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In this article we will discuss about the relation between compressive and tensile strength of concrete.

The theoretical compressive strength of concrete is eight times larger than its tensile strength. This implies a fixed relation between the compressive and tensile strength of concrete. In fact there is a close relation but not a direct proportionality. The ratio of tensile to compressive strength is lower for higher compressive strengths.

Experimental results also have shown that concrete in compression and tension (both direct tension and flexural tension) are closely related but the relationship is not of direct proportionality type. The ratio of tensile strength to compressive strength depends upon the strength of concrete. Thus higher the compressive strength, higher the tensile strength, but the rate of increase of tensile strength is of decreasing order. The tensile strength of concrete is more sensitive to improper curing than the compressive strength.

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**This may be due to the following two reasons:**

(a) Formation of inferior quality gel due to improper curing.

(b) Development of more shrinkage cracks due to improper curing. The uses of pozzolanic materials have shown the increase in tensile strength.

The Central Road Research Institute Delhi has carried out extensive study for establishing the relation between tensile and compressive strength of concrete for the construction of concrete roads. Fig.13.7.

**On the basis of test data of the study, CRRI has suggested the following relation between flexural strength and compressive strength of concrete: **

y = 11 x – 3.4

where y is the compressive strength of concrete in MPa and x its flexural strength. This relation depends on the size of coarse aggregate. The strength is found to vary with the nature and size of aggregate.

**The relation is reproduced below: **

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(a) For 20 mm Crushed Stone y = 15.3 x – 9.0

(b) For 20 mm Natural Graval y = 14.3 x – 10.4

(c) For 40 mm Crushed Stone y = 9.9 x -5.5

(d) For 40 mm Natural Graval y = 9.8 x – 2.5

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**The flexural strength of concrete was found to be 8 to 11% of the compressive strength of concrete of higher strength concrete of the order of 25 MPa (250 kg/cm ^{2}) and 9 to 12.8% for concrete of strength less than 25 MPa (250 kg/cm^{2}) see Table 13.1: **

The ratio of flexural strength to compressive strength was found higher for 40 mm maximum size aggregate than that of 20 mm max sized aggregate. In general the ratio was found slightly higher for natural gravel than for crushed stone.

It has been observed that the flexure strength or modulus of rupture is obtained much lower by two point methods than central point loading as shown in Fig.13.8.

The test can be performed as per IS 516-1959 or 1964 on a beam specimen of 10 x 10 x 50 cms within a span of 40 cms. The modulus of rupture is given by the relation.

*f*b = 2p x a/bd^{2}

where,

*f*b = modulus of rupture

p = load applied on specimen in kg,

a = distance between the crack and the nearest support

b = width of beam in cms

d = depth of specimen at failure point in cms

The value of modulus of rupture varies from 11% to 23% of the compressive strength of the same concrete. The average value may be assumed as 15% of the compressive strength, the use of angular aggregate or rough textured aggregate results in higher modulus of rupture than smooth textured aggregates.

**Thus the modulus of rupture is higher than direct tensile strengths for the same concrete due to the following reasons: **

1. Assumption of the shape of the stress block. In the calculation of modulus of rupture, it is assumed that stress is proportional to the distance from the neutral axis of the beam, while the shape of the actual stress block under loads nearing failure is known to be non-triangular, but parabolic. The modulus of rupture thus overestimates the tensile strength of concrete and gives a higher value than would be obtained in direct tensile test or splitting test.

2. The accidental eccentricity in a direct tensile test results in a lower apparent strength of the concrete.

3. Under direct tension the entire volume of specimen is subjected to the maximum stress, so that the probability of a weak element occurring is high.

4. The maximum fibre stress reached may be higher due to the fact that the propagation of a crack is blocked by less stressed material nearer to the neutral axis. However the actual values may vary depending on the properties of concrete.

**Relation between compressive, tensile strength and modulus of rupture is shown in Table 13.2 below:**

**Central Point and Two or Third Point Loading****:**

The Central Road Research Institute Delhi has carried out extensive study to find out the relation between central point loading and third point loading value of modulus of rupture. The ratio of span to depth of specimen was kept constant.

**On the basis of experimental data, they established the following relation: **

x_{1} = x_{2} + 0.72

Where,

x_{1} = flexural strength of concrete in MPa under central point loading

x_{2} = flexural strength of concrete in MPa under third point loading

During the study it was observed that central point loading gave higher average value of flexural strength than third point loading irrespective of the size of the specimen.

**The higher strength obtained in central point loading may be due to the following facts: **

1. The beam is subjected to the maximum stress at the predetermined point not necessarily the weakest.

2. The span to depth ratio of specimen was kept 4. The alteration in this ratio was found to alter the flexural strength. The change in the span to depth ratio by 1% induced 3% change in the flexural strength when tested by third point loading and 2.5% change when tested by central point loading. With the increase in span to depth ratio, the flexural strength was found to decrease.

3. The rate of stress application was found to influence the flexural strength to a great extent. If the rate of application of stress is increased from the standard rate of application of 0.7 MPa per minute, the flexural strength was found to increase upto 25%. The increase was found more with the leaner mixes.

There are many empirical relations between the tensile and compressive strength of concrete.

**One of the most common relations is given by the following relation: **

Tensile strength = K (compressive strength)^{n}.

The value of K may be taken as 6.2 for gravel and 10.4 for crushed aggregate. The average value for both may be taken as 8.3 and the value of n may vary from 0.5 to 0.75.

The I.S. 456-2000 has suggested the following relation between the compressive strength and flexural strength of concrete.

Flexural strength = 0.7 √*f*_{ck}

where *f*_{ck} is the compressive strength cylinder of concrete in MPa (N/mm^{2}).

**The relationship between the compressive and tensile strengths of concrete suggested by Association of Portland cement laboratories is shown in Table 13.3. below: **

**Factors Affecting Relation or Ratio of Tensile and Compressive Strengths:**

**Following factors affect the ratio of tensile and compressive strengths:**

**1. Properties of Coarse Aggregate:**

The properties of coarse aggregate affect the cracking of concrete very much. It has been observed that vertical cracking in a specimen subjected to uniaxial compression starts under a load equal to 50 to 75% of the ultimate load. The stress at which the cracks form depends to a great extent on the properties of coarse aggregate. It has been observed that concrete made from smooth surfaced gravel cracks at much lower stress than concrete made with rough and angular crushed rock aggregate. This may be due to mechanical bond and shape of the coarse aggregate.

Further it has been observed that the properties of coarse aggregate affect tensile strength or the cracking load in compression more than the compressive strength of concrete The influence of the type of coarse aggregate on the strength of concrete varies in magnitude and depends on the water/ cement ratio of the mix. For water/cement ratio below 0.4, the use of crushed aggregate has resulted in increase in strength upto 38% than when gravel is used.

With water/cement ratio above 0.5, the influence of water/cement ratio falls off and at 0.65 water/cement no difference in strength is observed. The properties of coarse aggregate have been found to have little effect on direct and splitting tensile strength, but flexural strength is found greater with the use of crushed and angular crushed rock aggregate than the rounded gravel.

**2. The Properties of Fine Aggregate:**

The properties of fine aggregate also influence the ratio of tensile to compressive strength of concrete.

**3.** The grading of aggregate also has been found to influence this ratio.

**4. Effect of Moisture:**

The moisture condition of concrete influences the relation between the flexural and compressive strength. If one concrete is cured continuously in water and the other is curved wet and then stored in dry environment and tested. It is found that the dry concrete gave greater compressive strength than the continuously wet cured concrete. The direct and splitting tensile strengths are not affected in similar manner. However flexural strength of drying concrete is found lower than that of wet concrete. This may be due to the development of shrinkage cracks in the concrete.

**5. Age of the Concrete at the Time of Testing:**

The increase in tensile strength after one month is slower than the compressive strength. Hence the age of specimen affects this ratio.

**6. Methods of Testing:**

**Tensile strength is determined by the following methods: **

(a) Direct tension method.

(b) Splitting tension method.

(c) Flexural tension method.

All the three methods give different results. The lowest value is given by the direct tension method. Thus values are given in order as direct tension < splitting tension < Flexural tension.

**7. Size of Specimen:**

The smaller the size of test specimen, lesser volume of concrete is subjected to tensile stress, resulting in less chances of a weak element to be subjected to tensile stress and ultimately resulting in failure. On the other hand, larger the size of specimens, greater is the volume of concrete having greater chances of a weak element and ultimately resulting in failure, thus larger the specimen, greater the chances of failure.

Secondly both the splitting and flexural test methods involve a non-uniform stress distribution which reduces the propagation of a crack, resulting in delay of ultimate failure on the other hand the stress distribution in the direct tension test is uniform, so that once a crack is formed, it can propagate quickly throughout the section of the specimen.

**8. Effect of Inadequate Curing:**

Tensile strength of concrete is found more sensitive to the inadequate curing than compressive strength. This may be due to the non -uniform shrinkage of flexural test beams. As flexural test beams are very sensitive to shrinkage cracks and are very serious for the test.

**Difficulties in Carrying Out Direct Test of Concrete Strength: **

**To carry out the uniaxial tensile test is difficult due to the following problems: **

**1. Problem of Gripping the Specimen:**

Proper and satisfactory gripping equipment is not available to grip the test specimen i.e., the gripping equipment should be such that the premature failure does not take place near the ends of the attachment. Generally there is a tendency of the specimen to break near the ends, which cause error in the results.

**2. Due to Eccentricity in the Applied Load:**

There should be no eccentricity in the applied load. However there is always some eccentricity present in the applied load system. The stresses are changed due to the eccentricity of loading, which may introduce major error on the stresses developed regardless of size and shape of the specimen. Therefore direct tensile test is not standardized and rarely used.

Thus due to these difficulties the tensile strength is determined by indirect methods as flexure test and splitting test. However these test methods give higher values of the tensile strength than the true tensile strength determined by uniaxial loading.

**As stated above different test methods yield numerically different results in the following order: **

(a) Direct or uniaxial tension yields the lowest value of tensile strength.

(b) Splitting tension test yields higher value than direct tension method, but lower than flexural tension test.

(c) Flexural tension test yields the greatest value of tensile strength of all the methods i.e. direct tension < splitting tension < flexural tension.

**Reasons of Difference in Results: **

**Following are the reasons for the difference in the value of tensile strength given by different methods:**

1. With the usual size of the laboratory specimen, the volume of concrete subjected to tensile stress decreases in the same order as listed above i.e., area subjected in direct tension < splitting tension area < flexural tension area. Thus statistically there is a greater chance of a weak element in the larger volume than a smaller volume and thus more failure in larger volume.

2. Both the splitting and flexural test methods involve non uniform stress distribution which restrict the propagation of a crack and thus delay the ultimate failure. On the other hand in the direct test the stress distribution is uniform so that once a crack is formed, it may propagate quickly through the section of specimen. The relation between the compressive and tensile strength of concrete is shown in Fig. 13.4.