- The Point Loads Are Placed At The Fixed Positions Shown In The Figure And They Are Live Loads A B 1 B At C Sc Ast Cro 1 (25.85 KiB) Viewed 41 times
The point loads are placed at the fixed positions shown in the figure and they are live loads. A B 1₁ b at C 'SC Ast Cross section 1/₂ W C (centre) D f'c f'ct.f fsy E = 200 GPa Ec= = 28600 MPa = 38 Mpa = 3.5 MPa = 500 MPa 1₁ E
Variables 3 m 2.5 m a 45 mm b 550 mm c 350 mm Asc 800 mm² Ast 1₁ 12 3200 mm²
Q1) The beam carries an unfactored uniformly distributed dead load (including self-weight) of G = 4 kN/m and an unfactored uniformly distributed live load of Q = 0 kN/m. Two concentrated unfactored live loads of P = 4 kN are applied at the positions shown in the figure. (Note: The load factor for G is 1.2, and for P is 1.5. There is no uniformly distributed live load.) 1.1 Find the maximum moment of the beam. Mmax 1.2 Find the neutral axis distance from the top surface. If the section is governed by the linear elastic regime, the transformed area method dn = 2 y A/ Σ A, is valid. d₁ = 1.3 Find the cracking moment. Mc = 1.4 Does this section crack? kNm (1 mark) (No answer given) (1 mark) mm (1 mark) kNm (2 mark)
Q2) Now, the live load increases gradually and the moment at the critical section just exceeds the cracking moment (Mcr), but the compressive section of the concrete is still under the linear elastic region. Please be reminded that in the flexural design, the crack of the concrete section starts where the tensile stress reaches the tensile strength. It is assumed that the cracks then propagate rapidly up to the entire tension section (up to the neutral axis) and this cracked concrete section cannot resist the tension. It should be noted that in reality, concrete sections between the primary cracks can still resist some tensile stresses as shown in the figure, which should be considered in the displacement design. However, in flexure design, we design the critical section where the largest moment occurs and thus it is reasonable to have such a primary crack in this critical section. 2.1 Find the new neutral axis in the cracked section under the elastic region on the compressive section and also find the cracked second moment of areas (Icr). The transformed area method (d₁ = Σy₁ A₁/A,) is still valid. However, we may need to solve the quadratic equation for calculating the neutral axis distance from the top surface. b Asc N.A__ Ast Stress profile (concrete) Cross section before cracking b a Asc N.A →→ cracks Ast Stress profile (concrete) Cross section just after cracking
dn = Icr= M secondary crack mm (2 marks) x 109 mm (2 marks) Neutral axis primary cracks M -concrete in tension
2.2 The applied loads keep increasing and the concrete section is now under the non-linear regime. The neutral axis moves upward and the concrete compressive section has the curvilinear distribution of concrete stresses, which require the integration of the stress to compute the neutral axis and the ultimate nominal moment. To avoid the complex calculation, the Whitney stress block is introduced during the class. It should be noted that this ultimate strength stage is a condition of the flexural design. Thus, when we design the beam in terms of bending, we can only consider this ultimate condition directly without the previous linear stage. Find the neutral axis distance from the top fibre (dn) at the ultimate strength stage and the ultimate nominal moment (M₁) that this concrete section can carry using the given section dimensions and materials properties. b a Cross section before cracking dn = A EZ N.A. Ast Stress profile (concrete) M* = mm (2 marks) b (No answer given) + (1 mark) kNm (2 marks) Does the beam satisfy the design requirement? NA Z cracks Ast Cross section just after cracking kNm (1 mark) Stress profile (concrete) b M₂= Note: In flexural design, the ultimate moment the section occurs when the compressive fibre reached the ultimate concrete strain (cu = 0.003). The nominal ultimate moment is computed from the property of the concrete section and independent to the applied loads. A.. N.A + cracks A.. 2.3 The beam now has to carry an unfactored dead load (including self-weight) of G = 15 kN/m and an unfactored live load of Q = 15 kN/m. Two concentrated unfactored live loads of P = 15 kN are applied at the positions shown in the figure. Determine if the design moment strength (M) exceeds the factored design moment M*. $ = 0.85 Stress profile (concrete) Cross section non-linear