computational methods in reinforced concrete

Computational Engineering for Reinforced ConcreteStructures

U. Huler-Combemailto:[email protected]

Institute of Concrete Structureshttp://www.tu-dresden.de/biwitb/mbau

Technische Universitt Dresdenhttp://www.tu-dresden.de

PRELIMINARY DRAFT

State

April 4, 2013

IIntroductory Remarks

The following notes1 serve as accompanying study material for the lecture ComputationalEngineering for Reinforced Concrete Structures of the Institute of Concrete Structures,Technische Universitt Dresden.

The notes are not yet finished. Some existing chapters deserve a completion. More chap-ters not yet included are under work. Thus, a Preliminary Draft is given.

The actual state has been worked out according to best knowledge. Nevertheless, it maycontain formal errors or wrong facts. Corresponding hints are highly appreciated by theauthor ( mailto:[email protected]).

1 2013( Ulrich Huler-Combe. All rights preserved.

mailto:[email protected] State April 4, 2013

Contents

1 Finite Elements in a Nutshell 11.1 Modeling Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Finite Element Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Material Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Weak Equilibrium and Discretization . . . . . . . . . . . . . . . . . . . . . . . 71.6 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7 Numerical Integration and Solution Methods for Algebraic Systems . . . . . . 14

2 Uniaxial Structural Concrete Behavior 202.1 Short Term Stress-Strain Behavior of Concrete . . . . . . . . . . . . . . . . . 202.2 Long Term Effects - Creep, Shrinkage and Temperature . . . . . . . . . . . . . 262.3 Strain-Rate Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.5 Reinforcing Steel Stress-Strain Behavior . . . . . . . . . . . . . . . . . . . . . 332.6 Bond between Concrete and Reinforcing Steel . . . . . . . . . . . . . . . . . . 342.7 Reinforced Tension Bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.8 Tension Stiffening for Reinforced Tension Bar . . . . . . . . . . . . . . . . . . 382.9 Cyclic Loading of Reinforced Tension Bar . . . . . . . . . . . . . . . . . . . . 40

3 2D Structural Beams and Frames 423.1 General Cross Sectional Behavior . . . . . . . . . . . . . . . . . . . . . . . . 42

3.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.1.2 Linear elastic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 443.1.3 Cracked reinforced concrete behavior . . . . . . . . . . . . . . . . . . 45

3.1.3.1 Compressive zone and internal forces . . . . . . . . . . . . . 453.1.3.2 Linear concrete compressive behavior . . . . . . . . . . . . 473.1.3.3 Nonlinear concrete compressive behavior . . . . . . . . . . . 49

3.2 Equilibrium of Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Structural Beam Elements for 2D . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 System Building and Solution Methods . . . . . . . . . . . . . . . . . . . . . 603.5 Further Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.1 Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.2 Temperature and Shrinkage . . . . . . . . . . . . . . . . . . . . . . . 693.5.3 Tension Stiffening . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.5.4 Prestressing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763.5.5 Shear stiffness for reinforced cracked concrete sections . . . . . . . . . 81

3.6 Application Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6.1 Transient Dynamics of Beams . . . . . . . . . . . . . . . . . . . . . . 833.6.2 More Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

II

CONTENTS III

4 Strut-and-Tie Models 884.1 Linear Elastic Panel Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 884.2 Truss Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3 Computation of Plane Elasto-Plastic Truss Models . . . . . . . . . . . . . . . 904.4 Ideal Plastic Truss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 964.5 Application Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 Multiaxial Concrete Material Behavior 1045.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.2 Some Basics of Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . 1055.3 Basic Linear Material Behavior Description . . . . . . . . . . . . . . . . . . . 1105.4 Basics of Nonlinear Material Behavior . . . . . . . . . . . . . . . . . . . . . . 114

5.4.1 Tangential Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.2 Stress Limit States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4.3 Phenomenological Approach for Biaxial Anisotropic Stress-Strain Re-

lation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1195.5 Isotropic Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.6 Isotropic Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.7 Microplane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1305.8 Localization and Regularization . . . . . . . . . . . . . . . . . . . . . . . . . 1305.9 Long Term Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.10 Short Term Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6 Deep Beams 1336.1 Limit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1336.2 2D Crack Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1406.3 2D Modeling of Reinforcement and Bond . . . . . . . . . . . . . . . . . . . . 1456.4 Biaxial Concrete Material Laws . . . . . . . . . . . . . . . . . . . . . . . . . 1516.5 Further Aspects and Application Case Studies . . . . . . . . . . . . . . . . . . 152

7 Slabs 1547.1 Cross Sectional Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

7.1.1 Kinematic Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1547.1.2 Linear elastic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 1567.1.3 Reinforced cracked sections . . . . . . . . . . . . . . . . . . . . . . . 157

7.2 Equilibrium of slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.3 Structural Slab Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

7.3.1 Area coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1627.3.2 A triangular Kirchhoff element . . . . . . . . . . . . . . . . . . . . . . 162

7.4 System Building and Solution Methods . . . . . . . . . . . . . . . . . . . . . 1647.5 Reinforcement Design with linear elastic internal forces . . . . . . . . . . . . . 168

8 Shells 1778.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1778.2 Approximation of Geometry and Displacements . . . . . . . . . . . . . . . . . 1778.3 Approximation of Deformations . . . . . . . . . . . . . . . . . . . . . . . . . 1798.4 Shell Stresses and Material Laws . . . . . . . . . . . . . . . . . . . . . . . . . 1818.5 System Building . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1838.6 Slabs and Beams as a Special Case . . . . . . . . . . . . . . . . . . . . . . . . 1858.7 Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1868.8 Reinforced Concrete Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189


IV CONTENTS

A ConFem 196

B Transformations of coordinate systems 197

C Linear regression analysis applications 199C.0.1 Determination of moments in triangular slab elements . . . . . . . . . 199

D Numerical Integration of Elastoplastic Material Laws 201

E ACCESS lectures in summer term 2013 204

State April 4, 2013 mailto:[email protected]

Chapter 1

Finite Elements in a Nutshell

1.1 Modeling Basics

See Fig. 1.1

Figure 1.1: Modeling Basics according to [CF08, Fig. 7.31]

Reality of interest

Conceptual model

Computational model

1.2 Finite Element Basics

Subdivision of area of interest into smaller units

elements

nodes

Interpolation of displacement fields elementwise

Trial functions displacement compatibility

Integral equilibrium at nodes

1

2 1.3 Elements

Figure 1.2: FE basic idea

1.3 Elements

Basic concepts

Coordinates

* global x = ( x y z )T

* local r = ( r s t )T

* element number I and local coordinate r in element

Displacement variables

* translation u = ( u v w )T

* rotation = ( x y z )T

Deformation variables

* 1D strain (small strain)

=u

x(1.1)

* 2D strain (small strains Voigt notation)

=(x y xy

)T=

(u

x

v

y

u

y+v

x

)T(1.2)

* Curvature (small deflections, 2D)

=2w

x2=yx

(1.3)

Generalized force variables, e.g.

* with Cauchy stress tensor where dw = d gives a mechanical workincrement per unit volume in a body.

* M with bending moment M where dw = M d gives a mechanical workincrement per unit length of a 2D bending bar.

Interpolation

* Coordinatesx = N(I, r) xI (1.4)

with vector xI collecting global coordinates of nodes of element I and rowvector N collecting shape functions.


3* Displacementsu = N(I, r) uI (1.5)

with vector uI collecting global displacements of nodes of element I and rowvector N collecting shape functions.

* Deformations, e.g. small 2D engineering strains

= B(I, r) uI (1.6)with vector uI collecting global displacements of nodes of element I and matrixB collecting derivatives of shape functions with respect to global coordinates.

Index I within (I, r) will be omitted in the following, as the functions N,B are thesame for all elements of one (isoparametric) element type.

Bar elements

Kinematic assumption

* cross section displacements constant in longitudinal direction Interpolation functions linear 1D, 2 nodes, 2 degrees of freedom

x(r) =[

12(1 r) 12(1 + r)

] ( xI1xI2

)u(r) =

[12(1 r) 12(1 + r)

] ( uI1uI2

) = dudx =

[ 12 12 ] drdx ( uI1uI2)

=[ 12 12 ] 2LI

(uI1uI2

) (1.7)

with a bar length LI = xI2 xI1 and a Jacobian

J =x

r=LI2

(1.8)

Interpolation functions linear 2D, 2 nodes, 4 degrees of freedom

(x(r)y(r)

)=

[12(1 r) 0 12(1 + r) 0

0 12(1 r) 0 12(1 + r)]

xI1yI1xI2yI2

(u(r)v(r)

)=

[12(1 r) 0 12(1 + r) 0

0 12(1 r) 0 12(1 + r)]

uI1vI1uI2vI2

(1.9)

Small strain is taken in the bar direction, i.e. in a rotated coordinate system. Therotation angle (counterclockwise positive) and the transformation matrix are givenby

cos =xI2 xI1

LI, sin =

yI2 yI1LI

, T =

[cos sin sin cos

](1.10)

with a bar length LI =

(yI2 yI1)2 + (xI2 xI1)2. Reusing Eq. (1.7)3 we get

=2

LI

[ 12 12 ] [ cos sin 0 00 0 cos sin]

uI1vI1uI2vI2

(1.11)


4 1.3 Elements

Spring elements

Kinematic assumption

* Displacement difference between two nodes irrespective of their distance. Thetwo nodes share the same location as special but common case.

"Interpolation" function 1D

u =[ 1 1 ] ( uI1

uI2

)(1.12)

"Interpolation" function 2D

(uv

)=

[ 1 1 0 00 0 1 1

]

uI1vI1uI2vI2

(1.13) 2D continuum elements (e.g. plate with loading in plane panel, deep beam)

Kinematic assumptions

* Displacements u(x, y), v(x, y) are continuous functions.

* Lateral displacement w is w = 0 ( plane strain) or determined from a condi-tion z = 0 ( plane stress)

Interpolation functions 4-node quadrilateral

x(r, s) =4

J=1

NJ xIJ , y(r, s) =

4J=1

NJ yIJ , u(r, s) =

4J=1

NJ uIJ , v(r, s) =

4J=1

NJ vIJ

(1.14)with

NJ(r, s) =1

4(1 + rJr)(1 + sJs) (1.15)

with the local node coordinates rJ , sJ . For the following we need the Jacobian

J =

[xr

yr

xs

ys

], J = det J (1.16)

which is easily derived from Eq. (1.14)1,2 and relates the partial derivatives of afunction with respect to local and global coordinates1(

rs

)= J

( xy

)

( xy

)= J1

(rs

)(1.17)

Small engineering strains with Eqns. (1.2), (1.14), (1.17)2 x(r, s)y(r, s)xy(r, s)

= ur rx + us sxvr ry + vs sy

ur

ry +

us

sy +

vr

rx +

vs

sx

=

4J=1

14

[rJ(1 + sJs)

rx + (1 + rJr)sJ

sx

]uIJ4

J=114

[rJ(1 + sJs)

ry + (1 + rJr)sJ

sy

]vIJ4

J=114

{[rJ(1 + sJs)

ry + (1 + rJr)sJ

sy

]uIJ

+[rJ(1 + sJs)

rx + (1 + rJr)sJ

sx

]vIJ

}

(1.18)

which after evaluation leads to a form like Eq. (1.6).1Closed forms of J and J1 are available for the 4-node quadrilateral, see e.g. [???].


5 2D beam elements

Displacement variables

* Transverse displacement / deflection w of reference axis rotation angle of reference axis

* Rotation angle of cross section

Kinematic assumptions

* Plane cross sections remain plane

* Bernoulli beam: rotation of reference axis = rotation of cross section

* Timoshenko beam: Rotation of cross section decouples from rotation of refer-ence axis and shear angle serves as independent variable

Interpolation functions Bernoulli beam, see Page 56. Interpolation functions Timoshenko beam, see Page 58.

Plate elements (loading perpendicular to plane slab) Displacement variables Kinematic assumptions Kirchhoff plate (thin plate) small displacements, mid-surface without strains, ro-

tation of cross section can be derived from deflection of mid-surface

Reissner-Mindlin plate (thick plate)

3D continuum elements

Additional aspects concerning elements

Continuity of displacements or deformations along element boundaries Element locking Independent interpolation of displacement, deformations, generalized forces


6 1.4 Material Behavior

1.4 Material Behavior

In the following, a material is considered as homogeneous.

Material behavior is constituted by a force response in reaction to a deformation.

Linear elastic material law

Uniaxialx = E x (1.19)

with Youngs modulus E. plane strain xy

xy

= E(1 )(1 + )(1 2)

1 1 01 1 00 0 122(1)

xy

xy

(1.20)with Youngs modulus E and Poissons ratio . This is a subset of the triaxialisotropic linear elastic Hooke law.

plane stress xyxy

= E1 2

1 0 1 00 0 12

xy

xy

(1.21)with Youngs modulus E and Poissons ratio ensuring z = 0 for every combina-tion x, y, xy

2D bendingM = EJ (1.22)

with Youngs modulus E and cross-sectional moment of inertia J . Eqns. (1.19)-(1.21) are a special case of

= C (1.23)with the constant material stiffness matrix C.

Uniaxial perfect elasto-plastic (as a simple example for a nonlinear constitutive equation) physical nonlinearity

x =

{E (x xp) for xp x xpsignx fy otherwise

(1.24)

andxp = x for |x| = fy (1.25)

with a yield stress fy and an internal state parameter xp. This value is the actual remain-ing strain upon unloading, i.e. x = 0 for x xp.

In case of nonlinear material equations at least an incremental form

d = CT d or = CT (1.26)should exist, where the tangential material stiffness might depend on stress, strain, internalstate variables.

On occasion the flexibility is needed, as counterpart of stiffness, i.e.

= D (1.27)


71.5 Weak Equilibrium and Discretization

Weak forms of equilibrium condition including dynamic parts

Preceding remark: given a point on a boundary either a kinematic boundary con-dition or a force boundary condition (zero force is also a condition!) has to beprescribed for this point.

Body (continuum) in space with a volume V , a specific mass %, prescribed dis-tributed loads p, prescribed displacements u on the surface partAu, prescribed trac-tions t on surface part At (Au together with At give the whole surface A), accelera-tion u, virtual displacements u and associated virtual strains

VT dV +

VuT u %dV =

VuT p dV +

At

uT t dA (1.28)

under the conditionsu = u onAu, u = 0 onAu (1.29)

Unixial bar along 0 x LLx xAdx+

Lu u %Adx =

Lu px dx+ [u t]

L0 (1.30)

under the conditions

u0 = u0, u0 = 0 or uL = uL uL = 0 (1.31)

with a cross section area A and a load per length px, whereby the formulation ofthe last term indicates the boundary term of a partial integration where traction isprescribed at either x = 0 or x = L (or none, but not both at the same time).

2D Bernoulli beam along 0 x LLw w mdx+

LM dx =

Lw p dx [M ]L0 + [wQ]L0 (1.32)

with a distributed mass m per length and a distributed load p per length. with cor-responding pairs (, M) and (w, Q). Only one quantity out of a pair can be pre-scribed at a boundary. Furthermore, at least two kinematic boundary conditionsshould be given with at least one deflection w0 and/or wL.

All these forms are variations of the principle of virtual displacements and immediatelyallow for physical nonlinearities.


8 1.5 Weak Equilibrium and Discretization

Discretization

The following is performed using Eq. (1.28). Other weak formulations use the sameprocedure. The general approach of GALERKIN is followed here.

Small displacements are used, if not otherwise stated.

Steps of discretization

1. Interpolation approach for displacements in the spatial domain, e.g. with Eq. (1.14)3,4

* An infinite number of degrees of freedom is reduced to a finite number ofnodal values of degree of freedom ( discretization).

* This leads to discretized strains e.g. with Eq. (1.18) and further to stresses with a constitutive law, e.g. Eq. (1.20) or (1.21) in the linear elastic case.

2. Interpolation approach for virtual displacements

* The same interpolation Eq. (1.14)3,4 is used as for the displacement (approach of BUBNOV-GALERKIN).

* This leads to discretized virtual strains with Eq. (1.18).

3. Evaluation of integrals

* This is performed on a element by element baseVI

T dV = uTI fI , fI =VI

BT dVVI

uT u %dV = uTI MI uI , MI =VI

NT N %dVVI

u p dx = uTI pI , pI =VI

NT p dVAtI

uT t dA = uTI tI , tI =AtI

NT t dA(1.33)

with an element index I . For integration methods see Section 1.7.

Figure 1.3: Schematic flow of nonlinear calculation

4. Assembling of element contributions to the whole system, see Figure 1.2.


9* Regarding, e.g. internal nodal forces f , it consists of entries for every de-gree of freedom of every global node. On the other hand, every discretiza-tion should have a table, which connects every element to the global nodesbelonging to it.

* This table relates the position of the entries of fI to a position in f .

* As a node generally gets contributions from more than one element, thevalue of an entry in fI has to be added to the corresponding entry in f . Thisis symbolically described by

VT dV = uT f =

IuTI fI (1.34)

* The same argumentation holds for uI ( u), uI ( u), MI ( M),pI ( p), tI ( t).

* Regarding arbitray values of u a spatially discretized system

M u + f(u) = p + t (1.35)finally results. This is a system of ordinary differential equations of sec-ond order in time t. It might be nonlinear due to nonlinear dependence ofinternal nodal forces f on nodel displacements u.

The linear case = C leads to internal nodal forcesfI =

VI

BT C B dV uI = KI uI (1.36)

see Eqns. (1.33)1, (1.6), with a constant element stiffness matrix KI . Assemblingleads to a system stiffness matrix K

f(u) = K u (1.37)and regarding Eq. (1.35) to

M u + K u = p + t (1.38)which embodies a system of linear ordinary differential equations of second orderin time t.

The system Eqns. (1.35) or (1.38) should be supported with appropriate kinematicboundary conditions to prevent rigid body displacements.

Tangential stiffness

The system tangential stiffness matrix is needed for the solution of the system andfurthermore reveals characteristic properties of a system, i.e. in particular its stabil-ity properties.

Element tangential stiffness

dfI =fIuI duI = KTI duI or fI = KTI uI (1.39)

withKTI =

VI

BT uI

dV =

VI

BT CT B dV (1.40)

see Eqns. (1.33)1, (1.26) (1.6), and a system tangential stiffness KT

df = KT du or f = KT u (1.41)


10 1.6 Convergence

1.6 Convergence

The linear elastic, quasistatic small displacement case is regarded in the following. Givena linear material law

= C (1.42)weak, integral equilibrium Eq. (1.28) for a structural problem can be written as

VT C dV =

VuT p dV +

At

uT t dA (1.43)

with appropriate displacement boundary conditions preventing rigid body motions, givenvalues for p, t and arbitrary virtual displacements u.

The strains , are derived from the displacements u, u by a differential oper-ator depending on the type of the structural problem under consideration.

The interpolation functions u, , u, belong to an appropriate function space2 Hdefined over the problem domain V ( space occupied by the the structure).

The boundary A of V is composed of Au and At, i.e. A = At Au and At Au = 0. u is prescribed on Au ( Dirichlet conditions) and t on At ( Neumannconditions) whereby t = n with the boundarys normal n.

The mathematical model of the structural problem Eq. (1.43) can be written as

a(u,v) = (f ,v) v H (1.44)

with a symmetric, bilinear a(, ), a linear (f , ) and v formally replacing u . Symmetry

a(u,v) = a(v,u) (1.45)

Bilinearitya(1u1 + 2u2,v) = 1a(u1,v) + 2a(u2,v)a(u, 1v1 + 2v2) = 1a(u,v1) + 2a(u,v2)

(1.46)

Linearity(f , 1v1 + 2v2) = 1(f ,v1) + 2(f ,v2) (1.47)

A norm maps a function v into a non-negative number. Sobolev norms ||v||i of order iare used in this context [Bat96, 4.3.4,(4.76)]. It is assumed that i = 1 is appropriate forthe following. It can then be shown that a has the properties

Continuity3

M > 0 : |a(v1,v2)| M v11 v21 v1,v2 H (1.48)

Ellipticity > 0 : a(v,v) v21 v H (1.49)

where M, depend on problem type and material values but not on v1,v2,v.

Due to Eq. (1.49) a(v,v) 0, i.e. a is a norm and may be physically interpreted asenergy, i.e. it is twice the internal strain energy.

2Square integrable functions fulfilling the kinematic boundary conditions, see, e.g., [Bat96, 4.3.4].3 means: it exists a (real number).


11

It can be shown that the problem Eq.(1.44) i.e. determine a function u H such thatEq. (1.44) is fulfilled for all v H has a unique solution u, see, e.g., [Bat96, 4.3]. Thissolution is exact for the mathematical model of the structural problem.

Discretization uses functions uh,vh Hh of a subset Hh H based upon the conceptof meshes with elements and nodes, see Section 1.3.

A uniform mesh4 of elements is assumed and h is a mesh size parameter, e.g. adiameter or length of a generic element.

The approximate solution uh Hh of Eq. (1.44) is determined bya(uh,vh) = (f ,vh) vh Hh (1.50)

The difference between approximate and exact solution gives the error

eh = u uh (1.51)

The approximation uh is known for Hh given, the error eh has to be estimated.

Some properties of the approximate solution

Orthogonality of error, see [Bat96, (4.86)]

a(eh,vh) = 0 vh Hh (1.52)

Energy of approximation is smaller than exact energy, see [Bat96, (4.89)]

a(uh,uh) a(u,u) (1.53)

Energy of error is minimized, see [Bat96, (4.91)]

a(eh, eh) a(u vh,u vh) vh Hh (1.54)

Combining Eqs. (1.49), (1.54), (1.48) leads to

eh21 = u uh21 a(eh, eh) = infvhHh

a(uvh,uvh) M infvhHh

u vh21(1.55)

where inf is infimum, the largest lower bound5. This is rewritten as

u uh1 c d(u, Hh), d(u, Hh) = infvhHh

u vh1 , c =M/

(1.56)d is a "distance" of functions in Hh to the exact solution u, c depends on the struc-tural problem type and the values of its parameters, but not on Hh.

Convergence means uh u or u uh1 0 with mesh size h 0. This can generally be reached with an appropriate selection of function spaces Hh

whereby reducing the distance d(u, Hh), see Eq. (1.56).

A more precise statement is possible utilizing interpolation theory. It introduces theinterpolant6 ui Hh of the exact solution u.

4The following considerations may be transferred to non-uniform meshes, see [Bat96, 4.3.5].5u vh1 ,vh Hh is a subset of real numbers. infvhHh u vh1 is the largest number less or equal to

the numbers in this subset.6u and ui coincide at nodes, but generally not apart from nodes. Generally it is ui 6= uh.


12 1.6 Convergence

Complete polynomials7 of degree k are used for discretization and interpolation,respectively. Interpolation theory estimates the interpolation error with

u ui1 c hk uk+1 (1.57)

with the mesh size h and a constant c which is independent of h, see [Bat96, (4.99)].uk+1 is the k + 1-order Sobolev norm of the exact solution.

On the other hand it must be infvhHh u vh1 u ui1 as ui Hh. Usingthis and Eqs. (1.56), (1.57) yields

u uh1 cc hk uk+1 (1.58)

The value cc can be merged to c, which depends on the structural problem type andthe values of its parameters, but not on h. A further merger of c and uk+1 leadsto the well known formulation

u uh1 c hk (1.59)

where c depends on the structural problem type, the values of its parameters and thenorm of the exact solution.

Conditions for convergence, see also [Bat96, 4.3.2]

A basic prerequisite is integrability of all quantities. This leads to requirementsfor the integrands of the energy a and the Sobolov norms8, which are uh,vh,u orderivatives thereof.

* It corresponds to the requirement of compatibility of finite element interpolationfunctions along inter element borders.

According to Eq, (1.59) a sequence of approximate solutions uh with h 0 willconverge9 with respect to u uh1 if k 1.

* The case k = 1 is covered by the so called patch test, i.e. the ability to modelfields with constant first derivatives of finite element interpolation functions inarbitrary element configurations, see, e.g., [BLM00, 8.3.2]).

The convergence rate will be higher for larger values of k, i.e. if the finite elementinterpolation has a higher order of completeness.

Limitations

The coefficient c may become so large under certain conditions that accepable solu-tions, i.e. a sufficiently small values ||u uh||, cannot be reached with realizablevalues h small enough.

* A particular occurence is given with locking of approximate solutions with in-compressible or nearly incompressible materials.

Extended weak forms7A polynomial in x, y is complete of order 1 if it includes x, y, complete of order 2 if of order 1 and including

x2, xy, y2, complete of order 3 if complete of order 2 and including x3, x2y, xy2, y3 and so on.8Sobolev norms are built from integration of squares of functions and squares of their derivatives.9Converge with respect to first order Sobolev norm u uh1 may not be sufficient if generalized strains are

derived from higher derivatives of displacements, e.g. with beams, slabs, shells. The theory has to be extended forthis case.


13

Eqs. (1.43,1.44) are weak forms of displacement based methods, as a solution isgiven by a displacement field. Strains and stresses are derived from this solution.

Extended weak forms allow to involve fields for stresses and strains as independentsolution variables. Most prominent are the principles of Hu-Washizu and Hellinger-Reissner [Bat96, 4.4.2].

* An abstract extended problem definition analogous to Eq. (1.50) given by, see[Bat01, (16)]

a(uh,vh) + b(h,vh) = (f ,vh) vh Hhb(wh,uh) c(h, eh) = 0 wh Wh (1.60)

in which a, c are symmetric bilinear forms, b is a bilinear form, f is a linearform, Hh,Wh are appropriate functions spaces, uh Hh, h Wh are theapproximate solutions.In most cases h stands for an independent field of strains or stresses.

* The foregoing theory concerning convergence has to be extended, see [Bat01].Such an extension includes the widely referenced inf-sup condition.

These approaches may solve, e.g., locking problems.

Nonlinear problems

The foregoing considerations relate to linear problems. They cannot be strictlyapplied to nonlinear probems physically nonlinear and/or geometrically nonlin-ear. But the conclusions to be drawn regarding element selection and discretizationshould also be considered for nonlinear problems.


14 1.7 Numerical Integration and Solution Methods for Algebraic Systems

ni i i0 0.0 2.01 0.57735 02691 89626 1.02 0.77459 66692 41483 0.55555 55555 55556

0.0 0.88888 88888 888893 0.86113 63115 94053 0.34785 48451 374540.33998 10435 84856 0.65214 51548 62546

......

...

Table 1.1: Sampling points and weights for Gaussian numerical integration (up to 15 digitsshown)

1.7 Numerical Integration and SolutionMethods for Algebraic Sys-tems

Integration of Eqs. (1.34) for the two-dimensional case as an example

Integration of isoparametric elementsVI

f(x, y) dV =

y2y1

x2(y)x1(y)

f(x, y) tdxdy =

+11

+11

f(r, s) J(r, s) tdrds

(1.61)with the determinant J of the Jacobian, see Eq. (1.16)2, and a constant thickness t.

Numerical integration +11

+11

f(r, s) J(r, s) tdrds = t

nii=0

nij=0

ij f(i, j) J(i, j) (1.62)

with integration order ni, sampling points and weighting factors .

Gaussian scheme

* Sampling points and weighting factors see Table 1.1. Generally we haveni

i=1 i =2. Also for other integration schemes.

* Integration accuracy: an integration order ni gives exact (within the scope ofnumerical accuracy) results for polynoms of order 2ni + 1, e.g. a uniaxialintegration of order 1 with two sampling points integrates exactly a polynom oforder 3.

Other schemes: Simpson, Newton-Cotes, Lobatto Integration of triangular elements

* Remains to be added.


15

Solution method for quasistatic problems

Regarding Eq. (1.35) we have to solve

r(u) = f(u) p = 0, p = p + t (1.63)

with internal nodal forces f depending on displacements u and external nodal loadp, which are assumed to be independent of u.

The general case is nonlinear dependence of f on u. So the solution of Eq. (1.63)has to be determined with an iteration with a sequence u(0) . . .u().

Looking at an arbitrary iteration step () we have r(u()) 6= 0 and seek for a cor-rection u. A linear Taylor-expansion is used as a basic approach

r(u() + u) r(u()) + K()T u, K()T = ruu=u()

= fuu=u()

= 0(1.64)

for the Newton-Raphson method with the tangential stiffness matrix KT . This leadsto

u = [K()T ]1 r(u())u(+1) = u() + u

(1.65)

and an (hopefully) improved value u(+1). Iteration may stop if r(u(+1)) 1and u 1 with a suitable norm . The method generally has a fast conver-gence, but is relatively costly. In every step () the tangential stiffness matrix has tobe computed and a LU-decomposition has to be performed.

Other iteration methods use variants of the iteration matrix (modified Newton Raph-son, secant methods like BFGS)

Iterative methods like Newton-Raphson generally are embedded in an incrementallyiterative scheme, so loading is given as a history: p = p(t). Often we choose0 t 1 for the load history time10, without restrictions to generality.

* We look at discrete time values ti, t0 = 0 and have pi = p(ti). This is knownbefore a solution.

* Accordingly, we have ui = u(ti), fi = f(ui). These are not known before asolution, except u0, f0. We assume r0 = f0 p0 = 0.

* The solution starts with t1, where u1 has to be determined. This is done withan iteration sequence u(0)1 . . .u

()1 , e.g. with the Newton-Raphson method and

u(0)1 = u0.

* A converged u1 is used as a base for t2 and so on.

10Load history time is different to real time.



Solution method for transient problems, in particular creep problems

Extended constitutive law = CT + (1.66)

with the tangential material stiffness CT , see Eq. (1.26), and an additional termdepending on stress and strain, see e.g. Section 2.2. We have to solve for (t),whereupon (t), (t) are given as functions of time.

A numerical method is used with the trapezoidal rule for derivatives of stress andstrain

i+1 = i + t [i+1 + (1 ) i]i+1 = i + t [i+1 + (1 ) i] (1.67)

with time discretization parameters t, . Eqs. (1.66), Eq. (1.67)1 lead to

i+1 = i + (1 ) t(CT,i i + i

)+ t

(CT,i+1 i+1 + i+1

)(1.68)

The parameters , t rule stability and accuracy of the temporal discretization.Regarding numerical methods for systems of first order differential equations see[BSMM00, 19.4], [Hug00, 8].

In the following, two approaches will be considered: (1) the implicit Euler approachwith = 1, (2) the explicit Euler approach with = 0. Regarding Eq. (1.67)2, theexpicit approach leads to

i+1 = i + CT,i (i+1 i) + ti (1.69)and the implicit to

i+1 = i + CT,i+1 (i+1 i) + ti+1 (1.70)The latter may be a nonlinear equation in case depends on and/or .

A particular case is given with

= V W (1.71)with a constant Kelvin viscosity V and a constant Maxwell viscosity W, see e.g.Section 2.2. Thus, i+1 = V i+1 W i+1 and the implicit scheme isi+1 = i + CT,i+1 (i+1 i) + tV i+1 tW i+1

= [I + tW]1 [i + CT,i+1 (i+1 i) + tV i+1]= [I + tW]1 [i + [CT,i+1 + tV] (i+1 i) + tV i]

(1.72) Internal nodal forces according to Eq. (1.33)1

fi+1 =

V

BT i+1 dV = fi + KT,i+1 u + fi+1 (1.73)

withfi =

V B

T i dVfi+1 = t

V B

T i+1 dVKT,i+1 =

V B

T CT,i+1 B dV(1.74)

This is embedded in a time stepping scheme, compare Eq. (1.63)

ri+1 = fi+1 pi+1 = 0 KT,i+1 u = pi+1 fi fi+1 (1.75)Starting with initial conditions f0, f0 and a given load p(t) displacements may bedetermined time step by time step. But Eq. (1.75) might be nonlinear in a time stepand need an iteration, see Eq. (1.65).


17

Solution methods for dynamic problems

Based on Eq. (1.35) we have in analogy to Eq. (1.63)

r = M u + f p(t) = 0, p(t) = p(t) + t(t) (1.76)

where the loading p is a prescribed function of the time t. The displacements u(t)and all derived values (velocities u(t), accelerations u(t), internal nodal forces f )are unknown before solution. Eq. (1.76) is discretized in the spatial domain, but notyet in the time domain, i.e. it is system of ordinary differential equations of 2ndorder in time.

This problem needs appropriate boundary conditions and initial conditions for thedisplacements u0 = u(0) and velocities u0 = u(0).

A solution may be computed with a discretization in time via a difference scheme.A widespread approach is given with the Newmark method

ui+1 = ui + t[ui+1 + (1 )ui

]ui+1 = ui + t ui + t

2[ui+1 + (

12 )ui

] (1.77)with ui+1 = u(ti+1), ui+1 = u(ti+1), ui+1 = u(ti+1) a time step length t =ti+1ti and integration parameters , . Eqns. (1.77) are solved for the accelerationand velocity in time step i+ 1. We get

ui+1 =1

t2[ui+1 ui+1] (1.78)

with an auxiliary quantity

ui+1 = ui + t ui +t2

2(1 2) ui (1.79)

and the velocity

ui+1 =

t[ui+1 ui] +

(1

)ui + t

(1

2

)ui (1.80)

Finally, dynamic equilibrium Eq. (1.76) is applied for time step i+ 1 with the accel-eration according to Eq. (1.78):

r =1

t2M [ui+1 ui+1] + fi+1 pi+1 = 0 (1.81)

With given parameters , ,t, a given previous state ui, ui, ui, given mass matrixM and load Pi+1, Eq. (1.81) has to be solved for ui+1, where the dependence offi+1 on ui+1 is crucial and might be nonlinear.

We apply the Newton-Raphson method Eq. (1.65). The Jacobian matrix is givenwith

A()T =

1

t2M +

f

u

u=u

()i+1

=1

t2M + K

()T (1.82)

leading to an iteration scheme

u(+1)i+1 = u

()i+1 +

[A

()T

]1 (pi+1 f ()i+1 1t2 M [u()i+1 ui+1])

(1.83)



In the linear case with

f()i+1 = K u()i+1, A()T = A =

1

t2M + K (1.84)

the Eq. (1.83) simplifies to

ui+1 = A1

(pi+1 +

1

t2M ui+1

)(1.85)

with no iteration necessary, compare [Bat96, 9.2.4].

Numerical integration parameters , rule consistency and numerical stability ofthe method.

* Stability means that an amount of error introduced in a certain step due to afinite time step length t is not is not increased in the subsequent steps.

* Consistency means that the iteration scheme converges to the differential equa-tion for t 0.

Stability and consistency are necessary to ensure that the error of the numericalmethod remains within some bounds for a finite time step length tGenerally a choice = 14 , =

12 is reasonable for the Newmark method to reach

consistency and stability.


Bibliography

[Bat96] BATHE, K.J.: Finite Element Procedures. Englewood Cliffs, New Jersey : PrenticeHall, 1996

[Bat01] BATHE, K.J.: The inf-sup condition and its evaluation for mixed finite elementmethods. In: Computers and Structures 79 (2001), S. 243252

[BLM00] BELYTSCHKO, T. ; LIU, W.K. ; MORAN, B.: Nonlinear Finite Elements for Con-tinua and Structures. Chichester : John Wiley & Sons, 2000

[BSMM00] BRONSTEIN ; SEMENDJAJEW ; MUSIOL ; MUEHLIG: Taschenbuch der Mathe-matik. 5. Auflage. Frankfurt/Main : Verlag Harri Deutsch, 2000

[CF08] CEB-FIP: Practitioners guide to finite element modelling of reinforced concretestructures. Bd. Bulletin Nr. 45. Lausanne : International Federation for StructuralConcrete fip, 2008

[Hug00] HUGHES, T.R.: The Finite Element Method - Linear Static and Dynamic FiniteElement Analysis. Mineola, New York : Dover Publications, Inc., 2000

19

Chapter 2

Uniaxial Structural Concrete Behavior

2.1 Short Term Stress-Strain Behavior of Concrete

Short term behavior means, that loading is applied within a few minutes, hours, or days,and that one regards the immediate reaction of the structural material.

Time-dependent behavior, e.g. creep and shrinkage, is not considered.

Unconstrained compression

Unconstrained spatially homogeneous loading Stress-strain behavior under monotonic loading, see Fig. 2.1a

* Linear part

* Nonlinear hardening part

* Nonlinear softening part

Figure 2.1: a) Uniaxial compressive stress strain behavior b) mesoscale view

* Analytical description Approach by Saenz, see [CS94, 8.8.1]

=Ec0

1 +(Ec0Ec1 2)

c1

+(

c1

)2 (2.1)with initial Youngs modulus Ec0 of concrete, its secant modulus Ec1 =fc/c1 at compressive strength fc (unsigned), strain c1 (signed) at strength.With = c1 Eq. (2.1) leads to = fc.

20

21

The derivative with respect to ( tangential material stiffness) is givenwith

Et =d

d=

Ec0

(1 2

2c1

)(

1 +(Ec0Ec1 2)

c1

+ 2

2c1

)2 (2.2)which leads to Et = Ec0 for = 0, furthermore Et = 0 for = c1 andEt < 0 for < c1, || > |c1|.

Alternatives by Modelcode 90 [Com93, 2.1.4.4.1] and DIN 1045-1 [din08,9.1.5, 8.5.1].

These approaches may all be classified as phenomenological, i.e. free parameters of a polynomial form are chosen such, that character-

istic points of measured stress-strain behavior are reproduced.

Mesoscale view with a distinction of

* aggregates,

* mortar,

* interface layer between aggregates and mortar,

* highly inhomogeneous stress distribution within a specimen with internallateral tensile stresses locally.

Failure modes

* Lateral splitting / cracking with respect to compressed longitudinal direction

* Volume expansion

* Diffuse failure

Unconstrained tension

Figure 2.2: a) Uniaxial tensile stress strain behavior b) mesoscale cracking

Homogeneous end loading of a uniform bar Stress-strain behavior, see Fig. 2.2a

* Linear part

* Nonlinear hardening part

* Nonlinear softening part


22 2.1 Short Term Stress-Strain Behavior of Concrete

Mesoscale view, see Fig. 2.2b, with successive states

* micro cracking,

* crack bridges,

* macro cracking.

Figure 2.3: Scheme of localization

Localization and crack band

* Weak cross section within specimen where tensile strength is reached

* Narrow band of high strains crack band* Strain distribution c(x) across crack band with x1 x x2 and a crack band

width bw = x2 x1* Softening of crack band increasing strains with decreasing stresses* Unloading of specimen areas beyond crack zone

Snap back behavior of whole bar depending on ratio of crack band widthto total length

Localized failure in contrast to diffuse failure* Process band ending up in macro crack with zero stress and zero strain beyond

crack band.

* The whole process may be unstable under quasistatic conditions.

Fictitious crack widthw =

x2x1

c(x) dx = bw c (2.3)

with mean strain c in the crack band.

* In the following, c const. is assumed within the crack band for simplifica-tion.

* Accordingly, the fictitious crack width, see Eq. (2.3), is given with

w = bw c (2.4)

* Furtmermore, this assures a c-relation in the softening part of Fig. 2.2a anda critical strain cr with (cr) = 0.

* Regarding Eq. (2.4), a c-relation may be transformed into a w-relationwith a fictitious critical crack width wcr with (wcr) = 0.


23

Crack energy

* Energy in a material element within crack band depending on c or w

g =

c1

() d =1

bw

w0(w) dw (2.5)

with a lower bound e.g. 1 = ct and () according to Fig. 2.2a.

* With c = cr orw = wcr Eq. (2.5) leads to the specific crack energy gf (relatedto volume), which corresponds to the shaded area in Fig. 2.2a.

* Sum along crack band leads to crack energy (related to surface)

Gf = bw gf = bw

cr1

() d = wcr

0(w) dw (2.6)

* Gf measures energy dissipation due to creation of new surfaces.

* Gf is assumed as (constant) material parameter, so that Eq. (2.6) leads to aconstraint for a w-relation or c-relation, respectively.

Unloading to stress free state with a reduction of stresses and strains

Reduction of stiffness damage Remaining strains in stress free state plasticity

Unloading is hard to realize for softening structural elements, as it requires total displace-ment control of softening areas.

Example 2.1 Simple concrete tensile bar with localization

Figure 2.4: Example 2.1 a) geometry scheme b) force-displacement curve

Geometry and Discretization, see Fig. 2.4a

Bar Length L = 0.5 m, cross sectional area Ac = 0.1 0.1 m2 Discretization with Ne = 500 1D-bar elements with two nodes, see Eq. (1.7).

Material

Concrete grade C40 according to [Com93, 2.1]


24 2.1 Short Term Stress-Strain Behavior of Concrete

Initial Youngs modulus Ec = 36 000 MN/m2, tensile strength fct = 3.5 MN/m2

Material model

Damage model with gradient damage, see [HC07].

Boundary conditions

Left end with zero displacements,right end with prescribed displacement. Nonlocal damage assumed as zero at both ends ( failure in mid point).

Solution method

Incrementally iterative with arc length control of load step size and Newton-Raphsoniteration within each increment.

Results

Figure 2.5: Example 2.1 strain distribution a) load step A b) load step B

Load displacement curve see Fig. 2.4b.

* Initial elastic part followed by increasing damage

* Limit load where the stresses reach the tensile strength of the material

* Softening part with increasing displacements and decreasing load Bar has two types of material behavior in its longitudinal direction. Hard-

ening and softening tangential material stiffness. In the softening part thebar elongates with decreasing stresses ( localization) in the hardeningpart it becomes shorter due to reduced stresses.

* Snap back part with decreasing displacements and decreasing load Totally, it shortens in the example case, but this depends on the ratio of

length of softening part to hardening part. Longer bars show a more pronounced snap back behavior.

Strain distribution along bar

* Before limit load see Fig. 2.5a. Due to prescribed zero nonlocal damage onboth ends strain moderately increases in the mid range of the bar. Stress is = 2.90 MN/m2 and mean strain = 0.94 104.

* After limit load in the snap back range see Fig. 2.5b. Strains strongly increasein a short mid range and otherwise decrease due to the load decrease. Stress is = 1.64 MN/m2 and mean strain = 1.08 104.


25

end example 2.1

Remarks

In a real specimen the localization will not center exactly in the mid point of a bar,but in the weakest cross section. This cross section will arise due to the stochasticvariation of material strength. Its cross-sectional strength and location cannot beexactly determined, but only with statistical parameters.


26 2.2 Long Term Effects - Creep, Shrinkage and Temperature

2.2 Long Term Effects - Creep, Shrinkage and Temperature

Besides the immediate reaction of a structural material, its deferred reaction upon an ap-plied loading is regarded. This is typically creep or relaxation, respectively. The followingapproach is chosen for the uniaxial strain depending on time t

(t) =

t0J(t, )() d, 0 t (2.7)

for a creeping material exposed to a uniaxial load history (). The compliance functionJ(t, ) is specific for every material type. Eq. (2.7) describes linear creep.

The following form has proven to be appropriate for the compliance function

J(t, ) =

N=1

J(t, ), J(t, ) =1

E()

(1 e[y()y(t)]

), y() = (/)

q

(2.8)with material parameters , q and material functions E(). The parameter has adimension of time and the function E a dimension of stress.

The case N = 1, E1 = E = const., 1 0 gives J(t, ) = 1/E with Eq. (2.8),and with Eq. (2.7) a linear elastic law (t) = C (t).

Combining Eqns. (2.7), (2.8) leads to

(t) =N

=1(t), (t) =

t0J(t, ) () d (2.9)

Determination of J literature

Figure 2.6: a) Kelvin-Voigt element, Maxwell element b) Simple chains

Simplifications are used in the following, i.e. E = const., q = 1. Using Eqns. (2.8),(2.7) leads to

(t) =(t)

E e

t

E

t0() e

d (2.10)

with

(t) =(t)

E+

1

e t

E

t0() e

d e

t

E() e

=

1

e t

E

t0() e

d

(2.11)


27

Thus, the strain Eq. (2.10) fulfills the differential equation

(t) + E (t) = (t), = E (2.12)

with an initial condition (0) = 0. Eq. (2.12) describes a Kelvin-Voigt element with aspring and a damper in parallel, see Fig. 2.6a.

An alternative is given with the Maxwell element with a spring and a damper inseries, see Fig. 2.6a.

Eq. (2.9)1 yields a Kelvin-Voigt chain, i.e. a number of Kelvin-Voigt elements in series.These systems of springs and dampers add up to rheological models.

A simple Kelvin-Voigt chain N = 2, = 0 . . . 1, 0 = 0 is considered

E0 0(t) = (t), 1 1(t) + E1 1(t) = (t) (2.13)

see Fig. 2.6b, with

(t) = 0(t) + 1(t), (t) = 0(t) + 1(t) (2.14)

First of all, we have

1(t) = (t) (t)E0

, 1(t) = (t) (t)E0

(2.15)

Thus, Eq. (2.13)2 after a few calculations leads to

(t) +E0 + E1

1(t) = E0 (t) +

E0E11

(t) (2.16)

We introduce a final stiffness 1/E = 1/E0 +1/E1, considerE0 as an initial stiffness, andintroduce a creep coefficient defined by E = E0/(1 + ). This leads to E1 = E0/and Eq. (2.16) may be reformulated as

(t) +(1 + )E01

(t) = E0 (t) +E201

(t) (2.17)

Thus, prescribing a constant stress (t) = 0, = 0 with an initial strain (0) = 0/E0leads to a strain varying with time

(t) =0E0

[1 +

(1 e t

)], =

E01

(2.18)

The asymptotical final value is asym = (1+)0/E0, the creep portion is 0/E0. Thisprovides a method to describe 1, as the creep portion has a characteristic time t? upon apart with 0 < 1 of the asymptotic creep part. This leads to

1 e t? = = ln(1 )t?

(2.19)

The values E0, t?, , may be determined based on experiments. This results in 1 anda differential visco-elastic material law Eq. (2.17) to describe creep.


28 2.2 Long Term Effects - Creep, Shrinkage and Temperature

Incremental material law

See Section 1.4, Eq. (1.26). This has to be extended with respect to viscous parts,like e.g. Eq. (2.17). For following uniaxial applications the form

(t) = E0[(t) + (t)

] [1 + ] (t), = E01

= lnt?

(2.20)

is chosen.

Shrinkage and temperature

Temperature and shrinkage impose strains independent from loads. Uniaxial tem-perature strains, e.g., are given by

T = T T (2.21)

with thermal expansion coefficient T and a temperature change T . Concreteshrinkage strains cs mainly depend on time, humidity conditions and ratio of sur-face to volume, see [din08, 9.1.4] or [Com93, 2.1.6.4.4].

Stress is induced by total, measured strain less imposed strains. Thus, in the uniaxiallinear elastic case stress is given by

= E ( I) , I = T + cs (2.22)

This is transferred to the case including linear creep with Eq. (2.20)

(t) = E0

[(t) I(t) +

[(t) I(t)

]] [1 + ] (t)= E0

[(t) I(t)

]+ E0

[(t) I(t)

] [1 + ](t) (2.23)or

(t) = CT[(t) I(t)

]+ V

[(t) I(t)

]W (t) (2.24)with

CT = E0, V = E0, W = [1 +

](2.25)

where I(t), I(t) should be a known function of time.

A transient problem arises first of all without inertial terms , which needs somecare regarding integration of systems equations in time, see Section (1.7).

Example 2.2 Simple concrete tensile bar with creep and imposed strains

Geometry and Discretization

Bar Length L = 1.0 m, cross sectional area Ac = 1.0 m2. Discretization withNe = 5 1D-bar elements with two nodes, see Eq. (1.7).

As a homogeneous state along longitudinal direction is considered in this exam-ple, the length of the bar is irrelevant and one element would be sufficient.

Material

Concrete with an initial Youngs modulusE0 = 30 000 MN/m2 and a tensile strengthfct = 3.5 MN/m

2.


29

Material creep is assumed with = 2.0 and t? = 100 [d] for = 0.5, i.e. half oftotal creep occurs after 100 days for a constant stress load. With Eqns. (2.20), (2.25)this leads to

= 0.006931 1/d, CT = 30 000 MN/m2, V = 207.94 MN/m2d, W = 0.020794 1/d

(2.26)

Boundary conditions

Displacement of left node is prescribed with zero in all following cases.

Solution method

Incrementally iterative with Newton-Raphson iteration within each increment, ifneccessary. The method is described on Page 16.

Following time discretization values are chosen: implicit and t = 10 days. Aperiod of 500 days is regarded.

Figure 2.7: a) Strain depending on time b) Stress depending on time

Case 1: validation with stress loading 0 = 3.0 MN/m2 constant in time

Computed strain depending on time see Fig. 2.7a. An exact solution is given withEq. (2.18) for this case. Differences between exact solution and numerically com-puted solution are small and are not visible in the Figure.

Case 2: prescribed immediate strain 0 = 3.0/30 000 = 0.0001 constant in time

Computed stress depending on time see Fig. 2.7b. In exact technical terms, this is not creep anymore but relaxation.

Case 3: prescribed imposed contraction

A slow contraction of 0.15 is prescribed over a period of t = 100 d linear intime and then hold constant. As total strain is prescribed with zero, a tensilestress is induced, which would lead to tensile failure without relaxation. Stress withrelaxation is shown in Fig. 2.7b.

end example 2.2

While analytical, exact solutions are availabe for cases 1, 2, the numerical approach isnecessary for arbitrarily prescribed loads or displacements. Furthermore, more complexcreep models, see e.g. [Mal69, 6.4], [JB01, 28, 29], can only be solved with numericalmodels.


30 2.3 Strain-Rate Effects

2.3 Strain-Rate Effects

Remains to be added.


31

2.4 Cracks

This is strongly connected to the notions of localization, crack band and fictitious crackwidth, see Page 21.

We consider an instant with a fictitious crack width w given according to Eq. (2.3). Dueto equilibrium reasons a constant cross sectional stress occurs along the crack band,which gives a pair , w. Considering all instants of tension softening a relation (w) canbe derived, i.e. a stress across crack band depending on fictitious crack width.

Figure 2.8: Constitutive law for cohesive crack model

Cohesive crack model (general spatial view)

Fictitious crack

* Implies a geometric plane corresponding to one of the two crack surfaces.

* This plane supports a reference system, where the distance between the cracksurfaces is measured by a normal crack width component wx and two slidingcrack width components wy, wz .

A high idealization in the area of e.g. crack bridges.

Crack-tractions tx, ty, tz Cauchy stress tensor projected onto crack plane* Crack-tractions tx, ty, tz are transferred across crack plane.

Material law connecting crack-traction and crack-width

* A common format is

tx = fn(wx), ty = fs(wy), tz = fs(wz) (2.27)

with different laws fn for the normal component and fs for the sliding compo-nent.

* This material law uses crack tractions as force variable and crack widths asdeformation variable, compare page 2

* All material frameworks like plasticity, damage etc. may be used.


32 2.4 Cracks

A common simplified approach: Set fs = 0. fn is given by a cut off and a scalingof Fig. 2.2a, which results in Fig. 2.8. This reflects the uniaxial case.

The cohesive crack model is a macroscopic model for the complex crack develop-ment process.

Smeared crack model for the simplified approach

Approach for strain of a uniaxial black box element1 with a length Lc with one crack

=1

Lc[(Lc bw) m + bw c] = (1 ) m + c, = bw

Lc(2.28)

with the strain m of the uncracked material, see Fig. (2.9). For c, bw see Eq. (2.4).

Figure 2.9: Smeared crack concept

Connection to stresses / forces

* The strain part u is connected to a homogeneous material, see Section 1.4,leading to material stresses.

* The strain part c or w is connected to a crack or discontinuity, leading to crackstresses, see Eq. (2.27)1.

* Both stress parts are connected through equilibrium, e.g. have the same valuein an uniaxial element.

If element length is adjusted to crack band width bw, i.e. Lc = bw from Eqns. (2.4),(2.28) we get a crack width

w = Lc (2.29)

In case of = 0, u = 0 adjusting of element length is not necessary, i.e. w = Lc for any selection of Lc.

The smeared crack model is a model for the fictitious crack preserving continuityof displacements, which is important for ordinary finite elements. There may bealternative models for the fictitious crack, i.e. for the implemention of the cohesivecrack model.

1We cannot see inside.


33

2.5 Reinforcing Steel Stress-Strain Behavior

Uniaxial Compression / Tension, see 2.10a, b

Linear range Transition range Yielding range Unloading Reloading Cyclic loading Energy dissipation

Figure 2.10: a) Uniaxial stress-strain behavior for reinforcing steel b) hardening behavior

Simple uniaxial constitutive law with isotropic hardening

Stress-strain law

s =

{Es (s p) if p fyEs p +

fyEs

sign fy else(2.30)

with Youngs modulus Es of reinforcing steel, its yield stress fy, the sign functionsign and the actual plastic strain p as internal state parameter.

Evolution law for internal state parameter

p = sfy = ET |s|

}if s s > 0 and |s| = fy (2.31)

with a hardening or tangential modulus ET . Eqns. (2.30), (2.31) cover loading,unloading and reloading for tension and compression for uniaxial ideal elastoplasticbehavior. The yield stress fy may increase due to hardening.

Hardening, see Fig. 2.10b

Isotropic Kinematic

Codes

DIN 1045-1 [din08, 9.2, 9.3], Modelcode 90 [Com93, 2.2, 2.3].


34 2.6 Bond between Concrete and Reinforcing Steel

2.6 Bond between Concrete and Reinforcing Steel

Bond mechanisms, see Fig. 2.11a, with

reinforcement ribs,

conical struts,

circumferential ties,

complex triaxial problem.

Figure 2.11: a) Triaxial bond mechanisms b) Bond law

Modeling and model variables, compare page 2

Uniaxial orientation along rebar direction.

Kinematic variable slip s with a dimension of length. Force variable bond stress with a dimension of stress. Bond law = f (s).

* All constitutive frames like plasticity, damage etc. may basically be used.

Bond flow t = U with a rebar cirumference U and dimension force per length.

Bond laws

Monotonous loading.

* See e.g. [Com93, 3.1.1].

* A smoothed version = f (s), see Fig. 2.11b. Characterized by maximum bond stress max, a corresponding slip s1, a

residual bond stress f and a corresponding slip s2. This is composed of a quadratic, cubic and linear polynom with continuous

derivatives at the connection points.

* As long as monotonous loading is considered, approaches with plasticity anddamage approaches are dispensable.

Unloading and Reloading

* Remains to be added.


35

2.7 Reinforced Tension Bar

A finite element model

Concrete

* Element type: 1D linear bar element, see Eq. (1.7).

* Constitutive law: uniaxial linear elastic, see Eq. (1.19) with a limited tensilestrength fct

=

{Ec , fctEc0, else

(2.32)

Concrete with Cracks

* Smeared crack approach, see Page 32.

Reinforcing steel

* Element type: 1D linear bar element, see Eq. (1.7).

* Constitutive law: uniaxial elastoplastic, see Eqns. (2.30), (2.31).

Bond

* Element type: 1D spring element, see Eq. (1.12), with slip s instead of u.

* Constitutive law: bond law, see Fig. 2.11b.

Geometry scheme see Fig. 2.12a. Boundary conditions / loading

* Base with fixed zero displacement on one end.

* Prescribed displacement on the other end.

Figure 2.12: a) Geometry scheme of reinforced tension bar b) force-displacement curve ofexample 2.3

Solution methods

Incrementally iterative. Equilibrium iteration within each loading increment withNewton-Raphson method, see Page 15.

Computation of crack width

See Eq. (2.29) and the following remark. As cracks are assumed as stress free imme-diately after cracking we have w = LI with the length LI of the cracked elementand its computed strain .


36 2.7 Reinforced Tension Bar

Example 2.3 Simple reinforced concrete tension bar

Model input values

Bar length L = 1.0 m, cross sectional area of concrete Ac = 0.1 0.1 m2, rein-forcement 1 16, As = 2.01 cm2, circumference Us = 5.02 cm.

Discretization with two-node bar elements, see Eq. (1.7), length LI = 0.01 m.

* 100 Elements for concrete with 101 nodes, 100 Elements for reinforcementwith 101 nodes, 101 bond elements connecting concrete nodes and reinforce-ment nodes.

* Totally 202 nodes.

Material parameters, see Table 2.1

concreteYoungs modulus Ec MN/m2 35 000tensile strength fct MN/m2 3.5

reinforcing steelYoungs modulus Es MN/m2 200 000yield strength fsy MN/m2 500

bondstrength max MN/m2 6.0slip at strength mm 0.1residual strength res MN/m2 3.0slip at residuum mm 1.0

Table 2.1: Material parameters of RC tensile bar example 2.3

* Concrete material model see Eq. (2.32).

* Reinforcement material model see Eqns. (2.30), (2.31).

* Bond model see see Fig. 2.11b. This corresponds to a material behavior perunit surface of bond. To gain the corresponding values for a whole spring, amultiplication by Us LI is neccessary.

Boundary conditions

* Zero displacement reinforcement node on left boundary, prescribed displace-ment uR = 2.4 mm for reinforcement node on right side incrementally appliedin 100 steps corresponding to a mean strain mean = 2.4 .

Results: monotonic loading

Load-displacement curve see Fig. 2.12b

* state I: Uncracked

* state IIa: Ongoing cracking. Each sudden load decrease corresponds to a crack.Stiffness is reduced after each crack.

* state IIb: Final cracking state before rebar yielding.

* state III: Limit state with rebar yielding with a limit load Pu = Asfsy =0.1005 MN at a displacement uR = 2.14 mm


37

Figure 2.13: Example 2.3 a) concrete stresses b) steel stresses

* Stiffness of bare reinforcement EA = 40 MN. Limit load of Pu = 0.1005 MNapplied on steel alone leads to a displacement uR = 2.5 mm compared to theactual displacement of uR = 2.14 mm ( tension stiffening effect).

Stress distributions along bar at beginning limit state (uR = 2.14 mm).

* Concrete stresses see Fig. 2.13a with zero stresses at a crack. Three cracksaccording to three peaks in the load-displacement curve, see Fig. 2.12b.

* Reinforcement stresses see Fig. 2.13b with peak stresses at a crack. Yieldstrength is reached at each crack.

* In between cracks forces are transferred among concrete and reinforcement bybond stresses, see Fig. 2.14a. Bond stresses have maximum absolute values incracks and change sign across a crack.

Figure 2.14: Example 2.3 a) bond stresses b) displacements

Displacements at beginning limit state

* See Fig. 2.14b. Displacements of concrete and reinforcement are different ata bars cross section due to flexible bond. The difference results in slip, whichconnected to bond stresses, see Section 2.6.

Crack widths by Eq. (2.29), where element length LI and crack band width arechosen to be equal. Thus, in a cracked concrete element crack width is given bythe displacement difference of the end nodes. According to Fig. 2.14b here we gettypically w 0.4 mm in the beginning limit state.

end example 2.3


38 2.8 Tension Stiffening for Reinforced Tension Bar

2.8 Tension Stiffening for Reinforced Tension Bar

Mean strain of tension bar

There is some contribution of concrete between cracks with reduced reinforcementstresses between cracks, see Fig. 2.13b.

It is assumed that reinforcement has not yet reached its yield limit. Therefore, rein-forcement strain has the same shape as the reinforcement stress.

We consider the mean strain of the reinforcement which corresponds to the meanstrain of the tension bar.

* Mean strain refers to some reference length in the center area of the tension barspanning several cracks. It results from the integration of strains in the referencelength ( displacement of reference length) divided by reference length itself.

Stress depending on mean strain

Stress of tension bar is given by reinforcement stress in cracks. This value alsocorresponds to the force of the tension bar.

Stress depending on mean strain corresponds to the course of Fig. 2.12b. It shows ahigher stiffness compared to the stiffness of the reinforcement alone. The differenceis called tension stiffening effect.

Stabilized cracking is distinguished from ongoing cracking with single cracks in the fol-lowing. For a discussion of crack types see e.g. [HCH09].

Figure 2.15: Models for a) cracking b) tension stiffening

Estimation of tension stiffening effect for stabilized cracking

The mean value of reinforcement strain between cracks may be estimated with

sm =1

Es(sr t s) (2.33)

with reinforcement stress sr in a crack, reinforcement stress difference s fromcrack to minimum value between cracks, and a parameter t for the shape of re-inforcement stress distribution, see Fig. 2.15a. For a discussion of t see e.g.[HCH09]. Common values are assumed in the range 0.4 t 0.6.


39

In case of stabilized crack pattern the stress difference is given by

s =fct%eff

(2.34)

with the concrete tensile strength fct and the effective reinforcement ratio %eff .Eqns. (2.33), (2.34) lead to

sr = Es sm + tfct%eff

(2.35)

Eqn. (2.35) leads to a shift of the pure reinforcement stiffness to the left, see Fig. 2.15a,i.e. a given mean strain has a higher stress with tension stiffening. This is limited bythe yielding plateau of the reinforcement.

Ongoing cracking with single cracks

In case of single cracks a suitable reference length is hard to determine, as crackspacing is irregular and areas affected by cracks alternate with such not affected bycracks which have a full utilization of concrete.

The envelope2 of the a stress-strain curve is relatively flat. Starting with the initialreinforcement stress sr,i = fct/%eff in a crack immediately after cracking and acorresponding mean strain fct/Ec immediately before cracking an assumption

sr = k

(sm fct

Ec

)+

fct%eff

(2.36)

is made with a free parameter k. This line should meet the line Eq. (2.35) at a stressvalue sr,i leading to

k = Ec 1

t n%eff , n =EsEc

(2.37)

whereby experience shows that 1.3 can be assumed. The strain sm belonging to the meeting point can be determined with Eq. (2.35),

which leads tosm =

1

Es

fct%eff

( t) (2.38)

A simplified version might straighten the initial kink, see Fig. 2.15b.

This leads tosr =

sr,ism

sm (2.39)

in the initial range.

Further references: [Com93, 3.2], [DAf03, zu 8.5.1].

2Envelope to saw tooth behavior during ongoing cracking.


40 2.9 Cyclic Loading of Reinforced Tension Bar

2.9 Cyclic Loading of Reinforced Tension Bar

Reinforcement should not yield in case of cyclic loading. Thus, linear elastic behavior ofreinforcement will be assumed. Furthermore, a stabilized cracking state is assumed.

First unloading

Initially along loading path of stabilized crack state Upon further unloading crack closure resistance follows as rough crack surfaces do

not fit exactly together. Residual displacements / strains remain compared to uncracked state in stress free

state.

Figure 2.16: Model for cyclic loading

Reloading

Degradation of bond initially lower reloading stiffness compared to final unload-ing stiffness.

After full utilization of concrete same tangential stiffness as reinforcement. But theconnection point / area to this tangential stiffness is shifted compared to that of theprevious unloading path.

Further unloading-reloading cycles follow the same pattern with lower initial reloadingstiffness compared to previous final unloading stiffness. But this effect becomes lesspronounced.

After a large number of loading cycles the stress-strain behavior of tensions bar shouldconverge against stress-strain behavior of reinforcement alone, with some horizontal shiftdue to crack closure effect.

A corresponding stress-strain relation is given by

sr = Es sm s,c (2.40)see Fig. 2.15b, but the value s,c is hard to estimate.

Total crack closure w = 0 is connected with ein eigenstress state, i.e. concrete com-pressive stresses and reinforcement tensile stresses which upon integration equili-brate and have no resulting tensile bar force and stress, respectively.

See also Fig. (2.16) and [Com93, ?].


Bibliography

[Com93] COMITE EURO-INTERNATIONAL DE BETON: CEB-FIP Model Code 1990. London: Thomas Telford, 1993

[CS94] CHEN, W.F. ; SALEEB, A.F.: Constitutive Equations for Engineering Materials,Volume 1: Elasticity and Modeling. 2. Auflage. Amsterdam : Elsevier Science B.V.,1994

[DAf03] DAFSTB: Erluterungen zu DIN 1045-1 / Deutscher Ausschuss fr Stahlbeton.Berlin, 2003 (Heft 525). Forschungsbericht

[din08] DIN 1045-1: Tragwerke aus Beton, Stahlbeton und Spannbeton. Teil 1: Bemessungund Konstruktion. : DIN 1045-1: Tragwerke aus Beton, Stahlbeton und Spannbeton.Teil 1: Bemessung und Konstruktion, August 2008

[HC07] HUSSLER-COMBE, U.: Zur Verwendung von Stoffgesetzen mit Entfestigung innumerischen Rechenverfahren. In: Bauingenieur 82 (2007), S. 286298

[HCH09] HUSSLER-COMBE, U. ; HARTIG, J.: Rissbildung von Stahlbeton bei Zwang-beanspruchungen. In: Bauingenieur 84 (2009), S. 546556

[JB01] JIRAZEK, M. ; BAZANT, Z.: Inelastic Analysis of Structures. 1. New York : JohnWiley & Sons, 2001

[Mal69] MALVERN, L. E.: Introduction to the Mechanics of a Continuous Medium. 1. Au-flage. Englewood Cliffs, New Jersey : Prentice-Hall, 1969

41

Chapter 3

2D Structural Beams and Frames

3.1 General Cross Sectional Behavior

3.1.1 Kinematics

Straight reference axis along beam direction

Coordinate system x, z with x in the longitudinal direction and z in the transversedirection, see Fig. 3.1.

Bernoulli-hypothesis ( cross sections remain plane during deformation) Displacements

w(x, z) = w(x, 0)= w(x)

u(x, z) = u(x) z (x)= u(x) z

[w(x)

x (x)

] (3.1)with a cross section rotation angle (x), a shear angle (x) and a displacementu(x), w(x) of the reference axis.

u(x,z)z1

z2

x,u

w

w(x,z)

z,w

w

Figure 3.1: Beam kinematics

Strains, compare Eqns. (1.1)-(1.3)

x(x, z) =u

x= ux z

[2wx2 x

]xz(x, z) =

u

z+w

x= wx + + wx =

(3.2)

42

3.1.1 Kinematics 43

In the following a notation /x = , 2 /x2 = is used for abbreviation. The overbars on u, w will be omitted in the following. To simplify the notation u,w will

be written instead.

Considering shear deformations, deformations are advantageously defined as

(x) =u

x, (x) = = w (3.3)

With Eq. (3.2) this leads to strains

x(x, z) = (x) z (x) (3.4)

linearily varying along the beam height with extremal values at the top and bottom of thecross section.

In the following , , are chosen as deformation variables.

strain of the reference axis curvature of deformed cross sections1 shearing angle of deformed cross sections relative to reference axis

To describe material behavior, deformation variables have to be connected to force vari-ables, which are moment M , normal force N and shear force Q in case of 2D beams.Generally, the following dependencies are assumed

M = M(, ), N = N(, ), Q = Q() (3.5)

Basics of beam theory look quite simple, but there are some little inconsistencies.

A shear strain xz , which is constant over the cross section leads to non-vanishingshear stress at the lower and upper side of a beam. But this contradicts the localequilibrium conditions.

On the other hand, a parabolic or other nonlinear course of shear stresses accordingto equilibrium conditions with linear normal stresses leads to a curved course ofshear strains with vanishing values on top and bottom sides.

These contradictions can be resolved with plate theory. Plane beam theory is itslimiting case or a very useful approximation, respectively.

1This is different compared to the curvature of the deflectionw of the reference axis. Both are related by Eq. (3.3).


44 3.1 General Cross Sectional Behavior

3.1.2 Linear elastic behavior

We consider Eq. (1.21) with y = 0, i.e. y = x, and gain as normal stressesx(x, z) = E x = E [(x) z (x)] (3.6)

and as shear stresses with the shear modulus G

xz(x) = Gxz, G =E

2(1 + )(3.7)

Internal forces

N =

z2z1

x bdz = E

z2z1

bdz (x) E z2z1

z bdz (x)

M = z2z1

x z bdz = E z2z1

z bdz (x) + E

z2z1

z2 bdz (x)

Q =

z2z1

xz bdz = G

z2z1

xz bdz = G

z2z1

bdz (x)

(3.8)

The shear correction factor is used to compensate the difference between meanshearing strain/stress over the cross section and the shearing strain/stress , G inthe reference point with z = 0.

In case of a rectangular cross section shape with a reference axis through the midpoints it is = 5/6. This is only valid in the linear elastic case. Other cross sectionshapes have other values .

Section properties with cross-sectional area A, sectional modulus S and second momentof area J

A =

z2z1

bdz, S =

z2z1

z bdz, J =

z2z1

z2 bdz (3.9)

Generally it is assumed, that the reference axis x coincides with the centre of area, whichhas a condition

S =

z2z1

z bdz = 0 (3.10)

which formally simplifies a lot in the linear elastic case, but is not mandatory.

Finally, in the linear elastic case under the condition of Eq. (3.10) we get the well knownrelations2

N = EA, M = EJ , Q = GA (3.11)

or material stiffness which equals the tangential material stiffness

C = CT =

EA 0 00 EJ 00 0 GA

(3.12)or in case of a theory neglecting shear deformations

C = CT =

[EA 00 EJ

](3.13)

which altogether is the simple linear elastic form of Eqns. (3.5).

2Sign for moment is different compared to classical structural beam. The difference results from a differentorientation of the z-axis, see Fig. 3.1.


3.1.3 Cracked reinforced concrete behavior 45

3.1.3 Cracked reinforced concrete behavior

3.1.3.1 Compressive zone and internal forces

For cross sections of reinforced concrete structures the linear elastic relations3 hold onlyuntil the tensile strength of the concrete is reached.

In the following cracked sections are treated, i.e. beam strains x exceed the tensile limitstrain ct of concrete.

More kinematic items

The reference axis is placed in the centre of the cross section, thus the bottom sidehas a coordinate z = z1 = h/2 and the top side z = z2 = h/2.

We look at the strain 1 at the bottom side z1 = h/2 and 2 at the top side z2 = h/2with the cross section height h. Eq. (3.4) leads to

1 = z1 = + h2, 2 = z2 = h

2 (3.14)

and further to =

1 2h

, h = z2 z1 (3.15)Correct signs for strains have to be considered, e.g. 1 > 0, 2 < 0 in bending withcompression on the top side.

The coordinate of a line (in width direction) with a given strain x = is determinedwith Eq. (3.4)

z =

(3.16)

e.g. the zero line = 0 is determined with

z0 =

(3.17)

The vertical z-coordinates of the concretes tensile or compressive limit strain maybe determined in the same way.

Strains in cracked concrete sections are fictitious values regarding concrete. The lower reinforcement has a coordinate zs1 = h/2+d1 und the upper reinforce-

ment zs2 = h/2 d2, where d1, d2 give the edge distances of both reinforcements.Eq. (3.4) leads to reinforcement strains

s1 = (h

2 d1

), s2 = +

(h

2 d2

) (3.18)

Perfect bond is implicitely assumed.

3The reinforcement part has then to be considered with a weighting factor n = ES/Ec.



Concrete compressive zone

Edge strains according to Eq. (3.14) and zero line Eq. (3.17) determine the size ofthe concrete compressive zone:

z0 < h/2 and 1 < 0 cross section totally under compressionz0 < h/2 and 1 0 totally under tensionh/2 z0 h/2 and 2 < 0 upper bending compressive zoneh/2 z0 h/2 and 1 < 0 lower bending compressive zonez0 > h/2 and 2 < 0 totally under compressionz0 > h/2 and 2 0 totally under tension

Cases with compression or tension of total cross section may also include bendingmoments and normal forces. Concrete contribution is assumed within a cross sectionrange zc1, zc2:

cross section totally under compression zc1 = z1, zc2 = z2totally under tension no concrete contributionupper bending compressive zone zc1 = z0, zc2 = z2lower bending compressive zone zc1 = z1, zc2 = z0

To consider a concretes restricted compressive limit strain or an tensile limit strainlarger zero due to a tensile strength the value z0 has to be replaced according to Eq. (3.16) with appropriate values chosen for .

Resulting internal forces

Normal forceN = As1 s1 +As2 s2 +

zc2zc1

c bdz (3.19)

where uniaxial reinforcement stresses s1, s2 and conrete stresses c are functionsof x, see Eq. (3.4). Correct stress signs have to be considered.

Bending moment

M = As1s1 zs1 As2s2 zs2 zc2zc1

c z bdz (3.20)

for sign conventions see Fig. 3.3.

A variable cross section width b(z) may be regarded within a numerical integration ofEqns. (3.19), (3.20), (3.43), (3.44).



3.1.3.2 Linear concrete compressive behavior

A linear stress-strain behavior is assumed for concrete in its compressive zone

= Ec , Ec =

{Ec 00 else

(3.21)

excluding a tensile strength. A limited tensile strength may easily be incorporated.

A cross section with height h is considered. A beam strain , given, a compressive zoneis determined according to Page 46. The compressive zone has an extension zc1 z zc2 with a height hc = zc2 zc1 and edge strains

ce = B (3.22)with

ce =

(c1c2

), B =

[1 zc11 zc2

], =

(

)(3.23)

This leads to edge stresses

ce = Ec ce = E

c B , ce =

(c1c2

)(3.24)

Following cross sectional values will be used

Ac =

zc2zc1

bdz, Sc =

zc2zc1

z bdz, Jc =

zc2zc1

z2 bdz (3.25)

Extremal stresses at zc1, zc2 are given by c1, c2 with a linear course in between accord-ing to Eq. (3.21). They are interpolated with

c(z) =c1zc2 c2zc1

zc2 zc1 +c2 c1zc2 zc1 z (3.26)

Thus, concrete contributions to internal forces are given by

Nc =

zc2zc1

c(z) bdz =c1zc2 c2zc1

hcAc c1 c2

hcSc

Mc = zc2zc1

c(z)z bdz = c1zc2 c2zc1hc

Sc +c1 c2

hcJc

(3.27)

see also Eq. (3.8), orc = A ce = A B ce (3.28)

with

c =

(NcMc

), A =

[Ac ScSc Jc

], B =

1

hc

[zc2 zc11 1

](3.29)

whereby c and ce have to be clearly distinguished in the following!

Eq. (3.21) is applied to stresses ce, ce at zc1, zc2, i.e. leading to a material stiffness

c = C (3.30)with

C = Ec A B B = Ec A = Ec[

Ac ScSc Jc

](3.31)

In case of rectangular cross sections with width b and a height h this evaluates to Ac =b(zc2 zc1), Sc = b2(z2c2 z2c1), Jc = b3(z3c2 z3c1), in case of zc1 = h/2, zc2 = h/2 toAc = bh, Sc = 0, Jc = bh

3/12. Contrarily to Eq. (3.11) the Eq. (3.30) couples a normalforce Nc to and a bending moment Mc to .



As the compressive zone limits zc1, zc2 may change due to change of in , the relationEq. (3.30) is basically nonlinear. To consider this effect Eq. (3.28) is written as(

NcMc

)=

[Ncc1

Ncc2

Mcc1

Mcc2

](c1c2

)+

[Nczc1

Nczc2

Mczc1

Mczc2

](zc1zc2

)(3.32)

orc = A ce + Az zc (3.33)

with A according to Eqs. (3.28,3.29). To simplify Az a linear variation of width b isassumed with b1 = b(zc1), b2 = b(zc2). This yields

Az =1

6

c1b2 2c1b1 2c2b2 c2b1 c1b2 + 2c1b1 + 2c2b2 + c2b1(b2zc1 + zc2b2 + zc1b1)c2+(b2zc1 zc2b1 + 3zc1b1)c1

(b2zc1 3zc2b2 zc2b1)c2(zc1b1 + zc2b2 + zc2b1)c1

(3.34)

The values zc1, zc2 stand for zero lines given by

z0 =

, z0 =

2 (3.35)

see Eq. (3.17), or upper or lower edges of the concrete compression zone. The followingcases have to be considered:

1. Dominating bending with lower compression zone zc1 = h/2, zc2 = z0 < h/2and zc1 = 0, c2 = 0, c2 = 0 and(

zc1zc2

)=

( h2

),

(zc1zc2

)=

[0 01 2

](

)(3.36)

2. Dominating normal forces with fully compressed cross section zc1 = h/2, zc1 =0, zc2 = h/2, zc2 = 0, hc = h and(

zc1zc2

)=

( h2h2

),

(zc1zc2

)=

[0 00 0

](

)(3.37)

3. Dominating bending with upper compression zone zc1 = z0 > h/2, zc2 = h/2and zc2 = 0, c1 = 0, c1 = 0 and(

zc1zc2

)=

(h2

),

(zc1zc2

)=

[1 20 0

](

)(3.38)

Anyway, we setzc = Bz (3.39)

We use Eqs.(3.22,3.39) and (3.24) to formulate the edge strains and stresses

ce = B zc = B Bz = (B Bz) ce = E

c ce = E

c (B Bz) (3.40)

This yields a tangential material stiffness together with Eq. (3.33)

c = Ec A (B Bz) + Az Bz

=[Ec A (B Bz) + Az Bz

] = CT

(3.41)

also considering the change of the extension of the concrete compressive zone.

A reinforcement can be superposed.



3.1.3.3 Nonlinear concrete compressive behavior

Material behavior

Concrete uniaxial stress c see Eq. (2.1) or alternatives Modelcode 90 [Com93,2.1.4.4.1] and DIN 1045-1 [din08, 9.1.5, 8.5.1].

Reinforcement uniaxial stress s see Eq. (2.30) or alternatives Modelcode 90 [Com93,2.2.4] and DIN 1045-1 [din08, 9.2.3, 8.5.1]

Tangential material stiffness for bending moment and normal force

Strain dependence on deformation variables according to Eq. (3.4) leads tox

= 1,x

= z (3.42) With Eq. (3.19)

N

= As1

s1x

x

+As2s2x

x

+

zc2zc1

cx

x

bdz

= As1s1x

+As2s2x

+

zc2zc1

cx

bdz

N

= As1

s1x

x

+As2s2x

x

+

zc2zc1

cx

x

bdz

= As1 s1x

zs1 As2 s2x

zs2 zc2zc1

cx

z bdz

(3.43)

With Eq. (3.20)M

= As1 s1

x

x

zs1 As2 s2x

x

zs2 zc2zc1

cx

x

z bdz

= As1s1x

zs1 As2s2x

zs2 zc2zc1

cx

z bdz

=N

M

= As1 s1

x

x

zs1 As2 s2x

x

zs2 zc2zc1

cx

x

z bdz

= As1s1x

z2s1 +As2s2x

z2s2 +

zc2zc1

cx

z2 bdz

(3.44)Basically the derivatives of the longitudinal stresses s1/x, s2/x, c/xare needed, the latter varying with z in the limits zc1, zc2, see Fig. (3.1), wherebydependency of integration limits zc1, zc2 on , has been neglected.

The tangential material stiffness, see Eq. (1.26), is given with

CT =

[N

N

M

M

](3.45)

which again couples normal force to and moment to . Linear elastic, symmetric system as special case with s1/x = s2/x =Es, c/x = Ec and

bdz = A,

z bdz = 0,

z2 bdz = J .

To include shear forces Eq. (3.45) has to be extended w

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