Appendix - Aalborg Universitetprojekter.aau.dk/projekter/files/213944465/Appendix_Project_C...2 Appendix B Analytical analysis Singly reinforced beam ... (1984) for concrete applications.

1

0.00E+00

2.00E+08

4.00E+08

6.00E+08

8.00E+08

1.00E+09

1.20E+09

0 0.02 0.04 0.06 0.08 0.1 0.12

Mo

me

nt

[N*m

m]

Rotation [rad]

Moment-Rotation

AbaqusData

AnalyticalMoment

ElasticMoment-Rotation

Appendix

1 Appendix A Test

The result of the test performed in order to validate the approach used in this master thesis.

Angle of rotation formula in elastic behaviour for the corresponding static system:

𝐿 = 600 mm Young Modulus

Moment of inertia

E 200 103

N

mm2

Ib h

3

126.3 10

9 mm

4

2

The formula was used to find the elastic moment in respect to the rotation applied to the beam.

Moments Elastic

0

63000000

126000000

220500000

346500000

472500000

598500000

724500000

850500000

976500000

Analytical calculations

Yield stress

Length

Area

Moment

fy 30N

mm2

L 6000mm

h 600mm Height

Widthb 350mm

A h b 2.1 105

mm2

M1

2fy A

h

2

M 9.450 108

N mm

3

Abaqus data

Rotation Moments

0 0

0.001 59143400

0.002 1.18E+08

0.0035 2.07E+08

0.0055 3.25E+08

0.0075 4.44E+08

0.0095 5.62E+08

0.0115 6.8E+08

0.0135 7.89E+08

0.0155 8.02E+08

0.0175 8.14E+08

0.0195 8.26E+08

0.0215 8.38E+08

0.0235 8.5E+08

0.0255 8.62E+08

0.0275 8.74E+08

0.0295 8.86E+08

0.0315 8.98E+08

0.0335 9.1E+08

0.0355 9.23E+08

0.0375 9.35E+08

0.0395 9.43E+08

0.0415 9.46E+08

0.0435 9.47E+08

0.0455 9.48E+08

0.0475 9.48E+08

0.0495 9.48E+08

0.0515 9.48E+08

0.0535 9.48E+08

0.0555 9.48E+08

0.0575 9.48E+08

0.0595 9.48E+08

0.0615 9.49E+08

0.0635 9.49E+08

0.0655 9.49E+08

0.0675 9.49E+08

4

0.0695 9.49E+08

0.0715 9.49E+08

0.0735 9.49E+08

0.0755 9.49E+08

0.0775 9.5E+08

0.0795 9.5E+08

0.0815 9.5E+08

0.0835 9.5E+08

0.0855 9.5E+08

0.0875 9.5E+08

0.0895 9.5E+08

0.0915 9.5E+08

0.0935 9.51E+08

0.0955 9.51E+08

0.0975 9.51E+08

0.0995 9.51E+08

0.1 9.51E+08

5

2 Appendix B Analytical analysis

Singly reinforced beam

Radius of reinforcement

Area of steel

Yield strength of steel reinforcement

Compressive strength of concrete

Height of RC Beam

Concrete cover

Ultimate strain in concrete

Effective depth

Young modulus of elasticity

Yield strain

Neutral axis

Strain of steel

radius 16mm

As 4 radius2

3.217 103

mm2

fy 460N

mm2

fc 30N

mm2

30 MPa

h 600mm

b 350mm Width

c 25mm

c 0.0035

d h c 16mm 559mm

Es 200GPa 2 105

MPa

yfy

Es0.002

xAs fy

0.8fc b176.169mm

sd x

xc 0.007606

Mn As fy d1

20.8x

7.229 108

N mm

6

Doubly reinforced concrete beam analysis

Aria of steel in tension

Radius of reinforcement

Area of steel in tension

Aria of steel in compression

Steel proprieties

Yield strength of steel reinforcement

Modulus of elasticity

Yield strain of concrete

Concrete proprieties

Compressive strength of concrete

Ultimate strain of concrete

Height of RC Beam

Concrete cover

Effective depth

Depth to compression reinforcement

rt 16mm

Ast 4 rt2

3216.991mm2

rc 8mm

Asc 2 rc2

402.124mm2

fy 460N

mm2

Es 200 103

N

mm2

yfy

Es0.002

fc 30N

mm2

30 MPa

c 0.0035

h 600mm

c 25mm

b 350mm

d h c rt 559mm

d' rc c 33mm

7

From https://www.wolframalpha.com/ it is found that x=150.03 mm

Neutral axis

Length of stress distribution

Verify the strains if they have reached yielding

Nominal moment

xAst fy Asc Es 0.00273

0.8fc b150.03mm

x1 x0.8 120.024mm

s'c x d'( )

x0.00273 s' y

sd x

x

c 0.00954 s y

y 0.5x1 60.012mm

Cs' fy Asc 1.85 105

N

Ts fy Ast 1.48 106

N

Mn Cs' y d'( ) Ts d y( ) 7.434 108

N mm

8

3 Appendix C

Mohr-Coulomb criterion the theory is taken from Niels Saabye Ottosen, Matti Ristinmaa. The

Mechanics of Constitutive Modelling. 2005.

Failure characteristics for concrete

We assume that 𝜎2 is of minor importance

The most simple expression of this form is then provided by a linear relation between the

principal stresses

Where k and m are material parameters. Requiring that this expression should predict the

uniaxial compressive strength value 𝜎𝑐, the stress state (𝜎1, 𝜎2, 𝜎3) = (0,0, −𝜎𝑐) should fulfil the

above equation and we find :

9

This so-called Coulomb criterion was suggested by Coulomb (1776) and is the oldest criterion

ever proposed.

Alternative formulation of the Coulomb criterion:

10

11

12

In order to further elucidate the properties of the Coulomb criterion, we consider its predictions

for plane stress conditions. From (8.38), the results shown in Fig. 8.25 are easily obtained

(note that in this figure the usual convention of 𝜎1 > 𝜎2 > 𝜎3 has been abandoned). It appears

that the predicted uniaxial tensile strength becomes 𝜎𝑡 = 𝜎𝑐/𝑘.

Due to its simplicity, the Coulomb criterion is widely used in analytical applications, cf. for instance

Chen (1975) for soil applications and Nielsen (1984) for concrete applications. In numerical

applications, however, its use is impeded by the comers of the surface, cf. Fig. 8.24. By

calibration of the parameter k, the criterion can be used to model a large variety of

materials.

Concrete damage plasticity

This part is taken from Dassault Systèmes. Abaqus Analysis User's Manual. 2013.

The concrete damaged plasticity model in Abaqus:

provides a general capability for modeling concrete and other quasi-brittle materials in all types of structures (beams, trusses, shells, and solids);

uses concepts of isotropic damaged elasticity in combination with isotropic tensile and compressive plasticity to represent the inelastic behavior of concrete;

can be used for plain concrete, even though it is intended primarily for the analysis of reinforced concrete structures;

can be used with rebar to model concrete reinforcement; is designed for applications in which concrete is subjected to monotonic, cyclic,

and/or dynamic loading under low confining pressures; consists of the combination of nonassociated multi-hardening plasticity and

scalar (isotropic) damaged elasticity to describe the irreversible damage that occurs during the fracturing process;

allows user control of stiffness recovery effects during cyclic load reversals; can be defined to be sensitive to the rate of straining; can be used in conjunction with a viscoplastic regularization of the constitutive

equations in Abaqus/Standard to improve the convergence rate in the softening regime;

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