedformulation; (b) examination of its applicability when differentparameters are varied: Poisson ratio, proportion between the sec-tion and column height (L/h0, Fig. 2), and cross-section shape.The first one is achieved by comparing the results produced byself-developed 2D FE software with those of the analysis per-formed with SAP 2000 [43] commercial 3D FE software and/or withanalytical results when available; the 2D sectional analysis FE codeis written in Matlab [44]. For the second one, the effect of Poissonratio and geometrical proportion was examined using threecolumns: (i) rectangular cross-section, L/h0 = 8.3 (m = 0, m = 0.25),

(ii) rectangular section, L/h0 = 0.8 (m = 0.2), (iii) spillway pier,L/h0 = 0.4 (m = 0). The analyzed section was located for all cases atLs = L/4, Fig. 2. The applied loads are: Vx, Vy and P–ex–ey; at thesection level they correspond to a three-dimensional loading state,P–Vx–Vy–Mx–My–T. The 2D FE software employs the quadratic tri-angular element (LST), 1 DOF/node (DOF = degree of freedom);the mesh is produced with the open-source three-dimensional gridgenerator Gmsh [45]. In SAP2000 the finite element is the standardsolid element with 8 nodes (3 DOF/node) with incompatible modesactivated. All three DOF are fully fixed at bottom end for all col-umns; at the other end all DOF are free. The loads were appliedon the rectangular top section for examples 1, 2 and at elevation8.535 m block B1 for example 3. The planar distribution for normalstresses (P–ex–ey) Eq. (1) and the parabolic distribution for shearstresses (Vx,Vy) Eq. (47) were analytically integrated on eachelement of the mesh to obtain nodal forces. By analyzing theresults near the boundary, it is clearly noticed that the 2D modelperformed a lot better than the 3D model for that region, asexpected. The most representative stress distributions rz, sxz, syz

are discussed in the following paragraphs. The 3D plots in Figs.4–6 are produced with the 2D sectional analysis FE code.

Studies for the mesh density were conducted in order to checkthe influence of the mesh on the final results. The number of finiteelements to mesh the analyzed section is not critical as long as: (a)a minimum of 10 FE is used on each edge of the section; (b) theshorter edge is divided in at least 2 segments (when there is animportant difference between the longer and the shorter edge);(c) quadratic triangle are used; and (d) the equilibrium and theboundary conditions is ‘‘numerically’’ satisfied when are checkedafter the analysis. Frontal meshing algorithm is preferred for hav-ing high quality elements [45].

4.1. Rectangular section pier with a single sloped boundary

The first example addresses the ‘‘classical’’ example of a rectan-gular cross-section with one sloped boundary (d side), Fig. 4. Forthis case, the hypothesis that the normal stress distribution rz islinear is fully satisfied (also, expected because of large L/h0 = 8.3 ra-tio). The cross-section mesh has 25 (13) equally distributed nodesin the x (y) direction, regardless the elevation; along the pier height201 nodes are equally distributed to keep the ratio of the FE sidesclose to 1. The 3D model has 195 975 DOF while the 2D sectionalanalysis model has only 7 811 DOF. Nevertheless, by comparingthe stress variation in Fig. 4(b)–(d) there is no noticeable differencebetween the results provided by the 3D vs. 2D sectional analysismodel for all load conditions: biaxial shear Vx, Vy or eccentric axialforce P (the maximum difference is less than 1%). As mentionedearlier, near the boundary the results found by the 2D model arecloser than those of the 3D model to the analytical (from equilib-rium) value �rz � tan(an), Eq. (22).

4.2. Influence of Poisson ratio

The previous example was repeated with a value of Poisson ra-tio v non-null, v = 0.25. For this case, also, there is practically nodifference between the results of the 2D and 3D models. Conceptu-ally, the example with v = 0 corresponds to the solution where onlythe equilibrium equations were integrated; this is the commonsolution found in literature. The compatibility requirements andequilibrium equations are necessary to formulate the problemwhen v – 0. The effect of m on the shear stresses is globally smallbut locally the difference can be up to 50% if compared with thecase v = 0, Fig. 4(d4). Thus, in general the Poisson ratio effect can-not be neglected especially when the local contribution of theshear stresses (sxz and syz) is a concern.

Fig. 4. Rectangular section pier with a single sloped boundary: (a) geometry and notations, and structural response for applied forces (b) Vx = 100 kN, (c) Vy = 100 kN, and (d)P = �2000 kN, ex0.1 m, ey = 0.05 m.

36 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

4.3. Rectangular section buttress with unsymmetrical slopes

The second example is adapted from a real concrete buttressdam in Morocco (Tamzaourt, dam height 86 m), Fig. 5. Both sidesare sloped with a tangent of 0.5 (u side) respectively 0.6 (d side);

v = 0.2. Because the ratio L/h0 is small (0.8) the column heightis smaller than the section height) it is expected that the distribu-tion of normal stress rz is non-linear (e.g. Fig. 5(d3)). Note that forthe analysis of hydraulic structures the assumption of linear varia-tion for normal stress rz is generally accepted. The cross-section

Fig. 5. Concrete buttress (unsymmetrical boundary conditions): (a) geometry and notations, and structural response for applied forces (b) Vx = 1000 kN, (c) Vy = 1000 kN, and(d) P = �1000 kN, ex = 1 m, ey = 1 m.

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 37

mesh has 131 (11) equally distributed nodes in the x (y) direction,regardless the elevation; along the pier height 101 nodes areequally distributed to keep the ratio of the FE sides close to 1.The 3D model has 436 623 DOF while the 2D sectional analysis

model has only 8 043 DOF (this large number of DOF for the 3Dmodel is necessary because of the high ratio between the sectionheight and depth: 104 � 5 m). The differences between the shearstresses sxz computed with the 3D and 2D models are more

Fig. 6. Pier (squat wall) for hydraulic structures: (a) geometry and notations, and structural response for applied forces (b) Vx = 1000 kN, (c) Vy = 1000 kN, and (d)P = �1000 kN, ex = 1 m, ey = 1 m.

38 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

pronounced, mainly because of the nonlinearity of rz. These differ-ences can be acceptable in practice as the maximum shear stressmagnitude is obtained by both models. The differences betweenthe two models are small for syz. The stress distributions with/

without warping torsion are practically identical (the buttressslopes are close to one another). The section behaves more like athin-walled section concentrating the warping function variationnear the ends.

Fig. 7. Along the height variation for torsion analysis variables: (a) torsion constant J, (b) warping constant Cw, (c) shear center, (d) twist angle, and (e) resistant (TSV,TW) andapplied torsion (Ttot) for load case Vy = 1000 kN of example 3.

L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41 39

4.4. Pier (squat wall) for hydraulic structures

The last example (Fig. 6) demonstrates the capability of the pro-posed formulation to evaluate the shear distribution in more diffi-cult conditions: (a) three-dimensional loads, (b) very small ratioL/h0 (0.4) generating an important effect of the restraints of thepier base, and (c) non-linear normal stress rz. The column is a realspillway pier from an existing hydroelectric facility in Québec,Canada. The analyzed section, a typical spillway section with oneaxis of symmetry, has the bounding box of 18.85 m � 2.44 m; thedownstream is sloped with a tangent of 0.17; v = 0, Fig. 6(a). Themesh for the base cross-section is detailed in Fig. 6(a3). It is dividedon 9 blocks on each block the nodes being equally distributed. Themesh has 13 nodes in the y direction and 47 equally distributednodes along the pier height to keep the ratio of the FE sides closeto 1. The number of nodes in the x direction for each block is: 87(B1), 4–7 (B2), 15 (B3, B4), 8 (B5, B6), 58 (B7, B8), and 5 (B9) fora total of 1217 nodes for a typical cross-section. Along the pierheight the nodes situated on the left of the line LvFE, Fig. 6(a3), keepthe same x and y position while those located on the right of thisline move to the left linearly with respect to the slope change.On the upper part of the pier, where the section is reduced, thenodes that fall outside of the reduced section are simply discardedfrom the mesh. The 3D model has 141 456 DOF while the 2D sec-tional analysis model has only 13 788 DOF. The variation for thetorsional constants, the shear center, the twist angle as well asthe resistant and applied torsion for the load case Vy = 1000 kNare depicted in Fig. 7. For syz the differences between the 2D and3D models are negligible. The shear stress sxz computed with the2D model captures the variation pattern of the 3D model withacceptable difference. For this case the use of the warping torsionin the formulation improves significantly the shear distribution.The proposed method is therefore suitable even for structuralelements having large shear deformations like the case of squatwalls.

5. Summary and conclusions

This paper presents a mathematical and finite element formu-lations for elastic sectional analysis of variable section elementssubjected to three-dimensional loads (axial force, biaxial shearforce, bending moment, and torsion) producing arbitrary defor-mations (‘‘wall’’ and ‘‘plate’’ behaviors). The element has arbitraryunidirectional longitudinal variability and arbitrary uniaxial sym-metrical cross section. Regular as well as thin-walled sections canbe analyzed. The formulation consists in the determination of the

normal and shear stress distributions by solving two boundaryvalue problems with appropriate boundary conditions: one forshear without torsion and the other one for pure torsion. The for-mulation is based on the equilibrium and compatibility equationsfollowing a linear material law with the inclusion of Poisson ratio.The proposed method was validated using three examples rang-ing from a regular column to a squat wall. The main conclusionsare:

(a) The analysis of a variable section element can be accom-plished by doing a 2D sectional analysis for an equivalentconstant section element by extending the differential for-mulation of Euler–Bernoulli beam theory including warping.This shows practical significance of major interest for theanalysis of hydraulic structures because their safety assess-ment can be first efficiently evaluated with series of sec-tional analyses. Hence, the proposed method can be usedas a powerful tool to estimate more sophisticated 3D modelsresponse.

(b) The mathematical formulation is analytical giving the samelevel of accuracy as the one of the input normal stress: whenthe normal stress is exact, the shear stresses are exact to;when it includes an approximation the shear stressesinclude it to. Thus, for the elements where the assumptionof linear distribution for normal flexural stress is satisfiedthe proposed sectional approach computes results within1% of 3D finite element (FE). If this assumption is not math-ematically fulfilled (bulk and squat walls) the approximationin normal stress is propagated in the shear stresses compu-tation. However, even for this case the stress distributionsare very similar to 3D FE while the stress intensity showscertain variations. The assumption of linear normal stressis typically used for the analysis of hydraulic structuresbeing aware that for some structures it is only an estimationof the stress field.

(c) The 2D sectional analysis is a viable alternative to the full 3DFE analysis even facing deep sections such as the case ofsquat-walls subjected to bending, shear, and non-uniformtorsion. Considerable reduction in computational effort isnoticed because the analysis employs surface 2D meshinstead of much more elaborate volumetric 3D mesh.

(d) Analyzing in detail the proposed formulation, one canextrapolate that the framework of the proposed method isgeneral enough to handle various extensions: arbitrary sec-tion/spatial variation, non-linear normal stress variation,boundary element formulation, etc.

40 L. Stefan, P. Léger / Computers and Structures 90–91 (2012) 28–41

(e) The proposed method adds no further complications if com-pared with the analysis of constant section elements. Itoffers the advantage of generality and accessibility allowinga direct implementation in existing finite element code forsectional analysis of constant section elements subjectedto arbitrary three dimensional load combinations.

Currently our research directions are oriented to the extensionof the present method to non-linear material for cracked variablesection elements.

Acknowledgements

The financial support provided by the Quebec Fund for Researchon Nature and Technology, and the Natural Science and Engineer-ing Research Council of Canada is acknowledged.

Appendix A

The Appendix A presents the detailed proof for the shear stressresultants (Eqs. (35) and (36)) using two integral properties de-tailed in Appendix B.

H1 ¼Z

A

@ðxrzÞ@z

dA ¼ �EIydr1z

dz¼ ���!

use:ðA:7ÞVx �

PA� ddz

ZA

xdA� �

þ RMy

Iy� ddz

ZA

x2dA� �

¼ � � � ���!use:ðB:1Þ

Vx þPA�Z

Cx � tanðanÞds

� RMy

Iy�Z

Cx2 � tanðanÞds ðA:1Þ

H2 ¼Z

A

@ðxsxzÞ@x

þ @ðxsyzÞ@y

dA

¼ �������!Green’s theorem

ZC

xðsxzlþ syzmÞds ¼Z

Cx � �rz � tanðanÞds ¼ � � �

¼ � PA�Z

Cx � tanðanÞdsþ RMy

Iy�Z

Cx2 � tanðanÞds

þ RMx

Ix�Z

Cxy � tanðanÞds|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0ðsymmetryÞ

ðA:2Þ

) H1 þ H2 ¼ Vx ðA:3Þ

H3 ¼Z

A

@ðyrzÞ@z

dA ¼ �EIxdr2z

dz¼ Vy þ

RMx

Ix� ddz

ZA

y2dA� �

¼ . . . ���!use ðB:2Þ

Vy �RMx

Ix�Z

Cy2 � tanðanÞds ðA:4Þ

H4 ¼Z

A

@ðysxzÞ@x

þ @ðysyzÞ@y

dA

¼ �������!Green’s theorem

ZC

yðsxzlþ syzmÞds ¼Z

Cy � �rz � tanðanÞds

¼ � � � ¼ � PA�Z

Cy � tanðanÞds|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}0ðsymmetryÞ

þRMy

Iy�Z

Cxy � tanðanÞds|fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

0ðsymmetryÞ

þ RMx

Ix�Z

Cy2 � tanðanÞds ðA:5Þ

) H3 þ H4 ¼ Vy ðA:6Þ

xg ¼R

A xdAA) dxg

dz

����z¼0¼ 1

A� ddz

ZA

xdA� �

� 1

A2 �Z

AxdA|fflfflffl{zfflfflffl}0

�dAdz

¼ 1A� ddz

ZA

xdA� �

ðA:7Þ

Appendix B

ddz

ZA

f ðxÞdA� �

¼ ddz

Z hdðzÞ

huðzÞf ðxÞ

Z y2ðxÞ

y1ðxÞ

dydx¼ ddz

Z hdðzÞ

huðzÞf ðxÞ � lðxÞdx �����!

Leibniz’s rule� � �Z hdðzÞ

huðzÞ

@

@zf ðxÞlðxÞð Þ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

0ðnot a z functionÞ

dx

þ f ðhdÞ|ffl{zffl}fd

lddhd

dz|{z}�tanðadÞ

� f ðhuÞ|fflffl{zfflffl}fu

ludhu

dz|{z}tanðauÞ

¼�ðfdld tanðadÞþ fulu tanðauÞÞ¼ �� �

¼�Z

Cd

f ðxÞtanðanÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}constant onCd

dsþZ

Cu

f ðxÞtanðanÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}constant onCu

ds

0B@

þZ

C�Cd�Cu

f ðxÞtanðanÞds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0ðtanðanÞ¼0Þ

1CCCA¼�

ZC

f ðxÞtanðanÞds ðB:1Þ

ddz

ZA

f ðyÞdA� �

¼ ddz

Z hdðzÞ

huðzÞ

Z y2ðxÞ

y1ðxÞf ðyÞdy|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}

FðxÞ

dx

¼ ddz

Z hdðzÞ

huðzÞFðxÞdx������!

Leibniz’s rule� � �Z hdðzÞ

huðzÞ

@FðxÞ@z|fflffl{zfflffl}

0

dx

þFðhdÞdhd

dz|{z}�tanðadÞ

�FðhuÞdhu

dz|{z}tanðauÞ

¼�Z y2d

y1d

f ðyÞtanðadÞdy

þZ y2u

y1u

f ðyÞtanðauÞdy

!¼ �� �

¼�Z

Cd

f ðyÞtanðanÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Cd¼lineðy1d ;y2dÞ

ds

0B@ þ

ZCu

f ðyÞtanðanÞ|fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}Cu¼lineðy1u ;y2uÞ

dsþZ

C�Cd�Cu

f ðyÞtanðanÞds|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0ðtanðanÞ¼0Þ

1CCCA

¼�Z

Cf ðyÞtanðanÞds ðB:2Þ

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