Theory of structures - Appunti

Curved Beams and Parabolic Arch

In the end, substituting in the equation of σ we get:

θ R R N M M N M− − − −σ = E ε = E y =(η y χ) = y + yθ θ 2R + y R + y A A R I AR A RR R N R + y M R + y M N M M Ry− − − −= y =R + y A R AR R I A A R I R + yR R

We can see that σ has a non-linear pattern as ε .θ θ2.

1. Parabolic Arch

Let us study the parabolic arch in the figure:

We call f the maximum height of the arch. The analytic equation of the beam is: 2z z −y = 4 f +l l

Since the system is one time redundant we will make it isostatic and we add a redundant actionX.

The inclination α is: 4 fdy − −= (2 z l)tan α = 2dz l

Theory of Structures 212. Curved beams 2.1. Parabolic arch

Doing equilibrium we get: N l0− −→ −N cos

we make several approximations to get the solution for X:

• −→ ≃if h/R < 0.25 I IR
• MN , χ =if h/R < 0.10 we can assume the same constitutive law for straight beams
• η = E A EITand t = ∗G A
• →if our arch is slender enough, shear strain is negligible t = 0
• ≃if α is small we can neglect N (sin α 0)
• IA and I = because this is the only way to obtain an analyticalwe assume A = 0 0cos α

The internal work reads: ls ZZ X N M + X M X N M + X M dz1 0 1 1 0 1L = N + M ds = N + M =i 1 1 1 1E A E I E A E I cos α0 0l l 2 Z ZX N M + X M cos α y pl p X1 0 1 2 2− −= N + M dz = X z z + y dz1 1E A E I E A E I 2 2 E I0 0 0 0 00 0

Let us analyze the three components of the integral separately: ll l l 2 Z Z Z1 1 l 4 f2 − −cos α dz = arctan (2 z l) =dz = dz =2 2 28 f l1 + tan α h i4 f0 0 0 − − 01 + (2 z l)2l2 l 4 f ≃= arctan l4 f

Theory of Structures 222. Curved beams

2.1. Parabolic arch

Doing this approximation if f /l < 0.1 the error is of the 5% and if f /l < 0.2 the error is of the 15%.

ll 4 5 33 Z 2 z z p l fpl p z2 − −−y − + =z dz = 2 p l f 2 3 42 2 3 l 4 l 5 l 150 0ll 2 5 4 2 3 2 Z 16 f z 2 l z l z 8 f l2 −y dz = + =4l 5 4 3 150 0

The principle of virtual work gives us:

2 2 3 1 l 4 f 1 8 f l p l f−X arctan + =0E A 4 f l E I 15 15 E I0 0 0pl 1X = = plh i 8 f8 f [1 + ξ]I l151 + 0 l3l 32 A f0 2A l2

We define the slenderness of the arch as λ = :0I 02 15 l l1 4 fξ = arctan2 28 λ f 4 f l

Figure 2.5: Variation of X with f /l and λ

Looking at the graph on the left we can see that for increasing values of f /l the unknown X tends2p l → →to and for f /l 0 also X 0. Looking at the graph on the right we can see that the8 fmaximum shifts towards left for increasing values of λ.−→ ≃λ = 69 X 3.13 p

is the differential governing equation of the problem. We need 4 boundary conditions and we start observing that we can exploit the double symmetry of the problem

The bending moment is: P P s M (s) = X + R sin α = X + R sin² 2 R

The differential equation is: 2 d u u M 1 P sr r+ = χ = = X + R sin² 2ds R E I E I 2 RR R² 2 2

Since s = R α we can say that ds = R dα :2 2 2 d u 1 P d u R P1 r r−→+ u = X + R sin α + u = X + R sin αr r2 2 2R dα E I 2 dα E I 2R R

The solution for this differential equation is: 2 3X R P R−u = A cos α + B sin α + α cos αr E I 4 E IR R

The boundary conditions are u (0) = 0, u (π/2) = 0, φ(0) = 0 and φ(π/2) = 0.

s sZd u ud u s rs − −→ −= = u = u dαs rds R dα R α 32 P RX R−A − α + (α sin α + cos α) + C

= sin α + B cos αs E I 4 E IR R2 3 dφ d u 1 P R1 r −A −+ u = cos α B sin α += (2 sin α + α cos α) + A cos α+r2 2 2ds R dα R 4 E I R2 3 X R P R X PR−+B sin α + α cos α = + sin αE I 4 E I E I 2 E IR R R R2Z PR XR P R CX − −φ = + R dα = α cos αE I 2 E I E I 2 E I RR R R RαSubstituting boundary conditions we have: 33 P RP R −→+ C B =u (0) = 0 = B +s 4 E I 4 E IR R2 3 3X R P R P Rπ π π−A − −→u (π/2) = 0 = + + C A =s E I 2 4 E I 2 E I 8R R R2 3P R C P R− − −→ −φ(0) = 0 = C =2 E I R 2 E IR RXR C PRπ − −→ −φ(π/2) = 0 = X =E I 2 R πRIn the end the solution is: 3 1 X αP R π − −cos α + sin α cos αu =r E I 8 4 π 4R3 P R 1 α α 1π −−u = sin α + cos α + + sin αs

3.1 Mindlin-Reissner model

In this model we consider shear strain in order to have the more general plate theory we can have. The main hypothesis is: any segment orthogonal to the mid surface of the plate remains straight and undeformed after the process. Since plates are bodies in the tridimensional space we define the 3 axis x, y and z and we call u, v and w the 3 components of displacement in the space. We also define the