frames

Structural Engineering DesignReinforced Concrete Framed Buildings

Continuity + FramesContinuity + Frames

Eur Ing Prof Costas Georgopoulos CEng FICE FHEA FCS FIStructE

Continuous beamsGeneral ConsiderationsSagging moments at the spans i.e. beam and slab

act together - designed as a T-beam.Hogging moments at the supports i.e. beam is

always designed as a rectangular section.At an exterior column the beam reinforcing bars At an exterior column the beam reinforcing bars

must anchor within the column. Top slab reinforcement must pass over the beam

reinforcement and still have a cover.Moment of resistance of a T-beam is greater than

that of a rectangular beam i.e. moment redistribution is advantageous

Methods of AnalysisElastic Analysis Moment Distribution

Stiffness Method Computer Programs using

the Stiffness Method (e.g. QSE)(e.g. QSE)

Frame or sub-frame analysis

Design Coefficients From EC2 Table 15.3

Methods of Analysis

Arrangement of Bending Reinforcement

Arrangement of Shear Reinforcement

Moment Redistribution

Stress/strain relationship for an ideally elastic-plastic material

Moment/curvature relationship for an ideally elastic-plastic beam


Elastic

Moment

at C

Elastic Bending Moment Diagram

at C


By increasing Q the moment at C reaches MpFurther increase of Q will not increase Mp at CThe moments at B and D reach MpThe structure becomes a mechanism and collapses


A B C D E

MB = Qul/4 - MC/2 but MC = MB = MPMP = Qul/4 - MP/2 therefore MC = MP = Qul/6 and if it had remained elastic MC = 3Qul/16 and the ratiob= (Qul/6) / (3Qul/16) = 0.889

Moment RedistributionThe designer has a number of choices i.e. provide a

beam of uniform MP or one with 0.5MP at C and 1.25MP at B and D or any other combination

For the above the moment redistribution ratio is:b= (Qul/12) / (3Qul/16) = 0.444

Moment RedistributionFor a brittle/elastic materialSudden rapture occurs at C

when Mu =3Qul/16 is reachedThe beam splits into two The beam splits into two

simply supported beams.The moments at B and D are

equal to Qul/4 i.e. greater than Mu of the beam, therefore

The whole structure collapses without warning


An under-reinforced beam would develop adequate ductility in plastic hinge regions and therefore plastic regions and therefore plastic design could be used.

The precise shape of M against 1/r depends on the type of reinforcement, steel ratio and for doubly reinforced beams on ( - )

Moment Redistribution (example)

Moment RedistributionWhy not more than 30%

Region ab is under a sagging moment at ULS but under a hogging moment at SLS

In other words, at ULS no top reinforcement is required, and therefore cracks would develop there at SLS.

To guard against that Mu 70% of Me

Classification of slabsOne-way spanningbetween beams or walls

ly / lx > 2

Two-way spanningTwo-way spanningbetween beams or walls ly / lx 2

Flat slabs on columnsand edge beams or wallswith no interior beams

ly / lx 2

Methods of AnalysisElastic Analysis Strips spanning one way or

a grid with strips spanning both ways.

Elastic plate theoryFinite Element Analysis Finite Element Analysis

Design Coefficients

Plastic Analysis

From EC2 (obtained from yield line analysis).

Yield line analysis (Johansen) Strip method (Hillerborg)

One-way spanning slabs

Continuous one-way spanning solid slabs using EC2 design coefficients

Continuous one-way spanning ribbedslabs using EC2 design coefficients

Design as a tee-beam with effective width (at the span) equal to the actual distance between the ribs.

Two-way spanning solid slabsusing Table 8 of TCC How2 slabs

lx

msy = synlx2

n=1.35Gk+1.5Qkmsx = sxnlx2

lx

The bending moments apply to the 1m wide middle strips only

Stair slabs - Building Regulations

Transverse Stair slabs

Longitudinal Stair slabs

Openings through slabs(Hillerborg Strip Method)

Slab Arrangement

Slab Arrangement - 2way spanning

Slab Arrangement - 1way spanning

Slab Arrangement - flat

RC Frames

Horizontal Load PathsFrame AnalysisFrame AnalysisShear WallsRobustness and Design of Ties

Horizontal Load Paths in Structures

Concept of relative stiffness.

The column axial stiffness k2 is much greater than the k2 is much greater than the beam bending stiffness k1.

Therefore the column will generally carry the greater portion of the load (in fact over 90% of P in most practical cases).


Concept of relative stiffness.The horizontal thrust is transferred to the

foundation principally by deep beam action in the shear wall rather than bending in the portal frame.

Horizontal Load Paths in StructuresTwo-storey skeletal

r.c.frame with rigidly connected members and pinned supports.

The horizontal force F1 applied at I is principally transmitted to the foundations by a frame action in each of the three plane frames.


The largest proportion of the force F1 is transferred directly through the central through the central frame GHIJKL.

The central frame therefore deforms more than the outer frames.


The slab effectively acts as a deep beam and forces each frame to deform by the same to deform by the same amount.

The applied load is distributed in equal proportions to the three plane frames.


The deformation of the entire structure is uniform and is uniform and is significantly smaller than the deformations of the skeletal frame without the slab.


The two external frames are replaced by concrete or masonry panels.panels.

Most of the load is transferred by the slab (deep beam) to the stiffer external panels (cantilever shear walls).

No frame action.


R.C. elevator core.Loads are applied to

floor slab levels.Greater portion of Greater portion of

forces F1 and F2 are carried by the stiffer solid member ABCD (cantilever shear wall)

Symmetry can prevent torsion problems.

Analysis of Frames

Lateral stability in E-W direction is provided by shear walls (non-sway frames).

Lateral stability in N-S direction is achieved through frame action (sway frames)

Analysis of Non-sway Frames

Analyse by computer, orSimplify by dividing into a set of sub-frames, orAssume beams are continuous over columns.

Analysis of Non-sway FramesSub-frame method

Sub-frames can be analysed by hand using moment distribution method.

Analysis of Non-sway FramesContinuous beam method

Continuous beams on simple supports.Moments to columns using models as above.

Analysis of Non-sway FramesSub-frame versus continuous beam

Max sagging moment for FJ more conservative using continuous beam method.

Analysis of Sway Frames

For vertical loads same as non-sway framesFor horizontal loads assume p.o.c. at the c.o.m.

Analysis of Sway FramesExample

Walls in RC Buildings

1 Internal non-load bearing partitions

Framed (1,2,3)

Walls Buildings

bearing partitions2 External curtain walls3 Stability internal or

external walls4 Load bearing walls for

both vertical and horizontal loads

Shear walls - no frames (1,2,4)

Combined (1,2,4)

Stocky Braced Shear Walls

1 Mainly Axial or Transverse Moment

Column Charts from EC2

Load Design

Transverse Moment and Uniform Axial

3 In-plane Moments and Axial Load

4 Axial Load, Transverse and In-plane Moments

EC2

Interaction Chart (or elastic stress distribution or end zones - moments)

3 Stages (in-plane, transverse, combined)

Stocky Braced Shear Walls

Based on the assumptions for the design of beamsdesign of beams

Straight wall with uniform reinforcement

Uniform steel distribution running the full length of the wall

Robustness and Design of Ties

Robustness and Design of Ties

In accordance with the Approved Document A of Document A of Building Regs vertical ties in columns and walls should be provided in all buildings that fall into Class 2B and 3

frames

Documents

beam reinforcement

rectangular beam

beam of uniform mp

moment redistributionfor

moment of resistance

moment redistribution

reachedthe beam splits

plastic regions