Cracking Moment
Cracking moment represented as (Mcr) is well-defined as the moment, that is once surpassed reasons the cracking of concrete. Intended for the design of the deflection in the concrete at cracking moment, the moment is considered at the point, the failure will happen in concrete.
Phases to Find the Cracking Moment Are as Assumed Under
1. Compute the value of the modulus of rupture (fr) with the method
fr = 7.5√ƒc
therefore, the word fr is the compression strength in concrete.
2. Compute value of distance yt by means of the specified formula
Used for rectangular section, the value of yt is assumed as under.
yt = h/2
Here, his vertical height of rectangular beam and yt is distance from center of gravity of beam to the extreme fiber of the tension side.
3. Calculate the value of the Ig.
For the rectangular cross-section, Ig is calculated as given below.
Ig = bh3/12
Here, b is width of concrete beam, h is height of beam, and Ig is a moment of the beam.
4. Calculate the moment of cracking (Mcr).
Use the formulae given below to calculate the moment of cracking (Mcr).
Mcr = (Æ’r.Ig)/yt
Here, the term (fr) is compression strength in concrete, yt is the distance between center of gravity of the section and extreme fiber in tension side, and Ig is the moment of the beam.
Cracking of Concrete
Deformation of concrete happens in concrete owing to uneven loads, temperature alteration or freezing and thawing. Cracking of concrete happens due to numerous causes identical owing to constructional program, shrinkage, respectively.
Afterward hardening, cracks happen owing to chemical reactions, thermal changes respectively.
The key details of concrete cracks are as beneath.
- The precise quantity of water is essential to be further in the mixture to whole the chemical reaction or hydration. If the concrete dry out earlier than conclusion of the reaction, it reasons cracking.
- Lack of control joint or control joint with wrong dimension of thickness causes cracking.
- When the extra quantity of water freezes in winter, the volume enlarges, and cracking happens.
- Dissimilar concrete will have dissimilar strengths. If the concrete mix with essential strength is not bent throughout the job, it can reason cracking.
- Once an extra of water is additional to the concrete growing its w/c ratio, the strength of the concrete also will decrease. Throughout summer, vaporization of additional water takings place that reasons shrinkage. Owing to this shrinkage, cracking in concrete take place.
Precautionary procedures to evade cracking are as follows.
- By means of the true quantity of water in concrete-water mix paste.
- Appropriate shrinkage as well as temperature reinforcement at control joints.
Types of Cracks in Concrete
Different types of cracks are as follows.
1. Cracks Owed to Overburdening
Pre-mix concrete originates through their particular strength measured pounds per square inch, which designates the maximum pressure that can proceed beforehand being crushed. Nevertheless, overburdening of slabs happens rather frequently in housing areas.
In circumstances where the original ground has to develop soft, its strength allows a section of the slab to be lacking downwards. Heavy vehicle parking spots are the greatest probable to see such cracks.
2. Expansion Crack
After subject to heat, concrete slabs have a tendency to enlarge outward. Lack of space to expand causes slabs to develop cracks. Nevertheless, expansion cracks are typically dealt with throughout the planning and concrete distribution phase of the structure.
Expansion joints are introduced between slabs that can captivate pressure from expansion and therefore pre-empt and evade cracks.
3. Premature Drying
Once a concrete slab fails moisture rapidly, it can lead to cracks. The cracks seem when the top coating of the slab rapidly fails moisture certainly, approaching a spider-web.
Crusting cracks seem during the stamping process when the top layer is dry for embedding patterns. Together these two types might look unattractive, but they are mostly inoffensive for the structural strength of the slab.
4. Plastic Shrinkage Crack
Beforehand the concrete hardens, it is measured plastic and encompasses a noteworthy amount of water. Occasionally puddles of water, upon drying, leaving overdue voids inside the slab.
As a result of such voids, the slab’s surface is prone to cracking when subjected to strain. Plastic shrinkage cracks do not damage the foundation of the slabs until they are prominently visible.
5. Heaving Crack
Heaving cracks are also another form of temperature-related crack. Heaving cracks form as a result of the slab condensing due to exposure to severe cold weather.
Whenever the weather gets back to normal, the slab back to its original form. However, this shape shift often results in the creation of heaving cracks.
6. Settling Cracks
The underlying ground’s structural failure will result in settling cracks. When a massive tree is taken down, the roots rot and leaving large voids in the earth, softening it.
A shaky foundation provides inadequate protection for the concrete slab and the above steel bar that surrounds it, resulting in settling cracks.
Calculating of Cracking Moment for Hollow Rectangular Beam
The area of reinforcement as a fraction of the overall cross-sectional area of a beam is very small (usually 2% or less), and its effect on beam properties is virtually non-existent as long as the beam is not cracked.
As a result, depending on the gross properties of the beam’s cross-section, an estimated estimate of the bending stresses in such a beam can be achieved.
The tension in the concrete at some point y from the cross section’s neutral axis can be calculated using the flexure theorem, where M is the bending moment equal to or less than the section’s cracking moment and Ig is the cross section’s gross moment of inertia:
f = My/Ig
ACI Code states that the cracking moment of a section may be determined with ACI Equation given below. The cracking moment is as follows:
Mcr = (Æ’r.Ig)/yt
Where Æ’r is the modulus of rupture of the concrete.
yt is the distance from the centroidal axis of the section to its extreme fiber in tension.
Example: Assuming the concrete is uncracked, compute the bending stresses in the extreme fibres of the beam for a bending moment of 25 ft- k. the normal weight concrete has an fc of 4000 psi and a modulus of rupture fr = 7.5 ( 1.0 ) Â = 474 psi. determine cracking moment of the section.
Solution:
Bending stress
Ig = (1/12)Â b h3
Where, b = 12 inch and h = 18 inch.
Ig =Â (1/12) x 12 x 183
Ig = 5834 in 4
f = My/Ig
where,
M = 25 ft-k
M = 25000 ft-lb
Multiplying 25000 ft-lb by 12 in/ft to obtain in-lb as shown here,
f = [(12 x 25000 x 9)/5832]= 463 psi.
Cracking moment
Mcr = (Æ’r.Ig)/yt
Mcr = (474 x 5832)/9
Mcr = 25.6 ft-k.
Calculating of Cracking Moment for T Beam
If the t beam is uncracked, calculate the stress in concrete at top and bottom of extreme fibres under a positive bending moment of 80 ft-k. if fc = 3000 psi and normal weight of concrete is used, what is the maximum uniformly distributed load the beam can carry if it is used as a simple beam with 24 ft span without exceeding the modulus of rupture of concrete?
Given : bf = 60; hf = 5; h = 32;
Y = {[bÆ’ x hÆ’ x (0.5 hÆ’)] + [bÆ’ x (h – hÆ’) x ( hÆ’ – {0.5 x ( h – hÆ’)})] }/ {[ bÆ’ x hÆ’ ]+ [bÆ’ x (h – hÆ’)]}
Y = {(60 x 5 x2.5) + [(12 x 27) x (5 + 13.5)]}/{(60 x 5) + (12 x 27)}
Y = 10.81 in.
The moment of inertia is,
Ig = {bÆ’.hÆ’3/12} + { bÆ’.hÆ’ [(Y-0.5 hÆ’)2 + ( [bw (h-hÆ’)3]/12)+(bw (h-hÆ’)] x [ Y – hÆ’ – (0.5 [h – hÆ’])]2}Â
Ig = {60 x 53 /12} + { 60x 5 [(10.81-(0.5 x5))2 + ( [12 (32-5)3]/12)+(12 (32-5)] x [ 10.81 – 5 – (0.5 [32 – 5])]2}Â
Ig = 60185 in 4.
The stress in the top and bottom fibre under the given moment of 80 ft- k is:
Fbottom = Mc/I
Fbottom= (80 x 12 (32 – 10.81)) /60185
Fbottom= 0.338 K/in 2
Fbottom= 338 lb/ in 2
Ftop= Mc/ I
Ftop=Â (80 x 12 x 10.81) / 60185
Ftop= 0.172 K/in 2
Ftop= 172 lb/ in 2
The modulus of rupture fr of usual weight concrete through fc = 3000 psi
Fr = 7.5√ƒc
Fr= 7.5 ( 1.0 )
Fr = 411 lb/in 2
The moment that reasons a stress equivalent to the modulus of rupture is:
Mcr = (Æ’r.Ig)/c
Mcr = (411 x 60185)/(32-10.81)
Mcr= 1167.34 in-lb
Mcr = 97.28 ft-k
The uniformly distributed load on a simple span that causes moment is
w = 8M/l2
w = (8 x 97.28) /242
w= 1.351 k/ft
w= 1351 lb/ft
Cracking Moment Calculation Example for Inverted T Beam
Unless the beam is inverted, at that time the word c is used to compute Mcr = 10.81
Mcr = (Æ’r.Ig)/c
Mcr = (410 x 60186)/10.81
Mcr = 228272 in–lb
Mcr = 190.69 ft-k
The uniformly distributed load on a simple span that causes moment is
w = 8M/l2
w = (8 x 190.69) /242
w= 2.648 k/ft
w= 2648 lb/ft
FAQ
What is cracking moment in concrete beams?
Cracking moment (Mcr) is the bending moment at which the tensile stress in concrete reaches its tensile strength, causing the concrete to crack.
How is cracking moment calculated?
Cracking moment can be calculated using the formula: Mcr = (fr * Ig) / yt, where fr is the modulus of rupture of concrete, Ig is the moment of inertia of the beam’s cross-section, and yt is the distance from the centroid to the extreme fiber on the tension side.
Why is cracking moment important in concrete design?
Cracking moment is crucial in concrete design as it helps engineers ensure that the structure can withstand anticipated loads without developing cracks that could compromise its integrity or aesthetics.
What factors influence cracking moment?
Factors influencing cracking moment include the modulus of rupture of the concrete, the dimensions of the beam (height and width), and the distribution of reinforcement within the beam’s cross-section.
How can cracking in concrete be prevented or minimized?
Cracking in concrete can be prevented or minimized by using appropriate reinforcement, controlling the water-cement ratio during mixing, providing adequate curing, and designing joints to accommodate shrinkage and thermal expansion.
What are the consequences of exceeding the cracking moment in concrete design?
Exceeding the cracking moment can lead to visible cracks in the concrete, which may compromise its structural integrity, increase maintenance costs, and affect the aesthetics of the structure.
How does temperature affect cracking moment in concrete?
Temperature variations can influence cracking moment by causing thermal expansion and contraction of concrete, which can induce internal stresses leading to cracking if not properly controlled or accounted for in design.
Can cracking moment vary for different types of concrete and loading conditions?
Yes, cracking moment can vary based on the type of concrete mix used (normal weight, lightweight, etc.) and the loading conditions applied to the structure. It is essential to consider these factors during the design phase to ensure structural durability.
What are the implications of cracking moment in reinforced concrete beams versus plain concrete beams?
In reinforced concrete beams, cracking moment affects both the concrete and the reinforcement, influencing the overall behavior under load. Plain concrete beams, without reinforcement, tend to exhibit more pronounced cracking and reduced load-carrying capacity near the cracking moment.
How is cracking moment tested or verified in concrete structures?
Cracking moment is typically verified through theoretical calculations using established formulas and principles of structural mechanics. Experimental testing and finite element analysis can also be employed to validate these calculations in practical applications.
