<content> PROGRAM 1: STIFFNESS METHOD: To obtain Moments of Beams, Uppe Columns & Lower Columns </content>

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Sub Frame Analysis - Stiffness Method

Example 1 - Page 25
Worked Examples for the Design of Concrete Structures to Eurocode 2
By TONY THRELFALL - CRC Press - Taylor & Francis Group

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t.0 0 time :

Define kNm kN m * :

Assume Young's Modulus for RC: This is not really required, but used only for consistent units in intermediate calculations. We can use any numerical value. Ex E = 1 MPa OR E = 1

Assume E 40 MPa * :

Joint Moment Sign Convention
Clockwise + ve
Anticlockwise - ve

<content> PROGRAM 1: STIFFNESS METHOD: To obtain Moments of Beams, Uppe Columns & Lower Columns </content>

Program 1: Support / Column Moments for All Cases Using Stiffness Method

For All 3 CASES

ff LL k BMM S.BEAM 422422 mat : FF1 FF 1 el : FF2 FF 2 el : FF3 FF 3 el : 31 mat fem 112 / FF1 LL 2^* vectorize FF2 LL 2^* vectorize FF3 LL 2^* vectorize 13 mat transpose * : j 1 fem rows range : MM j el fem j el 1 el - fem j el 1 el fem j el 2 el - fem j el 2 el fem j el 3 el - fem j el 3 el 41 mat : ST j el k 1 -^MM j el * - : M1 j el fem j el 1 el - fem j el 1 el 21 mat E I.beam * LL 1 el / S.BEAM * ST j el 1 el ST j el 2 el 21 mat * (+ (: M2 j el fem j el 2 el - fem j el 2 el 21 mat E I.beam * LL 2 el / S.BEAM * ST j el 2 el ST j el 3 el 21 mat * (+ (: M3 j el fem j el 3 el - fem j el 3 el 21 mat E I.beam * LL 3 el / S.BEAM * ST j el 3 el ST j el 4 el 21 mat * (+ (: -------------------------------- M.UC j el ST j el 4 E * I.col * H.UC / * : M.LC j el ST j el 4 E * I.col * H.LC / * : M1 M2 M3 M.UC M.LC augment 131 line :

f1 f2 L1 L2 L3 k BMM S.BEAM 422422 mat : fem 112 / f1 L1 2^* f1 L2 2^* f1 L3 2^* 31 mat f1 L1 2^* f2 L2 2^* f1 L3 2^* 31 mat f2 L1 2^* f1 L2 2^* f2 L3 2^* 31 mat 13 mat transpose * : j 1 fem rows range : MM j el fem j el 1 el - fem j el 1 el fem j el 2 el - fem j el 2 el fem j el 3 el - fem j el 3 el 41 mat : ST j el k 1 -^MM j el * - : M1 j el fem j el 1 el - fem j el 1 el 21 mat E I.beam * L1 / S.BEAM * ST j el 1 el ST j el 2 el 21 mat * (+ (: M2 j el fem j el 2 el - fem j el 2 el 21 mat E I.beam * L2 / S.BEAM * ST j el 2 el ST j el 3 el 21 mat * (+ (: M3 j el fem j el 3 el - fem j el 3 el 21 mat E I.beam * L3 / S.BEAM * ST j el 3 el ST j el 4 el 21 mat * (+ (: -------------------------------- M.UC j el ST j el 4 E * I.col * H.UC / * : M.LC j el ST j el 4 E * I.col * H.LC / * : M1 M2 M3 M.UC M.LC augment 121 line :

Beam Stiffness

FEM

Joint Moments

Rotations at Joints

Beam (BF) Support Moments at B & F

Beam 1 Support Moments at 1 & 2

Beam (FK) Support Moments at F & K

Beam 2 Support Moments at 2 & 3

Beam 3 Support Moments at 3 & 4

Upper Col Moments

Lower Col Moments

<content> PROGRAMS 2,3,4,5,6: To obtain SF, Max Span Moments, X_max </content>

Program to find SF, Max Span Moment and Distance to Max Span Moment

PROGRAM 2: For any Loading Case Find SF, Max Span BM X_max M# f L# Param j 1 f rows range W j el f j el L# j el * : V1 j el 0.5 W j el * M# j el sum L# j el / (- : V2 j el W j el V1 j el - : M.max_span j el V1 j el 2^2 f j el * / M# j el 1 el + : X_max j el V1 j el f j el / : 51 line for V1 V2 M.max_span X_max 14 mat 21 line :

W f L * ≡

V1 W 2 / M.L M.R + (L / + ≡

V2 W V1 - ≡

M.max_span V1 2^2 f * / M.L + ≡

X.max V1 f / ≡

PROGRAM 3: Calls PROGRAM 2 find SF, Max Span BM & X_max for ALL Loading Cases M# ff L# Find_All j 1 M# rows range M_case j el M# j 13 range el : B j el M_case j el ff j el L# Param : 21 line for M_case B 31 line :

PROGRAM 4: Extract All SFF res SF_All j 1 res rows range A j el res j el 112 range el : j 1 ≡ B A 1 el : B B A j el stack : if 21 line for B 21 line :

PROGRAM 5: Extract ALL Max Span BMM res Span_M_All j 1 res rows range A j el res j el 13 el : 11 line for A 21 line :

PROGRAM 6: Extract All X-max res X_Max_All j 1 res rows range A j el res j el 14 el : 11 line for A 21 line :

DATA

Column dimensions b.col 300 mm * : h.col 300 mm * : H.UC 3.5 m * : H.LC 3.5 m * : 41 line

Beam dimendions L.AB 6 m * : L.BC 7.2 m * : L.CD 6 m * : b.beam 300 mm * : h.beam 450 mm * : 51 line

Define L L.AB L.BC L.CD 31 mat :

Calculation of Moments of Inertia : All Columns have same sizes & All Beams have same sizes

I.col 112 / b.col * h.col (3^* : mm 4^6.75 108^*

I.beam 112 / b.beam * h.beam (3^* : mm 4^2.28 109^*

Max

Min

F1 67.5 kN m / * :

F2 35.8 kN m / * :

Define Load Patterns for All 3 Cases

FF F1 F1 F1 31 mat F1 F2 F1 31 mat F2 F1 F2 31 mat 13 mat : kN m / 67.5 67.5 67.5 31 mat 67.5 35.8 67.5 31 mat 35.8 67.5 35.8 31 mat 13 mat

1. Stifness Coeff due to unit rotation at A (Node 1)

K.11 4 E * I.col * H.UC / 4 E * I.col * H.LC / + 4 E * I.beam * L.AB / + : kNm 122.4643

K.21 2 E * I.beam * L.AB / : kNm 30.37

K.31 0 :

K.41 0 :

2. Stifness Coeff due to unit rotation at B (Node 2)

K.12 2 E * I.beam * L.AB / : kNm 30.37

K.22 4 E * I.col * H.UC / 4 E * I.col * H.LC / + 4 E * I.beam * L.AB / + 4 E * I.beam * L.BC / + : kNm 173.0893

K.32 2 E * I.beam * L.BC / : kNm 25.31

K.42 0 :

3. Stifness Coeff due to unit rotation at C (Node 3)

K.13 0 :

K.23 2 E * I.beam * L.BC / : kNm 25.31

K.33 4 E * I.col * H.UC / 4 E * I.col * H.LC / + 4 E * I.beam * L.BC / + 4 E * I.beam * L.AB / + : kNm 173.09

K.43 2 E * I.beam * L.CD / : kNm 30.37

4. Stifness Coeff due to unit rotation at D (Node 4)

K.14 0 :

K.24 0 :

K.34 2 E * I.beam * L.CD / :

K.44 4 E * I.col * H.UC / 4 E * I.col * H.LC / + 4 E * I.beam * L.CD / + : kNm 122.46

Global Stiffness Matrix

K K.11 K.12 K.13 K.14 K.21 K.22 K.23 K.24 K.31 K.32 K.33 K.34 K.41 K.42 K.43 K.44 44 mat : kNm 122.46 30.37 00 30.37 173.09 25.31 00 25.31 173.09 30.37 00 30.37 122.46 44 mat

Find Support / Column Moments for all 3 Cases. Using PROGRAM 1

UC
End

Beam

Call PROGRAM 1 M.all FF L K BMM :

AB
BA

BC
CB

CD
DC

Upper
Col

Lower
Col

M.all kNm 97.8 - 267.5 21 mat 284.6 - 284.6 21 mat 267.5 - 97.8 21 mat 48.9 8.6 8.6 - 48.9 - 41 mat 48.9 8.6 8.6 - 48.9 - 41 mat 112.8 - 215.5 21 mat 172.4 - 172.4 21 mat 215.5 - 112.8 21 mat 56.4 21.6 - 21.6 56.4 - 41 mat 56.4 21.6 - 21.6 56.4 - 41 mat 36.9 - 193.8 21 mat 263.2 - 263.2 21 mat 193.8 - 36.9 21 mat 18.5 34.7 34.7 - 18.5 - 41 mat 18.5 34.7 34.7 - 18.5 - 41 mat 35 mat

Case 1

Case 2

Case 3

0 time t.0 - 0.03 s *

Upper Column Moments: (from Column 4 of M all )

Lower Column Moments: (from Column 5 of M all )

M.UC_all M.all 1 M.all rows range 4 el : kNm 48.9 8.55 8.55 - 48.9 - 41 mat 56.38 21.58 - 21.58 56.38 - 41 mat 18.46 34.67 34.67 - 18.46 - 41 mat 31 mat

M.LC_all M.all 1 M.all rows range 5 el : kNm 48.9 8.55 8.55 - 48.9 - 41 mat 56.38 21.58 - 21.58 56.38 - 41 mat 18.46 34.67 34.67 - 18.46 - 41 mat 31 mat

1. Support Moments for Loading Cases

BEAM

AB
BA

BC
CB

CD
DC

M_Case1 M.all 113 range el : kNm 97.8 - 267.48 21 mat 284.58 - 284.58 21 mat 267.48 - 97.8 21 mat 13 mat

CASE 1

M_Case2 M.all 213 range el : kNm 112.75 - 215.51 21 mat 172.36 - 172.36 21 mat 215.51 - 112.75 21 mat 13 mat

CASE 2

M_Case3 M.all 313 range el : kNm 36.93 - 193.83 21 mat 263.16 - 263.16 21 mat 193.83 - 36.93 21 mat 13 mat

CASE 3

2. All SF, Max Span BM and X_max Cases using PROGRAM 3

Res_All M.all FF L Find_All : 1.74 105^* kg m * s 2^/ * 2.43 105^* kg m * s 2^/ * 2.31 105^* kg m * s 2^/ * 31 mat 2.31 105^* kg m * s 2^/ * 2.43 105^* kg m * s 2^/ * 1.74 105^* kg m * s 2^/ * 31 mat 1.27 105^* kg m 2^* s 2^/ * 1.53 105^* kg m 2^* s 2^/ * 1.27 105^* kg m 2^* s 2^/ * 31 mat 2.58 m * 3.6 m * 3.42 m * 31 mat 14 mat 1.85 105^* kg m * s 2^/ * 1.29 105^* kg m * s 2^/ * 2.2 105^* kg m * s 2^/ * 31 mat 2.2 105^* kg m * s 2^/ * 1.29 105^* kg m * s 2^/ * 1.85 105^* kg m * s 2^/ * 31 mat 1.42 105^* kg m 2^* s 2^/ * 59627.23 kg m 2^* s 2^/ * 1.42 105^* kg m 2^* s 2^/ * 31 mat 2.75 m * 3.6 m * 3.25 m * 31 mat 14 mat 81249.87 kg m * s 2^/ * 2.43 105^* kg m * s 2^/ * 1.34 105^* kg m * s 2^/ * 31 mat 1.34 105^* kg m * s 2^/ * 2.43 105^* kg m * s 2^/ * 81249.87 kg m * s 2^/ * 31 mat 55274.82 kg m 2^* s 2^/ * 1.74 105^* kg m 2^* s 2^/ * 55274.82 kg m 2^* s 2^/ * 31 mat 2.27 m * 3.6 m * 3.73 m * 31 mat 14 mat 31 mat

Extract: Shear Forces, Max Span Moments and X_max - Using PROGRAMS 4, 5, 6

PROGRAM 4

PROGRAM 5

PROGRAM 6

SFF Res_All SF_All :

MM_Span Res_All Span_M_All :

XX_Max Res_All X_Max_All :

SFF kN 174.2 243 230.8 31 mat 230.8 243 174.2 31 mat 185.4 128.9 219.6 31 mat 219.6 128.9 185.4 31 mat 81.2 243 133.6 31 mat 133.6 243 81.2 31 mat 32 mat

MM_Span kNm 127 152.8 12731 mat 141.8 59.6 141.8 31 mat 55.3 174.2 55.3 31 mat 31 mat

XX_Max m 2.58 3.6 3.42 31 mat 2.75 3.6 3.25 31 mat 2.27 3.6 3.73 31 mat 31 mat

0 time t.0 - 0.08 s *

All Beam Support Moments

M_Sup_All M.all 13 range 13 range el :

M_Sup_All kNm 97.8 - 267.48 21 mat 284.58 - 284.58 21 mat 267.48 - 97.8 21 mat 112.75 - 215.51 21 mat 172.36 - 172.36 21 mat 215.51 - 112.75 21 mat 36.93 - 193.83 21 mat 263.16 - 263.16 21 mat 193.83 - 36.93 21 mat 33 mat

For All Cases : Arrange Beam Moments, Span Moments, Shear Forces, Max Span Moments and X_Max

M_Sup M_Sup_All transpose vectorize stack : kNm 97.8 - 267.5 12 mat 284.6 - 284.6 12 mat 267.5 - 97.8 12 mat 112.8 - 215.5 12 mat 172.4 - 172.4 12 mat 215.5 - 112.8 12 mat 36.9 - 193.8 12 mat 263.2 - 263.2 12 mat 193.8 - 36.9 12 mat 33 mat

Support Moments

M.UC M.UC_all transpose vectorize stack : kNm 48.9 8.6 8.6 - 48.9 - 14 mat 56.4 21.6 - 21.6 56.4 - 14 mat 18.5 34.7 34.7 - 18.5 - 14 mat 31 mat

------------ Upper Col Moments

M.LC M.LC_all transpose vectorize stack : kNm 48.9 8.6 8.6 - 48.9 - 14 mat 56.4 21.6 - 21.6 56.4 - 14 mat 18.5 34.7 34.7 - 18.5 - 14 mat 31 mat

------------ Lower Col Moments

SF1 SFF transpose vectorize stack :

SF SF1 vectorize : kN 174.2 230.8 185.4 219.6 81.2 133.6 32 mat 243243 128.9 128.9 24324332 mat 230.8 174.2 219.6 185.4 133.6 81.2 32 mat 13 mat

----------- Shear Forces

M_Span MM_Span transpose vectorize stack : kNm 127 152.8 12713 mat 141.8 59.6 141.8 13 mat 55.3 174.2 55.3 13 mat 31 mat

----------- Max Span Moments

X_Max XX_Max transpose vectorize stack : m 2.58 3.6 3.42 13 mat 2.75 3.6 3.25 13 mat 2.27 3.6 3.73 13 mat 31 mat

----------- Dist to Max BM

0 time t.0 - 0.09 s *