5 Structural Analysis and Design of Steel Railway
5.2.1.2 Influence Lines for Maximum Effects of Moving Loads on Statically Determinate SuperstructuresStatically Determinate Superstructures
Influence lines facilitate both the appropriate placement of loads and determination of maximum effects in steel beam and girder superstructures (shear forces and bending moments), trusses (axial forces), and arches (axial forces, shear forces, and bending moments). Influence lines may be constructed for moving load analysis of statically determinate superstructures by moving a unit concentrated load across the super-structure and determining the value of an effect at each location. The construction of
Structural Analysis and Design of Steel Railway Bridges 163
L
A B
C a
a/L (L–a)/L
1
Shear at location C
Shear at location A
FIGURE 5.5 Influence lines for shear at locations C and A for concentrated moving loads applied directly to the superstructure.
influence lines may be simplified by determining the value of the effect at locations where changes in the influence line will occur (i.e., supports, panel points, hinges, etc.) and joining those locations with straight lines.∗
5.2.1.2.1 Influence Lines for Maximum Shear Force and Bending Moment in Simply Supported Beam and Girder Spans
5.2.1.2.1.1 Maximum Shear Force (with Loads Applied Directly to the Superstructure) The influence lines for shear force at location C and at the end of the simple span (location A) are shown inFigure 5.5.They are developed by determin-ing the shear force at location C and reaction at the end of the simple span (location A) with a unit load placed at locations A, B, and C.
5.2.1.2.1.2 Maximum Shear Force (with Loads Applied to the Superstruc-ture through Transverse Members) The influence line for shear in panel BC of the simply supported span is shown inFigure 5.6.It is developed by determining the shear force at locations B and C with a unit load placed at locations A, B, C, and D;
where, n is the number of panels, nLis the number of panels left of panel BC, and nR
∗ For axial force, shear force and bending moment in statically determinate structures, influence lines comprise straight-line segments. However, for deflections this is not the case.
164 Design of Modern Steel Railway Bridges
L
A B C
sp
nR/n
nL/n Shear in panel BC d1
D
FIGURE 5.6 Influence line for shear in panel BC for concentrated moving loads applied to the superstructure at transverse members.
is the number of panels right of panel BC.
d1= (nR/nL)sp
(1+ (nR/nL)) = spnR
2 ((nR/nL)+ 1), (5.17) L= n(sp)= (nL+ nR+ 1)(sp).
5.2.1.2.1.3 Maximum Bending Moment (with Loads Applied Directly to the Superstructure) The influence lines for bending at location C and at the center of the simple span are shown inFigure 5.7.They are developed by determining the bending moment at location C and at the center span with a unit load placed at locations A, B, and C.
5.2.1.2.1.4 Maximum Bending Moment (with Loads Applied to the Superstructure by Transverse Members) The influence lines for moment in panel BC (at distance d2from B) and at location C of the simple span are shown in Figure 5.8.They are developed by determining the bending moments at locations B and C with a unit load placed at locations A, B, C, and D. As shown in Figure 5.8, a reduction in bending moment occurs for superstructures loaded through transverse members.
5.2.1.2.1.5 Maximum Floorbeam Reactions for Loads on Simply Supported Stringers The influence line for floorbeam reaction at location C assuming simply supported stringer spans is shown inFigure 5.9.It is developed by determining the shear forces at locations B, C, and D with a unit load placed at locations B, C, and D.
Since stringer spans are generally relatively short, the location of concentrated loads for maximum floorbeam reaction is usually quite obvious by inspection.
Structural Analysis and Design of Steel Railway Bridges 165
L
A B
C a
a(L–a)/L
Moment at location C
Moment at center L/4
L/2
FIGURE 5.7 Influence lines for bending at location C and the center for concentrated moving loads applied directly to the superstructure.
L
A B C
sP
Moment in panel BC
Moment at C a
a(L – a)/L d2
nRsP(a – sP+ d2)/L
nLsP(L – a + sP– d2)/L
C
FIGURE 5.8 Influence lines for moment in panel BC and at location C for concentrated moving loads applied at transverse members.
166 Design of Modern Steel Railway Bridges
FIGURE 5.9 Influence line for floorbeam reaction at location C.
Example 5.3
Determine the maximum bending moment per rail under axle NP= 5 of the Cooper’s E80 load on a 60 ft long DPG span. The moment at the center, C, is assumed to be near the maximum bending moment in both location and magnitude.
The influence line for the center span bending moments is shown in Figure E5.3.
The ordinates of the influence lines are
a= (7/30)15 = 3.50
Structural Analysis and Design of Steel Railway Bridges 167
d= (25/30)15 = 12.50 e= 60/4 = 15.00
f = (21/30)15 = 10.50 g= (16/30)15 = 8.00 h= (10/30)15 = 5.00 i= (5/30)15 = 2.50
MC= 20(3.50) + 40(7.50 + 10.00 + 12.50 + 15.00) + 26(10.50 + 8.00 + 5.00 + 2.50) = 2546 ft-kips.
Example 5.4
Determine the bending moment per rail at location C under axle NP= 3 of the Cooper’s E80 load on a 60 ft long through plate girder span with a floor system comprising floorbeams and 20 ft long stringers.
The influence line for bending moments at location C is shown in Figure E5.4.
The ordinates of the influence lines are a= (7/20)13.33 = 4.67 b= (15/20)13.33 = 10.00
30′ 30′
A 13′ B
5′ 20′
C 5′
8′ 5′ 5′ 9′ 5′ 6′ 5′
a
b c
d e
f g h i
40 k 40 k 20 k
40 k 40 k
26 k
26 k 26 k 26 k
FIGURE E5.4
168 Design of Modern Steel Railway Bridges
c= 40(20)/60 = 13.33 d= (35/40)13.33 = 11.67
e= (30/40)13.33 = 10.00 f = (21/40)13.33 = 7.00 g= (16/40)13.33 = 5.33 h= (10/40)13.33 = 3.33 i= (5/40)13.33 = 1.67
MC= 20(4.67) + 40(10.00 + 13.33 + 11.67 + 10.00) + 26(7.00 + 5.33 + 3.33 + 1.67) = 2345 ft-kips.
5.2.1.2.2 Influence Lines for Maximum Axial Forces in Statically Determinate Truss Spans
The influence lines developed in Section 5.2.1.2.1 for shear force and bending moment are useful in the construction of axial force influence lines for truss web and chord members, respectively. In addition, the consideration of the moving load effect at panel points simplifies the construction of axial force influence lines.
Influence lines for chord members may be constructed by considering free body diagrams and equilibrium of moments. Influence lines for web members may be constructed by considering free body diagrams and equilibrium of forces. The con-struction of influence lines for axial forces in members of simply supported truss spans is illustrated by examples of a Pratt truss and a Parker truss, respectively, in Examples 5.5 and 5.6.∗
Example 5.5
Construct influence lines for the 156.38 ft eight-panel Pratt through truss in Figure E5.5. The influence lines are constructed by locating unit loads at appropriate locations and using the method of sections or the method of joints.
Determine influence lines for the reactions and members U1–U2, U3–L3, L1–L2, L3–L4, U1–L1, and U1–L2.
Section 1-1 may be isolated to determine the forces in members U1–U2 (Figure E5.5b), L1–L2 (Figure E5.5c),and U1–L2 (Figure E5.5d).
In Figure E5.5a: with unit load at L1 and taking moments about L2, the force in U1–U2= [(1/8)(6)(19.55)]/27.25 = −0.54 (compression direction to balance reaction moment about L2).
In Figure E5.5b: with unit load at L2 and taking moments about L2, the force in U1–U2= [(2/8)(6)(19.55)]/27.25 = −1.08 (compression direction to balance reaction moment about L2).
∗ These truss forms are often used for medium span steel railway bridges.
Structural Analysis and Design of Steel Railway Bridges 169
8 @ 19.55' = 156.38'
27.25' U1
L1 L2 U2
L3 L4 U3
–1 –1
Reaction at R0
Reaction at R8 FIGURE E5.5a
8 @ 19.55' = 156.38'
27.25'
U1 U2
1
–0.54
–1.08
Member U1–U2 L2
1 L1
FIGURE E5.5b
InFigure E5.5c:with unit load at L1 and taking moments about U1, the force in L1–L2= [(1/8)(7)(19.55)]/27.25 = +0.63 (tension direction to balance reaction moment about U1).
In Figure E5.5c: with unit load at L2 and taking moments about U1, the force in L1–L2= [(2/8)(7)(19.55) − (1)(19.55)]/27.25 = +0.54 (tension direction to balance reaction moment about U1).
InFigure E5.5d:with unit load at L1 and summing horizontal forces in panel 1–2, the force in U1–L2= −(−0.54 + 0.63)(19.552+ 27.252)1/2/19.55= −0.15.
In Figure E5.5d: with unit load at L2 and summing horizontal forces in panel 1–2, the force in U1–L2= −(−1.08 + 0.54)(19.552+ 27.252)1/2/19.55= +0.93.
Section 2-2 may be isolated to determine the forces in member L3–L4 (Figure E5.5e).
170 Design of Modern Steel Railway Bridges
U1
8 @ 19.55' = 156.38'
27.25' 1
+0.63
L2 L1
1
+0.54
Member L1–L2
FIGURE E5.5c
U1
8 @ 19.55' = 156.38'
27.25'
+0.93 L2 1
–0.15 1
Member U1–L2 L1
FIGURE E5.5d
In Figure E5.5e: with unit load at L3 and taking moments about U3, the force in L3–L4= [(3/8)(5)(19.55)]/27.25 = +1.35 (tension direction to balance reaction moment about U1).
InFigure E5.5e:with unit load at L4 and taking moments about U3, the force in L3–L4= [(4/8)(5)(19.55) − (1)(19.55)]/27.25 = +1.08 (tension direction to balance reaction moment about U1).
Section 3-3 may be isolated to determine the forces in member U3–L3 (Figure E5.5f ).
In Figure E5.5f: with unit load at L3 and summing vertical forces in panel 3–4, the force in U3–L3= +3/8 = +0.38.
In Figure E5.5f: with unit load at L4 and summing vertical forces in panel 3–4, the force in U3–L3= +1/2 − 1 = −0.50.
Structural Analysis and Design of Steel Railway Bridges 171
8 @ 19.55' = 156.38'
27.25'
L3 L4 2
Member L3–L4 +1.08 +1.35
2 U3
FIGURE E5.5e
8 @ 19.55' = 156.38'
27.25'
L3 U3
L4 3 3
Member U3–L3 –0.50
+0.38 FIGURE E5.5f
The forces in member U1–L1(Figure E5.5g)may be determined by the method of joints by locating unit loads at L0, L1, and L2.
The hanger U1–L1 is loaded only when moving loads are in adjacent panels of the hanger.∗
Example 5.6
Construct influence lines for members U1–U2, U1–L2, and U2–L2 in the 240 ft six-panel curved-chord Parker through truss inFigure E5.6.The influence lines
∗ There are also increased impact effects for through truss hangers due to the short live load influence line (see Chapter 4).
172 Design of Modern Steel Railway Bridges
8 @ 19.55' = 156.38'
27.25'
Member U1–L1 +1.00
L1 U1
FIGURE E5.5g
are constructed by using the method of sections and locating unit loads at appropriate locations.
ap= 40
(36/28)− 1− 40 = 100 ft,
hp=
ap+ 2(40)
⎛
⎜⎝ 28 (ap+ 40)2+ 282
⎞
⎟⎠ = 35.3 ft,
βp= 45◦− tan−1 28
ap+ 40 = 33.7◦, Lp=
(ap+ 40)2+ 282cos(βp)= 118.8 ft.
Considering Section 1-1 in Figure E5.6: with unit load at L2 and taking moments about L2, the force in U1–U2= [−(4/6)(2)(40)]/35.3 = −1.51 (compression direction to balance reaction moment about L2) (Figure E5.7).
6 @ 40' = 240'
28' U1
L2 U2
1 ap
hp
PO L1
Lp Bp
36'
2 2
L3 1
FIGURE E5.6
Structural Analysis and Design of Steel Railway Bridges 173
U1
L2 U2
28' 36'
Member U1–U2
Member U1–L2 1.51
–0.48
0.56
Member U2–L2 1.42
–0.63 6 @ 40' = 240'
FIGURE E5.7
Considering Section 1-1 in Figure E5.6: with unit load at L1 and taking moments about P0, the force in U1–L2= [−(1/6)(240 + 100)]/
118.8= −0.48.
Considering Section 1-1 in Figure E5.6: with unit load at L2 and taking moments about P0, the force in U1–L2= (4/6)(100)/118.8 = 0.56 (Figure E5.7).
Considering Section 2-2 in Figure E5.6: with unit load at L2 and taking moments about P0, the force in U2–L2= 2/6(100 + 240)/80 = 1.42.
Considering Section 2-2 in Figure E5.6: with unit load at L3 and taking moments about P0, the force in U2–L2= −1/2(100)/80 = −0.63 (Figure E5.7).
The distance, hp, in Figure E5.6 illustrates the effect of the “modified panel shear” created by the sloped chord, which participates in resisting the panel shear force.
Influence lines for other chord and web members of the truss may be constructed in a similar manner by applying unit loads at panel points and determining axial forces in members by the method of sections or the method of joints.
174 Design of Modern Steel Railway Bridges
5.2.1.2.3 Influence Lines for Maximum Effects in Statically Determinate Arch Spans
Many steel railway arches are designed as three hinged to impose statically deter-minate conditions (Figure 5.10a). Statically determinate arches are often simpler to fabricate and erect, and are not subjected to temperature or support displace-ment induced stresses. The construction of influence lines for statically determinate arches can be made efficient by understanding the relationships between arch reac-tions, internal forces (shear, bending, and axial), and the influence lines obtained in Section 5.2.1.2.1 for shear and bending in simply supported spans.
5.2.1.2.3.1 Maximum Bending Moment, Shear Force, and Axial Force (with Moving Loads Applied Directly to the Arch) For the moving concentrated load, P= 1, a distance xPfrom support A in Figure 5.10a,
RA= L− xP
L , (5.18a)
RB= xP
L. (5.18b)
B C
D
A
HB
L
hD h
aD
HA
RB RA
xP
P = 1
1
Influence line for RA
Influence line for HA L/(4h) E
ϕD
FIGURE 5.10a Three-hinged arch rib with concentrated moving loads applied directly to the rib.
Structural Analysis and Design of Steel Railway Bridges 175
D
A hD
aD
HA
RA
MD
VD PD
FIGURE 5.10b Free body diagram of arch rib from support A to point D.
Therefore, the influence line for the vertical components of the arch reactions, RA
and RB, will be the same as those for a simply supported beam of length, L, as shown in Figure 5.10a.
If moments are taken about the arch crown pin (point C),∗
HA(h)= RA
L 2
. (5.19)
Since RA(L/2) is the bending moment at point C in a simply supported span, the influence line for horizontal thrust reaction, HA, is proportional (by the arch rise, h) to this simple span bending moment as shown in Figure 5.10a. Therefore, the criteria for the position of Cooper’s load for maximum bending moment (see Section 5.2.1.1.3) can be used for the determination of maximum horizontal thrust.
The arch reactions may now be used to determine the internal shear force, bending moment, and axial force influence lines for the arch rib. From Figure 5.10b, the bending moment, MD, at a location, D, is
MD= RA(aD)− HA(hD). (5.20) Equation 5.20 indicates that the influence line for bending moment in the arch rib at location D can be obtained by subtracting the ordinates for the influence line for HA(Figure 5.10a) multiplied by the distance hDfrom the ordinates for simple beam
∗ It is the inclusion of the crown pin that enables this equilibrium equation to be written; thereby illustrating the benefits of statically determinate design and construction.
176 Design of Modern Steel Railway Bridges
Influence line for MD LhD/(4h)
aD(L–aD)/L
A
B
D E C
FIGURE 5.11 Influence line for bending moments at location D in three-hinged arch rib.
bending at location, D, described by RA(aD). The construction of this influence line is shown in Figure 5.11. The ordinates (shaded areas) may be plotted on a horizontal line for ease of use in design.
FromFigures 5.10aandb,the shear force, VD, at a location, D, is
VD= RAcosφD− HAsinφD. (5.21) Equation 5.21 indicates that the influence line for shear force in the arch rib at location D can be obtained by subtracting the ordinates for the influence line for HAmultiplied by sinφD from the ordinates for simple beam shear at D multiplied by cosφD. The construction of this influence line is shown inFigure 5.12.Again, the ordinates (shaded areas) may be plotted on a horizontal line for ease of use in design. Location E is the position of the moving load that creates no shear force or bending moment in the arch at location D (Figure 5.10a).
From Figures 5.10a and b, the axial force, FD, at a location, D, is
FD= −RAsinφD− HAcosφD. (5.22) Equation 5.22 indicates that the influence line for axial force at location D in the arch rib can be obtained by adding the ordinates for the influence line for HA multiplied by cosφD to the ordinates for simple beam shear at D multiplied by sinφD. The construction of this influence line is shown inFigure 5.13.Again, the ordinates (shaded areas) may be plotted on a horizontal line for ease of use in design.
Influence line for VD [(L–aD)/L]cosϕD
[aD/L]cosϕD
L/(4h)sinϕD
A
B
C
D E
FIGURE 5.12 Influence line for shear forces at location D in three-hinged arch rib.
Structural Analysis and Design of Steel Railway Bridges 177
Influence line for FD [(L–aD)/L]sinϕD
L/(4h)cosϕD
A
B
C D
[aD/L]sinϕD
FIGURE 5.13 Influence line for axial force at location D in three-hinged arch rib.
5.2.1.2.3.2 Maximum Bending Moment, Shear Force, and Axial Force [with Loads Applied to the Arch by Transverse Members (Spandrel Columns or Walls)] Medium- and long-span steel railway bridges can be economically constructed of three-hinged arches with the arch rib loaded by spandrel members (Figure 5.14).Influence lines for arch spans with spandrel columns or vertical posts
B C
D
A
HB
L
hD h
aD
HA
RB RA
xP
P = 1
1
Influence line for HA
L/(5h) E
ϕD
Influence line for RA
FIGURE 5.14 Three-hinged arch rib with concentrated moving loads applied to the rib at transverse members (e.g., spandrel columns).
178 Design of Modern Steel Railway Bridges
Influence line for MD A
B
D E C
FIGURE 5.15 Influence line for bending moments at location D in three-hinged arch rib.
can be developed from influence lines for directly loaded arches in a manner analo-gous to simple spans with transverse members (floorbeams) (see Sections 5.2.1.1.2 and 5.2.1.1.4).
For example, with a pin at location C, the influence line for bending moment at D will be of the general form shown in Figure 5.15. The influence lines for other internal forces can be determined in a similar manner.
5.2.1.2.3.3 Maximum Axial Forces with Moving Loads on a Statically Deter-minate Trussed Arch Long-span steel railway bridges can be economically constructed of three-hinged arches with the arch rib replaced by a truss. The techniques used in Section 5.2.1.2.3.1 to determine maximum effects are useful for the construc-tion of influence lines for trussed arches. The crown hinge is designed to achieve static determinacy with a bottom chord pin and top chord sliding arrangement∗ as shown in Example 5.7 andFigure E5.8.
Example 5.7
Determine the influence line for member U1–U2 in the 400 ft eight-panel deck trussed arch in Figure E5.8.
The force in the chord U1–U2 can be determined using Equation 5.20 by considering Section 1.1 and taking moments about L2.
FU1–U2= MD
yD = RA(aD)− HA(hD)
yD = RA(aD)− (L/4h)(hD)
yD .
The ordinate of the influence line at L2 provides RA(aD)= (6/8)(100) = 75.
This component of the influence line is related to vertical reaction, RA. The ordinate of the influence line at L4 provides (L/4h)(hD)= (400/
4(150))(45+ 55) = 66.67. This component of the influence line is related to horizontal thrust reaction, HA.
∗ Thereby, rending the force in one central top chord member as zero.
Structural Analysis and Design of Steel Railway Bridges 179
1
1 U1 U2
L2
8 @ 50 ft = 400 ft
25 ft
150 ft 135 ft
55 ft 45 ft
RA
1.00 yD
0.89
Sliding mechanism
Pin, typ.
L4
aD = 100 ft
hD = 55 + 45 = 100 ft
FIGURE E5.8
With yD= 175 − 45 − 55 = 75 ft, the influence line for axial force in chord U1–U2 (shown by the shaded area in Figure E5.8) can be determined by the superposition of the influence lines for RAand HA.
5.2.1.2.4 Influence Lines for Maximum Effects in Statically Determinate Cantilever Bridge Spans
Long-span steel railway bridges may also be economically constructed as cantilever bridges (see Chapter 1). The economical relative lengths of the cantilever arm, Lc, anchor, La, and suspended, Ls, spans will vary with live to dead load bending moment ratio. For the relatively high live to dead load bending moment ratios of steel railway superstructures, typical La/Lcvalues of between 1 and 2 are used depending on the suspended span length, Ls. In steel railway superstructures, Lc/Ls values typically range from 0.4 to 2. The relative lengths of the cantilever arm, anchor, and suspended spans may also vary based on site conditions that dictate the location of piers at a crossing (see Chapter 3). Influence lines for cantilever superstructures may also be constructed by consideration of unit loads traversing the bridge. The ordinates of the influence lines are readily determined by calculation of the reaction, bending moment, and shear due to unit loads at locations where the influence lines change direction.
5.2.1.2.4.1 Cantilever Bridge Span Influence Lines [with Loads Applied Directly to the Superstructure] Influence lines for reactions at locations A and B, bending moment in the anchor span at location E and at location F in the cantilever
180 Design of Modern Steel Railway Bridges
A B C
E F D
Moment at E
Moment at F Suspended span Cantilever span
Anchor span
Reaction at A
Reaction at B
La Lc Ls
Shear at E
Shear at F
FIGURE 5.16 Influence lines for reactions, bending moments, and shear forces in anchor and cantilever spans with loads applied directly to the superstructure.
span may be constructed by considering effects of unit loads placed at locations A, B, and C as shown qualitatively∗in Figure 5.16.
5.2.1.2.4.2 Cantilever Bridge Span Influence Lines [with Loads Applied to the Superstructure by Transverse Members (Floorbeams)] Influence lines for shear force and bending moment in the anchor span panel point A1–A2 and in the cantilever span panel point C2–C3 can be constructed by considering
∗ Qualitative influence lines are useful in both manual and electronic calculations of maximum or minimum effects to determine the approximate location of live load.
Structural Analysis and Design of Steel Railway Bridges 181
A B C D
A1
Moment at A2
Moment at C2 Suspended span Cantilever span
Anchor span
A2 C2 C3
Shear in panel A1–A2
Shear in panel C2–C3
FIGURE 5.17 Influence lines for shear forces and bending moment in anchor and cantilever spans with panel points.
the effects of unit loads placed at locations A, A1, A2, B, C2, C3, and C (also shown qualitatively in Figure 5.17).
For long-span railway superstructures it is of further efficiency to utilize truss spans in cantilever bridges. The influence lines developed in Figure 5.17, in con-junction with those developed in Sections 5.2.1.2.1 and 5.2.1.2.2, are useful in the construction of axial force influence lines for cantilever bridge truss web and chord members. Also, as usual, the consideration of the moving load effect at panel points simplifies the construction of axial force influence lines. Influence lines constructed in this manner for axial forces in cantilever bridge truss members are shown in Example 5.8.
Example 5.8
Construct influence lines for members in panel point 2–3 in the anchor arm of the cantilever truss bridge inFigure E5.9.
By inspection and placement of unit loads at L0, L2, L3, and L6 and consider-ing the hconsider-inge at the end of the cantilever and suspended spans, the influence lines for axial forces in L2–L3, U2–U3, and U2–L3 are shown in Figure E5.9. Influ-ence lines for axial force in other members of the trusses may be constructed in a similar manner.
182 Design of Modern Steel Railway Bridges
L2
Symmetrical about center line
L3
Axial force L2–L3 L6
U2 U3
Reaction at L0
Reaction at L6
0.33
1.33 1.00
1.00 12 @ 25' = 300'
1.11
15'
0.56
1.11
0.74
0.43
0.65
0.43 L0
30'
Axial force U2–U3
Axial force U2–L3
FIGURE E5.9
5.2.1.3 Equivalent Uniform Loads for Maximum Shear Force and