A Novel Stiffness/Flexibility-Based Method for Euler-Bernoulli

Applied Mathematical Modelling 40 (2016) 7627–7655

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Applied Mathematical Modelling

journal homepage: www.elsevier.com/locate/apm

A novel stiffness/ﬂexibility-based method for Euler–Bernoulli/Timoshenko beams with multiple discontinuities and singularities

∗ Reza Khajavi

Earthquake Research Center, Ferdowsi University of Mashhad, VakilAbad Blvd., Mashhad, Iran

a r t i c l e i n f o a b s t r a c t

Article history: Nonprismatic beams have vast and various applications in mechanical and structural sys- Received 17 July 2015 tems; thus, much research is dedicated to develop well-performed stiffness matrices for

Revised 28 February 2016 beams with different forms of section changes and singularities. This paper introduces Accepted 23 March 2016 stiffness matrices by the use of ﬂexibility and stiffness methods. For this purpose, the Available online 1 April 2016 spring model of the beam element is introduced. This model provides an innovative phys-

Keywords: ical interpretation for the beam element. It can also be used to develop ﬁnite elements

Euler–Bernoulli beam for both Euler–Bernoulli and Timoshenko beams with different combinations of tapering, Discontinuity singularity and discontinuity. In this model, beam sections are represented by some ap- Flexibility propriate virtual springs; their connection in series/parallel gives the ﬂexibility/stiffness Stiffness matrix of the beam element. The obtained matrices are in general forms and applicable to Timoshenko beam different beam conditions. ©2016 Elsevier Inc. All rights reserved.

1. Introduction

Beam member, as one of the most applicable components in building construction, has attracted much research interest in the structural engineering community. Before using computers in structural analysis, engineers had to use 1D simplified beam models. Today, in spite of the availability of high-performance computing processors, analysis of 2- or 3-D structural building models are still computationally expensive, and thus not available for every-day engineering routines. Decades of engineering experience, indeed, have proved the efficiency and practicality of 1D beam theories for structural modeling. To this date, almost all design codes and standards for steel and concrete buildings are developed according to 1D beam theories. The two Euler–Bernoulli (EB) and Timoshenko (T) beam theories are the most practical ones used for 1D beam modeling. EB formulation assumes that cross sections remain planar after deformation, due to negligible shear strains. This assumption is only valid for long beams. T formulation accounts for shear strains; it is specially used for the analysis of short beams, where shear strains could not be ignored. Researchers have developed finite element solutions for the differential equations of EB and T beam theories. Finite Element Method (FEM) is a powerful numerical tool which has greatly made use of the fast-advancing performance of computers. When FEM is applied to any of beam differential equations, an algebraic system of equations with a well-defined stiffness matrix is developed for the beam element.

∗ Tel.: + 98 51 38804374; fax: + 98 51 38791191. E-mail address: [email protected] , [email protected] http://dx.doi.org/10.1016/j.apm.2016.03.029 S0307-904X(16)30154-8/© 2016 Elsevier Inc. All rights reserved. 7628 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Alongside the primary efforts to find stiffness matrices for the beam elements, the concept of stiffness and flexibility methods emerged [1] . In the stiffness method, interpolating functions are defined for element displacement, out of which other unknown fields (e.g. deformation and internal moment) are derived. In the flexibility method, moment field is assumed as independent, and is represented by appropriate interpolating functions. In the stiffness (flexibility) method, the equilibrium (kinematic) equations are defined in their weak forms and the kinematic and constitutive (equilibrium and constitutive) equations are used in their strong forms. The stiffness method is typically used for computer implementation; however, for nonprismatic (and inelastic) beams, the stiffness matrix obtained by the stiffness method is not a suitable option. In fact, displacement interpolating functions cannot follow complex displacement variations along nonprismatic beams. To overcome such problem, some authors examined improved interpolation functions [2,3] . Stiffness matrices were developed for beams with certain tapering patterns by integrating exact governing differential equations [4–9] . These stiffness matrices have exact explicit forms, but they are complicating in their implementation and representation (especially for complicated patterns of tapering and discontinuity). As an alternative, some other authors developed the stiffness matrix by the flexibility method [10–14] . As mentioned before, in the flexibility method, one considers interpolation functions for the independent field of internal forces. In most applied cases, variations in the internal forces are much simpler than displacements; thus, the stiffness matrices obtained by flexibility methods turn out more efficient than those obtained from stiffness methods. However, no absolute judgment could be made about the supremacy of one over the other; as it seems that each is more efficient in estimating a specific unknown field [15] . In the literature, stiffness- and flexibility-based finite element methods are characterized by their independent fields as mentioned before; however, this paper presents a new interpretation of the two methods by introducing the spring model of the beam element. Engineers are well accustomed to the concept of the spring; they often use it as a handy tool to simplify complex structural behaviors and models. It is characterized for its simple configuration, versatility, adaptability, and its ability to represent any combination of springs with a single one. In spring modeling, the connection of the springs can be categorized into parallel or series configurations. Two or more springs are in series if the external force/moment applied to the ensemble is exactly transferred to each spring. In such a configuration, the deformation of the ensemble thus equals the sum of those of the individual springs. If the deformation of the ensemble is the same for all of the constituent springs and the force/moment of the ensemble is the sum of those of the individual springs, they are said to be parallel. In this paper, it is shown that stiffness/flexibility-based finite element methods, used for the EB and T beam theories, could be redefined according to the concept of cross-sectional springs. Some preliminaries are introduced in Section 2 . The stiffness matrices for EB and T beam elements with multiple tapering and abrupt discontinuities are then obtained in Sections 3 and 4 respectively. The proposed method can efficiently deal with singularities (slope discontinuities); this will be verified in Section 5 by several numerical experiments. In short, the proposed method is characterized by its simplicity, generality (in dealing with different types of tapering, discontinuity and singularity), and new physical understanding of the beam theory.

2. Formulation

In the following, a linear elastic finite element in the domain ⊂ R d with d ∈ {1, 2, 3} is considered under static loading. The unknown fields of the element are assumed to be displacement, u˜ ∈ C k (R d ) , k ∈ { 0 , 1 } , strain, ε˜ ∈ C k (R d ) , k ∈ {−1 , 0 , 1 } , and stress, σ˜ ∈ C 1 (R d ) , k ∈ {−1 , 0 , 1 } ( C k (R d ) is the real function space with k -order continuity). In the stiffness method, the independent field of approximate displacement u is described as a linear combination of k d some generalized functions Nˆ q . They constitute a basis for the degree- n polynomial space P n () ⊂ C (R ) :

u = Nˆ q qˆ . (2.1) Here, qˆ is the vector of generalized coordinates. The u -dependent strain ε u is derived from displacement according to the kinematic law:

u ε = L Nˆ q qˆ = Bˆ q qˆ , (2.2)

where L is the kinematic differential operator and Bˆ q is the strain-generalized coordinates matrix. Since rigid body motions of geometrically-linear ﬁnite elements do not produce any deformation [16] , Bˆ q can be partitioned as:

Bˆ q = [ B qr | B q ] = [ 0 nr | B q ] , (2.3a)

u qr ε = [ 0 | B ] , (2.3b) nr q q with nr representing the number of rigid body motions. q r involves generalized coordinates for the rigid body motions, and q represents energy-conserving strain states. The elastic constitutive law is indicated as:

σu = C . ε u, (2.4) R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7629

T with C being the constitutive matrix. Pre-multiplying the above equation by B q and after substituting from Eqs. (2.2) and ( 3.3 b), an integration over the domain gives:

T σu = T . , Bq d Bq C Bq d q (2.5) or < σ> = , Pq Kqs q (2.6) < σ> where Pq and Kqs are the generalized internal force vector and the generalized stiffness matrix, respectively. The subscript s in Kqs implies that the matrix is obtained by the stiffness method. Note that in Eq. (2.5), Bq is a basis for the space of the approximate strain field. In the flexibility method, the independent field of stress σ is written as a linear combination of generalized functions b q . 1 d These functions constitute a basis for the polynomial space of P n () ⊂ C (R ) :

σ = σ , bq q (2.7) σ where q is generalized kinetic coordinates. The constitutive law is: σ − ε = C 1 . σ, (2.8)

σ T where ɛ is the σ-dependent strain field. Pre-multiplying the above equation by b q and integrating over the spatial domain gives: σ − T ε = T 1 . σ , bq d bq C bq d q (2.9) or < ε > = σ , Eq Fq q (2.10) = T −1 < ε > = T ε σ where Fq bq C bq d is the flexibility matrix, and Eq bq d is the generalized strain vector. Therefore, − < ε > σ = 1 . q Fq Eq (2.11) Using a dimensional static matrix R , one gets: − − T σ = T 1 1 , R q R Fq R R Eq < ε > ∴ = , pq Kqf q (2.12) < ε > where q denotes the vector of generalized coordinates, and is a weighted integration of stress-dependent strains. pq is the generalized internal force vector, and Kqf is the generalized stiffness matrix. The subscript f implies that the stiffness matrix is derived by the flexibility method. The dimensional static matrix R is employed to make the inverse of F q dimensionally consistent with Kqs in the stiffness method; it is obtained as [17]:

= T . R Bq bq d (2.13)

Once the generalized stiffness matrix Kq is obtained based on the stiffness method Eq. (2.6) or the ﬂexibility method Eq. (2.12) , the stiffness matrix for nodal degrees of freedom (DOFs) K is retrieved by the following transformation:

= ψ T ψ, K Kq (2.14)

ψ is a submatrix of the transformation matrix ψˆ which relates qˆ to nodal displacements D :

= ψˆ , qˆ D

q ψ (2.15) r = r D. q ψ

The form of ψˆ depends on the special basis selected for the formulation [18] .

3. Implementation for EB beam element

= { ( , , ) ∈ × | ∈ − L , L ≡ ⊂ , ( , ) ∈ ⊂ 2 } A linear elastic EB beam element on the spatial domain : x y z L A x [ 2 2 ] L R y z A R under static loading is considered, where L is the beam length. It is assumed that the neutral axis is straight or nearly straight such that arching action is negligible and axial, shear and bending behaviors are decoupled [19] . A can arbitrarily change along L . This change may involve different types of continuous tapering and abrupt discontinuities such as singularities (ideal hinges). Also, the low-stress zones around discontinuity sections [19] and local taper effects [20] are neglected. −1 −1 Variations in A are described by C ≡ s ( x ) and C ≡ f (x ) = s (x ) for the stiffness and flexibility methods, respectively. Since axial and bending behaviors are assumed decoupled, no axial analysis is involved. The beam element has two nodes; 7630 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 each node contains two nodal DOFs: a deflection and a rotation. Deflection u ≡ w ( x ), curvature ɛ ≡ κ( x ), and moment σ ≡ M ( x ) are the three unknown fields of the EB beam element, which are related through the kinematic relation:

κ(x ) = ∂ xx w (x ) , (3.1)

( ∂ xx is the second derivative with respect to x ), and the constitutive law: M(x ) = s (x ) κ(x ) , (3.2)

κ(x ) = f (x ) M(x ) . (3.3)

3.1. Stiffness method

As EB beam element has four degrees of freedom/generalized coordinates, the independent deflection field is such that: 1 w 1 w (x ) ⊆ P 4 (R ) . Since L ≡ ∂ xx , the deflection-dependent curvature field is: κ (x ) ⊆ P 2 (R ) ; therefore, the following basis functions are selected for interpolating curvature field along the element:

B q = [1 x − x 0 ] , (3.4)

x0 denotes an arbitrary point on L , which is referred to as pivot node henceforth. Substituting Eq. (3.4) into Eq. (2.5), the stiffness matrix (in the selected basis and with the pivot node x0 ) becomes: ⎡ ⎤ s (x ) dx Sym. ⎢ ⎥ L x0 = ⎢ ⎥ . Kqs ⎣ ⎦ (3.5) 2 s (x ) (x − x 0 ) dx s (x ) (x − x 0 ) dx L L Note that this matrix involves entries for the strain states which store strain energy. Entries corresponding to rigid body 0 × 0 motions are zero, and must be included as: [ nr nr ] ; thus, the complete stiffness matrix is obtained. Also, the generalized x0 0 Kqs internal force vector is found as: T < σ>, x0 = ( ) ( ) . ( − ) . Pq M x dx M x x x0 dx (3.6) L L The matrix and the vector introduced in Eqs. (3.5) and ( 3.6 ) are related by Eq. (2.6) . The generalized coordinate vector is = κ κ T κ κ q [ 0 00 ] . According to the selected Bq in Eq. (3.4) and referring to Eq. (2.3b), 0 and 00 represent constant curvature and linear curvature (or curvature-gradient) along the EB beam element. Eq. (3.5) provides physical interpretation for the concept of the stiffness matrix of the EB beam element. It shows how

the stiffness matrix is dependent on section stiffnesses and stiffness distribution along the element. The ﬁrst diagonal entry, s (x ) dx, shows that the stiffness matrix is partly established by the sum of stiffnesses at all sections. One can imagine L each section as an equivalent spring with its force and displacement indices considered as moment M and curvature κ respectively. The stiffness of the equivalent (bending) spring is s ( x ) ≡ EI ( x ) where E is the modulus of elasticity and I ( x ) is κ the second moment of inertia at section x. Now, if the element is subjected to the unit constant curvature 0 , each section (spring) undertakes a moment (force index) that is consistent with its own stiffness s ( x ). The beam in the assumed strain state can thus be considered as an ensemble of parallel springs, as shown by Fig. 1(a). Thus, the stiffness of the ensemble ( ) κ = is: s x dx. Note that the deformation shown in Fig. 1(a) displays unit curvature (i.e. 0 1 for all sections), and should L not be confused with the translational rigid body motion. The non-diagonal entry of the stiffness matrix in Eq. (3.5) , s (x ) (x − x 0 ) dx, is the ﬁrst moment of the beam stiff- L ness around anarbitrary pivot node. If the pivot node coincides with the beam’s center of stiffness, the non-diagonal entry becomes zero ( s (x ) (x − x 0 ) dx = 0), and the stiffness matrix becomes diagonal. Borrowing from Linear Algebra nomen- L clature, each diagonal entry of this diagonal stiffness matrix is referred to as an eigenstiffness . Note that the eigenstiffness is different from the well-known eigenvalue of the stiffness matrix. 2 The second diagonal entry of the stiffness matrix in Eq. (3.5) , s (x ) (x − x 0 ) dx, represents the contribution of sec- L tional stiffness distribution to the beam stiffness. This entry can also be considered as the sum of stiffnesses of bending- κ gradient springs. These springs model beam reaction against the strain state of curvature-gradient (linear curvature) 00 . In Fig. 1 (b), this strain state is shown by the element rotation, and should not be confused with the rigid body rotation κ of the element. The stiffness for each such spring is obtained from s(x). If the element is subjected to 00 , then for each κ = κ − = κ − equivalent bending spring: 00 (x x0 ) and M(x) s(x) 00 (x x0 ). The moment of M(x) around the pivot node is the − = κ − 2 force index of the bending-gradient spring, i.e. M(x) (x x0 ) s(x) 00 (x x0 ) . The stiffness of the bending-gradient spring at any section will thus be:

( )( − ) M x x x0 2 = s (x ) (x − x 0 ) . (3.7) κ00 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7631

Fig. 1. Representation of EB beam element by parallel equivalent springs: (a) equivalent bending springs which give the ﬁrst beam eigenstiffness (deformation of springs against constant curvature κ 0 is shown); (b) deformation of springs against linear curvature κ 00 ; (c) equivalent bending-gradient springs which give the second beam eigenstiffness.

Bending-gradient springs are shown as rotational springs in Fig. 1(c). Since the springs are in parallel, the stiffness for 2 this strain state is s (x ) (x − x 0 ) dx. L

3.2. Flexibility method

To ﬁnd the stiffness matrix of EB beam element by the ﬂexibility method, the following set of basis functions b q is 1 selected for the space of independent moment M(x ) ⊆ P 2 (R ) :

b q = [1 x − x 0 ] . (3.8) Beam moment is thus linearly interpolated. From Eq. (2.9) , the flexibility matrix (in the selected basis and with the pivot node x0 ) is obtained as: ⎡ ⎤ f (x ) dx Sym. ⎢ ⎥ L x0 = ⎢ ⎥ , Fq ⎣ ⎦ (3.9) 2 f (x ) (x − x 0 ) dx f (x ) (x − x 0 ) dx L L and the generalized strain vector becomes: T < ε >, x0 = κ( ) κ( ) . ( − ) . Eq x dx x x x0 dx (3.10) L L The first diagonal entry of the flexibility matrix, f (x ) dx, is the sum of section flexibilities. One may consider a beam L element, formulated by the flexibility method, as an infinite number of equivalent springs which are connected in series.

Thus, the flexibility of this infinite set (i.e. the beam element) is obtained by adding all spring (or section) flexibilities. The ( )( − ) non-diagonal entry f x x x0 dx, is the first moment of beam flexibility around a pivot node x0 . If the pivot node L coincides with the beam’s center of flexibility, the non-diagonal entry becomes zero; thus, the flexibility matrix becomes diagonal. Each diagonal entry is named as eigenflexibility . Note that beam’s centers of stiffness and flexibility may not coincide, since distributions of stiffness and flexibility along the element are not generally identical. 2 The second entry of the flexibility matrix in Eq. (3.9) , f (x ) (x − x 0 ) dx, is the second moment of beam flexibility. It L shows how section flexibility distribution affects element flexibility. Since in the flexibility method, equivalent springs are supposed to be in series, this entry is obtained by subjecting all springs to the same shear force V0 ; thus:

∀ A in L : κ(x ) = f (x ) V 0 (x − x 0 ) . (3.11) 7632 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

The moment of curvature for each section then becomes:

2 κ(x ) (x − x 0 ) = f (x ) V 0 (x − x 0 ) . (3.12)

2 Therefore, the ﬂexibility of the new equivalent spring is f (x ) (x − x 0 ) . This spring relates shear force to the moment of curvature. For the whole beam element: 2 κ(x ) (x − x 0 ) dx = f (x ) (x − x 0 ) dx . V 0 . (3.13) L L

σ = T Note that q [M0 V0 ] where M0 and V0 are moment and shear at the pivot node. Once the flexibility matrix is found by Eq. (3.9) , the stiffness matrix is obtained by inversing the flexibility matrix; however, the obtained stiffness matrix is not dimensionally consistent with that of Eq. (3.5) . Thus, a dimensional static matrix R = is required. From Eqs., (3.4) and (3.8), and for x0 0, R becomes: ⎡ ⎤ L 0 = ⎣ ⎦ . R L 3 (3.14) 0 12 Substituting the above matrix and the flexibility matrix of Eq. (3.9) into Eq. (2.12) , the flexibility-based stiffness matrix

K x0 is obtained. qf

3.3. EB beam element with an internal singularity

3.3.1. Stiffness method

( = ς ) ∈ − L , L A nonprismatic EB-beam element with an ideal hinge at xH H L L ≡[ 2 2 ] is considered. The section stiffness function is introduced as:

s (x ) = s˜ ( x ) [1 − ηδ(ς − ς H )] , (3.15) where s˜ ( x ) is the beam stiffness function with no singularity included. As will be shown later, η, is a kind of normalization parameter and is deﬁned as below: 2 2 η = s˜ ( x ) (x − x˜C .S. ) dx s˜ ( x ) (x − x H ) dx, (3.16) L L where x˜C .S. is the stiffness center of the beam without singularity. As a simple example, the uniform EB beam element with an internal hinge is considered. Section moment of inertia is

I0 . By substituting the stiffness function given by Eq. (3.15) into Eq. (3.5), it is obtained that: ⎡ ⎤ E I 0 L (1 − η) −E I 0 L [ x 0 + η( x H − x 0 )]

x0 = ⎣ ⎦ . Kqs 3 (3.17) EI0 L 2 2 −E I L [ x + η( x − x )] + E I L [ x − η( x − x ) ] 0 0 H 0 12 0 0 H 0 The transformation matrix: ⎡ ⎤ 12 x 1 6 x 12 x 1 6 x 0 − + 0 − 0 + 0 ⎢ L 3 L L 2 L 3 L L 2 ⎥ ψ = ⎣ ⎦ , (3.18)

12 6 − 12 6 L 3 L 2 L 3 L 2 is used in Eq. (2.14) to ﬁnd the stiffness matrix K from the generalized stiffness matrix in Eq. (3.17) . In Appendix B , it is shown how ψ is found.

L 2/ 12 An interesting case is where x 0 = − (i.e. pivot node coincides with the stiffness center, x C.S. ), for which the stiffness xH matrix in Eq. (3.17) becomes: ( − η) x EI0 L 1 0 K S = . (3.19) q 0 0

Such representation for the stiffness matrix of the singular EB beam element offers straightforward explanations for the ambiguous factor η. According to the proposed formulation, the normalization factor η is found based on the two following conditions: −η Condition 1: First moment of the hinge stiffness ( EI0 L) around an appropriate pivot node x0 , equilibrates that of the beam stiffness resultant (EI0 L) around x0 (see Fig. 2(a)):

E I 0 L (0 − x 0 ) = ηE I 0 L ( x H − x 0 ) . (3.20) R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7633

Fig. 2. Pivot node located on: (a) beam’s stiffness center, and (b) beam’s flexibility center; first moment of stiffness/flexibility around stiffness/flexibility center is zero.

Box 1.

Condition 2: Second moment of the hinge stiffness around x0 equilibrates that of the beam stiffness resultant around x0 : 3 EI0 L 2 2 + E I L (0 − x ) = ηE I L ( x − x ) . (3.21) 12 0 0 0 H 0 2 / = − L 12 η From the two above equations, x0 gets equal to xC.S. , and is found such that: xH 2 1 x x = 1 − H = 1 + H . (3.22) 2 η x 0 L / 12 It can be shown that the above value for η is the same as that given by Eq. (3.16) . Also note that the second eigenstiffness in Eq. (3.19) , which corresponds to the curvature-gradient strain state, is zero. This shows that there is no stiffness for this strain state as a consequence of singularity. The proposed procedure for ﬁnding stiffness-based stiffness matrix of an EB beam with multi-discontinuity and singularity is outlined in Box 1 .

3.3.2. Flexibility method To represent the EB beam element with an ideal hinge, Palmeri and Cicirello [21] recommended using the following section flexibility: ˜ f (x ) = f (x ) [1 + αδ(ς − ς H )] , (3.23) instead of the section stiffness given by Eq. (3.15) . This resolves the problem of negative stiffness at the hinge point which is physically unjustifiable. Here, α is an index for the severity of damage in x H . α =0 represents an undamaged section; α → ∞ denotes an ideal hinge with no flexural rigidity. Now, by substituting Eq. (3.23) into Eq. (3.9) , the following flexibility matrix is found: ⎡ ⎤ L L (1 + α) [ α x H − (1 + α) x 0 ] ⎢ E I 0 E I 0 ⎥ x0 = . Fq ⎣ ⎦ (3.24) 3 L L L 2 2 [ α x H − (1 + α) x 0 ] + [ x 0 + α( x H − x 0 ) ] E I 0 12 E I 0 E I 0 7634 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Box 2.

The stiffness matrix K x0 is then determined from Eq. (2.12) . The transformation matrix as given by Eq. (3.18) should be qf used. Let α → ∞ , the flexibility-based stiffness matrix of the element with an internal hinge is obtained. It is the same as the one developed by the stiffness method. α = If x0 1+ α xH , i.e. pivot node coincides with the flexibility center xC.F. as shown by Fig. 2(b), the stiffness matrix obtained by the flexibility method becomes: ⎡ ⎤ E I L 0 0 ⎢ + α ⎥ 1 xF = ⎢ ⎥ . Kqf ⎣ ⎦ (3.25) 1 + α E I L 3 0 0 + α + ας 2 12 1 12 H

α → ∞ → For an ideal hinge: ; thus x0 xH , and the stiffness matrix becomes: ⎡ ⎤ 0 0

xH ⎣ 3 ⎦ K = 1 E I L . (3.26) q 0 0 + ς 2 12 1 12 H

The proposed procedure for ﬁnding the ﬂexibility-based stiffness matrix of the EB beam element with multi-discontinuity and singularity is outlined in Box 2 .

4. Implementation for the Timoshenko (T) beam element

= { ( , , ) ∈ × | ∈ − L , L ≡ ⊂ , ( , ) ∈ ⊂ 2 } A linear elastic T beam element on the spatial domain : x y z L A x [ 2 2 ] L R y z A R under static loading is considered. A can have smooth or abrupt variations along longitudinal domain L . For T beam ( ) ≡ sb x 0 −1 ≡ elements, both bending and shear contributions are considerable. Thus, constitutive matrices: C [ ( )] and C 0 ss h x f (x ) 0 −1 −1 [ b ] describe variations of along . s ( x ), s ( x ), f (x ) = s (x ) and f (x ) = s (x ) represent bending stiffness, ( ) A L b s h b b s h sh 0 fs h x shear stiffness, bending flexibility, and shear flexibility at a section with abscissa x , respectively. The T beam element with two end nodes, each having a deflection and a rotation DOF, is considered. Displacement u ≡ [ w ( x ) θ( x )] T (with θ as rotation), strain ɛ ≡ [ κ( x ) γ ( x )] T , and stress σ ≡ [ M ( x ) V ( x )] T are the three unknown fields of the T beam element.

4.1. Stiffness method

The two following cases, TSM1 (Timoshenko Stiffness-based Method 1) and TSM2 (Timoshenko Stiffness-based Method 2), are possible:

1. TSM1: For the T beam element (having four DOFs/generalized coordinates), rigid body deﬂection and rotation, along κ γ with the two strain states of constant curvature 0 and constant shear strain 0 should satisfy ﬁnite element convergence criterion. Therefore, the T element, in its simplest form, is described as: R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7635

Fig. 3. Representation of T beam element by parallel equivalent springs; (a) bending springs, (b) shear springs, and (c) shear-gradient springs obtained from shear springs.

q ε = Bˆ qˆ = [ 0 I] r = B q , q q q κ 1 0 κ ∴ = 0 , (4.1) γ 0 1 γ0

where I is the identity matrix. Using the constitutive matrix C , one obtains from Eq. (2.5) that: ⎧ ⎫ ⎡ ⎤ ⎪ ( ) ⎪ ( ) ⎨ M x dx⎬ sb x dx 0 L ⎢ L ⎥ κ0 = ⎣ ⎦ . (4.2) ⎪ ⎪ γ0 ⎩ ( ) ⎭ ( ) V x dx 0 ssh x dx L L For the T beam model, two springs (a bending spring just like the EB beam model, and a shear spring) are assigned to any section. In the stiffness method, bending springs are in parallel ( Fig. 3 (a)); thus, beam bending stiffness is the sum of bending stiffnesses of all sections. Shear springs are also considered in parallel ( Fig. 3 (b)), and the shear stiffness can thus be obtained by adding all section shear stiffnesses. Note that the deformations displayed in Fig. 3 (a) and (b) represent constant-curvature and constant-shear strain states, and should not be confused with rigid body motions. 2. TSM2: In the derivation of Eq. (4.2) , kinematic coupling of shear strain and curvature is neglected. However, when it is considered by the following relations:

κ(x ) = ∂ x θ(x ) , (4.3)

γ (x ) = θ(x ) − ∂ x w (x ) . (4.4)

B in Eq. (4.1) should be modiﬁed as: q 1 0 B = . (4.5) q − x x0 sh 1

Here, x0 sh is the pivot node for the shear beam. Substituting Eq. (4.5) into Eq. (2.5), one obtains that: ⎧ ⎫ ⎡ ⎤ ⎪ ⎪ ⎪ M(x ) dx + V (x ) (x − x ) dx⎪ ( ) + ( ) ( − )2 . ⎨ 0 sh ⎬ sb x dx ssh x x x0 sh dx sym L L ⎢ L L ⎥ κ0 = . ⎣ ⎦ ⎪ L ⎪ γ0 ⎪ ⎪ ( )( − ) ( ) ⎩ V (x ) dx ⎭ ssh x x x dx ssh x dx 0sh L L L (4.6)

κ According to Eq. (4.6), when the T beam element is subjected to the constant curvature 0 , both bending and shear- gradient springs get activated. Shear-gradient spring, as shown in Fig. 3 (c), is a spring with a stiffness equal to the γ = γ / | = , ratio of the shear-force moment to the shear strain-gradient, 00 d dx x x0 sh

V (x )(x − x ) 0 = ( ) ( − ) 2. ssh x x x0 sh (4.7) γ00 7636 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Derivation of the above stiffness from shear stiffness is illustrated in Fig. 3 (c). It is similar to the procedure that was employed for the bending-gradient springs in Section 3.1 . From the kinematic matrix in Eq. (4.5) , one can conclude γ ≡ κ κ that 00 0 for the T beam elements. As a consequence, when the T beam element undergoes 0 , both bending and shear-gradient springs get activated. Therefore, the stiffness for this strain state is the sum of the two stiffnesses shown in Eq. (4.6) . When pivot node x coincides with the shear stiffness center x . . , non-diagonal entries 0 sh C Ssh become zero. Since both convergence (presence of constant curvature and shear strain) and the consistency (consistency of constant curvature with shear strain-gradient) conditions must be satisﬁed, the only available options for the stiffness-based T beam element are those given by Eqs. (4.2) and ( 4.6 ). Note that bending-gradient springs cannot be considered for the two-noded T beam elements.

4.2. Flexibility method

The following interpolation functions b q are considered for moment and shear ﬁelds: − 1 x x0 b b = . (4.8) q 0 1

Using Eq. (2.9), the following flexibility matrix (for the selected basis and the pivot node x0 b ): ⎡ ⎤ ( ) . fb x dx Sym ⎢ ⎥ x L F 0b = ⎣ ⎦ , (4.9) q ( ) ( − ) ( ) ( − )2 + ( ) fb x x x0 b dx fb x x x0 b dx fsh x dx L L L and generalized strain vector: T < ε >, x0 = κ( ) γ ( ) + κ( ) . ( − ) , Eq x dx x x x x0 dx (4.10) L L are obtained. Eq. (4.9) explains how flexibility, and its corresponding stiffness matrix, is influenced by sectional bending/shear flexibilities and bending flexibility distribution along the element. In flexibility method, unlike stiffness method ( Eq. (4.6) ), both shear and bending stiffness distributions contribute to shear deformations. Once the flexibility matrix in Eq. (4.9) is found, the stiffness matrix is calculated from Eq. (2.12) . It can then be substituted in Eq. (2.14) with R = L I , and: ⎡ ⎤ 1 1 0 − 0 ⎢ L L ⎥ ψ = ⎣ ⎦ , (4.11)

1 1 − 1 1 L 2 L 2 to ﬁnd the stiffness matrix of the T beam element. For constant bending and shear ﬂexibility distributions, the stiffness e matrix of type KE as reported by Felippa [22] is obtained.

4.3. T beam element with an internal singularity

Singularity (slope discontinuity) along the T beam is generally studied by the same procedures used in the EB beams (see e.g., [ [23] , [24] ]). To ﬁnd the stiffness matrix of the T beam element with an internal singularity, we used bending stiffness/ﬂexibility functions introduced in Section 3.3 . For the sake of simplicity and better understanding, the uniform T

( = ς ) ∈ − L , L beam element with an ideal hinge at xH H L L ≡[ 2 2 ] is considered. Without loss of generality, the results can be extended to multiple discontinuities along the nonprismatic T beam element. For the stiffness method, sb (x) is obtained

( ) = ˜ ( ) = −1 from Eq. (3.15), with s˜ x EI0 ; for the flexibility method, Eq. (3.25) is used, with f x EI0 . Shear stiffness is supposed to be GA0 for all sections. Closed-form stiffness matrices for different T beam formulations are reported in Table 1. It can be seen from Table 1 that the stiffness matrix given by TSM2 suffers rank redundancy; i.e. the rank of the matrix is wrongly equal to 2 rather than 1. Therefore, TSM2 formulation cannot represent a T beam element with an internal ideal hinge. The proposed procedure for finding stiffness- and flexibility-based stiffness matrices of T beams with multi-discontinuity and singularity are outlined in Box 3 and Box 4 , respectively.

5. Numerical examples

5.1. Nonprismatic elements without hinge

The two following examples: R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7637

Table 1 Stiffness matrices for different types of T beam element with an internal singularity.

Diagonal x0 Diagonal Method Fq Kq Kq L L + α 0 ( − )2 ( − ) 0 0 EI0 EI0 x0 xH x0 xH Flexibility (FM) 12 EI0 GA 0 L × 3 + ( 2 + 2 ) 12 E I 0 G A 0 L 1 L L L 12EI0 GA0 L 12xH + + . . ( x − x ) 1 0 + ( 2 + 2 ) 0 xH xC F b 0 H 12EI0 GA0 L 12x 12 EI0 GA0 EI0 H E I 0 L (1 − η) 0 0 0 Stiffness (TSM1) – 0 G A 0 L 0 GA0 L 3 GA0 L 2 2 G A L 3 E I 0 L (1 − η) + + G A 0 L x −G A 0 L x 0 0 Stiffness (TSM2) – 12 0 0 12 − 0 G A L GA0 L x0 GA0 L 0

Box 3.

Box 4.

1. A multi-span beam with tapered members (without hinge) 2. A portal frame with tapered columns and a uniform beam (without hinge) are considered to verify the validity of the method for dealing with nonprismatic elements with no hinges. The two structures are shown in Fig. 4 . The members are modeled as EB beam elements. All nonprismatic members follow the tapering

( ) = ( + ( − x )2 ) pattern of Fig. 4(c) with a varying moment of inertia as in: EI x EI 1 7 1 L . Elastic modulus is assumed to be E = 210 GPa, and axial stiffness is assumed to be extremely large for the case of simplicity. Any kind of buckling is prevented. It is also assumed that the initial curvature of the tapered element does not violate the EB assumption, due to the large centroid axis curvature radius-to-depth ratio. The structures are analyzed under the speciﬁed loadings, as illustrated in Fig. 4 (a) and (b), by the following methods:

1. FEM using the proposed stiffness-based stiffness matrix 2. FEM using the proposed ﬂexibility-based stiffness matrix 3. Analytical solution of the governing differential equations (exact responses) 7638 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 4. (a) Multi-span beam with tapered members (without hinge); (b) portal frame with tapered columns and uniform beam (without hinge); (c) tapering pattern of nonprismatic members in (a) and (b).

Fig. 5. Deflection, moment and shear diagrams for multi-span beam of Fig. 4 (a) under: (a) concentrated mid-span loading, and (b) uniform loading (red: exact; solid blue: proposed flexibility-based method; dashed blue: proposed stiffness-based method). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The deflection, moment and shear diagrams for the two structures are shown in Figs. 5 and 6 . Fig. 5 (a) and (b) pertain to two load cases: concentrated loads P = 50 kN at midspans (as shown in Fig. 4 (a)), and a uniform load p = 10 kN/m. The results for both structures show that deflections have good agreement with the exact responses, especially for the flexibility-based method with concentrated loads. The results of moments and shears deteriorate for the stiffness-based method; however, the results given by the flexibility-based method agree well with the exact ones. It is notable that for concentrated loading, the flexibility-based method gives exact results. R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7639

Fig. 6. Deflection, moment and shear diagrams for portal frame of Fig. 4 (b) (red: exact; solid blue: proposed flexibility-based method; dashed blue: proposed stiffness-based method). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7. EB/T beam with multiple discontinuities and an internal hinge.

5.2. Nonprismatic beam element with multiple discontinuities and an internal singularity

This example is aimed at disclosing the abilities and disabilities of the proposed stiffness and flexibility methods for beams with multiple discontinuities and singularities. The EB/T nonprismatic beam with multiple discontinuities and an internal singularity (ideal hinge) is considered as shown in Fig. 7 . The depth of the rectangular section of the beam varies linearly from 2 h to h along the first half of the beam. At the midpoint of the second half, section depth drops abruptly from h to h /2. An ideal bending hinge is located at the mid of Seg.2. Boundary conditions are displayed in the figure. The beam has a depth of h = 0.4 m, a thickness of t = 0.1 m, an elastic modulus of E = 210 GPa, and a length of L = 8 m for EB formulation and L = 2 m for T formulation. It is analyzed under the two following loads: (1) concentrated moment M = 100 kNm exerted at the hinged end, and (2) uniform distributed load p = 10 kN/m. The following four methods are examined:

1. FEM using the proposed stiffness-based stiffness matrix; the whole beam is considered as a unit finite element. 2. FEM using the proposed flexibility-based stiffness matrix; the whole beam is considered as a unit finite element. 3. Exact analytical solution of the governing differential equation. Exact results (for the concentrated loading) may also be obtained by FEM using exact stiffness matrices of the four members Seg.1, Seg.2-1, Seg.2-2 and Seg.3 (Franciosi and Mecca [25] for EB case, and Frieman and Kosmatka [2,11] for T case). 4. Finite element modeling of the beam as a plane-stress 2D membrane shell. 4-node bilinear quadrilateral elements are used for discretization. The ideal hinge at the middle of Seg.2 is modeled as a thin crack extending nearly over the whole depth (except for a small fraction of the beam section).

Deflections and rotations at discontinuity positions (obtained by the above methods) are compared in Table 2 (for the EB case). Fig. 8 (a) and (b) show the deformed shapes obtained by the above methods for the concentrated moment and distributed loadings respectively. In the proposed flexibility method (method 2), displacements are found using the shape functions introduced by Shooshtari and Khajavi [18] . These flexibility-based shape functions are shown in Fig. 9 . The effect of the hinge can be seen as slope discontinuities on the shape functions. It is clearly seen from Fig. 8 (a) and Table 2 that the proposed flexibility method (method 2) and the exact method (method 3) give the same results for the concentrated moment loading; however, this is not the case for the uniform distributed loading (As shown by Fig. 8 (b) and Table 2 ). There is a large discrepancy between the deformed shapes obtained by methods 2 and 3, which is due to their different equivalent nodal loads. Note that while method 2 employs just one element (with one active DOF), method 3 uses 4 elements (with 8 active DOFs). From Fig. 8 and Table 2 , one may conclude that the proposed flexibility-based stiffness matrix offers satisfactory results for nodal values. It also performs well for cases 7640 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Table 2 Results of analysis for the nonprismatic EB beam element with multiple discontinuities and an internal singularity ( Section 5.2 ).

Method x = L /2 x = 5 L /8 x = 3 L /4 x = L

Deflection Rotation Deflection Rotation Rotation Deflection Rotation Rotation (before hinge) (after hinge) mm rad mm rad rad mm rad rad

Concentrated moment Proposed stiffness method −2.786 −0.0 0 070 −3.264 −0.0 0 022 −0.0 0 022 −3.134 0.0 0 052 0.00279 Proposed ﬂexibility method −1.893 −0.00104 −3.034 −0.00119 −0.00175 −4.734 −0.00160 0.00792 FEM (4 exact EB elements) −1.893 −0.00104 −3.034 −0.00119 −0.00175 −4.734 −0.00160 0.00792 FEM (membrane elements) −1.117 −0.0 0 055 −1.912 −0.0 0 057 −0.00109 −2.976 −0.0 0 080 0.00492 Distributed loading Proposed stiffness method −1.486 −0.0 0 037 −1.741 −0.0 0 012 −0.0 0 012 −1.671 0.0 0 028 0.00146 Proposed ﬂexibility method −0.398 −0.0 0 022 −0.637 −0.0 0 025 −0.0 0 037 −0.994 −0.0 0 034 0.00166 FEM (4 exact EB elements) −1.851 −0.0 0 091 −2.816 −0.0 0 099 0.0 0 042 −2.380 0.0 0 047 0.00166 FEM (membrane elements) −1.083 −0.0 0 045 −1.522 −0.0 0 032 −0.0 0 029 −1.755 0.0 0 0 08 0.00126

Fig. 8. Deformed shapes of the EB beam introduced in Fig. 7 ( Section 5.2 ) under: (a) end moment loading and (b) distributed uniform loading, obtained by different methods. where concentrated nodal forces or moments are only applied. The deformed shape given by the proposed stiffness method does not have any slope discontinuity, since its shape functions cannot include the hinge. The method, however, considers the hinge effect in the stiffness matrix (through Eqs. (3.5) and ( 3.15 )), and consequently in the nodal results. Deﬂections and rotations at discontinuity positions for the T beam are reported in Table 3 . Again, it is seen that, for the concentrated loading, the results of methods 2 and 3 are exactly the same. Fig. 10 (a) shows the deformed shapes of the T beam. The results of methods 2 and 3 exactly coincide; however, this is not the case, just like the EB one, for the distributed R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7641

Fig. 9. Flexibility-based shape functions for the EB beam in Fig. 7 ( Section 5.2 ).

Table 3 Results of analysis for the nonprismatic T beam element with multiple discontinuities and an internal singularity ( Section 5.2 ).

Method x = L /2 x = 5 L /8 x = 3 L /4 x = L

Deflection Rotation Deflection Rotation Rotation Deflection Rotation Rotation (before hinge) (after hinge) mm rad mm rad rad mm rad rad

Concentrated moment Proposed stiffness method −0.016667 0.0 0 0 033 −0.015625 0.0 0 0 042 0.0 0 0 042 −0.012500 0.0 0 0 050 0.0 0 0 067 Proposed ﬂexibility method −0.152987 −0.0 0 0260 −0.236790 −0.0 0 0298 −0.0 0 0291 −0.319005 −0.0 0 0254 0.002127 FEM (4 exact EB elements) −0.152987 −0.0 0 0260 −0.236790 −0.0 0 0298 −0.0 0 0291 −0.319005 −0.0 0 0254 0.002127 FEM (membrane elements) −0.067149 −0.0 0 0132 −0.122475 −0.0 0 0222 −0.0 0 0230 −0.194585 −0.0 0 0330 0.0 0110 0 Distributed loading Proposed stiffness method −0.0 0 0556 0.0 0 0 0 011 −0.0 0 0521 0.0 0 0 0 014 0.0 0 0 0 014 −0.0 0 0417 0.0 0 0 0 017 0.0 0 0 0 02 Proposed ﬂexibility method −0.002159 −0.0 0 0 037 −0.003342 −0.0 0 0 042 −0.0 0 0 041 −0.004502 −0.0 0 0 036 0.0 0 0 030 FEM (4 exact EB elements) −0.010 0 07 −0.0 0 0 014 −0.014245 −0.0 0 0 015 0.0 0 0 011 −0.011756 0.0 0 0 011 0.0 0 0 030 FEM (membrane elements) −0.008087 −0.0 0 0 014 −0.012527 −0.0 0 0 019 0.0 0 0 0 02 −0.010807 0.0 0 0 0 09 0.0 0 0 023

loading, as is shown by Fig. 10 (b). It is evident from the figures that the results of the stiffness method (TSM1 in Section 4.1 ) have drastically deteriorated for the T beam. In general, the stiffness-based matrices might not be good options for modeling beams with multiple discontinuities, especially when singularities are included. This is nearly obvious, since modeling the beam with parallel virtual springs is intuitively far from reality. For the proposed flexibility-based method (method 2), the deformed shape is found using the shape functions introduced by Shooshtari and Khajavi [18] . The flexibility-based shape functions of the considered T beam are shown in Fig. 11. Fig. 12 (a)–(d) illustrate moment and shear diagrams for the two cases of end moment and uniform loading for both EB and T beam formulations. For all cases, shear and moment results obtained by the stiffness method are not satisfactory. The results of the flexibility method exactly coincide with the analytical solutions for the end moment loading; however, for the uniform loading, some large discrepancies are observed for EB and especially T beams. This is due to the fact that higher-order statements in analytical solutions may not be represented by the flexibility-based shape functions. Thus, for distributed loading, refined meshing by stiffness-based finite elements might be recommended; however, such refined meshes may not give satisfactory results especially for shear, as shown by Figs. 13 (a)–(c) and 14 (a)–(c). In these figures, the series of converged deflection, curvature, moment and shear diagrams for the stiffness-based formulations (EB, TSM1, and TSM2) are illustrated. In brief, the proposed flexibility method is best for the analysis of beams with multiple discontinuities/singularities or when high computational cost is a main concern. The approach is now extended to the beams with cracks. A crack, instead of a hinge, is considered at x H . For both EB and T formulations, the following stiffness, as suggested by Bilello [26] , is associated to the rotational spring at the crack position:

E h 2 t 0 . 9 (β − 1)2 k = . (5.1) 12 β (2 − β) 7642 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 10. Deformed shapes of the T beam introduced in Fig. 7 ( Section 5.2 ) under: (a) end moment loading, and (b) distributed uniform loading, obtained by different methods.

Fig. 11. Flexibility-based shape functions for the T beam in Fig. 7 ( Section 5.2 ). R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7643

Fig. 12. Moment and shear diagrams obtained for the beam of Fig. 7: (a) EB beam under end moment loading; (b) EB beam under uniform loading; (c) T beam under end moment loading; (d) T beam under uniform loading (red: exact; solid blue: proposed ﬂexibility-based method; dashed blue: proposed stiffness-based method). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.) 7644 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 13. Deﬂection, curvature, moment and shear diagrams obtained from the convergence test for the 4-segmented beam of Fig. 7 under end moment loading. (The rows of diagrams are associated with {Seg.1:1 element, Seg.2-1:1 element, Seg.2-2:1 element, Seg.3:1 element}, {2 elements, 1 element,1 element, 1 element}, {4 elements, 1 element, 1 element, 2 elements}, {8 elements, 2 elements, 2 elements, 4 elements}, {16 elements, 4 elements, 4 elements, 8 elements}): (a) EB formulation; (b) TSM1 formulation; (c) TSM2 formulation. R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7645

Fig. 13. Continued

β = d / h , in which d is the crack depth. Using this stiffness, the following value for α (see Eq. (3.25) ) is obtained [21] : h β (2 − β) α = . (5.2) L 0 . 9 (β − 1)2 For the stiffness method, an appropriate value for η ( Eq. (3.15) ) is found by setting the damage parameter introduced by Caddemi and Caliò [27] equal to the above value of α: η h β (2 − β) = . (5.3) 1 − η A L 0 . 9 (β − 1)2

η = 1 For the special case of an ideal hinge (with no rotational spring attached), A [28]. Now, from Eq. (3.16), it can be written that: 2 2 A = s˜ ( x ) (x − x H ) dx/ s˜ ( x ) (x − x˜C .S. ) dx . (5.4) L L From Eqs. (5.3) and ( 5.4 ), the following expression can be defined for η, for the case of a crack with depth ratio β at location x H : 1 η = . (5.5) ( ) ( − )2 s˜ x x xH dx . (β− ) 2 L + L 0 9 1 2 h β (2 −β) s˜ (x ) (x −x˜C .S. ) dx L In the present example, a crack is assumed at the same position where the hinge was considered. β is set to 0.5. The results of the EB case by the proposed stiffness and flexibility methods, as well as those of 2D model, are reported in Table 4 , and shown in Fig. 15 (a) and (b) for end moment and uniform loading cases, respectively. Shape functions obtained by the flexibility-based method are shown in Fig. 16 . A simple comparison between shape functions in Figs. 9 and 16 shows a decrease in slope discontinuity for the latter. From Tables 4 and 2 , it can be seen that the end rotation has not much differed from that of the hinge case; however, slope difference at the crack location has decreased, especially for the case of end moment loading. For both hinge and crack cases, the flexibility method overestimates the nodal rotation, and the stiffness method underestimates it; no exact rule governs the results of the interior points. 7646 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 14. Deﬂection, curvature, moment and shear diagrams obtained from the convergence test for the 4-segmented beam of Fig. 7 under uniform loading. (The rows of diagrams are associated with {Seg.1:1 element, Seg.2-1:1 element, Seg.2-2:1 element, Seg.3:1 element}, {2 elements, 1 element, 1 element, 1 element}, {4 elements, 1 element, 1 element, 2 elements}, {8 elements, 2 elements, 2 elements, 4 elements}, {16 elements, 4 elements, 4 elements, 8 elements}): (a) EB formulation; (b) TSM1 formulation; (c) TSM2 formulation. R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7647

Fig. 14. Continued

Table 4 Results of analysis for the nonprismatic EB beam element with multiple discontinuities and an internal crack ( β = 0.5) ( Section 5.2 ).

Method x = L /2 x = 5 L /8 x = 3 L /4 x = L

Deflection Rotation Deflection Rotation Rotation Deflection Rotation Rotation (before crack) (after crack) mm rad mm rad rad mm rad rad

Concentrated moment Proposed stiffness method −2.113 -0.0 0 053 −2.476 −0.0 0 017 −0.0 0 017 −2.377 0.0 0 040 0.00211 Proposed ﬂexibility method −2.050 −0.00114 −3.315 −0.00134 −0.00154 −4.822 −0.00142 0.00790 FEM (membrane elements) −1.100 −0.0 0 061 −1.754 −0.0 0 094 −0.00102 −2.952 −0.00124 0.00491 Distributed loading Proposed stiffness method −1.127 −0.0 0 028 −1.320 −0.0 0 0 09 −0.0 0 0 09 −1.268 0.0 0 021 0.00112 Proposed ﬂexibility method −0.448 −0.0 0 025 −0.724 −0.0 0 029 −0.0 0 034 −1.053 −0.0 0 031 0.00172 FEM (membrane elements) −1.142 −0.0 0 044 −1.511 −0.0 0 032 −0.0 0 030 −1.706 −0.0 0 017 0.00126

Results for the T beam element with the same crack at the mid of Seg. 2 are given in Table 5 . The deformed shapes obtained by different methods for the two cases of end moment and uniform loading are shown in Fig. 17 (a) and (b), respectively. Similar results as of the EB case are also obtained for the T beam. Moment and shear diagrams, obtained by the three methods, are shown in Fig. 18 (a)–(d). The results agree to those of cases with singularity ( Fig. 12 (a)–(d)). In brief, the following results are concluded based on this numerical example:

1. Generally, the results of the proposed stiffness method are not satisfactory, unless nodal loading is only applied and nodal values are only concerned. For multiple discontinuous beams, and especially when singularities are included, flexibility method is preferred to the stiffness one. 2. When nodal loading is only present, the proposed flexibility method gives exact results for both nodal and interior points of the beam element. 3. For cases with interior loading, nodal values given by the proposed flexibility method are satisfactory; however, the interior values are far from exact results, since the flexibility method may not correctly lump the interior load at the nodes. This is also the case when there is no singularity [18] ; but singularity intensifies such phenomenon. 7648 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 15. Deformed shapes of the EB beam introduced in Fig. 7 ( Section 5.2 ), with a crack substituted for the singularity (with β = 0.5), under: (a) end moment loading and (b) distributed uniform loading, obtained by different methods.

Fig. 16. Flexibility-based shape functions for the EB beam in Fig. 7 ( Section 5.2 ) with a crack substituted for the singularity (with β = 0.5).

Table 5 Results of analysis for the nonprismatic T beam element with multiple discontinuities and an internal crack ( β = 0.5) ( Section 5.2 ).

Method x = L /2 x = 5 L /8 x = 3 L /4 x = L

Deflection Rotation Deflection Rotation Rotation Deflection Rotation Rotation (before crack) (after crack) mm Rad mm rad rad mm rad rad

Concentrated moment Proposed stiffness method −0.015565 0.0 0 0 031 −0.014593 0.0 0 0 039 0.0 0 0 039 −0.011674 0.0 0 0 047 0.0 0 0 062 Proposed ﬂexibility method −0.152747 −0.0 0 0260 −0.236367 −0.0 0 0297 −0.0 0 0293 −0.318866 −0.0 0 0255 0.002127 FEM (membrane elements) −0.001839 −0.0 0 0 0 09 −0.004877 −0.0 0 0152 −0.0 0 0274 −0.138349 −0.0 0 060 0 0.0 010 0 0 Distributed loading Proposed stiffness method −0.0 0 0519 0.0 0 0 0 01 −0.0 0 0486 0.0 0 0 0 01 0.0 0 0 0 01 −0.0 0 0389 0.0 0 0 0 02 0.0 0 0 0 02 Proposed ﬂexibility method −0.002154 −0.0 0 0 0 04 −0.003333 −0.0 0 0 0 04 −0.0 0 0 0 04 −0.004496 −0.0 0 0 0 04 0.0 0 0 030 FEM (membrane elements) −0.006513 −0.0 0 0 010 −0.008635 −0.0 0 0 010 0.0 0 0 0 05 −0.009456 0.0 0 0 0 01 0.0 0 0 018 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7649

Fig. 17. Deformed shapes of the T beam introduced in Fig. 7 ( Section 5.2 ), with a crack substituted for the singularity (with β = 0.5), under: (a) end moment loading and (b) distributed uniform loading, obtained by different methods.

It is also notable that the discrepancies between the responses of exact 1D analysis with those of 2D shell model are due to the initial assumptions made in the two formulations. Such discrepancies are mostly caused by the low-stress areas around discontinuous sections [19] and local taper effects [20] , which are totally ignored in 1D models.

5.3. Portal frame with nonprismatic and singular members

This numerical example is devised based on the results of the previous example. It was concluded that the proposed procedure is eﬃcient for cases where no interior loading is present, or when nodal values are only concerned. The portal frame shown in Fig. 19 is considered. The two columns are identical. Each column consists of a uniform segment (with a constant EI ) and a tapered one (with cubic s ( x ); the wider end with 8 EI ). The beam member consists of two identical tapered segments (just like the ones for the columns), and a uniform mid segment (with a constant EI ) with a hinge included. Other required data are shown in Fig. 19 . EB formulation is considered for all members, and any kind of buckling is prevented. All members are assumed to be axially inextensible. The analysis is performed by the use of:

1. 3 proposed stiffness-based beam elements (Total DOFs = 12). 2. 3 proposed ﬂexibility-based beam elements (Total DOFs = 12). 3. 8 exact 1D frame elements, separated by discontinuities along the members (Total DOFs = 27). Stiffness matrices for nonprismatic elements are adopted from Franciosi and Mecca [25] . 4. 11,370 plane-stress membrane 2D quadrilateral elements, each having 4 nodes with 8 DOFs.

2 = L ( + ) In Table 6, nodal horizontal translation ux2 ux3 , in the form of EI CM M CP PL , and rotations of nodes 2, 3 and 4, in

L ( + ) the form of EI CM M CP PL , are reported for all four above methods; CM and CP are coefficients obtained by analysis. As shown in Table 6 , the results of the flexibility method coincide with the exact 1D solutions (method 3). The results of the stiffness method are again far from exact ones. Note that the exact results by the flexibility method are obtained using only 12 DOFs, while for the exact solution, 27 DOFs are used. Obviously, for large-scale structures, such performance of the 7650 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 18. Moment and shear diagrams obtained for the beam of Fig. 7 (with a crack substituted for the singularity with β = 0.5): (a) EB beam under end moment loading; (b) EB beam under uniform loading; (c) T beam under end moment loading; (d) T beam under uniform loading (red: exact; solid blue: proposed ﬂexibility-based method; dashed blue: proposed stiffness-based method). (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

proposed procedure is considerably favorable. It is also notable that the results of the flexibility method coincide well with those of 2D solution. Fig. 20 (a) compares different deformed shapes of the hinged portal frame subjected to lateral load P = 100 kN. Sections at uniform parts are rectangular with thickness t = 0.1 m and depth h = 0.2 m. Along tapered parts, they linearly vary up to h = 0.4 m. The elastic modulus is E = 210 GPa, and all member lengths are set to L = 9 m. It is seen that the flexibility method predicts deformation efficiently well and the results correspond with those of 2D solution. While stiffness method ignores slope discontinuity at the hinge location, such discontinuity is successfully modeled by the flexibility method. As pointed out in the previous example, the interior values given by the stiffness method might not be acceptable, especially when singularities are included. Fig. 20 (b) shows the results for the bending stress at the top fiber of the horizontal frame R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7651

Fig. 19. Portal frame with nonprismatic and singular members.

Table 6 Results of analysis for the portal frame by different methods ( Section 5.3 ).

DOF 2D Exact (1D) Flexibility method Stiffness method (membrane elements) Franciosi and Mecca [25] (11,370 elements) (8 Elements) (3 elements) (3 elements)

C M C P C M C P C M C P C M C P Value Value Value Value Value Error % Value Error % Value Error % Value Error %

− − − − ux 2(3) 0.0542 0.0599 0.0576 0.0623 0.0576 0.00 0.0623 0.00 0.0408 29.17 0.0433 30.50 θ 2 0.1373 −0.0426 0.1451 −0.0576 0.1451 0.00 −0.0576 0.00 0.0930 35 .91 −0.0408 29.17 θ 3 −0.0507 0.0081 −0.0517 0.0078 −0.0517 0.00 0.0078 0.00 −0.0387 25 .15 0.0089 14.10 θ 4 0.1406 −0.1157 0.1474 −0.1198 0.1474 0.00 −0.1198 0.00 0.1283 12 .96 −0.1008 15.86

member. As expected, the results offered by the stiffness method are not satisfactory; however, ﬂexibility method performs well and predicts values close to the 2D ﬁnite element results.

6. Conclusion

The paper proposes a new interpretation of the stiffness and ﬂexibility matrices (the spring model); based on this,:

1. EI and EI −1 functions are defined for beams with singularities or cracks. Then, stiffness- and flexibility- based stiffness matrices are developed. 2. Flexibility-based shape functions, as introduced by Shooshtari and Khajavi [18] for multi-nonprismatic EB and T beams, are extended to beams with singularities or cracks. 3. The proposed flexibility method is proved to be more efficient than the stiffness method. For cases with concentrated loading, flexibility method gives exact results. Even fine meshes of stiffness-based beam elements may not give good results for shear.

The proposed procedure for developing stiffness- and ﬂexibility-based stiffness matrices for nonprismatic and/or singular EB and T beam elements is characterized by the following features:

1. Simplicity: It offers simple representations for the stiffness- and ﬂexibility-based stiffness matrices. Such simplicity is due to the basis functions considered for strain ﬁelds. There are researches in the literature which offer exact solutions to the differential equations of EB or T nonprismatic beams; however, they are limited to special types 7652 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

Fig. 20. (a) Deformed shapes of the portal frame of Fig. 19 ( Section 5.3 ), obtained by different methods, and (b) top-ﬁber bending stress diagrams for the horizontal member of the portal frame, obtained by different methods.

of discontinuities (e.g. special tapering types [5] , or abrupt section changes and singularities without tapering [4] ); their presentations are rather complex, and a few of them are formulated in the framework of FEM. Also, note that integration in the proposed procedure is performed on the nonprismatic beam element; however, in most methods, differential equations are integrated which generally results in complex integrands. 2. Physical interpretation: The procedure presents a clear physical interpretation for the stiffness and flexibility methods. According to this interpretation, the beam element stiffness matrix is dependent on bending and shear stiffnesses/flexibilities of all sections, as well as their distributions along the beam. The procedure shows that entries of the stiffness matrix are the zero-, first-, and second-order moments of stiffness/flexibility distributions. A spring- model is also introduced which associates stiffness/flexibility methods with virtual parallel/series springs. 3. Generality: The proposed procedure can develop finite element models for beams with multiple patterns of discontinuity, tapering, and singularity. To the author’s knowledge, no such general method has yet been suggested. 4. Computational efficiency: The procedure enables one to consider the whole beam segments (aligned in one direction) as a single finite element. This effectively reduces number of DOFs. Therefore, less computational effort is required, without considerable loss of accuracy (especially when using flexibility method).

Acknowledgment

I would like to acknowledge the help of Dr. Peyman Pourmoghaddam and Mr. Javad Khajavi for their proofreadings, which profoundly improved the composition of this manuscript. R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7653

Appendix A

Notation Most of the symbols used in the context are categorized and listed as follows:

Fields u displacement field w deflection field θ rotation field ε strain field κ curvature field γ shear strain field σ stress field M bending moment field V shear force field Functions s section stiffness f section flexibility EI section bending stiffness GA section shear stiffness δ Dirac delta function Matrices C constitutive matrix N interpolation matrix B strain–displacement matrix K stiffness matrix F flexibility matrix R dimensional matrix transformation matrix I identity matrix Vectors d nodal displacement vector q generalized displacement vector e internal deformation vector p internal force vector Parameters η hinge stiffness parameter α hinge flexibility parameter γ discontinuity intensity parameter (from Biondi and Caddemi [28] ) A mathematical parameter (from Bagarello [29] ) Coordinates spatial domain x, y, z spatial coordinates ξ non-dimensional spatial coordinate L beam length Operators L kinematic differential operator matrix ∂ derivative operator Subscript represents for

( . ) 0 ( . ) at pivot node ( . ) 00 derivative of ( . ) at pivot node ( . ) C.S. ( . ) at the center of stiffness ( . ) C.F. ( . ) at the center of ﬂexibility ( . ) H ( . ) at the hinge location ( . ) r ( . ) corresponding to rigid body motion ( . ) q ( . ) corresponding to generalized coordinates ( . ) s ( . ) corresponding to stiffness method ( . ) f ( . ) corresponding to ﬂexibility method ( . ) b ( . ) corresponding to bending ( . ) sh ( . ) corresponding to shear ( . ) A ( . ) corresponding to beam section ( . ) L ( . ) corresponding to beam length Superscript represents for ( . ) ( . ) dependent of ( can be u , ε , σ, ...) > ( . ) < ( . ) dependent of an integral of Overscript represents for (·) exact value of ( . ) (·) complete representation of the partitioned ( . ) (·) ( . ) with no hinge included In this Appendix, it is shown how ψ is derived for Eq. (3.18) (EB beam) and Eq. (4.11) (T beam). For EB beam, element = θ θ = nodal displacement vector D [wi i w j j ] and the vector of generalized coordinates, introduced in Eq. (2.1), qm [ w m θm κm κmm ] , are related by the transformation ψ mD as follows [30] : 7654 R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655

q m = ψ mD D , (B.1) where, ⎡ ⎤ L L 2 L 3 1 − − ⎢ 2 8 48 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ L L ⎥ 0 1 − ⎢ ⎥ ⎢ 2 8 ⎥ ψ = . (B.2) mD ⎢ ⎥ ⎢ L L 2 L 3 ⎥ ⎢ 1 ⎥ ⎢ 2 8 48 ⎥ ⎣ ⎦ L L 2 0 1 2 8

The subscript m stands for the element center x m = 0, and the subscripts i and j represent element nodes. If q 0 =

[w 0 θ0 κ0 κ00 ] is the generalized coordinates evaluated at the pivot node x0 , then the generalized coordinates qm and q0 are related to each other by the following relationship:

q 0 = ψ 0 m q m , (B.3) where, ⎡ ⎤ 0 0 0 0 ⎢ 0 0 0 0 ⎥ ψ 0 m = ⎣ ⎦ , (B.4) 0 0 1 −x 0 0 0 0 1

q0 and D are thus related as below: ˆ q 0 = ψ D , (B.5) where, ⎡ ⎤ 0 0 0 0 ⎢ 0 0 0 0 ⎥ ⎢ ⎥ ψ ˆ −1 ⎢ 12x0 1 6x0 12x0 1 6x0 ⎥ r ψ = ψ 0 m ψ = − + − + = . (B.6) mD ⎢ 3 2 3 2 ⎥ ψ ⎣ L L L L L L ⎦

12 6 − 12 6 L 3 L 2 L 3 L 2

For T beam, the vector of generalized coordinates q m = [ w m θm κm γm ] and the element nodal displacement vector D are related by the transformation matrix [30] : ⎡ ⎤ L L 1 − 0 ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ L ⎥ ⎢ 0 1 − 0 ⎥ ⎢ 2 ⎥ ψ mD = . (B.7) ⎢ ⎥ ⎢ L L ⎥ 1 0 − ⎢ ⎥ ⎣ 2 2 ⎦ L 0 1 0 2

κ γ ψ = 02 02 Since in T formulation, and are kinematically decoupled, 0 m [ ] and thus: 02 I2 ⎡ ⎤ 0 0 0 0 ⎢ 0 0 0 0 ⎥ ⎢ ⎥ − 1 1 ψˆ = ψ 1 = ⎢ 0 − 0 ⎥ . (B.8) mD ⎢ ⎥ ⎣ L L ⎦

1 1 − 1 1 L 2 L 2 References

[1] J.H. Argyris , Energy theorems and structural analysis, Part 1, Aircr. Eng 26 (1954) 383–387 . [2] Z. Friedman , J.B. Kosmatka , An improved two-node Timoshenko beam finite element, Comput. Struct. 47 (3) (1993) 473–481 . [3] N. Ganesan , R.C. Engels , Timoshenko beam finite elements using the assumed modes method, J. Sound Vib. 156 (1) (1992) 109–123 . [4] B. Biondi , S. Caddemi , Euler–Bernoulli beams with multiple singularities in the flexural stiffness, Eur. J. Mech. A Solids 26 (2007) 789–809 . [5] D.L. Karabalis , D.E. Beskos , Static, dynamic and stability analysis of structures composed of tapered beams, Comput. Struct. 16 (6) (1983) 731–748 . R. Khajavi / Applied Mathematical Modelling 40 (2016) 7627–7655 7655

[6] S.Y. Lee , Y.H. Kuo , Static analysis of nonuniform Timoshenko beams, Comput. Struct. 46 (5) (1993) 813–820 . [7] J.M. Ortuzar , A. Samartin , Some consistent finite element formulations of 1-D beam models: a comparative study, Adv. Eng. Softw. 29 (7–9) (1998) 667–678 . [8] J. Petrolito , Stiffness analysis of beams using a higher-order theory, Comput. Struct. 55 (1) (1995) 33–39 . [9] X. Tang , Shape functions of tapered beam-column elements, Comput. Struct. 4 (5) (1993) 943–953 . [10] Z. Friedman , J.B. Kosmatka , Exact stiffness matrix of a nonuniform beam-I. Extension, torsion, and bending of a Bernoulli–Euler beam, Comput. Struct. 42 (5) (1992) 671–682 . [11] Z. Friedman , J.B. Kosmatka , Exact stiffness matrix of a nonuniform beam-II. Bending of a Timoshenko beam, Comput. Struct. 49 (3) (1993) 545–555 . [12] R.L. Taylor , F.C. Filippou , A. Saritas , F. Auricchio , A mixed finite element method for beam and frame problems, Comput. Mech. 31 (2003) 192–203 . [13] A. Tena-Colunga , Stiffness formulation for nonprismatic beam elements, J. Struct. Eng. ASCE 122 (12) (1996) 1484–1489 . [14] L. Vu-Quoc , P. Léger , Efficient evaluation of the flexibility of tapered I-beams accounting for shear deformations, Int. J. Numer. Methods Eng. 33 (1992) 553–566 . [15] K.D. Hjelmstad , E. Taciroghlu , Mixed variational methods for finite element analysis of geometrically nonlinear, inelastic Bernoulli-Euler beams, Com- mun. Numer. Meth. En. 19 (2003) 809–832 . [16] Y. Yang , H. Chiou , Rigid body motion test for nonlinear analysis with beam elements, J. Eng. Mech. ASCE 113 (9) (1987) 1404–1419 . [17] K.D. Hjelmstad , E. Taciroghlu , Mixed methods and flexibility approaches for nonlinear frame analysis, J. Constr. Steel Res. 58 (2002) 967–993 . [18] A. Shooshtari , R. Khajavi , An efficient procedure to find shape functions and stiffness matrices of nonprismatic Euler–Bernoulli and Timoshenko beam elements, Eur. J. Mech. A. Solids 29 (2010) 826–836 . [19] N. El-Mezaini , C. Balkaya , E. Citipitioglu , Analysis of frames with nonprismatic beam elements, J. Struct. Eng. ASCE 117 (6) (1991) 1573–1592 . [20] D.H. Hodges , J.C. Ho , W. Yu , The effect of taper on section constants for in-plane deformation of anisotropic strip, J. Mech. Mater. Struct. 3 (3) (2008) 425–440 . [21] A . Palmeri , A . Cicirello , Physically-based Dirac’s delta functions in the static analysis of multi-cracked Euler–Bernoulli and Timoshenko beams, Int. J. Solids Struct. 48 (2011) 2184–2195 . [22] C.A. Felippa , The Amusing History of Shear Flexible Beam Elements Report No. CU-CAS-05-01, Department of Aerospace Engineering Sciences and Center for Aerospace Structures, University of Colorado at Boulder, 2005 . [23] G. Juarez , A.G. Ayala , Finite element variational formulation for beams with discontinuities, Finite Elem. Anal. Des. 54 (2012) 37–47 . [24] A. Yavari , S. Sarkani , J.N. Reddy , On nonuniform Euler–Bernoulli and Timoshenko beams with jump discontinuities: application of distribution theory, Int. J. Solids Struct. 38 (2001) 8389–8406 . [25] C. Franciosi , M. Mecca , Some finite elements for the static analysis of beams with varying cross section, Comput. Struct 69 (1998) 191–196 . [26] Bilello, C. (2001), Theoretical and Experimental Investigation on Damaged Beams Under Moving Systems. Ph.D. thesis, Universita‘ degli Studi di Palermo, Palermo, Italy. [27] S. Caddemi , I. Caliò, Exact solution of the multi-cracked Euler-Bernoulli column, Int. J. Solids Struct. 45 (16) (2008) 1332–1351 . [28] B. Biondi , S. Caddemi , Closed form solutions of Euler-Bernoulli beams with singularities, Int. J. Solids Struct. 42 (2005) 3027–3044 . [29] F. Bagarello , Multiplication of distribution in one dimension: possible approaches and applications to δ-function and its derivatives, J. Math. Anal. Appl. 196 (1995) 885–901 . [30] J.O. Dow , A Unified Approach to the Finite Element Method and Error Analysis Procedures, Academic Press, 1999 .