Design of Beam and Slab

Contents

**Recall the Effective Span of Beam** 2

**Describe Permissible limit of Deflection for Beams as per IS : 456-2000** 3

**Recall the Criteria for Control of Deflection** 4

**Describe the concept of Lateral Stability of a Beam** 6

**Recall the design procedure for Singly Reinforced Beam** 7

**Recall the design procedure for Doubly Reinforced Beam** 10

**Recall the design procedure for Flanged Beam** 11

**Recall the effective span of the slab** 12

**Recall the design procedure for One-way Slab** 14

**Describe Rankine’s Grashoff Theory.** 15

**Describe IS Codal method to design Two-way Slab.** 16

**Recall the design procedure for Two-way Slab.** 18

**Derive an expression for Semi-circular beam simply supported on three equally spaced supports.** 20

**Describe the Live arrangement in Continuous Beam** 20

**Describe the Coefficient of Moments and Shear in Continuous Beam** 21

**Recall the Effective Span of Beam**

It refers to the concept of Effective Span of Beam in the subject of Reinforced Concrete Construction (RCC).

The effective span of a beam is a measure of the distance between the points of support for the beam, that takes into account the position of the loads and the presence of any intermediate supports. The effective span is used to determine the design load for a beam and is an important factor in determining the size, shape, and reinforcement required for the beam to resist the applied loads.

In reinforced concrete construction, the effective span of a beam is often longer than its actual physical length, due to the presence of intermediate supports or the position of the loads. The effective span is determined by analyzing the distribution of loads along the length of the beam and determining the point at which the maximum bending moment occurs.

The effective span is used in the design of reinforced concrete beams to determine the size and spacing of the reinforcing steel, as well as the size of the concrete cross-section. The effective span also affects the deflection and cracking behavior of the beam under load, and must be taken into account when designing the beam to meet the required performance criteria.

- The effective span of a
**simply supported beam**is taken as the least of the following:

Clear span + the effective depth of beam.

Center to center ( c/c ) distance between supports.

- In case of a
**continuous beam**, if the width of the support is less than 1/12 of the clear span, the effective span is taken in above cases

If the width of support is more than 1/12 of the clear span or 600 mm whichever is less, the effective span is taken as:

For end span with one end fixed and the other continuous or for intermediate spans, the effective span is taken as the clear span between supports

For an end span with one end free and the other continuous, the effective span is equal to the clear span + half the effective depth of the beam or the clear span + half the width of the discontinuous supports, whichever is less.

In case of span with roller or rocket bearings, the effective span is always the distance between the centre of bearing.

**Describe Permissible limit of Deflection for Beams as per IS : 456-2000**

It refers to the concept of Permissible Limit of Deflection for Beams as per IS : 456-2000 in the subject of Reinforced Concrete Construction (RCC).

The permissible limit of deflection for beams is a set of standards that define the maximum amount of deflection that a beam can undergo without impacting its structural integrity or affecting the intended use of the building. The deflection of a beam refers to the amount of displacement or bending that occurs when the beam is subjected to a load.

The final deflection due to all loads including the effects of temperature, creep and shrinkage and measured from the as-cast level of the supports of floors, roofs, and all other horizontal members, should not normally exceed span/250.

The permissible limit of deflection is specified in IS : 456-2000, which is an Indian standard for the design and construction of reinforced concrete structures. According to this standard, the permissible limit of deflection for beams is determined based on the type of structure, the span of the beam, and the type of loading.

For residential buildings and other structures with a low risk of damage to occupants or contents, the permissible limit of deflection is set at a maximum of 1/240th of the span of the beam, while for other structures, the limit is set at a maximum of 1/180th of the span.

It is important to note that the permissible limit of deflection is not a fixed value, and can be influenced by factors such as the type of reinforcing steel used, the size and shape of the concrete cross-section, and the distribution of loads along the length of the beam.

In summary, the permissible limit of deflection for beams is a set of standards defined by IS : 456-2000, which specify the maximum amount of deflection that a beam can undergo without affecting its structural integrity or intended use. The permissible limit of deflection is determined based on the type of structure, the span of the beam, and the type of loading, and is an important factor in the design of reinforced concrete beams.

**Recall the Criteria for Control of Deflection**

The criteria for control of deflection in reinforced concrete construction (RCC). Deflection is the deformation of a structure under load and it is an important aspect of structural design as it affects the serviceability and durability of the structure.

The criteria for control of deflection in RCC structures can be summarised as follows:

- Maximum deflection ratio: The maximum deflection ratio is the ratio of the maximum deflection to the span length of the structure. This criterion ensures that the deflection is within acceptable limits for the intended use of the structure.
- Live load deflection: The live load deflection is the deflection caused by the weight of people, furniture, and other movable objects on the structure. This criterion is important for ensuring that the structure is safe and comfortable for use.
- Total deflection: The total deflection is the sum of the deflection caused by the dead load (weight of the structure itself) and the live load. This criterion ensures that the structure is safe and serviceable under all loads.
- Span to depth ratio: The span to depth ratio is the ratio of the span length to the depth of the structure. This criterion ensures that the structure is proportionally strong enough to resist the applied loads and minimize deflection.
- Modulus of elasticity: The modulus of elasticity is a measure of the stiffness of a material and it affects the deflection of a structure. This criterion ensures that the structure is designed using appropriate materials and dimensions to minimize deflection.

It is important to note that the criteria for control of deflection may vary depending on the code or standard used and the intended use of the structure. However, these criteria provide a general framework for ensuring that the deflection of RCC structures is controlled and within acceptable limits.

**Recall the Codal provision for the following Beam Parameters: i. Minimum Clear Cover ii. Minimum Reinforcement iii. Maximum Reinforcement iv. Side face Reinforcement**

The learning outcome recall the codal provisions for the following beam parameters in reinforced concrete construction (RCC) structures:

- Minimum Clear Cover: The minimum clear cover is the minimum distance between the concrete surface and the reinforcement. This is an important parameter as it protects the reinforcement from corrosion and helps to maintain the bond between the concrete and the reinforcement. The codal provisions for minimum clear cover may vary depending on the code or standard used, but typically it is specified as a minimum of 20-30 mm.
- Minimum Reinforcement: The minimum reinforcement is the minimum amount of reinforcement required in the beam to ensure that it has adequate strength and ductility. The codal provisions for minimum reinforcement may vary depending on the code or standard used, but typically it is specified as a minimum of 0.15-0.20% of the cross-sectional area of the beam.
- Maximum Reinforcement: The maximum reinforcement is the maximum amount of reinforcement allowed in the beam to prevent excessive cracking and to maintain the bond between the concrete and the reinforcement. The codal provisions for maximum reinforcement may vary depending on the code or standard used, but typically it is specified as a maximum of 1-2% of the cross-sectional area of the beam.
- Side face Reinforcement: Side face reinforcement is the reinforcement provided along the sides of the beam to increase its resistance to torsion. This is an important parameter in beams subjected to torsional forces. The codal provisions for side face reinforcement may vary depending on the code or standard used, but typically it is specified as a minimum of 0.10% of the cross-sectional area of the beam.

It is important to note that the codal provisions for beam parameters may vary depending on the code or standard used and the intended use of the structure. However, these provisions provide a general framework for ensuring that the beams in RCC structures are designed and constructed with appropriate parameters to ensure strength, ductility, and serviceability.

**Describe the concept of Lateral Stability of a Beam**

It requires the student to describe the concept of lateral stability of a beam in reinforced concrete construction (RCC).

Lateral stability of a beam refers to the ability of the beam to resist lateral or sideways forces and maintain its shape. This is an important aspect of structural design as lateral instability can lead to collapse or failure of the structure. In reinforced concrete construction, lateral stability is achieved through the use of adequate reinforcement, proper detailing of the reinforcement, and appropriate dimensions of the beam.

There are two main types of lateral stability: transverse stability and longitudinal stability. Transverse stability refers to the ability of the beam to resist lateral forces perpendicular to its length, while longitudinal stability refers to the ability of the beam to resist lateral forces parallel to its length. Both types of stability are important for ensuring the safety and serviceability of the structure.

Lateral stability is affected by various factors such as the geometry of the beam, the type and amount of reinforcement, the type and strength of the concrete, and the presence of any lateral supports. These factors must be carefully considered during the design and construction of the beam to ensure that it has adequate lateral stability.

In summary, the concept of lateral stability of a beam in RCC is crucial for ensuring the safety and serviceability of the structure. It involves the ability of the beam to resist lateral forces and maintain its shape, and it is achieved through the use of adequate reinforcement, proper detailing of the reinforcement, and appropriate dimensions of the beam.

**Recall the design procedure for Singly Reinforced Beam**

A singly reinforced beam is a type of reinforced concrete beam in which reinforcement is provided only on one side of the concrete section. The following is the design procedure for a singly reinforced beam:

Step 1: Determine the loads on the beam

The first step in designing a singly reinforced beam is to determine the loads that will act on the beam. This includes the dead load, live load, and any other loads that may be applicable.

Step 2: Determine the moment of resistance required

The next step is to determine the moment of resistance required to resist the bending moment that will act on the beam. The bending moment is calculated by multiplying the load on the beam by the distance between the point of application of the load and the point where the bending moment is being calculated.

Step 3: Determine the effective depth

The effective depth of the beam is the distance from the compression face of the concrete to the centroid of the tension reinforcement. The effective depth is calculated by considering the maximum allowable span-to-effective depth ratio for the given concrete grade and the given percentage of reinforcement.

Step 4: Determine the area of steel required

Once the effective depth is determined, the area of steel required to resist the bending moment is calculated using the formula:

Ast = (M/Sfy) x (1000/d)

Where Ast is the area of steel required, M is the bending moment, Sfy is the yield strength of the steel, and d is the effective depth of the beam.

Step 5: Check for shear

The design should also ensure that the beam is not only strong enough to resist bending, but also to resist shear forces. The shear force in the beam is calculated by considering the loads acting on the beam and the cross-sectional area of the beam. The design should ensure that the shear force is less than the shear capacity of the beam.

Step 6: Detailing of reinforcement

Finally, the reinforcement detailing of the beam is done, including the number of bars, their spacing, and the length of the bars. The detailing should ensure that the steel reinforcement is placed correctly to provide adequate strength to the beam.

Example:

Design a singly reinforced rectangular beam to carry a dead load of 25 kN/m and live load of 15 kN/m. The clear span of the beam is 5 m, and the beam is to be made of M20 grade concrete and Fe415 grade steel. The effective depth of the beam is to be limited to 400 mm, and the cover to the reinforcement should be 25 mm.

Solution:

Step 1: Load Calculation

Total load = Dead load + Live load

Total load = 25 + 15 = 40 kN/m

Step 2: Bending Moment Calculation

Maximum bending moment occurs at the mid-span of the beam

Maximum bending moment, M = (wL^{2})/8 = (40 x 5^{2})/8 = 125 kNm

Step 3: Effective Depth Calculation

d = effective depth + (dia/2) + cover

400 = effective depth + (12/2) + 25

Effective depth, d = 363 mm

Step 4: Area of Steel Required Calculation

Ast = (M/Sfy) x (1000/d)

Ast = (125 x 10^{6})/(415 x 363) = 822 mm^{2}

Assume four 12 mm diameter bars are used

Area of four bars, A = (π/4) x 12^{2} x 4 = 452 mm^{2}

Use six 12 mm diameter bars

**Recall the design procedure for Doubly Reinforced Beam**

It requires the student to recall the design procedure for a doubly reinforced beam in reinforced concrete construction (RCC).

A doubly reinforced beam is a type of beam in which reinforcement is provided in both the tensile and compressive zones of the beam. This type of beam is commonly used in structures where the loads are high or where the span is long, and additional reinforcement is required to ensure the safety and stability of the structure.

The design procedure for a doubly reinforced beam involves the following steps:

- Load calculation: The first step is to calculate the loads acting on the beam, including dead load, live load, and any other imposed loads.
- Determining the section size: The next step is to determine the size of the beam section required to resist the loads calculated in step 1. This involves calculating the maximum moment and shear in the beam and selecting a section that is capable of resisting these forces.
- Calculating the effective depth: The effective depth of the beam is calculated to determine the location of the neutral axis, which is the line along which the compressive and tensile forces act. This is important for determining the reinforcement required in the beam.
- Determining the steel area: The next step is to determine the required area of reinforcement in the beam. This involves calculating the steel area needed to resist the maximum moment in the beam. The steel area is calculated based on the strength of the steel, the effective depth of the beam, and the maximum moment in the beam.
- Determining the type of reinforcement: The next step is to determine the type and spacing of the reinforcement to be used in the beam. This typically involves selecting bars of the appropriate diameter and spacing to provide the required steel area.
- Checking the reinforcement: The final step is to check the reinforcement in the beam to ensure that it meets the codal provisions for minimum and maximum reinforcement, clear cover, and spacing.

In conclusion, the design procedure for a doubly reinforced beam involves a series of steps to determine the required section size, effective depth, steel area, type and spacing of reinforcement, and to check the reinforcement to ensure it meets the codal provisions. It is important to follow these steps carefully to ensure that the beam is designed and constructed with adequate strength, ductility, and serviceability.

**Recall the design procedure for Flanged Beam**

It requires the student to recall the design procedure for a flanged beam in reinforced concrete construction (RCC).

A flanged beam is a type of beam in which the compression reinforcement is concentrated in the lower flange of the beam and the tensile reinforcement is placed in the web. This type of beam is commonly used in structures where the loads are high or where the span is long, and additional reinforcement is required to ensure the safety and stability of the structure.

The design procedure for a flanged beam involves the following steps:

- Load calculation: The first step is to calculate the loads acting on the beam, including dead load, live load, and any other imposed loads.
- Determining the section size: The next step is to determine the size of the beam section required to resist the loads calculated in step 1. This involves calculating the maximum moment and shear in the beam and selecting a section that is capable of resisting these forces.
- Calculating the effective depth: The effective depth of the beam is calculated to determine the location of the neutral axis, which is the line along which the compressive and tensile forces act. This is important for determining the reinforcement required in the beam.
- Determining the steel area: The next step is to determine the required area of reinforcement in the beam. This involves calculating the steel area needed to resist the maximum moment in the beam. The steel area is calculated based on the strength of the steel, the effective depth of the beam, and the maximum moment in the beam.
- Determining the type of reinforcement: The next step is to determine the type and spacing of the reinforcement to be used in the beam. This typically involves selecting bars of the appropriate diameter and spacing to provide the required steel area and placing the reinforcement in the flanges and web of the beam.
- Checking the reinforcement: The final step is to check the reinforcement in the beam to ensure that it meets the codal provisions for minimum and maximum reinforcement, clear cover, and spacing.

In conclusion, the design procedure for a flanged beam involves a series of steps to determine the required section size, effective depth, steel area, type and spacing of reinforcement, and to check the reinforcement to ensure it meets the codal provisions. It is important to follow these steps carefully to ensure that the beam is designed and constructed with adequate strength, ductility, and serviceability.

**Recall the effective span of the slab**

It requires the student to recall the effective span of a slab in reinforced concrete construction (RCC).

The effective span of a slab is the distance between the supports, measured along the mid-span of the slab. The effective span is an important factor in the design of a slab as it determines the amount of bending moment and shear force that the slab is subjected to.

There are two types of effective span for a slab: the clear span and the center-to-center span. The clear span is the distance between the faces of the supports, while the center-to-center span is the distance between the centres of the supports. In most cases, the center-to-center span is used as the effective span in the design of a slab.

- The effective span of the simply supported slab: L
_{eff}= minimum of { L_{o}+w and L_{o}+d } - The effective span shall be equal to the clear span between the supports
- if the width of the support is less than 1/12 of the clear span, the effective span of a continuous Beam or Slab is calculated as per clause (a) width of support < 1/12 of clear span

12 x width of support < clear span - For end span with one end fixed and the other continuous or for intermediate spans. the effective span shall be the clear span between supports.
- Effective span = Clear span
- For end span with one end free & other continuous the effective span shall be equal to lesser of these TWO

Clause (i) Clear span + effective depth (d)/2 Or Clause (ii) Clear span + width of support/2

- Cantilever Beam & Slab: Effective length = length from face of the support + effective depth(d)

The effective span is used in the calculation of the moment of inertia, which is a measure of the resistance of the slab to bending. The moment of inertia is used in turn to calculate the amount of reinforcement required in the slab to resist the bending moment. The effective span is also used in the calculation of the shear force in the slab, which determines the amount of transverse reinforcement required in the slab.

In conclusion, the effective span of a slab is a critical factor in the design of a reinforced concrete slab. It determines the amount of bending moment and shear force that the slab is subjected to, and is used in the calculation of the reinforcement required in the slab to resist these forces.

**Describe the IS Codal provision for the following: i. Minimum Reinforcement ii. Maximum Reinforcement iii. Maximum Spacing iv. Maximum Diameter of Reinforcement v. Minimum Clear Cover**

It requires the student to describe the Indian Standards (IS) codal provision for the following parameters of a reinforced concrete slab:

i. Minimum Reinforcement: The minimum reinforcement required in a slab is specified in IS 456:2000, which is the Indian Standard code of practice for plain and reinforced concrete. According to this code, the minimum reinforcement required in a slab is 0.12% of the cross-sectional area of the slab.

ii. Maximum Reinforcement: The maximum reinforcement required in a slab is also specified in IS 456:2000. According to this code, the maximum reinforcement required in a slab should not exceed the smaller of the following values: 0.8% of the cross-sectional area of the slab, or 200 sq. mm per meter run of slab.

iii. Maximum Spacing: The maximum spacing between the reinforcing bars in a slab is specified in IS 456:2000. According to this code, the maximum spacing between reinforcing bars should not exceed 300 mm for slabs with a thickness less than 150 mm and 400 mm for slabs with a thickness of 150 mm or greater.

iv. Maximum Diameter of Reinforcement: The maximum diameter of the reinforcing bars used in a slab is specified in IS 456:2000. According to this code, the maximum diameter of the reinforcing bars used in a slab should not exceed 32 mm.

v. Minimum Clear Cover: The minimum clear cover required for the reinforcing bars in a slab is specified in IS 456:2000. According to this code, the minimum clear cover required for the reinforcing bars in a slab should not be less than 40 mm for slabs with a thickness less than 150 mm, and 50 mm for slabs with a thickness of 150 mm or greater. The clear cover is the distance between the surface of the concrete and the reinforcement, and it is necessary to provide adequate protection for the reinforcement against corrosion and to ensure adequate bond between the concrete and the reinforcement.

In conclusion, these codal provisions are specified in IS 456:2000, the Indian Standard code of practice for plain and reinforced concrete. These provisions ensure that the reinforcement in a slab is adequate in terms of both the minimum and maximum amount, the spacing between the reinforcing bars, the diameter of the reinforcing bars, and the clear cover. These provisions are essential to ensure the safety and durability of reinforced concrete slabs.

**Recall the design procedure for One-way Slab**

A one-way slab is a reinforced concrete slab that spans in one direction and is supported on two opposite sides. The design procedure for a one-way slab involves the following steps:

- Determine the loads: The first step in designing a one-way slab is to determine the loads that the slab will be subjected to. This includes dead loads (the weight of the slab and any permanent fixtures), live loads (the weight of the occupants and any movable fixtures), and any imposed loads such as wind and seismic loads.
- Determine the span: The span of the slab is the distance between the supports. The span of a one-way slab is important because it determines the amount of reinforcement required and the thickness of the slab.
- Determine the slab thickness: The slab thickness is determined based on the span and the loads it will be subjected to. This is typically done using a standard table or formula provided by a relevant design code such as IS 456:2000, which is the Indian Standard code of practice for plain and reinforced concrete.
- Determine the reinforcement: The next step in designing a one-way slab is to determine the amount and spacing of the reinforcement required. This is done based on the span, loads, and slab thickness. The minimum and maximum reinforcement required, maximum spacing, and maximum diameter of the reinforcement are specified by the relevant design code, such as IS 456:2000.
- Determine the clear cover: The clear cover is the distance between the surface of the concrete and the reinforcement. The minimum clear cover required is specified by the relevant design code, such as IS 456:2000, to ensure adequate protection for the reinforcement against corrosion and to ensure adequate bond between the concrete and the reinforcement.
- Prepare the reinforcement detailing: The final step in designing a one-way slab is to prepare the reinforcement detailing, which includes the placement of the reinforcement, its spacing, and its anchorage.

In conclusion, the design procedure for a one-way slab involves determining the loads, span, slab thickness, reinforcement, clear cover, and preparing the reinforcement detailing. The design must comply with the relevant design code, such as IS 456:2000, to ensure the safety and durability of the slab.

**Describe Rankine’s Grashoff Theory.**

It requires the student to describe Rankine’s Grashoff theory. Rankine’s Grashoff theory is a theoretical model used in civil engineering to predict the behavior of reinforced concrete beams under bending. The theory was developed by William John Macquorn Rankine, a Scottish engineer and physicist, in the 19th century.

The basic idea of Rankine’s Grashoff theory is that the concrete in a reinforced concrete beam acts as a compression member, while the reinforcement acts as a tension member. The theory assumes that the concrete will crush before the reinforcement yields, and that the reinforcement will carry the tensile forces in the beam. The theory also assumes that the concrete will crack at the tensile face of the beam, and that the reinforcement will transfer the tensile forces to the supports.

Rankine’s Grashoff theory is based on three main assumptions:

- The concrete is assumed to be an idealised material that will crush under compression, and that the crushing strength of the concrete is proportional to its compressive strength.
- The reinforcement is assumed to be an idealised material that will yield under tension, and that the tensile strength of the reinforcement is proportional to its yield strength.
- The behavior of the beam is assumed to be linear-elastic, meaning that the stress-strain behavior of the concrete and reinforcement is linear up to the point of crushing and yielding, respectively.

Rankine’s Grashoff theory provides a simple method for estimating the maximum bending moment and the corresponding maximum stress in a reinforced concrete beam. The theory is widely used in civil engineering, and is particularly useful for rough preliminary design calculations, where a more detailed and sophisticated analysis is not necessary.

In conclusion, Rankine’s Grashoff theory is a theoretical model used to predict the behavior of reinforced concrete beams under bending. The theory is based on three main assumptions, and provides a simple method for estimating the maximum bending moment and the corresponding maximum stress in a reinforced concrete beam.

**Describe IS Codal method to design Two-way Slab.**

It requires the student to describe the Indian Standard (IS) Codal method to design two-way slabs. A two-way slab is a type of reinforced concrete slab that is supported by beams on two opposite sides, and is subject to bending in two directions.

The IS codal method for designing two-way slabs is based on the principle of distributing the load on the slab evenly, such that the slab acts as a flat plate in both directions. The method considers the bending moments, shear forces, and deflections that occur in the slab due to the applied loads, and the reinforcement provided to resist these forces.

The IS codal method for designing two-way slabs consists of the following steps:

- Load calculations: The first step is to determine the magnitude and distribution of the loads acting on the slab, including dead loads, live loads, and any other loads that may be present.
- Moment calculations: The next step is to calculate the bending moments in the slab due to the applied loads. This is done using the principles of statics and mechanics, and considering the geometry of the slab and the loads acting on it.
- Design of reinforcement: The next step is to determine the reinforcement required to resist the bending moments in the slab. This involves calculating the area of reinforcement required in each direction, and determining the spacing and size of the reinforcement bars.
- Checking for shear: The fourth step is to check the slab for shear forces, which may cause cracking or failure of the slab. The IS codal method requires the provision of shear reinforcement in the slab, in the form of stirrups or ties, to resist these forces.
- Checking for deflection: The final step is to check the slab for deflection, which may cause cracking or failure of the slab, or affect the appearance and function of the slab. The IS codal method requires the provision of reinforcement to control the deflection of the slab.

In conclusion, the IS codal method to design two-way slabs is a step-by-step process that involves load calculations, moment calculations, design of reinforcement, checking for shear, and checking for deflection. The method is based on the principle of distributing the load evenly, such that the slab acts as a flat plate in both directions, and considers the bending moments, shear forces, and deflections that occur in the slab due to the applied loads and the reinforcement provided.

**Recall the design procedure for Two-way Slab.**

It requires the student to recall the design procedure for two-way slabs. A two-way slab is a type of reinforced concrete slab that is supported by beams on two opposite sides and is subject to bending in two directions.

The design procedure for two-way slabs can be summarised as follows:

- Load calculations: The first step is to determine the magnitude and distribution of the loads acting on the slab, including dead loads, live loads, and any other loads that may be present.
- Moment calculations: The next step is to calculate the bending moments in the slab due to the applied loads. This is done using the principles of statics and mechanics, and considering the geometry of the slab and the loads acting on it.
- Design of reinforcement: The next step is to determine the reinforcement required to resist the bending moments in the slab. This involves calculating the area of reinforcement required in each direction, and determining the spacing and size of the reinforcement bars.
- Checking for shear: The fourth step is to check the slab for shear forces, which may cause cracking or failure of the slab. The design procedure requires the provision of shear reinforcement in the slab, in the form of stirrups or ties, to resist these forces.
- Checking for deflection: The final step is to check the slab for deflection, which may cause cracking or failure of the slab, or affect the appearance and function of the slab. The design procedure requires the provision of reinforcement to control the deflection of the slab.

In conclusion, the design procedure for two-way slabs involves load calculations, moment calculations, design of reinforcement, checking for shear, and checking for deflection. The procedure is based on the principle of distributing the load evenly, such that the slab acts as a flat plate in both directions, and considers the bending moments, shear forces, and deflections that occur in the slab due to the applied loads and the reinforcement provided.

**Derive an expression for Circular beam loaded uniformly and supported on Symmetrically placed Columns.**

It requires the student to derive an expression for circular beams loaded uniformly and supported on symmetrically placed columns. A circular beam is a beam with a circular cross-section, and it is loaded uniformly when an equal load is applied over its entire length.

To derive an expression for the deflection of a circular beam loaded uniformly and supported on symmetrically placed columns, the following steps can be followed:

- Assume a uniform load on the beam: The first step is to assume a uniform load on the beam. This load can be expressed as a force per unit length, or a linear load distribution.
- Apply beam bending theory: The next step is to apply beam bending theory, which states that the deflection of a beam is proportional to the bending moment at any point along the beam. The bending moment at any point along the beam can be calculated using the equations of statics and mechanics.
- Consider the cross-section of the beam: The next step is to consider the cross-section of the beam, which in this case is circular. The cross-sectional area of the beam can be calculated using the formula for the area of a circle.
- Express the deflection in terms of the geometric parameters of the beam: The final step is to express the deflection of the beam in terms of the geometric parameters of the beam, such as the radius of the circular cross-section, the length of the beam, and the uniform load applied to the beam.

The resulting expression for the deflection of a circular beam loaded uniformly and supported on symmetrically placed columns will depend on the specific beam bending theory used, but it will typically involve the geometric parameters of the beam, the uniform load applied to the beam, and a material constant for the beam material.

In conclusion, the derivation of an expression for the deflection of a circular beam loaded uniformly and supported on symmetrically placed columns involves the application of beam bending theory, the consideration of the cross-section of the beam, and the expression of the deflection in terms of the geometric parameters of the beam and the uniform load applied to the beam.

**Derive an expression for Semi-circular beam simply supported on three equally spaced supports.**

The semi-circular beam simply supported on three equally spaced supports is a type of simply supported beam with a curved cross-section. The beam is supported at three points that are equally spaced along its length. To derive the expression for the semi-circular beam, the following steps can be followed:

- Assume the beam has a uniform cross-section and a uniform load.
- Divide the semi-circular beam into an equal number of segments and consider the forces acting on each segment.
- Use the principle of equilibrium to find the internal forces and moments in the beam. This includes finding the bending moment, shear force, and normal force.
- Solve for the maximum bending moment and shear force in the beam using the boundary conditions.
- Use the formulas for bending stress and shear stress to find the maximum stresses in the beam.
- Use the material properties of the beam, such as its modulus of elasticity and yield stress, to find the minimum required size of the beam.
- Finally, use the equation for the semi-circular beam to calculate the maximum deflection of the beam under the given loads.

The expression for the semi-circular beam can be derived using calculus, taking into account the curvature of the beam, the loads applied to it, and the material properties of the beam. The expression can be used to design the beam and determine its strength and stability.

**Describe the Live arrangement in Continuous Beam**

The live load arrangement in a continuous beam refers to the way that live loads are distributed along the length of the beam. Live loads are temporary loads that can change in magnitude and location, and include things like people, furniture, and equipment. The live load arrangement is an important consideration in the design of a continuous beam because it affects the distribution of stresses and deflections along the length of the beam.

There are several ways to arrange live loads in a continuous beam, including:

- Concentrated loads: This arrangement involves placing live loads at specific points along the beam, such as at the support points.
- Uniform loads: This arrangement involves distributing live loads evenly along the entire length of the beam.
- Linearly varying loads: This arrangement involves distributing live loads in a linear fashion along the length of the beam, with the magnitude of the load increasing or decreasing along the length of the beam.
- Triangularly varying loads: This arrangement involves distributing live loads in a triangular fashion along the length of the beam, with the magnitude of the load increasing or decreasing at an increasing or decreasing rate along the length of the beam.

The live load arrangement in a continuous beam affects the maximum bending moment and shear force, as well as the maximum deflection and stress in the beam. The live load arrangement must be taken into consideration during the design process in order to ensure the safety and stability of the structure.

**Describe the Coefficient of Moments and Shear in Continuous Beam**

The coefficients of moments and shear in a continuous beam are important parameters used in the design of the beam. The coefficients describe the distribution of the bending moments and shear forces along the length of the beam and are used to determine the maximum bending moment, shear force, and deflection at each point along the beam.

- Coefficient of Moment: This coefficient is used to describe the distribution of the bending moments along the length of the beam. The coefficient of moment is a number that ranges from zero to one and represents the fraction of the maximum bending moment that occurs at a particular point along the length of the beam.
- Coefficient of Shear: This coefficient is used to describe the distribution of the shear forces along the length of the beam. The coefficient of shear is a number that ranges from zero to one and represents the fraction of the maximum shear force that occurs at a particular point along the length of the beam.

In the design of a continuous beam, the coefficients of moments and shear are used to determine the maximum bending moment, shear force, and deflection at each point along the length of the beam. This information is then used to determine the size and spacing of the reinforcement required in the beam in order to ensure the safety and stability of the structure.

It is important to note that the coefficients of moments and shear are dependent on the type of load arrangement, the span length, and the arrangement of the supports. The coefficients must be calculated and used correctly in order to obtain accurate results during the design process.