Review of structural analysis

REVIEW OF BASIC STRUCTURAL ANALYSIS

PRESENTED BY:HASSAN ABBA MUSA1401306006

#1 - INTRODUCTION TO STRUCTURAL ANALYSIS

Structural Analysis is the analysis of a given structure subject to some given loads, and the idea is to predict the response of the structure (system), thus there are some inputs referred to as ‘’stimulus’’ and an outputs referred to as ‘’response’’. The Structural analysis is the application of solid mechanics to predict the response (Interms of forces & displacements) of a given structure (existing or proposed) that subjected to a specified loads.

LOADS(INPUTS)

RESPONSE(INPUTS)

STRUCTURE (SYSTEM)

Loads are classified into 2 parts: Dead loads (Self-weight; of the components of the

structure). Imposed loads & forces (Live, Wind, Snow, Rain, and

Temperature, Erection loads, Seismic forces & others).The Structural Analysis is only ascertained correct when the following requirements were satisfied;- Equilibrium Forces. Compatibility of Displacement. Force/Displacement Relations. Therefore, In every forms of the structures, the structural Engineer has to consider the following structural considerations and purposes.

Stability, Strengths, Stiffness, Economy, as well as Aesthetic aspects of the structure.

The following Softwares are available

Orion, STAAD III, ANSYS, AXIS VM, ETABS, SAP, Etc

#2 – DIFFERENT TYPES OF SUPPORTS

Roller, Pinned, Hinged, Fixed, Link, Ball and Socket, Rigid Support, Spring Support.

FORCES

#2 – DIFFERENT TYPES OF STRUCTURES

Frame Structure, Truss Structure, Shell Structure, Arch Structure, Suspension Structure, Mass Structure, & Composite Structure.

#3 – LIST OF EQUILIBRIUM EQUATIONSFor state of static equilibrium to exist, it is necessary that the combined resultant effect of the system of forces shall be neither a Force nor a Couple; otherwise there will be tendency for motion of a body. However, many 3D structures are idealized as series of 2D components: If the combined resultant effect of a general system of forces

acting on a planar structures is not be equivalent to a resultant force, the algebraic sum of all components must be equal to Zero and the algebraic sum of all components must to be equal to Zero, i.e .

If the combined resultant effect is not to be equivalent to a couple, the algebraic sum of the moment of all forces about any axis parallel to the Z-reference axis and normal to the plane of the structure must be equal to zero, i.e

#4 – DETERMINATE AND INDETERMINATE OF A STRUCTURE

If the number of unknown forces is identical to the number of equilibrium equations, the problem is said to be Statically Determinate since the unknown forces can be determined directly from the equilibrium.

If, however the number of unknown forces exceeds the number of equilibrium equations, the problem is said to be Statically Indeterminate to a degree equal to this excess.

#5 – TECHNIQUES USED IN ANALYSING STRUCTURE

Macaulay’s Method Moment Area Method Conjugate Beam Method Virtual Work Method Unit load Method Influence line theory The Three Moment Equation (Clapeyron’s Theorem) Stiffness & Flexibility Method Slope Deflection Method Moment Distribution Method

MACAULAY’S METHOD This is a method suggested by W. H. Macaulay to relate the stiffness, radius of curvature, deflection and the bending moments in a beam by integration methods. Beam Deflections using successive integration;

Consider an infinitely small Section, dx, of the above loadedbeam;

MOMENT AREA METHOD This is a method suitable for calculating slope &

deflection at selected points on a beam. It is also effective for calculating the deflections of

beams with various cross sections. The simplest way to evaluate the fixed -end moments,

etc, will often be by the use of the Moment area method. There are two theorems associated with the moment area methods:

First Moment Area Theorem: “The difference in slope between two points on a beam is equal to the area of the M/EI diagram between the two points.”

Second Moment Area Theorem: “The moment about a point A of the M/EI diagram between points A and B will give the deflection of point A relative to the tangent at point B.”

First Moment Area Theorem

Second Moment Area Theorem

#6 – INTRODUCTION TO FLEXIBILITY/STIFFNESS METHOD

Flexibility Method (Force Method of Analysis) In this method, the degree of statical indeterminacy is initially determined. Thereafter, a number of releases equal to the degree of statical indeterminacy is introduced, each release being made by the removal of an external or internal force. Hence, the forces on the original indeterminate structure are calculated as the sum of the correction forces (redundants) and forces on the released structure. Flexibility Matrix [ f] ;- [f]{F} = {}D represents inconsistencies in deformation while {F} represents the redundants, Elements represent prescribed displacements at their respective coordinates. The column vector { - D} thus depends on the external loading.

Stiffness Method (Displacement Method of Analysis) The displacement method can be applied to statically determinate or indeterminate structures, but is more useful in the latter, particularly when the degree of statical indeterminacy is high. In this method, one must first determine the degree of kinematic indeterminacy. A coordinate system is then established to identify the location and direction of joint displacements. Restraining forces equal in number to the degree of kinematic indeterminacy are introduced at the co-ordinates to prevent the displacement of the joints. Stiffness Matrix [S];- {D} = {-F}The elements of the vector {D} are the unknown displacements. The elements of the matrix [S] are forces corresponding to unit values of displacements. The column vector {F} depends on the loading on the structure.

#3 – MATRIX OPERATIONS

Let A, B, and C be mxn matrices. We haveProperties involving Addition. 1. A+B = B+A2. (A+B)+C = A + (B+C)3. A+0 = A  where is the mx zero-matrix (all its entries are equal to 0);4. A+B = 0, if and only if B = -A.