Homework esame Theory of Structures

Theory of Structures + Stability of Structures

February 4^th, 2021(the solution should be delivered within February 4^th, 2021, h. 13:00)

Theory of Structures

Problem T1

The problem is referred to a parabolic arch, whose geometry is depicted in the figure. The material is homogeneous (E = 30 GPa, v = 0.2) and the keystone section is square (side length 150 cm). The variation of the cross-section properties with respect to the arch inclination is A = A₀/cos α, I = I₀/cos α, where A₀ and I₀ are the keystone cross-section area and inertia, respectively. The external load is represented by a concentrated force (P = 150 kN). The student should solve the problem by making use of symbolic variables and, finally, should obtain the numerical values of the reaction forces and the diagrams of shear, axial force and bending moment.

Problem T2

The above figure represents a multi-cellular section with uniform thickness 50 mm (all the measurements are in mm). The students should obtain the stress field for a shear force V = 200 kN, applied in the upper left corner of the middle line. Moreover, the position of the shear center should be computed. G = 18 GPa.

Theory of Structures + Stability of Structures

February 4^th, 2021

(the solution should be delivered within February 4^th, 2021, h. 13:00)

Theory of Structures

Problem T1

The problem is referred to a parabolic arch, whose geometry is depicted in the figure. The material is homogeneous (E = 30 GPa, ν = 0.2) and the keystone section is square (side length 150 cm). The variation of the cross-section properties with respect to the arch inclination is A = A_o/cos α, I = I_o/cos α, where A_o and I_o are the keystone cross-section area and inertia, respectively. The external load is represented by a concentrated force (P = 150 kN). The student should solve the problem by making use of symbolic variables and, finally, should obtain the numerical values of the reaction forces and the diagrams of shear, axial force and bending moment.

Problem T2

The above figure represents a multi-cellular section with uniform thickness 50 mm (all the measurements are in mm). The students should obtain the stress field for a shear force V = 200 kN, applied in the upper left corner of the middle line. Moreover, the position of the shear center should be computed. G = 18 GPa.

THEORY OF STRUCTURES

PROBLEM T1

P = 150 kN

1. REDUNDANT STRUCTURE:
   • Add a redundant action in B substituting the hinge with a roller
2. CASE 0: external load:

H_AO = 0

Focus on a part:

Equilibrium

∑H = 0: H_AO = 0

∑V = 0: V_AO + V_BO = P → V_AO = ³/₄ P

∑M_A = 0: V_BO L - P ^e/₄ = 0 → V_BO = P/₄

Check ∑M_B = 0: ³/₄ PL - P. ³/₄ e = 0 ✓

N_o = ∫ - ³/₄ P sen α, z < ^e/₄

     (³/₄ P + P) sen α, z ≥ ^e/₄

T_o = ∫ - ³/₄ P cos α, z < ^e/₄

     (¹/₄ P + P) cos α, z ≥ ^e/₄

M_o = ∫  ³/₄ P z    z < ^e/₄

     [³/₄ P z - P(z - ^e/₄)]   z ≥ ^e/₄

3) CASE 1: Horizontal unitary thrust x̅ = 1

EQUILIBRIUM

∑H=0: H_A1 - 1

∑V=0: V_A1 + V_B1 = 0

V_A1 = V_B1 = 0

Focus on a part

N₁ = 1cosα

T₁ = -1senα

M₁ = 1·y

4) PRINCIPLE OF VIRTUAL WORK:

L_e = 1 · δ_HB

because, since in real structure we have an hinge in B, we can't have any horizontal displacement.

L_i = ∫_ξ^ξ (N_t M_t I_t t + M_t x_t) ds

FOR AN ARCH:

• M = N_RED / EA
• t = I / GA_mod
• χ = M / EI_red - N_red / REA

where

• N_red = reduced axial
• A_mod = modified area
• I_red = reduced inertia

IF B/R < 0.10, arch is sufficiently slender:

R(z) = [1+(y'(z))²]^3/2 / |y''(z)|

R / R_min = 0.275 < 0.10 →

R_min = R(z - ξ/2) = [1+0]^3/2 / | - δ / ξ²