Engineering Thesis Giuseppe Di Palma

UNIVERSITÀ DEGLI STUDI DI SALERNO

FACOLTÀ DI INGEGNERIA

CORSO DI LAUREA IN INGEGNERIA CIVILE

Tesi di Laurea in

Tecnica delle Costruzioni

A NEW TIMOSHENKO-BASED ANALYTICAL

MODEL FOR STEEL-CONCRETE COMPOSITE BEAMS

IN PARTIAL INTERACTION

RELATORE CANDIDATO

Prof. Ing. Ciro Faella Giuseppe Di Palma

CORRELATORE Matr. 163/000542

Dott. Ing. Enzo Martinelli

Anno Accademico 2007/2008

In the Name of Allah, the Most Gracious, the Most Merciful

« Read!, In the Name of your Lord Who has created,

He has created man from a clot ,

Read!, and your Lord is the Most Generous,

Who has taught by the pen,

He has taught man that which he knew not » .

The Noble Qur’an, Surat XCVI, 1‐5

Ai miei cari Genitori:

senza i loro sacrifici, la loro pazienza e il loro incoraggiamento

questa Tesi non sarebbe stata mai scritta.

Grazie.

i

Sommario

1.Introduction 1 1.1State of the art 1

2.A Timoshenko‐based model for composite beams in partial interaction 7 2.1Key geometric and mechanical properties of the composite cross section 7 2.2Model equations 8

2.2.1 Equilibrium equations 8 2.2.2 Constitutive laws 11 2.2.3 Global equilibrium equation 13 2.2.4 Compatibility equation throughout the interface 13 2.2.5 Equilibrium equation throughout the interface 14 2.2.6 Stress‐strain law for shear connection 15

3.Outline of the governing equations 16 3.1The system of three equations in three unknown functions 16 3.2Displacement formulation 18

3.2.1 Deducing the problem dimensions 19 3.2.2 Differential equation in terms of deflection 22 3.2.3 Deriving the other parameters 25

3.3Extended Newmark’s equation in terms of curvature 30

4.Solution in the elastic range 36 4.1Composite beam under axial force 36 4.2Composite beam in bending 37

4.2.1 Non‐redundant beams in bending 40 4.2.2 Boundary conditions for non‐redundant beams 42 4.2.3 Redundant beams in bending 47 4.2.4 Boundary conditions for redundant beams 48

Giuseppe Di Palma 163/000542

ii

5.Stiffness matrix 55 5.1Identification of the problem 55 5.2Coefficients of the stiffness matrix 57

5.2.1 General procedure for deriving the integration constants 57 5.2.2 Stiffness matrix: the first column 62 5.2.3 Stiffness matrix: the second column 69 5.2.4 Stiffness matrix: the third column 78 5.2.5 Completing the stiffness matrix 90

5.3Vector of the external nodal force and vector nodal forces equivalent to distributed action 91

5.3.1 Vector of the external nodal forces 91 5.3.2 Vector nodal forces equivalent to distributed actions. 91

6.Applications 117 6.1Simply‐supported composite beam 117

6.1.1 Solutions in terms of displacements 120 6.1.2 Comparisons between Timoshenko and Bernoulli model 123 6.1.3 Solution by matrix method 136

6.2Continuous composite beam 151 6.2.1 Analytical solution 151 6.2.2 Comparison between Timoshenko and Bernoulli model 158 6.2.3 Solution by matrix method 170 6.2.4 Solutions in terms of forces 185 6.2.5 Comparisons between Timoshenko model and Bernoulli model 186

7.Conclusions 198

8.Bibliography 199

1

1. Introduction

Structural behaviour of steel‐concrete composite beams and

structures is generally influenced by several phenomena related to the

behaviour of steel and concrete as well as the behaviour of shear

connectors.

The present thesis is aimed to derive stiffness matrix of the composite beam

under sufficiently general hypotheses. In particular, after a through

examination of previous works in the scientific literature, various

contributions can be found, and a complete analytical derivation of the

stiffness matrix for composite beams in partial interaction behaving to

Bernoulli theory, has been already formulated, starting from the original

Newmark theory.

1.1 State of the art

Timoshenko [1] developed a theory for composite beams with two

bonded materials using Bernoulli‐Euler beam theory for each component

and constraining transverse displacements to be equal. Newmark et al. [2]

established the governing equations for elastically connected steel‐concrete

beams neglecting uplift and friction. Adekola [3] extended this work by

including uplift and frictional effects. He proposed a finite‐


2

difference procedure for solving the differential equation for uplift and

axial forces. Robinsion and Naraine [4] addressed the issue of whether the

forces at the interface act on the concrete slab or pull on the steel beam.

Cosenza and Mazzolani [5] proposed a new solution procedure that is

suitable for general loading conditions and McGarraugh and Baldwin [6]

used a simple analytical model to prove that the strength of a composite

girder with partial interaction can be derived by nonlinear interpolation of

the beam strength for the extreme cases of no interaction and full

interaction. For the study of the nonlinear behavior of composite members

the existing studies can be grouped into the following two categories: (1)

Finite‐element models utilizing beam, plate, shell, or brick finite elements

to represent in great detail the constituents of the composite structural

element (such models are rather complex, very computationally intensive,

and limited to monotonic loads); and (2) 1D beam elements that capture

salient features of the nonlinear behavior of composite girders within the

framework of Navier‐Bernoulli beam theory. Within the latter category

proposed models can be grouped into three categories: (1) Full composite

action models based on displacement interpolation functions with fiber

discretization of the cross section and uniaxial stress‐strain relations of the

constituent materials, as proposed by Mirza and Skrabek [7] for the

analysis of composite columns under uniaxial bending and El‐Tawil et al.

[8] under biaxial bending; (2) models of the partial composite action

between concrete and steel based on displacement interpolation functions

for the concrete and steel component of the composite element, which

Chapter I ‐ Introduction

3

readily supply the relative longitudinal or transverse displacement at the

interface; in this category belong the study by Daniel and Crisinel [9] for

composite beams under monotonic loads, the study by Amadio and

Fragiacomo [10] for the effect of concrete creep and shrinkage in composite

beams, the study by Hajjar et al. [11] for concrete‐ filled, steel tube columns,

and the study by Salari et al. [12] for composite beams under cyclic loads;

and (3) recent models that attempt to overcome the limitations of

displacement‐based models by the use of force interpolation functions

(flexibility formulation); interest in this type of nonlinear model increased

after the work by Ciampi and Carlesimo [14], who are the first to propose a

consistent implementation of the flexibility formulation of a nonlinear

Bernoulli beam element within the framework of a general purpose

nonlinear analysis program. The selection of suitable force interpolation

functions that strictly satisfy element equilibrium is rather straightforward

for the case of a nonlinear Bernoulli beam element; in this case the fiber

discretization of the cross section affords a convenient means of describing

the complex hysteretic response of members under cyclic loading histories

(Spacone et al. [15]). Difficulties arise, however, in the selection of force

interpolation functions that strictly satisfy equilibrium for cases that

involve interaction between beam displacements and internal forces.

Examples are an anchored reinforcing bar, a prestressed concrete girder, a

steel‐concrete girder with partial composite action, and a slender column.

Attempts to extend the advantages of a force‐based flexibility formulation

to these cases have been recently reported (Yassin [16]; Ayoub and Filippou


4

[17]; Monti et al. [19]; Neuenhofer and Filippou [20]; Salari et al.). With the

exception of the study by Neuenhofer and Filippou, which is, however,

limited to linear elastic material behavior, the other studies resort to ad hoc

assumptions for overcoming the difficulty of deriving force interpolation

functions that strictly satisfy equilibrium. The formulations by Yassin,

Monti et al., and Ayoub and Filippou limit the interaction between the two

components to the end nodes of the element and assume a linear

interpolation of bond or friction forces in between. This requires a small

element size for accurate local response eliminating one of the advantages

of the flexibility‐based formulation (Neuenhofer and Filippou). To

overcome this weakness Salari et al. introduced higher order bond force

distribution functions. The formulation, however, lacks clarity about the

relation between the slip distribution in the element and the element end

displacements. Analytical results reveal interelement discontinuities of slip

displacements in violation of variational principles. It is also not clear that

the element can be extended to accommodate distributed element loads. In

view of the limitations of the displacement formulation (Neuenhofer and

Filippou) and the difficulty of selecting force interpolation functions that

strictly satisfy equilibrium for problems with strong interaction between

displacements and internal forces, Ayoub and Filippou recently proposed a

consistent mixed formulation of the anchored reinforcing bar problem with

independent interpolation functions for the axial displacements and the

reinforcing steel stresses. This formulation combines the advantages of the

Chapter I ‐ Introduction

5

displacement and force formulations while overcoming most of their

limitations.

More recently, Wu and Xu [22],[24] considered the Timoshenko kinematics

for both the connected members in order to derive a more general model

under similar hypotheses. Three applications were proposed in the

mentioned papers:

• simulation of the behaviour of a beam in bending considering shear

flexibility;

• buckling analysis of the axially loaded beam‐columns taking into

account second order effects due to flexural deflection;

• dynamics and vibration analysis of composite beams.

The authors provided some examples and the deflection values at the

midspan for all the previously mentioned three cases. The value of the

Euler critical load has been also provided regarding the buckling of the

beam. This work is the starting point of the present thesis whose final result

consists in deriving the close‐form expression for the stiffness matrix,

including the shear flexibility effects .

Ranzi and Zona [23] worked at the same problem, including the shrinkage

effects using the Volterra equation and they solved it by approximating

with a linear function.

A recent research by Sakr and Sakla [26] deals with beams with incomplete

connections, in presence of cracking. The distributed effects are taken into

account both in the cracked and uncracked stage. The non‐linear behaviour

of the connection is modelled according to Ollgard as usually


6

accepted also within the previously mentioned papers. The stress strain

relationship is the Volterra one which includes a stress function at a generic

instant, so that it can be only numerically solved step by step. As far as the

stress‐strain relationship of steel, two Young moduli have to be used: one

for the steel beam and one for the internal reinforcement of the concrete

slab. It can be seen that the setting of this method is just the same of the

FEM analysis. It has been observed a substantial influence of the long‐term

deformability of connection, especially for simply supported beams.

Finally, more advanced studies try to model the stress‐strain relationship of

the materials with non‐linear functions and take into account the

distributed effects of the cracking.

The present work frames itself into the linear analysis of composite beams

with flexible connection, and its purposes are to give a further contribution

about shear strains and stresses.

7

2. A Timoshenko‐based model for composite

beams in partial interaction

The key features of a brand‐new model for simulating the behaviour

of steel‐concrete composite beams in partial interaction, looking after the

shear flexibility of both concrete slab and steel beam will be proposed.

2.1 Key geometric and mechanical properties of the

composite cross section

The typical cross section of a steal concrete composite beam is

represented in the Figure 2.1.

Figure 2.1. Properties of the cross section.


8

A geometric centroid of the section as a whole can be easily defined and its

position can be referred to the centroids of steel beam and concrete slab

through the following equations:

2 2 1 11 1 1 2 2 2 1 2

1 1 2 2 1 1 2 2,E A E AE A y E A y y h y h

E A E A E A E A= ⇒ = =

+ + (2.1)

2.2 Model equations

The general equations of the mentioned model will be formulated in

the present section.

2.2.1 Equilibrium equations

The governing differential equations will be derived considering the

external actions and the internal (generalized) stresses represented in

Figure 2.2 .

Figure 2.2. An infinitesimal element of the composite beam.

Chapter II –A Timoshenko‐based model for composite beams in partial interaction

9

Among the former ones, the inertial actions can be defined as follows:

,

,

d tt

d tt

F AvM I

ρρ ϕ

= −= −

(2.2)

where:

1 1 2 2

1 1 2 2

A A AI I I

ρ ρ ρρ ρ ρ

= += +

(2.3)

Second order effects are also considered in this stage. Figure 2.3 shows that

the axial force in Timoshenko beam is directed as the axis of the beam

therefore it is not orthogonal to the cross section because of the shear

flexibility (see the equation (2..4)):

, xvγ ϕ= + (2.4) As a result, the angle between the axial direction and the horizontal one is

defined through the function v(x); according to Figure 2.3 the second

derivative of deflection v(x) is always non‐positive (the x‐ axe is positive

towards the right and the deflection is positive towards the bottom)

therefore the second derivative is negative.

Figure 2.3. Calculation of the force FN.


10

Therefore the angle β is estimated as:

, xxv dxβ = − (2.5)

and in case of infinitesimal angles we have :

senβ β= (2.6) Therefore, the vertical force results as ,xx-N v dx (directed towards the top).

Therefore assuming that the forces are positive towards the bottom we get

as a result: N ,xxF = N v dx . According to the hypothesis of small

deformation, the dynamic equilibrium of forces in vertical direction is

eased to the following equation:

0i i dF F∑ + = (2.7) and replacing the forces represented in Figure 2.2:

, , ,

, , ,

0x xx tt

x tt xx

Q Q Q dx Nv dx qdx Av dxQ q Av Nv

ρρ

− + + + + − = ⇒= − + −

(2.8)

and being N=F:

, , ,x tt xxQ q Av Fvρ= − + − (2.9)

where the infinitesimal parts of the second order are eliminated.

According to the hypothesis of small deformation and the dynamic

equilibrium condition of moments, we have:

0i i dM M∑ + = (2.10) replacing the moments of forces in Figure 2.2:

,, 0ttxM M M dx Qdx mdx I dxρ ϕ− + + − + − = (2.11)

and simplifying:

, ,x ttM m I Qρ ϕ= − + + (2.12)

where the infinitesimal parts of the second order are eliminated.


11

2.2.2 Constitutive laws

General constitutive (stress‐strain) relationships have to be

introduced to formulate the composite beam model.

Anelastic deformation can be also considered in those relationships for the

sake of generality.

In particular, Figure 2..4 shows the following imposed strain components:

• thermal‐induced anelastic strain, reproduced by a linear field

throughout the composite beam depth;

• shrinkage in concrete slab.

The shrinkage axial deformation is assumed with the positive sign in case

of the extension according to the usual conventions of mechanics.

Figure 2.4. The anelastic distributed effects.

The expressions of bending moments of steel beam and concrete slab can

be stated as follows:

• on the concrete slab:

1 1 1 ,( )x TM E I ϕ ∆= −Θ (2.13)


12

• on the steel beam:

2 2 2 ,( )x TM E I ϕ ∆= −Θ (2.14)

Considering the (generalized) stress‐strain relationship for shear, the

following formula can be introduced.

,( )xQ KGA v ϕ= + (2.15)

As a basic feature of the present model, shear force and the corresponding

(generalized) strain are not defined for either the steel beam or the concrete

slab, but deal with the composite cross section as a whole; the shear

stiffness of the beam section is defined as follows:

1 1 1 2 2 2KGA K G A K G A= + (2.16) For the sake of simplicity, the following assumption will be considered for

bending stiffness and other parameters:

1 1 2 2

1 1 2 2 2

1 1 2 2

2 21 1 2 2 1 1 2 2

1 1 2 2 1 1 2 2

21 1 2 22

1 1 2 2

1 1 2 2

1 1 2 2

s

s s

s

EI E I E IE A E AEI EI hE A E A

EI k E A E A h E A E A EI EAk E A E A EI E A E A k

E A E A hkE A E A EI

E A E AEAE A E A

α

α

Σ = +

= Σ + =+

⎛ ⎞Σ + Σ= + =⎜ ⎟Σ +⎝ ⎠

⎛ ⎞+= +⎜ ⎟Σ⎝ ⎠

=+

(2.17)


13

2.2.3 Global equilibrium equation

Considering the beam in Figure 2..5, two equilibrium equations can

be stated as follows, including the normal stresses and the bending

moments for concrete slab and steel beam.

Figure 2.5. Global equilibrium.

The equilibrium in horizontal direction leads to the following relationship:

1 2N N N F= + = (2.18)

while the equilibrium of bending moments can be stated as follows :

1 2 1 2-M M M N h Fy= + + (2.19)

2.2.4 Compatibility equation throughout the interface

Considering Figure 2.6, we observe that rotation is common for

both steel beam and concrete slab. As a result of the interlayer slip we

have:

2 1 1 2 2 1- - - - -su u u h h u u hϕ ϕ ϕ= = (2.20)


14

Figure 2.6. The common rotation of the cross section.

and the kinematic assumptions of the model lead to the following

relationships:

• on the concrete slab:

1

1, 1 11 1

-x sh TNu yE A

ε ε ∆= = + Θ (2.21)

• on the steel beam:

22, 2 2

2 2x T

Nu yE A

ε ∆= = +Θ (2.22)

2.2.5 Equilibrium equation throughout the interface

An interlayer distributed force Qs arises at the interface between steel

profile and concrete slab. This interlayer distributed shear force should be

related to the two normal stresses in N1 and N2 on concrete slab and steel

beam, respectively.


15

Figure 2.7. Equilibrium in the interface.

According to Figure 2.7 in the concrete slab we have:

1 1 1,- 0x sN N N dx Q dx+ + + = ⇒ 1, -x sN Q= (2.23)

and in the steel beams we have:

2 2 2,- - 0x sN N N dx Q dx+ + = ⇒ 2, x sN Q= (2.24)

2.2.6 Stress‐strain law for shear connection

According to Hooke rule the interlayer distributed shear force is

determined by the linear expression:

s s sQ k u= (2.25)

depending on the interlayer displacement us.

16

3. Outline of the governing equations

The equations derived in the previous section can be condensed and

simplified to obtain the key set of simultaneous equations describing the

behaviour of shear flexible composite beams in partial interaction.

3.1 The system of three equations in three unknown

functions

Deriving the expression of the slip (2.20) and introducing the

definitions of normal strains (2.21)‐(2.22) we have:

, 1, 2, 2 1 , 2

2 2

1 1 11 , 2

1 1 2 2 1 1

1 , 12 2 1 1 2 2

,

-

1 1

s x xxs x x T

s s

sh T x T

sh T x

sh T x

Q N Nu h yk k E A

N F N Ny h yE A E A E A

Fy h NE A E A E A

h h

ε ε ϕ

ε ϕ

ε ϕ

ε ϕ

∆

∆ ∆

+ ∆

+ ∆

= = − = − − = +Θ +

−⎛ ⎞− + Θ − = +Θ − +⎜ ⎟⎝ ⎠

⎛ ⎞− Θ − = − + +⎜ ⎟⎝ ⎠

− Θ −

(3.1)

from which we get:

1, 1 ,2 2 1 1 2 2

1 ,2 2 1 1 2 2

1 1

1 1

xx s sh T x

ss s sh s T s x

FN k N h hE A E A E A

F k N k k k h k hE A E A E A

ε ϕ

ε ϕ

+ ∆

− ∆

⎛ ⎞⎛ ⎞= − − + − Θ − =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞= − + + + Θ +⎜ ⎟⎝ ⎠

(3.2)

and as a result:

Chapter III ‐ Outline of the governing equations

17

1

1, ,2 2

sxx s s sh s T s x

F k NN k k k h k hE A EA

ε ϕ− ∆= − + + Θ + (3.3)

Furthermore, introducing the expressions of bending moments (2.13) and

(2.14) into the equation (2.19) we obtain :

1 1 , 2 2 , 1 2( ) ( )x T x TM E I E I N h Fyϕ ϕ∆ ∆= −Θ + −Θ − + (3.4)

from which we have:

1 2

, x TM N h Fy

EIϕ ∆

+ −= +Θ

Σ (3.5)

introducing the equation (3.5) into (3.4) we get :

1 1 21,

2 2

1 1 2

2 2

sxx s s sh s T

ss T s s sh s

Fk N M N h FyN k k k hE A EA EI

Fk N M N h Fyk h k k k hE A EA EI

ε

ε

∆

∆

+ −⎛ ⎞= − + + + +Θ +⎜ ⎟Σ⎝ ⎠+ −

− Θ = − + + +Σ

(3.6)

and the following expression can be finally obtained after some

mathematical simplification:

2

21, 1

2 2

11 sxx s s s sh

h k hM hyN k N k F kEA EI EI E A EIε

⎛ ⎞ ⎛ ⎞− + = − + +⎜ ⎟ ⎜ ⎟Σ Σ Σ⎝ ⎠⎝ ⎠ (3.7)

The definitions (2.17) can be introduced obtaining the following equation:

221, 1

2 2

1sxx s s sh

k hM hyN N k F kEI E A EI

α ε⎛ ⎞− = − + +⎜ ⎟Σ Σ⎝ ⎠ (3.8)

Deriving (2.19) and introducing (2.12) the following equation can be

obtained:

, 1, 2, 1, ,x x x x ttM M M N h m I Qρ ϕ= + − = − + + (3.9)

that is:


18

1 1 , , 2 2 , , 1, ,

, , 1, ,

( ) ( )( )

xx T x xx T x x tt

xx T x x tt

E I E I N h m I QEI N h m I Q

ϕ ϕ ρ ϕϕ ρ ϕ

∆ ∆

∆

−Θ + −Θ − = − + +Σ −Θ − = − + +

(3.10)

and the following equation can be finally obtained:

, , , 1,xx T x tt xEI EI I N h m Qϕ ρ ϕ∆ −Σ −Σ Θ − + = (3.11)

Finally, deriving equation (2.15), through the (2 .9) one gets:

, , , , ,( )x xx x tt xxQ KGA v q Av Fvϕ ρ= + = − + − (3.12)

that is finally:

, , , ,( )xx x xx ttKGA v Fv Av qϕ ρ−+ + = − (3.13)

To sum up briefly, the governing equations of the problem result being

listed in the following set of simultaneous equations:

221, 1

2 2

, , , 1,

, , , ,

1

( )

sxx s s sh

xx T x tt x

xx x xx tt

k hM hyN N k F kEI E A EI

EI EI I N h m Q

KGA v Fv Av q

α ε

ϕ ρ ϕ

ϕ ρ

∆ −

−

⎧ ⎛ ⎞− = − + +⎜ ⎟⎪ Σ Σ⎝ ⎠⎪⎪Σ −Σ Θ − + =⎨⎪⎪

+ + = −⎪⎩

(3.14)

The above set of the three equations involves the unknown functions N1(x),

φ(x) e v(x).

3.2 Displacement formulation

The three equations obtained in the previous section can further be

worked and simplified to obtain a single differential equation in terms of

(generalized)displacement.


19

3.2.1 Deducing the problem dimensions

The obtained set of equations can be reduced into the equation

depending on the function v(x).The first two equations (3.14) can be

simplified by eliminating function N1(x). Indeed, considering equation

(3.11) the following equation can be derived:

( )

11, ,

2 2

, , , , ,

, , , ,

, ,1

sxx s s sh s T s x

xxx T xx ttx x x

xxx T xx ttx x

tt xx

F k NN k k k h k hE A EA

EI EI I m Qh h h h h

EI EI I mh h h h

q Av Fvh

ε ϕ

ϕ ρ ϕ

ϕ ρ ϕ

ρ

− ∆

∆

∆

= − + + Θ + =

⎛ Σ Σ Θ ⎞= − − + − =⎟⎜ ⎠⎝Σ Σ Θ⎛= − − + +⎜

⎝⎞− − + − ⎟⎠

(3.15)

in which N1 can be obtained as follows :

1 ,2 2

, , , , , ,

ss sh s T s x

s

xxx T xx ttx x tt xx

EA F kN k k h k hk E A

EI EI I m q Av Fvh h h h h h h

ε ϕ

ϕ ρ ϕ ρ

∆

∆

⎛= − + Θ − +⎜⎝

Σ Σ Θ ⎞+ − − + + − + ⎟⎠

(3.16)

Deriving N1 twice with respect to the abscissa x, we have:

,1, , , ,

, , , , , ,

xxxxxxx s sh xx s T xx s xxx

s

T xxxx ttxxx xxx xx ttxx xxxx

EA EIN k k h k hk h

EI I m q Av Fvh h h h h h

ϕε ϕ

ρ ϕ ρ

∆

∆

Σ⎛= − + Θ − + +⎜⎝

Σ Θ ⎞− − + + − + ⎟⎠

(3.17)

From the last two equations we have:


20

,, , ,

, , , , , ,

,2,

2 2

, , , , ,

xxxxxs sh xx s T xx s xxx

s

T xxxx ttxxx xxx xx ttxx xxxx

s xxxs sh s T s x

s

T xx ttx x tt xx

EA EIk k h k hk hEI I m q Av Fv

h h h h h hEA F k EIk k h k hk E A hEI I m q Av Fvh h h h h h

ϕε ϕ

ρ ϕ ρ

ϕα ε ϕ

ρ ϕ ρ

∆

∆

∆

∆

Σ⎛− + Θ − + +⎜⎝

Σ Θ ⎞− − + + − + +⎟⎠

Σ⎛− − + Θ − + +⎜⎝

Σ Θ ⎞− − + + − +

[

]

,2 2

, , , , ,,

, 22

2 2

1 0

s sx T s sh s T

s

xxx T xx ttx x tts x

xxs s sh

k h EA F kEI EI h k k hEI k E A

EI EI I m q Avk hh h h h h h

Fv hyFy k F kh E A EI

ϕ ε

ϕ ρ ϕ ρϕ

ε

∆ ∆

∆

+⎟⎠

⎛− Σ −Σ Θ − − + Θ +⎜Σ ⎝Σ Σ Θ

− + − − + + − +

⎞ ⎛ ⎞+ + + + − =⎟ ⎜ ⎟Σ⎠ ⎝ ⎠

(3.18)

The following equation can be obtained through suitable simplifications:

, , ,

2 2, ,

2, ,

2,

2

xxxxx ttxxx ttxxs s s

xxx ttxs s

tt xxxxs s

xxs

EI EA I EA AEAvhk hk hk

EI EA I EA I hEAhk hk EI

AEA AhEA F EAv vhk EI hk

F EA F hEAvhk EI

ρ ρϕ ϕ

ρ ρϕ α ϕ α

ρ ρα

α

α

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Σ+ − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞Σ

+ − + − +⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞

+ − + +⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎛ ⎞

+ − + +⎜ ⎟Σ⎝ ⎠

−2

2 2 2 2 2 2

31 1 2 22

,

s s

s s sx

EA EAk h k FE A E A EI E A

E I k h E I k h k h EAhEAEI EI EI

α ϕ

⎛ ⎞− − +⎜ ⎟Σ⎝ ⎠

⎛ ⎞− − + + + =⎜ ⎟Σ Σ Σ ⎠⎝


21

( )

2 2

22

,

32 2

,

,

, ,

,

s s s

ssh s sh xx

s

sT s T xx

s

T xxxxs

EA hEA EA EA hEAq q xx m xhk EI hk hk EI

EA h EAkm xxx EA k EAhk EI

k h EA EI EAEAh k hEI hk

EA EIhk

α α

ε α ε

α α∆ ∆

∆

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + − + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞⎛ ⎞+ − − − − − − +⎜ ⎟⎜ ⎟ Σ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞Σ+Θ − − +Θ − +⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎠⎝

⎛ ⎞Σ+Θ ⎜ ⎟

⎝ ⎠

(3.19)

The last two terms of the left‐hand member are both zero. Compacting the

other elements we have :

, , ,

2 2, ,

2, ,

2 2,

xxxxx ttxxx ttxxs s s

xxx ttxs s

tt xxxxs s

xxs

EI EA I EA AEAvhk hk hk

EI EA I EA I hEAhk hk EI

AEA AhEA F EAv vhk EI hk

F EA F hEA EAv qhk EI hk

ρ ρϕ ϕ

ρ ρϕ α ϕ α

ρ ρα

α α

⎛ ⎞ ⎛ ⎞ ⎛ ⎞Σ− − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞Σ

− + − +⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞

+ − + +⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎛ ⎞

− − =⎜ ⎟Σ⎝ ⎠

( )

2

2, , ,

, , ,

s

s s s

sh xx T xx T xxxxs s

hEAEI

EA EA hEA EAq xx m x m xxxhk hk EI hk

EI EA EA EIEAhk hk

α

ε α∆ ∆

⎛ ⎞− +⎜ ⎟Σ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞Σ Σ+ −Θ +Θ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

(3.20)

Observing that :

2 2

s s

EA hEA EA EIhk EI hk EI

α α⎛ ⎞ Σ

− =⎜ ⎟Σ⎝ ⎠ (3.21)


22

and multiplying equation (3.20) by ks h/ ΣEI EA the following equation can

be derived:

( )

( )

2 2, , , , ,

22

, , ,

2

, , ,

, , ,1

xxxxx ttxxx ttxx xxx ttx

tt xxxx xx

x T xx sh xx

xx xxx T xxxx

I A IvEI EI EI

A Fv v FvEI EI EI

q m EI EA hEI

q m EIEI

ρ ρ ρϕ ϕ α ϕ α ϕ

ρ αα

α ε∆

∆

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− − − + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞+ + − =⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

= + −Θ + +

− + −Θ ΣΣ

(3.22)

The above equation is equivalent to the first two ones reported in (3.14)

and has to be completed by the third one for obtaining the first solution.

3.2.2 Differential equation in terms of deflection

The simultaneous differential equations have to be reduced for

deriving the only differential equation similarly to the problem of

Timoshenko beam. From the third equation of the system (3.14) we have :

( ), , , ,1

x tt xx xxq Av Fv vKGA

ϕ ρ −= − + − (3.23)

The following equation can be derived by introducing equation (3.23) into

the (3.22):


23

[ ]

[{ ] }

[ ]{

} [ ]

, , , ,

, , , ,

2, , , ,

2, , , , ,

2

1

1

1

1

xxxx ttxxxx xxxxxx xxxxxx

ttxx ttttxx xxxxtt xxxxtt

ttxx xx ttxx xxxx

xxxx tt tttt xxtt xxtt

q Av Fv vKGAI q Av Fv vEI KGAA v q Av FvEI KGA

Iv q Av Fv vEI KGA

AEI

ρ

ρ ρ

ρ α ρ

ρα ρ

ρα

⎧ ⎫− + − − +⎨ ⎬⎩ ⎭

− + − + − − +Σ

− − − + − +Σ

⎧ ⎫− + − + − − +⎨ ⎬⎩ ⎭

+

( )

( )

2

, , ,

2

, , ,

, , ,1

tt xxxx xx

x T xx sh xx

xx xxx T xxxx

Fv v FvEI EI

q m EI EA hEI

q m EIEI

α

α ε∆

∆

+ − =Σ

= + −Θ + +

− + −Θ ΣΣ

(3.24)

A few easy simplifications lead to the following equation in terms of

deflection:

, ,

2 2

, ,

2 2, ,

2,

1 1

1

1

xxxxxx xxxxtt

ttttxx xxtt

tttt tt

xxxx

F A I Fv vKGA KGA EI KGA

A I A A I Fv vKGA EI EI KGA EI KGA

A I Av vKGA EI EI

F FvKGA EI

ρ ρ

ρ ρ ρ α ρ α ρ

ρ ρ ρα α

α

⎡ ⎤⎛ ⎞ ⎛ ⎞+ − + + +⎜ ⎟ ⎜ ⎟⎢ ⎥Σ⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤⎛ ⎞ ⎛ ⎞+ + + + +⎜ ⎟ ⎜ ⎟⎢ ⎥Σ Σ⎝ ⎠ ⎝ ⎠⎣ ⎦

⎛ ⎞ ⎛ ⎞− − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎡ ⎛ ⎞− + +⎜ ⎟⎢ Σ⎝ ⎠⎣

( )

( )

2,

2

, , ,

, , ,

2, , , ,

1

1

xx

x T xx sh xx

xx xxx T xxxx

xxxx xx ttxx tt

FvEI

q m EI EA hEI

q m EIEI

I Iq q q qKGA EI EI

α

α ε

ρ ρα

∆

∆

⎤+ =⎥

⎦

= − + −Θ + +

+ + −Θ Σ +Σ

⎛ ⎞− − − +⎜ ⎟Σ⎝ ⎠

(3.25)


24

The above equation represents in the dynamic field the equation of the

deflection in presence of the constant axial force and of the finite shear

stiffness ( that is F≠0 and [(KGA L2)/EI] ≠ ∞).

In the static field the above equation is simplified as follows:

( )

( ) ( )

2 2, , ,

2

, , ,

2, , , , ,

1 1

1 1

xxxxxx xxxx xx

x T xx sh xx

xx xxx T xxxx xxxx xx

F F F Fv v vKGA KGA EI EI

q m EI EA hEI

q m EI q qEI KGA

α α

α ε

α

∆

∆

⎡ ⎤⎛ ⎞ ⎛ ⎞+ − + + + =⎜ ⎟ ⎜ ⎟⎢ ⎥Σ⎝ ⎠ ⎝ ⎠⎣ ⎦

= − + −Θ + +

+ + −Θ Σ − −Σ

(3.26)

As the axial force F=0 , the following equation can be derived:

( )

( ) ( )

22

, , , , ,

2, , , , ,

1 1

xxxxxx xxxx x T xx sh xx


v v q m EI EAhEI

q m EI q qEI KGA

αα ε

α

∆

∆

− = − + −Θ + +

+ + −Θ Σ − −Σ

(3.27)

If no shear flexibility is assumed, the Timoshenko equation usually reduces

to the Bernoulli’s one and the following equation can be obtained as F=0:

( )

( )

22

, , , , ,

, , ,1


xx xxx T xxxx

v v q m EI EAhEI

q m EIEI

αα ε∆

∆

− = − + −Θ + +

+ + −Θ ΣΣ

(3.28)

Finally, equation (3.28) reduces to the usual differential equation of the

Bernoulli beam as a completely stiff connection is considered ( Lα →∞ ) :

( ), , , ,1

xxxx x T xx sh xxv q m EI EAhEI

ε∆= + −Θ + (3.29)


25

while, as absent connection occurs ( 0Lα → ), the equation of the beam

with flexural stiffness ΣEI is obtained:

( ), , , ,1

xxxxxx xx xxx T xxxxv q m EIEI

∆= + −Θ ΣΣ

(3.30)

As concrete slab is connected to steel beam , for Lα →∞ , shrinkage axial

deformation causes the bending moment . Therefore, such shrinkage axial

deformation appears in the equation of bending (3.29). However, no

bending moment arises by shrinkage in concrete for the case of 0Lα → .

As a result, shrinkage axial deformation freely occurs in concrete slab

(which is not constrained by steel beam through any friction or connection)

and it does not appear in equation of the flexion.

3.2.3 Deriving the other parameters

The following statements are valid only in the static field .

Equation (2.19) in terms of bending moment M can be transformed as

follows:


26

, ,2 2

, ,

, , ,, ,

,2 , 1

sxx xx T

s

s xx xx s sh s T

xx T xx xxxxx xxxx

xxxxxx

s

q F hEA F kM EI v vKGA KGA k E A

q Fk h v v k k hKGA KGA

EI q F EI m qv vh KGA KGA h h hFv EA EI FFy vh k KGA

v

ε

∆

∆

∆

⎛ ⎞ ⎡= Σ − − − −Θ − +⎜ ⎟ ⎢⎝ ⎠ ⎣⎛ ⎞− − − − − + Θ +⎜ ⎟⎝ ⎠

Σ Σ Θ⎛ ⎞+ − − − − + + +⎜ ⎟⎝ ⎠

⎡ Σ ⎤⎤ ⎛ ⎞+ + = + +⎜ ⎟⎢ ⎥⎥⎦ ⎝ ⎠⎣ ⎦

+

[ ] ( )

2,

2

, ,

22 2

,

1 1xxs

xx xs s s

T sh

T xxs

F F EA FEI EAhKGA KGA k

EI EAh EA EI EA EAq q mKGA KGA k KGA k k

F EA hF Fy EI EA hE A

EI EAk

ε∆

∆

⎡ ⎤⎛ ⎞ ⎛ ⎞−Σ + − + − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦⎡ ⎤ ⎡ ⎤Σ Σ ⎛ ⎞+ − − − + + − +⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎝ ⎠⎣ ⎦⎣ ⎦⎡ ⎤+ − + +Θ − + +⎢ ⎥⎣ ⎦

⎡ ⎤Σ+Θ ⎢ ⎥

⎣ ⎦

(3.31)

The term involving F results as zero; compacting the remaining terms we

obtain:

( ) ( )

, ,

, ,

,

1 1xxxx xxs

xx xs s s s

T sh T xxs

EA EI F FM v v EIk KGA KGA

EA F EI EA EI EA EAq q mk KGA k KGA k k

EI EAEI EA hk

ε∆ ∆

⎡ Σ ⎤ ⎡⎛ ⎞ ⎛ ⎞= + − + +⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣⎛ ⎞Σ⎤ ⎛ ⎞ ⎛ ⎞+ − + + − +⎜ ⎟⎜ ⎟ ⎜ ⎟⎥⎦ ⎝ ⎠ ⎝ ⎠⎝ ⎠

⎛ ⎞Σ−Θ + +Θ ⎜ ⎟

⎝ ⎠

(3.32)

and finally:


27

( )

, ,2 2

, ,2 2 2

,2 2

1 1xxxx xx

xx x

sT sh T xx

EI F F EI FM v v EIKGA KGA EI

EI EI EI EIq q mKGA EI KGA EI

EI k h EIEIEI

α α

α α α

εα α

∆ ∆

⎡ ⎤⎡ ⎤⎛ ⎞ ⎛ ⎞= + − + + +⎜ ⎟ ⎜ ⎟⎢ ⎥⎢ ⎥ Σ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎛ ⎞ ⎛ ⎞ ⎛ ⎞

− + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞−Θ + +Θ⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎝ ⎠

(3.33)

The normal stress in concrete slab N1, and the normal stress in the steel

beam N2, are obtained from the following equation which is derived by

substituting (3.23) into (3.16):

1 22 2

, , ,, ,

, , ,

ss sh s T

s

xx xx xxxxs xx xxxx

T xx x xx

EA F kN F N k k hk E A

q Fv EI q Fvk v h vKGA h KGA

EI m q Fvh h h h

ε ∆

∆

⎧= − = − + Θ⎨⎩

+ Σ +⎛ ⎞ ⎛ ⎞− − − + − − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Σ Θ ⎫− + + + ⎬⎭

(3.34)

that is reorganizing:

,1 2

2 2

2, ,

,

,

1 1

1

s xxs sh s T

T xx x sxxxx

xx s

EI F k EI qN k k hEI E A h KGA

EI F EI m q k hvh KGA h h h KGA

F Fv k hh KGA

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ⎛ ⎞− + − + + + +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

⎫⎡ ⎤⎛ ⎞+ + + =⎬⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎭

(3.35)

Equation (2.12) can be written in the static field. From this equation Q can

be derived. Substituting equation (3.33) into Q expression one obtains:


28

, , 1,

, ,, ,

,, ,2

2, , ,

,

,

1 1

1

xx T x x

x xxxxxx T x

xxxs sh x s T x

T xxx xx x sxxxxx

xxx s

Q EI EI m N hq FvEI v EI mKGA

EI EI qh k k hEI h KGA

EI F EI m q k hvh KGA h h h KGA

F Fv k hh KGA

ϕ

εα

∆

∆

∆

∆

= Σ −Σ Θ + − =

+⎛ ⎞= Σ − − −Σ Θ + +⎜ ⎟⎝ ⎠

⎧ Σ− − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ⎛ ⎞− + − + + + +⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

⎡ ⎛ ⎞+ + +⎜ ⎟⎝ ⎠

⎫⎤⎬⎢ ⎥

⎣ ⎦⎭

(3.36)

simplifying the above equation the following expression can be derived :

, ,2

2

2

, 2

2, ,2 2

2

1 1

1

1

xxxxx xxx

s

sx

ss T x sh x

EI F FQ v v EIKGA KGA

EI F Fh k hEI h KGA

EI EI k hqKGA EI KGA

EI h k EIEI k h mEI EI

EIEI

α

α

α

εα α

α

∆

⎧⎛ ⎞⎛ ⎞ ⎛ ⎞= + + −Σ + +⎨⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎩⎫⎡ ⎤⎛ ⎞− + + +⎬⎜ ⎟⎢ ⎥Σ ⎝ ⎠⎣ ⎦⎭

⎡ ⎤⎛ ⎞Σ− + + +⎢ ⎥⎜ ⎟Σ ⎝ ⎠⎣ ⎦⎛ ⎞ ⎛ ⎞− Σ + Θ + + +⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠⎛ ⎞−⎜ ⎟Σ⎝ ⎠

,, ,2 2

xxxxx T xxx

EI EI qmKGAα α

∆⎛ ⎞ ⎛ ⎞+ Θ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

(3.37)

and finally one obtains:


29

( )

, ,2 2

, , ,2 2

,, ,2 2 2

1 1xxxxx xxx

sx T x sh x

xxxxx T xxx

EI F F F EIQ v EI vKGA KGA EI

EI EI h k EIq EI mKGA EI EIEI EI EI qmEI KGA

α α

εα α

α α α

∆

∆

⎡ ⎤⎛ ⎞⎛ ⎞ ⎛ ⎞= + − + + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎢ ⎥Σ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦⎡ ⎤ ⎛ ⎞− + − Θ + + +⎜ ⎟⎢ ⎥Σ Σ⎣ ⎦ ⎝ ⎠⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + Θ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(3.40)

The interlayer distributed shear force Qs throughout the interface is

obtained from equation (2.25):

, ,2

,, , ,

,,

1

1

1

ss x s T x

xxxxx s s sh x T xxx

xxxxxxxx

EI k hQ q k hEI KGA h

F F m EIv k h kKGA h h h

EI q EI Fvh KGA h KGA

θα

ε

∆

∆

⎧ ⎛ ⎞= − + + +⎨ ⎜ ⎟Σ ⎝ ⎠⎩⎡ ⎤ Σ⎛ ⎞+ + + − + − Θ +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

⎫Σ ⎛ Σ ⎞⎡ ⎤− − + ⎬⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠⎭

(3.39)

The rotational displacement can be deduced from the following expression

which is obtained by inverting equation (2.15):

, xQ vKGA

ϕ = − (3.40)

and rotational displacement can be finally derived as a function of the other

known parameters:


30

( )

, ,2

,,2 2

, , ,

2 2

, ,

2

1 1xxxxx xxx

xx

xxx xx T x

sh x s T xxx

EI F EI Fv vKGA KGA KGA KGA

EI F q EI EIvEI KGA KGA KGA EI

q EI m m EI EIKGA KGA KGA KGA EI KGA

EI k h EIKGA EI KGA

ϕα

α α

α α

εα α

∆

∆

⎛ ⎞ ⎛⎡ ⎤ ⎡ ⎤= + − + +⎜ ⎟ ⎜⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝⎝ ⎠⎞ ⎛ ⎞+ − − + +⎟ ⎜ ⎟Σ Σ⎝ ⎠⎠

⎛ ⎞ Θ⎛ ⎞+ + − − +⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠Θ

+ +Σ 2

⎛ ⎞⎜ ⎟⎝ ⎠

(3.41)

In the static field the differential equation of deflection can be finally stated

as follows:

( ) (

) ( )

22

, , ,

2

, , , , ,

2, , ,

1 1

1

1

xxxxxx xxxx xx

x T xx sh xx xx xxx

T xxxx xxxx xx

F F F Fv v vKGA KGA EI EI

q m EI EAh q mEI EI

EI q qKGA

αα

α ε

α

∆

∆

⎛ ⎞⎡ ⎤ ⎡ ⎤+ − + + + =⎜ ⎟⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎝ ⎠

= − + −Θ + + + +Σ

−Θ Σ − −

(3.42)

3.3 Extended Newmark’s equation in terms of

curvature

Deriving an extension of the well‐known Newmark equation in terms

of curvature under the more general hypotheses introduced for

formulating the present model is the final objective of the present section.

Since curvature , xϕ χ= is, in the static field, described by the expression

(3.12):


31

,,

xxxx

q Fv vKGA

χ +⎛ ⎞= − −⎜ ⎟⎝ ⎠

(3.43)

the following equation can be derived:

, 1xxF qvKGA KGA

χ⎛ ⎞− − = +⎜ ⎟⎝ ⎠

(3.44)

and finally:

,

1xx

qKGA qKGAv F KGA F

KGA

χ χ+ += − = −

++ (3.45)

The final expression of bending moment M can be finally obtained by

introducing equation (3.45) in (3.33):

( )

( )

( )

( )

, ,

2

2

, ,2 2 2

,2 2

, ,

2

xx xx

xx x

sT sh T xx

xx xx

EI KGA FKGA qMKGA F KGA

EI KGA FKGA q EI FKGA F KGA EI


EI k h EIEIEI

KGA q EIKGA

χα

χα

α α α

εα α

χα

∆ ∆

+⎛ ⎞+= − +⎜ ⎟+ ⎝ ⎠

+⎛ ⎞++ + +⎜ ⎟+ Σ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞−Θ + +Θ =⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎝ ⎠+

= − +( )

( )( )

( )

,2 2 2

, ,2 2 2

xx

sx T sh T xx

KGA q EIKGA

KGA q EI F EI EI EIq qKGA F EI KGA EI KGA

EI EI k h EIm EIEI EI

χ

χα α α

εα α α

∆ ∆

++

+ ⎛ ⎞ ⎛ ⎞+ − + + +⎜ ⎟ ⎜ ⎟+ Σ Σ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− −Θ + +Θ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ ⎝ ⎠⎝ ⎠ ⎝ ⎠

(3.46)

and, consequently, we obtain the following expression:


32

( ) ( )

( )

,,2 2

2 2

, ,2 2 2

,2 2

xxxx

xx x

sT sh T xx

EI EI EIM q EI qKGA KGA

KGA EI F q EI FKGA F EI KGA F EI


EI k h EIEIEI

χ χα α

χα α

α α α

εα α

∆ ∆

= − − + + +

+ + ++ Σ + Σ

⎛ ⎞ ⎛ ⎞ ⎛ ⎞− + + − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞−Θ + +Θ⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎝ ⎠

(3.47)

Therefore, multiplying (3.47) by ( )2 KGA F EIα + Σ we obtain:

( ) ( )

( ) ( )

( ) ( )

,,

2 2

, ,2 2 2

2,2 2

xxxx

xx x

sT sh T xx

q EIEI KGA F EI KGA F EIKGAq EIEI KGA F EI KGA F EIKGA

KGA EI F q EI F


EI k h EIEI KGA F EIEI

χ

χ α α

χ

α α α

ε αα α

∆ ∆

− + Σ − + Σ +

+ + Σ + + Σ +

+ + +

⎧ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎪− + + − +⎨ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎪ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎩⎫⎛ ⎞ ⎪⎛ ⎞−Θ + +Θ + Σ⎬⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎪⎝ ⎠ ⎭

( ) 2M KGA F EI α

=

= + Σ

(3.48)

and finally one obtains:


33

( ) ( )

( ) ( )

( )

( ) ( )

2,

2,

,2 2

, 2 2

2, 2

xx

xx

xx

sx T sh

T xx

EI KGA F EI KGA EI F

q EI q EI EIKGA F EI KGA FKGA KGA

EI EI EIq EI F q qKGA EI KGA

EI EI k hm EIEI EI

EI KGA F EI M KGA F EI

χ α χ χ

α

α α

εα α

α αα

∆

∆

− + Σ − =

Σ= − + Σ + + +

⎧ ⎛ ⎞ ⎛ ⎞⎪+ + − + + +⎨ ⎜ ⎟ ⎜ ⎟Σ⎪ ⎝ ⎠ ⎝ ⎠⎩⎛ ⎞ ⎛ ⎞

− −Θ + +⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠⎫⎛ ⎞+Θ + Σ − + Σ⎬⎜ ⎟

⎝ ⎠⎭

( )

( )

2

2, ,

22

,,2 2 2

2 2,2 2

xx xx

xxx T

ssh T xx

EI F EIq EI EI q q EI EIKGA

q EI EI F qEIq EI F KGA F EIKGA KGA

q EI q EI EIm EIEI KGA EI

EI k h EI M KGA EI M EI FEI

α

α α

α α α

ε α αα α

∆

∆

=

Σ= − Σ − + Σ +

Σ ⎡+ + + + Σ − +⎢⎣

− + − −Θ +Σ Σ

⎤⎛ ⎞ ⎛ ⎞+ +Θ − Σ − Σ⎥⎜ ⎟ ⎜ ⎟Σ ⎝ ⎠⎝ ⎠ ⎦

(3.49)

Hence, the following equation in terms of curvature can be finally derived

for the shear‐flexible composite beams:

( ) ( )

( )

( )

2,

, ,

2

2, ,

1

1

xx

x T xx

sh

shx T xx T

EI KGA F KGA F

KGA EI q m EIEI

M EA hTEI EI

M EA hF EI m EIEI EI EI

χ α χ χ

εα

εα

∆

∆ ∆

− Σ + − =

⎡= − Σ + −Θ Σ +⎢Σ⎣⎤⎛ ⎞+ +Θ∆ − +⎜ ⎟⎥⎝ ⎠⎦

⎡ ⎤⎛ ⎞− Σ −Θ Σ + +Θ −⎜ ⎟⎢ ⎥Σ ⎝ ⎠⎣ ⎦

(3.50)


34

Such an equation extends the one reported by Faella et al. [25] based on the

work by Newmark et al. [2].

The above equation turns into the standard one as F=0 :

( )2, , ,

2

1xx x T xx

shT

q m EIEI

M EA hEI EI

χ α χ

εα

∆

∆

− = − + −Θ Σ +Σ

⎛ ⎞− +Θ −⎜ ⎟⎝ ⎠

(3.51)

which represents Newmark equation supposing that the shear stiffness is

finite (Timoshenko model).

Such equation takes into account the presence of the following loads:

• distributed vertical load

• distributed bending moment

• shrinkage axial deformation

• thermal induced anelastic strain.

The effect of the shear flexibility , in the equation (3.51), is present in the

definition of the curvature (3.43).

Finally, the equation (describing the curvature of the beam with the

flexural stiffness EI) is obtained on the condition that the connection has its

infinite rigidity ( Lα →∞ ). In concrete slab shrinkage axial deformation is

present resulting in the consequent bending moment:

shT

M EA hEI EI

εχ ∆⎛ ⎞= +Θ −⎜ ⎟⎝ ⎠

(3.52)

If connection has zero stiffness ( 0Lα → ) the following condition can be

easily derived and assured:


35

( ), , ,1

xx x T xxq m EIEI

χ ∆= − + −Θ ΣΣ

(3.53)

Even here we observe that for Lα →∞ the shrinkage axial deformation is

also present in curvature equation, while for 0Lα → such a shrinkage

axial deformation does not result in curvatures, but only in relative

displacements.

36

4. Solution in the elastic range

The equation formulated in the previous sections will be solved

within the linear range. General boundary and restraint conditions will be

considered throughout the present chapter.

4.1 Composite beam under axial force

Restraint conditions need to be formulated with reference to the

composite beam considering relative slip as further displacement

components along with the usual ones (transverse displacement, rotational

displacement). A non‐zero axial force F is then considered on the composite

section.

Figure 4.1. Beam loaded with a normal force.

The only reacting restraint is the horizontal simple support that is

restrained in the centroid of the composite cross section. Imposing the

equilibrium and the compatibility we obtain the stresses in concrete slab

and in steel beam as noted in the system below:

Chapter IV ‐ Solution in the elastic range

37

1 21 1 2 2

1 21 21 1 2 2 1 1 2 2

1 1 2 2

,N N F

E A E AN F N FN N E A E A E A E AE A E A

+ =⎧⎪ ⇒ = =⎨ + +=⎪⎩

(4.1)

Consequently, no relative slips occur as an axial force F is applied on the

composite section and shared into the two parts N1 and N2 related to

concrete slab and steel beam respectively.

4.2 Composite beam in bending

Before analysing the bending problem, we should observe that the

composite beam kinematics is based on one more parameter in comparison

with a simple beam.

The kinematic quantity us finds its static counterpart in the mutual reaction

of the dual restraint: horizontal mutual pendulum between the concrete

slab and the steel beam. The bending problem is approached in this section.

The simultaneous equations (3.14) involve three unknowns, namely

deflection v(x) , rotational displacement φ(x) and normal stress N1(x).

In particular, it results that N1(x)=‐N2(x) as F=0.

Consequently the “slip force” S(x) can be considered as a new stress,

perfectly dual of the slip between two parts of the section. The above

definition conceptually simplifies and complete the correspondence

between nodal force S and displacement us.


38

Figure 4.2. Slip force S(x) in interface.

More precisely, the slip force in interface is defined as S(x)=N1(x)=‐N2(x)

and the possible boundary conditions reported in Figure 4.3 can be

imposed at the composite end of the beam.

Figure 4.3. The chart of the simple restraints of the flexional problem.


39

The slip force can be applied at the free end at the slip and in this case the

distribution of stresses is obtained by applying the principle of

superposition.

Figure 4.4. R0 distribution on the cross section .

It is observed that R0 causes two normal stresses and two bending

moments. However, bending moment is equal to zero on the composite

section. In fact, the superposition effects principle can be applied (see

Figure 4.4).


40

1

2

''

o

o

N RN R

=⎧⎨ = −⎩

(4.2)

1 21 1 2 2

1 21 2

1 1 2 2

'' , ''o

o o

M M R hE I E IM R h M R hM M EI EI

E I E I

+ =⎧⎪ ⇒ = =⎨ Σ Σ=⎪⎩

(4.3)

Consequently the stresses in the cross section can be obtained as follows:

1 1 1

2 2 2

1 1 1 11 1 1

2 2 2 22 2 2

' '' 0' '' 0

' '' 0

' '' 0

o o

o o

o o

o o

N N N R RN N N R R

E I E IM M M R h R hEI EIE I E IM M M R h R hEI EI

= + = + == + = − + = −

= + = + =Σ Σ

= + = + =Σ Σ

(4.4)

The bending moment is equal to 0 as it has been stated before :

1 1 2 2

1 2 1 0o o oE I E IM M M N h R h R h R hEI EI

= + − = + − =Σ Σ

(4.5)

4.2.1 Non‐redundant beams in bending

In case of simply supported beams the bending moment is always

known throughout equilibrium. This bending moment results by q(x),

m(x), as well as by the nodal forces M0 and F0. The shrinkage axial

deformation and thermal curvature as well as the restraint displacements

do not result in the stresses being the non‐redundant beam.

Therefore subsists the following equation:


41

0 0 0( , , , , )( ) 0sh T sv uM x ε ϕ∆Θ = (4.6)

Consequently, the total bending moment M(x) can be obtained as follows :

( ) ( )( ) ( ) ( ) ( ) ( )q x m x Mo FoM x M x M x M x M x= + + + (4.7)

and is always known a priori. The equation in terms of bending moments

can be applied:

, ,2 2

, , ,2 2 2 2

xxxx xx

sxx x T sh T xx

EI EI EIM v v EI qKGA EI

EI EI EI k h EIq m EIKGA EI EI

α α

εα α α α

∆ ∆

⎛ ⎞= − − + +⎜ ⎟Σ⎝ ⎠

+ − −Θ + +ΘΣ Σ

(4.8)

which can be solved against deflection v(x) as follows:

( ) ( )

2 2, ,

2, , ,

1 1

shxxxx xx T

x T xx xx

M EA hv vEI EI

q m EI q qEI KGA

εα α

α

∆

∆

⎛ ⎞− = +Θ −⎜ ⎟

⎝ ⎠

+ + −Θ Σ + −Σ

(4.9)

the general integral is:

1 2 3 4( ) ( ) cosh( ) ( )pv x C senh x C x C x C v xα α= + + + + (4.10) being vp(x) a peculiar integral of the complete equation which can be

founded on condition that the functions M(x) as well as the static and

kinematic loads are known, and :

C1, C2, C3 and C4 are the integration constants obtainable by four

boundary conditions.


42

4.2.2 Boundary conditions for non‐redundant beams

The integration constants can be derived by imposing the relevant

boundary conditions deriving from the restraints.

4.2.2.1 Vertical support

Figure 4.5. Vertical support.

The kinematic boundary condition results in a simple condition in terms of

end transverse displacement:

( )o ov x v= (4.11) The static boundary condition results in a simple condition in terms of end

slip force:

,

2

, ,,

2

,

( )( ) ( ) ( )

( ) ( )( )

( ) 1 ( )

xx oo s sh o s T o

T xx o x oxxxx o

o sxx o s o

EI EI q xS x k x k x hEI h KGAEI EI x m xv xh h h

q x k h v x k h Rh KGA

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

Σ Σ Θ− − +

⎫⎛ ⎞ ⎪+ + + =⎬⎜ ⎟⎪⎝ ⎠ ⎭

(4.12)


43

4.2.2.2 Simple rotational restraint

Figure 4.6. Simple rotational restraint.


end rotational displacement:

( )

, , ,2

, ,

2 2

, , ,

2 2

,

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(

o xxxxx o xxx o x o

x o xxx o o

xx o T x o sh x o s

T xxx

EI EIx v x v x v xKGA KGA

q x EI EI q x EI m xKGA KGA EI KGA KGA KGA

m x EI x x EI k hEIKGA EI KGA KGA EI

x

ϕα

α α

εα α

∆

∆

⎛ ⎞ ⎛ ⎞= − − +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞− + + + +⎜ ⎟⎜ ⎟Σ⎝ ⎠ ⎝ ⎠Θ⎛ ⎞− − + +⎜ ⎟Σ Σ⎝ ⎠

Θ+ 2

)oo

EIKGA

ϕα

⎛ ⎞ =⎜ ⎟⎝ ⎠

(4.13)

The static boundary condition results in a simple condition in terms of end

slip force:


44

}

,

2

2, ,

,

,

( )( ) ( ) ( )

( ) ( ) ( )( ) 1

( )

xx oo s sh o s T o

T xx o x o o sxxxx o

xx o s o

EI EI q xS x k x k x hEI h KGA

EI EI x m x q x k hv xh h h h KGA

v x k h R

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ− − + + + +⎜ ⎟

⎝ ⎠+ =

(4.14)

4.2.2.3 Mutual horizontal pendulum

Figure 4.7. Mutual horizontal pendulum.


end slip:

, ,2

,, , ,

,,

1( ) ( ) ( )

( )( ) ( ) ( )

( ) ( )

ss o x o s T x o

s

xx oxxx o s s sh x o T xxx o

xxx oxxxxx o so

EI k hu x q x k h xEI k KGA h

m x EIv x k h k x xh h

EI q x EIv x uh KGA h

α

ε

∆

∆

⎡ ⎛ ⎞= − + + Θ +⎜ ⎟⎢Σ ⎝ ⎠⎣Σ

+ − + − Θ +

Σ Σ ⎤− − =⎥⎦

(4.15)


45

4.2.2.4 Free end

Figure 4.8. Free end.


slip force:

}

,

2

2, ,

,

,

( )( ) ( ) ( )

( ) ( ) ( )( ) 1

( )

xx oo s sh o s T o


xx o s o



v x k h R

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ− − + + + +⎜ ⎟

⎝ ⎠+ =

(4.16)

Since the beam in bending is non‐redundant , there are always four

boundary conditions:

( )o oS x R= or ( )s o sou x u= at ends;

the other two or each of them are: ( )o ov x v= ,or one of them is

( )o ov x v= and the other one as: ( )o oxϕ ϕ= .

The possible boundary conditions for non‐redundant beams are

represented in the following Figures .


46

Figure 4.9.Non‐redundant beams (free slip at both ends).

Figure 4.10. Non‐redundant beams (free slip at one end).


47

Figure 4.11. Non‐redundant beams (restrained slip at both ends).

4.2.3 Redundant beams in bending

Since bending moment cannot be determined through simple

equilibrium conditions, equation (3.27) should be solved considering

bending moment as a further unknown:

( )

( ) ( )

22

, , , , ,

2, , , , ,

1 1



v v q m EI EA hEI

q m EI q qEI KGA

αα ε

α

∆

∆

− = − + −Θ + +

+ + −Θ Σ − −Σ

(3.27)

The general integral of such an equation results as follows: 3 2

1 2 3 4 5

6

( ) ( ) cosh( )( )p

v x C senh x C x C x C x C xC v x

α α= + + + + ++ +

(4.17)

where vp(x) is still a particular solution of the complete equation, obtained

from the functions of external loads.

Six constants C1, C2, C3, C4, C5 and C6 are obtained from six boundary

conditions which are presented in the following paragraph.


48

4.2.4 Boundary conditions for redundant beams

4.2.4.1 Vertical support

Figure 4.12. Vertical support.


end transverse displacement:

( )o ov x v= (4.18) The static boundary condition results in a simple condition in terms of end

slip force:

}

,

2

2, ,

,

,

( )( ) ( ) ( )

( ) ( ) ( )( ) 1

( )

xx oo s sh o s T o


xx o s o



v x k h R

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ− − + + + +⎜ ⎟

⎝ ⎠+ =

(4.19)

The second static boundary condition results in a simple condition in terms

of end bending moment:


49

, ,2 2

, ,2 2

,2 2

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

o xxxx o xx o o

xx o x o T o

ssh o T xx o o

EI EI EIM x v x v x EI q xKGA EI

EI EIq x m x x EIKGA EI

EI k h EIx x MEI

α α

α α

εα α

∆

∆

⎛ ⎞= − − + +⎜ ⎟Σ⎝ ⎠

+ − −Θ +Σ

+ +Θ =Σ

(4.20)

4.2.4.2 Simple rotational restraint

Figure 4.13. Simple rotational restraint.


slip force:

}

,

2

2, ,

,

,

( )( ) ( ) ( )

( ) ( ) ( )( ) 1

( )

xx oo s sh o s T o


xx o s o



v x k h R

εα

∆

∆

⎧ Σ= − + Θ − +⎨Σ ⎩

⎛ ⎞Σ Σ Θ− − + + + +⎜ ⎟

⎝ ⎠+ =

(4.21)


50


end rotational displacement:

( )

, , ,2

, ,

2 2

, , ,

2 2

,

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

(

o xxxxx o xxx o x o

x o xxx o o

xx o T x o sh x o s

T xxx

EI EIx v x v x v xKGA KGA

q x EI EI q x EI m xKGA KGA EI KGA KGA KGA

m x EI x x EI k hEIKGA EI KGA KGA EI

x

ϕα

α α

εα α

∆

∆

⎛ ⎞ ⎛ ⎞= − − +⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞− + + + +⎜ ⎟⎜ ⎟Σ⎝ ⎠ ⎝ ⎠Θ⎛ ⎞− − + +⎜ ⎟Σ Σ⎝ ⎠

Θ+ 2

)oo

EIKGA

ϕα

⎛ ⎞ =⎜ ⎟⎝ ⎠

(4.22)


of end shear stress:

( )

( )

, ,2

, ,2

, , ,2 2 2

,

2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

o xxxxx o xxx o

x o T x o o

ssh x o xx o T xxx o

xxx oo

EIQ x v x EI v x

EI EI q x EI x m xKGA EIh k EI EI EIx m x xEI EIEI q x F

KGA

α

α

εα α α

α

∆

∆

⎛ ⎞= − +⎜ ⎟⎝ ⎠

⎡ ⎤− + − Θ + +⎢ ⎥Σ⎣ ⎦⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + Θ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞+ =⎜ ⎟⎝ ⎠

(4.23)


51

4.2.4.3 Mutual horizontal pendulum

Figure 4.14. Mutual horizontal pendulum.


end slip:

, ,2

,, , ,

,,

1( ) ( ) ( )

( )( ) ( ) ( )

( ) ( )

ss o x o s T x o

s

xx oxxx o s s sh x o T xxx o

xxx oxxxxx o so

EI k hu x q x k h xEI k KGA h

m x EIv x k h k x xh h

EI q x EIv x uh KGA h

α

ε

∆

∆

⎡ ⎛ ⎞= − + + Θ +⎜ ⎟⎢Σ ⎝ ⎠⎣Σ

+ − + − Θ +

Σ Σ ⎤− − =⎥⎦

(4.24)


bending moment:

, ,2 2

, ,2 2

,2 2

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( )

o xxxx o xx o o

xx o x o T o

ssh o T xx o o

EI EI EIM x v x v x EI q xKGA EI

EI EIq x m x x EIKGA EI

EI k h EIx x MEI

α α

α α

εα α

∆

∆

⎛ ⎞= − − + +⎜ ⎟Σ⎝ ⎠

+ − −Θ +Σ

+ +Θ =Σ

(4.25)


52


of end shear stress:

( )

( )

, ,2

, ,2

, , ,2 2 2

,

2

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( )

o xxxxx o xxx o

x o T x o o

ssh x o xx o T xxx o

xxx oo

EIQ x v x EI v x

EI EI q x EI x m xKGA EIh k EI EI EIx m x xEI EIEI q x F

KGA

α

α

εα α α

α

∆

∆

⎛ ⎞= − +⎜ ⎟⎝ ⎠

⎡ ⎤− + − Θ + +⎢ ⎥Σ⎣ ⎦⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ − + Θ +⎜ ⎟ ⎜ ⎟ ⎜ ⎟Σ Σ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎛ ⎞+ =⎜ ⎟⎝ ⎠

(4.26)

The beam in bending is redundant and six boundary conditions are always

need for solving the problem.

In each end there are three boundary conditions:

( )o oS x R= or ( )s o sou x u= ;

( )o ov x v= or ( )o oQ x Q= ;

( )o oxϕ ϕ= or ( )o oM x M= .

Therefore, so‐called the possible one time redundant schemes are the

subsequent ones:

Figure 4.15. One time redundant beams (free slip at both ends).


53

Figure 4.16. One time redundant beams (free slip at one end).

Figure 4.17. One time redundant beams (restrained slip at both ends).

The possible two time redundant schemes are the subsequent ones:

Figure 4.18. Two time redundant beam (free slip at both ends).


54

Figure 4.19. Two time redundant beam (free slip at one end).

Figure 4.20. Two time redundant beam(restrained slip at both ends).

55

5. Stiffness matrix

The closed‐form expressions of the stiffness matrix deriving by the

shear‐flexible beam model will be derived in the present section.

5.1 Identification of the problem

Let us define nodal displacement vector of the beam as:

{ }1 1 1 2 2 2, , , , ,Ts sD v u v uϕ ϕ= (5.1)

and the nodal forces vector of the beam:

{ }1 1 1 2 2 2, , , , ,Ts sF V C H V C H= (5.2)

obtained from the difference of:

0T T T

EF F F= − (5.3)

between the vector of the external forces ,which are applied on the nodes

FE, and the reactive forces vector F0 connected with the kinematically

determinated beam (in other words connected with clamped nodes).

Therefore, it is possible to write equilibrium equation of the beam as:

0T TEK D F F= − (5.4)

Consequently, the 6∙6 stiffness matrix of the beam model has been already

defined .The i‐th column of this matrix is represented by the group of 6

nodal forces corresponding to a displacement vector whose i‐th component


56

is one and the other zero.

Displacements and nodal forces which are both dual with each other, have

the positive sign indicated Figure 5.1 and can be derived as follows.

Figure 5.1. Positive convention for nodal forces and displacements .

1

2

3

4

5

6

(0) /(0) /

(0) /( ) /( ) /( ) /

i i

i i

i i

i i

i i

i i

K Q DK M DK S DK Q L DK M L DK S L D

= −⎧⎪ = −⎪⎪ =⎨ =⎪⎪ =⎪

= −⎩

(5.5)

after having defined with Di the displacement i‐th component of the

displacement vector D.

Chapter V ‐ Stiffness matrix

57

5.2 Coefficients of the stiffness matrix

5.2.1 General procedure for deriving the integration constants

The term of the stiffness matrix can be derived by solving the

differential equation imposing nodal displacements in each component as

the others are forced to zero.

Figure 5.2. Beam in bending under general restraint condition.

The distributed loads are equal to zero, because the only external actions

are the restraint displacements and the particular integral vp(x) is equal to

zero, so: 3 2

1 2 3 4 5 6( ) ( ) cosh( )v x C senh x C x C x C x C x Cα α= + + + + + (5.6)

The following derivates of such function, until the derivate of the order 6,

result as:


58

( )( )( )( )( )

2, 1 2 3 4 5

2, 1 2 3 4

3, 1 2 3

4, 1 2

5, 1 2

( ) cos ( ) h( ) 3 2

( ) ( ) cosh( ) 6 2

( ) cos ( ) h( ) 6

( ) ( ) cosh( )

( ) cos ( ) h( )

x

xx

xxx

xxxx

xxxxx

v x C h x C sen x C x C x C

v x C senh x C x C x C

v x C h x C sen x C

v x C senh x C x

v x C h x C sen x

α α α

α α α

α α α

α α α

α α α

= + + + +

= + + +

= + +

= +

= +

(5.7)

The expressions of relevant displacements and stresses can be derived as

follows:

( ), ,2 xxxxx xxxEIQ v EI vα⎛ ⎞= −⎜ ⎟⎝ ⎠

(5.8)

( ), ,2 xxxx xxEIM v EI vα

⎛ ⎞= −⎜ ⎟⎝ ⎠

(5.9)

, ,2 2

sxx xxxx

EI k h EIS v vEI hα α

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠ (5.10)

, , ,2 xxxxx xxx xEI EIv v v

KGA KGAϕ

α⎛ ⎞ ⎛ ⎞= − −⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠ (5.11)

, ,2 2s xxx xxxxxs

EI h EIu v vEI k hα α

⎛ ⎞⎛ ⎞= − + ⎜ ⎟⎜ ⎟Σ⎝ ⎠ ⎝ ⎠ (5.12)

The following boundary conditions in terms of generalized displacements

can be introduced for evaluating the constants involved in the above

expressions:


59

1

1

1

2

2

2

( 0)( 0)( 0)

( )( )( )

s s

s s

v x vx

u x uv x L vx L

u x L u

ϕ ϕ

ϕ ϕ

= =⎧⎪ = =⎪⎪ = =⎨ = =⎪⎪ = =⎪

= =⎩

(5.13)

and the following set of equations can be obtained by introducing the

displacement definitions in equations (5.8)÷(5.12):

( ) ( )

( ) ( )

( ) ( )

2 6 1

31 5 1

31 3

12

3 21 2 3 4 5 6 2

3 21 2 3 4

5 2

3 31 2 3

2 22

6

6

6 3 2

6

s

s

C C vEIC C CKGAEI C EI h C uks h EI

C senh L C cosh L C L C L C L C vEIC C cosh L C senh L C L C LKGACEI C EI C EI h Ccosh L C senh L uks h ks h EI

α ϕ

αα

α α

α α α α

ϕ

α αα αα

+ =⎧⎪⎪− − − =⎪⎪Σ⎪ − =

Σ⎪⎪ + + + + + =⎨⎪⎪− − − − − +

− =

Σ Σ+ − =

Σ⎩

⎪⎪⎪⎪⎪

(5.14)

Consequently, those constants can be easily related to the values of the

imposed nodal displacements. The following simplification will be

introduced in the definitions of the constant expressions.

( ) ( ) ( )1 / 2ctgh L tgh Lsenh L

α αα

⎡ ⎤− =⎢ ⎥

⎣ ⎦ (5.15)

For the sake of brevity, all the constants are expressed as a function of C3.


60

2

1 1 33 5

6s ss

k h EI k hC u CEI EI EIα α

= +Σ Σ Σ

(5.16)

( )( )

( ) ( )

2 1 23 3

2

35

6 1

s ss s

s

k h ctgh L k hC u uEI EI senh L

EI k h ctgh L CEI EI senh L

αα α α

αα α

= − + +Σ Σ

⎡ ⎤− −⎢ ⎥Σ Σ ⎣ ⎦

(5.17)

( ) ( ) ( ) ( )( )( )

( )

1 2 1 21 2 33 2

3

32

2 2 / 2

2 / 2121 12 1

ss s

v v L tgh Lk hu uL L EI L

CL tgh LEI EI

KGA L EI L

ϕ ϕ α αα

α αα

⎡ ⎤− + −− − + ⎢ ⎥

Σ ⎢ ⎥⎣ ⎦=⎡ ⎤−⎛ ⎞ ⎛ ⎞+ + − ⎢ ⎥⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠ ⎢ ⎥⎣ ⎦

(5.18)

( ) ( )( )

( )( )

( )( )

1 24 12 2

12 32

2

3 34

1 1

1 6

26 2 1

ss

ss

s

v v ctgh Lk hC uL EI L L Lsenh L

ctgh Lk h EIu CEI L L L senh L L KGA L

ctgh LEI k hC L CEI EI L L L senh L

αα α α α

α ϕα α α α

αα α α α

⎡ ⎤−= − + − + + +⎢ ⎥Σ ⎣ ⎦

⎡ ⎤− − + + +⎢ ⎥Σ ⎣ ⎦

⎡ ⎤− + − + +⎢ ⎥Σ Σ ⎣ ⎦

(5.19)

2

5 1 1 3 32 4

6 6s ss

k h EI k h EIC u C CEI EI EI KGA

ϕα α

= − − − −Σ Σ Σ

(5.20)


61

( )( )

( ) ( )

6 1 2 13 3

2

35

6 1

sss s

s

k hctgh Lk hC v u uEI senh L EI

EI k h ctgh L CEI EI senh L

αα α α

αα α

= − + +Σ Σ

⎡ ⎤+ −⎢ ⎥Σ Σ ⎣ ⎦

(5.21)

The nodal forces can be easily evaluated starting from equation (5.6)

through equations (5.8)‐(5.10) for x equal to 0 and the space length L:

3(0) 6Q EI C= − (5.22)

3( ) 6Q L EI C= − (5.23)

4(0) 2M EI C= − (5.24)

4 3( ) 2 6M L EI C EI LC= − − (5.25)

2

2 42

2(0) sEI EI k hS C Ch EIα

αΣ

= − +Σ

(5.26)

( ) ( )( )

( )

2

1 2

3 42

( ) cos

6 2s

EIS L C senh L C h Lh

EI k h C L CEI

α α α

α

Σ= − + +

+ +Σ

(5.27)

Once the six force components have been derived, the stiffness matrix can

be directly calculated by solving equation (5.14) considering displacement

vector in which the i‐th component is the only non‐zero one.


62

5.2.2 Stiffness matrix: the first column

Considering a nodal displacement vector D=[v1,0,0,0,0,0] in equation

(5.14) the terms of the first column can be determined by equations

(5.22)÷(5.27).The constant Ci can be derived by solving those equations to

obtain the following results:

( )( )

( )

3

1 1

32

12 1

2 / 2121 12 1

EIEIL

C vL tgh LEI EI

KGA L EI L

α

α αα

⎛ ⎞−⎜ ⎟Σ⎝ ⎠=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.28)

( )( )

( )( )

3

2 1

32

/ 212 1

2 / 2121 12 1

tgh LEIEI L

C vL tgh LEI EI

KGA L EI L

αα

α αα

⎛ ⎞−⎜ ⎟Σ⎝ ⎠= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.29)

( )( )

33 1

32

2

2 / 2121 12 1

LC vL tgh LEI EI

KGA L EI Lα α

α

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.30)

( )( )

24 1

32

3

2 / 2121 12 1

LC vL tgh LEI EI

KGA L EI Lα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.31)


63

( )( )

3 2

5 1

32

12 1 1

2 / 2121 12 1

EI EIL KGA EIC v

L tgh LEI EIKGA L EI L

α

α αα

⎡ ⎤⎛ ⎞+ −⎜ ⎟⎢ ⎥Σ⎝ ⎠⎣ ⎦= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.32)

( )( )

( )( )

3

6 1

32

/ 212 1

12 / 2121 12 1

tgh LEIEI L

C vL tgh LEI EI

KGA L EI L

αα

α αα

⎧ ⎫⎛ ⎞⎪ ⎪−⎜ ⎟Σ⎪ ⎪⎝ ⎠= +⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

(5.33)

Hence, the following nodal forces can be derived:

( )( )

( )( )

2

1

32

(0)

2 / 26

2 / 2121 12 1

EI EISh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.34)

( )( )

( )( )

2

1

32

( )

2 / 26

2 / 2121 12 1

EI EIS Lh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.35)


64

( )( )

21

32

6

(0)2 / 2121 12 1

EILM v


α αα

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.36)

( )( )

21

32

6

( )2 / 2121 12 1

EILM L v


α αα

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.37)

( )( )

31

32

12

(0)2 / 2121 12 1

EILQ v


α αα

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.38)

( )( )

31

32

12

( )2 / 2121 12 1

EILQ L v


α αα

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.39)

The particular case of Bernoulli beams can be derived by the above

expression in the limit that the ratio 2KGAL

EI→∞ :


65

( )( )

( )( )

212

3

2 / 26

lim (0)2 / 2

1 12 1KGALEI

L tgh LLEI EIS v

h L L tgh LEIEI L

α αα

α αα

→∞

−

−Σ= −

⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.40)

( )( )

( )( )

212

3

2 / 26

lim ( )2 / 2

1 12 1KGALEI

L tgh LLEI EIS L v

h L L tgh LEIEI L

α αα

α αα

→∞

−

−Σ=

⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.41)

( )( )

2

21

3

6

lim (0)2 / 2

1 12 1KGALEI

EILM vL tgh LEI

EI Lα α

α→∞

=⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.42)

( )( )

2

21

3

6

lim ( )2 / 2

1 12 1KGALEI

EILM L vL tgh LEI

EI Lα α

α→∞

= −⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.43)

( )( )

2

31

3

12

lim (0)2 / 2

1 12 1KGALEI

EILQ vL tgh LEI

EI Lα α

α→∞

= −⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.44)


66

( )( )

2

31

3

12

lim ( )2 / 2

1 12 1KGALEI

EILQ L vL tgh LEI

EI Lα α

α→∞

= −⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.45)

Furthermore, the case of full shear interaction (namely, no slip occurrence

or rigid connection) can be derived from the general model at hand by

forcing Lα →∞ ; the following relationships can be derived for the

Bernoulli beam:

212lim lim (0) 6

L KGALEI

EI EIS vh Lα →∞

→∞

−Σ= −

(5.46)

212lim lim ( ) 6

L KGALEI

EI EIS L vh Lα →∞

→∞

−Σ=

(5.47)

212

6lim lim (0)L KGAL

EI

EIM vLα →∞

→∞

= (5.48)

212

6lim lim ( )L KGAL

EI

EIM L vLα →∞

→∞

= − (5.49)

213

12lim lim (0)L KGAL

EI

EIQ vLα →∞

→∞

= − (5.50)


67

213

12lim lim ( )L KGAL

EI

EIQ L vLα →∞

→∞

= − (5.51)

On the contrary, the general model reproduces the case of absent

interaction as 0Lα → :

20lim lim (0) 0L KGAL

EI

Sα →

→∞

= (5.52)

20lim lim ( ) 0L KGAL

EI

S Lα →

→∞

= (5.53)

2120

6lim lim (0)L KGAL

EI

EIM vLα →

→∞

Σ=

(5.54)

2120

6lim lim ( )L KGAL

EI

EIM L vLα →

→∞

Σ= −

(5.55)

2130

12lim lim (0)L KGAL

EI

EIQ vLα →

→∞

Σ= −

(5.56)

2130

12lim lim ( )L KGAL

EI

EIQ L vLα →

→∞

Σ= −

(5.57)

Finally, coming back to the general definition of stiffness terms in equation

(5.5), the terms of the first column of the stiffness matrix are listed below:


68

( )( )

11 3

32

12 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.58)

( )( )

21 2

32

6 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.59)

( )( )

( )( )

31 2

32

2 / 26

2 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.60)

( )( )

41 3

32

12 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.61)

( )( )

51 2

32

6 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.62)


69

( )( )

( )( )

61 2

32

2 / 26

2 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.63)

5.2.3 Stiffness matrix: the second column

Considering a nodal displacement vector D=[0,φ1,0,0,0,0] in equation

(5.14) the terms of the second column can be determined by equations

(5.22)÷(5.27).The constant Ci can be derived by solving those equations to


( )( )

( )

2

1 1

32

6 1

2 / 2121 12 1

EIEIL

CL tgh LEI EI

KGA L EI L

α αϕ

α αα

⎛ ⎞−⎜ ⎟Σ⎝ ⎠= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.64)

( )( )

( )( )

2

2 1

32

/ 26 1

2 / 2121 12 1

tgh LEIEI L

CL tgh LEI EI

KGA L EI L

αα α

ϕα α

α

⎛ ⎞−⎜ ⎟Σ⎝ ⎠=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.65)


70

( )( )

23 1

32

1

2 / 2121 12 1

LCL tgh LEI EI

KGA L EI L

ϕα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.66)

( )( )

4

1

32

12

312 / 2121 12 1

CL


ϕα α

α

=

⎧ ⎫⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.67)

{

( )( )

( )

5

22

1

32

1

6 6 1

2 / 2121 12 1

C

EI EIKGA L EIL


αϕ

α αα

= − +

⎫⎡ ⎤⎛ ⎞ ⎪+ −⎢ ⎥⎜ ⎟Σ⎝ ⎠ ⎪⎢ ⎥⎣ ⎦+ ⎬⎛ ⎞−⎛ ⎞ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎭

(5.68)

( )( )

( )( )

2

6 1

32

/ 26 1

2 / 2121 12 1

tgh LEIEI L

CL tgh LEI EI

KGA L EI L

αα α

ϕα α

α

⎛ ⎞−⎜ ⎟Σ⎝ ⎠= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.69)

The consequent nodal stresses result as:


71

( )( )

( )( )

1

32

(0)

2 / 23

12 / 2121 12 1

EI EISh L

L tgh LL


α αα

ϕα α

α

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.70)

( )( )

( )( )

1

32

( )

2 / 23

12 / 2121 12 1

EI EIS Lh L

L tgh LL


α αα

ϕα α

α

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.71)

( )( )

1

32

(0)

312 / 2121 12 1

EIML


ϕα α

α

= −

⎧ ⎫⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.72)


72

( )( )

1

32

( )

312 / 2121 12 1

EIM LL


ϕα α

α

= −

⎧ ⎫⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.73)

( )( )

2

1

32

6(0)

12 / 2121 12 1

EIQL


ϕα α

α

=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.74)

( )( )

2

1

32

6( )

12 / 2121 12 1

EIQ LL


ϕα α

α

=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.75)



EI→∞ :


73

( )( )

( )( )

2

1

3

lim (0)

2 / 23

12 / 2

1 12 1

KGALEI

EI EISh L

L tgh LL

L tgh LEIEI L

α αα

ϕα α

α

→∞

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

i

i (5.76)

( )( )

( )( )

2

1

3

lim ( )

2 / 23

12 / 2

1 12 1

KGALEI

EI EIS Lh L

L tgh LL

L tgh LEIEI L

α αα

ϕα α

α

→∞

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

i

i (5.77)

( )( )

21

3

3lim (0) 12 / 2

1 12 1KGALEI

EIML L tgh LEI

EI L

ϕα α

α→∞

⎧ ⎫⎪ ⎪⎪ ⎪

= − +⎨ ⎬⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

(5.78)


74

( )( )

21

3

3lim ( ) 12 / 2

1 12 1KGALEI

EIM LL L tgh LEI

EI L

ϕα α

α→∞

⎧ ⎫⎪ ⎪⎪ ⎪

= − −⎨ ⎬⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

(5.79)

( )( )

212

3

6 1lim (0)2 / 2

1 12 1KGALEI

EIQL L tgh LEI

EI L

ϕα α

α→∞

=⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.80)

( )( )

212

3

6 1lim ( )2 / 2

1 12 1KGALEI

EIQ LL L tgh LEI

EI L

ϕα α

α→∞

=⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.81)




Bernoulli beam:

21lim lim (0) 4

L KGALEI

EI EISh Lα

ϕ→∞

→∞

−Σ=

(5.82)

21lim lim ( ) 2

L KGALEI

EI EIS Lh Lα

ϕ→∞

→∞

−Σ= −

(5.83)


75

21

4lim lim (0)L KGAL

EI

EIMLα

ϕ→∞

→∞

= − (5.84)

21

2lim lim ( )L KGAL

EI

EIM LLα

ϕ→∞

→∞

= (5.85)

212

6lim lim (0)L KGAL

EI

EIQLα

ϕ→∞

→∞

= (5.86)

212

6lim lim ( )L KGAL

EI

EIQ LLα

ϕ→∞

→∞

= (5.87)



21lim lim (0)

L KGALEI

EI EISh Lα

ϕ→∞

→∞

−Σ=

(5.88)

21lim lim ( )

L KGALEI

EI EIS Lh Lα

ϕ→∞

→∞

−Σ=

(5.89)

21

3lim lim (0)L KGAL

EI

EI EIMLα

ϕ→∞

→∞

+ Σ⎛ ⎞= −⎜ ⎟⎝ ⎠

(5.90)


76

21

3lim lim ( )L KGAL

EI

EI EIM LLα

ϕ→∞

→∞

− Σ⎛ ⎞= −⎜ ⎟⎝ ⎠

(5.91)

212

6lim lim (0)L KGAL

EI

EIQLα

ϕ→∞

→∞

Σ=

(5.92)

212

6lim lim ( )L KGAL

EI

EIQ LLα

ϕ→∞

→∞

Σ=

(5.93)



( )( )

12 2

32

6 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.94)

( )( )

22

32

312 / 2121 12 1

EIKL


α αα

=

⎧ ⎫⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.95)


77

( )( )

( )( )

32

32

2 / 23

12 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.96)

( )( )

42 2

32

6 12 / 2121 12 1

EIKL L tgh LEI EI

KGA L EI Lα α

α

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.97)

( )( )

52

32

312 / 2121 12 1

EIKL


α αα

=

⎧ ⎫⎪ ⎪⎪ ⎪− +⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.98)

( )( )

( )( )

62

32

2 / 23

12 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪− +⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.99)


78

5.2.4 Stiffness matrix: the third column

Considering a nodal displacement vector D=[0,0,us1,0,0,0] in equation

(5.14) the terms of the second column can be determined by equations

(5.22)÷(5.27). The constant Ci can be derived by solving those equations to


( )( )

( )( )

1 3

3

1

32

2 / 26 1

12 / 2121 12 1

s

s

k hCEI

L tgh LEIEI L

uL tgh LEI EI

KGA L EI L

α

α αα

α αα

=Σ

⎧ ⎫⎛ ⎞−⎛ ⎞⎪ ⎪− ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎪ ⎪⎝ ⎠−⎨ ⎬⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.100)

( ){

( )( )

( )

( )( )

2 3

3

1

32

2 / 26 1 / 2

2 / 2121 12 1

s

ks hC ctgh LEI

L tgh LEI tgh LEI L

uL tgh LEI EI

KGA L EI L

αα

α αα

α

α αα

= − +Σ

⎫⎛ ⎞−⎛ ⎞ ⎪− ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠ ⎪⎝ ⎠+ ⎬⎛ ⎞−⎛ ⎞ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎭

(5.101)

( )( )

( )( )

3

3 1

32

2 / 2

2 / 2121 12 1

s

L tgh Lks hEI L

C uL tgh LEI EI

KGA L EI L

α αα

α αα

⎛ ⎞−⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.102)


79

( ) ( )

( )( )

( )( )

4

1

- 22

- 2 / 2 1818 -1 3 2

- 2 / 2121 12 -12 3

s

tgh aL aL tgh aLks hCaL aLEI a L

aL tgh aLEI EISEI KGA LaL

uaL tgh aLEI EI

SEIKGA L aL

⎧ ⎛ ⎞= +⎨ ⎜ ⎟

⎩ ⎝ ⎠Σ

⎫⎛ ⎞⎛ ⎞ ⎪⎜ ⎟ +⎜ ⎟ ⎪⎜ ⎟⎝ ⎠ ⎪⎝ ⎠⎬

⎛ ⎞⎛ ⎞ ⎪⎛ ⎞⎜ ⎟⎜ ⎟+ + ⎪⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎪⎝ ⎠ ⎝ ⎠⎭

i

i

(5.103)

( )( )

( )( )

( )

5 2

2 2

1

32

2 / 26 61

12 / 2121 12 1

s

s

k hCEI

L tgh LEI EIEI KGA L LL

uL tgh LEI EI

KGA L EI L

α

α ααα

α αα

=Σ

⎧ ⎫⎛ ⎞⎛ ⎞−⎛ ⎞⎪ ⎪− +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎪ ⎪⎝ ⎠⎝ ⎠− +⎨ ⎬⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i

(5.104)

( ){

( )( )

( )

( )( )

6 3

3

1

32

2 / 26 1 / 2

2 / 2121 12 1

s

s

k hC ctgh LEI

L tgh LEI tgh LEI L

uL tgh LEI EI

KGA L EI L

αα

α αα

α

α αα

= +Σ

⎫⎛ ⎞−⎛ ⎞ ⎪− ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠ ⎪⎝ ⎠− ⎬⎛ ⎞−⎛ ⎞ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎭

(5.105)

The consequent nodal stresses result as:


80

( )( )

( )( )

( )

( )( )

12

2 2

3

32

2

(0)

2 / 212 1

2 / 2121 12 1

12 1

sEI EIS uh L

L tgh LL EI EItgh L EI EIL

L tgh LEI EI EIKGA L EI EIL

L senh LEI EIEI EI L

α ααα α

α αα

α αα

•−Σ

=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

( )( )

( )

( )

32

2

2

13

2 / 2121 12 1

12 1

2 /121 12 1

senh L


EI EI EI EI LEI KGA L EI EI tgh L

L tgh LEI EIKGA L EI

α

α αα

αα

α α

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

( )( )3

2 EIEILα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.106)


81

( )( )

( )( )

( )

( )

12

2 2

3

32

( )

2 / 212 1

2 / 2121 12 1

cos12 1

sEI EIS L uh L

L tgh LL EI EIsenh L EI EIL


L L senEI EIEI EI

α ααα α

α αα

α α

•−Σ

=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

( )( ) ( )

( )( )

( )

2

32

2

2

16

2 / 2121 12 1

12 1

121 12 1

h LL senh L


EI EI EI EI LEI KGA L EI EI senh L

LEI EIKGA L EI

αα α

α αα

αα

α

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

( )( )3

2 / 2tgh L EIEIL

αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.107)

( )( )

( )( )

1

32

(0)

2 / 23

12 / 2121 12 1

s

EI EIMh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.108)


82

( )( )

( )( )

1

32

( )

2 / 23

12 / 2121 12 1

s

EI EIM Lh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.109)

( )( )

( )( )

2

1

32

(0)

2 / 26

2 / 2121 12 1

s

EI EIQh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.110)

( )( )

( )( )

2

1

32

( )

2 / 26

2 / 2121 12 1

s

EI EIQ Lh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.111)



EI→∞ :


83

( )( )

( )( )

( )

( )( ) ( )

2 2

2 2

3

3

2

lim (0)

2 / 212 1

2 / 21 12 1

12 1

KGALEI

EI EISh L


L tgh LEI EIEI EIL

L senh LEI EIEI EI L senh L

α ααα α

α αα

α αα α

→∞

−Σ=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞− +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( )( )

1

3

13

2 / 21 12 1

suL tgh LEI EI

EI EILα α

α

⎫⎛ ⎞⎪⎜ ⎟⎜ ⎟ ⎪⎪⎝ ⎠⎬

⎡ ⎤⎛ ⎞− ⎪Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎪⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.112)

( )( )

( )( )

( )

( ) ( )( )

2 2

2 2

3

3

2

lim ( )

2 / 212 1

2 / 21 12 1

cos12 1

KGALEI

EI EIS Lh L


L tgh LEI EIEI EIL

L L senh LEI EIEI EI L s

α ααα α

α αα

α α αα

→∞

−Σ=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠⎪ ⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( )( )

( )

1

3

16

2 / 21 12 1

senh L

uL tgh LEI EI

EI EIL

α

α αα

⎫⎛ ⎞⎪−⎜ ⎟⎜ ⎟⎪⎪⎝ ⎠⎬

⎡ ⎤⎛ ⎞− ⎪Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ − ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎪⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.113)


84

( )( )

( )( )

2

1

3

lim (0)

2 / 23

12 / 2

1 12 1

KGALEI

s

EI EIMh L

L tgh LL

uL tgh LEI

EI L

α αα

α αα

→∞

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

i

i

(5.114)

( )( )

( )( )

2

1

3

lim ( )

2 / 23

12 / 2

1 12 1

KGALEI

s

EI EIM Lh L

L tgh LL

uL tgh LEI

EI L

α αα

α αα

→∞

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎪ ⎪⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠⎩ ⎭

i

i

(5.115)

( )( )

( )( )

212

3

2 / 26

lim (0)2 / 2

1 12 1

sKGALEI

L tgh LLEI EIQ u

h L L tgh LEIEI L

α αα

α αα

→∞

−

−Σ=

⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.116)

( )( )

( )( )

212

3

2 / 26

lim ( )2 / 2

1 12 1

sKGALEI

L tgh LLEI EIQ L u

h L L tgh LEIEI L

α αα

α αα

→∞

−

−Σ=

⎛ ⎞−⎛ ⎞+ − ⎜ ⎟⎜ ⎟⎜ ⎟Σ⎝ ⎠⎝ ⎠

(5.117)


85

The above general expressions basically reduce to the ones derived by

Faella, Martinelli and Nigro (2002) within the framework of the Newmark

theory as 2KGAL

EI→∞ .




Bernoulli beam:

2lim lim (0)L KGAL

EI

Sα →∞

→∞

= +∞ (5.118)

( )2

2

12lim lim ( ) 2 sL KGAL

EI

EI EIS L u

EI h Lα →∞→∞

−Σ= − (5.119)

( )2

1lim lim (0) 4 sL KGAL

EI

EI EIM u

h Lα →∞→∞

−Σ= −

(5.120)

( )2


EI

EI EIM L u

h Lα →∞→∞

−Σ=

(5.121)

( )2

12lim lim (0) 6 sL KGAL

EI

EI EIQ u

h Lα →∞→∞

−Σ=

(5.122)


86

( )2


EI

EI EIQ L u

h Lα →∞→∞

−Σ=

(5.123)



2120

lim lim (0) sL KGAL

EI

EI EIS uh Lα →

→∞

−Σ=

(5.124)

2120

lim lim ( ) sL KGAL

EI

EI EIS L uh Lα →

→∞

−Σ=

(5.125)

21

0lim lim (0) sL KGAL

EI

EI EIM uh Lα →

→∞

−Σ= −

(5.126)

21

0lim lim ( ) sL KGAL

EI

EI EIM L uh Lα →

→∞

−Σ= −

(5.127)

20lim lim (0) 0L KGAL

EI

Qα →

→∞

= (5.128)

20lim lim ( ) 0L KGAL

EI

Q Lα →

→∞

= (5.129)




87

( )( )

( )( )

13 2

32

2 / 26

2 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.130)

( )( )

( )( )

23

32

2 / 23

12 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.131)


88

( )( )

( )( )

( )

( )( )

33 2

2 2

3

32

2

2 / 212 1

2 / 2121 12 1

12 1

EI EIKh L



L senh LEI EIEI EI L senh

α ααα α

α αα

α αα

−Σ=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( )( )

( )

( )( )

( )

32

2

32

13

2 / 2121 12 1

12 1

2 / 2121 12 1

L




α

α αα

αα

α αα

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

EIEI

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.132)

( )( )

( )( )

43 2

32

2 / 26

2 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.133)


89

( )( )

( )( )

53

32

2 / 23

12 / 2121 12 1

EI EIKh L

L tgh LL


α αα

α αα

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.134)

( )( )

( )( )

( )

( ) ( )

63 2

2 2

3

32

2 / 212 1

2 / 2121 12 1

cos12 1

EI EIKh L



L L senh LEI EIEI EI

α ααα α

α αα

α α α

−Σ= −

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( ) ( )( )

( )

( )

2

32

2

2

16

2 / 2121 12 1

12 1

2121 12 1

L senh L



L tgEI EIKGA L EI

α α

α αα

αα

α

⎛ ⎞−⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

( )( )3

/ 2h L EIEIL

αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.135)


90

5.2.5 Completing the stiffness matrix

The three remaining columns of the stiffness matrix could be derived

through the same procedure followed for the first three ones. However, for

the sake of brevity, an alternative procedure has been put in place in this

work, looking after both geometric and structural symmetry.

K14=K41, K15=K51, K16=K61, K24=K42, K25=K52, K26=K62, K34=K43, K35=K53, K36=K63 .

The diagonal elements at the last three positions, being the direct effects,

have necessarily the positive sign (the stiffness matrix serves in order to

define the elastic potential energy of the system which is a quadratic

definite positive form). The diagonal elements also have the expressions

which coincide with the expressions of the elements in the diagonal in the

first three positions:

K44=K11, K55=K22, K66=K33 .

Finally we have:

K54= ‐K21, K64= ‐K31, K65= K32

• after observing that both bending moment and slip force at the end

2, which are caused by the lowering in the extreme 1, have the

sign opposite to the bending moment and opposite to the slip force

at the end 1 (on condition that the bending moment and the slip

force are nodal forces);

• and after noticing that the slip force in the extreme 2, which is

caused by the rotation of the extreme 2 ,is of the same sign as

regards the slip force in the extreme 1 (on condition that the slip


91

force at the extreme 2 is the nodal force). All this is caused by the

symmetry of the structural system.

For the symmetry of the stiffness matrix we have:

K45=K54, K46=K64, K56=K65 .

5.3 Vector of the external nodal force and vector

nodal forces equivalent to distributed action

The nodal actions equivalent to the actions distributed throughout

the beam axis will be evaluated in the present paragraph.

5.3.1 Vector of the external nodal forces

The vector FE of the equivalent external nodal forces is in this case

equal to the null vector , since the beam results restrained at both its nodes

with respect to all the degrees of freedom. Consequently, no stress results

in the beam by applying external nodal forces:

{ }1 1 1 2 2 2, , , , , 0TE E E Es E E EsF V C H V C H= = (5.136)

5.3.2 Vector nodal forces equivalent to distributed actions.

As far as the vector of forces equivalent to the distributed actions F0

it can be derived following a procedure substantially similar to those

considered for evaluating the terms of the stiffness matrix. Obviously, the

terms of the vector depend on the shape and nature of the loads applied


92

throughout the beam axis. In this case all the distributed constant loads are

assumed (Figure 5.3): sh sh T Tq(x)= q, m(x)= m, (x)= , (x)=ε ε ∆ ∆Θ Θ .

However, the imposed displacements are: s01 02 01 02 01 02u , us , v , v , , ϕ ϕ .

Figure 5.3. Beam loaded with external actions.

Let us now deal with the vector of nodal forces by calculating the two

contributions due to the both imposed nodal displacements and distributed

loads. However, the forces are being transformed into the nodal actions of

the perfect clamp according to the following information (Figure 5.1):

01

01

01

02

02

02

(0)(0)

(0)( )( )

( )

V QC MHs SV Q LC M LHs S L

= −⎧⎪ = −⎪⎪ =⎨ =⎪⎪ =⎪

= −⎩

(5.137)


93

5.3.2.1 Nodal forces due to imposed nodal displacements

The nodal forces due to imposed nodal displacements are formally

similar to the terms of the stiffness matrix. In fact, if we substitute the

constraining normalized displacements s1 2 1 2 1 2u , us , v , v , , ϕ ϕ with the

external loads s01 02 01 02 01 02u , us , v , v , , ϕ ϕ , the expressions can be rewritten

as following:

• nodal forces caused by v01

( )( )

( )( )

01 1 2

1

32

( )

2 / 26

2 / 2121 12 1

s o

o

EI EIH vh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.138)

( )( )

01 1

21

32

( )6

2 / 2121 12 1

o

o

C vEIL v


α αα

=

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.139)

( )( )

01 1

31

32

( )12

2 / 2121 12 1

o

o

V vEIL v


α αα

=

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.140)


94

( )( )

( )( )

02 1 2

1

32

( )

2 / 26

2 / 2121 12 1

s o

o

EI EIH vh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.141)

( )( )

02 1

21

32

( )6

2 / 2121 12 1

o

o

C vEIL v


α αα

=

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.142)

( )( )

02 1

31

32

( )12

2 / 2121 12 1

o

o

V vEIL v


α αα

=

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.143)

• nodal forces caused by v02

( )( )

( )( )

01 2 2

2

32

( )

2 / 26

2 / 2121 12 1

s o

o

EI EIH vh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.144)


95

( )( )

01 2

22

32

( )6

2 / 2121 12 1

o

o

C vEIL v


α αα

=

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.145)

( )( )

01 2

32

32

( )12

2 / 2121 12 1

o

o

V vEIL v


α αα

=

= −⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.146)

( )( )

( )( )

02 2 2

2

32

( )

2 / 26

2 / 2121 12 1

s o

o

EI EIH vh L

L tgh LL

vL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.147)

( )( )

02 2

22

32

( )6

2 / 2121 12 1

o

o

C vEIL v


α αα

=

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.148)


96

( )( )

02 2

32

32

( )12

2 / 2121 12 1

o

o

V vEIL v


α αα

=

=⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.149)

• nodal forces caused by φ01

( )( )

( )( )

01 1

1

32

( )

2 / 23

12 / 2121 12 1

s o

o

EI EIHh L

L tgh LL


ϕ

α αα

ϕα α

α

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.150)

( )( )

01 1

1

32

( )

312 / 2121 12 1

o

o

EICL


ϕ

ϕα α

α

=

⎧ ⎫⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.151)

( )( )

01 1 2

1

32

6( )

12 / 2121 12 1

o

o

EIVL


ϕ

ϕα α

α

= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.152)


97

( )( )

( )( )

02 1

1

32

( )

2 / 23

12 / 2121 12 1

s o

o

EI EIHh L

L tgh LL


ϕ

α αα

ϕα α

α

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.153)

( )( )

02 1

1

32

( )

312 / 2121 12 1

o

o

EICL


ϕ

ϕα α

α

= −

⎧ ⎫⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.154)

( )( )

02 1 2

1

32

6( )

12 / 2121 12 1

o

o

EIVL


ϕ

ϕα α

α

=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.155)


98

• nodal forces caused by φ02

( )( )

( )( )

01 2

2

32

( )

2 / 23

12 / 2121 12 1

s o

o

EI EIHh L

L tgh LL


ϕ

α αα

ϕα α

α

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.156)

( )( )

01 2

2

32

( )

312 / 2121 12 1

o

o

EICL


ϕ

ϕα α

α

= −

⎧ ⎫⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.157)

( )( )

01 2 2

2

32

6( )

12 / 2121 12 1

o

o

EIVL


ϕ

ϕα α

α

= −

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.158)


99

( )( )

( )( )

02 2

2

32

( )

2 / 23

12 / 2121 12 1

s o

o

EI EIHh L

L tgh LL


ϕ

α αα

ϕα α

α

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.159)

( )( )

02 2

2

32

( )

312 / 2121 12 1

o

o

EICL


ϕ

ϕα α

α

=

⎧ ⎫⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.160)

( )( )

02 2 2

2

32

6( )

12 / 2121 12 1

o

o

EIVL


ϕ

ϕα α

α

=

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.161)


100

• nodal forces caused by us01

( )( )

( )( )

( )

01 1 12

2 2

3

32

( )

2 / 212 1

2 / 2121 12 1

12 1

s so soEI EIH u uh L



L seEI EIEI EI

α ααα α

α αα

α

−Σ=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( )( ) ( )

( )( )

( )

2

32

2

2

13

2 / 2121 12 1

12 1

121 12 1

nh LL senh L



LEI EIKGA L EI

αα α

α αα

αα

α

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

( )( )3

2 / 2tgh L EIEIL

αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.162)

( )( )

( )( )

01 1

1

32

( )

2 / 23

12 / 2121 12 1

so

so

EI EIC uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.163)


101

( )( )

( )( )

01 1 2

1

32

( )

2 / 26

2 / 2121 12 1

so

so

EI EIV uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.164)

( )( )

( )( )

( )

02 1 12

2 2

3

32

( )

2 / 212 1

2 / 2121 12 1

c12 1




LEI EIEI EI

α ααα α

α αα

α

−Σ= −

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( ) ( )( ) ( )

( )( )

( )

2

32

2

2

os 16

2 / 2121 12 1

12 1

121 12 1

L senh LL senh L



EI EIKGA L EI

α αα α

α αα

αα

⎛ ⎞−−⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

⎛ ⎞ ⎛+ + −⎜ ⎟ Σ⎝ ⎠

( )( )3

2 / 2L tgh L EIEIL

α αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞− ⎪Σ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎪⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.165)


102

( )( )

( )( )

02 1

1

32

( )

2 / 23

12 / 2121 12 1

so

so

EI EIC uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.166)

( )( )

( )( )

02 1 2

1

32

( )

2 / 26

2 / 2121 12 1

so

so

EI EIV uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.167)


103

• nodal forces caused by us02

( )( )

( )( )

( )

01 2 22

2 2

3

32

( )

2 / 212 1

2 / 2121 12 1

c12 1




LEI EIEI EI

α ααα α

α αα

α

−Σ= −

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( ) ( )( ) ( )

( )( )

( )

2

32

2

2

os 16

2 / 2121 12 1

12 1

121 12 1

L senh LL senh L



EI EIKGA L EI

α αα α

α αα

αα

⎛ ⎞−−⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

⎛ ⎞ ⎛+ + −⎜ ⎟ Σ⎝ ⎠

( )( )3

2 / 2L tgh L EIEIL

α αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞− ⎪Σ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟⎪⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.168)

( )( )

( )( )

01 2

2

32

( )

2 / 23

12 / 2121 12 1

so

so

EI EIC uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

⎧ ⎫−⎪ ⎪⎪ ⎪−⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.169)


104

( )( )

( )( )

01 2 2

2

32

( )

2 / 26

2 / 2121 12 1

so

so

EI EIV uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ= −

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.170)

( )( )

( )( )

( )

02 2 22

2 2

3

32

( )

2 / 212 1

2 / 2121 12 1

12 1




L seEI EIEI EI

α ααα α

α αα

α

−Σ=

⎧ ⎛ ⎞−Σ Σ⎛ ⎞ ⎛ ⎞⎪ + −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎝ ⎠ ⎝ ⎠⎪ ⎝ ⎠ +⎨⎡ ⎤⎛ ⎞−⎛ ⎞⎪ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎪ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎪⎣ ⎦⎩

−Σ Σ⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+

i

i

( )( ) ( )

( )( )

( )

2

32

2

2

13

2 / 2121 12 1

12 1

121 12 1

nh LL senh L



LEI EIKGA L EI

αα α

α αα

αα

α

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠ +⎡ ⎤⎛ ⎞−⎛ ⎞ Σ⎛ ⎞ ⎛ ⎞⎢ ⎥+ + − ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞Σ Σ Σ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− +⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎣ ⎦+

−⎛ ⎞ ⎛ ⎞+ + −⎜ ⎟⎜ ⎟ Σ⎝ ⎠⎝ ⎠

( )( )3

2 / 2tgh L EIEIL

αα

⎫⎪⎪⎪⎬

⎡ ⎤⎛ ⎞ ⎪Σ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎪⎜ ⎟ ⎝ ⎠⎢ ⎥⎝ ⎠ ⎪⎣ ⎦ ⎭

(5.171)


105

( )( )

( )( )

02 2

2

32

( )

2 / 23

12 / 2121 12 1

so

so

EI EIC uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

⎧ ⎫−⎪ ⎪⎪ ⎪+⎨ ⎬

⎛ ⎞−⎛ ⎞⎪ ⎪⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟⎪ ⎪⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠⎩ ⎭

i

i (5.172)

( )( )

( )( )

02 2 2

2

32

( )

2 / 26

2 / 2121 12 1

so

so

EI EIV uh L

L tgh LL

uL tgh LEI EI

KGA L EI L

α αα

α αα

−Σ=

−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

i

i (5.173)

After obtaining these expressions, the contribution of nodal imposed

displacements to the vector of equivalent nodal forces can be easily derived

01 1, 2, 1, 2, 1, 2 01 1 01 2 01 1

01 2 01 1 01 2

01 1, 2, 1, 2, 1, 2 01 1 01 2 01 1

01 2 01 1 01 2

01 1, 2, 1, 2, 1,

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

(

o o o o so so o o o

o so so

o o o o so so o o o

o so so

s o o o o so

V v v u u V v V v VV V u V uC v v u u C v C v CC C u C uH v v u

ϕ ϕ ϕϕ

ϕ ϕ ϕϕ

ϕ ϕ

= + + ++ + +

= + + ++ + +

2 01 1 01 2 01 1

01 2 01 1 01 2

02 1, 2, 1, 2, 1, 2 02 1 02 2 02 1

02 2 02 1 02 2

02 1, 2, 1, 2, 1, 2 02 1 02

) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

( ) ( ) (

so s o s o s o

s o s so s so

o o o o so so o o o

o so so

o o o o so so o

u H v H v HH H u H u

V v v u u V v V v VV V u V uC v v u u C v C v

ϕϕ

ϕ ϕ ϕϕ

ϕ ϕ

= + + ++ + +

= + + ++ + +

= + 2 02 1

02 2 02 1 02 2

02 1, 2, 1, 2, 1, 2 02 1 02 2 02 1

02 2 02 1 02 2

) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )

o o

o so so

s o o o o so so s o s o s o

s o s so s so

CC C u C uH v v u u H v H v HH H u H u

ϕϕ

ϕ ϕ ϕϕ

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪

+ +⎪⎪+ + +⎪

= + + +⎪⎪+ + +⎩

(5.174)


106

5.3.2.2 Nodal forces due to distributed loads

As far as the actions caused by both static and kinematic distributed

loads, the differential equation of the deflection (3.27) can be written as

follows:

2 2, ,xxxxxx xxxx

qv vEI

α α− = − (5.175)

The general integral in such equation results as :

( ) ( ) 3 21 2 3 4 5 6

4

( ) cosh

24

v x C senh x C x C x C x C x C

q xEI

α α= + + + + + +

+

(5.176)

The other displacement components can be derived by equations (3.39) and

(3.41):

( ) ( )( ) ( )3 31 2

6( ) cosh

1

ss s

s

C EI EIEIu x C x C senh xk h k h

q x EIk h EI

α α α−ΣΣ

= + − +

Σ⎛ ⎞− −⎜ ⎟⎝ ⎠

(5.177)

( ) ( ) 21 2 3 4 5

33

( ) cosh 3 2

66

x C x C senh x C x C x C

q x q x m EI CEI KGA KGA KGA

ϕ α α α α= − − − − − +

− − + − (5.178)

After obtaining these expressions, the equivalent nodal forces can be

derived by solving the following system of simultaneous equations:


107

( 0) 0( 0) 0( 0) 0

( ) 0( ) 0( ) 0

s

s

v xx

u xv x Lx L

u x L

ϕ

ϕ

= =⎧⎪ = =⎪⎪ = =⎨ = =⎪⎪ = =⎪

= =⎩

(5.179)

whose explicit shape can be written as follows:

( )

( ) ( )

( ) ( )

( ) ( )

2 6

31 5

3 31

3 21 2 3 4 5 6

4

3 21 2 3 4

3

5

3 31

2

06 0

60

0246 3 2

06

6

C CEIC mC CKGA KGA

EI EI CEI Cks h ks h

C senh L C cosh L C L C L C L C

q LEIEIC C cosh L C senh L C L C LKGA

q L q L mCEI KGA KGA

EI C EIcosh L C senh Lks h ks h

α

α

α α

α α α α

α αα α

+ =

− − − + =

−ΣΣ− =

+ + + + + +

+ =

− − − − − +

− − − + =

Σ Σ+ +

−( ) 3

1 0EI EI C q L EIks h ks h EI

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪

−Σ⎪ Σ⎛ ⎞− − =⎜ ⎟⎪ ⎝ ⎠⎩

(5.180)

The relevant expressions of the constants Ci can be evaluated introducing

the following definition:


108

( ) ( )( )

( )12 - 2 / 2

3 212

B EI EI KGA L tgh L

L EI EI KGA L

α α

α

= Σ − +

+ Σ + (5.181)

the constants result expressed as follows:

( )11 24- - 32

q mC EI EI LBEI EIα

⎧ ⎫= Σ +⎨ ⎬

Σ⎩ ⎭ (5.182)

( )( )

( )2

24 / 21 - 32 / 2

mtgh LqC EI EI LBEI EI tgh L

α

α α

⎧ ⎫⎪ ⎪= Σ −⎨ ⎬Σ⎪ ⎪⎩ ⎭

(5.183)

( )3

32 cosh / 2

12EI L m Lq LC

EI Bα αΣ

= − + (5.184)

( )2 3 2

4 2

1 12 12 7224

EI EIL EI L mC qEI KGA EI EI B

αα

⎧ ⎫−Σ⎛ ⎞ Σ⎪ ⎪= − − −⎨ ⎬⎜ ⎟Σ⎪ ⎪⎝ ⎠⎩ ⎭ (5.185)

( )

( )

5 2

2

22 2

242

EI EI qL m qLCEI EI KGA

L EI EI EI EI KGA mKGA B

α

α α

−Σ += + +

Σ

⎡ ⎤Σ + −Σ⎣ ⎦−

(5.186)

( )( )

61 24

32 / 2

q mC EI EI LBEI EI tgh Lα α

⎧ ⎫⎪ ⎪= −Σ − +⎨ ⎬Σ⎪ ⎪⎩ ⎭

(5.187)


109

In this case generalized force expressions result from equations (3.33),

(3.35), (3.38): 2

3 42 2( ) 6 22

T sh

q q x EI EIM x C x EI C EI qKGA EI

EI EIEIh

α α

ε∆

⎛ ⎞= − − − − + +⎜ ⎟Σ⎝ ⎠−Σ

−Θ +

(5.188)

3( ) 6Q x C EI q x m= − − + (5.189)

( ) ( )( )

( ) ( )( )

2

22

1 2 3 4

41 2

1( )

cosh 6 22

cosh

ss T s sh

s

EI k hS x q k h k hEI KGA h

q xk h C sen x C x C x CEI

EI qC sen x C xh EI

εα

α α α

α α α

∆⎧ ⎛ ⎞= + + Θ − +⎨ ⎜ ⎟Σ ⎝ ⎠⎩

⎡ ⎤+ + + + + +⎢ ⎥

⎣ ⎦Σ ⎫⎡ ⎤− + + ⎬⎢ ⎥⎣ ⎦⎭

(5.190)

As a consequence, the expressions of stress on the beam ends are obtained

as follows:

42 2(0) 2

T sh

q EI EIM C EI qKGA EI

EI EIEIh

α α

ε∆

⎛ ⎞= − − + +⎜ ⎟Σ⎝ ⎠−Σ

−Θ + (5.191)

2

3 42 2( ) 6 22

T sh

q q L EI EIM L C L EI C EI qKGA EI

EI EIEIh

α α

ε∆

⎛ ⎞= − − − − + +⎜ ⎟Σ⎝ ⎠−Σ

−Θ +(5.192)


110

3(0) 6 0Q C EI m= − + (5.193)

3( ) 6Q L C EI q L m= − − + (5.194)

2

2 42 4 2

1(0)

2

ss T s sh

s

EI k hS q k h k hEI KGA h

EI qk h C C Ch EI

εα

α α

∆⎧ ⎛ ⎞= + + Θ − +⎨ ⎜ ⎟Σ ⎝ ⎠⎩

Σ ⎫⎡ ⎤⎡ ⎤+ + − + ⎬⎣ ⎦ ⎢ ⎥⎣ ⎦⎭

(5.195)

( ) ( )( )

( ) ( )( )

2

22

1 2 3 4

41 2

1( )

cosh 6 22

cosh

ss T s sh

s

EI k hS L q k h k hEI KGA h

q Lk h C sen L C L C L CEI

EI qC sen L C Lh EI

εα

α α α

α α α

∆⎧ ⎛ ⎞= + + Θ − +⎨ ⎜ ⎟Σ ⎝ ⎠⎩

+ + + + + +

Σ ⎫⎡ ⎤− + + + ⎬⎢ ⎥⎣ ⎦⎭

(5.196)

Replacing the value of the constants evaluated in (5.180) by the former

expressions, we obtain the equivalent nodal forces :


111

( ) ( )( ) ( )

( ) ( ) ( )( )

( )( )

2

2

32

(0)

2 / 21 1 6

12 / 2

2 / 26

2 / 2121 12 1

sh TS EA h EA

L tgh Lq L EIh EI L tgh L

L tgh Lm EI EIKGA L h L


ε

α α

α α

α αα

α αα

∆= − +Θ +

⎛ ⎞−Σ⎛ ⎞+ − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎡ ⎤−

−Σ ⎢ ⎥⎣ ⎦−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.197)

( ) ( )( ) ( )

( ) ( ) ( )( )

( )( )

2

2

32

( )

2 / 21 1 6

12 / 2

2 / 26

2 / 2121 12 1

sh TS L EA h EA




ε

α α

α α

α αα

α αα

∆= − +Θ +

⎛ ⎞−Σ⎛ ⎞+ − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎡ ⎤−

−Σ ⎢ ⎥⎣ ⎦+

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.198)

( )

( )( )

2

32

(0)12

6

2 / 2121 12 1

shT

EI EI q LM EIh

EI mKGA L


ε

α αα

∆−Σ

= −Θ − +

+⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.199)


112

( )

( )( )

2

32

( )12

6

2 / 2121 12 1

shT

EI EI q LM L EIh

EI mKGA L


ε

α αα

∆−Σ

= −Θ − +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.200)

( )( )

2

32

(0)2

12

2 / 2121 12 1

q LQ m

EI mKGA L


α αα

= + +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.201)

( )( )

2

32

( )2

12

2 / 2121 12 1

q LQ L m

EI mKGA L


α αα

= − + +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.202)

It is worth noticing that shear connection stiffness (namely, the αL

parameter) only affects the terms corresponding to the interface force S(0)

and S(L). Moreover, shear stiffness only affects the contribution of the

distributed bending moment m(x) in all the above terms. The general

expressions derived for the terms of the vector of nodal forces can be

specialized to the case of the Newmark model (namely, to the case of

Bernoulli‐behaving beams) by solving the following limits:


113

( ) ( )( ) ( )

2

2

2

lim (0)

2 / 21 1 6

12 / 2

sh TKGA LEI

S EA h EA


ε

α α

α α

∆

→∞

= − +Θ +

⎛ ⎞−Σ⎛ ⎞+ − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(5.203)

( ) ( )( ) ( )

2

2

2

lim ( )

2 / 21 1 6

12 / 2

sh TKGA LEI

S L EA h EA


ε

α α

α α

∆

→∞

= − +Θ +

⎛ ⎞−Σ⎛ ⎞+ − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

(5.204)

( )2

2

lim (0)12

shT

KGA LEI

EI EI q LM EIh

ε∆

→∞

−Σ= −Θ −

(5.205)

( )2

2

lim ( )12

shT

KGA LEI

EI EI q LM L EIh

ε∆

→∞

−Σ= −Θ −

(5.206)

2lim (0)

2KGA LEI

q LQ m→∞

= + (5.207)

2lim ( )

2KGA LEI

q LQ L m→∞

= − + (5.208)

in which the shear stress and the bending moment at the nodes do not

depend on the connection rigidity. Therefore, such results remain identical

also for the connection of infinite rigidity as well as for the case of absent


114

interaction. On the contrary, interface forces depend on both shear

interaction parameter αL and shear stiffness.

2

2

lim lim (0) 112

sh TL KGA L

EI

q L EIS EA h EAh EIα

ε ∆→∞

→∞

Σ⎛ ⎞= − +Θ + −⎜ ⎟⎝ ⎠

(5.209)

2

2

lim lim ( ) 112

sh TL KGA L

EI

q L EIS L EA h EAh EIα

ε ∆→∞

→∞

Σ⎛ ⎞= − +Θ + −⎜ ⎟⎝ ⎠

(5.210)

20lim lim (0) sh TL KGA L

EI

S EA h EAα

ε ∆→

→∞

= − +Θ (5.211)

20lim lim ( ) sh TL KGA L

EI

S L EA h EAα

ε ∆→

→∞

= − +Θ (5.212)

The components of the vector of nodal forces can be evaluated by equation

(5.137) as follows:

( )( )

01

2

32

( , , , )2

12

2 / 2121 12 1

sh Tq LV q m m

EI mKGA L


ε

α αα

∆Θ = − − +

+⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.213)


115

( )

( )( )

2

01

32

( , , , )12

6

2 / 2121 12 1

shsh T T

EI EI q LC q m EIh

EI mKGA L


εε

α αα

∆ ∆−Σ

Θ = − +Θ + +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.214)

( ) ( )( ) ( )

( ) ( ) ( )( )

( )( )

01

2

2

32

( , , , )

2 / 21 1 6

12 / 2

2 / 26

2 / 2121 12 1

s sh T sh TH q m EA h EA




ε ε

α α

α α

α αα

α αα

∆ ∆Θ = − +Θ +

⎛ ⎞−Σ⎛ ⎞+ − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎡ ⎤−

−Σ ⎢ ⎥⎣ ⎦−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.215)

( )( )

02

2

32

( , , , )2

12

2 / 2121 12 1

sh Tq LV q m m

EI mKGA L


ε

α αα

∆Θ = − + +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.216)


116

( )

( )( )

2

02

32

( , , , )12

6

2 / 2121 12 1

shsh T T

EI EI q LC q m EIh

EI mKGA L


εε

α αα

∆ ∆−Σ

Θ = −Θ − +

−⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.217)

( ) ( )( ) ( )

( ) ( ) ( )( )

( )( )

02

2

2

32

( , , , )

2 / 21 1 6

12 / 2

2 / 26

2 / 2121 12 1

s sh T sh TH q m EA h EA




ε ε

α α

α α

α αα

α αα

∆ ∆Θ = −Θ +

⎛ ⎞−Σ⎛ ⎞− − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎡ ⎤−

−Σ ⎢ ⎥⎣ ⎦−

⎛ ⎞−⎛ ⎞ ⎛ ⎞+ + − ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟Σ⎝ ⎠⎝ ⎠ ⎝ ⎠

(5.218)

Finally, the vector of nodal forces can be obtained by summing the two

contributions due to imposed nodal displacements and distributed actions.

01 01 1, 2, 1, 2, 1, 2 01

01 01 1, 2, 1, 2, 1, 2 01

01 01 1, 2, 1, 2, 1, 2 01

02 02 1, 2, 1, 2, 1, 2 02

( ) ( , , , )( ) ( , , , )

( ) ( , , , )( ) ( , ,

o o o o so so sh T

o o o o so so sh T

s s o o o o so so s sh T

o o o o so so

V V v v u u V q mC C v v u u C q mH H v v u u H q mV V v v u u V q m

ϕ ϕ εϕ ϕ εϕ ϕ εϕ ϕ ε

∆

∆

∆

= + Θ= + Θ= + Θ= +

02 02 1, 2, 1, 2, 1, 2 02

02 02 1, 2, 1, 2, 1, 2 02

, )( ) ( , , , )

( ) ( , , , )

sh T

o o o o so so sh T

s s o o o o so so s sh T

C C v v u u C q mH H v v u u H q m

ϕ ϕ εϕ ϕ ε

∆

∆

∆

⎧⎪⎪⎪⎨ Θ⎪⎪ = + Θ⎪

= + Θ⎩

(5.219)

117

6. Applications

The differential equation in terms of transverse deflections v(x) can be

utilized in the present section to present a possible application of the

presented models to the case of simply supported beam and continuous

beam in partial interaction.

6.1 Simply‐supported composite beam

Figure 6.1.Simply supported composite beam.

The first application deals with the beam in Figure 6.1. The functions

expressing displacements and forces are represented as follows :

, , ,2( ) xxxxx xxx xEI EIx v v v

KGA KGAϕ

α= − − (6.1)


118

, ,2 2( )s xxx xxxxxs

EI h EIu x v vEI k hα α

⎛ ⎞= − +⎜ ⎟Σ⎝ ⎠ (6.2)

, ,2 xxxxx xxxEIQ v EIvα

= − (6.3)

, ,2 2xxxx xxEI EI EIM v EIv q

KGA EIα α⎛ ⎞

= − − +⎜ ⎟Σ⎝ ⎠ (6.4)

2

, ,2 1 ss xx xxxx

EI EI q k hS k hv vEI h h KGAα

⎧ ⎫⎛ ⎞Σ⎪ ⎪= − + +⎨ ⎬⎜ ⎟Σ ⎪ ⎪⎝ ⎠⎩ ⎭ (6.5)

The only relevant functions in this case are v(x) and S(x); the boundary

conditions are as follows:

( 0) 0( 0) 0( ) 0( ) 0

v xS xv x LS x L

= =⎧⎪ = =⎪⎨ = =⎪⎪ = =⎩

(6.6)

These conditions result in a unique solution for the differential equation of

the deflection given by (6.4):

22 2

, ,1

xxxx xxMv v qEI EI KGA

αα α⎛ ⎞

− = + +⎜ ⎟Σ⎝ ⎠ (6.7)

which is a particular case of the Newmark equation (3.51) in terms of

deflection v(x) through the (3.43).

The general integral for that equation:

Capitolo VI ‐ Applications

119

( ) ( )1 2 3 4( ) cosh ( )pv x C senh x C x C x C v xα α= + + + + (6.8)

where vp(x) is a particular solution of the complete equation, related to the

moment equation M(x). This one can be obtained from the vertical and

rotational equilibrium conditions:

2( )2 2qL qM x x x= − (6.9)

Consequently, the differential equation in terms of deflection can be

written as follows:

2 2 2 22

, ,1

2 2xxxx xx

x Lv v q xEI EI EI KGA

α α αα⎛ ⎞

− = − + + +⎜ ⎟Σ⎝ ⎠ (6.10)

Based on the expression (6.9) of bending moments the solution of such an

equation can be assumed as follows: 2 2 4 3 2( ) ( )pv x x A x B x C A x B x C x= + + = + + (6.11)

that is, substituting in differential equation:

( ) 24 3 2

2( )24 12 2

pEI EIq q L qv x x x x

EI EI EI EI KGAα

α−Σ⎡ ⎤

= − − +⎢ ⎥Σ⎣ ⎦ (6.12)

hence the integral can be written:

( ) ( )

( )

41 2 3 4

23 2

2

( ) cosh24

12 2

qv x C senh x C x C x C xEI

EI EIq L qx xEI EI EI KGA

α α

αα

= + + + + +

−Σ⎡ ⎤− − +⎢ ⎥Σ⎣ ⎦

(6.13)

and the relevant derivatives can be finally evaluated:


120

( ) ( )

( )

22 2

, 1 2

2

2

cos2 2

xxq x q L xv C senh x C h xEI EI

EI EIqEI EI KGA

α α α α

αα

= + + − +

−Σ⎡ ⎤− +⎢ ⎥Σ⎣ ⎦

(6.14)

( ) ( )4 4, 1 2cosxxxx

qv C senh x C h xEI

α α α α= + + (6.15)

6.1.1 Solutions in terms of displacements

Considering the above expressions of v(x) and the relevant boundary

conditions the following simultaneous equations can be written to define

the constants Ci :


121

( )

( ) ( )

( )

( )

2 4

22

22 2

24

2

4

1 2 3 4

2 2

2

2 21 22

0

1 0

cosh24

02

cos

s

s

s

C C

EI EIEI qk h CEI EI EI KGA

q EI q k hCEI h h KGA

q LC senh L C L C L CEI

EI EIq LEI EI KGA

EI k h C senh L C hEI

ααα α

α

α α

αα

α α αα

+ =

⎧ ⎛ ⎞−Σ⎡ ⎤⎪ − + +⎜ ⎟⎨ ⎢ ⎥⎜ ⎟Σ Σ⎣ ⎦⎪ ⎝ ⎠⎩⎫⎛ ⎞Σ ⎪⎛ ⎞− + + + =⎬⎜ ⎟⎜ ⎟

⎝ ⎠ ⎪⎝ ⎠⎭

+ + + − +

−Σ⎡ ⎤− + =⎢ ⎥Σ⎣ ⎦

+Σ

( )({( )

( ) ( )

2

2

4 41 2

2

cos

1 0s

L

EI EIqEI EI KGA

q EIC senh L C h LEI h

q k hh KGA

α

αα

α α α α

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪

+⎪⎪⎪ ⎞−Σ⎡ ⎤⎪− + +⎟⎢ ⎥ ⎟⎪ Σ⎣ ⎦ ⎠⎪

Σ⎪ ⎛ ⎞− + + +⎜ ⎟⎪ ⎝ ⎠⎪⎪ ⎫⎛ ⎞⎪+ + =⎪ ⎬⎜ ⎟

⎪⎝ ⎠⎪ ⎭⎩

(6.16)

which can be evaluated as follows:

( ) ( )1 4 / 2EI EI

C q tgh LEI EI

αα

−Σ= −

Σ (6.17)

( )2 4

qC EI EIEI EIα

= −ΣΣ

(6.18)


122

( ) ( )2 2

3 2

12 12

24

qL EI EI KGA EI EI KGALC

EI EI KGA

α

α

⎡ ⎤−Σ + Σ +⎣ ⎦=Σ

(6.19)

( )4 4

qC EI EIEI EIα

= − −ΣΣ

(6.20)

Finally, the deflection v(x) can be written as follows:

( ){( )( ) ( )

( )( ) ( ) ( ) ( ) }

44

2 2

2 2 2

( ) 1224

2 24 *

* 12

24 cosh / 2

qv x EI EI L x xEI EI KGA

KGA L x x EI KGA L x x

L Lx x

EI EI KGA x senh x tgh L

αα

α α

α

α α α

⎡= Σ − +⎣Σ

⎤ ⎡+ − + − + Σ + −⎣⎦⎤⎡ ⎤− + + − +⎣ ⎦⎦

+ −Σ −⎡ ⎤⎣ ⎦

(6.21)

It is easy to check that the above general expression basically reduces to the

Newmark’s one (based on Bernoulli theory for which 2KGAL

EI→∞ )and to

the Bernoulli equation as Lα →∞ :

3

1 2 4 30 ,24q LC C C CEI

= = = = (6.22)

The analytical expression of the midspan deflection is provided as follows:

( )( ){ [

( ) ( ) ( ) ( )

4

24 2

2 4

( / 2) *384

* 48 8 * 384

148 5 384cosh / 2

qv LEI EI KGA

EI KGA L EI L KGA EI KGA

L L EI EI KGAL

α

α α

α αα

=Σ

− + Σ + +Σ +

⎫⎪⎤− + + −Σ ⎬⎦ ⎪⎭

(6.23)


123

Obviously, due to the symmetry of the beam, the deflection value is the

maximum throughout the beam axis. Once again, to check what has been

shown, in the Timoshenko case we obtain:

4 2

4 2

5( / 2)384 8

5( / 2) 0384 8

q L q Lv L se LEI KGAq L q Lv L se LEI KGA

α

α

= + →∞

= + →Σ

(6.24)

while in the Bernoulli’s one:

4

4

5( / 2)384

5( / 2) 0384

q Lv L se LEIq Lv L se LEI

α

α

= →∞

= →Σ

(6.25)

The above limit values will be considered as reference to compare the

general results obtained by equation (6.23).

6.1.2 Comparisons between Timoshenko and Bernoulli model

In order to evaluate the influence of the shear effects (other than the

rigidity of the connection ) of deflection, case‐study in table 6.1 has been

assumed denoting with 1 the properties of concrete and 2 those of steel.


124

E1 30000 N/mm2 E2 210000 N/mm2 ν 0,2 ‐ ν 0,2 ‐ G1 12500 N/mm2 G2 87500 N/mm2 A1 150000 mm2

A2 (IPE 600) 15600 mm2 I1 281250000 mm4

I2 (IPE 600) 920800000 mm4 h 375 mm L 1000 mm ΣEI 2,01806E+14 N*mm2 EA 7776000000 N EI 1,29531E+15 N*mm2 K1 0,833 ‐ K2 0,464 ‐ KGA 2196000000 N q 50 N/mm

Table 6.1. Geometric and mechanical properties.

The shear factor of the slab has been assumed equal to 5/6 (rectangular

cross‐section) and the shear factor of steel beam web has been taken as the

web‐to‐total area ratio. The values of the stiffness ks related to seven values

of αL to be assumed for the analysis, as reported in Table 6.2.


125

αL 0 α 0 mm‐1 ks 0 N/mm2 αL 1 α 0,0001 mm‐1 ks 8 N/mm2 αL 10 α 0,001 mm‐1 ks 817 N/mm2 αL 20 α 0,002 mm‐1 ks 3267 N/mm2 αL 30 α 0,003 mm‐1 ks 7351 N/mm2 αL 50 α 0,005 mm‐1 ks 20420 N/mm2 αL 70 α 0,007 mm‐1 ks 40023 N/mm2

Table 6.2. Connection rigidity ks.

The seven values of αL, the deflection plots with a finite KGA (equal to the

one of the beam under consideration) and infinite KGA are reported in

order to compare their influence on the shear deformability and on the

total deflection and to understand the values of the deflection varying the

connection stiffness.

The first diagram in Figure 6.2, shows the various deflection which can be

qualitatively compared:


126

• infinite KGA (2KGAL

EI→∞ ),

• finite KGA (2KGAL

EI∈ ),

the latest ones, with equal αL factor, are a bit bigger as the beams are

affected by the slip.

The Bernoulli deflections are represented by the continuous line, while the

Timoshenko’s one by the discontinuous one (Figure 6.2).

In the Figures 6.3÷6.9 the deflections are compared for various values of

the αL parameter ranging from 0 to 70.


127

Figure 6.2. Deflection of the beam in the cases examinated.


128

Figure 6.3. (αL)=0.


129



130



131



132



133



134



135

Table 6.3 collects the values of the maximum deflection at midspan

reported for each of the values of αL evaluated for both shear‐flexible or

infinitely stiff section.

v max v max Δv/v mm mm %

KGA finito KGA infinito

aL=0 aL=0 32,5 32,2 0,87

aL=1 aL=1 30,0 29,7 0,95

aL=10 aL=10 7,71 7,4 3,69

aL=20 aL=20 5,9 5,6 4,78

aL=30 aL=30 5,5 5,3 5,08

aL=50 aL=50 5,4 5,1 5,26

aL=70 aL=70 5,3 5,0 5,31

Table 6.3. Values of the maximum deflection.


136

It can be seen considering that the Δv/v ratio between the Timoshenko

and Bernoulli deflection is as large as shear connection is stiff.

6.1.3 Solution by matrix method

An alternative approach for solving simply‐supported beams can be

followed by using the stiffness matrix formulated in the present work. The

beam can be subdivided in “elements” and the deflection can be developed

through interpolation with a larger precision increasing the number of

finite elements (ten elements have been considered herein).

The following cases have the same connection rigidity ks of the previous

ones. Deflection derived for the cases of Timoshenko and Bernoulli beams

are represented in Figure 6.10÷6.23 for different values of αL.


137

Figure 6.10. (αL)=0 (Timoshenko).


138



139



140



141



142



143



144

Figure 6.17. (αL)=0 (Bernoulli).


145



146



147



148



149



150



151

It can be seen that there is a good convergence among the results obtained

by an analytical approach and those derived by the matrix one.

6.2 Continuous composite beam

Figure 6.24. Continuous composite beam with two equal spans.

The case‐study considered in the following is represented in Figure

6.24. Symmetry will be considered in the analyses of the beam which will

be obtained by both analytical and matrix approach.

6.2.1 Analytical solution

The differential equation (5.175) has to be solved with respect to the

constants C1÷C6 by using the displacement function and the relevant

boundary conditions. General integral of equation (5.176) is:


152

( ) ( ) 3 21 2 3 4 5 6

4

( ) cosh

24

v x C senh x C x C x C x C x C

q xEI

α α= + + + + + +

+

(5.176)

and boundary conditions:

( 0) 0( 0) 0( 0) 0

( ) 0( ) 0( ) 0s

v xS xM xv x Lx L

u x Lϕ

= =⎧⎪ = =⎪⎪ = =⎨ = =⎪⎪ = =⎪

= =⎩

(6.26)

or:


153

( ) ( )

( )

( )

( ) ( )

( )

2 6

2 42 4 2 2

2

4 22 2 42

2

3 21 2 3 4 5 6

4

31

0

2

0

2

0

0246

C C

EI EI q EIC C Ch EI h

EI EIEIqEI h h KGA

EI qC EI C CEI

EI EIqKGA EI

C senh L C cosh L C L C L C L C

q LEIEIC C cosh L CKGA

α αα

α

α αα

α

α α

α α α

+ =

−Σ⎧ ⎛ ⎞+ − + +⎨ ⎜ ⎟⎝ ⎠⎩⎫−Σ⎛ ⎞⎪+ + =⎬⎜ ⎟Σ ⎪⎝ ⎠⎭

⎛ ⎞+ − + +⎜ ⎟⎝ ⎠

⎛ ⎞− + =⎜ ⎟Σ⎝ ⎠

+ + + + + +

+ =

− − − ( )

( ) ( )

( )

22 3 4

3

5

3 31

2

3

3 2

06

61 0

senh L C L C L

q L q LCEI KGA

EI C EIcosh L C senh Lks h ks hEI EI C q L EIks h ks h EI

α

α αα α

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪ − − +⎪⎪⎪− − − =⎪⎪Σ Σ

+ +⎪⎪⎪ −Σ Σ⎛ ⎞⎪− − − =⎜ ⎟⎪ ⎝ ⎠⎩

(6.27)

The unknown constants can be determined by solving the above

simultaneous equations and the definition of the parameter P can be

introduced:

{( ) ( )( ) }

2 2

( ) *

* 3 3 cosh( )

3 ( )

P L EI

EI EI KGA EI EI KGA L L

EI EI EI KGA senh L

α

α α

α

=

⎡ ⎤−Σ + Σ + +⎣ ⎦− −Σ

(6.28)


154

( )( ) ( )( )( ) ( )

( ) ( )( )

1 4

2 2

4

2 2

1 *8

* 24

12 *

5 24 cosh( )

8( ) 3 3 ( )

q EICP EI

EI EI KGA

L EI EI EI EI KGA

L EI KGA EI EI KGA L

L EI EI KGA EI EI KGA L senh L

α

α α

α α

α α α

⎛ ⎞= −⎜ ⎟Σ⎝ ⎠

− −Σ +⎡⎣

+ Σ + −Σ +

+ Σ + −Σ +

⎤− −Σ + Σ + ⎦

(6.29)

2 4 1q EICEI EIα

⎛ ⎞= −⎜ ⎟Σ⎝ ⎠ (6.30)

( ) ( )(( )( ) ( ) )

( ) ( )

23

42

* 8 4 *16

* cosh( )

8 1 ( ) ( )

qC EI EI KGA LP

EI EI EI EI KGA L EI KGA L

EI EI KGA L senh L

αα

α α α

α α

⎡= − −Σ +⎣

Σ + −Σ + Σ +

+ −Σ + ⎤⎦

(6.31)

4 2

1 1 12 2

EIC qKGA EI EIα

⎡ ⎤⎛ ⎞= − + −⎜ ⎟⎢ ⎥Σ⎝ ⎠⎣ ⎦ (6.32)


155

( ) ( ){( ) ( )( )( )

( ) ( )( )

( )

5 3

2 2 2

2 2 2

2 2 4 2

2 2 2 4

2

2

*48

* 144 72 *

* 2 6( ) *

24 5 ( ) *

* 72 30 cosh( )

24 6 3 *

* 2

2( ) 3

qCEI KGA P

EI EI KGA EI EI EI KGA

EI KGAL L EI EI EI KGA

EI KGA L L EI

EI EI KGA L KGA L L

EI EI KGA EI EI KGA EI

EI KGAL

L EI EI KGA

α

α

α α

α α

α

α

α

=Σ

⎡ −Σ + −Σ⎣

Σ + + −Σ Σ

+ + Σ

⎤+ + +⎦⎡− −Σ −Σ + Σ⎣

+ +

+ −Σ ( )( ) }2 23 ( )EI EI KGAL senh Lα α ⎤+ Σ + ⎦

(6.33)

6 4 1q EICEI EIα

⎛ ⎞= − −⎜ ⎟Σ⎝ ⎠ (6.34)

Once more, the general model reduces to the Newmark’s one as

2KGALEI

→∞ and a further reduction to the case of complete interaction

can be obtained as Lα →∞ :

3

1 2 4 6 3 50 , ,16 48q L q LC C C C C CEI EI

= = = = = − = (6.35)

whose solution is in good agreement with the one in terms of deflection for

a monolithic beam with EI stiffness and infinite shear stiffness.

By introduction the constants in equation (5.176) the deflection is obtained:


156

( ) ( )({( ) ( )(

( ) (( )( ) ( ))

( ) ( )(( )))

4

2 2 4

2 2 6 2 2

2 2 2 2

2

2 2

( ) *48

* 48 cosh( ) cosh( ) *

* 144 6 *

* 24 5 72

2 6 5

24 6 3

2 2

qv xEI KGA P

EI EI x L L x

EI EI KGA EI EI EI KGA L x

EI KGA L L x x EI L x EI

KGA L L x L x EI KGA L L x x

EI EI KGA EI EI KGA L x

EI KGA L L x x

α

α α α

α

α

α

=Σ

⎡−Σ + −⎣

− −Σ + −Σ Σ

⎡ ⎤+ + − + Σ +⎣ ⎦

+ − + + + − +

− −Σ Σ − +

+Σ + − + ( )

) ( ) ( )((( )) (

( ) ( )( ))

( ) ( )( )(( )

2 2

2 2 2

2 2 2

4 4

2 4 3 3 4

4

144 *

* ( ) 6 4 6

* 6 3 24*

* 12

5 * ( ) ( )*

* 24 12 2 8 4

12 2

EI EI KGA

senh x EI EI KGA x EI EI KGA

EI EI KGA L KGA x

EI EI KGA EI EI EI EI KGA L

EI KGA L senh x senh L

EI KGA x L x L x Lx x

EI EI x L x K

α α

α

α α

α α α

α α

α

−Σ

+ −Σ − −Σ +

+ Σ − − + + −

−Σ + Σ + −Σ +

+ Σ +

−Σ + − + + − + +

+ Σ − + + ( )( )( )( ) ( )( ) )) }

2

2 2

2 2

8 3 3 ( )

GA x L x

L EI EI KGA EI EI KGA L senh x

α

α α α

+ − + +

⎤− −Σ + Σ + ⎥⎦

(6.36)

The analytical expression of deflection in the midspan (that is not the

maximum in the considered scheme) is as follows:


157

( )

( )(

( )

( )

4

2 2 6 2 2 2

6 2 4 6 2 2 6

24 2

2 2

( / 2) *48

* 48 cosh( / 2) cosh( ) *2

* 144 36

63 14 2

213 244

324 62

11

qv LEI KGA P

LEI EI L L

EI EI KGA EI EI L

EI EI KGA L EI KGA L

KGA LEI EI EI KGA L EI

EI EI KGA EI EI EI KGA L

α

αα α

α

α α

α

α

=Σ

⎧ ⎡ ⎛−Σ +⎨ ⎜⎢ ⎝⎣⎩

− −Σ + Σ +

+ Σ + Σ +

⎛ ⎞+ −Σ Σ + +⎜ ⎟

⎝ ⎠⎛− −Σ Σ − +⎜⎝

+ ( ) )

( ) ( ( )

( ) ( ) ( )(

( )( ) )( )(

( ) ( ) ( )

22 2

3

3

2 2 2 4 4

4 2

2 4

144 ( / 2)4

6 12 12 *

11* 242

12 5 *

* ( / 2) 24 9

57916

EI KGA L EI EI KGA senh L

EI EI KGA EI EI L L KGA

EI EI EI KGA L EI EI KGA

EI EI EI EI KGA L EI KGA L

senh L KGA EI EI EI EI L

L KGA EI EI L

α

α α

α

α α α

α α

α α

⎞⎞Σ + −Σ +⎟⎟⎠⎠

+ −Σ − Σ −

−Σ − Σ + − −Σ +

+ Σ + −Σ + Σ

+ −Σ − Σ +

− −Σ −

( )(

( )) ) ) }2 2

8 3

3 ( / 2) ( )

EI KGA

L EI EI KGA

EI EI KGA L senh L senh L

α

α α α

Σ +

− −Σ +

⎤+ Σ + ⎥⎦

(6.37)


158

6.2.2 Comparison between Timoshenko and Bernoulli model

In order to evaluate the influence of the shear effects (other than the

rigidity of the connection ) on the deflection, the characteristics of the beam

in Table 6.4 have been assumed (denoting with 1 the characteristics of the

concrete and 2 of the steel).


A2 (IPE 600) 15600 mm2 I1 281250000 mm4


Table 6.4. Geometric and mechanical properties.

The shear factor of the slab is assumed equal to 5/6 (rectangular cross

section) and equal to the area of the web to the total area of the section ratio

(T‐shape section) for the steel beam.


159


A2 (IPE 600) 15600 mm2 I1 281250000 mm4


Table 6.5. Connection stiffness assumed in the case‐study.

In order to evaluate the influence of the shear flexibility on deflection, the

values of the deflections are reported against parameter αL for both the

cases of finite KGA and for infinite KGA.

The first diagram in Figure 6.25 shows deflections values throughout the

beam axis.

Table 6.5 reports the values of both stiffness ks and interaction parameter

αL adopted in the following analysis.

Continuous line refers to the case of infinite shear stiffness, while the

discontinuous one is related to the case of finite value of KGA (Figure 6.25).


160

In the following Figure 6.26÷6.32, the deflections obtained for different

cases of αL ranging from 0 to 70 are reported comparing the case of finite

and infinite shear stiffness. Only the solution of one span has been

represented for the sake of symmetry.


161

Figure 6.25. Deflection of the beam in the cases examinated.


162



163



164

Figure 6.28. (αL)=10.


165

Figure 6.29. (αL)=20.


166

Figure 6.30. (αL)=30.


167

Figure 6.31. (αL)=50.


168

Figure 6.32. (αL)=70.


169

Table 6.6 reports the maximum deflection values and the relative difference

derived by assuming either finite or infinite shear stiffness of the cross

beam.

v max v max Δv/v cm cm %


aL=0 aL=0 13,7 13,4 2,37

aL=1 aL=1 13,2 12,9 2,45

aL=10 aL=10 4,5 4,2 7,08

aL=20 aL=20 3,1 2,7 10,55

aL=30 aL=30 2,7 2,4 11,93

aL=50 aL=50 2,5 2,2 12,85

aL=70 aL=70 2,5 2,1 13,32

Table 6.6. Values of maximum deflection.


170

It can be seen that when the connection stiffness increases the difference of

the deflections between the Bernoulli and Timoshenko model increases.

6.2.3 Solution by matrix method

An alternative approach for solving continuous beam can be followed

by using the stiffness matrix formulated in the present work. The beam can

be subdivided in elements and the deflection can be developed through

interpolation with a larger precision increasing the number of finite

elements (ten elements have been considered herein for every span).

The following cases have the same connection rigidity ks of the previous

ones. Deflection derived for the cases of Timoshenko and Bernoulli beams

are represented in Figure 6.33÷6.46 for different values of αL.


171



172



173



174



175



176



177



178



179



180



181



182



183



184



185

6.2.4 Solutions in terms of forces

Equations (6.29)÷(6.34) relating the integration constants Ci to the

nodal force and displacements can be now utilized for post‐processing the

solution obtained in terms of nodal displacements.

Once the constants C1,C2,C3,C4,C5 and C6 are known , it is possible to rich

the functions of interest. In particular, we pay the attention on the bending

moment in order to study its variation between the case of finite KGA and

infinite KGA .Bending moments can be expressed as follows:

, ,2 2xxxx xxEI EI EIM v EIv q

KGA EIα α⎛ ⎞

= − − +⎜ ⎟Σ⎝ ⎠ (6.38)

that is, in terms of constants of integration :

( ) ( )

( ) ( )

4 41 22

22 2

1 2 3 4

2

cosh

cosh 6 22

EI qM C senh x C xEI

q xEI C senh x C x C x CEI

EI EIqKGA EI

α α α αα

α α α α

α

⎡ ⎤= + + +⎢ ⎥⎣ ⎦⎡ ⎤

− + + + + +⎢ ⎥⎣ ⎦

⎛ ⎞− +⎜ ⎟Σ⎝ ⎠

(6.39)

Simplifying the above equations:

2

3 42 1 6 22

q EI q x EIM EIC x EIC qEI KGAα

⎛ ⎞= − − − − − −⎜ ⎟Σ⎝ ⎠

(6.40)

and substituting the value of the constants, we have:


186

( )( )

( )(

( )( ) ( ) )( ) ( ) ( ) ( )( )

2

2

2

2

42 2 2

12

3 88

4 *

*cosh 8 1

qx q EI q EIMEI KGA

q EI EI EI EI KGAEI KGA

xq EI EI EI KGAPEI EI EI EI KGA L L EI KGA

L EI EI KGA L senh L

α

α

α

α

α α α

α α α

⎛ ⎞= − − − − +⎜ ⎟Σ⎝ ⎠

Σ + −Σ+ +

Σ

⎡− − −Σ +⎣

− Σ + −Σ − Σ

⎤+ −Σ + ⎦

(6.41)

A final check about the limit behaviour of the solution can be carried out in

terms of bending moments M(x), which directly reduces to the case of

monolithic Bernoulli beam as 2KGAL

EI→∞ and Lα →∞ :

2

38 2q L q xM x= − (6.42)

6.2.5 Comparisons between Timoshenko model and Bernoulli

model

In order to evaluate the influence of the shear flexibility on bending

moment, the values of the bending moments are reported against

parameter αL for both the cases of finite KGA and for infinite KGA.

The first diagram in Figure 6.47 shows bending moment values throughout

the beam axis; continuous line refers to the case of infinite shear stiffness,

while the discontinuous one is related to the case of finite value of KGA.


187

In the following Figure 6.48÷6.54, the bending moments obtained for

different cases of αL ranging from 0 to 70 are reported comparing the case

of finite and infinite shear stiffness. Only the solution of one span has been

represented for the sake of symmetry.

In the following graphs, the bending moments are expressed in KNm.


188

Figure 6.47. Bending moment.


189



190



191

Figure 6.50. (αL)=10.


192

Figure 6.51. (αL)=20.


193

Figure 6.52. (αL)=30.


194

Figure 6.53. (αL)=50.


195

Figure 6.54. (αL)=70.


196

Also the minimum value of the bending moment for finite KGA and

infinite KGA with fixed the seven values of αL are reported.

Mmin Mmin ΔM/M KNm KNm %


aL=0 aL=0 623 625 0,28

aL=1 aL=1 606 609 0,36

aL=10 aL=10 564 573 1,52

aL=20 aL=20 596 606 1,68

aL=30 aL=30 605 615 1,71

aL=50 aL=50 610 621 1,73

aL=70 aL=70 612 623 1,73

Table 6.7. Values of the minimum bending moment.


197

Figure 6.55.

Increasing the rigidity of the connection the percentage difference between

the moment under Bernoulli hypothesis and the one under Timoshenko

hypothesis increases in turn (Figure 6.55); but this variation is less

pronounced compared with the one of the deflections. It means that the

static regime, compared with the deformative one, is less affected by the

difference between the two models.

Moreover, in the case examinated, (αL)=70 can be already considered

coinciding with the case of infinite connection even as regards the bending

moment, as the percentage variation becomes constant from (αL)=70.

It can be seen in the table 6.7.

198

7. Conclusions

• Formulation of an analytical model for shear flexible composite

beams in partial interaction, representing a generalization of the

well‐known Newmark model;

• derivation of the stiffness matrix and the vector of equivalent nodal

forces in closed‐form;

• comparison between the stiffness matrix derived for shear flexible

beams with the ones derived by the Bernoulli‐based Newmark

model; the key parameters influencing the difference between the

two models have been also emphasized through various case‐

studies;

• application of both the analytical model and the matrix‐approach

for solving two relevant case‐studies.

The following issues are among the possible future development of the

proposed model:

• numerical implementation to reproduce the non‐linear behaviour of

materials of shear connection;

• further generalization of the analytical model for the case of

completely free rotations for the two connected parts.

199

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[27]. Nunzio Scibilia “Strutture miste acciaio‐calcestruzzo legno‐calcestruzzo” Dario

Flaccovio Editore (2001).

[28]. Ciro Faella “Metodi di analisi delle strutture intelaiate” Cues Editore (2002).

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Engineering Thesis Giuseppe Di Palma

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