CE 331, Fall 2010 Influence Les for Trusses 1 / 7 An fluence le shows how the force a partcular member changes as a concentrated load s moved along the structure. For eample, say we are desgng the seven panel parallel chord roof truss shown below. We ve determed that the dead load causes the mamum compressve force the top chord at mdspan (member M11 ). art of the lve load conssts of a 5000 lb ar condtoner unt. Unfortunately, the locaton of the ar condtoner s not specfed at the tme we are desgng the roof truss. We need to desgn the truss to support the ar condtoner at any locaton (nodes N10 through N15). 8 ft 10 ft Typ Fgure 1. 7 panel parallel chord roof truss A brute force approach would be to apply the 1500 lb load at Node N10, calculate the bar force Member M11, move the load to Node N11, calculate the bar force Member M11, and so on for all s possble load locatons. Ths process s llustrated Fgure 2. The structural analyss program RISA was used to calculate the bar forces. In RISA, compresson s postve, whch s opposte to the sgn conventon used your Mechancs of Materals (MoM) class where compresson s negatve. We wll use the MoM sgn conventon our calculatons ths semester. Note that the mamum force Member M11 (10,714 lb) occurs when the weght of the ar condtoner s appled to Node N13.
CE 331, Fall 2010 Influence Les for Trusses 2 / 7 Fgure 2. Member aal forces (lbs) due to unt loads at s dfferent jots. Compresson = + ve.
CE 331, Fall 2010 Influence Les for Trusses 3 / 7 The aal forces Member M11 due to the 5000 lb ar condtoner are summarzed the 2 nd column of Table 1 below. The forces M11 (zero) when a load s appled at the truss ends (N9 and N16) were added. In the thrd column, the aal forces Member M11 due to a unt load are lsted. These forces, when plotted, form an fluence le for the aal force Member M11, shown Fgure 3. Table 1. Forces Member M11 (MoM sgn conventon) Load at Jot f_11, lb f_11/ 5000 lb = I N9 0 0.000 N10 2679 0.536 N11 5357 1.071 N12 8036 1.607 N13 10714 2.143 N14 7143 1.429 N15 3571 0.714 N16 0 0.000 Fgure 3. Influence le for Member M11 2.50 Influence Value 2.00 1.50 1.00 0.50 0.00 N9 N10 N11 N12 N13 N14 N15 N16 Locaton of Concentrated Load Each value of the fluence le shown Fgure 3 requred a separate structural analyss. Ths same fluence le can be calculated wth one analyss usg the equaton shown below (derved elsewhere from Bett s Law of Recprocal Deflectons.) Where Δ = f Δ = the deflecton at Jot due to the deformaton of member = the deformaton of member f = the force member " due to a concentrated load at Jot = the concentrated load at Jot (1) For ths eample, = Jots N9 through N16 and " = Member M11. We can calculate the rato on the rght hand sde of Equaton 1 by calculatg the rato on the left hand sde. The rato on the left hand sde can be calculated for = N9 through N16 one step usg RISA. In theory, could be any number, but t s easest f we let be 1. We cause Member M11 to elongate by 1 RISA by applyg a temperature change to the member. The magntude of the temperature s calculated as follows.
CE 331, Fall 2010 Influence Les for Trusses 4 / 7 = α ΔT L 1" = (6.5 10 ΔT = 1282 o 6 F / o F) ( ΔT ) (10 ft 12 / ft ) Ths temperature load s appled to Member M11 RISA and the resultg node deflectons are calculated. Fgure 4 shows the load appled to Member M11, the resultg deflected shape of the truss, and a table of jot deflectons. Note that the Y deflectons for Jots N9 through N16 equal the fluence values, I, shown Column 3 of Table 1. (a) (b) (c) Fgure 4. (a) Temperature load to cause 1 elongaton of Member M11, (b) deflected shape of truss due to 1 elongaton of Member M11, and (c) jot deflectons due to 1 elongaton of Member M11. Another way to elongate a truss member one ch s to apply a par of loads to the jots on ether end of the member. The magntude of the loads s calculated below for ths eample.
CE 331, Fall 2010 Influence Les for Trusses 5 / 7 = L A E L = 10 (1 = ft A= 2.96 12 2 E = 29,000 / ft ks )(2.96 2 120 = 120 )(29,000 ks ) = 715.3 k Fgure 5 below shows the loads appled to the jots on ether end of Member 11, the resultg deflected shape, and a table of jot deflectons. The deflected shape and jot deflectons are dentcal wth those shown Fgure 4.
CE 331, Fall 2010 Influence Les for Trusses 6 / 7 (a) (b) (c) Fgure 5. (a) Loads to cause 1 elongaton of Member M11, (b) deflected shape of truss due to 1 elongaton of Member M11, and (c) jot deflectons due to 1 elongaton of Member M11.
CE 331, Fall 2010 Influence Les for Trusses 7 / 7 The procedure to calculate the mamum compressve chord force a truss s llustrated below. 1. Calculate the mamum compressve chord force due to dead loads and note the member whch ths occurs. ma. chord force due to dead load = 50 k and occurs Member 11. f D 11 = 50 k C 2. Draw the fluence le for Member A usg a structural analyss program. Apply a par of 715.3 k loads to the jots on ether end of Member 11. The resultg vertcal deflectons are the fluence values, I. Jot I N9 0.000 N10 0.536 N11 1.071 N12 1.607 N13 2.143 N14 1.429 N15 0.714 N16 0.000 3. Apply the lve load to cause the mamum compressve force Member A. Equaton 1 from prevous page: For ths eample: Δ f =, f Δ k 2.143 11 13 = 13 = ( 5 ) = 1 = f 10. 72 The sgn conventon for Eqn 1 s shown below: I Δ k and Δ : = + ve f and : tenson/elongaton = + ve Therefore, the lve load causes a compressve force Member 11. f 11 L =10.70 k C Note, we use the fluence dagram to decde where to place the lve load to cause mamum total force the member of terest. Sce the bar force due to dead load was compressve, we want to place the lve load to cause a compressve bar force due to lve load. 4. Calculate the total force due to factored loads and check the strength of the member. u = 1.2 f D 11 + 1.6 f L 11 = 1.2 (50 k C) + 1.6 (10.7 k C) = 77.1 k C. u = 77.1 k < 81 k = φ n, OK A smlar procedure can be used to check the mamum dagonal force.