Basis of Structural Design Course 3 Structural action: trusses and beams Course notes are available for download at https://www.ct.upt.ro/studenti/cursuri/stratan/bsd.htm Arch Truss rafter tie Linear arch supporting a Relieving of support concentrated force: large spreading: adding a tie spreading reactions at supports between the supports Truss forces . Truss members connected by pins: axial forces (direct stresses) only . Supports: – one pinned, allowing free rotations due to slight change of truss shape due to loading – one roller bearing support ("simple - (C) - (C) support") - allowing free rotations and lateral movement due to + (T) loading and change in temperature . Forces in the truss: – tie is in tension (+) – rafters are in compression (-) Truss forces . If more forces are present within the length of the rafter bending stresses - - . To avoid bending stresses, + diagonal members and vertical - - - - posts can be added + + . More diagonals and posts can be added for larger spans in order to avoid bending stresses Alternative shape of a truss . For a given loading find out the shape of a linear arch (parabolic shape) . Add a tie to relieve spreading of supports . Highly unstable shape Alternative shape of a truss . Add web bracing (diagonals and struts) in order to provide stability for the pinned upper chord members . If the shape of the truss corresponds to a linear arch web members are unstressed, but they are essential for stability of the truss . Reverse bowstring arches: – advantage: longer members are in tension – disadvantage: limited headroom underneath Truss shapes . Curved shape of the arch: difficult to fabricate trusses with parallel chords . Trusses with parallel chords: web members (diagonals and struts) carry forces whatever the loads . Pratt truss: – top chord in compression – bottom chord and diagonals in tension – economical design as longer members (diagonals) are in tension Truss shapes . Howe truss: – top chord in compression – bottom chord in tension – diagonals in compression . Warren truss: – top chord in compression – bottom chord in tension – diagonals in tension and compression – economy of fabrication: all members are of the same length and joints have the same configuration Truss joints . Pinned joints statically determinate structures member forces can be determined from equilibrium only . Rigid joints small bending stresses will be present, but which are negligible due to the triangular shape . Traditionally trusses are designed with pinned joints, even if members are connected rigidly between them Space trusses . The most common plane truss consists of a series of triangles . The corresponding shape in three dimensions: tetrahedron (a) . The truss at (b) is a true space truss – theoretically economical in material – joints difficult to realise and expensive . Two plane trusses braced with cross members are usually preferred Statically indeterminate trusses . Indeterminate trusses: large variety . Example (a): cross diagonals in the middle panel, so that one of the diagonals will always be in tension . Example (b): Sydney Harbour Bridge, Australia - both supports pinned Beams . Beam: a structure that supports loads through its ability to resist bending stresses . Leonardo da Vinci (1452-1519): the strength of a timber beam is proportional to the square of its depth . Leonhard Euler and Daniel Bernoulli were the first to put together a useful theory around 1750 Beams: analogy with trusses . Forces in a Pratt truss loaded by a unit central force . Forces in a Howe truss . Forces in a truss with double diagonals (reasonable estimate) Beams: analogy with trusses . Chords: – The forces in the top and bottom chord members in any panel are equal, but of opposite signs, and they increase with the distance from the nearest support – Chords have to resist the bending moment, proportional to the distance from the nearest support . Diagonals: – The forces in the diagonal members are equal, but opposite in sign, and have the same values in all panels – Diagonals have to resist the shear forces, the same in all panels Beams: analogy with trusses . Bending and shear deformations in a truss Steel plate girder . Steel plate girder: heavy flanges and thin web welded together, and reinforced by transversal stiffeners . Unit vertical force at the midspan . Top flange: in compression . Bottom flange: tension . Web: shear, with principal tension and compression stresses similar to those in a truss . After web buckling, only tensile loads are resisted by the web, plate girder acting as a Pratt truss Beams: bending action . Top flange in compression linear variation of . Bottom flange in tension normal stress . Normal stress proportional to distance from the neutral plane . Simplifications: – Thin web, thick flanges web has a small contribution to the bending resistance (ignore it) – Normal stress can be considered uniform on flanges Beams: bending action . Moment resistance – Idealised double T beam: M = Ad/2 – Rectangular beam of the same area and depth: M = bd2/6 = Ad/6 . The best A/2 arrangement F = ·(A/2) of material for bending d d M = ·A·d/2 resistance: F = ·(A/2) away from A/2 the neutral axis F = ·(0.5d·b/2) d M = ·A·d/6 A 2d/3 F = ·(0.5d·b/2) b Beams: bending action . Examples of efficient location of material for bending resistance – light roof beams (trusses) – hot-rolled and welded girder Beams: bending action . Examples of efficient location of material for bending resistance – panel construction Beams: bending action . Examples of efficient location of material for bending resistance – corrugated steel sheet Beams: bending action . Examples of efficient location of material for bending resistance – castellated joist Beams: bending action . Examples of efficient location of material for bending resistance – columns requiring bending resistance in any direction: tubular sections Beams: shear stresses . Simply supported beam of uniform rectangular cross- section loaded by a concentrated central force W: – can carry a moment M = bd2/6 – has a deflection . If the beam is cut in two parts along the neutral plane: – sliding takes place between the two overlapped beams – the two overlapped beams can carry a moment M = 2[b(d/2)2/6] = bd2/12, half of the uncut beam – the deflection of the two overlapped beams is 4 Beams: shear stresses . In the uncut beam stresses should be present along the neutral plane to prevent sliding of the lower and upper halves of the beam: shear stresses . Smaller stresses would be required to keep the unity of action if the beam were cut above the neutral plane . Shear stresses – parabolic variation in a rectangular cross-section – carried mainly by the web, on which they can be considered to be constant for a steel double T beam Structural shapes . Simply supported beam subjected to a uniformly distributed load . The "perfect" use of material for bending A/2 resistance in a beam with idealised double T cross- section (M = Ad/2): parabolic variation of A/2 height Structural shapes . Simply supported truss subjected to a uniformly distributed load . The "perfect" use of material for "bending" action: parabolic variation of height Structural shapes . Bridge with a simply supported central span and two cantilevered sides . The shape of the truss must resemble the bending moment diagram in order to make efficient use of material in upper and bottom chords . Quebec railway bridge Structural shapes . Forth bridge, Scotland . Anghel Saligny bridge, Romania.