WORKSHEET 4 BEAMS

Q1 tributary area 600mm 2m tributary area for joist = 2 x 0.6 = 1.2 m2 Given that floor joists are at 600mm centres and span 2.0m between bearers, what is the tributary area for one joist? tributary area 600mm 2m tributary area for joist = 2 x 0.6 = 1.2 m2

Q2 Given a floor 18 m x 18 m with columns on a 6m x 6m grid, what is the tributary area for: 6m (i) an internal column 6 x 6 = 36 m2 6m (ii) a column on the edge 6 x 3 = 18 m2 6m (ii) a corner column 3 x 3 = 9 m2

Q3 Given the values in the Building Principles Notes for the Dead Loads of materials (P17), determine the dead load of the roof/ceiling construction shown below 6mm corrugated fibre cement sheet - 0.11kN/m2 13mm plasterboard ceiling - 0.22kN/m2 100 x 50 hardwood rafters @ 600mm crs - 11 kN /m3 We are after a 1sq m of roof, but the rafters are at 600mm centres so that 1m width of roof will contain 1.67 rafters (1 / 0.6). 0.6 1.67 1.67 x 0.6 = 1.0 Another way of doing this is to say that 1sqm can be achieved by an area 0.6 wide x 1.67 long (1 / 0.6). Weight of rafter 1.67m long = 0.1 x 0.05 x 1.67 x 11 = 0.09 kN Weight of 1 sq m fibre cement = 1 x 0.11 = 0.11 kN Weight of 1 sq m plasterboard = 1 x 0.22 = 0.22 kN Total weight of roof/ceiling per sq m = 0.42 kN / m2 = 0.42 kPa

Q4 The roof above spans between roof trusses which are at 2.5 m centres and span 10m 2.5m a) sketch the layout described and indicate the tributary area for one truss 2.5m 10.0m b) what is the total load on one truss? (neglecting the self-weight of the truss) Tributary area = 2.5 x 10 = 25 m2 Total load = 25 x 0.42 = 10.5 kN c) what is load per metre on one truss? Note: We have neglected the self-weight of the truss Load per metre = 10.5 / 10 =1.05kN / m

Q5 What are the two main types of stress involved in beam action? a) bending b) shear

Q6 In buildings: a) which of the above two (bending & shear) is more important? bending b) why? have bigger spans relative to loads. In the design of machines have short spans with heavy loads and shear more important.

Q7 What does shear force do to: a) timber beams? can cause horizontal splitting along grain b) steel beams? not so critical - make sure don’t exceed allowable shear stress c) concrete? tends to cause diagonal tension cracks near supports

Q8 How is shear resisted in concrete beams: a) steel reinforcement at 450 b) stirrups

+ - Q9 What is the sign convention for Bending Moment Diagrams for: a) sagging? positive - b) hogging? negative

Q10 a) What does a Shear Force Diagram tell you? the values of the shear force along the beam you can see where the maximum shear force occurs b) What does a Bending Moment Diagram tell you? the values of the bending moment along the beam you can see where the maximum bending moment occurs and whether it is positive or negative

Q11 For each of the Following Loading Conditions a) Sketch the deflected shape and note where positive and negative bending moments are expected to occur b) Find the reactions c) Draw the Shear Force Diagrams d) Find the maximum bending moment(s) and draw the Bending Moment Diagrams draw the diagrams approximately to scale (i.e. in proportion) and mark significant values make use of symmetry and standard Bending Moment coefficients where appropriate

Q11 A & B A B + Deflected Shape + SFD BMD 16 kN 2m 4m 4m UDL 5kN/m WL/4 = 16 x 4 / 4 = BMD +10 kNm wL2/8 = 5 x 4 x 4 / 8 =

Q11 C & D - - C D Deflected Shape SFD BMD 2m 10 kN UDL 5kN/m 2m R =10 kN W = w x L = 5 x 2 = - 10 kN SFD +10 kN +10 kN -10 kNm -wL2/2 = -5 x 2 x 2 / 2 = BMD -WL = -20 kNm

Q11 E SFD + Deflected Shape BMD 5m 20kN 2m 1m A B C D +16 kN -24 kN TL = 20 + 20 = 40 kN For reactions Moments about A RR x 5 = 20 x 2 +20 x 4 = 120 RR = 24 kN RL = 16 kN Moment at B = 16 x 2 = 32 kNm Moment at C = 24 x 1 = 24 kNm 24 kN Deflected Shape + 16 kN 32 kNm 24 kNm BMD

Q11 F SFD - BMD Deflected Shape 2m 10kN 1m 5kN A B C +15 kN +5 kN TL = 10 + 5 = 15 kN For reactions Moment at A = 10 x 1 + 5 x 2 = 20 kNm Moment at A = - 20 kNm Moment at B = - 5 x 1 = - 5 kNm - Deflected Shape 15 kN -20 kNm -5 kNm BMD

Q11 G - SFD + BMD 4m 2m 20kN 5kN A C B +12.5 kN -7.5 kN -5 kN TL = 20 + 5 = 25 kN For reactions Take Moment at C RL x 4 = 5 x 6 + 20 x 2 = 70 RL = 17.5kN RR = 7.5kN Moment at A = -5 x 2 = -10 kNm Moment at B = 7.5 x 2 = 15kNm WL/4 = 20x4/4 = 20kNm 20 kNm -10 kNm BMD +15 kNm 7.5 kN + 17.5 kN - Deflected Shape

Q11 H - SFD + BMD 4m UDL 5kN/m 2m A B C -10 kN +12.5 kN -7.5 kN 30kN TL = 5 x 6 = 30 kN For reactions Take Moment at C RL x 4 = 30 x 3 = 90 RL = 22.5kN RR = 7.5kN Moment at A = -10 x 1 = -10 kNm Moment at B = 7.5 x 2 - 5 x 2 x 1 = 15 - 10 = 5 kNm WL/8 = 20x4/8 = 10kNm -10 kNm +5 kNm +~5.6 kNm 10 kNm BMD 7.5 kN 10kN 20kN + 22.5 kN - Deflected Shape

Q11 H (cont.) BMD (cantilever) wL2/2 = -10 kNm BMD (Simply Supported) BMD(Comb)

Q11 I - - SFD (Cantilevers) Cantilevers SFD Simply Supported (Simply UDL 5kN/m 2m SFD (Cantilevers) +10 kN -10 kN RL = 10 RR = 10 - 10 kN 10kN Cantilevers +10 kN -10 kN SFD (Simply Supported) RL = 10 RR = 10 + 10 kN 20kN Simply Supported +10 kN -10 kN SFD (Combined) RL = 20 RR = 20 - 20 kN 20kN 10kN Combined

Q11 I (cont.) BMD (cantilevers) BMD (Simply Supported) BMD(Comb) -10 kNm -10 kNm wL2 / 2 = 5 x 2 x 2 / 2 = 10 kNm BMD (Simply Supported) +10 kNm wL2 / 8 = 5 x 4 x 4 / 8 = 10 kNm BMD(Comb)