Basis of Structural Design

Course 3

Structural action: trusses and beams

Course notes are available for download athttps://www.ct.upt.ro/studenti/cursuri/stratan/bsd.htm

Arch

Linear arch supporting aconcentrated force: largespreading reactions at supports

rafter

tie

Relieving of supportspreading: adding a tiebetween the supports

Truss

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 support") - allowing free rotations and lateral movement due to loading and change in temperature

Forces in the truss:– tie is in tension (+)

– rafters are in compression (-)

- (C) - (C)

+ (T)

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

Bottom flange in tension

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

linear variation of normal stress

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 arrangement of material for bending resistance: away from the neutral axis

d

F = ·(A/2)

F = ·(A/2)

d M = ·A·d/2

A/2

A/2

d

b

F = ·(0.5d·b/2)

F = ·(0.5d·b/2)

2d

/3 M = ·A·d/6A

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 resistance in a beam with idealised double T cross-section (M = Ad/2): parabolic variation of height

A/2

A/2

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 Salignybridge, Romania