Evaluation of Shear and Diagonal Tension in Plain Concrete

To the Graduate School: The members of the Committee approve the thesis of Cristy L. Guenther presented on July 10, 2007.

____________________________________ Jennifer E. Tanner, Chair

____________________________________ Charles W. Dolan

____________________________________ David E. Walrath

APPROVED:

Jay A. Puckett, Head, Department of Civil and Architectural Engineering

Don A. Roth, Dean, The Graduate School

Guenther, Cristy L., Evaluation of Shear and Diagonal Tension in Plain Concrete. M.S., Department of Civil and Architectural Engineering, August 2007.

Abstract Shear strength is an important property in civil engineering materials and structures. Several practical concrete structures result in loading conditions with high shear and low moment such as corbels, corbels, brackets, composite floor systems systems and bridge decks. Shear failure in concrete concrete is undesirable due to the brittle nature nature of failure. In addition, it remains remains difficult to predict accurately, despite all the research in the area. Experimental and analytical work has been conducted to investigate a modified Iosipescu test as a means of measuring the direct shear strength of plain concrete beams. The Iosipescu loading configuration results in shear failure in a predefined plane, and the failure path is completely contained within the region of high shear. While other researchers have argued different failure modes for similar loading configurations, it is concluded from this study that high shear stress along the failure plane causes high principal tension. Finite element analyses show that the failure planes observed in the specimens are defined by the orientation of principal principal tensile stress. Combined with the experimental experimental results, these finite element analyses provide a means to predict failure based on the magnitude and orientation of the principal tensile stress.

Evaluation of Shear and Diagonal Tension in Plain Concrete by Cristy Louise Guenther

A thesis submitted to the Department of Civil and Architectural Engineering and the Graduate School of The University of Wyoming in partial fulfillment of the requirements for the degree of 

MASTER OF SCIENCE in CIVIL ENGINEERING

Laramie, Wyoming August, 2007

UMI Number: 1446905

UMI Microform 1446905 Copyright 2007 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code.

ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, MI 48106-1346

To my family for all of their support

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Acknowledgements I would like to thank my advisor Dr. Jennifer Tanner for encouraging me to go to graduate school, and for her guidance along the way. Thanks to Tyler Robison for all his assistance assistance in construction and testing. Thanks to JVI Neoprene for providing neoprene neoprene for this project. I would like to thank fellow current current and former students Matthew Matthew Olsen, Patrick  Lindblom, Cody Parker, Eric Anderson, Christina Behrens, Emre Insel, Kyle Eyre, Jiangang Deng, Yuan Li, and Zach Gutierrez for making the office a better place, and John Coombs for making graduate graduate school fun and always always keeping me in line. Thanks to my best friends friends Katie and Rachel and my boyfriend boyfriend Jason for always being there there when I needed them. Thanks to the professors in the department of Civil and Architectural Engineering for making a difference in my life. Thanks to everyone everyone in the machine machine shop for all of their their assistance. assistance. I would like to thank  thank  the members of my committee including Dr. Jennifer Tanner, Dr. Charles Dolan and Dr. David Walrath. Last but not least, I would like to thank my family family for all they have done for me; I wouldn’t have made it this far without them.

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Table of Contents 1

Introduction Introduction and Background Background .................................................. ................................................................................................. ............................................... 1 1.1 Resear Research ch Signi Significa ficance.......... nce................ ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ........... .... 5 1.2 Literat Literature ure Review.... Review........... .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. .............. ............. ......... ... 6 1.2.1 Develop Developmen mentt of Iosipes Iosipescu cu Shear Shear Test....... Test.............. ............. ............. .............. ............. ............. .............. .............. ............. ........ .. 6 1.2.2 Iosipes Iosipescu cu Beam Beam Tests Tests on Concrete Concrete ............. .................... .............. ............. ............. .............. ............. ............. .............. ............. ...... 7 2 Analytical Analytical Models.................................................................... Models............. .................................................................................................... ............................................. 13 2.1 Split Split Cylind Cylinder er....... .............. ............. ............. .............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. ........... .... 14 2.1.1 Model Model ............. .................... .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 14 2.1.2 Results...... Results............ ............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 15 2.2 Iosipes Iosipescu cu Beam Beam ............. .................... ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 17 2.2.1 Model Model ............. .................... .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 19 2.2.2 Results...... Results............ ............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 21 2.3 Long Lon g Beam....... Beam............. ............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 28 2.3.1 Long Lon g Beam Beam Model Model 1 .............. ..................... ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. ............ ..... 28 2.3.2 Model Model 1 Results Results ............. .................... ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 30 2.3.3 Long Lon g Beam Beam Model Model 2 .............. ..................... ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. ............ ..... 33 2.3.4 Model Model 2 Results Results ............. .................... ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 33 2.4 Theory Theory of of Elastic Elasticity.. ity......... .............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 35 2.4.1 Model Model ............. .................... .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 35 2.4.2 Results...... Results............ ............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 36 3 Experimental Experimental Testing............................................................... Testing........ .................................................................................................... ............................................. 38 3.1 Testing Testing Program....... Program.............. .............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 38 3.2 Test Test Proce Procedure duress ............. .................... ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 39 3.2.1 Compre Compressio ssion n Cylinde Cylinders............ rs................... ............. ............. .............. .............. ............. ............. .............. .............. ............. ............. .......... ... 39 3.2.2 Split Split Cylinde Cylinders rs .............. ..................... ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. .............. ............. ........ 39 3.2.3 Iosipes Iosipescu cu Beam Beam Test .............. ..................... ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. ............ ..... 41 3.2.4 Modulus Modu lus of Rupture Rupture....... .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 45 3.2.5 Split Split Prism...... Prism............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 46 3.3 Results...... Results............ ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. .............. .............. ............. ............. .......... ... 48 4 Discussion of Results.......................... Results.................................................................................. .................................................................................. .......................... 59 5 Conclusions....................... Conclusions.............................................................................. .................................................................................................... ............................................. 69 5.1 Finite Element Modeling General................................................................ ................. 69 5.2 Iosipescu Iosipescu Finite Element Model ................................................ ................................................................................... ................................... 69 5.3 Experime Experimental.......... ntal................. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ............. ......... .. 70 5.4 Final Final Conclu Conclusion sions..... s............ .............. .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 70 6 References............... References....................................................................... .............................................................................................................. ...................................................... 71 6.1 Cited Cited Refe Referenc rences es ............. .................... .............. ............. ............. .............. .............. ............. ............. .............. ............. ............. .............. .............. ............. ...... 71 6.2 Refere Referenced nced Standard Standardss and Codes........... Codes................. ............. .............. .............. ............. ............. .............. ............. ............. .............. ........... .... 72 Appendix A: A: Load, Shear, and and Moment Moment Calculations Calculations ................................................ ................................................................. ................. 73 Appendix B: Notation and Definitions Definitions ..................................................... ........................................................................................ ................................... 75 Appendix C: C: Iosipescu Iosipescu FEM Dimensions Dimensions and Loads Loads ................................................. .................................................................. ................. 76

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List of Figures Figure 1: Shear span-to-dep span-to-depth th ratio...................................................................................... .......... 2 Figure 2: Increase in shear shear strength strength based on depth to shear shear span ratio........................................ ratio... ..................................... 4 Figure 3: 3: Small shear shear span-to-depth span-to-depth conditions............................................................................. conditions............................................................................. 5 Figure 4: Specimen geometry geometry and loading for for Iosipescu shear test and ASTM D 5379............... 7 Figure 5: Loading, Loading, notch configuration, configuration, and failure path for different different researchers..................... researchers..................... 10 Figure 6: ASTM C496 .................................................. ......................................................................................................... ............................................................... ........ 14 Figure 7: 7: Different loading loading conditions conditions used used in split cylinder cylinder model model ........................................... 15 Figure 8: Horizontal stress along diameter diameter of split cylinder for different loading conditions..... conditions..... 16 Figure 9: Horizontal stresses stresses along diameter diameter of split cylinder normalized normalized to 2P/ LD................ 17 Figure 10: Loading for Iosipescu Iosipescu beam .................................................... ....................................................................................... ................................... 18 Figure 11: Centerline Centerline and diagonal diagonal failure plane to evaluate evaluate stresses stresses ......................................... 18 Figure 12: Stress components components and angle................................ angle...................................................................................... ...................................................... 19 Figure 13: Different loading loading conditions used used in Iosipescu beam model model ..................................... 20 Figure 14: Horizontal stress along centerline of Iosipescu beam for different loading conditions ....................................................................................................................................................... 21 Figure 15: Normal stresses along diagonal of Iosipescu beam for different loading conditions. 23 Figure 16: Shear stress along diagonal diagonal of Iosipescu Iosipescu beam for different different loading conditions conditions ....... 24 Figure 17: Shear stress along centerline centerline of Iosipescu Iosipescu beam for different different loading conditions conditions ..... 25 Figure 18: Stresses along along diagonal diagonal for Iosipescu beam beam modeled modeled with a unit thickness thickness .............. 26 Figure 19: Stresses along along diagonal for Iosipescu Iosipescu beam modeled using actual actual thickness thickness ............ 27 Figure 20: Long beam model.................................................................... model............ ........................................................................................... ................................... 29 Figure 21: Shear stresses stresses in beam section near concentrated concentrated load (not to scale) ........................ 31 Figure 22: Normal stresses stresses in beam section near concentrated concentrated load (not to scale) scale) ..................... 32 Figure 23: Shear stresses stresses in beam section near concentrated concentrated load (not to scale) ........................ 34 Figure 24: Beam geometry geometry and applied loads (Timoshenko (Timoshenko and Goodier Goodier 1970)........................ 1970) ........................ 35 Figure 25: Shear stresses stresses along line n-n for different values of b/c............................................. 36 Figure 26: Testing apparatus apparatus used for for splitting tensile tensile strength of of 4 in. cylinders ...................... 40 Figure 27: Test setup for splitting tensile tensile strength of 6 in. cylinders cylinders .......................................... 41 Figure 28: Modified Iosipescu test fixture.......... fixture.................................................................. ......................................................................... ................. 42 Figure 29: Free body body diagram diagram of test fixture fixture ..................................................... ............................................................................... .......................... 42 Figure 30: Loading, Loading, shear, and moment diagrams diagrams for Iosipescu Iosipescu beam test................................. test......................... ........ 43 Figure 31: Loading point setup....................................................... setup .................................................................................................... ............................................. 44 Figure 32: Neoprene used to compensate compensate for unparallel unparallel surfaces surfaces ............................................... 44 Figure 33: Modified Iosipescu Beam Test.......... Test.................................................................. ......................................................................... ................. 45 Figure 34: Modulus of rupture rupture beam beam ........................................................ ........................................................................................... ................................... 46 Figure 35: Test setup for splitting tensile tensile strength of a prism ..................................................... 47 Figure 36: Failure stress ratio ratio versus compressive compressive strength ................................................ ........................................................ ........ 48 Figure 37: 37: Differences Differences in cracks cracks based on aggregate aggregate size................................................... size ........................................................... ........ 50 Figure 38: Test results for pea gravel ................................................................. .......................... 51 Figure 39: Test Test results for 3/8 in. in. aggregate ....................................................... ................................................................................. .......................... 52 Figure 40: Test results for ¾ in. aggregate ........................................................................... ........ 53 Figure 41: Specimen Specimen size effect in split split cylinder test results.................... results....................................................... ................................... 54 Figure 42: Results for for 4 in. split split cylinders.......... cylinders.................................................................. ......................................................................... ................. 55 Figure 43: Iosipescu beam results................................. results........................................................................................ ............................................................... ........ 56 56

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Figure 44: Modulus of rupture test results results................................................ ................................................................................... ................................... 56 Figure 45: Split prism test results ................................................... ................................................................................................ ............................................. 57 Figure 46: Comparison Comparison of Iosipescu beam results........................................................................ results.................. ...................................................... 58 Figure 47: Iosipescu beam beam specimen specimen from Mix 1 shown before before testing and and at failure ............... 60 Figure 48: Iosipescu beam beam specimen specimen from Mix 6 shown before before testing and and at failure ............... 60 Figure 49: Loading, Loading, failure path, principal tensile stress surface surface plot, and direction of principal tensile stresses (psi)....................................................................................................................... 61 Figure 50: 50: Comparison Comparison of shear shear stresses in beam without without notches ............................................. 62 Figure 51: Shear stress along centerline centerline for different different notch depths............................................. 63 Figure 52: Shear stress along centerline for Bazant and Pfeiffer (1986) and Ingraffia and Panthaki (1985)............................................................................................................................. 64 Figure 53: Loading configuration, configuration, failure path, and principal tensile tensile stress surface plot (Ingraffia and Panthaki Panthaki 1985) ................................................................. ...................................................... 66 Figure 54: Loading configuration, configuration, failure path, and principal tensile tensile stress surface plots (Bazant and Pfeiffer 1986) ................................................. ........................................................................................................ ........................................................................ ................. 68 Figure 55: Free body diagram diagram of Iosipescu test brackets brackets and concrete specimen specimen (UW)............. 73 Figure 56: Loading, Loading, shear, and moment diagrams diagrams for Iosipescu Iosipescu beam test................................. test......................... ........ 74 Figure 57: Specimen Specimen and loading of Iosipescu Iosipescu beam test............... test............................................................ ............................................. 76 Figure 58: 58: Applied loads loads for Iosipescu Iosipescu beam beam FEM model.......................................................... model............. ............................................. 77 Figure 59: Loading configuration configuration and and dimensions used for Bazant Bazant and Pfeiffer FEM............... 78 Figure 60: Loading configuration configuration and and dimensions used for Ingraffia Ingraffia and Panthaki Panthaki FEM .......... 79

List of Tables Table 1: Comparison Comparison of maximum shear shear and bending bending stresses stresses in long beam............................. beam ............................. 31 Table 2: Comparison Comparison of maximum shear shear and bending bending stresses stresses in long beam............................. beam ............................. 34 Table 3: Experimental Experimental test test results..... results............................................................. ........................................................................................... ................................... 49

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1 Introduc Introductio tion n and Backgrou Background nd Shear strength is an important property in civil engineering materials and structures. Shear failure in plain concrete is the formation of a crack that forms when principal tensile stresses exceed the tensile strength, otherwise known as diagonal tension failure. In reinforced concrete design shear reinforcement begins to work after cracking. Shear failures are considered undesirable because they reduce the ductility of structural elements and may cause catastrophic failure if the concrete concrete and shear shear reinforcement reinforcement are not sufficient sufficient to carry the load. load. Shear failure failure remains difficult to predict accurately, despite all the research in the area. Shear failure in unreinforced concrete is a brittle failure and occurs suddenly with no advanced warning. With increasing load, tensile cracks form where the tensile stresses exceed tensile strength and will cause immediate failure of the element. A typical concrete element is a beam, which is generally generally reinforced with stee steell to stop the progression of cracks. cracks. Beams subject subject to bending stresses have longitudinal reinforcement effective in resisting tension near the tension face of the beam. Shear stresses increase increase in proportion to increasing loads, loads, and significant diagonal tensile stresses are created in regions of high shear forces and low moment, especially close to supports. Consequently, Consequently, concrete beams beams are provided with shear reinforcement reinforcement to increase the probability of a ductile failure controlled by yielding of flexural reinforcement rather than a brittle shear failure. Adequate designs designs must account for tensile stresses resulting resulting from shear alone or those resulting from combined shear and bending. For this reason it is important to have a model to predict the loads at which these cracks will form. There are two types of shear cracks in traditional simply supported reinforced concrete beams subject subject to gravity loads. loads. 1) Vertical flexural flexural cracks form form first and are distributed distributed in the center of the beam. Cracks in zones of moderate shear may propagate propagate diagonally forming forming

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flexure-shear cracks. 2) Cracks form perpendicular to diagonal tensile stresses at locations of  high shear and low moment. moment. Near the ends of a simply supported supported beam subject to gravity gravity loads these cracks cracks are oriented at 45˚. For beams with small shear shear span–to-depth span–to-depth ratios (Figure 1), the shear stresses are very high while the flexural stresses are very low. At a location of high shear, V , and small bending moment, M , there will be little to no flexural cracking before the formation

of shear cracks and a direct shear failure can occur. Generally, beams with smaller shear spanto-depth ratios have have greater shear strength than beams with large shear span–to-depth span–to-depth ratios. For beams subjected to concentrated loads, the major variable affecting the mode of failure is the distance from the load to the support, and the depth of the member.

Figure Figure 1: Shear span-to-depth span-to-depth ratio ratio

Chapter 11 of ACI 318 defines defines building code requirements requirements for shear and torsion. torsion. Unless designs use strut and tie models, the cross sections subject to shear are designed based on Equation (1), where V u is the factored shear force at the section, V n is the nominal shear strength, and  is the strength reduction factor.

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 V n

 V u

Equation (1)

The nominal shear strength is computed by Equation (2), where V c is the nominal shear provided by concrete and V s is the nominal shear strength provided by shear reinforcement.

V n

=

V c

+

V s

Equation (2)

Section 11.3 of ACI 318 covers the shear strength provided by concrete for nonprestressed members. members. For members members subject to shear shear and flexure only only V c can be calculated according to Equation (3), where f’c is the concrete compressive strength, bw is the width of the web, and d  is the distance from the compression face of the beam to the centroid of the tensile steel.

V c

=

2 f  'c bw d 

Equation (3)

Engineers may calculate V c using the more detailed calculation of Equation (4), where

w

is the

reinforcement ratio, V u is the factored shear at the section, and M u is the factored moment at the section.

V c

=

(1.9 f  ' c

+

2500  w

V u d  M u

)bw d   3.5 f  ' c

Equation (4)

The value of  V ud/M u is equivalent equivalent to the depth to shear shear span ratio. Figure 2 shows with an increasing depth to shear span ratio there is an increase in shear strength when calculating V c using Equation (4).

3

4.0

3.5

3.0 ) i s p ( ’c 2.5 f / h t g 2.0 n e r t s r 1.5 a e h S

Equation 4

1.0

2f'c 0.5

3.5f'c

0.0 0

1

2

3

4

5

6

7

8

Depth to shear span ratio

Figure Figure 2: Increase Increase in shear strength strength based on depth to shear span ratio ratio

There are conditions where small shear span–to-depth ratios occur in reinforced concrete members members such as squat shear walls. Other common situations situations occur in precast concrete structures structures particularly in regions near connections. connections. Shear forces must be transferred transferred across the interface between a beam and a composite slab or at the supports of precast elements such as corbels or brackets loaded near the column (Figure 3).

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P P

a.) Beam and composite slab

b.) Corbel loaded near column Figure 3: Small shear span-to-depth conditions

Although many researchers have created testing programs to predict the shear strength of  plain concrete beams, one of the major problems in shear strength studies is determining the concrete shear strength at a location of high shear and low moment (direct shear strength) as opposed to the shear strength determined by combined flexure and shear (flexure-shear strength). This study modified the Iosipescu test (Iosipescu 1967) to develop a method for measuring the shear strength strength of plain concrete in confined confined locations such as shown in Figure Figure 3. The Iosipescu Iosipescu loading scheme induces a high shear stress at the center of the specimen. The high shear region occurs where the moment approaches zero and results in direct shear failure of test samples.

1.1 Research Research Significa Significance nce Experimental and analytical work has been conducted to investigate a modified Iosipescu test fixture proposed by Ross (2000) as a means of measuring the shear strength of plain concrete beams. A final test procedure proposed by Ross (2000) has been further investigated as a standard test for determining shear resistance of plain concrete over a wide range of concrete

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mixtures. Iosipescu beam test results were compared to results from ASTM standardized tests for the tensile and compressive strength of concrete to confirm the need for a new standard. Finite element results are compared to experimental results to evaluate shear strength. The Iosipescu test provides a simple and effective method to experimentally confirm the confined shear capacity of plain concrete.

1.2 Literatur Literaturee Review Review Literature was reviewed regarding the origin of the Iosipescu shear test for metals, along with similar shear tests for plain concrete beams. The information information is summarized and discussed in the following sections.

1.2.1

Development Development of Iosipescu Iosipescu Shear Test

A test procedure was developed by Nicolae Iosipescu (1967) that produces failure of test specimens specimens under pure shear stresses. stresses. A pure shear load is generated using using a simple device that generates a shear force in a straight beam in the area of zero moment and results in failure of the specimen. specimen. Prior to this test, shear testing procedures existed, but none that produced produced failure of  the material material under shear stress alone. The solution of the problem problem of pure shear testing was established by Iosipescu on the basis of principles of strength of materials and the theory of  elasticity. elasticity. Iosipescu Iosipescu found it was necessary to weaken the specimen specimen in the desired failure section by means of angular notches. notches. Two 90˚ notches were cut to ¼ of the beam depth on both the top and bottom surfaces surfaces (Figure 4). A photoelastic study on on Plexiglas was used to experimentally experimentally confirm the model. Iosipescu believed this study indicated that using notches ensured maximum shear stresses remained constant through the cross section. First used for studies of shear strength in railway weldments, this test has subsequently been applied to other rolled metals and

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cast metals. metals. Iosipescu Iosipescu shear tests are now now applied to non-metal non-metal materials. materials. The V-notched V-notched beam test, ASTM D5379, established a standardized procedure for testing the shear properties of  composite materials. Composite materials are limited to fiber reinforced polymer laminated composites.

Figure 4: Specimen geometry and loading for Iosipescu shear test and ASTM D 5379

1.2.2

Iosipescu Iosipescu Beam Tests on Concrete Concrete

Iosipescu and Alexandrescu (1965) proposed a similar device to test concrete in pure shear. They believed believed that it was necessary to weaken the specimen specimen using notches to ensure ensure that the specimen fractured in pure shear. Two 90 degree notches cut to a depth a quarter of the height were made in each specimen. By weakening the test pieces, the authors believed that a pure shear state occurred and maximum shear stresses were uniformly distributed throughout the section. The specimens usually fractured along the centerline between notch tips (Figure 4). Specimens failed in pure shear, and because of the notches the failure plane was vertical along the centerline. Iosipescu concluded that this test method could be used for the pure shear testing of concrete. 7

A research program was conducted by Bazant and Pfeiffer (1986) 1 at Northwestern University to investigate investigate the shear fracture of concrete. In the terminology of fracture mechanics mechanics shear fracture fracture is referred to as Mode II failure. Similar to Iosipescu, Iosipescu, notched beams beams were loaded in a manner that produced concentrated shear forces and failure. The test specimens used were beams with rectangular cross section and a constant length-to-depth ratio of 8:3. Specimens of  various depths (1.5, 3, 6, and 12 inches) were tested while maintaining a constant thickness of  1.5 inches. All beams were cast from the same batch of concrete and symmetric notches (0.1 in. wide) were cut to a depth of one sixth the beam depth on the top and bottom surfaces at the center of each specimen. specimen. When loaded, cracks propagated propagated between the two notch tips resulting in failure of the specimen specimen (Figure (Figure 5a). The authors authors concluded that that shear fracture fracture exists. Tests were also repeated using wider shear spans and in these cases the cracks propagated from the notch tip in a direction normal normal to the maximum principal principal tensile stress. stress. Maximum loads measured from the specimens with a small shear span were found to be greater than the specimens with a wide shear span. Bazant and Pfeiffer (1986) believed that their test method was a good way to measure pure Mode II fracture. This conclusion was further supported by linear elastic finite element results. They believed believed that the cracks initiated from the notch tips and propagated continuously continuously toward the center, representing shear cracks. The behavior in the tests described by Bazant and Pfeiffer (1986) conflict with those observed in similar specimens tested at Cornell University (Ingraffia and Panthaki 1985) 2. Again, tests were performed with similar loading, but with only a single notch in the bottom 1

This journal article is based on conference proceedings from 2 nd Symposium on the Interaction of Non-Nuclear Munitions on Structures, Panama City Beach, FL, April 15-19, 1985. 2 Results discussed in this article article were originally published in 1981. Arrea, M., and Ingraffia, A.R., A.R., 1981, “Mixed Mode Crack Propagation in Mortar and Concrete”, Department of Structural Engineering Report 81-13, Cornell University

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center of the beam. beam. However, the principal principal loads were were applied further further from the notch notch tips. In all the tests performed, fractures initiated at the notch tip in a direction normal to the principal tensile stress (Figure (Figure 5b). Similar tests were performed performed on limestone and granite specimens specimens with comparable comparable behavior patterns. patterns. One of the important conclusions of these tests was that shear fracture did not occur in the specimens. Ingraffia Ingraffia and Panthaki conducted a linear elastic finite element analysis on the specimen geometry of Bazant and Pfeiffer (1986) to validate the findings. Finite element analysis showed the principal stresses in the region between notch tips were tensile, and the direction direction of the stresses was horizontal. horizontal. The diagonal diagonal region between the two center supports had uniform tensile stresses. The shear stress distribution shows that the minimum shear stress is in the center of the beam and maximum near the notch tips. To validate the finite element model the authors used theory of elasticity, arguing that by moving the center loads closer together together the intensity of shear stresses were decreased rather than increased. increased. It was concluded that the failure was similar to a splitting tensile strength test, and the failure mode was actually cracking due to principal tensile stresses rather than shear fracture.

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