This document summarizes key concepts related to structural analysis including:
- Effects of axial and eccentric loads on columns including direct stress, bending stress, and maximum/minimum stresses.
- Maximum and minimum pressures at the base of dams including calculations of total water pressure, eccentricity, and formulas for stress.
- Concepts related to retaining walls including total earth pressure, stability conditions of no overturning, no tension, and no sliding.
- Forces and stresses on chimneys and walls due to wind pressure including calculations of direct stress, bending moment, and extreme stresses.
The document discusses different types of loads acting on columns and how they induce stresses. It defines axial load, eccentric load, and eccentricity. It explains that eccentric loads produce both direct and bending stresses while axial loads only produce direct stresses. It provides equations to calculate the maximum and minimum stresses in a column under eccentric loading. An example problem is worked out calculating the maximum stresses in a T-section column loaded eccentrically. The document also discusses loads, eccentricity, and stresses on dams, retaining walls, and chimneys/walls loaded by wind pressure.
- A column subjected to an eccentric axial load experiences both direct compressive stress and bending stress. The maximum stress occurs on the edge of the column closest to the load and the minimum stress is on the opposite edge.
- For a dam, the total water pressure acts below the centroid and combined with the weight of the dam creates an eccentric load. This leads to maximum and minimum pressures at the base that depend on the eccentricity.
- For a retaining wall, it experiences an eccentric load due to the total earth pressure acting below the midpoint. This causes maximum and minimum pressures similar to a dam.
- A chimney or wall subjected to wind pressure on one side experiences direct stress due to its self weight and bending
1. The document discusses stresses in solids due to eccentric and combined loading, including bending and direct stresses.
2. It defines the core of a section as the area where a load can be applied without causing tensile stress. For a rectangular section, the core is a rhombus with diagonals of B/3 and D/3.
3. Wind loading on structures like walls and chimneys is also analyzed, calculating bending moments and resultant stresses. Maintaining compressive stresses only is important for structural integrity.
Liquid limit is the water content where the soil starts to behave as a liquid. Liquid limit is measured by placing a clay sample in a standard cup and making a separation (groove) using a spatula. The cup is dropped till the separation vanishes. The water content of the soil is obtained from this sample.
1) Poisson's ratio describes the strain in the perpendicular direction due to loading. Under uniaxial loads, the perpendicular strain is proportional to the parallel strain multiplied by Poisson's ratio. Under biaxial loads, the strains are calculated considering the effects of both loads.
2) Thermal expansion causes materials to expand or contract with changes in temperature. Temperature changes can also induce thermal stresses if expansion is constrained.
3) Composite members made of different materials experience stress based on their relative moduli of elasticity. The stress in each material is proportional to the other by their modular ratio.
6. Bending Moment combined with Axial Loads.pptxyishChaudhari
The document discusses bending moment with axial loads. It explains that when a member is subjected to both axial loading and transverse bending, it experiences both direct stress due to axial load and bending stress due to the bending moment. The stresses are calculated separately and then added together as they act normal to the cross-section. Eccentric loading is discussed along with examples of symmetrical and unsymmetrical columns with eccentric axial loads. Methods to calculate maximum and minimum stresses in such cases are presented.
This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
The document discusses different types of loads acting on columns and how they induce stresses. It defines axial load, eccentric load, and eccentricity. It explains that eccentric loads produce both direct and bending stresses while axial loads only produce direct stresses. It provides equations to calculate the maximum and minimum stresses in a column under eccentric loading. An example problem is worked out calculating the maximum stresses in a T-section column loaded eccentrically. The document also discusses loads, eccentricity, and stresses on dams, retaining walls, and chimneys/walls loaded by wind pressure.
- A column subjected to an eccentric axial load experiences both direct compressive stress and bending stress. The maximum stress occurs on the edge of the column closest to the load and the minimum stress is on the opposite edge.
- For a dam, the total water pressure acts below the centroid and combined with the weight of the dam creates an eccentric load. This leads to maximum and minimum pressures at the base that depend on the eccentricity.
- For a retaining wall, it experiences an eccentric load due to the total earth pressure acting below the midpoint. This causes maximum and minimum pressures similar to a dam.
- A chimney or wall subjected to wind pressure on one side experiences direct stress due to its self weight and bending
1. The document discusses stresses in solids due to eccentric and combined loading, including bending and direct stresses.
2. It defines the core of a section as the area where a load can be applied without causing tensile stress. For a rectangular section, the core is a rhombus with diagonals of B/3 and D/3.
3. Wind loading on structures like walls and chimneys is also analyzed, calculating bending moments and resultant stresses. Maintaining compressive stresses only is important for structural integrity.
Liquid limit is the water content where the soil starts to behave as a liquid. Liquid limit is measured by placing a clay sample in a standard cup and making a separation (groove) using a spatula. The cup is dropped till the separation vanishes. The water content of the soil is obtained from this sample.
1) Poisson's ratio describes the strain in the perpendicular direction due to loading. Under uniaxial loads, the perpendicular strain is proportional to the parallel strain multiplied by Poisson's ratio. Under biaxial loads, the strains are calculated considering the effects of both loads.
2) Thermal expansion causes materials to expand or contract with changes in temperature. Temperature changes can also induce thermal stresses if expansion is constrained.
3) Composite members made of different materials experience stress based on their relative moduli of elasticity. The stress in each material is proportional to the other by their modular ratio.
6. Bending Moment combined with Axial Loads.pptxyishChaudhari
The document discusses bending moment with axial loads. It explains that when a member is subjected to both axial loading and transverse bending, it experiences both direct stress due to axial load and bending stress due to the bending moment. The stresses are calculated separately and then added together as they act normal to the cross-section. Eccentric loading is discussed along with examples of symmetrical and unsymmetrical columns with eccentric axial loads. Methods to calculate maximum and minimum stresses in such cases are presented.
This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
1. The document defines key terms related to loads, stresses, strains, and elastic behavior of materials. It describes different types of loads, stresses, strains and their relationships based on Hooke's law.
2. Formulas are provided for calculating tensile stress, compressive stress, shear stress, elastic modulus, and deformation of tapered and composite bars.
3. The principles of superposition and self-weight induced stresses in cantilever beams are also summarized.
Structural Mechanics: Shear stress in Beams (1st-Year)Alessandro Palmeri
- The document discusses shear stress in beams, specifically focusing on Jourawski's formula for calculating shear stress.
- Jourawski's formula provides an approximate solution for the shear stress distribution over a beam cross-section using simple equilibrium considerations.
- The formula can be applied to solid rectangular sections, giving a parabolic shear stress distribution, but provides inaccurate results for T-section and I-section flanges.
- For T-sections and I-sections, the formula can be applied to the web, where it accurately models the shear stress as parabolic. The maximum shear stress occurs at the neutral axis.
This document discusses key concepts related to ship strength, including stress, strain, bending, shear, moment of inertia, and section modulus. It explains that stress is force per unit area, while strain is the deformation caused by an applied load. Bending leads to both tension and compression on opposite sides of the neutral axis. The moment of inertia and section modulus help determine the maximum stress in bending. Ship strength calculations are complicated by irregular cargo loads, buoyancy forces, and dynamic wave loading that cause varying shear forces and bending moments over time.
This document discusses the forces acting on gravity dams and their environmental impacts. It outlines various forces like water pressure, weight of the dam, uplift pressure, earthquake pressure, and wave pressure. It also explains how these forces are calculated. Regarding failure, it notes dams can fail through overturning, sliding, compression, or tension. The document concludes by covering environmental impacts of dam construction like pollution, and impacts of reservoirs like habitat destruction and sedimentation.
The document discusses methods for determining deflection and slope in beams. It defines terms like deflection, slope, flexural rigidity, and presents the differential equation of the elastic curve. Macaulay's method is described as a way to determine slope and deflection for complex beam loadings by integrating the bending moment expression while applying boundary conditions. Several example problems are given demonstrating the use of Macaulay's method to calculate deflection and slope at specific points for beams with various loading conditions.
The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
The document discusses plate bending and deflection under applied forces and moments. It provides equations to calculate deflection based on plate theory and stresses. Maximum deflections and stresses occur at different locations depending on boundary conditions and load placement. As an example, it analyzes deflection and yield pressure of a circular plate with fixed edges under uniform pressure, finding the center has maximum deflection. It also gives the working pressure calculation based on a factor of safety of 2.
This document provides information about shear stress and shear force in structural elements like beams. It defines shear stress and explains how to calculate it. It discusses the distribution of shear stress across different cross section shapes, including rectangular, circular, T-sections, and wide flange sections. It also describes how shear stress affects steel and concrete materials and what reinforcement is used to address problems caused by shear stresses.
The document discusses column buckling and failure modes. It defines a strut and column, and describes failure due to direct stress, buckling stress, or a combination. It provides differential equations to model column buckling based on the column's end conditions, including fixed-fixed, fixed-free, and fixed-hinged. Solutions provide the critical buckling load. The document also discusses Rankine's formula which relates crushing load and buckling load for a column's total capacity.
- Stress is defined as the internal force per unit area within a material. It can be tensile or compressive. Common types include normal stress and shear stress.
- Strain is a measure of deformation in a material under stress. Normal strain measures changes in length while shear strain measures changes in shape.
- The allowable stress for a material is less than its failure stress to ensure safety under loads. Factor of safety is defined as the ratio of failure stress to allowable stress.
The document discusses concepts of stress, including:
1. Stress is defined as the force per unit area acting on a surface or section. There are two main types: normal stress and shear stress.
2. To determine if a structure can safely support a load, both the internal forces and the material properties must be considered.
3. Allowable stress values lower than the actual failure stress are used in design, with factors of safety typically between 1-3 depending on the application. This ensures the structure does not fail under expected loading conditions.
The document discusses foundations and their design. It defines foundations as structures that transmit loads from superstructures to underlying soil or rock. Foundations are categorized as either shallow or deep depending on their embedment depth. Key factors in selecting a foundation type include loads, subsurface conditions, performance requirements, and materials. Foundation design involves checking bearing capacity, settlement, and structural integrity. Shallow foundations like spread and combined footings are further described in terms of their geometry, loading conditions, and structural design.
In this section the concept of stress will be introduced, and this will be applied to components that are in a state of tension, compression, and shear. Strain measurement methods will also be briefly discussed.
This document discusses hydrostatic forces on surfaces, including:
1. Forces act perpendicular to surfaces submerged in fluid.
2. Forces on plane surfaces can be calculated based on pressure, area, and depth. Center of pressure is below the geometric center.
3. Forces on curved surfaces are resolved into horizontal and vertical components based on the surface orientation at each point.
Columns are structural elements that transmit loads in compression from beams and slabs above to other elements below. Columns can experience both axial compression and bending loads. Biaxial bending occurs when a column experiences simultaneous bending about both principal axes, such as in corner columns of buildings. The biaxial bending method permits analysis of rectangular columns under these conditions. The document provides details on analyzing a sample reinforced concrete column for adequacy using the reciprocal load method to check that factored loads do not exceed design capacity. Diagrams are presented showing interaction surfaces and stress distributions for concentrically and eccentrically loaded columns.
Strain energy is a type of potential energy that is stored in a structural member as a result of elastic deformation. The external work done on such a member when it is deformed from its unstressed state is transformed into (and considered equal to the strain energy stored in it.
This document discusses various topics related to stresses, strains, and elasticity in mechanics of solids including Hooke's law, stress-strain diagrams, stresses in bars of different cross-sectional variations, deformation due to self-weight, and stresses in composite bars. Key concepts covered are stress, strain, elasticity, modulus of elasticity, stress-strain curves, prismatic and non-prismatic bars, deformation of uniform and varying cross-section bars, and calculating stresses and strains in composite bars.
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This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
This document discusses bending moment stress, which is the internal stress caused by bending moments in a beam. It defines bending moment stress and provides the formula to calculate it based on the moment acting, distance from the neutral axis, and moment of inertia. It also discusses uniaxial bending stress, which acts in one direction, and biaxial bending stress, which acts in two directions. Examples of each type are provided to illustrate stress distribution under different loading conditions.
1. The document defines key terms related to loads, stresses, strains, and elastic behavior of materials. It describes different types of loads, stresses, strains and their relationships based on Hooke's law.
2. Formulas are provided for calculating tensile stress, compressive stress, shear stress, elastic modulus, and deformation of tapered and composite bars.
3. The principles of superposition and self-weight induced stresses in cantilever beams are also summarized.
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- The document discusses shear stress in beams, specifically focusing on Jourawski's formula for calculating shear stress.
- Jourawski's formula provides an approximate solution for the shear stress distribution over a beam cross-section using simple equilibrium considerations.
- The formula can be applied to solid rectangular sections, giving a parabolic shear stress distribution, but provides inaccurate results for T-section and I-section flanges.
- For T-sections and I-sections, the formula can be applied to the web, where it accurately models the shear stress as parabolic. The maximum shear stress occurs at the neutral axis.
This document discusses key concepts related to ship strength, including stress, strain, bending, shear, moment of inertia, and section modulus. It explains that stress is force per unit area, while strain is the deformation caused by an applied load. Bending leads to both tension and compression on opposite sides of the neutral axis. The moment of inertia and section modulus help determine the maximum stress in bending. Ship strength calculations are complicated by irregular cargo loads, buoyancy forces, and dynamic wave loading that cause varying shear forces and bending moments over time.
This document discusses the forces acting on gravity dams and their environmental impacts. It outlines various forces like water pressure, weight of the dam, uplift pressure, earthquake pressure, and wave pressure. It also explains how these forces are calculated. Regarding failure, it notes dams can fail through overturning, sliding, compression, or tension. The document concludes by covering environmental impacts of dam construction like pollution, and impacts of reservoirs like habitat destruction and sedimentation.
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The document discusses stress and strain in materials. It introduces the key concepts of normal stress, shear stress, bearing stress, and thermal stress. Normal stress acts perpendicular to a cross-section, shear stress acts tangentially, and bearing stress occurs at contact points. The relationships between stress, strain, elastic modulus, and Poisson's ratio are explained. Methods for calculating stress and strain in axial loading, torsion, bending and combined loading are presented through examples. The stress-strain diagram is discussed to show material properties like yield strength and ductility.
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- Stress is defined as the internal force per unit area within a material. It can be tensile or compressive. Common types include normal stress and shear stress.
- Strain is a measure of deformation in a material under stress. Normal strain measures changes in length while shear strain measures changes in shape.
- The allowable stress for a material is less than its failure stress to ensure safety under loads. Factor of safety is defined as the ratio of failure stress to allowable stress.
The document discusses concepts of stress, including:
1. Stress is defined as the force per unit area acting on a surface or section. There are two main types: normal stress and shear stress.
2. To determine if a structure can safely support a load, both the internal forces and the material properties must be considered.
3. Allowable stress values lower than the actual failure stress are used in design, with factors of safety typically between 1-3 depending on the application. This ensures the structure does not fail under expected loading conditions.
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1. Subject name : structure analysis-1
Subject code : 2140608
Guided by :- Prof. Pritesh Rathod
Prof. Rajan lad
2. Name Enrollment Number
Gandhi Harshil R. 141100106018
Deshmukh Bhavik H. 151103106002
Kotila Jayveer V. 151103106008
Misrty Aditya P. 151103106009
Pandya Dhrumil D. 151103106010
3. (A). Column :-
Axial load : When load is acting along the longitudinal
axis of column. It produces compressive stress in column.
Eccentric load : A load whose line of action does not
coincide with the axis of a column. It produces direct
and bending stress in column.
Eccentricity (e) : The horizontal distance between the
longitudinal axis of column and line of action of load.
In axially loaded column e = 0
5. When short column is subjected to axial compressive
force, only direct stress(σo) is produced in the column.
Direct stress = σo = P/A
Eccentric load produces both direct stress (σo) and
bending stress (σb) in column.
Direct stress = σo = P/A
Bending stress = σb =M/z= M/I .y ∴ Z =I/Y
6. Maximum and Minimum Stresses :-
When a column is subjected to eccentric load, the edge of
column towards the eccentricity will be subjected to maximum
stress (σmax) and the opposite edge will be subjected to
maximum stress.
Maximum Stress (σmax) Minimum Stress (σmin)
σmax = direct stress + bending stress
= σo + σb
= P/A + M/Z
= P/A (1+ 6e/b)
σo = direct stress, σb = bending stress, M = Moment = P . e, e = eccentricity
Z = section modulus = , I = moment of Inertia, y = distance of extreme fibre
from c.g. of column.
σmin = direct stress - bending stress
= σo - σb
= P/A -M/Z
= P/A (1- 6e/b)
7. Stress distribution in Column :-
Stress distribution in column as the load (P) moves from
centre of column to the edge of column as shown in fig.
8. Limit of eccentricity (e limit):
The maximum distance of load from the centre of
column, such that if load acts within this distance there
is no tension in the column. The maximum distance is
called Limit of eccentricity.
When load is acting within e limit,
𝜎min will be compressive. (+ve)
When load is acting at the point of e limit,
𝜎min will be zero.
When load is acting beyond e limit,
𝜎min will be tensile (-ve)
9.
10.
11. (B). Dams and Retaining Wall :-
Maximum and Minimum pressure at the base of Dam :
12. (1) Weight of Dam:
Weight = cross sectional area of dam x density of
dam material
∴ W = (a + b) x (H/2)x ᵟ
where ᵟ = density of dam material in kN/m3
(2) Total water pressure on Dam:
Total water pressure = Area of water pressure diagram
∴ P =
1
2
x wh x h Where, w = density of water
∴ p=wh²/2 = 1000 kg/m3
= 10 kN/m3
13. (3) Eccentricity (e) :
Total water pressure (p) acts horizontally at height
h
3
from
the base of dam.
Total weight of dam (W) acts vertically downwards.
R is the resultant of P and W.
R = P2 + W2
Resultant (R) cut the base at point K.
distance JK = x = (p/w)x(h/3)
distance AJ =
a2+ab+b2
3 (a+b)
∴ d = AJ + JK
∴ eccentricity = e = d −(𝑏/2)
14. (4) Maximum and Minimum Pressure :
Maximum Pressure.
𝜎max = w/b(1 + 6e/b)
Minimum Pressure.
𝜎min = w/b(1 − 6e/b)
15. Retaining Wall :
A retaining wall is a structure used to retain soil(earth).
The basic difference between dam and retaining wall is that, a
dam retain water and subjected to water pressure while, a
retaining wall retain earth and subjected to earth pressure.
16. Total earth pressure :
P =
wh2
2
x Ka
∴ P =
wh2
2
x (
1 −sin∅
1+sin∅
)
Where,
Ka = active earth pressure coefficient
=
1 −sin∅
1+sin∅
∅ = Angle of repose of soil
Total earth pressure (P) acts at height
h
3
from the bases of
retaining wall.
17. Stability conditions for retaining
wall or Dam
A retaining wall or a dam is checked for the following conditions of
stability:-
No overturning
two major forces acting on retaining wall/dam are:-
(a) total earth/water pressure(P)
(b) weight of wall/ dam(W)
for no overturning
resisting moment > overturning moment
JB > JK
18. No tension at base
To avoid tension in the masonary at the base,
minimum stress should not be(-ve).
for no tension at base eccentricity should be less
than (b/6).
19. No sliding
total frictional force at the base of wall/dam
= Fmax = 𝜇. 𝑊
for no sliding = 𝜇. 𝑊 > P
if factor of safety = 1.5
(𝜇. 𝑊/P) > 1.5
No crushing at base
the material of wall/dam should be safe against
crushing at the base.
∴ 𝜎 max < 𝜎𝑐
𝜎𝑐 = permissible crushing stress
𝜎 max = maximum pressure at the base
26. Consider a chimney or wall having plan dimension b x d and
height h.
Let, 𝛿 = unit weight of chimney or wall
𝛿 =
total weight
volume
Where,
∴ 𝛿 = w/v A = base area
∴ w = 𝛿 x V
w = 𝛿 x A x h
∴ direct stress = σo =w/A
=(𝛿 ∗ 𝐴 ∗ 𝐻)/𝐴
σo = 𝛿 x h ….(i)
If there is a uniform horizontal wind pressure (p) acting
on a side of width b,
wind force = P = p x b x h
This force will induce a bending moment on the base.
27. This force will induce a bending moment on the base.
∴ M = P x (h/2)
Bending stress caused on the base due to moment,
σb =M/Z
The extreme stresses on the base are,
𝜎max = σo + σb
𝜎min = σo − σb
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