S07: Composite I-Girder with Time-Dependent Effects

The girder and deck system is supported by a pin on one end and a roller at the other (the supports are level with the centroid of the deck). The length of the span is 300 ft. The span is divided into 10 segments. The girder is along the x-axis with the z-axis parallel to the web. The members have an orientation angle of 90 degrees. (This is a 2D problem: TY, RX, and RZ are fixed as universal restraints.)

The member elements are not assigned a material because their materials are set in the Section Composer section. It has no member end offsets.

Because the reference axes are placed at a different location from the centroid, the composite centroid is shifted down from the joint-to-joint line automatically.

The composite beam will be constructed in two phases. The steel part will be constructed first. After the steel part deforms due to self-weight, the concrete slab will be added.

The cross-section is defined in LARSA Section Composer with a construction sequence state (State A) in which only the steel I-shape is active.

The distance from the reference axis to the composite centroid is computed by taking the weighted average of the centroid locations of the two shapes (relative to the reference axis), each weighted by the corresponding E*A:

LARSA Section Composer’s computation of the centroid location matches exactly.

Area and Izz

Hand calculations to verify A and Izz follow. When computing properties of composite sections, the area and moment of inertia properties of each shape are factored by the ratio of the modulus elasticity of the shape to the modulus of elasticity of the reference material (which in this example is the material of the rectangle). The total cross-sectional area is the sum of the factored areas of the two shapes. For computing the moment of inertia in z (Izz), the moment of inertia of each shape individually (as given from standard formulas) is transformed by adding the shape’s area times the square of the distance from the shape’s centroid to the section centroid. Each shape’s contribution to Izz is then factored by the ratio of its modulus of elasticity to that of the reference material, and the two contributions are summed.

LARSA Section Composer’s computations of the composite A and Izz match exactly.

Reactions

The weight of the two shapes and the total weight of the composite beam is computed in the following table:

This model is constructed in two stages. In the first stage, the I shape is constructed alone and self-weight is applied. In the second stage, the rectangule shape is constructed, making a composite section, and the additional self-weight from this shape is applied. The total Z reaction reported in LARSA 4D matches the total weight hand calculation above for the incremental reactions in each stage (corresponding to the weight of each piece alone) and the cumulative reaction at the end of the second stage.

Displacements

The maximum deflection at mid-span is given by d=5/384*(W*L³)/(EI), where L = 300 ft. The two incremental deflections and the cumulative deflection are given in the table below:

The two incremental displacements and the cumulative displacement computed by LARSA 4D match exactly.

Stresses

There is no strain at the compound centroid in a model with loading and restraints such as in this problem. Stresses are due entirely to the bending stress. Further, at the pin supports there is no bending stress.

The stress at mid-span can be computed by first computing the rotational strain (the slope of the strain curve along the member’s y-axis) which is given by r=M/(E*I), where M = WL/8. Because of the composite assembly of this structure, only the I shape takes the load of its self-weight, while both parts of the composite section take the load of the rectangle.

The rotational strain is converted into a stress at a fiber by multiplying it by the modulus of elasticity at the fiber and the distance from the fiber to the section centroid.

In the first stage, when the section is composed of the I shape alone, note that the beam centroid is at the I shape centroid, and thus stress point 3 is located at 2.5 ft from the section centroid. In the second stage, the fiber locations are relative to the composite centroid. The incremental stresses are reported below:

The stresses in LARSA 4D match up to rounding.

In a time-dependent analysis, the concrete slab will be affected by the time effect on elastic modulus while the steel part will remain unchanged. This necessitates a recomputation of the composite member properties. In this example we will assume the self-weight loading occurs 15 days after casting.

The effect of time is modeled by multiplying E by f = sqrt(age/(4.0+0.857 X age) ) where age is in days and is no greater than 28. The concrete E is multiplied by f = 0.943. As above, the location of the composite centroid relative to the reference line is determined by the following table:

Hand calculations for A and Izz follow as described above, except this time we treat the reference material as the time-adjusted concrete.

LARSA 4D does not report time-adjusted cross-sectional properties.

In LARSA 4D, a Material Time Effect record (with default settings) is applied to the Fc_4 material. The casting day of the composite member is set to day 0. Self-weight loading is applied on day 15. The CEB-FIP 90 analysis code is chosen, the time effect on elastic modulus is turned on (all other time effects are turned off), and a Time-Dependent Staged Construction Analysis is run.

The results in the first stage are unchanged.

Displacements

The maximum deflection at mid-span is given by 5/384*(W*L3)/(EI). W and L are as before. E is from the reference material, now the time-adjusted concrete property. The revised computation for incremental displacement in the second stage is:

Stresses

The reviesd stresses are:

The rotational strain is converted into a stress at a fiber by multiplying it by the modulus of elasticity at the fiber and the distance from the fiber to the section centroid.

In the first stage, when the section is composed of the I shape alone, note that the beam centroid is at the I shape centroid, and thus stress point 3 is located at 2.5 ft from the section centroid. In the second stage, the fiber locations are relative to the composite centroid. The incremental stresses are reported below:

The results in LARSA 4D are correct up to rounding.

To verify shrinkage, we will look at the total effect of shrinkage between days 15 and 400. Shrinkage, εS as given below according to CEB-FIP 90, is an additional strain added into the cross-section accumulated from time t0 to time t1. In this part, the time effect on elastic modulus is not included.

where TS = 3, BetaSC = 5, H= 2 * Area/Perimeter converted to milimeters, RH is the relative humidity (80%), and t0, t1 are the ages of the section in days. Fc_28 = 27.6 MN/m2 (576 kip/ft2).

Shrinkage in the composite section is due only to the effect of shrinkage in the concrete part. The area of the rectangle is 3.2 ft2 and the perimeter for this example is treated as 9.6 ft (which includes the unexposed perimeter at the interface of the shapes in order to match the perimeter used by LARSA 4D). H = 203.2 mm. Substutiting these values into the formula gives εS(15,400) = -1.33x10-4.

Because the two shapes that make up the cross-section are connected and governed by the assumption that plane sections remain plane, shrinkage will create self-equilibriating stress (also known as eigenstresses) in both the concrete and steel parts of the section.

The computation of self-equilibriating stress due to shrinkage involves combining two equal-and-opposite strains. The first component is the “external” component with magnitude εS in the concrete part (and zero in the steel part), and it is integrated across the fibers that make up the cross-section area to determine an equivalent force F = εS*E*A and moment M = F*cy imposed at the beam end at the composite centroid. This force and moment determines the actual deformation of the beam. In Ghali, Favre, and Elbadry (2002) example 5.2, the external component corresponds to the forces and moments that undo the artificial restraint on shrinkage.

Using the E and A values of the concrete part and the distance from the beam centroid to the centroid of the concrete part (the same as the offset of the beam centroid from the beam reference axis given in the first table), F = -222 kip, M = -522 kip-ft. The corresponding strain for the composite beam is -2.51x10-5 pure strain and -1.39x10-5 1/ft bending strain.

The second or “internal” component has magnitude -εS at any fiber in the concrete part and zero at any fiber in the steel part, but it does not cause deformation. It corresponds to Ghali, Favre, and Elbadry’s strain in the section when shrinkage is restrained.

Self-equilibriating stress at a fiber is the modulus of elasticity at the fiber multiplied by the sum of the strain due to the external component (a linear stress profile across the entire cross-section) and the strain due to the internal component at the fiber (a step function with value -εS in the concrete part and 0 in the steel part).

There are no continuity stresses in a model such as this. Stress is constant along the length of the beam.

The computation of total stress is given in the following table:

The results in LARSA 4D match up to rounding.