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Case Studies in Structural Engineering

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Strut effectiveness factor for reinforced concrete deep beams under dynamic loading conditions

Ammar N. Hanoona b *, M.S. Jaafara, Farzad Hejazia, F.N.A. Abd Aziza

a Department of Civil Engineering, Universiti Putra Malaysia, Malaysia b The engineering affairs department, Baghdad University, Iraq

CrossMark

ARTICLE INFO

Article history:

Received 27 March 2016

Received in revised form 1 August 2016

Accepted 9 August 2016

Available online 11 August 2016

Keywords: Deep beam

Strut effective strength Dynamic Strain rate Shear strength Strut-and-tie model

ABSTRACT

In a current study, a new strut effectiveness factor based on Strut-and-Tie Model (STM) is proposed to assess the ultimate shear strength of reinforced concrete (RC) deep beam subjected to dynamic loads. The derivation of the new effectiveness factor of concrete struts is based on Mohr-Coulomb criterion failure. Two types of concrete failure, diagonal splitting and concrete crushing failure modes, are proposed and examined. The modification of the proposed model is simulated in a MATLABSIMULINK environment. The proposed model exhibits efficiency in assessing dynamic shear resistance for deep beams. Moreover, a parametric study is then conducted to examine the effect of flexural reinforcement ratio, transverse reinforcement and shear-span to depth ratio on shear behavior of RC deep beams with consideration of the changes in strain rate. The proposed effectiveness factor is validated by utilizing the experimental results obtained from the literature and shows good accuracy for prediction the shear strength of reinforced concrete deep beams under different loading conditions.

© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

During their service life, reinforced concrete (RC) structures may suffer from various deteriorations, such as cracks, concrete spalling, large deformations, and collapse [5]. Such deteriorations are caused by several factors, including aging, corrosion of steel reinforcement, and environmental effects (earthquakes, impacts, and blasts) [11,14,28].

Under high loading rates the increase in the fracture energy and peak load are influenced due to the effect of inertia force. The strength and the elastic modulus of concrete were found to increase with the increasing the loading rates. Also the yield strength and the corresponding strain of steel increased with the increasing loading rates. To get proper design for all loading types, the studying of concrete behavior under wide range of strain rate (Fig. 1) is required [22].

Numerous studies have examined the behaviors of RC slender beams subjected to drop weight impact [18,19,23] and different loading rates [3,16,17,21]. According to the current design codes, such as ACI building code [1,2,8,10,13], a RC deep beam should be analysed utilizing the STM, which takes into consideration the complex stresses flow in D-regions. Based on the best author knowledge, studies on the ultimate strength and behavior of RC deep beams subjected to different loading rates are scarce. As a result, the necessity to modify methods for assessing the deep beams ultimate shear strength has become more significant in current literature topics. Therefore, this study aims theoretically to examine the ultimate shear

* Corresponding author at: Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia, Malaysia. E-mail addresses: civileng.ammar@gmail.com, structuraleng.ammar@gmail.com (A.N. Hanoon), msj@upm.my (M.S. Jaafar), farzad@fhejazi.com (F. Hejazi), farah@upm.edu.my (F.N.A.A. Aziz).

http://dx.doi.org/10.1016/j.csse.2016.08.001

2214-3998/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/ licenses/by-nc-nd/4.0/).

Blasts

Earthquake and induced shocks

Creep Quasi static Vehicle impacts Plane crush Hard impacts (missiles, rock falls)

H-1-1-1-1-1-1-1-1-1-1-1—► .

10'8 10"7 10"6 10'5 10'4 10"3 10-2 10-1 10° 101 102 1 o3 straJ

Fig. 1. Strain rate according to real loads [22].

capacity of RC deep beams under dynamic loading conditions, and suggests a new effectiveness factor which utilizes in the STM for designing ofRC deep beams subjected to different loading rates. This factor plays a significant role in the STM design of RC deep beams and shear capacity prediction.

Various factors affect on the value of the effectiveness factor, for instance material properties, beam size, steel reinforcement, and structure loading. According to MacGregor et al. [20] the effectiveness factor v differs from 0.25 to 0.85 based on the concrete strut in the plastic truss model utilized to estimate the deep beam capacity.

In the present study, the STM is extended to account for the ultimate shear strength of RC deep beams under varying loading rates. The proposed model considers the effect of the combined tensile strength of longitudinal and transverse reinforcements and the tensile strength of concrete. A linear failure criterion based on modified Mohr-Coulomb theory is adopted. A simple and refined interaction formula for predicting the ultimate dynamic shear strength of RC deep beams is derived. To imply and integrate the dynamic effect, the proposed constitutive relationships of concrete and reinforcing steel [16] are taken consideration. The proposed model considers the interaction between two failure modes, namely, diagonal splitting and concrete crushing of struts. A case study is also presented to verify the proposed method for deep beams subjected to varying loading rates. The analysis results indicated a good accuracy for prediction the ultimate shear strength of RC deep beams under dynamic loading conditions.

1.1. Research significance

Limited information has been recorded on RC deep beams subjected to dynamic loads in literature. Moreover, the previous proposed STM approaches for RC deep beams and design equations used for slender beams, are mostly over or under estimation due to it is empirical nature. A new strut effectiveness factor for deep beams under different loading rates is proposed, with a refined STM which used to assist in the appropriate design of such members.

1.2. Limitations and assumptions of the proposed model

To develop the applicability of the STM concept for RC deep beams under dynamic loads, the following assumptions and limitations are made:

• The proposed model is confined for simply supported RC deep beams.

• A uni-axial compressive stress f2 (Fig. 1) is applied to the concrete strut which inclined at an angle q with respect to the beam axis;

• The failure of shear tension caused by the inadequate anchorage of flexural reinforcement is not considered.

• The proposed model considers the effect of crushing and diagonal splitting concrete failure.

• The proposed model implies the dynamic effect by considering the constitutive relationships of concrete and steel reinforcement which proposed by Fujikake et al. [16]

• The proposed model implies iterative procedures to calculate the ultimate strength of deep beams under dynamic loading conditions.

2. Stress-strain relationships of concrete and steel reinforcing under dynamic loading

2.1. Material model for concrete

Concrete materials are sensitive to changes in strain rates. Under dynamic loading rates, both tensile and compressive concrete strengths significantly increase. In the current study, the strain effect, which can be accounted by utilizing a Dynamic Increase Factor (DIF). The DIF values proposed in the literature Fujikake et al. [16] are used for both concrete and steel materials. Fig. 2 shows the constitutive relationships for the stress-strain of concrete and steel reinforcement [16].

Reinforcing steel

Fig. 2. Constitutive relationships of concrete and reinforcing steel.

Fig. 3. Strut-and-tie model for simply supported deep beams.

For concrete in compression:

/'c,dyn t)]'

DIF = '

/'c,dyn refers to the dynamic concrete compressive strength (MPa), e is the strain rate, and f'c is the static concrete compressive strength (MPa).

Bs = 1.2 x 10-5 (s-1)

2.2. Material model for steel

For the reinforcing steel, the dynamic yield strengthfyjdyn can be calculated from Eq. (3) [16] with consideration of the changes in strain rate e.

DIF = % = (1.202 + 0.04loglOB) >/y

/y refers to the static yield strength of steel reinforcement.

2.3. Mechanisms of shear resistance in deep beams

By referring to Fig. 3, two stresses are created through the diagonal strut because of the applied load Vn (Fig. 3).

Fig. 4. Determination of Tensile Stress f1 at Bottom Nodal Zone.

1. Compressive stress f2 is created through a direction between t aed load and support. This stress causes the possible formation of a concrete crushing failure in the diagonal strut, which has to be resisted by the concrete compressive strength vf 'c [8] (CSA, 1994).

2. Transverse tensile stressf1 is created in a direction perpendicular to the diagonal strut. Consequently, the deep beam may fail by concrete splitting, which can be resisted by the longitudinal steel, transverse reinforcement, and concrete tensile strength.

The equilibrium forces at the bottom nodal zone (B) of the diagonal strut (Fig. 3) are concluded through the following

EFV = 0; F _ Vn (4)

y c = sin0, (4)

VFX = 0; T = (5)

^ tanft 1 ;

9c refers to the angle of the inclined strut and is defined by

tan 9c = — (6)

where a is the distance between the applied load and support, and is the lever arm of the longitudinal reinforcement to the center of the upper strut (Fig. 4) and can be defined as follows.

Zc = h - - | c 2 2

where, la and lc are the depth of the top and bottom nodal zones, respectively (Fig. 3). The compressive stressf2 can be calculated from Eq. (4) as follows:

f = = Vn (7)

Actr Actr sin 9c 1 ;

where, Actr is the diagonal strut cross-sectional area (mm2), and Vn is the applied load (kN).

To determine the principal tensile stress f1 (Fig. 3) perpendicular to the diagonal strut at the bottom nodal zone, first csider a deep beam [25]

fi=ATH(8)

where Ac is the deep beam cross-sectional area; and kT sin9c/ (Ac/sin 9c) is the average tensile stress across the diagonal strut due to the component of T in the principal tensile direction of the bottom nodal zone, T sin 9c (Fig. 4), and k is the stress distribution factor.

The stress distribution is nonlinear and cannot be determined directly by the assumptions of the beam theory, therefore assumptions are needed to find the stress distribution factors (k1 and k2), as displayed ig. here k1 and k2 refer to the stress distribution factors at the bottom and top nodal zones, respectively. Previous studies [24,29] found that the values of stress distribution factors k1 and k2 of 2 and 0, respectively. This indicated best agreement with experimental results of RC deep beams under static loading conditions. The bottom nodal zone experiences a biaxial tension-compression stress state, and

the compressive strength of concrete is reduced due to the softening effect of the tensile stress. Based on the modified from the Mohr-Coulomb theory [9], in this study the failure criterion at the bottom nodal zone is assumed as a linear interactive relationship between f and f2, as displayed in the next section.

2.4. Consideration of concrete softening effect

Three main approaches, which take into consideration the concrete softening effect under biaxial tension-compression, are used. These approaches are summarized in the following:

1. Many of the codes, including [1,2,6,8], adopt the concrete strength efficiency factors, thereby resulting in statistical test results. However, an argument exists in evaluating the factors. Moreover, for some specific conditions, these factors may be over- or underestimated because they are defined as empirical values.

2. Function expressions, such as fi =f (e1), are used to consider the influence of principal strain on the compressive strength [10,12,15]. This method seems to be more accurate, but adds complexity because of the simultaneous application of equilibrium conditions, compatibility equations, and stress-strain relationships.

3. Linear interactive failure criteria, such as modified Mohr-Coulomb theory [24], are utilized to account for the softening effect directly. Thus, the relationship of Eq. (9) is utilized in this research.

f. + J^ = 1 (9)

/ c,dyn

where J1 and /2 = respective principal tensile and compressive stresses at the nodal zone, and they represent the actual stress state; /'c,dyn = cylinder compressive strength under dynamic effect, and it represents the maximum compressive capacity in the/2 direction; and/t = tensile strength contribution of reinforcement, and concrete, and it represents the maximum tensile capacity in the /1 direction.

Consequently, Eq. (9) is not applicable as the bottom nodal zone experiences a biaxial compression-compression stress state. Although there is additional lateral confinement available at the bottom nodal zone, the authors suggest that the compressive stress/2 along the diagonal strut should not exceed/'c,dyn. Thus

/2 < /'c.dyn (10)

Furthermore, the top nodal zone also experiences a biaxial compression-compression stress state. Therefore, if the width of the top-loaded region is comparable to that of the bottom support region, Eq. (10) is sufficient to safeguard failure of the top nodal zone. Thus, no further consideration is given to the top node.

The denominator term/t in Eq. (9) is the combined tensile strength contribution of reinforcement, and concrete and it is given by

e kA/y,dyn sin0S 2Aw/yw,dyn sin (0s +^w)dw

/t = -T~T~-—ñ--1--—ñ--T + /ct (11a)

Ac/sin 0S Ac/sin 0S d

where As, and Aw are respective total areas of longitudinal and web reinforcement; /y,dyn and /yw,dyn, are respective yield strengths of longitudinal and web reinforcement, and /ct is tensile strength of concrete and is given by

fct = 0.5vf 'c.dyn (11b)

The first term in Eq. (11a) represents the tensile capacity of longitudinal steel reinforcement and is derived in a similar fashion as the termf1 in Eq. (8), except that the full strength of longitudinal reinforcement is used in place of T. Furthermore, the effect of longitudinal reinforcement (Asfydyn cos 0s) in the f2 direction has been ignored for simplicity. For a deep beam with a very small a/d ratio, this component will be insignificant, as cos 0s approaches zero. On the other hand, if the a/d ratio is relatively high, the deep beam is likely to fail due to excessive tensile stress in the f1 direction. In this case, failure is governed bthe first term in Eq. (9), and thus the term hasf'c little influence on the ultimate shear strength. Therefore, it is justifiable to neglect the contribution of longitudinal reinforcement to the compressive capacity in the f2 direction. The same assumption is made for web reinforcement.

The second term in Eq. (11a) represents the tensile capacity of inclined web reinforcement at an angle 0w to the horizontal axis in Fig. 4. It takes account of different positions and arrangements of web reinforcement, be it vertical, horizontal, inclined, or combined. From the geometry of the strut-and-tie model, the tensile force contribution of web reinforcement in the f1 direction is Awfyw,dyn sin (0s + 0w).The positional influence factor dw/d [25] is introduced to account for different levels of web reinforcement.

For simplicity, the second term in Eq. (11a) can be reduce to sin2(?s or AshWynsm ^, respectively, where AsV and

Ash refer the total areas of vertical and horizontal web reinforcement within the distance of spear span.

Fig. 5. Modified Coulomb Failure Criteria.

If the bottom nodal zone is subjected to the biaxial compression-compression stress state, the following equation can be derived from Eq. (10):

Vn <Astrf 'c,dynsin 6s 2.5. Strut effectiveness factor derivation

Numerous analytical models [7,26,27] have been proposed to take in consideration the effect of compression softening in RC cracking for tension-compression stress states. Based on the proposed STM, the equation that yields the concrete strut effectiveness factor v (Fig. 5) can be expressed as:

Fc vf'

7— = vJ C,dyn Astr

Substituting Eqs. (7), (8) and (13) into Eq. (9), the following equation can be obtained:

v = 1 -

kTs sin2 es

where Ts refers to the tension force in the bottom steel reinforcement and can be illustrated as esEsAs. It is noteworthy that the second term in Eq. (14) represents the contribution of the tension force of the bottom steel resolved in the direction of the diagonal strut, which negates a certain amount of compression force in the strut itself. Substituting Eq. (11a) into Eq. (14) and using the relationship between Ts and es, the following equation for the effectiveness factor is created,

kssEsAs sin2 6s

where Es and As are respectively the elastic modulus and cross sectional area of longitudinal reinforcement; ss is the strain in the longitudinal reinforcement; 0s is the inclined angle of the strut.

As shown in Eq. (12) the effectiveness factor is influenced by many variables, such as concrete compressive strength steel yield strength, and steel reinforcement ratios. However, the strut angle 0s is the main variable that affects the effectiveness factor u significantly.

The nominal shear strength Vn can be derived by substituting Eqs. (13) and (14) into Eq. (1) as follows:

Vn = vAstrf'C, dynsin 0s (16)

2.6. STM dimensioning

The cross-sectional area of the strut Astr is calculated by the following:

Astr = bw (l a cos 0s + lb sin 0s) (17)

where bw is the width of the beam, la is the bottom tie depth (mm), and lb is the support-bearing plate width (mm).

T depth of the top node lc can be determined considering the limit equilibrium of the top node. If a stress limit of 0.85J'c is imposed on the top node, the depth lc can be determined from Eq. (18).

l ___ (18)

c 0.85f'c,dynbw tan 0s. 1 ;

The lc value cannot be calculated at first. Consequently, 0s cannot be determined, which makes an iterative approach necessary. The value of the nominal shear strength Vn can then be calculated by utilizing Eq. (16) during the iterations. Fig. 6 displays the iterative steps to calculate the ultimate capacity of RC deep beams. A MATLAB simulation environmental has been used to simulate the ultimate shear strength of RC deep beams subjected to dynamic loads. At least four to five iterations are appropriate to obtain a convergence in the results.

2.7. Influence of stress distribution factor k on the prediction of ultimate strength Vn As displayed in Eq. (11a), the maximum tensile strength in thef1 direction is:

kAJy,dyn sin 0s 2Awfyw,dyn sin (0s + 0w)dw , f Jt Ac/sin 0s + Ac/sin 0s d +Jct.

where, k is the stress distribution factor, and it is assumed that the tensile stress transverse to the concrete strut is kpt. The beam ultimate shear strength can be expressed from Eq. (16)

Vn = vAstrf c,dyn sin 0s

From Eqs. (11a) and (16), it is observed that for RC deep beams without transverse reinforcement, and tensile contribution of concrete, the ultimate beam capacity is not dependent on stress distribution factor k. That leads to the shear strength Vn to be constant along the diagonal concrete strut. This occurred due to ignored the second and third part of Eq. (11a) and omitting k value. However, for optimal assessing of shear strength, the et offct, and transverse reinforcement is taken in consideration. In order to display the influence of k on the ultimate capacity of RC deep beams, a parametric study is undertaken. With a longitudinal reinforcement rf3.26%, and a/d of 1.0,150 imaginary RC deep beams are examined utilizing the proposed model. In addition, a wide range of strain rates of ( 0.004, 0.04, 0.4, and 21 /s), and concrete compressive strength of (20, 40, and 60MPa) have been used in the analysis. In this study the value of k is assumed equal to 2.5. The influence of k on the model prediction of the ultimate shear strength Vn can be expressed by (Vn - Vk=25) /Vk=25, where Vk=2 5 represents the shear strength obtained using the proposed model with k = 2.5. The plotted curves are shown in Fig. 7.

For increasing the strain rates, the ultimate shear strength predicted decreases quickly with k increasing. Similarly, the ultimate shear strength decreases more quickly for higher concrete compressive strengths as increasing of k factor (Fig. 5). Generally, for 0 < k < 2.5, the estimation is smaller than for k = 2.5, and greater when 2.5 < k < 10. As displayed in Fig. 7, the reduction in the ultimate shear strength is about 5% for k greater than 2.5, thus, it can be said that k = 2.5 proposed is sufficiently accurate for the proposed model estimations.

As displayed in Fig. 8, the strut effectiveness factor value influenced by stress distribution factor and the strain rate values. The results indicated an increasing in the effectiveness factor with decrease in the K factor. Moreover, there is a decreasing in the strut effectiveness factor value with increase the strain rates which may be occurring due to the crack growth.

3. Comparision of test results with case study

Scarce studies have been recorded in literature for RC deep beams subjected to dynamic loads. Thus, to evaluate the proposed model, a case study [4] was employed for verification purposes. This case study dealt with RC deep beams under dynamic loading conditions.

3.1. Case study description

A total of 12 RC deep beams were tested under point loading [4]. The experimental objective was to examine the influence of loading rates on ultimate shear strength and the behavior of RC deep beams. Fig. 9 shows the notation utilized to describe

the test parameters of each specimen and the relevant details. All the beam specimens had an overall depth of h = 250mm with an effective depth and beam width of d = 210mm and bw = 150mm, respectively (Fig. 6). The beam specimens comprised two 022mm bars as flexural reinforcement at top and bottom, whereas 06mm deformed bars were utilized as web reinforcements. The yield strength of longitudinal and web reinforcement are fy = 371MPa, and fyv = fyh = 442MPa, respectively. Table 1 describes the ultimate capacities of the RC deep beams according to the test results, and Fig. 10 presents the specimen designations.

f'c = 20 MPa

1 : 7 1

Stress distribution factor, k f'c =40 MPa

i i. 1

""" ■*

Stress distribution factor, k f'c =60 MPa

NL1 •..

- Strain rate=0.0004 1/s Strain rate=0.041/s

■Strain rate=0.41/s - Strain rate=2 1/s

— Strain rate= 0.0004 1/s ■•■ Strain rate= 0.04 1/s -Strain rate= 0.4 1/s Strain rate= 2 1/s

-Strain rate= 0.0004 1/s

.......Strain rate= 0.041/s

— • - Strain rate= 0.4 1/s Strain rate= 2 1/s

Stress distribution factor, k

f'c= 20 MPa

f'c= 40 MPa

f'c= 60 MPa

2 4 6 8

Stress distribution factor, k

Fig. 7. Influence of stress distribution factor k on the ultimate shear capacity estimations.

Fig. 8. Influence of stress distribution factor, k on the strut effectiveness factor prediction.

4. Results discussion

Table 2 demonstrates the statistical analysis of the ultimate strengths predicted by the proposed model once again match the experimental results well. The mean value of the prediction is 0.976 with a standard deviation of 0.172 and coefficient of variation of 0.176. Figs. 11 and 12 display the comparison of model predictions with the experiment performed in Adhikary et al. [4]. There is only one beam (Table 2) that has inaccurate prediction. This indicates that the proposed STM model is generally trustworthy. It is also remarkable that the mean value of the case study is very close to 1.0 (0.976).

S0=0%, S42=0.42% and S84=0.84%

Fig. 9. Test case and specimen designations.

Table 1

The ultimate strength of tested beam specimens [4].

Specimen Strain rates (1/s) Dynamic shear resistance (kN)

RC1.9 S0 0.0059 170.1

0.067 246.5

0.59 290.5

3.5 353.6

RC1.9 S42 0.0068 271.9

0.041 300.6

0.48 361.3

3.1 417

RC1.9 S84 0.0027 331.8

0.029 378.2

0.46 420.75

3.7 446.7

Table 2

Case study for prediction for PExp. [4].

Specimen Strain rates (1/s) Dynamic shear resistance (kN) Test Predict Test/Predict

RC1.9 S0 0.0059 170.1 286.81 0.59

0.067 246.5 306.42 0.80

0.59 290.5 334.35 0.87

3.5 353.6 365.71 0.97

RC1.9 S42 0.0068 271.9 301.71 0.90

0.041 300.6 318.25 0.94

0.48* 361.3 352.05 1.03

3.1 417 388.29 1.07

RC1.9 S84 0.0027 331.8 307.14 1.08

0.029 378.2 328.33 1.15

0.46 420.75 368.69 1.14

3.7 446.7 413.47 1.08

Mean 0.97

SD 0.16

C.O.V. 0.166

The coefficient of variation obtained from the results (0.176) displayed good accuracy and consistency for the values obtained. This was enough to consider the proposed STM model proper for assessment of the ultimate strength of RC deep beams under dynamic loading conditions, and different geometrical and material (steel and concrete strength) properties.

RC1.9-S84

Fig. 10. Dimensions of RC deep beams and layout of reinforcements of the case study [4].

5. Parametric study

5.1. Numerical simulation case studies

After validating the proposed strut and tie model with the experimental results, this section displays a parametric examination to present more information about the behavior of RC deep beams under varying loading rates. Various key parameters such as shear span to effective depth ratios, longitudinal main reinforcement ratios, web reinforcement ratios and strain rates, used to study the response of deep beam specimens. Figs. 13 and 14 illustrate the effect of shear span to effective depth ratios and the longitudinal reinforcement ratios on the ultimate strength capacity of RC deep beams, in which there is a decreasing in the ultimate strength with increase of the shear span to depth ratio. The parametric study was simulated in the MATLAB/Simulink environment.

Fig. 11. Comparison between experimental and predicted ultimate strength.

I Predicted, KN 286.81306.42334.35365.71301.71318.25352.05388.29307.:

Fig. 12. Comparison between the predicted ultimate shear strength and the experimental.

5.2. Effect of main reinforcement ratio on DIF

A total of 36 experimental RC deep beams were examined to demonstrate the effect longitudinal reinforcement on DIF of maximum resistance under various strain rates, as shown in Fig. 15. Three span to effective depth ratios (a/d: 0.95,1.43 and 1.9) were examined. As displayed in Fig. 15, there is a DIF increases with the increase of the flexural reinforcement ratio.

5.3. Effect of web reinforcement ratio on DIF

Fig. 11 displays the influence of web reinforcement on DIF of ultimate capacity of RC deep beam under various strain rates. From the results (Fig. 16), web reinforcement does not have a significant effect on DIF, but the important effect was on the ultimate strength of the RC deep beam. This is similar to an observation by Adhikary et al. [4].

5.4. Effect of shear span-to-effective depth ratio on DIF

In this case study, three span-to-effective depth ratios (a/d: 0.9, 1.43 and 1.9) were investigated. The changing in the ratio was for span length, whereas, the depth was kept constant for all specimens. Fig. 17 illustrates the influence of shear

Fig. 13. Effect of shear span-to-effective depth ratio on ultimate strength of RC deep beams.

Fig. 14. Ultimate strength of RC deep beams with effect of different longitudinal reinforcement ratio and shear span-to-depth ratio.

span-to-depth ratio on the DIF of ultimate strength of the deep beam specimens. The results exhibit increasing in DIF with the increase of shear span-to-depth ratio of all strain rates.

6. Summary and conclusions

A simple STM approach is extended to account for the effect of strain rates, main reinforcement, and web reinforcement on the behavior of RC deep beams. Linear interactive failure is used to imply the softening effect. Moreover, the MATLAB SIMULINK software is used in simulating the proposed model. The experimental results [4] were compared with the predictions of proposed model A new effectiveness factor for concrete struts was proposed based on the failure principles of Mohr-Coulomb. The proposed STM exhibits efficiency in assessing dynamic shear resistance for RC deep beams under varying loading rates, which could be implemented quite satisfactorily. The main conclusions of the current study are presented below.

• The dynamic shear resistance of RC deep beams increases with loading rates increase.

• The analysis results indicated that the web reinforcement does not have significant effect on DIF of ultimate strength. Whereas, the results exhibit increasing in DIF with the increase of shear span-to-depth ratio for all strain rates.

a/d=0.95

1.4 1.35

1.3 1.25

1.15 1.1 1.05 1

0.05 1.05 2.05 3.05

Strain Rate, 1/s

▲ ps=0.019 • ps=0.0238 Xps=0.0286

1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1

(a) a/d=1.43

✓ 4J ** ••• J

▲ ps=0.019 • ps=0.0238 Xps=0.0286

0.05 1.05 2.05 3.05 4.05

Strain Rate, 1/s

1.45 1.4 1.35 1.3 u_ 1.25 Q 1.2 1.15 1.1 1.05 1

(b) a/d=1.9

2.05 3.05 4.05 Strain Rate, 1/s

A ps=0.019 • ps=0.0238 X ps=0.0286

Fig. 15. Effect of longitudinal reinforcement ratios on DIF of ultimate shear resistance of RC deep beams: (a) a/d:0.95; (b) a/d:1.43; (c) a/d:1.9.

Fig. 16. Effect of transverse reinforcement on DIF of ultimate shear resistance of RC deep beams: (a) a/d:0.95; (b) a/d:1.43; (c) a/d:1.9.

ps=0.019

1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1

/ ..•jr

Aa/d=0.95 • a/d=1.43 Xa/d=1.9

0.05 1.05 2.05 3.05 4.05

Strain Rate, 1/s

ps=0.024

1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1

Aa/d=0.95 • a/d=1.43 Xa/d=1.9

0.05 1.05 2.05 3.05 4.05

Strain Rate, 1/s

1.45 1.4 1.35 1.3 1.25 1.2 1.15 1.1 1.05 1

ps=0.029

____ .;----x

, .'JO,

Aa/d=0.95 • a/d=1.43 Xa/d=1.9

0.05 1.05 2.05 3.05 4.05

Strain Rate, 1/s

Fig. 17.

Effect of shear span to effective depth ratios on DIF of ultimate shear resistance of RC deep beams: (a) rs:0.019; (b) rs:0.024; (c) rs:0.029.

• The comparison of results indicates that the proposed approach yields safe and appropriate estimates for RC deep beams with different strain rates.

• With mean 0.97, and coefficient of variation of 0.166, the proposed effectiveness factor for concrete struts was indicated an ability to provide a highly accurate estimation of the ultimate shear strength of deep beams subjected to dynamic loads. The new effectiveness factor proposed can be used as a guide line in the design process of STM under dynamic loads.

• For increasing strain rates values, the ultimate shear strength predicted increases slowly with stress distribution factor, k increasing.

Acknowledgements

The authors would like to express their gratitude to the support provided by the Universiti Putra Malaysia (UPM). The

first author also acknowledges a scholarship program from the Ministry of Higher Education, Iraq.

. A.1 Worked Example

The following example show the calculation involved in determining Vn for RC deep beams under loading rates effect.

. Example RCde lates effect [4]

T ultimate capacity of Beam (Rc1.9 S42) in Table 2 is to be calculated based. The geometrical properties of the beam are: a = 400mm, d = 210mm, h = 250mm, bw = 150mm, lb = 50mm; concrete strength: f'c = 40MPa; flexural reinforcement: As = 760mm2, fy = 371MPa; transverse reinforcement: Ah = 0mm2, fyh = 342MPa; Av = 126mm2, fyv = 342MPa, strain rate = 0.48s-1.

Answer:

la = 2 (h - d) = 2 (250 - 210) = 80mm

lc = 57.839mm (fromtheiteration)

h - 2 - k 250 - 80 - «.p tan 9S =-a-2 =-400- = 0 4553

a sin0S = 0.4273, and cos 9S = 0.9101

n 006[,og10(A)]105 / 0 48 V006[log1^1^)] .

f yn = fe(m) = 54 386MPa

fydyn = fy (1.202 + 0.04 log 10e) = 371 x (1.202 + 0.04 log 10 (0.48)) = 441.212MPa fyv,dyn = fy (1.202 + 0.04 log 10e) = 342 x (1.202 + 0.04 log 10 (0.4)) = 406.723MPa Astr = 150(80 x 0.91 +50 x 0.4273) = 14025mm2 Ac = bw h = 37500mm2

fct = 0.W54.386 = 3.9347N/mm2

, Msfy,dyn sin0s , 2Awfyw,dyn sin (0s + 0w)dw o oncyn

ft = -—!—■—7\--1--■ , •—7X--T + fct = 8.3125MPa

Ac / sin 0s Ac/sin 0s d

Vn = uAstfdjm sin 0s = - fcTs>sint Astr.fc,dyn sin 0s = 176.037kN

Pu = 2 x Vn = 2 x 176.037 = 352.0474kN In this worked example, the Ultimate capacity PExp from experiment is 361.3 kN [4]. Thus, the ratio of Pu/Pexp is 1.03.

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