SDOF Transformation Factors

Ingeania uses the following transformation factors in its Single Degree of Freedom calculator SDOF for the analysis of structural elements subjected to dynamic loading.

Load and Mass Distribution Resistance Domain Load Factor Mass Factor Load Mass Factor Maximum Resistance Linear Stiffness Equiv. Linear Stiffness Dynamic Reaction
Kl Km Klm Rmax kl ke V
Simply Supported Beam with Pin Pin Boundary Conditions and Distributed Mass and Distributed Force Elastic 0.64 0.50 0.78 8Mp/L 384EI/(5L3) 0.39R+0.11F
Plastic Bending 0.50 0.33 0.66 8Mp/L 0 0.38R+0.12F
Plastic Membrane 0.50 0.33 0.66 4Np/L 2Npymax/L
Simply Supported Beam with Pin Pin Boundary Conditions and Point Mass and Distributed Force Elastic 0.64 1.00 1.56 8Mp/L 384EI/(5L3) 0.50R
Plastic Bending 0.50 1.00 2.00 8Mp/L 0 0.50R
Plastic Membrane 0.50 1.00 2.00 4Np/L 2Npymax/L
Simply Supported Beam with Pin Pin Boundary Conditions and Distributed Mass and Point Force Elastic 1.0 0.49 0.49 4Mp/L 48EI/(L3) 0.78R-0.28F
Plastic Bending 1.0 0.33 0.33 4Mp/L 0 0.75R-0.25F
Plastic Membrane 1.0 0.33 0.33 4Np/L 2Npymax/L
Simply Supported Beam with Pin Pin Boundary Conditions and Point Mass and Point Force Elastic 1.0 1.0 1.0 4Mp/L 48EI/(L3) 0.78R-0.28F
Plastic Bending 1.0 1.0 1.0 4Mp/L 0 0.75R-0.25F
Plastic Membrane 1.0 1.0 1.0 4Np/L 2Npymax/L
Simply Supported Beam with Pin Pin Boundary Conditions and Distributed Mass and 2 Point Force Elastic 0.87 0.52 0.60 6Mp/L 56.4EI/(L3) 0.525R-0.025F
Plastic Bending 1.0 0.56 0.56 6Mp/L 0 0.52R-0.02F
Plastic Membrane 1.0 0.56 0.56 6Np/L 3Npymax/L
Simply Supported Beam with Pin Pin Boundary Conditions and 2 Point Mass and 2 Point Force Elastic 0.87 0.76 0.87 6Mp/L 56.4EI/(L3) 0.525R-0.025F
Plastic Bending 1.0 1.0 1.0 6Mp/L 0 0.52R-0.02F
Plastic Membrane 1.0 1.0 1.0 6Np/L 3Npymax/L
Load and Mass Distribution Domain Kl Km Klm Rmax kl ke V
Encastre Beam with Fix Fix Boundary Conditions and Distributed Mass and Distributed Force Elastic 0.53 0.41 0.77 12Mps/L 384EI/(L3) 307m1m1=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.25) 0.36R+0.14F
Elasto-Plastic Bending 0.64 0.50 0.78 8(Mps+Mpm)/L 384EI/(5L3) 0.39R+0.11F
Plastic Bending 0.50 0.33 0.66 8(Mps+Mpm)/L 0 0.38R+0.12F
Plastic Membrane 0.50 0.33 0.66 4Np/L 2Npymax/L
Encastre Beam with Fix Fix Boundary Conditions and Point Mass and Distributed Force Elastic 0.53 1.00 1.88 12Mps/L 384EI/(L3) 307m1m1=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.25) 0.50R
Elasto-Plastic Bending 0.64 1.00 1.56 8(Mps+Mpm)/L 384EI/(5L3) 0.50R
Plastic Bending 0.50 1.00 2.00 8(Mps+Mpm)/L 0 0.50R
Plastic Membrane 0.50 1.00 2.00 4Np/L 2Npymax/L
Encastre Beam with Fix Fix Boundary Conditions and Distributed Mass and Point Force Elastic 1.00 0.37 0.37 4(Mps+Mpm)/L 192EI/(L3) 0.71R-0.21F
Plastic Bending 1.00 0.33 0.33 4(Mps+Mpm)/L 0 0.75R-0.25F
Plastic Membrane 1.00 0.33 0.33 4Np/L 2Npymax/L
Encastre Beam with Fix Fix Boundary Conditions and Point Mass and Point Force Elastic 1.00 1.00 1.00 4(Mps+Mpm)/L 192EI/(L3) 0.71R-0.21F
Plastic Bending 1.00 1.00 1.00 4(Mps+Mpm)/L 0 0.75R-0.25F
Plastic Membrane 1.00 1.00 1.00 4Np/L 2Npymax/L
Encastre Beam with Fix Fix Boundary Conditions and Distributed Mass and 2 Point Force Elastic 0.80 0.41 0.51 9Mps/L 260EI/(L3) 212m1m1=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.25) 0.48R+0.02F
Elasto-Plastic Bending 0.87 0.52 0.60 6(Mps+Mpm)/L 56.4EI/(L3) 0.52R-0.02F
Plastic Bending 1.00 0.56 0.56 6(Mps+Mpm)/L 0 0.52R-0.02F
Plastic Membrane 1.00 0.56 0.56 6Np/L 3Npymax/L
Encastre Beam with Fix Fix Boundary Conditions and 2 Point Mass and 2 Point Force Elastic 0.80 0.64 0.80 9Mps/L 260EI/(L3) 212m1m1=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.25) 0.48R+0.02F
Elasto-Plastic Bending 0.87 0.76 0.87 6(Mps+Mpm)/L 56.4EI/(L3) 0.52R-0.02F
Plastic Bending 1.00 1.00 1.00 6(Mps+Mpm)/L 0 0.52R-0.02F
Plastic Membrane 1.00 1.00 1.00 6Np/L 3Npymax/L
Load and Mass Distribution Domain Kl Km Klm Rmax kl ke V
Propped Cantilever Beam with Pin Fix Boundary Conditions and Distributed Mass and Distributed Force Elastic 0.58 0.45 0.78 8Mps/L 185EI/(L3) 160m2m2=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.5) V1=0.26R+0.12F
V2=0.43R+0.19F
Elasto-Plastic Bending 0.64 0.50 0.78 4(Mps+2Mpm)/L 384EI/(5L3) 0.39R+0.11F±Mps/L
Plastic Bending 0.50 0.33 0.66 4(Mps+2Mpm)/L 0 0.38R+0.12F±Mps/L
Plastic Membrane 0.50 0.33 0.66 4Np/L 2Npymax/L
Propped Cantilever Beam with Pin Fix Boundary Conditions and Point Mass and Distributed Force Elastic 0.58 1.00 1.73 8Mps/L 185EI/(L3) 160m2m2=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.5) V1=0.375R
V2=0.625R
Elasto-Plastic Bending 0.64 1.00 1.56 4(Mps+2Mpm)/L 384EI/(5L3) 0.50R±Mps/L
Plastic Bending 0.50 1.00 2.00 4(Mps+2Mpm)/L 0 0.50R±Mps/L
Plastic Membrane 0.50 0.33 0.66 4Np/L 2Npymax/L
Propped Cantilever Beam with Pin Fix Boundary Conditions and Distributed Mass and Point Force Elastic 1.00 0.43 0.43 16Mps/(3L) 107EI/(L3) 106m2m2=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.5) V1=0.25R+0.07F
V2=0.54R+0.14F
Elasto-Plastic Bending 1.00 0.49 0.49 2(Mps+2Mpm)/L 48EI/(L3) 0.78R-0.28F±Mps/L
Plastic Bending 1.00 0.33 0.33 2(Mps+2Mpm)/L 0 0.75R-0.25F±Mps/L
Plastic Membrane 1.00 0.33 0.33 4Np/L 2Npymax/L
Propped Cantilever Beam with Pin Fix Boundary Conditions and Point Mass and Point Force Elastic 1.00 1.00 1.00 16Mps/(3L) 107EI/(L3) 106m2m2=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.5) V1=0.25R+0.07F
V2=0.54R+0.14F
Elasto-Plastic Bending 1.00 1.00 1.00 2(Mps+2Mpm)/L 48EI/(L3) 0.78R-0.28F±Mps/L
Plastic Bending 1.00 1.00 1.00 2(Mps+2Mpm)/L 0 0.75R-0.25F±Mps/L
Plastic Membrane 1.00 1.00 1.00 4Np/L 2Npymax/L
Propped Cantilever Beam with Pin Fix Boundary Conditions and Distributed Mass and 2 Point Force Elastic 0.81 0.45 0.55 6Mps/L 132EI/(L3) 122m3m3=EI/(L3)*(2Mps/(Mps+3Mpm)+0.5) V1=0.17R+0.17F
V2=0.33R+0.33F
Elasto-Plastic Bending 0.87 0.52 0.60 2(Mps+3Mpm)/L 56EI/(L3) 0.525R-0.025F±Mps/L
Plastic Bending 1.00 0.56 0.56 2(Mps+3Mpm)/L 0 0.52R-0.02F±Mps/L
Plastic Membrane 1.00 0.56 0.56 6Np/L 3Npymax/L
Propped Cantilever Beam with Pin Fix Boundary Conditions and 2 Point Mass and 2 Point Force Elastic 0.81 0.67 0.83 6Mps/L 132EI/(L3) 122m3m3=EI/(L3)*(2Mps/(Mps+3Mpm)+0.5) V1=0.17R+0.17F
V2=0.33R+0.33F
Elasto-Plastic Bending 0.87 0.76 0.87 2(Mps+3Mpm)/L 56EI/(L3) 0.525R-0.025F±Mps/L
Plastic Bending 1.00 1.00 1.00 2(Mps+3Mpm)/L 0 0.52R-0.02F±Mps/L
Plastic Membrane 1.00 1.00 1.00 6Np/L 3Npymax/L
Load and Mass Distribution Domain Kl Km Klm Rmax kl ke V
Cantilever Beam with Fix Free Boundary Conditions and Distributed Mass and Distributed Force Elastic 0.40 0.26 0.65 2Mp/L 8EI/(L3) 0.69R+0.31F
Plastic Bending 0.50 0.33 0.66 2Mp/L 0 0.75R+0.25F
Cantilever Beam with Fix Free Boundary Conditions and Point Mass and Distributed Force Elastic 0.40 1.00 2.50 2Mp/L 8EI/(L3) 0.50R+0.50F
Plastic Bending 0.50 1.00 2.00 2Mp/L 0 0.50R+0.50F
Cantilever Beam with Fix Free Boundary Conditions and Distributed Mass and Point Force Elastic 1.00 0.24 0.24 Mp/L 3EI/(L3) 1.74R-0.74F
Plastic Bending 1.00 0.33 0.33 Mp/L 0 1.50R-0.50F
Cantilever Beam with Fix Free Boundary Conditions and Point Mass and Point Force Elastic 1.00 1.00 1.00 Mp/L 3EI/(L3) 1.00R
Plastic Bending 1.00 1.00 1.00 Mp/L 0 1.00R
where:
m1=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.25)
m2=EI/(L3)*(1.5Mps/(Mps+2Mpm)+0.5)
m3=EI/(L3)*(2Mps/(Mps+3Mpm)+0.5)