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1.054/1.541 Mechanics and Design of Concrete Structures Prof. Oral Buyukozturk

Spring 2004 Massachusetts Institute of Technology Outline 11

1.054/1.541 Mechanics and Design of Concrete Structures (3-0-9) Outline 11 Yield Line Theory for Slabs ‰

Loads and load effects qdxdy

x y

Vy dx

Vx dy

∂Vx ⎞ ⎛ dx ⎟ dy ⎜ Vx + ∂x ⎝ ⎠

∂Vy ⎞ ⎛ dy ⎟ dx ⎜ Vy + ∂y ⎝ ⎠

z

dy

h dx Surface and shear forces

dx

x

x myx dx y

N y dx y

my dx mx dy

dy mxy dy

∂mx ⎞ ⎛ dx ⎟ dy ⎜ mx + ∂x ⎝ ⎠ ∂mxy ⎞ ⎛ dx ⎟ dy ⎜ mxy + ∂x ⎝ ⎠ ∂my ⎞ ⎛ dy ⎟ dx ⎜ my + ∂y ⎝ ⎠

N yx dx ∂N xy ⎞ ⎛ dx ⎟ dy ⎜ N xy + ∂x ⎝ ⎠

N x dy

∂N yx ⎞ ⎛ dy ⎟ dx ⎜ N yx + ∂y ⎝ ⎠

∂m yx ⎞ ⎛ dy ⎟ dx ⎜ m yx + ∂y ⎝ ⎠ Moments

∂N x ⎞ ⎛ dx ⎟ dy ⎜ Nx + ∂x ⎝ ⎠

N xy dy

∂N y ⎞ ⎛ dy ⎟ dx ⎜ Ny + ∂y ⎝ ⎠

Membrane forces

1 /9

1.054/1.541 Mechanics and Design of Concrete Structures Prof. Oral Buyukozturk

Spring 2004 Outline 11

o Load effects to be solved: Vx , Vy , mx , m y , mxy , m yx , N x , N y , N xy , N yx Æ Ten unknowns and six equations Æ Indeterminate problem: We need to include stress-strain relation for complete elastic solution. o The relative importance of the load effects is related to the thickness of the slab. Most reinforced and prestressed concrete floor slabs fall within “medium-thick” class, i.e., plates are

‰

ƒ

thin enough that shear deformations are small, and

ƒ

thick enough that in-plane or membrane forces are small.

Analysis methods: o Elastic theory o Elastic-plastic analysis – Finite element analysis (FEA) o Approximate methods of analysis o Limit analysis – Yield Line Theory – Lower & upper bound analysis

‰

Elastic theory o Lagrange’s fourth-order PDE governing equation of isotropic plates loaded normal to their plane:

∂4w ∂4w ∂4w q +2 2 2 + 4 = D ∂x 4 ∂x ∂y ∂y where

w = deflection of plate in direction of loading at point ( x , y ). q = loading imposed on plate per unit area, q ≈ f ( x, y )

D = flexural rigidity of plate, D = E = Young’s modulus

h = plate thickness 2 /9

Eh3 12(1 − µ 2 )

1.054/1.541 Mechanics and Design of Concrete Structures Prof. Oral Buyukozturk

Spring 2004 Outline 11

µ = Poisson’s ratio. o Navier’s solution of Lagrange’s equation using doubly infinite Fourier series: ∞

∞

w ( x, y ) = q ⋅ C ⋅ ∑∑ Amn sin m =1 n =1

mπ x nπ y sin a b

where a, b = lengths of panel sides m, n = integers

C , Amn = constants – Boundary conditions.

‰

Finite difference (FD) method o It replaces Lagrange’s fourth-order PDE with a series of simultaneous linear algebraic equations for the deflections of a finite number of points on the slab surface. Deflections, moments, and shears are computed.

‰

Finite element (FE) method o It utilizes discretization of the physical system into elements. Displacement

functions

are

chosen.

Exact

approximate equilibrium considerations are used.

‰

Approximate methods o Direct design method o Equivalent frame method o Assignment of moments

‰

Types of slabs o According to the structural action

3 /9

compatibility

and

1.054/1.541 Mechanics and Design of Concrete Structures Prof. Oral Buyukozturk

Spring 2004 Outline 11

–– One-way slabs –– Two-way slabs

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