The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
This document provides tutorials on mechanical principles and engineering structures. It focuses on tutorial 2 which covers reaction forces in pin-jointed framed structures. It defines pin joints and how they allow rotation. It distinguishes between struts, which are members in compression, and ties, which are in tension. It introduces Bow's notation for solving forces in framed structures by drawing force polygons at each joint. Worked examples demonstrate how to apply this method to determine the forces and whether each member is a strut or tie. Further practice problems are provided for the student to solve pin-jointed frames.
L15 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
This document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview and introduction to the method, which was developed by Prof. Hardy Cross in 1932. It then describes the basic principles through a 5 step process: 1) joints are locked to determine fixed end moments, 2) joints are released allowing rotation, 3) unbalanced moments modify joint moments based on stiffness, 4) moments are distributed and modify other joints, 5) steps 3-4 repeat until moments converge. Key terms like stiffness and carry-over factors are also defined.
Se presentan problemas resueltos donde se calculan desplazamientos de estructuras estáticamente determinadas aplicando el método de la estructura conugada
The document discusses various methods for measuring distances across obstacles during land surveying. It classifies obstacles as: 1) Chaining free but vision obstructed, 2) Chaining obstructed but vision free, and 3) Both chaining and vision obstructed. For each type, it provides examples and explains specific measurement techniques such as reciprocal ranging, using random lines, constructing right triangles, and prolonging the line beyond obstacles to determine distances.
The document discusses the double integration method for determining beam deflections. It defines beam deflection as the displacement of the beam's neutral surface from its original unloaded position. The differential equation relating the bending moment, flexural rigidity, and slope of the elastic curve is derived. This equation is integrated twice to obtain expressions for the slope and deflection of the beam in terms of the bending moment and constants of integration. Several examples are provided to demonstrate solving for the slope and maximum deflection of beams under different loading conditions using this method.
This document provides tutorials on mechanical principles and engineering structures. It focuses on tutorial 2 which covers reaction forces in pin-jointed framed structures. It defines pin joints and how they allow rotation. It distinguishes between struts, which are members in compression, and ties, which are in tension. It introduces Bow's notation for solving forces in framed structures by drawing force polygons at each joint. Worked examples demonstrate how to apply this method to determine the forces and whether each member is a strut or tie. Further practice problems are provided for the student to solve pin-jointed frames.
L15 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
This document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview and introduction to the method, which was developed by Prof. Hardy Cross in 1932. It then describes the basic principles through a 5 step process: 1) joints are locked to determine fixed end moments, 2) joints are released allowing rotation, 3) unbalanced moments modify joint moments based on stiffness, 4) moments are distributed and modify other joints, 5) steps 3-4 repeat until moments converge. Key terms like stiffness and carry-over factors are also defined.
Se presentan problemas resueltos donde se calculan desplazamientos de estructuras estáticamente determinadas aplicando el método de la estructura conugada
The document discusses various methods for measuring distances across obstacles during land surveying. It classifies obstacles as: 1) Chaining free but vision obstructed, 2) Chaining obstructed but vision free, and 3) Both chaining and vision obstructed. For each type, it provides examples and explains specific measurement techniques such as reciprocal ranging, using random lines, constructing right triangles, and prolonging the line beyond obstacles to determine distances.
L18 analysis of indeterminate beams by moment distribution methodDr. OmPrakash
The document discusses the moment distribution method for analyzing indeterminate beams. It begins with an overview of the method and some basic definitions. It then describes the step-by-step process, which involves (1) computing fixed end moments by assuming locked joints, (2) releasing joints causing unbalanced moments, (3) distributing unbalanced moments according to member stiffnesses, (4) carrying moments over to other joints, and (5) repeating until moments converge. Key terms discussed include stiffness factors, carry-over factors, and distribution factors.
4-Internal Loadings Developed in Structural Members.pdfYusfarijerjis
This document discusses analyzing internal loadings in structural members. It provides objectives of determining internal shear and moment at specified points and constructing shear and moment diagrams. Methods covered include using sections, sign conventions, and equilibrium equations to find reactions, shear and moment at a point. Shear and moment functions and diagrams are developed for beams and frames. The method of superposition is presented for combining loading cases to determine moment diagrams.
This document provides information about engineering mechanics and structural analysis. It includes:
1) An overview of the concepts of equilibrium of rigid bodies, statically determinate and indeterminate structures, and the conditions for each.
2) A description of the method of joints technique for analyzing plane trusses through applying equilibrium equations at each joint to determine member forces.
3) Worked examples that demonstrate applying the method of joints to solve for unknown member forces and reactions in various truss structures.
This document discusses concepts related to statics including:
1. It covers four main topics - the condition of equilibrium of coplanar concurrent forces, the concept of a free body diagram, the sine rule for triangles, and Lami's theorem.
2. Lami's theorem states that if three coplanar forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
3. Several examples are provided to demonstrate how to apply the sine rule and Lami's theorem to calculate tensions in strings and magnitudes of forces.
The moment distribution method is a structural analysis method for statically indeterminate beams and frames developed by Hardy Cross. It was published in 1930 in an ASCE journal.[1] The method only accounts for flexural effects and ignores axial and shear effects. From the 1930s until computers began to be widely used in the design and analysis of structures, the moment distribution method was the most widely practiced method.
This document discusses transverse shear stresses in beams. It begins by explaining how shear stresses develop within beams subjected to transverse loads and defines the internal shear force V. It then discusses how shear stresses cause shear strains that distort the beam's cross-section. The document proceeds to derive the shear formula that relates the shear stress to the internal shear force V and the beam's geometry. It provides examples of applying the shear formula to compute shear stresses in different beam cross-sections.
This document discusses the derivation and application of three-moment equations for analyzing statically indeterminate continuous beams. The key points are:
1) Three-moment equations relate the bending moments at three successive supports to the applied loads on adjacent spans. They allow continuous beams to be analyzed by treating each span as simply supported with end moments.
2) The equations are derived by writing compatibility equations at each interior support in terms of the left, center, and right bending moments.
3) Examples show how to set up and solve the three-moment equations to determine support reactions and draw shear and moment diagrams for continuous beams with various loading conditions.
Macaulay's method provides a continuous expression for the bending moment of a beam subjected to discontinuous loads like point loads, allowing the constants of integration to be valid for all sections of the beam. The key steps are:
1) Determine reaction forces.
2) Assume a section XX distance x from the left support and calculate the moment about it.
3) Insert the bending moment expression into the differential equation for the elastic curve and integrate twice to obtain expressions for slope and deflection with constants of integration.
4) Apply boundary conditions to determine the constants, resulting in final equations to calculate slope and deflection at any section. This avoids deriving separate equations for each beam section as with traditional methods
The document discusses various methods for analyzing beam deflection and deformation under loading, including:
1) Deriving the differential equation for the elastic curve of a beam and applying boundary conditions to determine the curve and maximum deflection.
2) Using the method of superposition to analyze beams subjected to multiple loadings by combining the effects of individual loads.
3) Applying moment-area theorems which relate the bending moment diagram to slope and deflection, allowing deflection calculations for beams with various support conditions.
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.
The document discusses bending stresses in beams. It begins by outlining simplifying assumptions made in deriving the flexure formula to relate bending stresses to bending moments. These assumptions include plane sections remaining plane and perpendicular to the deformed beam axis. The neutral axis is defined as the axis where longitudinal fibers experience no deformation.
The derivation of the flexure formula is shown. Flexural stresses are proportional to the distance from the neutral axis and bending moment. Procedures for determining stresses at given points, as well as maximum stresses, are provided. Sample problems demonstrate applying the flexure formula and finding maximum stresses for different beam cross sections.
This document is a solution to a physics problem set composed and formatted by E.A. Baltz and M. Strovink. It contains solutions to 6 problems using vector algebra and trigonometry. The document uses concepts like the law of cosines, dot products, cross products, and vector identities to break vectors into components and calculate angles between vectors. It also applies these concepts to problems involving vectors representing locations on a sphere and wind resistance problems for airplanes.
The document discusses methods for calculating deflections in structures, specifically the moment area method. It provides examples of using the moment area method to calculate slopes and deflections at various points along beams and frames by relating the bending moment diagram area to slope changes and vertical deflections using theorems. Sample problems are worked through step-by-step to demonstrate calculating slopes and deflections for beams under different loading conditions.
This document discusses beam design criteria and deflection behavior of beams. It covers two key criteria for beam design:
1) Strength criterion - the beam cross section must be strong enough to resist bending moments and shear forces.
2) Stiffness criterion - the maximum deflection of the beam cannot exceed a limit and the beam must be stiff enough to resist deflections from loading.
It then defines deflection, slope, elastic curve, and flexural rigidity. It presents the differential equation that relates bending moment, slope, and deflection. Methods for calculating slope and deflection including double integration, Macaulay's method, and others are also summarized.
- Saint-Venant's principle states that the stress and strain distribution on a cross-section of a loaded material will be independent of the applied load if the cross-section is located away from the point of load application.
- The principle of superposition allows breaking down structures into individual load cases and adding their effects to determine the total stress, strain, or deflection.
- Statically indeterminate structures require additional compatibility equations relating deformations to solve for member forces.
This document contains solutions to mechanics of solids problems involving deflection of beams. The first problem involves calculating the slope and deflection of a steel girder beam with given properties under a central load. Subsequent problems calculate reactions, slopes, and deflections of beams with various support conditions and loadings using concepts such as bending moment diagrams, integration, and the conjugate beam method. The last problem determines the magnitude of a propping force required to keep a beam with a uniform distributed load level at the center.
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.
This document provides conceptual information and formulas related to structural design for the Architect Registration Exam. It covers topics like:
1) Beam design concepts for wood, steel, and concrete beams including shear, bending, and deflection calculations.
2) Column design concepts for wood and steel columns including slenderness and strength calculations.
3) Additional structural topics like truss analysis methods, force and moment diagrams, stress/strain behaviors, and support conditions.
Diagrams and tables are included to illustrate structural concepts, and memory tricks are provided for topics like trigonometry functions and truss analysis.
The document discusses the moment distribution method for analyzing beams and frames. It defines key terms such as:
- Distribution factor (DF), which represents the fraction of the total resisting moment supplied by a member.
- Member stiffness factor, which is the moment required to rotate a member's end by 1 radian.
- Joint stiffness factor, which is the sum of the member stiffness factors at a joint.
It then outlines the steps to perform moment distribution: 1) determine member/joint stiffness, 2) calculate DFs, 3) compute initial member moments, 4) distribute moments at joints, and 5) carry moments over to other members. An example problem demonstrates applying these steps to determine member moments.
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3) Examples show how to set up and solve the three-moment equations to determine support reactions and draw shear and moment diagrams for continuous beams with various loading conditions.
Macaulay's method provides a continuous expression for the bending moment of a beam subjected to discontinuous loads like point loads, allowing the constants of integration to be valid for all sections of the beam. The key steps are:
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Topic3_Displacement Method of Analysis Frames Sideway.pptx
1. CED 426
Structural Theory II
Lecture 15
Displacement Method of Analysis:
Analysis of Frames: Sideway
Mary Joanne C. Aniñon
Instructor
2. Analysis of Frames:
Sidesways
• A frame will sidesway, or be
displaced to the side, when it or the
load acting on it is nonsymmetrical.
• For Fig. 10-18, the loading P causes
unequal moments 𝑀𝐵𝐶 and 𝑀𝐶𝐵
3. Analysis of Frames:
Sidesways
• 𝑀𝐵𝐶 tends to displace joint B to the
right whereas 𝑀𝐶𝐵 tends to
displace joint C to the left
• Since the two moments are not
equal, the resulting net
displacement is a sidesway of both
joint B and C to the right
4. Procedure for Analysis
• Degrees of Freedom
• Slope-Deflection Equations
• Equilibrium Equations
30 30
5. Step 1: Degrees of Freedom
1.a. Label all the supports and joints (nodes) in order to identify the spans of
the beam or frame between the nodes.
1.b. Draw the possible deflected shape of the structure.
1.c. Identify the number of degrees of freedom (angular displacement and
linear displacement)
1.d. Compatibility at the nodes is maintained provided the members that are
fixed connected to a node undergo the same displacement as the node
1.e. If these displacements are unknown, then for convenience assume they
act in the positive direction so as to cause clockwise rotation of the member
or joint.
6. Step 2: Slope-Deflection-Equations
2.a. Write the slope-deflection equations. The slope-deflection
equations relate the unknown internal moments at the nodes to the
displacements of the nodes for any span of the structure.
2.b. Calculate the FEM if a load exist on the span.
2.c. If the node has a linear displacement, calculate ψ = ∆/L for
adjacent spans
7. Step 2: Slope-Deflection-Equations
• Apply the slope deflection
equations (Eq. 10-8)
• However, if span at the end of a
continuous beam or frame is pin
supported apply Eq. 10-10 only to
the restrained end, thereby
generating one slope-deflection
equation for this span
• Note that Eq. 10-10 was derived
from Eq. 10-8 on the condition
that the end span of the beam or
frame is supported by pin or roller
8. Step 3: Equilibrium Equations
3.a. Write an equilibrium equation for each unknown degree of
freedom for the structure. Each of these equations should be
expressed in terms of the unknown internal moments as specified by
the slope-deflection equations.
3.b. For beams and frames, write the moment equation of equilibrium
at each support
3.c. For frames also write joint moment equations of equilibrium. If the
frame sidesways or deflects horizontally, column shears should be
related to the moments at the ends of column.
9. Step 3: Equilibrium Equations
3.d. Substitute the slope-deflection equations into the equilibrium
equations and solve for the unknown joint displacements.
3.e. The results are then substituted into the slope-deflection
equations to determine the internal moments at the ends of each
member
3.f. If any of the results are negative, they indicate counterclockwise
rotation; whereas positive moments and displacements create
clockwise rotation
11. Example 1
Step 1: Degrees of Freedom
1.a. Label all the supports and joints (nodes): Span AB, Span BC, and Span CD
1.b. Draw the possible deflected shape of the structure.
1.c. Identify the number of degrees of freedom: 𝜽𝑩 , 𝜽𝑪 and ∆
Note:
• Sidesway occurs since both the applied loading and the geometry of the frame are nonsymmetric.
• As shown in the figure, both joints B and C are assumed to be displaced an equal amount ∆.
𝜃𝐵 𝜃𝐶
∆𝐵 ∆𝐶
12. Example 1
Step 2: Slope-Deflection-Equation
𝑀𝑁𝐹 = 2𝐸(
𝐼
𝐿
)(2𝜃𝑁 + 𝜃𝐹 − 3
∆
𝐿
+ (𝐹𝐸𝑀)𝑁𝐹
Note:
• Since the ends A and D are fixed, the
equation below applies for all 3 spans of the
frame.
13. Example 1
Step 2: Slope-Deflection-Equation
𝑀𝑁𝐹 = 2𝐸(
𝐼
𝐿
)(2𝜃𝑁 + 𝜃𝐹 − 3
∆
𝐿
+ (𝐹𝐸𝑀)𝑁𝐹
2.a. Write the Slope Deflection Equation for Span AB:
Note:
• A = D = 0 because it is fixed supports.
• (FEM)AB and (FEM)BA are zero because
there the load is applied directly to joint
B., and A and D are fixed supports
2.b. Calculate the FEM for span AB
𝜓𝐴𝐵 = 𝜓𝐵𝐴 =
Δ
4
𝜓𝐷𝐶 =
Δ
6
(𝜓𝐴𝐵)(4) = (𝜓𝐷𝐶)(6) (𝜓𝐴𝐵) = 𝜓𝐵𝐴 =
6
4
(𝜓𝐷𝐶)
14. Example 1
Step 2: Slope-Deflection-Equation
𝑀𝑁𝐹 = 2𝐸(
𝐼
𝐿
)(2𝜃𝑁 + 𝜃𝐹 − 3
∆
𝐿
+ (𝐹𝐸𝑀)𝑁𝐹
2.a. Write the Slope Deflection Equation for Span BC:
Note:
• A = D = 0 because it is fixed supports.
• ΔB and ∆C are equal, therefore for span
BC, ∆ can be considered zero.
• (FEM)BC and (FEM)CB are zero because
there the load is applied directly to joint
B., and A and D are fixed supports
2.b. Calculate the FEM for span BC
15. Example 1
Step 2: Slope-Deflection-Equation
𝑀𝑁𝐹 = 2𝐸(
𝐼
𝐿
)(2𝜃𝑁 + 𝜃𝐹 − 3
∆
𝐿
+ (𝐹𝐸𝑀)𝑁𝐹
2.a. Write the Slope Deflection Equation for Span CD:
Note:
• A = D = 0 because it is fixed supports.
• (FEM)CD and (FEM)DC are zero because
there the load is applied directly to joint
B., and A and D are fixed supports
2.b. Calculate the FEM for span CD
𝜓𝐶𝐷 =
Δ
6
𝜓𝐷𝐶 =
Δ
6
(𝜓𝐶𝐷)(6) = (𝜓𝐷𝐶)(6) (𝜓𝐶𝐷) = (𝜓𝐷𝐶)
16. Example 1
Step 3: Equilibrium Equations
3.a. Equilibrium equations for each unknowns.
• These six equations contain nine unknowns.
𝑀𝐴𝐵 = 𝐸𝐼(0.5𝜃𝐵 − 2.25𝜓𝐷𝐶)
𝑀𝐵𝐴 = 𝐸𝐼(1𝜃𝐵 − 2.25𝜓𝐷𝐶
𝑀𝐵𝐶 = 𝐸𝐼(0.8𝜃𝐵 − 0.4𝜃𝐶
𝑀𝐶𝐵 = 𝐸𝐼(0.8 + 0.4𝜃𝐵
𝑀𝐶𝐷 = 𝐸𝐼(0.667𝜃𝐶 − 1𝜓𝐷𝐶
𝑀𝐷𝐶 = 𝐸𝐼(0.333𝜃𝐶 − 1𝜓𝐷𝐶
17. Example 1
Step 3: Equilibrium Equations
3.a. Equilibrium equations for each unknowns.
• The first 2 equilibrium equations come from moment equilibrium
at joints B and C. The free body diagram of segment of the frame at
B and C is shown below
+M
+V
-M
+V
-M
+V
+M
+V
18. Example 1
Step 3: Equilibrium Equations
3.a. Equilibrium equations for each unknowns.
• Since a horizontal displacement ∆ occurs, we will consider
summing forces on the entire frame in the x direction. This
yields (the 9th equation)
• The free body diagrams of span AB and CD are shown
VA
VB
-M
+V
+M
+V
20. Example 1
Step 3: Equilibrium Equations
3.d. Substitute the slope-deflection equations into
the equilibrium equations and solve for the unknown
joint displacements.
(1)
(2)
(3)
21. Example 1
Step 3: Equilibrium Equations
3.e. The results are then substituted into the slope-
deflection equations to determine the internal
moments at the ends of each member
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