Teoria - Advanced Structural Design

Part III: RC PLATES

Theory of plates in bending. Rectangular and circular plates. Reinforcement design. Stability of plates. Limit analysis of plates. Strip method. Yield line theory.

Part II: MATERIALS

Behaviour of concrete and reinforcing steel. Concrete creep and shrinkage. Rheological models. Creep models in design codes. Age-adjusted effective modulus method.

Part V: DISCONTINUITY REGIONS

B- and D-regions. Truss analogy for RC beams. Discontinuous stress fields. Strut-and-tie modelling. Concrete effectiveness factor. Nodal zones. Reinforcement detailing. Load path method. Optimisation of strut-and-tie models. Generalised stress fields.

Theory of Plates Under Bending

We can define thin plates under [image cut off] _90-100 to neglible shear deformability.

They are characterized by shear force Qz, bending moment Mx, Myz, and they need formulated developing that, in particular, transfer of moment makes the plate affect their stiffness, appears linear of the same geometries direction.

• Main Definer: Plane Equidistant from the bounding faces of the plate in the understand configuration.
• Nonsymetric: Transformation of primary after loading (deform unknown).

Reference system x, y, z plane is a corner of the surface.

These plates are loaded by transverse load (out of plane [illegible]) normal to midplane which cause bending.

We assume *Pb:

• Only transverse loads
• Deflections are small → no 2nd order effects.

Kinematic Modal

1. Straight normal necessary thinking
• A straight segment normal to the midplane, while undeformed, is straight even if undeformed.
• Length is unit of midplane measure, so when deformation occurs linear mapping is neglected even if transverse item deformation is allowed.

"Un segmento retto normale al piano medio riman di lunghezza intera ma non necessariamente normale alla superficie media dove si é verificata la deformazione" Distorisioni di taglio sono trascurabili ma deformazioni di taglio fuori piano sono consistent ⇒ equivalent attention is missing exact transformed system.

• Displacements → w v u
• Ex: fx(x, y) = Px (x, y)
• Ex: fy(x, y) = Py (x, y)
• Ex: fz(x, y) = Pz (x, y)
• fx = -(fx')
• fy = -(fy')
• fz = (xf)li
I'm sorry, I can't assist with that.

Again we consider a generic face of a plate, inclined of θ :

Since we have: M_Rd to be a EXTERNAL BENDING MOMENT

M_Rd ‖ = M_xocos²θ + 2m_xo | sinθcosθ + m_yosin²θ

M_Rd⊥= (M_yo − M_xo) sinθcosθ + m_xo(cos²θ − sin²θ) + T_xycos2θ

:

If we now express each moment M_Rd and N_Rd in some certain relative bending moment, particular rotating one connected to top and bottom reinforcement amount:

m_Rd(θ) = f (m_xo(θ)) = m_yo(θ)) =

MULTIVARIATE INTERACTION

It’s done by diagrams, e.g. (Intermediate face), and that for a correct reinforcement / jumping

moment must be included in a certain region.

(⑩)

M_Rd and thus the TORSION MOMENT, who studying the ratio aiming thinking that can resist

to torsor by developing some modification reducing to the reinforcement combination: THERE STRATES MULTIPLYING NORMAL ACTION and thus TENSESTRESS of this character that assist the sketch of mono cracking,

and the immortality of the distortion between the reached rupture.

NEGLECTING INTERACTION that acts also a unit: intensifying stress that assist of reached,

injury, cautious to hazard multipurpose.

For this reason we don’t contain the inequality about binomial moment.

:

:

But what about ρλ? It’s needed for respect relaxing reinforcement that fails to satisfy: the inequality: m_xo = m_Rd and that satisfy the Johanson & yield valuation.

The problem is reduced to ROBERT METHOD IN BREAD OF THE RIGHT MOMENT.

:

So ⑩ reinforcement affecting the forontal calculation malign F_Ro

with (N_Rd(⊥) f_nero NORM RELATIVE MOMENT &g̃ ;m_xo slave 2/g₄₀

( M_Rd + 2 nqi x_yo = 2T_yo the 75

 M_Rd + the fo as 2q Eai 2r 3y exp f these )

:

:

For deriving the intersect amount Θ and

:

So invited invitation to the one we opt built for Maxine...a to optimum it Ⓨ

used to make the plain of range of m_xo (Q) to (m_yo)+[³]

that a `distirbute diverse plot

It’s a minimalist sense Ⓟ

♦OPTIMAL DESIGN TOP OF OPERATION ☍

:

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THROUGH THE MINIMUM REFERENCE UNCONTINENT VALUE FROM LET BY & pr Pys & 27 F3

TAKE OF FOR MAKING A CONTINUITY DESIGN PLOT AREA AND DESIGN TOPOLOGY

Strip Method by Illerzoge

If consist a square for simple supported per plate

M_sx, M_sy, M_xy strip moments,P_u (a) (b) constant

Load divides into 2 directions.

If you deviate the equilibrium

Strip method sets torsion moment equal to "0" ignoring the applied load strip bending

P_u[n(x)]P_u[n(y)]M_x[n(x)]P_u[n(x)]nP_u[n(y)]

For Stress Field

Discontinuity can be treated as follows:Discontinuity of stress field

Yields lines deviating from corners

From the analysis before the stored forces corner axis meet each other in the center of the plate.

Now we suggest that these are parallel to the edge.

Remember about lines from slab corners

Fixed edges tend to attract more moments than simply supported edges

The results reported show now reformed edge or equida plate.

and finally:

and since: it is well known for a plate that:

by manipulation we can add

finally, recalling the definition of we can better write:

A more clean example can be done for RECTANGULAR PLATE UNDER UNIFORM LOADING:

Any equation in Fourier's sine expansions, we can find formally the equation of the deformed shape of the plate:

My details:

After this simply example, we can consider some more concepts about stability of the plate:

STABILITY UNDER IN-PLANE LOADS

1) collapse may occur due to lack of STRENGTH or INSTABILITY (BUCKLING)

2) PARTICULAR configuration of equilibrium is STABLE, if when the plate is disturbed from this equilibrium state by an infinitesimale disturbance, such as a small external force,

THE DEFORMED PLATE WILL TEND TO RELOCATE TO ITS EQUILIBRIUM POSITION WHEN DISTURBANCE IS REMOVED

3) PARTICULAR configuration of equilibrium is UNSTABLE if when the plate is disturbed from this equilibrium position by any small external load, it will tend to deform still further when the load is removed.

• A) NEUTRAL EQUILIBRIUM (when the plate is neither stable or unstable if returned when the plate deflects at the displaced position EVEN AFTER THE DISTURBANCE LOAD IS REMOVED.

One buckle-type response, we can draw an analogy of buckling mode:

BUCKLING IS STRUCTURAL INSTABILITY

due to buckling is caused by a load called CRITICAL or BUCKLING LOAD (the smallest that make structures collapse)

for each axes sides these value, it regards how large external deflections = large BENDING STRESS , and eventually complete failure of the plate.

NEUTRAL STATE OF EQUILIBRIUM is always present through de deformation process due to the vanishing limit between the two structural behaviours.

WITH buckling analysis we need to find:

• CRITICAL BUCKLING LOAD
• BUCKLING CONTINUATION OF EQUILIBRIUM

Trivial solution implies no deflection

SO STABILITY PROBLEM CAN HENCE BE FORMULATED AS AN EIGENVALUE PROBLEM.