This document provides an overview of beam and column design concepts. It discusses types of beam supports, beams, shear force and bending moment diagrams, stresses in beams from bending and shear, and beam deflection calculations. It also covers column buckling, including the Euler buckling formula and Johnson's equation. The document provides examples of calculating stresses, strains, deflections, and buckling loads for different beam and column scenarios.
ANALYSIS OF FRAMES USING SLOPE DEFLECTION METHODSagar Kaptan
slope deflection equations are applied to solve the statically indeterminate frames without side sway. In frames axial deformations are much smaller than the bending deformations and are neglected in the analysis.
Types of support system,Beam classification,methods for the analysis of indeterminate structures,analysis of indeterminate structures,Formula for Determination of Degree of Static Indeterminacy(DOSI),DOSI,Determine DOSI for the structure.
ANALYSIS OF FRAMES USING SLOPE DEFLECTION METHODSagar Kaptan
slope deflection equations are applied to solve the statically indeterminate frames without side sway. In frames axial deformations are much smaller than the bending deformations and are neglected in the analysis.
Types of support system,Beam classification,methods for the analysis of indeterminate structures,analysis of indeterminate structures,Formula for Determination of Degree of Static Indeterminacy(DOSI),DOSI,Determine DOSI for the structure.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
Introduction-Plastic hinge concept-plastic section modulus-shape factor-redistribution of moments-collapse mechanism.
Theorems of plastic analysis - Static/lower bound theorem; Kinematic/upper bound theorem-Plastic analysis of beams and portal frames by equilibrium and mechanism methods.
Class notes of Geotechnical Engineering course I used to teach at UET Lahore. Feel free to download the slide show.
Anyone looking to modify these files and use them for their own teaching purposes can contact me directly to get hold of editable version.
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
Stress and Strains, large deformations, Nonlinear Elastic analysis,critical load analysis, hyper elastic materials, FE formulations for Non-linear Elasticity, Nonlinear Elastic Analysis Using Commercial Finite Element Programs, Fitting Hyper elastic Material Parameters from Test Data
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
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Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
block diagram and signal flow graph representation
Beams And Columns
1. Subject:Subject: Machine Design and Industrial Drafting (2141907)Machine Design and Industrial Drafting (2141907)
Chapter: Design Of Beams And ColumnsChapter: Design Of Beams And Columns
Department Mechanical Engineering
Name of Subject Teacher
Dr. Divyang Patel
2. Team MembersTeam Members
Name Enrollment Number
Vinay PatelVinay Patel 170990119014170990119014
Dhananjay PatelDhananjay Patel 170990119015170990119015
Dhyey ShuklaDhyey Shukla 170990119016170990119016
22. Deflection by Bending MomentDeflection by Bending Moment
EquationEquation
22
)(
2
'' qxqLx
xMEI −==υ
1
32
'
64
C
qxqLx
EI +−=υ
21
43
2412
CxC
qxqLx
EI ++−=υ
0,
24
2
3
1 =−= C
qL
C
23. Deflection by Loading EquationDeflection by Loading Equation
L
xLq
qEI
)(0'''' −
−==υ
0,
2
)(
11
2
0'''
=+
−
= CC
L
xLq
EIυ
0,
6
)(
22
3
0''
=+
−
−= CC
L
xLq
EIυ
,
24
)(
3
4
0'
C
L
xLq
EI +
−
=υ
,
120
)(
43
5
0 CxC
L
xLq
EI ++
−
−=υ
25. Strain Energy of Pure BendingStrain Energy of Pure Bending
EI
ML
L
L
=== κ
ρ
θ
L
EI
EI
LM
U
22
22
θ
==
26. Strain Energy of BendingStrain Energy of Bending
L
EI
EI
LM
U
22
22
θ
==
∫∫ == dx
EI
EI
dxM
U 2''
2
)(
22
υ
27. Strain Energy of a Beam in ShearStrain Energy of a Beam in Shear
∫=
AG
dxCV
U
2
2
Rectangular: 1.2
Circular: 1.11
Thin-walled tubular, round: 2.00
Box section: 1.00
Structural section: 1.00
28. Strain Energy of BendingStrain Energy of Bending
22
)(
2
qxqLx
xM −=
EI
Lq
EI
dxM
U
2402
522
== ∫
qx
qL
xV −=
2
)(
AG
Lq
AG
dxV
U
202
2.1 322
== ∫
29. Castigliano’s TheoremCastigliano’s Theorem
i
i
P
U
∂
∂
=δ
When forces act on a elastic system
subject to small displacements, the
displacement corresponding to any
force, collinear with the force, is equal
to the partial derivative of the total
strain energy with respect to that
force. It can also be used to find
the displacement when no force is
applied at that point.
i
i
M
U
∂
∂
=θ
0| =
∂
∂
= iQ
i
i
Q
U
δ
30. Modified Castigliano’s TheoremModified Castigliano’s Theorem
∫∫ ∂
∂
=
∂
∂
=
∂
∂
= dx
P
M
EI
M
EI
dxM
PP
U
iii
i
2
2
δ
0MPxM −−=
1,
0
−=
∂
∂
−=
∂
∂
M
M
x
P
M
EI
LM
EI
PL
dx
P
M
EI
M
i
A
23
2
0
3
+=
∂
∂
= ∫δ
EI
LM
EI
PL
dx
M
M
EI
M
A
0
2
0 2
+=
∂
∂
= ∫θ
39. Design ConsiderationsDesign Considerations
• Stress – Yield Failure or Code Compliance
• Deflection
• Strain
• Stiffness
• Stability – Important in compressive members
• Stress and strain relationships can be studied with Mohr’s circle
Often the controlling factor for
functionality
40. Deflection [Everything’s a Spring]Deflection [Everything’s a Spring]
• When loads are applied, we have deflection
• Depends on
• Type of loading
• Tension
• Compression
• Bending
• Torsion
• Cross-section of member
• Comparable to pushing on a spring
• We can calculate the amount of beam deflection by
various methods
41. SuperpositionSuperposition
• Determine effects of individual loads separately and
add the results [see examples 4-2,3,4]
• Tables are useful – see A-9
• May be applied if
• Each effect is linearly related to the load that produces it
• A load does not create a condition that affects the result of
another load
• Deformations resulting from any specific load are not large
enough to appreciably alter the geometric relations of the
parts of the structural system
42. Deflection --- Energy MethodDeflection --- Energy Method
• There are situations where the tables are insufficient
• We can use energy-methods in these circumstances
• Define strain energy
•
• Define strain energy density**
• V – volume
• Put in terms of σ, ε
∫=
1
0
x
FdxU
dV
dU
=µ
∫
=
=
=
==
=
dV
E
U
dUdV
dV
dU
E
E
x
x
xx
xx
2
2
2
1
2
1
2
1
σ
µ
µ
σ
εσµ
εσ
43. Example – beam in bendingExample – beam in bending
)(
2
2
2
2
2
2
22
2
xf
EI
M
dAdxdV
dV
EI
yM
U
dV
E
U
I
My
x
=
=
=
=
=
∫
∫
σ
σ
( )
dx
EI
M
U
dx
EI
dAyM
dAdx
EI
yM
dV
EI
yM
U
dAyI
∫
∫
∫
∫∫
∫
=
===
=
2
2
)(
22
2
2
22
2
22
2
22
2
44. Castigliano’s TheoremCastigliano’s Theorem
• Deflection at any point along a beam subjected to n loads may be
expressed as the partial derivative of the strain energy of the
structure WRT the load at that point
• We can derive the strain energy equations as we did for bending
• Then we take the partial derivative to determine the deflection
equation
• Plug in load and solve!
• AND if we don’t have a force at the desired point:
• If there is no load acting at the point of interest, add a dummy load Q,
work out equations, then set Q = 0
i
i
F
U
∂
∂
=δ
45. Castigliano ExampleCastigliano Example
• Beam AB supports a uniformly distributed load
w. Determine the deflection at A.
• No load acting specifically at point A!
• Apply a dummy load Q
• Substitute expressions for M, M/ QA, and QA
(=0)
• We directed QA downward and found δA to be
positive
• Defection is in same direction as QA (downward)
Q
EI
wL
A
8
4
=δ
( )( )
EI
wL
dxxwx
EI
x
Q
M
wxxQxM
Q
U
L
A
A
A
A
A
8
1
)(
dx
Q
M
EI
M
4
0
2
2
1
2
2
1
A
L
0
=−−=
−=
∂
∂
−−=
∂
∂
=
∂
∂
=
∫
∫
δ
δ
Aδ
46. StabilityStability
•Up until now, 2 primary concerns
• Strength of a structure
• It’s ability to support a specified load without
experiencing excessive stress
• Ability of a structure to support a specified load
without undergoing unacceptable deformations
•Now, look at STABILITY of the structure
• It’s ability to support a load without undergoing a
sudden change in configuration
Material
failure
47. BucklingBuckling
• Buckling is a mode of failure that does not depend on
stress or strength, but rather on structural stiffness
• Examples:
49. BucklingBuckling
• The most common problem involving buckling is the design
of columns
• Compression members
• The analysis of an element in buckling involves establishing a
differential equation(s) for beam deformation and finding
the solution to the ODE, then determining which solutions
are stable
• Euler solved this problem for columns
50. Euler Column FormulaEuler Column Formula
•
• Where C is as follows:
2
2
L
EIc
Pcrit
π
=
C = ¼ ;Le=2L
Fixed-free
C = 2; Le=0.7071L
Fixed-pinned
C = 1: Le=L
Rounded-rounded
Pinned-pinned
C = 4; Le=L/2
Fixed-fixed
2
2
e
crit
L
EI
P
π
=
51. BucklingBuckling
•Geometry is crucial to correct analysis
• Euler – “long” columns
• Johnson – “intermediate” length columns
• Determine difference by slenderness ratio
•The point is that a designer must be alert to the
possibility of buckling
•A structure must not only be strong enough, but
must also be sufficiently rigid
54. Solving buckling problemsSolving buckling problems
• Find Euler-Johnson tangent point with
• For Le/ρ < tangent point (“intermediate”), use Johnson’s Equation:
• For Le/ρ > tangent point (“long”), use Euler’s equation:
• For Le/ρ < 10 (“short”), Scr = Sy
• If length is unknown, predict whether it is “long” or “intermediate”, use the
appropriate equation, then check using the Euler-Johnson tangent point once you have
a numerical solution for the critical strength
2
2
=
ρ
π
e
cr
L
E
S
y
e
S
EL 2
2π
ρ
=
2
2
2
4
−=
ρπ
ey
ycr
L
E
S
SS
55. Special Buckling CasesSpecial Buckling Cases
• Buckling in very long Pipe
2
2
L
EIc
Pcrit
π
=
Note Pcrit is inversely related to length squared
A tiny load will cause buckling
L = 10 feet vs. L = 1000 feet:
Pcrit1000/Pcrit10 = 0.0001
•Buckling under hydrostatic Pressure
56. Pipe in Horizontal Pipe BucklingPipe in Horizontal Pipe Buckling
DiagramDiagram
•
57. Far End vs. Input Load withFar End vs. Input Load with
BucklingBuckling
•
60. ImpactImpact
• Dynamic loading
• Impact – Chapter 4
• Fatigue – Chapter 6
• Shock loading = sudden loading
• Examples?
• 3 categories
• Rapidly moving loads of constant magnitude
• Driving over a bridge
• Suddenly applied loads
• Explosion, combustion
• Direct impact
• Pile driver, jack hammer, auto crash
Increasing
Severity
61. Impact, cont.Impact, cont.
• It is difficult to define the time rates of load application
• Leads to use of empirically determined stress impact factors
• If τ is time constant of the system, where
• We can define the load type by the time required to apply the load (tAL
= time required to apply the load)
• Static
• “Gray area”
• Dynamic
k
m
πτ 2=
τ3>ALt
ττ 3
2
1
<< ALt
τ
2
1
<ALt
62. Stress and deflection due to impactStress and deflection due to impact
• W – freely falling mass
• k – structure with stiffness (usually large)
• Assumptions
• Mass of structure is negligible
• Deflections within the mass are negligible
• Damping is negligible
• Equations are only a GUIDE
• h is height of freely falling mass before its release
• δ is the amount of deflection of the spring/structure
65. Energy balanceEnergy balance
• Fe is the equivalent static force necessary to
create an amount of deflection equal to δ
• Energy Balance of falling weight, W
( )
s
e
e
e
static
e
W
F
W
s
F
kF
skkW
FhW
δ
δ
δ
δ
δ
δδ
δδ
=
=
=
==
=+
2
1
++=
++=
=+
=+
s
e
s
s
s
s
h
WF
h
h
WhW
δ
δ
δδ
δ
δ
δ
δ
δ
δ
2
11
2
11
2
1
2
1
)(
2
2
66. Impact, cont.
• Sometimes we know velocity at impact rather than the
height of the fall
• An energy balance gives:
++=
++=
=
s
e
s
s
g
v
WF
g
v
ghv
δ
δ
δδ
2
2
2
11
11
2