Torsional damage of concrete beams with softening behaviour

theoretical and

'-~ 2~:. :i~;"

1

ELSEVIER

applied fracture mechanics Theoretical and Applied Fracture Mechanics 22 (1995) 63-70

Torsional damage of concrete beams with softening behaviour C.G. Karayannis Department of Civil Engineering, Democritus University of Thrace, Xanthi 67100, Greece

Abstract Analysed in this paper is the torsional damage of concrete beam with softening behaviour. Change in the local stiffness and dissipated strain energy density are determined as the torsional load or rotation is increased. The idealized stress-strain curve is bilinear with a positive and negative slope. Use is made of the equations of elasticity for torsion and isoparameric mapping with finite difference. Numerical results are obtained for the pure torsion of a rectangular beam and combined torsion/compression of an I-beam. Determined are the critical torques which tend to agree well with the test data.

1. Introduction Material damage is a gradual process where its original homogeneous state is disturbed with increasing load. A reduction in the local stiffness could occur in a homogeneous fashion. For concrete the uniaxial stress and strain response need not be monotonic even though the load is increased monotonically. That is, the concrete could continue to absorb energy even after the equivalent uniaxial stress has reached the maximum. The strain would continue to increase with decreasing stress. Such a behaviour has been referred to as softening. Stress and failure analyses of concrete with softening can be found in [1-3]. The method in [4] accounts for change in the local moduli of a material as it is damaged with each loading step. This work further explores the damage of concrete beams subjected to torsional loads. Gradual damage is associated with changes of the local moduli and the dissipated strain energy density.

Results of the two examples showed that the critical torques are in good agreement with the experiments.

2. Concrete behaviour Recent experiments have shown that concrete under tension acquired a postcracking softening behaviour. Even though the concrete has cracked, the concrete can still function while softening takes place. Determination of its remaining strength involves the assessment of damage due to deflection and cracking [1-7]. 2.1. Bilinear stress-strain curve

The traditional design approach is to terminate the use of a concrete structure when the maximum tensile stress, say O'm~,, reaches the tensile strength of the concrete O'er It is now well recognized that this would be evenly conservative

0167-8442/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved

SSDI 0 1 6 7 - 8 4 4 2 ( 9 4 ) 0 0 0 4 9 - 2

(L G, Karayannis / Theoretical and Applied Fracture Mechanics 22 (1995) 63- 70

64

2.2. Failure criteria T

~Ec j,"!'E°~ //

o~ . . . .

/ /

/

," __~ - ~ ,

~

/

// I

//, 0

Failure criteria can be assumed to determine the load carrying capacity of the concrete structure. Fig. 3 displays the Mohr failure envelope [8] which is simplified as straight lines BD and DE. The former is tangent to the circle Cj (compression failure) at 37 ° with the horizontal line and latter to C 2 (tension failure). For a stress state ((r, z), the strength criterion is given by

'

~

Test curve for pure

tension

~',

~

Idealized stress-strain curve

....

• E;¢tu

%,

~er

Strain

g

oct

Fig. 1. Typical test curve for concrete in tension.

- 0.1594 ~-c

oct

'

(2) and provides an underestimate of the concrete strength. The present work shall consider a bilinear stress-strain behaviour for the concrete as stresses be the dotted lines in Fig. 1. The solid curve corresponds to the test data in [3]. Identified in Fig. 2 are the locations of point u (~rct, Ecr) , a (~r, E~), and f (0, ecru). The secant modulus of elasticity is Ec~ = ~rct/Ecr. After the peak stress, the stress will decrease while the strain will continue to increase until the ultimate final strain Ect. is reached. Unloading can occur at any point aline the line 0a. A degradation of the stiffness or modulus prevails. Reloading would then follow the path Oaf. A p a r a m e t e r a

=

Ectu/Ecr

(l)

can be introduced to define the degree of softening; it will hence be referred to as the softening coefficient.

.l

where ~rc and ~rct are, respectively, the compressive and tensile strength. If reference is made to the tensile strength (rc~ obtained from the softening portion of the curve, then

• i .

-

1 +

--7

O-ct

•

(3)

oct

According to C16-C50 of the European standards (CEB MC 90), the concrete compressive strength ~rc varies from 2500 to 7500 psi. The tensile strength of concrete oct is given by [9] ,rot = cv'~rc ,

(4)

in which c takes the values of 5 to 6. It follows that ~rc

Vzrc

o'c~

c

-- =

= 8.33

/

~

to 17.32.

~

B

(5)

Internalfriction ang,e37°

//'~

)::/i

u

O

°

- - ~ l .

Ecs

.

.

.

.

.

.

O"

Strain ¢

Fig. 2. Idealized stress-strain curve for concrete in tension.

Fig. 3. Mohr's failure envelope as simplified by Cowan.

C.G. Karayannis /Theoretical and Applied Fracture Mechanics 22 (1995) 63-70 4x)

17.32 (compression)

>

r

"a-

i

3. Analytical considerations 2.0

1

¢/)

where E c and E are, respectively, the initial and current modulus of the concrete.

4.0

(

2.0

1,0

(Compression)

Consider the deformation field in elasticity for the case torsion. There prevail only two non-zero shear stresses zxz and zyz which can be expressed in terms of the stress function F as

1.0

"~

O.O

ezx = OF/Oy,

0-0

0.0

0.2

0.4

0,6

0.8

1.0

Fig. 4. Concrete failure stesses under biaxial loading.

0-x The normalized quantities r / o h are plotted against o'/tr~ in Fig. 4 for different ratios O'c/O'ct. Depending on the value of o-, ~" can differ by a factor of 3.5.

2.3. Dissipated strain energy density As the concrete material is damaged at each increment of loading, the local effective modulus and strain energy density will change accordingly. Referring to Fig. 2, the dissipated strain energy density ( d W / d V ) d at a is given by the area Oua. The entire strain energy density (dW/dV)~ correspondes to the area Ouf. In terms of the stresses and strains at u, a and f, both ( d W / d V ) a and ( d W / d V ) c can be written as d=

= ~O'ct/~ctu .

c The ratio D

(dW/dV)~ (dW/dV)c

(9)

~

+ ~-y G 0y ]

- 20,

(10)

where G may depend on x and y.

3.1. lsoparametric mapping The domain of interest is discretized by 8-node isoparametric elements. That is all physical elements (x, y) regardless of their configuration are mapped to the (sr, r/) coordinates such that - 1 < ~:<1 and - 1 < ~ 7 < 1 where the nodes of the mapped elements are located at ~ = + 1 and B = + 1. If (xi, Yi) are the nodal coordinates, then the coordinates of any point in an element can be obtained as 8

8

x = Y: (Nixi),

Y = E (/WiY/),

i=1

(11)

i=1

where N~ are the shape functions. The transformation matrix is given by

O'at~ctu)'

(6)

[Ox/a~

Ox/O~ ]

J = [ oy/O~

Oy/O'q "

(12)

Let (7)

serves as a measure of the degree of damage. Similarly, the change in the total moduli may be quantified by

K = ( E c - E ) / E c,

~zr = - OF/Ox.

If G denotes the shear modulus of elasticity and O the angle of twist per unit length, the governing equation is given by

Normal stress o'Io"c

-~

65

(8)

G =jTj=

g12] gz2J'

[gll [g21

(13)

where jT is the transpose. The reverse of the Eq. (13) is l

G-1

[g'n = [ g21

glz/ g22 ]"

(14)

C.G. Karayannis/ Theoretical and Applied Fracture Mechanics 22 (1995) 63-70

66

8 50 cm

y

( r~:)i, s = (g'2,)iO

F i , j + 1 -- F i ,j --,

2

C + , , j - C.-,,j + (g22)i.,

2

(19)

Hence, the governing equation can be approxim a t e d as I

{~lFi:l,j

J_

I +°12Fi-l,j+l +°~3Fi+l,j+l +°14Fi+l,j-I

+ ol5Fi_2,j + a 6 F i , j + 2 + + a7Fi+2, j + as/~i,j_2

+ agF, 0 = - 80aij,

where oq, a 2 . . . . a 9 are expressions of the quantities G, A, g',a, g'12 and g22 at the a p p r o p r i a t e nodes.

Fig. 5. Cross-section data of a rectangular beam.

It follows that the shear stress c o m p o n e n t s are given as

[ rn~ [ OF/8~ re~ I =G-'[ OF/O~7]

(20)

(15)

4. Rectangular beam Consider a rectangular b e a m with the crosssection 8.50 × 17.80 cm as shown in Fig. 5. T h e compressive strength is ~r~ -=- 0.88~rc<2o> = 32.5 M P a .

such that

I;~r:Y]=J[ %~]'r~]

(16,

If zl denotes the d e t e r m i n a n t of J, then Eq. (10) can be t r a n s f o r m e d into the (~, ~7) domain as

-~(%za)+~-~ (r¢zA) = -2GOA.

(17)

3.2. Finite difference Let (i, j) be the local coordinates of an elem e n t such that Eq. (17) may be discretized as

- (rezA)i_ ,,j = - 2GOal, j.

(18)

(2t)

In Eq. (21), (r~20)= o'~10)/1.12 is o b t a i n e d from the compression test of a 10 cm cube that gives ~rc<10) = 41.3 MPa. A value of c = 5.5 in Eq. (4) is selected and hence Oct = 5.5~cc = 2.60 M P a .

(22)

With an initial modulus of elasticity E c = 30.320 M P a and EJEcs---1.35 [3] an estimate value of Ecs---22.460 M P a is used. T h e shear modulus of rigidity is G c = 935.8 MPa. Numerical results are obtained for a = ectu/ecr = 1, 2, 3, 5 and 7, where a = 1 represents the

Table 1 Critical torques for rectangular beam with different softening coefficients Coefficient a

Torque T~ (kN cm)

Energy density ratio D (%)

Moduli ratio K (%)

F/,j+ , - Fi,j_ 1

1

2

2 3 5 7

81.0 100.8 107.9 116.8 121.8

0.3 1.7 2.7 3.6 3.8

0.3 2.8 6.0 12.1 16.4

T h e shear stress c o m p o n e n t s referred to (sc, 77) become ('rrtz)i, j = (gql)i,j

F,+,,j + (g'12)i,j

2

'

C G. Karayannis / Theoretical and Applied Fracture Mechanics 22 (1995) 63- 70

67

140

a-7

~

120

100

80

5/ 30 7.0

2

3

~' 20 3.0

¢D -i

|

60

a. 10

0

I--

40 /x~-

20

o

Experiment [10]

:

o.0 0 0.0

i

J

0.5

1.0

15

I r

1,0

1.5

Rotation x 104 (rad/cm)

i"

Rotation x 104 (rad/cm)

0.5

Fig. 8. Parameter K (%) against rotation for a rectangular beam.

Fig. 6. Torsional behaviour of a rectangular beam.

linear response with no softening. The results for the critical torque Tc, D and K are summarized in Table 1. For a = l , Tc = 8 1 kN cm which agrees with T - - 7 9 . 9 kN cm obtained from torsion theory of elasticity. Experiments [10] gave a value of 118 kN crn which is closest to T~ = 116.8 kN cm for a = 5 in Table 1. Fig. 6 gives a plot of the torque T versus the rotation of the beam for different a. The dotted curve corresponds to the experimental results in [10]. Displayed in Figs. 7

and 8 are respectively, the percentage of D in Eq. (7) and the percentage of K in Eq. (8) as a function of the beam rotation. Note that all curves tend to increase with increasing angle of rotation. Such a trend suggests that D and K could be correlated to damage as they increase with the load. Figs. 9 and 10 show, respectively, the field

O0

2.0

4.0

40

¢J 1<

6O

e.o

¢II

30

o o

12.0

20 14.0 10 -

a

~

7

160

0.0 0

r 0.0

20

40

6.0

80

j

O.S 10 t5 Rotation x 104 (md/cm)

Fig. 7. P a r a m e t e r D (%) against rotation for a rectangular beam.

Coordinate y (cm)

Fig. 9. Field distribution of parameter D (%) over a rectangular cross-section for a = 5.0 and angle of twist 0 . 7 5 x 1 0 - 4 rad/cm.

C.G. Karayannis / Theoretical and Applied Fracture Mechanics 22 (1995) 63-70

68 0.0

50 7

2.0 40 4,0

5

~J

Z

x

30

8.0

C

~

o I-.,

10.0

20

o 0

0

12.0

10

0=1

14.0 16.0

0.0

0.5

1.0

1.5

Rotation x 104 (rad/cm) 0.0

2D

4.0

60

8D

Fig. 12, Torsional behaviour of the l-beam.

Coordinate y (cm)

Fig. 10. Field distribution of parameter K (%) over a rectangular cross-section for a = 5.0 and angle of twist 0.75 × 10-4 rad/cm.

distribution of D (%) and K (%) over the rectangular b e a m cross-section.

5. I-Beam Fig. 11 shows the dimensions o f the l-beam [9] with a height of 60.96 cm, flange width of 35.56

~.~

35.56 cm

Y

cm and web width of 12.70 cm. T h e compressive strength is trc = 52.44 M P a and the tensile strength is ~rct = 3.31 MPa. T h e initial modulus of elasticity is E c = 35 560 M P a such that Ecs ~ E c is assumed. This corresponds to G c = 14817 MPa. A n axial force P = 967.0 kN prevails at a distance e = 12.42 cm from the centroidal axis. Table 2 summarizes the nummerical results for the critical torque, D (%) and K (%).The value Tc = 4470 kN cm for a = 5 in Table 2 agrees well with the test results of 4452 kN cm in [10]. Plots of D (%) and K (%) against the b e a m rotation are given in Figs. 13 and 14, respectively. Values of D and K corresponding to the critical torques are nearly the same, about 12%. Again, all the curves are increased with increasing load or rotation.

! •

2.5:~--

3

60.96 cm

30.48 cm --~..12.54

- t Fig. l 1. Cross-section data of the I-beam.

Table 2 Critical torques for I-beam with different softening coefficients. The energy density ratio D and the moduli ratio K are approximately 12% Coefficient a

Torque Tc (kN cm)

1

2861 3556 3875 4470 4758

2 3 5 7

C.G. Karayannis / Theoretical and Applied Fracture Mechanics 22 (1995) 63-70

60

increase with increasing torsional load that is indicative of the phenomenon of accumulative damage. The calculated values of the critical torques agreed with the experimental data by a proper selection of the softening coefficient that controls the character of decreasing stress as the concrete is further strained.

a=l

50

23

40

69

_E 2 0 -

Acknowledgements

10

0

0.0

i

i

0.5

i

1.0

1.5

Rotation x 10 -= (md/ern)

Fig. 13. Parameter D (%) against rotation for the I-beam.

The author wishes to deeply thank Professor G.C. Sih for his kind help in improving this paper. Also, he wishes to thank Professor E.E. Gdoutos for his valuable advice and encouragement during the preparation of this work.

6. Conclusions

Obtained numerically are the critical torques that could be applied to a rectangular beam in torsion and I-beam in torsion/extension. The material is assumed to be concrete with an ascending and descending portion of the uniaxial stress-strain curve. Such a behaviour has been observed experimentally and is idealized as straight lines with positive and negative slopes. Evaluated are the change in local moduli and dissipated strain energy density. They tend to

60

o=1

50

2 3

m

~- 2o 10

0

,

00

|

i

i

0.5

11)

1.5

Rotationx 104 (rad/cm)

•

Fig. 14. Parameter g (%) against rotation for the I-beam.

References [1] G.C. Sih, Mechanics of material damage in concrete, Fracture Mechanics of Concrete: Material Characterization and Testing, eds. A, Carpinteri and A.R. Ingraffea (Martinus Nijhoff (now Kluwer Academic), The Netherlands, 1984) pp. 1-29. [2] A. Carpinteri and G.C. Sih, Damage accumulation and crack growth in bilinear materials with softening: application of strain energy density theory, J. Theoretical and Applied Fracture Mechanics 1, 145-159 (1984). [3] V.S. Gopalaratnam and S.P. Shah, Softening response of plain concrete in direct tension, J. American Concrete Institute 82, 310-323 (1985). [4] G.C. Sih and P. Matir, A pseudo-linear analysis of yielding and crack growth: strain energy density criterion, Defects, Fracture and Fatigue, eds. G.C. Sih and J.W. Proram (Martinus Nijhoff, (now Kluwer Academic), The Netherlands, 1983) pp. 223-232. [5] D.A. Zacharopoulos et al, Crack growth in concrete slab resting on steel beam: softening and damage accumulation, J. Theoretical and Applied Fracture Mechanics 18, 65-71 (1992). [6] J.K. Kim and T.G. Lee, Nonlinear analysis of reinforced concrete beams with softening, J. Computers & Structures 44, 567-573 (1992). [7] G. Papakaliatakis and C.G. Karayannis, Structural capacity valuation of reinforced concrete elements using the strain energy density theory, Proc. Int. Conf. on Design Construction and Repair of Building Structures in Earthquake Zones, Dubrovnik, 1987. [8] H.J. Cowan, Strength of reinforced concrete under the action of combined stresses at the representation of the

70

C.G. Karayannis / Theoretical and Applied Fracture Mechanics 22 (1995) 63-70

criterion of failure by a space model, Nature (London) 169, 3-20 (1952). [9] A.H. Mattock and A.N. Wyss, Full-scale torsion, shear, and bending test of prestressed I-Girders, J. Prestressed Concrete Institute 23 (2), 22-41 (1978).

[10] R. Narayanan and A.S. Kareem-Palanjian, Torsion in beams reinforced with bars and fibers, J. Structural Engineering, ASCE 112, 53-66 (1986). [ll] V. Kalevras, Lessons of Reinforced Concrete, Xanthi (1981) (in Greek).