Frame & Subframe Analysis for Vertical and Lateral Load

INTRODUCTION TO FRAME & SUBFRAME ANAL ANA LYSIS FOR VERTICAL VERTICAL & LATERAL LOAD

Continuous Beam Design & Detailing  –  Frame analysis

LEARNING OUTCOME : By completing this chapter, students shall be able to: 1.0

classify braced and un-braced frame.

2.0 2.0

awar aware e on the the anal analys ysis is of brac braced ed conc concre rete te fram framed ed building for vertical load.

3.0

anal analy yze un-br n-brac ace ed concre ncrette fram framed ed bui buildi lding fo for wind load by cantilever method.

Continuous Beam Design & Detailing  –  Frame analysis

Introduction to Frame Structure Beam (300x600mm)





Building frames often consist of girders/beams that are rigidly connected to columns.

Slab 200mm, 250mm & 300mm thickness

Varies column size (500x500mm, 750x750mm, 900x900mm)

Frames are classified as braced or u nnbraced against sidesway. Figure 1: 3D Modeling of

Continuous Beam Design & Detailing  –  Frame analysis

Un-braced frames Sway frames/Sideways uninhibited.  Framed need to be designed to resist the lateral loads, since there is no bracing elements in the building.  Un-braced frame need to carry both vertical & lateral loads. 

Figure b: Rigid Frame/ Un-braced Frame

UNBRACED FRAMES Vertical loading

Lateral/Horizontal loading

Column (vertical element) Beam (horizontal element)

Continuous Beam Design & Detailing  –  Frame analysis

Braced

Braced frames 

Non-sway frames/Sideways inhibited.  No lateral/horizontal loads (eg. wind load) being transferred either to columns or beams.  Diagonal bracing, shear-walls, masonry infill walls, lift/elevator shafts and staircases core provide lateral stability to the structural frame of the building.  Braced frame could take only vertical loads. .

Figure a: Braced Frame

BRACED FRAMES Vertical load

Bracing type X Column (vertical element)

Beam (horizontal element)

Continuous Beam Design & Detailing  –  Frame analysis

Infilled with brickworks, blockwork or

precast panels. It’s subjected to lateral load the infill behaves as strut along its compression diagonal to brace the frame. Forces and stresses in the infill and frame. procedure – design of infill, design of Design procedure – frame, horizontal deflection

Continuous Beam Design & Detailing  –  Frame analysis

Shear wall structures are entirely

lateral load resistant and much stiffer horizontally horizontally than rigid frame. Restrict openings in walls. Shear wall structure performs well in earthquake because ductility is important in design of shear wall.

Figure d: Shear Wall Structure

Continuous Beam Design & Detailing  –  Frame analysis

Combination of shear walls and rigid frames. Usually use in reinforced concrete structure,

however use of steel in bracing and rigid frames offer advantage in horizontal interaction.

Figure e: Wall-frame Structure

Continuous Beam Design & Detailing  –  Frame analysis

Lateral resistance provided by very

stiff moment resisting frames that form tube around the perimeter of the building. Consists of very closely spaced columns (2-4 m apart) joined by deep spandrel girders. Suitable in reinforced concrete or steel construction with building from 40-100 stories high. Repetitive frame pattern, grid like façade – façade – relatively efficient, easy construction.

Figure g: Frame Frame - tube

Continuous Beam Design & Detailing  –  Frame analysis

Figure d: Hearst Tower, New York City

Figure a: Empire State Building

Figure b: Citicorp Building, Figure c: Mercantile Building, St Louis New York City

Figure 4(a-d): Braced Frame Examples

Continuous Beam Design & Detailing  –  Frame analysis

Lateral Stability in Frame Structure

Figure 5: Elements in Frame Structure

Figure 6: Frame structure provides lateral stability (Plan View)

 A

B

C

1

2

column un-braced in both direction

3

4

 A

B

C

SHEAR WALL 1

2

3

Column braced in y direction and unbraced unbraced in x-direction

4

 A

B

C

SHEAR WALL 1

2

column braced in both direction

3

4

Continuous Beam Design & Detailing  –  Frame analysis

Approximate Method of Frame Analysis Analysis

Figure 8b: 8b: Plane frame for analysis

Figure 8a: 8a: Actual 5-storeys Frame Structure

Continuous Beam Design & Detailing  –  Frame analysis

Approximate Method of Frame Analysis Analysis A. Monolithic Braced Frames Not Providing Lateral Stability

(1) Simplificati Simplification on into sub-frame The structural frame is divided into sub-frames sub-fra mes consisting of beams at one level and columns above and below that level with ends taken as fixed (or pinned).

 A

B

C

D

Continuous Beam Design & Detailing  –  Frame analysis

A. Monolithic Braced Frames

–   con’t

(2) Simplificati Simplification on for individual beams & columns  The simplified sub-frame consists of the beam to be designed, the columns attached to the the ends ends of the beam and the beams on either side if any. The column and beam ends remote r emote from the beam considered are taken as fixed and the stiffness of the beams on either side should be taken as half their actual size. The moments for design for an individual column may be found from the same sub-frame analysis provided that its central beam is the longer of the two beams framing into the t he column.

: Continuous Beam Design & Detailing  –  Frame analysis

A. Monolithic Braced Frames

–   con’t

(2) Simplification for individual beams & columns  K

K/2

Beam A-B 

K/2

K

K/2

B

Beam C-D 

K

K/2

 A

Beam B-C 

C

D

Continuous Beam Design & Detailing  –  Frame analysis

A. Monolithic Braced Frames

–   con’t

(3) Continuous Beam Simplification 

The beam at the floor considered may be taken as a continuous beam over supports providing no restraint to rotation. This gives more conservati conservative ve design design than the procedures procedures previously. previously. (4)Asymmetrically-loadedColumns 

This method is to be used us ed where the beam has been analyzed. The column moments can be calculated on the assumption: assumpti on: - column column and and beam beam ends ends remote remote from from the junc junctio tionn under under consideration are fixed. - that that the the beams beams have have half half their their actual actual stiffn stiffness ess..

Continuous Beam Design & Detailing  –  Frame analysis

A. Monolithic Braced Frames

–   con’t

(3) Continuous Beam Simplification & Asymmetrically-loaded Columns

 A

B

D

C

K/2

A

K/2

K/2

B&C 

K/2

Continuous Beam Design & Detailing  –  Frame analysis

A. Monolithic Braced Frames

–   con’t

Choice of Critical Loading Arrangement  It will normally be sufficient to consider con sider the following arrangement of vertical load: (a) (a) All All span span load loaded ed wit with h the the maximum design ultimate load (1.35Gk + 1.5Qk). (a) Altern Alternati ative ve spans spans loade loaded d with with the the maximum design ultimate load (1.35Gk + 1.5Qk) and all other spans loaded with the minimum design ultimate load (1.0Gk). max

max

max

max

min

max

min

max

min

(a)

(b)

3 cases of load arrangement in Frame analysis for vertical load

Continuous Beam Design & Detailing  –  Frame analysis

Approximate Method of Frame Analysis Analysis B. Rigid UnUn-braced Frames Providing Lateral Stability Where rigid frames provides lateral stability, they must be analyzed for horizontal & vertical loads. As an alternative to the complete 3D frame structure analysis, the Code gives the following method for sway frames of three or more approximately equal bays.

Gravity Load

W i   n  d  L   o  a  d 

Continuous Beam Design & Detailing  –  Frame analysis

Approximate Method of Frame Analysis Analysis B. Rigid UnUn-braced Frames The design is based on the more severe of the conditions: 1. elastic analysis or vertical loads only with maximum design load 1.35Gk + 1.5Qk 2. or the sum of the moments obtained from: (a) elastic analysis of subframes as define in section with all the beams loaded with 1.2 1.2Gk Gk + 1.2 Qk Qk (Ho (Horiz rizont ontal al loads loads are are ignored). (b) elastic analysis analysis of the complete frame assuming points of contra flexure at the centers of all beams and columns for wind load 1.2Wk only. only.

Continuous Beam Design & Detailing  –  Frame analysis

Un--braced Frame Un Analysis by

Cantilever Method

Continuous Beam Design & Detailing  –  Frame analysis

Lateral Loads on Building Frames: Cantilever Method Introduction The cantilever method is based on the same action as long cantilever beam subjected to a transverse load. 2. The lateral loads on frame tend to tip up the frame over, or cause a rotation of the frame about neutral axis. axis. 1.

(Note: the neutral axis is in horizontal plane that that passes through through the columns at each floor level.) 

3. To coun countera teract ct this this tippin tipping, g, the the axia axiall force forcess or stress stress in the columns will be TENSILE on one side and COMPRESSIVE on the other side of the neutral axis. 4. The cantil cantileve everr metho method d is theref therefore ore appro appropri priate ate if if the the frame frame is tall & slender, slender, or has columns with different cross-sectional area.

Continuous Beam Design & Detailing  –  Frame analysis

Lateral Loads on Building Frames: Cantilever Method Z

Assumption in Cantilever Method for Fixed-Supported Frame 1.

T   e n  s  i     l      e

 C   o m   p r    e  s   s  i     v   e

A hinge is placed at the center of each girders & columns (zero moment). Z

cont

Note: o hinge (zero M)  c   o n  t   

2. Neutral axis

The axi The axiaal str stre ess in a col colum umnn is is proportional to its distance from the centroid of the cross-sectional areas of the columns at a given floor level.

Continuous Beam Design & Detailing  –  Frame analysis

Lateral Loads on Building Frames: Cantilever Method Procedures in Frame Analysis by Cantilever Method: 1.

Place a hinge at the center of each girders & columns .

2.

Determine the centroid of the columns’ cross sectional areas by :  _   x 

 _   x A

   A

3.

Analyz Analyze e a sectio sectionn through through the hinge hinge at the top story. story. Determ Determine ine the axial force in each column . (Tips: the force is proportional to its distance from the centroid of the columns’ crosssectional areas.)

4. 5. 6.

Determine the remaining hinge forces (V&N). Repe Repeat at step step 3-4 3-4 for for lower lower leve levell of of the the fram frame. e. Draw Dr aw the bend bendin ingg mom momen entt dia diagra gram m of of the the fram frame. e.

Continuous Beam Design & Detailing  –  Frame analysis

Example 1: Determine (approximately) the reactions at the base of the columns of the frame shown shown in Figure below. The columns are assumed to have equal cross-sectional areas. Use the cantilever method of analysis. C

F

30kN 4m B

E

15kN

4m  A

D

6m

: Serviceabil Serviceability ity & Durabilit Durability y Requirements

Example 1 (cont) (i). Placed hinge hinge at the midpoint of columns and girders & determine the centroid of the columns’ cross sectional areas. 30kN

C

F I H

15kN

K

B

4m Centroid, x =3m

E J G

L

 A

6m

4m

D

6m

Plan View  _ 

 _ 

 x 

 x A  0( A)  6( A)  3m 2A  A

Continuous Beam Design & Detailing  –  Frame analysis

Example 1 (cont) (ii). Analyze level at top storey. 3m

C

30kN

F I

3m

H

K

B

15kN

E

2m

O

J G

3P = 10kN

L

 A

3P = 10kN

6m

D

a. Determine P +



 M o

 0   3P(3)  3P(3)  30( 2)  0  P  

b. Determine remaining hinge forces

Hx

3

10kN

Iy

 F y  0 I y  10kN  +

10

15kN

Ix



 M I  0   10(3)  2H x  0  H x  15kN 

 F

 0 I  15kN 

4m

Kx

 F   0  K   x

 x

 15kN 

4m

Continuous Beam Design & Detailing  –  Frame analysis

Example 1 (cont) 30kN

(ii). Analyze level BE. 3m

C

F I

3m

H 15kN

K

B

E J

4m

G

O 3P = 35kN

4m

L

 A

2m

6m

4m

D

3P = 35kN

a. Determine P +



 M o

 0  3 P (3)  3 P (3)  30(6)  15(2)  0  P  

35 3

b. Determine remaining hinge forces 10kN

15kN Jy +

Jx

Gx

 F  y  0  J  y  25kN   M  B  0  15(2)  G x (2)  25(3)  0  G x  22.5kN 

 F

 x

 0  J x  7.5kN 

Continuous Beam Design & Detailing  –  Frame analysis

Example 1 (cont) 30kN

b. Determine remaining hinge forces (cont)

C

F I

10kN

H

15kN

15kN

25kN

K

B

4m

E J G

L

 A

7.5kN

6m

Lx

 F 

 x

 0   L x  22.5kN 

35kN

(iii). Analyze support level. 35kN

35kN 22.5kN

22.5kN

22.5kN 22.5kN

45kNm 35kN

45kNm 35kN

D

4m

Continuous Beam Design & Detailing  –  Frame analysis

Example 1 (cont) (iv). Shear (kN) in Beams and Columns 10

15 15

4m

25

22.5 22.5

6m

4m

To calculate moment at: Beam , M = F x beam span x 0.5 Column, M = H x storey height x 0.5 Examples: Beam , M = 10KN x 6 m x 0.5 = 30KN Column, M = 22.5KN x 4m x0.5 = 45KN

Continuous Beam Design & Detailing  –  Frame analysis

Example 1 (cont) (iv). BMD (kNm) (kNm) in Building Frame Frame that subjected to lateral load 30

10

30

30 30

15

4m

15 75 25

30

45

45

30 75

22.5

22.5 45

6m

45

4m

Continuous Beam Design & Detailing  –  Frame analysis

Exercise 1 I

10kN

J

K

L

4m E

G

F

20kN

H

4m  A

4m  Area Column:

 A=24(10-3)m2

C

B

5m  A=16(10-3)m2

D

4m  A=16(10-3)m2

 A=24(10-3)m2

Problem: Draw the moment diagram for girder IJKL of the building frame. Use the cantilever method of of analysis. Each column has the cross-sectional area indicated.

EXERCISE 2 The concrete building frame shown in Figure below is subjected to an ultimate uniformly distributed horizontal load of 20 kN/m. All columns have typical dimensions, each of 250mm x 200mm. By using cantilever method, calculate the bending moment in beam BC. A

B

C

D 3.5 m

E

F

G

H 20 kN/m

I

L

K 

J

3.5 m

4.0 m

N

M

3..2 m

O

4m

P

7..5 m