Slope and Deflection | Strengths of Materials

SLOPE AND DEFLECTION CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin November 24, 2017March 6, 2022

SLOPE AND DEFLECTION

CLASS NOTES FOR MECHANICAL

ENGINEERING

SLOPE

It is angular shift at any point of the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by dy/dx or θ. Its units are

radians. There is a maximum limit for slope

for any loaded beam.

Deflection

It is the vertical shift of a point on the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by y or 𝜹. Its units are

mm. There is a limit for maximum

deflection for any loaded beam.

METHODS TO FIND SLOPE AND DEFLECTION

Double Integration Method: It is valid for finding slope and deflection for one load at a time. Thus it is time consuming.
Macaulay’s Method: Uses SQUARE BRACKETS. It is applicable for any number and any types of loads.
Superposition Method
Moment Area Method Or Graphical Method : Uses Two Mohr’s Theorems

MOHR’S FIRST THEOREM

For finding Slope.

Statement says that the difference of slopes between any two points on the loaded beam.

It is equal to the area of the BMD between those two points divided by Bending Stiffness EI.

θ_B –θ_A=AREA OF BMD BETWEEN A AND B/EI

MOHR’S SECOND THEOREM

For finding deflection.

Statement says that the difference of deflections between any two points on the loaded beam.

Difference is equal to Moment of the BMD area between those two points divided by Bending Stiffness EI.

This moment is to be about the first point say A.

δ_B —δ_A=Y_B–y_{A =}(AREA OF BMD BETWEEN A AND B) x X_B/EI

5.Strain Energy Method

DIFFERENT DIFFERENTIAL EQUATIONS OR THE BENT BEAM OR DEFLECTED BEAM

Exact differential equation of the deflected beam

(EI d²y/dx²)/[1+(dy/dx)²] =± M_x

Approximate Differential equation of the bent beam:

Neglecting (dy/dx)² since dy/dx is small.

we get approximate Differential equation of the bent beam.

The use of this approximate differential equation introduces maximum error of 4 % in maximum deflection. It is in comparison to value found by exact differential equation. It is also much simpler to use.

(EI d²y/dx²) =±M_x

SLOPE AND DEFLECTIONS FOR DIFFERENT LOADED BEAMS

Sr. No.	TYPE OF LOADED BEAM	MAXIMUM DEFLECTION AND ITS LOCATION ON THE BEAM	MAXIMUM SLOPE AND ITS LOCATION ON THE BEAM
1.	SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD AT THE CENTER	y_max = (1/48)WL³/EI It is at the center.	WL²/16EI and it is at the ends of the beam.
2.	SIMPLY SUPPORTED BEAM WITH A UDL OVER THE ENTIRE SPAN	y_max = (5/384)WL³/EI, W =w L It is at the center.	WL²/24EI, W =w L and it is at the ends of the beam.
3.	CANTILEVER BEAM WITH A CONCENTRATED LOAD AT THE FREE END	y_max = WL³/3EI It is at the free end.	WL²/2EI and it is at the free end
4.	CANTILEVER BEAM WITH A UDL OVER THE ENTIRE SPAN	y_max = WL³/8EI, W =w L It is at the free end.	WL²/6EI, where W =w L and it is at the free end
5.	SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD NOT AT THE CCENTER, (length ‘a’ > length ‘b’	(i) Deflection under the load will be y_W = Wa²b²/3EIL. (ii) Maximum deflection y_max= [Wb(L2—b²)^3/2]/9 and it will be at x= [(L² –b2)/3]^0.5 from the center towards left portion ‘a’ . It is irrespective of the load placed anywhere on the beam span.	—-

https://www.mesubjects.net/wp-admin/post.php?post=4290&action=edit MCQ Bending Stresses Beams

https://www.mesubjects.net/wp-admin/post.php?post=4363&action=edit MCQ slope Deflection

Laminated Spring | Springs | Strengths of Materials

LAMINATED SPRINGS CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin August 24, 2018February 19, 2022

LAMINATED SPRINGS CLASS NOTES FOR MECHANICAL ENGINEERING This type of spring is very common in use. There is an advantage over the coiled spring. It acts as a structural member. In addition it absorbs a shock. Used in cars, trucks, buses and trains. This type of spring has number of plates of constant…

Mohr's Circle | Strengths of Materials

MOHR’S STRESS CIRCLE CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin November 21, 2013March 10, 2022

MOHR’S STRESS CIRCLE CLASS NOTES FOR MECHANICAL ENGINEERING Every item in nature is under complex loading. Hence, it is under complex stresses. But a material fails in a simple manner. It may fail in simple tension/ compression/shear/strain energy/simple strain/shear strain energy. Mohr’s Stress Circle is a graphical method to determine simple stresses and strains from…

Strengths of Materials | Testing of Materials

MATERIAL TESTING & FAILURES CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin March 1, 2015February 14, 2023

MATERIALS TESTING & FAILURES CLASS NOTES FOR MECHANICAL TESTING Perform tests on materials to determine the followings: 1.Strength of the material 2.Internal / external defects TESTING FOR STRENGTH Testing tells to use a material under a load in which it has the highest strength. Use a material under a tensile load If strong in…

Strengths of Materials | Theories of Elastic Failure

THEORIES OF FAILURE CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin May 16, 2021March 15, 2022

THEORIES OF FAILURE CLASS NOTES FOR MECHANICAL ENGINEERING Failure means breaking or deforming. Failure of ductile material is entirely different from the failure of a brittle material under same type of load. There are five well known theories of elastic failures. These theories explain material behavior under simple or complex loaded condition. It is elastic…

Strengths of Materials | Thin shells

THIN SHELLS CLASS NOTES FOR MECHANICAL ENGINEERING

Byadmin September 12, 2018March 8, 2022

THIN CYLINDRICAL SHELL CLASS NOTES FOR MECHANICAL ENGINEERING A vessel is a thin if its thickness is less as compared to its diameter. Mathematically it is expressed as a thin shell if D/t. A vessel is a thin shell where stresses are assumed to be uniform. Uniform stresses mean stress at the inner to…

MCQ | Strain Energy | Strengths of Materials

STRAIN ENERGY MULTIPLE CHOICE QUESTIONS (MCQ) WITH ANSWERS

Byadmin May 4, 2015March 6, 2022

STRAIN ENERGY MULTIPLE CHOICE QUESTIONS (MCQ) WITH ANSWERS Strain energy is due strain. There is strain energy in tension, compression, shear and bending. It is equal to work done in deforming a body. Strain energy determines slope and deflection. There are principal strain energy and shear strain energy as theories of failure. These help in the…

SLOPE AND DEFLECTION

CLASS NOTES FOR MECHANICAL

ENGINEERING

SLOPE

It is angular shift at any point of the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by dy/dx or θ. Its units are

radians. There is a maximum limit for slope

for any loaded beam.

Deflection

It is the vertical shift of a point on the

beam between no load condition and

loaded beam. Its value is different at

different points on the length of the beam.

It is represented by y or 𝜹. Its units are

mm. There is a limit for maximum

deflection for any loaded beam.

Double Integration Method: It is valid for finding slope and deflection for one load at a time. Thus it is time consuming.

Macaulay’s Method: Uses SQUARE BRACKETS. It is applicable for any number and any types of loads.

Superposition Method

Moment Area Method Or Graphical Method : Uses Two Mohr’s Theorems

MOHR’S FIRST THEOREM

For finding Slope.

Statement says that the difference of slopes between any two points on the loaded beam.

It is equal to the area of the BMD between those two points divided by Bending Stiffness EI.

θB –θA=AREA OF BMD BETWEEN A AND B/EI

MOHR’S SECOND THEOREM

For finding deflection.

Statement says that the difference of deflections between any two points on the loaded beam.

Difference is equal to Moment of the BMD area between those two points divided by Bending Stiffness EI.

This moment is to be about the first point say A.

δB —δA=YB–yA =(AREA OF BMD BETWEEN A AND B) x XB/EI

5.Strain Energy Method

DIFFERENT DIFFERENTIAL EQUATIONS OR THE BENT BEAM OR DEFLECTED BEAM

Exact differential equation of the deflected beam

(EI d2y/dx2)/[1+(dy/dx)2] =± Mx

Approximate Differential equation of the bent beam:

Neglecting (dy/dx)2 since dy/dx is small.

we get approximate Differential equation of the bent beam.

The use of this approximate differential equation introduces maximum error of 4 % in maximum deflection. It is in comparison to value found by exact differential equation. It is also much simpler to use.

(EI d2y/dx2) =±Mx

SLOPE AND DEFLECTIONS FOR DIFFERENT LOADED BEAMS

Sr. No.

TYPE OF

LOADED BEAM

MAXIMUM DEFLECTION AND ITS LOCATION ON THE BEAM

MAXIMUM SLOPE AND

ITS LOCATION ON THE BEAM

1.

SIMPLY SUPPORTED BEAM WITH A CONCENTRATED LOAD AT THE CENTER

ymax = (1/48)WL3/EI

It is at the center.

WL2/16EI and it

is at the

ends of the beam.

2.

SIMPLY SUPPORTED BEAM WITH A UDL OVER THE ENTIRE SPAN

ymax = (5/384)WL3/EI,

W =w L

It is at the center.

WL2/24EI, W =w L

and it is at the ends

of the beam.

3.

CANTILEVER BEAM WITH A CONCENTRATED LOAD AT THE FREE END

ymax = WL3/3EI

It is at the free end.

WL2/2EI and it is

at the free end

4.

CANTILEVER BEAM WITH A UDL OVER THE ENTIRE SPAN

ymax = WL3/8EI, W =w L

It is at the free end.

WL2/6EI,

where W =w L

and it is at the

free end

5.

θ_B –θ_A=AREA OF BMD BETWEEN A AND B/EI

δ_B —δ_A=Y_B–y_{A =}(AREA OF BMD BETWEEN A AND B) x X_B/EI

(EI d²y/dx²)/[1+(dy/dx)²] =± M_x

Neglecting (dy/dx)² since dy/dx is small.

(EI d²y/dx²) =±M_x

y_max = (1/48)WL³/EI

WL²/16EI and it

y_max = (5/384)WL³/EI,

WL²/24EI, W =w L

y_max = WL³/3EI

WL²/2EI and it is

y_max = WL³/8EI, W =w L

WL²/6EI,

y_W = Wa²b²/3EIL.

y_max= [Wb(L2—b²)^3/2]/9

at x= [(L² –b2)/3]^0.5 from the center towards left portion ‘a’ . It is irrespective of the load placed anywhere on the beam span.