S.F.D. for CANTILEVER BEAMS

SHEAR FORCE DIAGRAMS OF THREE DIFFERENT TYPES OF CANTILEVER LOADING

CANTILEVER BEAM

This is the most common beam in our surroundings. It is supported at one end with Fixed Joint and is known as Fixed End. The other end remains without any support and known as Free End. At the fixed end, there are a vertical reaction (R_V), a horizontal reaction (R_H) and a reaction moment (M_R).

How To Draw the Shear Force Diagram of a Cantilever.

(i) replace the fixed joint by a vertical, a horizontal reaction force and a reaction moment.

(ii) then divide the beam into different segment depending upon the position of the loads on the beam.

(iii) take the left most segment of the beam and draw a movable section within the segment.

(iv) let the distance of the extreme left end of the beam from the movable section line be X

(v) let the upward (vertical) forces or reactions are positive and the downward forces are negative. Now the sum of the total vertical forces left to the section line is equal to the shear force at the section line at a distance X from the left most end of the beam.

(vi) as positive SF produces positive Bending Moment, hence if we multiply all the forces those are in the left side of the section line with the distances of each force from the section line added with concentrated moment (clockwise as +ve, anti-clockwise as -ve) we get bending moment. So the sum of the products of each force that is in the left side of the section with the distance of it from section line added with pure moment on this section is equal to the Bending Moment at the section line.

CANTI-LEVER BEAM

 

Draw shear force & bending moment diagrams and equations

 

Solution: At first we shall find the reaction of the canti-lever beam.

A canti-lever beam is a common type of beam which is supported on a single fixed joint at one end. A fixed joint can provide a horizontal reaction, a vertical reaction and a reaction moment. While finding reaction we should transform a distributive load (UDL, UVL) to their equivalent concentrated or point load. An equivalent load of a distributed load can be found by placing the total load at the centroid of the distributed load diagram.  

FREE BODY DIAGRAM (FBD) OF THE BEAM

SF and BM Equations:

 Section AB (0 ≤ X≤ 2)

SF = R_A = 130 kN

BM = ‒ M_R + R_AX = ‒ 720 + 130X kN.m

At X = 0; SF = 130 kN and BM = ‒ 720 kN.m

At X =2; SF = 130 kN and BM = ‒ 720 + 260 = ‒ 460 kN.m

Section BD (2≤ X≤ 6)

SF = R_A ‒  20(X‒2) = 130  ‒  20(X‒2)

BM = ‒ M_R + R_AX  ‒  {20(X‒2)²}/2

= ‒ 720 + 130X ‒  {20(X‒2)²}/2

 At X = 2;  SF = 130 kN and BM = ‒ 460 kN.m

At X = 6; SF = 130 ‒  80 = 50 kN and BM = ‒ 720 + 780 ‒ 160 = ‒ 100 kN.m

When a distributive load remains fully on the left side of the section line as it is in the above diagram, we should use an equivalent point load in the place of Distributive load of UVL and UDL.

Section DE (6≤ X≤ 8)

SF = R_A ‒  80 = 130  ‒  80 = 50 kN

BM = ‒ M_R + R_AX  ‒  80(X ‒ 4) = ‒ 720 + 130X ‒  80(X ‒ 4)

At X = 6; SF = 130 ‒  80 = 50 kN and BM = ‒ 720 + 780 ‒ 160 = ‒ 100 kN.m

At X = 8; SF = 130 ‒  80 = 50 kN and BM = ‒ 720 + 1040 ‒ 320 = 0 kN.m

SFD and BMD 

 

IC ENGINES AND COMBUSTION CHAMBER

IC ENGINES :

IC engines, or internal combustion engines, are engines in which combustion of fuel and air occurs within the engine cylinder, converting the chemical energy of the fuel into mechanical energy to perform work. The combustion chamber is a critical component of an IC engine, as it is the location where combustion occurs.

COMBUSTION CHAMBER:

The combustion chamber is typically located at the top of the cylinder in a reciprocating engine, or in the center of the combustion chamber in a rotary engine. It is designed to confine the fuel and air mixture to a small volume, allowing for efficient and controlled combustion.

The shape and size of the combustion chamber can have a significant impact on the performance and efficiency of the engine. The shape of the combustion chamber can affect the way that the fuel and air mixture is mixed and ignited, as well as the speed at which the flame front propagates through the mixture. The size of the combustion chamber can affect the compression ratio of the engine, which in turn affects the power output and fuel efficiency of the engine.

There are various types of combustion chambers used in IC engines, including the traditional spark ignition chamber and the compression ignition chamber. The spark ignition chamber is typically used in gasoline engines, where a spark plug is used to ignite the fuel and air mixture. The compression ignition chamber is typically used in diesel engines, where the fuel is ignited by the heat generated by compressing the air in the cylinder.

Overall, the design of the combustion chamber is a critical factor in the performance and efficiency of an IC engine, and careful attention must be paid to its design in order to optimize engine performance.

COMPONENTS OF A COMBUSTION CHAMBER:

The combustion chamber in an internal combustion engine is typically composed of several key components that work together to promote efficient combustion of the fuel and air mixture. The following are some of the common components of a combustion chamber:

• Cylinder Head:

The cylinder head is the top part of the engine cylinder that contains the combustion chamber. It is typically bolted onto the engine block and is responsible for sealing the combustion chamber and providing a mounting point for the valves, spark plugs, and fuel injectors.

• Piston:

The piston is a cylindrical component that moves up and down within the engine cylinder. It is responsible for compressing the air/fuel mixture and transmitting the force generated by combustion to the crankshaft.

• Valves:

The valves are located in the cylinder head and are responsible for controlling the flow of air and fuel into the combustion chamber and the flow of exhaust gases out of the engine. There are typically two types of valves: intake valves and exhaust valves.

• Spark Plug:

The spark plug is a small device that is used to ignite the fuel and air mixture in the combustion chamber. It generates an electrical spark that ignites the mixture and initiates the combustion process.

• Fuel Injector:

The fuel injector is responsible for delivering fuel into the combustion chamber in a precise and controlled manner. It typically uses a high-pressure fuel system to inject fuel into the combustion chamber at the correct time and in the correct amount.

• Combustion Chamber Walls:

The walls of the combustion chamber are typically made of high-strength materials such as steel or aluminum. They are designed to withstand the high temperatures and pressures generated by combustion and to provide a seal for the combustion gases.

• Intake and Exhaust Ports:

The intake and exhaust ports are openings in the cylinder head that allow air and fuel to enter the combustion chamber and exhaust gases to exit the engine. Overall, the components of a combustion chamber work together to promote efficient and controlled combustion of the fuel and air mixture, maximizing engine performance and efficiency.

DESIGNING CRITERIA OF A COMBUSTION CHAMBER:

The design of a combustion chamber in an internal combustion engine is a critical factor in determining the performance, efficiency, and emissions of the engine. The following are some of the key criteria that must be considered in the design of a combustion chamber:

• Air/Fuel Mixture:

The combustion chamber must be designed to provide proper mixing of air and fuel. This is necessary to ensure efficient combustion and minimize emissions.

• Flame Propagation:

The combustion chamber must be designed to promote fast and efficient flame propagation. This is necessary to ensure that the fuel is burned completely and to maximize power output.

• Compression Ratio:

The combustion chamber must be designed to achieve the desired compression ratio. This is important for determining the engine's power output and fuel efficiency.

• Combustion Efficiency:

The combustion chamber must be designed to promote complete combustion of the fuel. This is necessary to minimize emissions and maximize fuel efficiency.

• Turbulence:

The combustion chamber must be designed to promote turbulence in the air/fuel mixture. This is important for promoting efficient combustion and minimizing emissions.

• Wall Heat Transfer:

The combustion chamber must be designed to minimize heat transfer to the cylinder walls. This is important for reducing engine heat loss and maximizing power output.

• Knock Resistance:

The combustion chamber must be designed to resist engine knock. This is important for maximizing power output and engine efficiency.

• Emissions:

The combustion chamber must be designed to minimize emissions of pollutants such as nitrogen oxides (NOx), carbon monoxide (CO), and particulate matter (PM). This is important for meeting emissions regulations and minimizing environmental impact. Overall, the design of the combustion chamber is a complex process that requires consideration of multiple factors. Careful attention to these criteria is necessary to optimize engine performance and meet emissions regulations.

FAILURE CRITERIA OF COMBUSTION CHAMBER:

The failure of a combustion chamber in an internal combustion engine can be catastrophic and can result in engine damage, reduced performance, or even complete engine failure. The following are some of the common failure criteria of a combustion chamber:

• Overheating:

One of the most common failure modes of a combustion chamber is overheating. This can be caused by a variety of factors, such as a lean air/fuel mixture, excessive compression, or a malfunctioning cooling system. Overheating can cause cracking or warping of the combustion chamber, leading to leaks or even catastrophic failure.

• Detonation:

Detonation occurs when the fuel/air mixture in the combustion chamber detonates spontaneously instead of burning in a controlled manner. This can be caused by factors such as excessive compression, hot spots in the combustion chamber, or low-quality fuel. Detonation can cause the combustion chamber to deform or crack, leading to reduced engine performance or even complete engine failure.

• Pre-ignition:

Pre-ignition occurs when the fuel in the combustion chamber ignites before the spark plug fires. This can be caused by factors such as hot spots in the combustion chamber, high compression, or low-quality fuel. Pre-ignition can cause damage to the combustion chamber and other engine components, leading to reduced engine performance or even complete engine failure.

• Corrosion:

Corrosion can occur in the combustion chamber due to the corrosive nature of the fuel or the combustion process itself. Corrosion can weaken the walls of the combustion chamber, leading to cracks or other types of damage that can compromise engine performance.

• Mechanical Damage:

Mechanical damage to the combustion chamber can occur due to improper installation, poor maintenance, or external factors such as debris striking the engine. This type of damage can cause leaks or other types of damage that can affect engine performance or even cause complete engine failure. Overall, the failure of a combustion chamber can have severe consequences for engine performance and reliability. Regular maintenance and proper operation of the engine can help to prevent these failure modes and ensure the long-term reliability and performance of the engine.