# 1 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures STATICALLY DETERMINED PLANE BAR STURCTURES (BEAMS)

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1 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures STATICALLY DETERMINED PLANE BAR STURCTURES (BEAMS)

3 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Superposition principle An example of the use of superposition principle for determination of support reactions 2 kN 1 kN/m 4 MNm 2 m 4 m RLRL RPRP K  M K =0 Y  Y=0 +2·2 + +4 +1·4·(2+2+4/2) +(2+2+4)·R P = 0 32+8R P =0 R P = - 4 4 kN R L = 6 + R P R L = 6 – 4 = + 2+ R L - 2= 0– R P - 1·4 2 kN

4 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Superposition principle 4 kNm 1 kN/m R L1 =[2·6]/8=1,5 R L3 =[1·4 ·2]/8=1,0 R L2 =-4/8= - 0,5R P2 =0,5 R R1 =0,5 R P3 =3,0 +  M K =0  Y=0 K Y R L = 2,0 R R = 4,0 0,5 kN 2 kN + 1,5 kN 2 kN 1 kN 0,5 kN 3,0 kN + + = 4 kN 2 kN 1 kN/m 4 kNm 2 m 4 m

5 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Typical simple beams Pin-pointed (or simply supported) beam Beams Semi-cantilever beam Cantilever beam

6 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Fundamental assumptions used for drawing diagrams of cross sectional forces Beams 1. Diagram of bending moment will appear always on the side of a beam which is subjected to the tension caused by this bending moment. Therefore, no sign is necessary. 2. We will make use of q-Q-M dependence: 3. Shear and normal forces will be always marked with their sign according to the following convention: Q N o n n o Q N

7 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures P Pa/l Pb/l l ab M P P/2 l/2 Pl/4 M P/2 Q - + Q N o n n o Q N q0q0 q=d Q=c Q=dx+e M=dx 2 /2+ ex+f M=cx+b Typical beams Pab/l Q Pa/l + - Pb/l

8 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures P M ql/2 P P aa l P q l l/2 ql 2 /8 Q + - ql/2 M Pa P P + - Q Typical beams q=d Q=dx+e M=dx 2 /2+ex+f

9 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures M/l l ab M M Mb/l Ma/l Q M/l - Q - M M M Q + M M M

10 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures [1+(a/l)] 2 [ql/2] l a M Q + - M/l l a M Q M M - P P(1+a/l) Pa/l l a M Q Pa P Pa/l + - P(1+a/l) q [1-(a/l) 2 ][ql/2] qa qa 2 /2 (l/2)[1-(a/l) 2 ] Typical beams l a M M M/l -

11 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures [1+(a/l)] 2 [ql/2] l a M Q + - M/l l a M Q M M - P P(1+a/l) Pa/l l a M Q Pa P Pa/l + - P(1+a/l) q qa qa 2 /2 (l/2)[1-(a/l) 2 ] P·aP·a P M q·aq·a q·a 2 /2 Q N o n Cantilever beam

12 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures The concept of multiple co-linear beams Gerber’s beams

13 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Q  0, N  0 M= 0 Gerber’s beams

14 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Formal definition: the set of aligned simple bars, hinged together and supported in the way which assures kinematical stability Equilbrium equations X Y K A B  X = 0  Y = 0  M K = 0 M B = 0 M A = 0 1 equation 2 equations 4 unknown horizontal reactions 4 unknown vertictal reactions 2 equations Structure is statically indetermined with respect to horizontal reactions Number of unknown reactions: 4 horizontal + 4 vertical Gerber’s beams

15 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Number of unknown reactions to be found: 1 horizontal reaction from 1 equilibrium equation (or we accept indeterminancy with respect to normal cross-sectional forces) (2 + n) remaining reactions (vertical reactions, moment reactions) from 2 equilibrium equations and n equations of vanishing bending moment in hinges Number of hinges is determined from the second of the above condition: structures appears to be kinemtaically unstable if there are too many hinges, and hyper-stiff – if there are too little hinges). However, the location of hinges – even if their number is correct one – cannot be arbitrary! GOOD! WRONG! HYPER-STIFF UNSTABLE Gerber’s beams

16 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Partitioning of a Gerber’s beam into series of simple beams allows for better understanding of structure’s work! M Gerber’s beams

17 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Gerber’s beams SUPERPOSITION! + + + + + =

18 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Gerber’s beams SUPERPOSITION! MQ ++++

19 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures  stop

20 /15 M.Chrzanowski: Strength of Materials SM1-04: Statics 3: Statically determined bar structures Gerber’s beams

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