Seismic Analysis: The Importance of Modeling Cracked Section Stiffness

Regulatory references

This post focuses on a critical consideration in seismic design: modeling the stiffness of cracked sections. Specifically, the NTC18 code states in Section 7.2.6 that for reinforced concrete and masonry structures:

“When representing the stiffness…” → “When modeling structural element stiffness, the effects of cracking must be accounted for. In the absence of detailed analysis, the flexural and shear stiffness of masonry, reinforced concrete, and composite steel-concrete elements may be reduced to 50% of the corresponding uncracked stiffness, with due consideration for the limit state under investigation and the influence of sustained axial loads.“

Section 7.8.5.1.2 for masonry structures specifies:

“The stiffness of masonry elements shall be calculated considering both flexural and shear contributions. The use of cracked stiffness values is recommended; in the absence of more rigorous evaluations, cracked stiffnesses may be assumed as 50% of the uncracked values”

In essence, the code requires that when analyzing structures subject to seismic loading, the structural model must adequately represent both stiffness distribution and mass distribution to ensure that all significant deformation modes and inertial forces are properly captured.Therefore, the code recommends evaluating the stiffness of load-bearing elements with explicit consideration of cracking effects.

Unless more refined analyses of cracked sections are performed, the elastic flexural and shear stiffness properties of concrete and masonry elements may be reduced to 50% of the corresponding uncracked values.

Stiffness reduction factors are also addressed in other codes and technical literature which, based on extensive research, propose specific values that vary according to structural element type.

The American FEMA guidelines provide a noteworthy approach, establishing a reduction factor of 0.5 for flexural elements under low axial compression and increasing it to 0.7 for elements subjected to high compression (>50% of critical axial stress), as these elements experience less degradation from cracking.

Implementation in PRO_SAP

PRO_SAP enables users to specify, through the materials database, the percentage of stiffness to be considered by differentiating among axial, flexural, and shear stiffness. These values affect seismic analyses only.

For D2 elements (beam and column members), users can specify distinct cracked stiffness values for axial behavior, secant shear stiffness, and flexural stiffness. During stiffness matrix generation, reduced values of the cross-sectional area A, moment of inertia J, or shear area AV (if defined in the materials database) are incorporated. Cracked stiffness is not applied to foundation beams.

For D3 elements (walls, slabs, membrane elements, and floor systems), reduced values of the elastic modulus E and shear modulus G (if defined in the materials database) are incorporated into the stiffness matrix. Cracked stiffness is not applied to foundation slabs.

For item D2

For the D3 shell element

For element D3 membrane (always orthotropic as the isotropic is defined by E and ν)

It should be noted that the above tables also indicate the material wood, which is not subject to cracking. Even for wood, in fact, it is necessary to modify the stiffness, but increasing it to take into account the effects of an instantaneous load. In fact, the NTC2018 in paragraph 7.7.4 says that “In the analysis of the structure, the deformability of the connections must be taken into account. The values of the elastic modulus for “instantaneous actions” must be used, obtained from the average values of the elastic modulus of the resistant elements” while the UNI EN 1998-1:2005 in paragraph 8.4 specify the value of the elastic modulus E0 for instantaneous loads, increased by 10% compared to the short-term one. Therefore, the variation in stiffness also affects the wood material, albeit with causes other than cracking, i.e. the use of the elastic modulus for instantaneous actions.

But what does this mean in the analyses? Let’s see it by applying the value suggested by NTC18 to a structure modeled with PRO_SAP for the aforementioned cracked stiffness, and comparing the results with those obtained from the original model at full stiffness.

Consequences of using cracked stiffness

By way of example, the model of a reinforced concrete framed structure with poroton panels is used, in which there are also D3 shell-type elements for the modeling of the retaining walls in the basement and the stairwell. Linear dynamic analyses are performed first on the model with uncracked stiffness, and consecutively on a copy of the model where the stiffness is cracked, then factored by 0.5 using the command shown above.

The first result we can compare is the frequency of the modes of vibration:

Characteristic stiffnesses and frequencies are directly proportional, and exactly as you would expect the frequencies in the cracked stiffness model drop significantly as the periods increase. This involves a shift to the right of the period in the response spectrum to which a variation of the spectral acceleration acting on the structure can also be associated, depending on the area of the spectrum in which one is located.

In this specific case, the proper period of the structure at full stiffness is located in the constant velocity part of the response spectrum with T > Tc, so an increase in period leads to a decrease in spectral acceleration. A reduction in the acceleration acting on the structure also corresponds directly to a decrease in the shear action at the base, which we verify through the “Fx Actions – Statistics” command by selecting all the nodes at the base of the model with reference to the first seismic combination.

Another consideration we can make concerns displacements: regardless of the point of the spectrum in which we are, if we want to define stiffness as “quantity of force necessary to obtain a displacement of the unitary structure” it is an immediate consequence that a reduction in stiffness corresponds to an increase in displacement and so it is possible to verify through the command “Display results > Node translation“.

Let’s see an account of the results obtained and the checks performed, from which it is evident that the cracked stiffness has advantages for the SLV life-saving limit states as the stresses acting on the structure are significantly reduced. On the contrary, SLD damage limit state checks in terms of stiffness are negatively affected due to the greater deformability of the structure and therefore the high displacements achieved.

Summing up

• For structures subjected to seismic loading, it is recommended that the stiffness of load-bearing elements be evaluated accounting for cracking effects.
• In the PRO_SAP materials database, cracked stiffness can be specified by independently defining axial, flexural, and shear stiffness values.;
• For static or wind load cases, PRO_SAP uses the full uncracked stiffness
• The use of cracked stiffness properties results in:
  • Advantages in terms of internal forces, as the increase in natural periods leads to reduced spectral accelerations
  • Disadvantages in terms of deformations, as a more flexible structure exhibits larger displacements, thereby penalizing second-order effects and serviceability limit state drift verifications.

Eng. Antonio Limena

limena@2si.it