04/21/23 Secant Formula 1 of 62

Real engineers actually laugh at jokes about mathematicians.

04/21/23 Secant Formula 2 of 62

04/21/23 Secant Formula 3 of 62

04/21/23 Secant Formula 4 of 62

04/21/23 Secant Formula 5 of 62

Ideal pin-ended compression members

04/21/23 Secant Formula 6 of 62

04/21/23 Secant Formula 7 of 62

04/21/23 Secant Formula 8 of 62

04/21/23 Secant Formula 9 of 62

04/21/23 Secant Formula 10 of 62

04/21/23 Secant Formula 11 of 62

04/21/23 Secant Formula 12 of 62

EI

04/21/23 Secant Formula 13 of 62

04/21/23 Secant Formula 14 of 62

04/21/23 Secant Formula 15 of 62

04/21/23 Secant Formula 16 of 62

04/21/23 Secant Formula 17 of 62

(with )

04/21/23 Secant Formula 18 of 62

04/21/23 Secant Formula 19 of 62

04/21/23 Secant Formula 20 of 62

04/21/23 Secant Formula 21 of 62

04/21/23 Secant Formula 22 of 62

04/21/23 Secant Formula 23 of 62

04/21/23 Secant Formula 24 of 62

04/21/23 Secant Formula 25 of 62

04/21/23 Secant Formula 26 of 62

04/21/23 Secant Formula 27 of 62

04/21/23 Secant Formula 28 of 62

04/21/23 Secant Formula 29 of 62

Remember: 2 plus 2 is four, but adding a safety factor we will call it 5.

04/21/23 Secant Formula 30 of 62

04/21/23 Secant Formula 31 of 62

04/21/23 Secant Formula 32 of 62

04/21/23 Secant Formula 33 of 62

Did they know that?

04/21/23 Secant Formula 34 of 62

04/21/23 Secant Formula 35 of 62

04/21/23 Secant Formula 36 of 62

4

4

04/21/23 Secant Formula 37 of 62

04/21/23 Secant Formula 38 of 62

04/21/23 Secant Formula 39 of 62

04/21/23 Secant Formula 40 of 62

04/21/23 Secant Formula 41 of 62

Oops: Collapse of the Second Narrows Bridge, Vancouver

04/21/23 Secant Formula 42 of 62

04/21/23 Secant Formula 43 of 62

04/21/23 Secant Formula 44 of 62

4

04/21/23 Secant Formula 45 of 62

04/21/23 Secant Formula 46 of 62

04/21/23 Secant Formula 47 of 62

04/21/23 Secant Formula 48 of 62

04/21/23 Secant Formula 49 of 62

04/21/23 Secant Formula 50 of 62

04/21/23 Secant Formula 51 of 62

04/21/23 Secant Formula 52 of 62

04/21/23 Secant Formula 53 of 62

04/21/23 Secant Formula 54 of 62

04/21/23 Secant Formula 55 of 62

Pre-stressed Concrete

Stage 1Tendons and reinforcement are positioned in the beam mould.

Stage 2Tendons are stressed to about 70% of their ultimate strength.

Stage 3Concrete is cast into the beam mould and allowed to cure to the required initial strength.

Stage 4When the concrete has cured the stressing force is released and the tendons anchor themselves in the concrete.

04/21/23 Secant Formula 56 of 62

Prestressed Concrete

04/21/23 Secant Formula 57 of 62

Post-tensioned Beams Stage 1Cable ducts and reinforcement are positioned in the beam mould. The ducts are usually raised towards the neutral axis at the ends to reduce the eccentricity of the stressing force.

Stage 2Concrete is cast into the beam mould and allowed to cure to the required initial strength.

Stage 3Tendons are threaded through the cable ducts and tensioned to about 70% of their ultimate strength.

Stage 4Wedges are inserted into the end anchorages and the tensioning force on the tendons is released. Grout is then pumped into the ducts to protect the tendons.

04/21/23 Secant Formula 58 of 62

Post-tensioned Beams

04/21/23 Secant Formula 59 of 62

When the tensioning force is released and the tendons are anchored to the concrete a series of effects result in a loss of stress in the tendons. The effects are :

• relaxation of the steel tendons • elastic deformation of the concrete • shrinkage and creep of the concrete • slip or movement of the tendons at the anchorages during

anchoring • other causes in special circumstances, such as when

steam curing is used with pre-tensioning.

Total losses in prestress can amount to about 30% of the initial tensioning stress.

Loss of Prestress

04/21/23 Secant Formula 60 of 62

Arrgggh, it’s over!

04/21/23 Secant Formula 61 of 62

Burj Dubai Tower 800 metres (2,640

feet) tall Upward-spiral design

Exterior cladding system

Built of glass, aluminium, concrete,

steel Designed by

Skidmore, Owings & Merrill, Chicago

Total floor space of 500,000 sq m (5.35

million sq ft)

04/21/23 Secant Formula 62 of 62

Solution to Problem 3: