Nonlinear Thermo- Mechanics of Plates and Shallow Shells

Transcription

1 Nonlinear Thermo- Mechanics of Plates and Shallow Shells Payam Khazaeinejad 1, Asif S. Usmani 1, Omar Laghrouche 1 IIE, School of Engineering, The University of Edinburgh IIE, School of the Built Environment, Heriot-Watt University

2 Outline 1 3 Introduction Formulation Solution Numerical tests Conclusions Future work

3 Introduction Formulation Solution Numerical tests Conclusions Introduction Abstract A practical mathematical approach is developed to model the nonlinear behaviour of common shell structures (i.e. shallow shells, plates and beams) in response to large thermoelastic deformations. Purpose To provide benchmark solutions for numerical methods Fast computation of realistic thermo-mechanical analysis of simple shell structures (e.g. as part of Monte Carlo methods to account for uncertainty) To provide advanced basis functions for Trefftz-type computational approaches To understand the mathematical underpinnings of engineering concepts such as the compressive ring in tensile membrane action in slabs

4 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Kinematics relations Based on the von Karman theory for moderately large deflections, the nonlinear kinematics relations for shallow shells are given by ε xx ε yy γ xy = u x + 1 w x + w R + x w v y + 1 w y R y u y + v x + w w x y + w R xy mid surface strains ζ w x w y w xy curvatures Fig. 1. Configuration of the problem studied: (a) the Cartesian coordinates and (b) geometry of a shallow shell.

5 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Constitutive equations The inverse strain-stress relations including thermal strains are ε xx ε yy γ xy αθ αθ 0 = 1 E σ xx νσ yy σ yy νσ xx (1+ν)τ xy where θ is the increment of temperature and α is the coefficient of thermal expansion. The nonlinear constitutive equations can be expressed as σ xx σ yy τ xy = E(θ) 1 ν (θ) 1 ν(θ) 0 ν(θ) ν(θ) ε xx ε yy γ xy E(θ)α(θ) 1 ν(θ) θ(x,y,ζ) E(θ)α(θ) 1 ν(θ) θ(x,y,ζ) 0

6 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Governing equations: compatibility equation The compatibility equation: 4 F a x 4 +(a 1+a 33 ) 4 F x y +a 4 F 11 y 4 stress functions terms + 1 w R x y + 1 w R y x w R xy x y effect of initial curvatures 4 w b 1 x 4 (b 11+b b 33 ) 4 w x y b 4 w 1 y 4 effect of material nonlinearity Nx T Nx T Ny T Ny T a 1 x a 11 y a x a 1 y effect of thermal loads = w w w xy x y effect of nonlinear terms

7 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Governing equations: equilibrium equation The equilibrium equation: 4 w d 11 x 4 +(d 1+d 33 ) 4 w x y +d 4 w y 4 = q(x, y) uniform distributed load deflection terms + F w y x 1 + F w R x x y 1 F w R y xy x y 1 R xy effect of nonlinear terms and curvatures 4 F +c 1 x +(c 11+c 4 c 33 ) 4 F x y +c 4 F 1 y 4 effect of membrane bending coupling Mx T x Mx T y effect of thermal loads

8 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Governing equations: coefficients where the coefficient are defined by the following relations [A]= h h [B]= h h [D]= h h E(θ) 1 ν (θ) E(θ) 1 ν (θ) E(θ) 1 ν (θ) 1 ν(θ) 0 ν(θ) ν(θ) 0 ν(θ) ν(θ) 0 ν(θ) ν(θ) 1 ν(θ) 1 ν(θ) dζ ζdζ ζ dζ [a]=[a] 1, [b]=[a] 1 [B], [c]=[b][a] 1, [d]=[d] [B][A] 1 [B]

9 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Temperature-dependency of material properties The material properties are proposed to be temperature-dependent according to the Eurocode (British Standards Institution) and Standards Australian empirical expressions. 1+ θ, 0 ± C<θ<600 ± C; 000ln θ θ θ 53.5, 600± C θ 1000 ± C. Fig.. Temperature-dependency of material properties for steel section.

10 Introduction Formulation Solution Numerical tests Conclusions Formulation of the problem Boundary conditions Two types of in-plane boundary conditions: Case I: v=w=0 at x=0,a u=w=0 at y=0,b Case II: u=v=w=0 at four edges u = a 0 u x dx=0 P x=... v= b 0 v y dy=0 P y=... Fig. 3. Reaction loads at the plate edges for immovable boundary condition.

11 Introduction Formulation Solution Numerical tests Conclusions Solution When temperature field is only varying over the depth, the following functions satisfy the governing equations w(x, y) w mn 0 q(x, y) N T (x,y) = q mn 0 m=1 M T n=1 Nmn T sin(mπx/a) sin(nπy/b) 0 (x,y) M mn T 0 F(x, y) F mn P x y bh +P yx ah By taking account of the mathematical coupling between higher order terms, more than two indices must be considered for representing them. It increases the time and cost needed to solve the derived system of equations. Moreover, its contribution to the final results can be negligible as it is shown in numerical test section. Hence, the governing equations are simplified for one term approximations.

12 Introduction Formulation Solution Numerical tests Conclusions Solution To expand the trigonometric functions over infinite sums, expansion theorem is applied. If a generic function is expressed by Ψ(x,y)= ψ mn S mn (x,y) m=1n=1 For normalised modal function S mn over an orthogonal basis, we have ψ mn = Ψ(x,y),S mn(x,y) S mn (x,y),s mn (x,y) = 4 ab a 0 b 0 Ψ(x,y)S mn(x,y)dy dx Therefore, a cubic characteristic equation can be derived for determining the large deformations of shallow shells and plates.

13 Introduction Formulation Solution Numerical tests Conclusions Solution For plates, deflection can be obtained by solving the following equation: Aw 3 mn+b w mn +C=0 where A, B, and C are functions of material properties, geometry of the plate, and thermo-mechanical load parameters. B= A= 3Ehm4 n 4 H mn (m b +n a ) Eh 3 m 1(1 ν ) a + n + 8NT mnh mn m n b π (m b +n a ) 1 m P x π ab a + n P y b m C= a + n M T mn q mn b π π 4

14 Introduction Formulation Solution Numerical tests Conclusions Solution where H mn = 1+( 1)m +( 1) n ( 1) 3m ( 1) 3n 3( 1) m+n +( 1) 3m+n +( 1) m+3n 3mnπ N T mn M T mn q mn = 4( 1+( 1)m )( 1+( 1) n ) mnπ N T M T q Px P y = Ehπ w mn 8(1 ν ) b( m +ν n a a(ν m a + n ) b b ) + NT (1 ν) b a hf mn( 1+( 1) m )( 1+( 1) n ) mn n b m a

15 Introduction Formulation Solution Numerical tests Conclusions Numerical tests Example 1: A square plate under mechanical load The membrane stresses : Fig. 4. Convergence of the nonlinear dimensionless centre deflection (w/h) for a square plate under UDL.

16 Introduction Formulation Solution Numerical tests Conclusions Numerical tests Example : A square slab under thermal loading over the depth The membrane stresses : A 5 (m) 5 (m) slab subjected to thermal loading with 100 (mm) depth, Young s modulus of 40,000 (N/mm ), Poisson s ratio of 0.3. Temperature distribution causes an equivalent thermal gradient of θ,z =5 ( C/mm) and an equivalent thermal expansion of θ=00 ( C) is considered. Table 1. The maximum thermal deflection (mm) of a plate with boundary case I Analytical solution (Cameron and Usmani, 004) 148 ABAQUS (Cameron and Usmani, 004) 136 Present solution 137

17 Introduction Formulation Solution Numerical tests Conclusions Numerical tests Example 3: A square plate under thermo-mechanical loading The membrane stresses : Fig. 5. Comparative study for thermo-mechanical deformations of a plate with boundary case I.

18 Introduction Formulation Solution Numerical tests Conclusions Numerical tests Example 4: A square plate under UDL with boundary case II The membrane stresses : Table. Comparison of dimensionless nonlinear centre deflection and stresses (obtained by three series terms) with RBF (Radial Basis Function) method and ANSYS (Al-Gahtani, Naffa a, Eng Anal Bound Elem, 009) UDL Centredeflection, w/h Membranestress, σ m a /Eh Bendingstress, σ b a /Eh ANSYS RBF Present ANSYS RBF Present ANSYS RBF Present % error 0.78% error 0.87% error

19 Introduction Formulation Solution Numerical tests Conclusions Future work The future work is investigating the influence of arbitrary imposed temperature fields, especially full non-uniform temperature distribution (threedimensional analysis) such as those resulting from a fire impinging on the shells. The three-dimensional temperature distribution can be expressed as θ(x,y,z)= m=1 n=1 θ(z)sin(mπx/a)sin(nπy/b) The temperature distribution along the depth is obtained by solving the heat transfer equation associated with boundary conditions. θ(z)=θ bottom θ bottom θ top h h dz k(z) z h dz k(z)

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