Aalto University School of Engineering

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1 Aalto University School of Engineering Kul-4.4 Ship Structural Design (P) ecture 6 - Response of Web-frames, Girders and Grillages

2 Kul-4.4 Ship Structures Response ecture 5: Tertiary Response: Bending of plates and stiffeners, σ 3 Design Philosophy oads ectures ecture 6: Bending of web frames, girders and grillages, σ Response ectures Strength ectures ecture 7: Hull girder bending, torsion and shear, σ ecture 8: Ship vibrations, σ σ 3

3 Contents The aim is to understand how web-frames and girders are designed and the basic differences to the bending of stiffeners Motivation Concept of effective breadth Shear deformation in beams Finite Element Modelling D-modelling of a web frame Stiffness Interaction wtih primary members 3D-modelling of a web frame oad assembly Extent of the model Classification society approach iterature Hughes, O.F.; Ship Structural Design. SNAME, 988. Odqvist, Hållfasthetslära. Taggart: Ship Design and Construction Ch. VI. Schade, H.A., Effective Breadth of Stiffened Plating under Bending oads, SNAME, 95. DNV, Strength Analysis of hull Structures in Tankers. Class. Notes No 3.3. January 999. Hakala, M., ujuusopin elementtimenetelmä. Otakustantamo, 457. Kujanpää, J., Ropax-aluksen 3D-rakennemalli alkusuunnittelussa. Diplomityö, TKK, Konetekniikan osasto,. Ylinen, A, Kimmo- ja lujuusoppi.

4 Weekly Exercise Exercise 6: Web-frames and Grillage - Given..5 9:, Return 6..5 Use classification society rules to design all webframes and girders and give these in table and main frame Calculate the area mass of steel structure Report and discuss the work. Deck oads Rule DNV, Part x, Sec y, Pressure [kpa] Web-frames b e [mm] Profile 5 5 T45x7/ at5x5. Stress in Girder [MPa] 6

5 Motivation Transversal web frames Tie primary structural elements together Transfer the loads from stiffeners to primary elements Girders in longitudinal direction are similar to web frames in response terms Web frame spacing is several frame spacings (e.g. 3-4) Both horizontal and vertical loads have to be checked Usually the analysis is carried out using D-FEM with springs modelling the influence of primary members The buckling strength of different parts of the web frame has to be checked Pillar lines are used to reduce the span of the web frames The web frame deflection may become critical especially in case of hight strength steels Things to be considered in comparison to stiffeners Effective breadth Shear deflections

Effective Breadth Concept The deck plating is not full effective when the spacing of web frames or girders is large Caused by shear lag Should not be mixed with effective width due to buckling and

6 Effective Breadth Concept The deck plating is not full effective when the spacing of web frames or girders is large Caused by shear lag Should not be mixed with effective width due to buckling and shear lag due to large deflection of plates We would still like to use beam theory to calculate the web frame response σ x,, b e Effective breadth can be used to evaluate the flange for the girder Equal area σ x (y), b Stress at the intersection between beam and plate Equal areas for real and idealized stress distributions b σ e x, b = σ xdy be = b σ dy σ x x, b b e

7 Effective Breadth We consider a membrane-type of plate (bending stiffness neglible) attached to the beam (e.g. T-beam) Beam carries the vertical load q Only membrane stresses at the plate are considered b n y t When this assembly bends, the bending stiffness is larger than that of beam alone Smaller than that considering the full web frame spacing The combined effect is called effective breadth and it must be accounted in the analysis since is affects The deflection The normal and shear stress y, v z, w pp z pp x, u N Q M e e τ xy t dx q dx b n τxyt dx = σ dy σ x x τ xy = σ x σ dy x νσ y M + dm N + dn dx Q + dq

8 Determination of Effective Breadth The load on the beam (Fourier-series): q(x) = q cos π l x The stresses at beam/plate-intersection σ X = 3 + œν ν (π l ) C cos π l x σ Y = œ( π l ) C cos π l x σ x -integral becomes: k.36 kuorma q M-jakautuma k.55.9 P M-jakaut. σ X dy= œν π l cos π x l kuorma q P and the effective breadth: b n = l 4 π 3 + ν +ν ( œν) = l 4 π (3 œν) ( +ν) So it does not depend on x-coordinate (it can in general!). For steel this simplifies (ν =.3) to b n = k l =.36 l k.38.9 M-jakautuma k.5.95 M-jakaut. The effective breadth depends on load and boundary conditions. It is not necessarily constant along span. Rule of thumb: the more concentrtated the load, the less is the effective breadth.

9 Stiffened Deck The web frame spacing is typically,. m m b/l-ratio is always less than, There are several responses interacting and l/b-ratio has large impact to value of effective breadth Simply supported at mid-span b e b =+ 5! b # $ 3" l % & Clamped at mid-span b e b =+ 5! # b $ " l % & b b n b n σ X = b σ X dy b

10 Simplified Approach For example DNV The effective breadth is evaluated with use of: where b e = C b. b is web frame spacing, C factor that depends on load and boundary conditions C is defined with, a distance between zero values at bending moment diagram r is number of point loads in the web frame C,9,8,7,6,5,4,3,, r = 5 r Š 3 r = a/b

11 Section Modulus Determination with Design Curves I-profile, DNV

12 Analysis in Practise Boundary Conditions and oading r = Number of Point oads Across Span a = distance between zero bending moment b = web frame spacing! Appropriate BC s based on symmetry of loading and pillars!

13 Example of Section Modulus Calculation Same as in previous lecture - Modify for the T-beam - Include effective breadth

14 Shear Deflection Shear Stress in Beam Web-frame can have small /h which means that shea deformation can be significant The external load (F, q) on the beam causes shear force that is equal to the shear stress (τ) intgerated over the shear area (A) Q = τda which was excluded in the previous derivations This can cause additional deflection on beams Called shear deflection Significant in beam with low /h (<) Significant in composite beams z l x σ τ F

15 Shear Stress The stress resultant R due σ at left end is z z M M R = x da = da = da = I I σ η η z z Y Y z z where S is the static moment of shaded area: M I Y S z S = ηda z on the right end the stress resultant is z M + dm R + dr= ηda = I z so the difference is dm dr= I Y S Y M + dm I Y S R M z σ τ Q dx τ R+dR Q + dq x M + dm z z da b z z CG η y

16 Shear Stress This has to be in balance with the shear stress τ integrated over area bdx which gives: S τbdx = dm IY Taking into account the relation between bending moment and shear force gives dm = Q dx So the shear stress is: τ = QS I b Y The shear flow is q = τt = QS I Y

17 Example Symmetric I-profile The maximum of shear stress is calculated from QS τ = IYb The moment of inertia is: I = A Awh f h f + 3 and the static moment: Awh S = Af h f + 4 and the breadth of web b b = A w h

18 Example Symmetric I-profile Assuming that h h f gives shear stress at neutral axis as: 3 Q + 4λ τ max = + 6λ A w The ratio between maximum and average shear stress τ max 3 + 4λ µ = = Q / A w + 6λ which shows that when flange is zero λ = niin µ =,5 so we have case as in solid rectangular section λ infinite so µ=, which means that the shear stress is constant in infinitely thin web. µ-ratio as function of λ-ratio,5,5 µ,75,5, λ λ = A A f w

19 Shear Deflection The shear stress causes additional deflection, point load dv s µ Q = dx GA Distributed load d v s µ q = dx GA So the total deflection is d v dx d dx v d dx v M = EI q GA tot = b s µ + Slide 9

20 Shear Deflection Slide

21 Shear Deflection Effective web of Girder Rules for Ships, January 4 Pt.3 Ch. Sec.3 Page 4 Amended, see Pt. Ch. Sec.3, July 5 the cross-section considered = hn + hn.. If an opening is located at a distance less than hw/3 from the cross-section considered, hn shall be taken as the smaller of the net height and the net distance through the opening. See Fig.. b k.6 b.4 b. hn 3 σ rtf Fig. Effective width of curved face plates for alternative boundary conditions hn a<hw/3 47 The effective flange area of curved face plates supported by radial brackets or of cylindrical longitudinally stiffened shells is given by: hn 3 r t f + ks r A e = t f b f 3 r tf + sr ls hw tw (mm ) k, bf, r, tf is as given in 47, see also Fig.3. sr Fig. 4 Effective web area in way of openings = spacing of radial ribs or stiffeners (mm). 54 Where the girder flange is not perpendicular to the considered cross section in the girder, the effective web area shall be taken as: AW =. hn tw +.3 AFl sin θ sin θ (cm) bf r hn = as given in 53 AF l = flange area in cm = angle of slope of continuous flange tw = web thickness in mm. 5r θ See also Fig.5. θ AFI Fig. 3 Curved shell panel C 5 Effective web of girders 5 The web area of a girder shall be taken in accordance with particulars as given below. Structural modelling in connection with direct stress analysis shall be based on the same particulars when applicable. 5 Holes in girders will generally be accepted provided the shear stress level is acceptable and the buckling strength is sufficient. Holes shall be kept well clear of end of brackets and locations where shear stresses are high. For buckling control, see Sec.3 B3. 53 For ordinary girder cross-sections the effective web area shall be taken as: AW =. hn tw (cm) hn = net girder height in mm after deduction of cut-outs in hn tw Fig. 5 Effective web area in way of brackets C 6 Stiffening of girders. 6 In general girders shall be provided with tripping brackets and web stiffeners to obtain adequate lateral and web panel stability. The requirements given below are providing for an DET NORSKE VERITAS Slide

22 Finite Element Analysis of Web Frames and Girders D The response at σ level is carried out efficiently with D FEM with beams Beams can be used if the /h-ratio is larger than 5, so that local effects do not dominate The effective breadth and shear stiffness need to be considered Note! shear locking might occur in slender beams The influence of primary strength member stiffness is modelled with springs

23 Finite Element Analysis of Web Frames and Girders D/3D-Beam Element In case of plane analysis (D) the degrees of freedom (DOF) are Displacement u (in-plane) and v (deflection) The rotation θ For two noded element the DOF s are { U} = { u } T, v, θ, u, v, θ and corresponding forces { } { } T F = Fx, Fy, M z, Fx, Fy, M z [ k] The relation is simply {F}=[k]{u} y v u x F x u EA = EA F y θ v M z F y E, A, I, θ, EI 3 6EI EI 3 6EI 6EI 4EI 6EI EI EA EA 3 6EI v EI 3 6EI EI M z F x u 6EI EI 6EI 4EI

24 Finite Element Analysis of Web Frames and Girders D/3D-Beam Element The local {U } and global {U} displacement are related by transformation matrix [T] by {U } = [T] {U} The coordinate transformation is gives as [ T ] = cosϕ sin ϕ sin ϕ cosϕ cosϕ sin ϕ The stiffness matrix is obtained from similar operation [k] = [T] T [k ] [T] sin ϕ cosϕ y Global x, y and local x, y coordinate system x ' y ' ϕ x

25 Finite Element Analysis of Web Frames and Girders D/3D-Beam Element When the web frame dimensions are such that at the nodal region stiffness is very high, beam element with infinite rigidity is used End lengths r and r, For example area of brackets, In local coordinate system the stiffness is [k] = [S] T [k*] [S] where [k*] is obtained by subsituting real lenght with elastic length l. The displacements are obtained by multiplication with: [ ] = r r S

26 3D-Finite Element Model

27 Web-Frame Analysis of RoPax D-analysis (Kujanpaa, ) aiva suorassa aiva kallistuneena Z

28 Stresses (Kujanpää, ) NormaalijŠnnitys (σ Ny ) eikkausjšnnitys (τ Qz ) JŠnnitystaso

29 Grillage Analysis of Passenger Ship

30 3D FEM Global 3D FE-model, which gives displacements Entire model Half model, Stresses with sub model 3D model is created from 3D-structural model, e.g. NAPA Steel oad Class rules Motion calculation basic response Real part Imaginary part

31 Bulk Carrier

32 Summary Web-frames and girders are analyzed using the same beam theory as longitudinal and transversals, with extensions to account Shear deflections Upper flange is defined by effective breadth Often the stresses are excessive unless spans are reduced by Bulkheads Pillars etc Alternative is to increase the web-frame height and thus the deck spacing

Longitudinal strength standard

(1989) (Rev. 1 199) (Rev. Nov. 001) Longitudinal strength standard.1 Application This requirement applies only to steel ships of length 90 m and greater in unrestricted service. For ships having one or