Reinfo in forc rce ed Plate Design esig n Design sig n for fo r Mx y Twi Twisti sting ng Moment By John Li 9 April 2002
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Introduction Reinforced plate design is a simple extension of the beam design. However, the Lagrange’s 4th order partial differentiation equation for plate is difficult to solve except for simple geometry and support conditions. Furthermore, the twisting moment and the Poisson ratio add complexity to an already difficult problem. The many classical approaches that are still popular today mainly focus on deriving alternative moment fields for reinforcement design by eliminating the mxy twisting moment. Although there are many papers and texts on such approaches, most go into lengthy, complicated derivations instead of simplifying our understanding. This paper is an exercise to extract from many sources both the classical theory and the modern finite element method to present a simple physical understanding of the behavior of plate without going through involving mathematical formulation. The later sections of this paper address the confusion surrounding the m xy twisting moment and how to indirectly design for it. Ultimately, with software available today, reinforced plate design should be fast and easy. Detail derivations of theories are available in many texts and therefore not included in this paper. Also, this paper does no include the discussion on Poisson ratio whic h is also an important topic in pl ate design.
Structur al Mechanics All structural mechanics theories, including the beam and plate theories, must satisfy the following three conditions: 1. Stress-strain relation – material. 2. Equilibrium – force. 3. Compatibility – geometry. The stress-strain relation can usually be satisfied using design equations. However, an exact solution to satisfy both equilibrium and compatibility may be difficult and sometimes unnecessary. When given a choice/decision between th em, satisfyin g equilibr ium is essential to prevent collapse.
Beam Theory With the plane-remains-plane assumption and shear deformation excluded, the beam theory equations are simple:
4
Bending:
d w dx
4
=
q
; EI
Deflection ∝
w
Slope ∝
dw
Moment Curvature ∝
d w
Shear ∝
d w
dx 2
dx
2
3
dx
3
4
Load ∝
Torsion:
d w
θ =
dx
4
T GJ
It is important to realize the followings: 1. The placements of longitudin al reinforc ement and torsi onal stirrup not c oupled - governed by separate equatio ns. 2. Beam torsion results in circular shear stress.
Beam
Classi cal Plate Theory For plate, with the straight-line-remains-straight assumption and shear deformation excluded, the Lagrange’s plate equation:
Deflection ∝
w
Slope ∝
∂w ∂w , ∂ x ∂y
∂4w ∂4w ∂4w q + 2 2 2 + 4 = ; Moment Curvature ∝ ∂ x 4 ∂x ∂y ∂y D where D =
Eh3
12(1 − µ 2 )
∂2w ∂2 w , ∂ x 2 ∂y 2
Twisting Moment ∝
∂2w ∂ x∂y
Shear ∝
∂3w ∂3 w , ∂ x 3 ∂y3
Load ∝
∂4w ∂4 w , ∂ x 4 ∂y 4
Slab Twisting Shear
The term mxy, the twisting moment, represents the twist, that is, the rate of change of slope in the x-direction as one moves in the y-direction or vise versa. The twisting moment results in shear stress parallel to the plate surface except near the ends. Because of this shear flow difference, the reinforcement to prevent tor sional beam failure should not be confus ed with th e reinforc ement to pr event twi sting pl ate failure.
∂2w ∂2w ∂ 2 w ∂ 2 w ∂2w D(1 − µ ) With m x = − D( 2 + µ 2 ) , m y = − D( µ 2 + 2 ) & m xy = − ∂ x ∂y ∂ x ∂y ∂ x∂y
The Lagrange’s Equation written in terms of m x, my and mxy:
∂ 2 mxy ∂ 2 m y ∂ 2 m x +2 + = −q ∂ x 2 ∂x∂y ∂y 2 This is the most important equation providing invaluable physical insight into problem of reinforced plate design. It reveals that the load “ q” can be arbitrarily apporti oned between m x , m y and m xy for r einforcement design as long as the LHS of the equatio n is l arger than the RHS at all poi nts o f the plate system. It also point s out that design solutions are not unique. It is extremely imp ortant to n ote if the design moment fields are such that part of the load is c arried by the mxy term, the design cannot j ust ignore m xy as that would make the addition of the mx and m y terms smaller than th e loads. This interpretation of satisfying equilibrium with allowance to violate compatibility leads to the Lower Bound Method.
Lower Bound Theory The calculation of ultimate load using limit analysis methods is based on the redistribution of moment and shear when the elastic limit is exceeded. To determine the ultimate load of a given plate system, either a lower bound theory or an upper bound theory may be used. This paper will concentrate only on the lower bound theory because it is conservative and that it can be easily programmed into a finite element software. The lower bound theory assumes a moment fields at ultimate load such that: 1. The equilibrium condition is satisfied at all points in the plate system. 2. The plate is reinforced according to the assumed moment fields. 3. Satisfy boundary conditions. Actually, engineers use this theory all the time to find alternative design moments without knowing it: 1. Moment redistribution. 2. 1-way slab design – apportion loads only to mx. 3. 2-way moment coefficient chart method – apportion loads to mx and my. 4. Equivalent frame method – apportion loads to mx and my. 5. Strip methods including the Hillerborg’s strip method – apportion loads to mx and my. 6. Wood-Armer transformation method – geometric transformation to apportion the mxy term to the mx and my terms. Without the Lower Bound Theory, the validity of all these “assumptions”, charts, cuts, strips and transformation methods would be in doubt. The reasons for developing and using these procedures are: 1. Simplifying the search for alternative equilibrium design moment fields.
2. Elimination of the mxy twisting moment because it is much easier to design reinforcement for alternative m x and my fields.
Finite Element Method In 1960’s, the aircraft industry pioneers the finite element method in calculating stress and strain. This method is now included in many structural engineering software. Unfortunately, the concept of finite element is still not well understood by practitioners. Many still think that, for example, the plate finite element is just a “smaller” plate and that the nodal reactive moments m’x and m’y are the same as m y and my in classical plate theory. Well, they are not! Let’s take a look at the Lagrange’s equation again:
∂4 w ∂4 w ∂4 w q +2 2 2 + 4 = D ∂ x 4 ∂x ∂y ∂y The major problem is finding a unique displacement field w(x,y) that satisfies the equation at all points of the plate system. Once the displacement field has been solved, all quantities can be derived easily using previously stated formulae. Using the finite element method, researchers formulate elements that have the same stiffness as the real plate. Stiffnesses associated with out-of-plane degrees of freedom, r x, r y and dz are lumped at corner nodes resulting in nodal reactive forces m’x, m’y and v’ when strained. An assemblage of such plate elements forms a global stiffness matrix which would simulate the displacements against imposed loads. The solution improves as the element mesh is refined.
[ Stiffness _ Matrix]{ Displacements} = { Forces} The whole purpose of us ing fi nite element is to find t he displacement field. It is import ant to note that the set of nod al reactive forces are also in equilibr ium wi th external loads and therefore also valid f or reinforcement design.
Moment Fields for Reinfo rcement Design For plate systems with complex conditions or where more accurate designs are required, the finite element method would give an accurate solution of the elastic displacement field. Graphical plots and textural output
Displacement w(x,y)
Classical Theory
∂ 2 w ∂ 2 w m y = − D( µ 2 + 2 ) ∂ x ∂y ∂2w ∂2w m x = − D ( 2 + µ 2 ) ∂ x ∂y ∂2w m xy = − D (1 − ) ∂ x∂y Wood-Armer mux & muy
Finite Element Nodal Reactive Moment
⎧θ x ⎫ ⎧ m 'x ⎫ ⎡ K ⎤ ⎪⎨θ y ⎪⎬ = ⎪⎨ m ' y ⎪⎬ ⎣ ⎦ ⎪ w ⎪ ⎪ V ⎪ ⎩ ⎭ ⎩ ⎭ No mxy
can then be generated easily by substituting the displacement field into the classical formulae. For reinforcement design, two sets of moment fields, both in equilibrium with the external loads, are available using the classical theory or the finite element (nodal reactive moments) method.
Wood-Armer Formula Many designers prefer to use the classical route because they are more comfortable with the classical plate theory. The problem now is to find a set of orthogonal reinforcement to resist mx, my and mxy. The Wood-Armer Formula is most popular approach to convert m x, my and mxy to orthogonal design moments m ux and muy – again eliminating mxy. In essence, this method uses the Mohr circle geometric approach to derive optimum orthogonal design moments that would prevent yielding in all directions. The yield condition is based on the Johansen’s Yield Criterion.
Nodal Reactive Moments Method When using finite element method to solve for the displacement field, the stiffnesses of all elements are already calculated when assembling the global stiffness matrix. It is therefore more natural and easier to use the nodal reactive moments method for reinforcement design because these nodal reactive moments can easily be obtain by multiplying the solved displacement vectors with element stiffness matrixes.
Reinfo rcement Design Regardless of the approach to take for reinforcement design, because of the need to consider many load combinations, the right locations for reinforcement calculations present another problem. Manual point method of selecting all the maxima and minima for design would not only be very tedious, it would be too conservative and therefore not acceptable. Using section cuts of mx, my and mxy and then using the Wood-Armer formula to calculate mux and muy would again be very tedious. Furthermore, there is not guarantee that such cuts would result in the correct m ux and muy design envelops unless many closely spaced section cuts are used. Many classical methods use the strip approach because the reinforcement design is based on the total load on the strip and therefore total equilibrium, not point equilibrium, is maintained. The strip approach is also very suitable for finite element method because the design moment envelop along a strip can be easily obtained by adding the nodal reactive moments of the elements that make up the strip. It is important to note that when a strip is large, it would inevitably include reactive moments of different signs. Theoretically, the reactive moments of different sign should be integrated separately because they produce top and bottom reinforcements and should not cancel out each other.
Summary The shear flows and therefore failure modes for beam torsion and plate twisting are very different and therefore should be reinforced differently. The Lagrange’s equation for plate shows that m x, my and mxy are coupled and therefore, according to the Lower Bound Theory, allows the apportioning of loads carry by the m xy term to the mx and mx terms. For slab, this is very natural and can be easily achieved by increasing the orthogonal m x and mx reinforcement. Many conventional reinforced plate design methods, based on the Lower Bound Theory, had been formulated to find alternative m x and my moment design fields that satisfy the following equations: 2 ∂ 2 m x ∂ 2 m x ∂ m y ≥ − q or + ≥ −q ∂ x 2 ∂ x 2 ∂y 2
However, these conventional methods are limited to simple structural forms and loads. Fortunately, with high-speed computer, the finite element method can solve complicated plate design problem and produce two sets of equilibrium moment design fields: 1. Wood-Armer Method. 2. Nodal Reactive Moments. These methods would produce different but equally valid reinforcement layouts. However, the Wood-Armer Method would result in more reinforcement because it is based on BOTH the prevention of yielding in all directions AND the satisfaction of equilibrium whereas the Nodal Reactive Method is based on the satisfaction of equilibrium only. Finally, the use of strip method with separate integration of moments with different signs is recommended over section cut method and point method because it achieve total equilibrium, theoretical correctness and computational efficiency.
