CLASSICAL STRUCTURAL ANALYSIS OF STATICALLY DETERMINATE STRUCTURES

A structure is considered to be in equilibrium if, initially at rest, it remains at rest when subjected to a system of forces and couples. If a structure is in equilibrium, then all its members and parts are also in equilibrium. In this module we will direct our attention to the most common form of structure that the engineer will have to analyze, and that is one that lies in a plane and is subjected to a force system that lies in the same plane. We begin by discussing the importance of choosing an appropriate analytical model for a structure so that the forces in the structure may be determined with reasonable accuracy. Then the criteria necessary for structural stability are discussed. Finally, the analysis of statically determinate, planar, pin-connected structures is presented.

AREA-MOMENT METHOD FOR BEAMS AND PLANE FRAMES

The moment area method provides a semi graphical technique for finding the slope and displacement at specific points on the elastic curve of a beam. It uses the area of moment divided by flexural rigidity (EI) diagram of a beam to determine deflection and slope along the beam. It consists of two theorems that were developed by Otto Mohr and later stated formally by Charles E. Greene in 1873.

FIRST MOMENT AREA THEOREM

“The change in slope between any two points on the elastic curve equals the area of the M/ EI diagram between these two points.”

SECOND MOMENT AREA THEOREM

“The vertical deviation of the tangent at a point (A) on the elastic curve with respect to the tangent extended from another point (B) equals the “moment” of the area under the M/EI diagram between the two points (A and B). This moment is computed about point A (the point on the elastic curve), where the deviation tA/B is to be determined.”

DOUBLE INTEGRATION METHOD FOR BEAMS

One such method that is considered in deflection analysis is the double integration method. The method of integration, requires applying the load-displacement relationship, represented by the differential equation v"=M/EI. If the beam is statically indeterminate, M will be expressed in terms of position x and in terms of the unknown redundants. This method is fairly straightforward in its application, but it often involves considerable algebraic manipulation. Thus, its primary advantage is that it produces the equation of deflection everywhere along the beam.
To derive the relationships of rotation of beams to the displacement and slope of elastic curve, consider an initially straight beam that is elastically deformed by loads applied perpendicular to the beam’s x axis and lying in the x-v plane of symmetry for the beam’s cross-sectional area as shown in the figure on the right. The deformation of the beam is caused by both the internal shear force and bending moment due to the applied loads P and w. Since the beam has a length that is much greater than its depth, the greatest deformation will be caused by bending.
When the internal moment M deforms the element of the beam, each cross section remains plane and the angle between them becomes dθ. The arc dx that represents a portion of the elastic curve intersects the neutral axis for each cross section. The radius of curvature for this arc is defined as the distance ρ, which is measured from the center of curvature O′ to dx. Other than dx, any arc will be subjected to normal strain.

CONJUGATE BEAM METHOD

The conjugate-beam method was developed by H. Müller-Breslau in 1865. Essentially, it requires the same amount of computation as the moment-area theorems to determine a beam’s slope or deflection at a specific point. A conjugate beam refers to a fictitious beam whose length is similar of the real beam, but with loading equal to the bending moment of actual beam divided by flexural rigidity EI. The similarity of the relationships among load, shear force, and bending moment, as well as curvature, slope, and deflection in previous chapters are shown in the equations presented below:

Two theorems related to the conjugate beam can be stated, namely:

Theorem 1: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam.

Theorem 2: The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam.

SUPPORTS FOR CONJUGATE BEAMS

CASTIGLIANO’S FIRST THEOREM

Aside from using the method of virtual work to solve for deflection, Castigliano’s Theorem can be used. The two theorems were proposed in 1879 by Alberto Castigliano, an Italian railroad engineer, for which he outlined a method for determining the deflection or slope at a point in a structure, be it a truss, beam, or frame.

The theorem states the following:

“The partial derivative of the strain energy (U), considered as a function of the applied forces acting on a linearly elastic structure, with respect to one of these forces, is equal to the displacement in the direction of the force of its point of application”

CASTIGLIANO'S THEOREM FOR TRUSSES

CASTIGLIANO'S THEOREM FOR BEAMS AND FRAMES

UNIT LOAD METHOD/VIRTUAL WORK METHOD

The principle of virtual work (also known as unit-load method) was developed by John Bernoulli in 1717. It provides a general means of obtaining the displacement and slope at a specific point on a structure such as beam, frame, or truss. If a series of external loads P is applied to a deformable structure of any shape or size, it will cause internal loads u at points throughout the structure. It is necessary that the external and internal loads be related by the equations of equilibrium. As a consequence of these loadings, external displacements ∆ will occur at the P loads and internal displacements δ will occur at each point of internal load u.
In general, these displacements do not have to be elastic, and they may not be related to the loads; however, the external and internal displacements must be related by the compatibility of the displacements.

If the rotational displacement or slope of the tangent at a point on a structure is to be determined, a virtual couple moment M’ having a “unit” magnitude is applied at the point. As a consequence, this couple moment causes a virtual load uθ in one of the elements of the body.
Assuming that the real loads deform the element an amount dL, the rotation θ can be found from the virtual-work equation.

METHOD OF VIRTUAL WORK FOR TRUSSES

METHOD OF VIRTUAL WORK FOR BEAMS AND FRAMES

PARTIAL DERIVATIVE METHOD

For complicated loadings or for structures such as trusses and frames, it is suggested that energy methods be used for the computations. Most energy methods are based on the conservation of energy principle, which states that the work done by all the external forces acting on a structure, Ue, is transformed into internal work or strain energy, Ui, which is developed when the structure deforms.
If the material’s elastic limit is not exceeded, the elastic strain energy will return the structure to its undeformed state when the loads are removed.

EXTERNAL WORK-FORCE

EXTERNAL WORK-MOMENT

STRAIN ENERGY- AXIAL FORCE

STRAIN ENERGY-BENDING

METHOD OF VIRTUAL WORK

CASTIGLIANO'S THEOREM

CLASSWORK ASSIGNMENT #1-#6

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REFLECTION:

In this module, I have learned that stability is an essential precondition for a structure to be able to carry the loads it is subjected to, and therefore being suitable for structural analysis. Since structural analysis is based on solving the unknown forces (or displacements) within a structure using some equations, it is essentially the comparison of the equations and unknowns that determine the stability of a structural system. Statical determinacy of a structure is a concept closely related to its stability. Also, I have laerned that once a structure is determined to be stable, it is important to determine whether it remains in equilibrium; i.e., if it can be analyzed by the concepts of statics alone, particularly for hand calculation. Most importantly, I learned that it is necessary to establish the determinacy and stability of the structure. The equilibrium equations provide both the necessary and sufficient conditions for equilibrium. When all the forces in a structure can be determined strictly from these equations, the structure is referred to as statically determinate, and structures having more unknown forces than available equilibrium equations are called statically indeterminate.

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