ERT 348 Controlled Environment Design 1 STRUCTURAL ANALYSIS

Reaction on a support connection For rolled section For pin section For fixed section Fy Fx Fy Fx M Fy

To design a structure it is necessary to know in each member: Bending moments Torsion moments Shear forces Axial forces

Equilibrium The goal of the whole design process is to achieve an equilibrium of the forces acting upon a structure. Without equilibrium the building will move and that is not good! Equilibrium must be accomplished for the building as a whole and for all the parts or smaller assemblies within the building as well. For all of the forces acting downward due to gravity, an equal, opposite force called a reaction must be pushing up. All of the loads acting on a structure will ultimately accumulate in the foundation and must be met with an equivalent reaction from the earth below.

Force Forces are a type of quantity called vectors Defined by magnitude and direction Statement of equilibrium Net force at a point in a structure = zero (summation of forces = zero) Net force at a point is determined using a force polygon to account for magnitude and direction

Forces in Structural Elements 100 lb Compression 100 lb Tension

Forces in Structural Elements 100 lb Bending Torsion

Beams Actions Tension Compression

Column Tensile Failure Compressive Failure

SHEAR FORCE & BENDING MOMENT

Shear Force & Bending Moment Two parameters which are fundamentally important to the design of beams are shear force and bending moment. These quantities are the result of internal forces acting on the material of a beam in response to an externally applied load system.

Static Equilibrium * The assumed positive direction is as indicated. Since the externally applied force system is in equilibrium, the three equations of static equilibrium must be satisfied, i.e. +ve ↑ ΣFy = 0 The sum of the vertical forces must equal zero. +ve ΣM = 0 The sum of the moments of all forces about any point on the plane of the forces must equal zero. +ve → ΣFx = 0 The sum of the horizontal forces must equal zero. * The assumed positive direction is as indicated.

Equations of Equilibrium A structure or one of its members in equilibrium is called statics member when its balance of force and moment. In general this requires that force and moment in three independent axes, namely

Equations of Equilibrium In a single plane, we consider

Sign convention

Examples

Shear Force Diagram (SFD) The calculation carried out to determine the shear force can be repeated at various locations along a beam and the values obtained plotted as a graph; this graph is known as the shear force diagram. The shear force diagram indicates the variation of the shear force along a structural member.

Bending Moment Diagram Bending inducing tension on the underside of a beam is considered positive. Bending inducing tension on the top of a beam is considered negative.

Bending Moment Diagram Note: Clockwise/anti-clockwise moments do not define +ve or −ve bending moments. The sign of the bending moment is governed by the location of the tension surface at the point being considered. As with shear forces the calculation for bending moments can be carried out at various locations along a beam and the values plotted on a graph; this graph is known as the ‘bending moment diagram’. The bending moment diagram indicates the variation in the bending moment along a structural member.

Shear Force and Bending Moment Diagram If the variations of V & M are plotted, the graphs are termed the shear diagram and moment diagram Changes in shear= Area under distributed load diagram Changes in moment = Area under shear diagram

Shear and Moment Diagram

Shear and Moment Diagram

Relationship of Loading, Shear and Moment

Examples