BENDING MOMENT BENDING MOMENT STRESS STRESS

(UNI AXIAL & BI (UNI AXIAL & BI

AXIAL)AXIAL)SERAJUS SALEKIN SERAJUS SALEKIN

TANVIRTANVIRSTUDENT ID: STUDENT ID: 10.01.03.08910.01.03.089

What is Bending?What is Bending? Bending is the present participle of Bending is the present participle of

the word bend which means to the word bend which means to cause something to deflect from the cause something to deflect from the straight line or to make something straight line or to make something have a curved shape.have a curved shape.

What is Bending What is Bending Moment?Moment?

A bending moment is a measure of the A bending moment is a measure of the bending effect due to forces acting on a bending effect due to forces acting on a beam. It is a type of stress and is beam. It is a type of stress and is measured in terms of force and distance.measured in terms of force and distance.

What is Bending Moment What is Bending Moment Stress?Stress?

Bending moment stress is the Bending moment stress is the normal stress that is induced at a normal stress that is induced at a point in a body subjected to loads point in a body subjected to loads that cause it to bend. that cause it to bend.

An internal An internal tensile or compressive tensile or compressive stressstress developed in a beam in developed in a beam in response to curvature induced by an response to curvature induced by an external load.external load.

In Diagram, a simply supported beam loaded at the center. It deflects under

the load . When examining bending moments, horizontal forces act on the

cross sectional face of the beam section.

In Diagram 2, here is the left end section of the beam. It shows the horizontal forces along the top and bottom in Diagram 2a, but the forces act across the whole cross section as shown in the side view in Diagram 2b. The horizontal forces decrease from maximum at the outer edges to zero at the neutral axis (an axis running through the centroid of beam cross section).

The maximum "Bending Stress" at some location along the beam is equal

to the bending moment, M, at that location "times" the distance, y, from the neutral axis to the outer edge of the beam "divided" by the moment of inertia, I, of the beam cross sectional

area.σ =M y / I

Why Bending Moment Why Bending Moment Stress is necessary for Stress is necessary for

design?design? To verify the design for safety of the To verify the design for safety of the

structure and the users. structure and the users. To understand the results obtained To understand the results obtained

from design analysis, it is necessary to from design analysis, it is necessary to have the theoretical knowledge of the have the theoretical knowledge of the bending moment stresses developed bending moment stresses developed in various loading conditions.in various loading conditions.

What is Uniaxial What is Uniaxial Bending Stress?Bending Stress?

uni axial bending stress is the stress that is generated when a structure is bend about one axis.

The 3-point or 4- point bending setups represent this type of bending.

In 3 point and 4 point bending setups ,the sample is placed on parallel support roller and bended by another set of rollers.

During bending the top surface is in compressive stress state and the bottom surface is in a tensile stress state.

The bending stress can be calculated for small deflection,For 3 point bending, σ= 3FI/2bd^2For 4 point bending, σ= 3F/2bd^2(L-L1)Where, load span for L ,the width b ,thickness d and applied force F.

Practical Example of 3 Practical Example of 3 point loadingpoint loading

Practical Example of 4 Practical Example of 4 point loadingpoint loading

What is Biaxial Bending What is Biaxial Bending Stress?Stress?

In Biaxial bending test, stress state is biaxial. In comparison to uniaxial bending stress, biaxial bending often occurs in application.

There are various setups of biaxial bending test. Among them ring-on-ring test and ball-on-ring test commonly used.

Some other test methods are piston-on-3 balls test ,ball-on-3-ball test, 3-ball-on-3-balls test ,ball breaker test etc.

EXAMPLE OF BIAXIAL BENDING EXAMPLE OF BIAXIAL BENDING STRESSSTRESS

PROBLEM:PROBLEM:1)Find the stress distribution at the section 1)Find the stress distribution at the section ABCD for the block shown in figure if P=64 KNABCD for the block shown in figure if P=64 KN2)At the same section find the line of zero stress2)At the same section find the line of zero stress

EXAMPLE OF BIAXIAL BENDING EXAMPLE OF BIAXIAL BENDING STRESSSTRESS

SOLUTION:SOLUTION: Moment of inertia about Y-axis and Z-axis,Moment of inertia about Y-axis and Z-axis, IyIy= 150*300^3/12=337.5*10^6 = 150*300^3/12=337.5*10^6

mm^4mm^4 IzIz= 300*150^3/12=84.38*10^6 = 300*150^3/12=84.38*10^6

mm^4mm^4 Area of the section ABCD, Area of the section ABCD, AA=150*300=45000 mm^2=150*300=45000 mm^2 Moment about Y-axis and Z-axis,Moment about Y-axis and Z-axis, MyMy=Pz=64*10^3*150=9.6*10^6 N-mm=Pz=64*10^3*150=9.6*10^6 N-mm MzMz=Py=64*10^3*(75+75)=9.6*10^6 =Py=64*10^3*(75+75)=9.6*10^6

N-mmN-mm

EXAMPLE OF BIAXIAL BENDING EXAMPLE OF BIAXIAL BENDING STRESSSTRESS

Now we can calculate, P/A=64*10^3/4500064*10^3/45000 =1.42 MPa=1.42 MPa MyZ/Iy=(9.6*10^6 9.6*10^6

*150)/337.5*10^6*150)/337.5*10^6 =4.27 MPa=4.27 MPa MzY/IzMzY/Iz=(9.6*10^6 =(9.6*10^6

*75)/84.38*10^6*75)/84.38*10^6 =8.53 MPa=8.53 MPa

EXAMPLE OF BIAXIAL BENDING EXAMPLE OF BIAXIAL BENDING STRESSSTRESS

Now stress at a point can be determined,

σ= -P/A± MyZ/Iy ±MzY/IzStress at point A,B,C,D can be calculated

σ at A = -1.42-4.27-8.53=-14.22 MPaσ at B = -1.42+4.27-8.53=-5.68 MPa

σ at C = -1.42+4.27+8.53=+11.38 MPaσ at D = -1.42-4.27+8.53=+2.84 MPa

EXAMPLE OF BIAXIAL BENDING EXAMPLE OF BIAXIAL BENDING STRESSSTRESS

Here in this example, we can see that due to the Here in this example, we can see that due to the application of load, here occurs biaxial bending application of load, here occurs biaxial bending situation. In this problem we observed that on A situation. In this problem we observed that on A side here occurs maximum compressive stress=-side here occurs maximum compressive stress=-14.22MPa and on C side here occurs maximum 14.22MPa and on C side here occurs maximum tensile stress=+11.38MPa.tensile stress=+11.38MPa.

Practical example of Biaxial Practical example of Biaxial bendingbending

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