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CVEN2303 – STRUCTURAL ANALYSIS AND MODELLING WEEK 1 - INTRODUCTION LECTURE 1 NOTES STRUCTURAL ANALYSIS 

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Statically Indeterminate Structures: three equations of equilibrium are not enough. o Flexibility Methods:  Equilibrium  Material Properties (Stress-Strain) o Stiffness Methods:  Geometry (Compatibility) Plane Trusses and Frames only. Some assumptions are: o Loads are Static  Loads are applied slowly to avoid unnecessary variants. o Small Displacements  Deflection is much smaller than the height and length of the beam (100x). o Linear Elastic Behaviour  Linear material has a linear stress-strain relationship. Elastic material is where pressure is applied to an object but when released it returns to its original position. Aim is: o To calculate Internal Forces (M, N, S). o To calculate Deformations.

TYPES OF STRUCTURES Trusses:

Frames:

Cables and Arches:

Arches:

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Domes:

STRUCTURAL ELEMENTS    

Rods Beams Columns Stocky/Slender/Buckling

STRUCTURAL IDEALISATION

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DETERMINACY, INTEDETERMINACY, STABILITY Equations of Equilibrium:

Plane Trusses:

Determinacy and Stability:

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A truss is considered to be stable when these two conditions exist: o At least three reaction forces that restrain the horizontal and vertical movement of the truss, and the lines of action of the reactions do not intersect. o The truss is composed from triangular or cross shapes.

REVIEW OF INTERNAL FORCE DIAGRAMS   

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Axial Forces (AFD) Shear Forces (SFD) Bending Moments (BMD)

NOTATION

MATHMATICAL RELATIONS

ZERO SHEAR FORCE = MAXIMUM BENDING MOMENT

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BASIC DIAGRAMS

EXAMPLE 1 – BEAM WITH TRIANGULAR LOAD

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EXAMPLE 2: BEAM WITH CONCENTRATED MOMENT

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EXAMPLE 3 – TYPICAL FRAME

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EXAMPLE 4 – FRAME WITH TRIANGULAR LOADING

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EXAMPLE 5- FRAMES WITH INCLINED MEMBERS

WEEK 2 – CONJUGATE

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BEAM METHOD

LECTURE NOTES - DEFLECTIONS OF BEAMS USING CONJUGATE BEAM METHOD DEFLECTIONS AND ROTATIONS FOR BASIC CASES

CONJUGATE BEAM-METHOD

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The conjugate beam is loaded with the M/EI diagram derived from the load q on the real beam. Its equilibrium solution gives the slopes and deflections of the real beam. We can state 2 theorems related to the conjugate beam: o The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam. o The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam. When drawing the conjugate beam, it is important that the shear & moment developed at the supports of the conjugate beam account for the corresponding slope & displacement of the real beam at its supports.

If the real beam is statically determinate then the conjugate beam is also statically determinate. The opposite is not true! For consistency of the solution: if the bending moment is causing tension at the lower side of the real beam, then it will be applied as an equivalent force of M/EI acting downwards on the conjugate beam, and vice versa. As a result: a moment that is causing tension at the lower side of the conjugate beam means a downloads deflection of the real beam, and vice versa.

EXAMPLE 1 – CANTILEVER BEAM

EXAMPLE 2 – SIMPLY SUPPORTED BEAM WITH A POINT LOAD

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EXAMPLE 3 - SIMPLY SUPPORTED BEAM WITH UDL

EXAMPLE 4 – BEAM WITH VARIABLE EI

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EXAMPLE 5 – BEAM WITH INTERMEDIATE HINGES

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EXAMPLE 6 – APPLICATION TO STATICALLY INDETERMINATE BEAMS

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WEEK 3 – THE PRINCIPLE OF WORK

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